D. Atwood et al. / Physics Reports 347 (2001) 1}222
1
CP VIOLATION IN TOP PHYSICS
David ATWOODa, Shaouly BAR-SHALOMb, Gad EILAMc, Amarjit SONId a
b
Department of Physics and Astronomy, Iowa State University, Ames, IOWA 50011, USA INFN, Sezione di Roma and Department of Physics, University of Roma I, La Sapienza, Roma, Italy c Physics Department, Technion-Institute of Technology, Haifa 32000, Israel d Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO
Physics Reports 347 (2001) 1}222
CP violation in top physics David Atwood , Shaouly Bar-Shalom , Gad Eilam, Amarjit Soni Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA INFN, Sezione di Roma and Department of Physics, University of Roma I, La Sapienza, Roma, Italy Physics Department, Technion-Institute of Technology, Haifa 32000, Israel Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA Received September 2000; editor: J.V. Allaby Contents 1. Introduction 2. General discussion 2.1. De"nitions of discrete symmetries C, P and T 2.2. CP-violating observables: categorizing according to ¹ , 2.3. Partial rate asymmetries and the CP-CPT connection 2.4. Resonant = e!ects and CP violation in top decays 2.5. E!ective Lagrangians and observables 2.6. Optimized observables 2.7. The naked top 2.8. Elements of top polarimetry 3. Models of CP violation 3.1. CP violation and the standard model 3.2. Multi-Higgs doublet models 3.3. Supersymmetric models 4. Top dipole moments 4.1. Theoretical expectations 4.2. Arbitrary number of Higgs doublets and a CP-violating neutral Higgs sector 4.3. Expectations from 2HDMs with CP violation in the neutral Higgs sector 4.4. Expectations from a CP-violating charged Higgs sector
4 7 5. 7 9 13 6. 19 20 23 28 29 31 31 39 52 64 64 65 67 72
7.
8.
4.5. Expectations from the MSSM 4.6. Top dipole moments } summary CP violation in top decays 5.1. Partial rate asymmetries 5.2. Partially integrated rate asymmetries 5.3. Energy asymmetry 5.4. -polarization asymmetry 5.5. CP violation in top decays } summary CP violation in e>e\ collider experiments 6.1. e>e\PttN 6.2. e>e\PttN h, ttN Z, examples of tree-level CP violation 6.3. e>e\PttN g 6.4. CP violation via == fusion in e>e\PttN N C C CP violation in pp collider experiments 7.1. ppPttN #X: general comments 7.2. ppPttN #X: general form factor approach and the CEDM of the top 7.3. 2HDM and CP violation in ppPttN #X 7.4. SUSY and CP violation in ppPttN #X CP violation in ppN collider experiments 8.1. ppN PttN #X 8.2. ppN PtbN #X 8.3. ppN PtbN h#X, a case of tree-level CP violation
75 80 85 86 102 103 104 105 106 106 119 135 146 149 149 150 154 162 164 164 171 180
E-mail addresses:
[email protected] (D. Atwood),
[email protected] (S. Bar-Shalom),
[email protected] (G. Eilam),
[email protected] (A. Soni). Much of the work in this review was done while S. Bar-Shalom was at Physics Department, University of California, Riverside, CA, USA. 0370-1573/01/$ - see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 1 2 - 5
D. Atwood et al. / Physics Reports 347 (2001) 1}222 9. CP violation in collider experiments 9.1. PX: general comments 9.2. PttN and the top EDM 9.3. PttN and s-channel Higgs exchange in a 2HDM 10. CP violation in >\ collider experiments 10.1. >\PttN 10.2. CP violation in the #avor changing reaction >\PtcN
183 184 186 189 194 194
11. Summary and outlook 12. Note on literature survey Acknowledgements Appendix A. One-loop C functions Appendix B. Abbreviations References
3 205 212 212 213 213 214
202
Abstract CP violation in top physics is reviewed. The standard model has negligible e!ects, consequently CP violation searches involving the top quark may constitute the best way to look for physics beyond the standard model. Non-standard sources of CP violation due to an extended Higgs sector with and without natural #avor conservation and supersymmetric theories are discussed. Experimental feasibility of detecting CP violation e!ects in top quark production and decays in high energy e>e\, , >\, pp and pp colliders are surveyed. Searches for the electric, electro-weak and the chromo-electric dipole moments of the top quark in e>e\PttM and in ppPttM X are described. In addition, other mechanisms that appear promising for experiments, e.g., tree-level CP violation in e>e\PttM h, ttM Z, ttM and in the top decay tPb and CP C C O violation driven by s-channel Higgs exchanges in pp, , >\PttM , etc. are also discussed. 2001 Published by Elsevier Science B.V. PACS: 11.30.!j
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1. Introduction Violations of the CP (charge conjugation combined with parity) symmetry are of great interest in particle physics especially since its origin is still unclear. Better understanding of this (so far) rare phenomenon can lead to new physics which may explain both the origin of mass and the preponderance of matter over anti-matter in the present universe. Indeed, reactions that violate CP are such a scarce resource that in over 30 years the only con"rmed examples of CP violation are those found in the decay of the K -meson. * The "rst experimental observation of CP violation was in 1964 by Christenson et al. [3], who observed a non-vanishing rate for the decay K P2 [4], * Br(K P2)"3.00$0.04;10\ . *
(1.1)
Since the dominant decay of K is to a 3 state of CP"!1, the above decay to a manifestly * CP"#1 state clearly violates this symmetry. Another example of CP violation which is well established in K is the di!erence between * (K Pl>l \) and (K Pl\l >), l"e, [4]: * * (K Pl>\)!(K Pl\>) * * "(3.27$0.012);10\ . (K Pl>\)#(K Pl\>) * *
(1.2)
All of these observations of CP violation in the K system can be explained by the CP violation * in the mixing of the neutral K mesons. Thus, K "((1# )K!(1! )KM )/(2(1# ) , *
(1.3)
K "((1# )K#(1! )KM )/(2(1# ) , 1
(1.4)
where the experimental value of is [4] "(2.263$0.023);10\, arg( )"(43.49$0.08)3 .
(1.5)
As is well known, this mixing can be accommodated in the standard model (SM) with three generations where the CP violation originates through a phase in the Cabibbo Kobayashi Maskawa (CKM) [5,6] matrix as will be discussed in some detail in Section 3.1. The SM further predicts that there is an additional CP violation in K P parameterized by * the quantity . The prediction is that / "O(10\); the theoretical di$culties in determining the hadronic matrix element prevent us from making a more precise estimate, see e.g., [7}14]. Experimentally, Re( / ) may be measured via [15]
1 , Re( / )K 1! 6 >\ For excellent recent books on CP violation see Refs. [1,2].
(1.6)
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where GHH K 5 * , " (1.7) GH GHH K 5 1 and H is the relevant weak interaction Hamiltonian. 5 After some two decades of intensive e!orts, new and quite dramatic experimental developments have recently taken place that we would now brie#y like to mention. First of all, let us recall that a few years ago the CERN experiment NA31 gave the result [16]: Re( / )"(23$6.5);10\ ,
(1.8)
appreciably di!erent from zero. On the other hand, the Fermilab experiment E731 found it completely consistent with zero [17,18]: Re( / )"(7.4$6);10\ .
(1.9)
For the past many years improved experiments have been underway, at CERN (experiment NA48), and at FNAL (KTEV) with an expected accuracy of about O(10\), KTEV has recently announced their new results on / , based upon analysis of 20% data collected so far [19]: Re( / )"(28$4.1);10\ .
(1.10)
For the world average one would now "nd [18] Re( / )"(21$5);10\ ,
(1.11)
thus, conclusively establishing that / O0. Such a non-vanishing value formally lays to rest the phenomenological superweak model [20] of CP violation as it unambiguously predicts / "0. However, unless the computational challenges presented by strong interactions can be overcome, it is unlikely that the measured value of / would con"rm or refute the SM in any reliable fashion. Experiments involving B-mesons are more likely to have a quantitative bearing on the SM. Just as the SM indicates that the natural size of CP asymmetries in K physics is O(10\}10\), it also strongly suggests that the e!ects in the B system are much bigger; in many cases CP asymmetries are expected to be tens of percents. This expectation renders the B system ideal for a precise extraction of the CKM phase and, indeed for a thorough quantitative test of the SM through a detailed study of the unitarity triangle [21,22]. The asymmetric and symmetric B factories currently in the early stages of running at KEK, SLAC and Cornell and hadron machines, should have a very important role to play in confronting the experimental results with the detailed predictions of the SM. A recent CDF result [23] for CP violation in BPJ/K , though crude at 1 the moment, indicates that CP violation may indeed be large in the b system. Experiments at FNAL have decisively [24,25] demonstrated that the mass of the top quark is extremely large, i.e., the D0 and CDF average is now m &174 GeV [26,27]. This has some very R important consequences. First, the top rapidly undergoes two-body weak decay: tPb#=, with a time scale of about 10\ s, which is shorter by an order of magnitude than the typical QCD time scale necessary for hadronic bound states to be formed [28]. Thus, unlike the other "ve quarks, the top does not form hadrons. It means that the dynamics of top production and decay does not get
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masked by the complications of non-perturbative, bound state physics, i.e., the `brown mucka. All of the CP violation phenomena relevant to the top are therefore of the `directa type. We should think of the top quark as an elementary fermion. For example, it therefore is sensible to ask for its dipole moment [29}32]. Unlike the other quarks the spin of the top quark becomes an extremely important observable. Indeed the decays of the top quark become very e!ective analyzers of its spin, see e.g., [33}35]. The SM predicts, however, that CP-violating e!ects in t-physics are very small. This is primarily due to the fact that its large mass in comparison to the other quarks renders the Glashow}Iliopoulos}Maiani (GIM) [36] cancellation particularly e!ective [37,38]. This being the case, what then is the motivation for the study of CP violation in the top quark system? There are two related reasons why one might expect to "nd such e!ects. First of all there is another important example of CP violation which the SM fails to explain, namely the excess of matter over anti-matter in the universe. It was shown by Sakharov [39] in 1967 that CP violation is one of the necessary conditions for baryon number asymmetry to appear in the early universe; baryon}anti-baryon asymmetry can be dynamically generated at early stages after the big bang even if the universe was `borna symmetric, provided that: (i) C and CP are violated, (ii) there are baryon number violating interactions and (iii) there is a deviation from thermal equilibrium. The basic idea is that, if CP is violated, then baryons and anti-baryons interact with di!erent rates at some point in the early universe. However, the CP violation due to the SM appears too weak to drive such an asymmetry [40,41]. In many cases, extensions of the SM such as the Two Higgs doublet model (2HDM) or the SUperSYmmetric (SUSY) extensions of the SM are able to supply the CP violation required to produce such a baryon asymmetry in the early universe. In fact, in some models [42,43], it is precisely the couplings of the top quark to CP-violating phases in beyond the SM physics which drive baryogenesis. Thus, the study of CP violation in top quark interactions in the laboratory could shed light on these primordial processes. The second motivation for investigating CP violation in top quark physics is that in many extensions of the SM, CP violation in the top quark can be particularly large. Indeed, because the SM contribution to CP violation in the top quark is so small, any observation of such e!ects would be a clear evidence of physics beyond the SM. The argument here parallels the search for the weak neutral current, in the 1970s, by looking for parity violation in deep-inelastic-scattering. The point is that the existing theory of the time, namely QED, could not cause parity violation in deepinelastic-scattering. Such an e!ect became an unambiguous signature for the existence of the weak neutral current. Since various extensions of the SM entail new CP-violating phase(s), we should seek the optimal strategies for searching each type of new phase. In this context, we "rst recall, what has been emphasized on the preceding page, that the b-quark is very sensitive to the CKM phase of the SM. Existing literature has revealed that top physics is very sensitive to several di!erent types of new phases. Upcoming high-energy colliders of the next decade can therefore serve as excellent laboratories for searching for new physics in top quark systems, and in particular, for studying CP-violating e!ects associated with those new CP-odd phases. The upgraded Tevatron pp collider (runs 2 and 3) at Fermilab which will be able to produce about 10}10 ttM /year, the CERN pp large hadron collider (LHC) will produce about 10}10 ttM /year, and about 10}10 ttM /year are expected at a future e>e\ next linear collider (NLC).
D. Atwood et al. / Physics Reports 347 (2001) 1}222
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A CP-odd phase due to an extended neutral Higgs sector or vertex corrections arising in other extensions of the SM can endow the top quark with a large dipole moment form factor. Such an e!ect could be detected both at an e>e\ collider, such as the NLC or hadron colliders such as the LHC. A CP-violating phase in the neutral Higgs sector also causes large CP asymmetries in the reactions e>e\PttM H and e>e\PttM , both of which should be a prime target for the NLC. C C Moreover, CP violation in the neutral Higgs sector and in supersymmetry can have interesting e!ects in single top production at the upgraded Tevatron pp collider at Fermilab and in ttM pair production at the LHC. The transverse polarization of the in the three-body top decay tPb is extremely sensitive to a new phase from a charged Higgs sector in multi-Higgs doublets models (MHDMs). Finally, CP-odd phases in SUSY models have also interesting e!ects in partial rate asymmetry (PRA) in tP=>b versus tM P=\bM . These processes and others will be discussed in the subsequent sections. We will not consider CP violation phenomena in which the top quark is virtual rather than an external particle. It su$ces to recapitulate that, in the SM, CP violation is often dominated by the virtual top quarks in the loops. Let us also comment that the discovery of the top with the measurement of m , and the progress in determination of the CKM matrix elements as well as R considerable progress in theory, has in#uenced our understanding of CP violation in K and B physics within the SM [13] and beyond the SM [44]. Furthermore, for the electric dipole moments (EDMs) of the electron and the neutron [45], the virtual top quark also plays a crucial role in extended Higgs sector scenarios.
2. General discussion 2.1. Dexnitions of discrete symmetries C, P and ¹ Let us now review the de"nitions of the discrete symmetries C, P and ¹ and recall a few basic facts concerning their manifestation in relativistic quantum "eld theory. Under the parity transformation, P, the spatial coordinate axes are reversed, i.e., Px"!x. Thus for an ingoing particle, X, in a speci"c momentum and spin state X; P, S , the action of parity is to reverse the momentum, leaving the spin "xed as angular momentum is an axial vector de"ned by a cross product. Hence, PX; P, S PX;!P, S .
(2.1)
Under the Wigner de"nition of time reversal, ¹, the sign of both momenta and spins are reversed, and also, due to the anti-unitary nature of ¹, and states are interchanged. Thus, ¹X; P, S PX;!P,!S .
(2.2)
Under charge conjugation, C, each particle is replaced by its anti-particle, and so CX; P, S PXM ; P, S , where XM means that all charges and other additive quantum numbers are reversed.
(2.3)
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It can be shown that local relativistic quantum "eld theories with the usual spin-statistics relations are invariant under the combined action of all three of these symmetries where [46}49]: CPTX; P, S "XM ; P,!S . Thus, such a theory violates ¹ if and only if it violates CP, where
(2.4)
CPX; P, S PXM ;!P, S , (2.5) and in this sense CP and ¹ violation are equivalent. Other well-known consequences of the CPT theorem are that masses of particles and antiparticles are the same, m "m M , and the total widths of particles and anti-particles are also equal, 6 6 " M . Note that it does not follow from CPT that decay rates to speci"c "nal states are the same. 6 6 In fact, partial width di!erences, i.e., a non-zero value of (XPA),(XPA)!(XM PAM ) ,
(2.6)
is a form of CP violation that we will discuss in more detail in Section 2.3. Clearly, it follows from " M that 6 6 (XPA)"0 , (2.7) where the sum is over all possible "nal states. The relationship between (XPA) and the other "nal states which compensate for it will also be discussed in detail in Section 2.3. In this report we are largely concerned with the violation of discrete symmetries in decay and scattering experiments. We, therefore, need to consider the implementation of CP and ¹ on the S-matrix. For C and P this is straightforward. Consider the initial state of n particles G i"P , P ,2, S , S ,2 , (2.8) ? @ ? @ and the "nal state of n particles D f "P , P ,2, S , S ,2 , (2.9) so that the S-matrix element is f Si"S
. DG The transformations P and C follow from the single-particle transformation
(2.10)
f "!P ,!P ,2, S , S ,2 , (2.11) . f "P , P ,2, S , S ,2 , (2.12) ! and likewise for i , i . The transformation of the S-matrix element under these symmetries is . ! thus . S S P , DG D. G. ! S . S P DG D! G!
(2.13) (2.14)
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and thus !. S S P . (2.15) DG D!. G!. The nature of time reversal, however, requires the interchange of and states so that the e!ect on the S-matrix will be anti-unitary. Thus, f "!P ,!P ,2,!S ,!S ,2 , 2 i "!P ,!P ,2,!S ,!S ,2 , 2 ? @ ? @ such that
(2.16) (2.17)
2 S . (2.18) S P DG G2D2 Needless to say, because of the interchange of initial and "nal states, accelerator-based experiments seldom test ¹ directly. These symmetries, as they are de"ned in S-matrix theory are fundamental in that if the Lagrangian and the vacuum states respect C, P or ¹, then the corresponding symmetry of the S-matrix will apply. We will also "nd it useful to consider the symmetry ¹ } `naivea time reversal , } for which this is not true. The de"nition of ¹ is to apply ¹ to the initial and "nal states without , interchanging them 2, S . (2.19) S P D2 G2 DG Thus, ¹ is a `symmetrya which can be tested in accelerator-based scattering experiments, but, as , we shall see in the following section, it only corresponds to `truea time reversal (¹) operation at tree-level in perturbation theory. It is nonetheless useful in categorizing the various modes of CP violation. 2.2. CP-violating observables: categorizing according to ¹ , It is useful to divide CP-violating observables into two categories (see e.g., [29,32,35,50,51]), those that are even under `naivea time reversal (¹ ) and those that are odd. Recall that ¹ is , , de"ned as a transformation which reverses the momenta and spins of all particles without the interchange of the initial and "nal states. This contrasts with true time reversal, ¹, in that under ¹ initial and "nal states are also interchanged. The symmetry ¹ is not a fundamental symmetry like C, P and ¹ since the S-matrix under , ¹ need not follow from the transformation properties of the Lagrangian. Nevertheless, it is , a useful tool for categorization and as we shall presently show, observables which are CP-odd and ¹ -odd, i.e., are CP¹ -even, may assume non-zero expectation values in the absence of "nal state , , interaction (FSI) e!ects. In particular, tree-level processes in perturbation theory may lead to non-vanishing expectation values for these operators. On the other hand, CP-odd ¹ -even (i.e., , CP¹ -odd) operators may only assume non-zero expectation values if such FSI e!ects are present , giving a non-trivial phase to the Feynman amplitude. Such a phase, often called a strong phase or absorptive phase, may arise in several ways. It may be present in a loop diagram if the internal particle(s) can be on-shell. An interesting variation of this, which we will consider in Section 2.4, is
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in the propagator of an unstable particle where the strong phase is the phase of the Breit}Wigner amplitude. Non-perturbative rescattering of "nal state particles can also give a strong phase though this is more of interest in CP studies in B and K physics. Indeed, ¹ is useful in understanding when CP-conserving observables depend on an absorptive , phase. In particular, a CP-even ¹ -odd observable (typically a CP-even triple product correlation , of momenta and/or spins) will only assume an expectation value if FSI e!ects are present. Thus, for instance, if one is looking for a CP-odd, ¹ -odd e!ect, one must have data both on the process of , interest and its CP conjugate (if they are di!erent) in order to distinguish from the possible background of CP-even ¹ -odd e!ects (see e.g., [52]). , In order to understand the role of ¹ , let us consider the unitarity relations of the S-matrix , (implied by conservation of probabilities). Following a derivation analogous to the optical theorem [53] we write the S-matrix in terms of the scattering amplitude T S"1#iT ,
(2.20)
where, for a given transition iPf, T is related to the `reduced scattering amplitudea, , by f Ti"(2)( p !p ) f i . D G Substituting Eq. (2.20) into the unitarity relation SRS"1 we obtain
(2.21)
T !TH "i TH T , (2.22) DG GD LD LG L where we denote aTb,T . In terms of this becomes ?@ !H "i(2) (p !p )H . (2.23) DG GD L G LD LG L Let us now assume that there are no rescattering e!ects and that (to the order of approximation considered) i and f are stable states so that " "0. Thus, for each possible intermediate state GG DD n, the rhs of Eq. (2.23) vanishes. Therefore, in the absence of rescattering, is hermitian "H . GD DG Now, if is CP invariant, then by the CPT theorem it is also ¹ invariant. Thus, f i" i f " f i H 2 2 2 2 and, therefore,
(2.24)
(2.25)
f i" f i . (2.26) 2 2 In fact, this equation means precisely that the modulus of f i is invariant under ¹ . Since, in the , absence of rescattering, the expectation value of any operator depends only on f i, Eq. (2.26) implies that if CP is conserved, then only ¹ -even operators can have a non-vanishing expectation , value. What we have shown therefore is that in the absence of rescattering e!ects (i.e., Im()"0, in which case the requirement of CPT invariance leads e!ectively to conservation of the scattering amplitude under CP¹ ) and in the absence of CP violation, ¹ -odd observables have zero , , expectation value. Thus, if such a ¹ -odd observable O has a non-zero expectation value, either CP ,
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Table 1 Transformation properties under ¹ and CP and presence or absence of "nal state interactions (FSI). Here Y,FSI , present and N,FSI absent ¹ ,
CP-violating
CP-conserving
even odd
Y N
N Y
is violated and O is CP-odd (i.e., CP¹ -even) or there are rescattering e!ects present (implying , CPTOCP¹ ) and O is CP-even (i.e., CP¹ -odd). Conversely, let us suppose O is CP-odd and , , ¹ -even (i.e., CP¹ -odd). Again Eq. (2.26) implies that this operator can only assume a non-zero , , expectation value if rescattering e!ects are present. These properties of the operators are summarized in Table 1. From Table 1 we see that another consequence of Eq. (2.26) is that a ¹ -odd signal is only , a de"nite signal for ¹ violation and hence of CP violation in the absence of rescattering e!ects. To con"rm the CP-even or CP-odd nature of such a reaction one must therefore compare data from iPf with the charge conjugate channel iM PfM to explicitly verify CP violation or else rule out rescattering e!ects in some other way. Recall from the de"nition of time reversal that the spatial components of vectors representing momenta and spins are reversed. Thus, an observable is ¹ -odd if it is proportional to a term of the , form (v , v , v , v ), where v are 4-vectors representing spins or momenta of initial and "nal state G particles and is the Levi-Civita tensor. Consequently, ¹ -odd signals can only be observed in , reactions where there are at least four independent momenta or spins that can be measured. There are two important venues for the investigation of CP violation that we will deal with extensively. The "rst one is when CP nonconservation appears in decays of a particle and the second, is to search for scattering processes that can give rise to CP violation. The latter consist of two di!erent possibilities: either the CP-violating e!ect is due to the subsequent decay of the particle which is produced in the scattering process or the CP nonconservation is driven by an intrinsic property of the scattering mechanism itself. An observable which is CP-odd and ¹ -even, thus requiring an absorptive phase (as was shown , above), and which is widely used in the case where the CP e!ect appears in decays of a particle is called PRA (partial rate asymmetry). This observable is non-vanishing when a particle A decays to a state B with a partial width (APB) whereas the partial width of the conjugate process, i.e., (AM PBM ) is di!erent from (APB). Thus, de"ning
(APB)!(AM PBM ) , , .0 (APB)#(AM PBM )
(2.27)
it is easy to see that is odd under CP and CP¹ . For to receive non-vanishing .0 , .0 contributions, at least two amplitudes with di!erent (CP-even) absorptive phases as well as with di!erent CP-odd phases must contribute to APB. To see this explicitly, let us de"ne M and M to be the two possible amplitudes contributing to APB M,M(APB)"M e P e B #M e P e B , M M ,M(AM PBM )"M e\ P e B #M e\ P e B , (2.28)
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where are CP-odd phases that change sign as one goes from APB to AM PBM , and are CP-even G G phases that can arise due to FSI. It is then easy to see that M!M M "!4M M sin( ! )sin( ! ) . (2.29) Clearly , being proportional to (M!M M ), will vanish if the two amplitudes do not have .0 a relative absorptive phase, i.e., ! O0 as well as a relative CP-odd phase, i.e., ! O0. Of course does not violate CPT [54] as the requirement of CPT applies only to total widths .0 (A)"(AM ). This equivalence (i.e., (A)"(AM )) is due to the fact that the absorptive phase of (APB) emanates from rescattering through an on-shell intermediate state C and vice versa, i.e., the absorptive phase for (APC) will emanate from the on-shell intermediate state B. This fact is an example of the well-known `CP-CPT connectiona [55}58]. The states B and C are referred to as compensating processes. Of course, one can in general have this compensation act between several "nal states. It is sometimes useful to de"ne a slightly di!erent asymmetry which also requires dealing with partial rates. This asymmetry is called the partially integrated rate asymmetry (PIRA) and is de"ned as (APB)! (AM PBM ) .' , (2.30) , .' .'0 (APB)# (AM PBM ) .' .' where is the partially integrated width for APB obtained by integrating only part of the full .' kinematic range of phase-space. Often such asymmetries can be larger than since the portion .0 of the "nal states not included in the integral may themselves be the compensating process. For example, in [59] it was shown that detecting CP violation e!ects in the process tPb through O is more e$cient than through as the former is driven by tree-level diagrams and the .'0 .0 latter by 1-loop diagrams. A related observable which is also CP-odd and ¹ -even is the energy asymmetry , E ! E M G , (2.31) , G # E # E M G G where E is the average energy of a particle i in a decay of the `parenta particle and E M is the G G average energy of the corresponding anti-particle iM in the decay of the conjugate state of the `parenta particle. Such an asymmetry becomes relevant when the decay involves three or more particles in the "nal state and may be regarded as a weighted PRA. A further generalization of the above constructions of CP-odd ¹ -even observables is by , considering combinations of dot products (thus being even under ¹ ) of measurable momenta or , spin vectors. Examples of such CP-odd ¹ -even observables will be given in the following sections. , As an example of the various types of operators discussed above let us consider the reaction
e\(p )#e>(p )Pt(p , s )#tM (p , s ) , (2.32) C C R R R R where p are 4-momenta and s are spins. Clearly, no ¹ -odd observable can be constructed G G , without the Levi-Civita tensor, , and since the momenta satisfy p #p "p #p , one needs to use C C R R the spins of the t and tM to construct such observables. Indeed no non-trivial CP-odd observable can
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be constructed without knowing the spins either, so top polarimetry is essential for the study of CP violation in this reaction (see Section 2.8). As an example of a ¹ -even CP-odd operator consider in the c.m. frame of the e>e\: , O "(p !p ) ) (s !s )"!(P !PM ) ) (S !SM ) . C C R R C C R R
(2.33)
Clearly this is ¹ -even and also C-even. It is P-odd since S , SM are axial vectors while P , PM are , R R C C polar vectors. Thus, since O is CP-odd and CP¹ -odd, it will have an expectation value if both CP , violation and rescattering e!ects are present. Consider now the operator O " ?@ABp p s s J(P !PM ) ) (S ;SM ) . C? C@ RA RB C C R R
(2.34)
Clearly O is ¹ -odd and C-even but also P-odd. This CP-odd observable can therefore give , a signal of CP violation without the necessity of rescattering e!ects. Finally, the operator O " ?@ABp p (s !s ) (p #p ) J(P !PM ) ) [(S !SM );(P #PM )] , C? C@ R RA R RB C C R R R R
(2.35)
is ¹ -odd, P-even and C-even and so provides a signal of CP-conserving rescattering e!ects. , Similarly many more operators can be constructed with the symmetries of the above. Some consideration of the physics to be tested for is helpful in selecting the operator most useful in measuring possible CP-violating e!ects. We will consider that in more detail in Section 2.6. 2.3. Partial rate asymmetries and the CP-CPT connection As mentioned in the previous section, one of the most interesting and widely studied observable for testing CP violation is the di!erence between the partial width of a reaction, (APB), from that of the conjugate reaction, (AM PBM ). Thus, if (APB)O(AM PBM ), then CP is violated in the decay. In practice, it is better to work with the corresponding dimensionless ratio in Eq. (2.27), .0 called the PRA. Its use, speci"cally in the context of heavy quarks and the SM, "rst appeared in [60]. Since the CPT theorem demands that particle and anti-particle have identical life-times (or total widths), that theorem imposes important restrictions on the form of CP-violating PRAs; these were "rst recognized by GeH rard and Hou [56,57]. In perturbative calculations, which lead to PRAs, if all the diagrams are systematically included, the requirement of CPT } that the total rate and its conjugate be identical } should be manifest order by order. The compensating processes should also be evident as the internal states of loop diagrams. When simpli"cations are used in such calculations the constraint of the CPT theorem provides an important consistency check. Furthermore, these restrictions can be used to greatly facilitate the calculations of the PRA for a compensating process. A general formalism for maintaining the CPT constrains in calculating PRAs was given in [58]. In particular, it was shown that if one de"nes a partial width di!erence, into a particular "nal state, I, as ,(PPI)!(PM PIM ) , '
(2.36)
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where P and PM are the decaying particle and its anti-particle, respectively, then equality of total widths (P)"(PM ) implies "! . ' ( ($' More speci"cally, in perturbation theory one can write at a given order
(2.37)
" (J) ' ' ($'
(2.38)
(J)"! (I) . ' (
(2.39)
and
Here (J) denotes a contribution to the partial width di!erence into the "nal state I which is ' driven by the "nal state J, and conversely, (I) is the contribution to the particle width di!erence ( into the "nal state J being driven by the "nal state I. Due to Eq. (2.39), summing up the partial width di!erences over all the "nal states, one gets "0 , (2.40) ) ) where K runs over all "nal states, I, J,2. Thus the requirements of CPT are automatically satis"ed. Two important conclusions that can also be drawn are: (a) Rescattering of a state on to itself cannot give rise to PRA. This follows immediately from Eq. (2.39) by setting J"I. (b) The knowledge of the PRA into some "nal state I that is driven by the absorptive cut across a "nal state J, can be used to deduce the PRA into the "nal state J arising over a corresponding cut across the state I. As mentioned before, such two processes are often called `compensating processesa. Let us "rst illustrate how these considerations apply to PRAs in b decays. Consider, for example, the process bPscc . The lowest order non-vanishing contribution to the PRA arises here, at order , from the interference of the tree graph in Fig. 1(a) with the penguin graph in Fig. 1(b) Q which has an absorptive cut across the u quark line. The compensating process is then bPsuu where, for this process, the absorptive part, Im(loop), is driven by a cut on the c quark line in the loop. Thus, (suu )# (scc )"0 , QSS QAA
(2.41)
where the compensating nature of these two processes is illustrated in Fig. 1(c). At this point, it is instructive to discuss in some detail how the cancellation follows from the Feynman diagrams combined in Fig. 1(c). Let us denote by ¹ , ¹ the tree-level contributions to QAA QSS bPscc , bPsuu , respectively. Likewise, let us denote by PO , PO the penguin contributions to QAA QSS bPscc , bPsuu , respectively, with q"u, c, t being the intermediate quark in the penguin. We also denote the conjugate amplitudes as ¹M , ¹M , PM O and PM O . QSS QAA QSS QAA
D. Atwood et al. / Physics Reports 347 (2001) 1}222
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Fig. 1. (a) Tree-level diagram for bPscc , (b) 1-loop (order ) penguin diagram for bPscc , and (c) a diagrammatic Q description for the compensating nature of the contribution from bPscc and bPsuu to the PRAs. For bPscc : the dashed line indicates an absorptive cut along inner particles (uu ) in the penguin diagram and the solid line refers to an external phase-space cut (cc ). For bPsuu the role of the cuts are reversed: the dashed line is the external uu phase-space cut and the solid line indicates the cc absorptive cut in the penguin contribution.
These amplitudes may be represented in terms of their magnitude and phase as follows ¹ "e (A ¹ , ¹M "e\ (A ¹ , QAA QAA QAA QAA ¹ "e (S ¹ , ¹M "e\ (S ¹ , QSS QSS QSS QSS PO "e (O e HAO PO , PM O "e\ (O e HAO PO , QAA QAA QAA QAA PO "e (O e HSO PO , PM O "e\ (O e HSO PO . (2.42) QSS QSS QSS QSS Here is the CP-odd weak phase which has its origin in the Lagrangian. In particular, the SM O gives "arg(<
(suu )" 2Re(¹ PSH )!Re(¹M PM SH ) dPh(bPscc ) QAA QAA QAA QAA QAA "4 ¹ PS sin( ! )sin(A ) dPh(bPscc ) , QAA QAA A S S (scc )" 2Re(¹ PAH )!Re(¹M PM AH ) dPh(bPsuu ) QSS QSS QSS QSS QSS
"!4 ¹ PA sin( ! )sin(S) dPh(bPsuu ) , QSS QSS A S A where dPh( ) indicates an integral over the phase-space of the decay.
(2.43)
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D. Atwood et al. / Physics Reports 347 (2001) 1}222
Referring to Fig. 1(c), we see that the expression
¹ PS sin(A ) dPh(bPscc ) , QAA QAA S
(2.44)
is the interference of the absorptive part of PS (indicated by the dashed line) with the tree-level QAA graph (shown to the right of the solid line). It is identical to
¹ PA sin(S) dPh(bPuu s) , QSS QSS A
(2.45)
since all the couplings are the same and we have just interchanged the role of the internal phase-space of the uu cut (dashed line) with the external phase-space cc cut (solid line). That is, for the PRA in the decay bPsuu , the dashed line represents the external phase-space cut whereas the solid line indicates the absorptive cc cut. We thus conclude that (suu )"! (scc ) , (2.46) QAA QSS as required. The same kind of argument may be applied at higher orders in perturbation theory as discussed in [56,57] in the case of b decays, and is clearly an elaboration of the discussion above. For a further discussion of this example in the S-matrix formalism see [58]. Let us now consider the analogous example from top decays. PRA in channels of the type tPuddM , cddM ,2 (d"d, s or b) arise through interference of the `treea graph in Fig. 2(a) with the penguin in Fig. 2(b). Since m '(m #m ), the = is on-shell and the =-propagator in Fig. 2(a) is R 5 @ complex, as the = has an appreciable width. So the diagram shown in Fig. 2(a) has, in fact, an imaginary part (indicated by the dashed line) due to the =-width, which can dominate over the imaginary part of the penguin diagram depicted in Fig. 2(b). In fact, the 2-loop diagram corresponding to Fig. 2(b) with a `bubblea on the internal =-propagator can dominate over the 1-loop diagram.
Fig. 2. Feynman diagrams contributing to the decays tPuddM (d"d, s or b and u"u or c) and to the decay tPde : (a) C `Tree-levela diagram for tPuddM (with an imaginary part from the unstable on-shell =), (b) 1-loop (order ) penguin Q diagram for tPuddM , (c) The `reala tree-level diagram for tPde , (d) 2-loop contribution to tPde with an absorptive C C cut along the intermediate du lines, and (e) A diagrammatic description for the compensating nature of the contribution of the 1-loop;1-loop processes to a PRA in tPuddM and the tree;2-loop processes to a PRA in tPde (see text for C discussion).
D. Atwood et al. / Physics Reports 347 (2001) 1}222
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The CP-CPT connection has two implications here, discussed by Soares [61]. Firstly, in calculating the PRA into a given channel such as tPcbbM , part of the total =-width that arises from =PcbM must be subtracted away as it represents rescattering of the "nal state onto itself (see Eq. (2.38)). Numerically, the most important consequence here is for the channel tPuddM as then the process =PudM is not Cabibbo suppressed and indeed contributes about 1/3 to the =-width. A second interesting implication of the CP-CPT connection, pointed by Soares [61], in particular Eq. (2.39), is that the PRA of leptonic decays of the top (e.g., tPde ) can be deduced C from the PRA into its hadronic decays (e.g., tPdcdM ). In the language of Eq. (2.39) these two reactions are the compensating processes of each other, i.e., I"dcdM and J"de> in Eq. (2.39) C leading to (2.47) C (dcdM )"! M (de ) . BAB C BCJ The relevant Feynman diagrams that contribute to a PRA in the leptonic top decay tPde are C depicted in Fig. 2(c) (tree-level diagram) and Fig. 2(d) (2-loop diagram with an absorptive cut along the inner du lines). A graphic illustration of how the compensating nature between the channels tPdudM and tPde comes about (i.e., Eq. (2.47)) is shown in Fig. 2(e). In this "gure the dashed line C indicates the absorptive cut that, due to the Cutkosky rule, is responsible for the FSI phase in the decay tPdudM , and the solid line indicates the separation between the two contributing diagrams to the PRA in tPdudM . Similarly, for the decay tPde the roles of the dashed and solid lines are C reversed such that the dashed line indicates the separation between the two contributing diagrams (shown in Figs. 2(c) and 2(d)) and the solid line represents the necessary absorptive cut for the PRA in the decay channel tPde . Notice that a direct calculation of the PRA in the leptonic top decays C would have required a calculation of a 2-loop graph (Fig. 2(d)). So this application of the CPT theorem is especially noteworthy as through Eq. (2.39), the necessary calculation gets simpli"ed to dealing with two 1-loop graphs (namely Fig. 2(a) and (b)). Unfortunately, all the PRAs resulting from such interferences between the =-propagator and the penguin are much too small to be of experimental interest, at least in the SM. Before we "nish this discussion on the CPT constraints on PRAs, let us emphasize two important points. First, it should be very clear that these constraints are imposed only on a single CP-violating observable, namely PRA. Indeed, in general, many other (¹ -even) CP-violating , observables can be constructed (e.g., energy asymmetry, helicity asymmetry,2) which are similar to PRA in that they all require CP-conserving FSI phase(s). CPT constraints are very speci"c; they do not a!ect any of these other observables other than PRA. In particular, the structure of CPT, that the rescattering of a "nal state onto itself cannot give rise to an asymmetry, is speci"c to PRA only. Rescattering of a "nal state onto itself can give rise to important non-trivial experimental implications for all other CP-violating observables; rescattering graphs can certainly cause observable helicity or energy asymmetry. Another way of viewing this situation, in light of the CPT theorem, is to regard each portion of the phase-space or each polarization as a separate "nal state. Thus, one portion of phase-space or polarization state can compensate for another. As a speci"c example of this, consider the Schmidt}Peskin e!ect namely the helicity asymmetry [33]: N(t tM )!N(t tM ) 0 0 N " * * *0 all ttM
(2.48)
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induced in the reaction ppPttM #X. Here N(t tM ) represents the number of pairs of left-handed * * t and left-handed tM produced in the inclusive reaction. In a 2HDM with CP-violating phase(s) in the neutral Higgs sector, a non-vanishing value of N arises from interference of tree-level diagrams *0 with 1-loop diagrams involving neutral Higgs exchanges in the loops (see Fig. 47 in Section 7). In this case, the CP-conserving FSI phase required by N is provided by an absorptive cut which *0 explicitly arises from ttM PttM , i.e., rescattering of a "nal state onto itself. So a very interesting CP-violating e!ect, namely the helicity asymmetry in Eq. (2.48) } a CP-odd ¹ -even observable , (similar to PRA with respect to this classi"cation) } arises from self-rescattering of a "nal state (we will return to a more detailed discussion of the Schmidt-Peskin e!ect in Section 7). Thus, in general, in discussing CP violation phenomena or in calculations of CP violation e!ects, self-rescattering graphs should not automatically be discarded, unless of course one is speci"cally calculating PRAs. In particular, for the speci"c case discussed above, we may regard t tM and t tM as compensating * * 0 0 "nal states. An important, although somewhat obvious consequence is that even when PRAs are vanishingly small or exactly zero, other CP-odd observables can be non-vanishing and can have interesting observational consequences. Another interesting example arises from CP-violating phase originating from a charged Higgs sector, as in the Weinberg model [62] with three doublets of Higgs "elds. Important CP-violating e!ects arise, for example, in the leptonic decay tPb from the interference of the SM graph (with O = exchange) with the tree-level charged Higgs exchange graph (see Figs. 22(a) and 22(c) in Section 5). For this case, let us "rst consider the PRA (tPb)!(tM PbM ) . " O (tPb)#M (tM PbM )
(2.49)
For simplicity, let us assume that m > 'm ; then a CP-conserving absorptive part required for & R arises from the =-boson `bubblea which contains all possible states other than as required O O by CPT. However, because of the spin zero nature of the Higgs the =>}H> interference has non-vanishing contributions only for the scalar part of the =. This argument is most readily seen if one uses the Landau gauge for calculating this interference (for more details see Section 5). In that case, the scalar and vector components of the = propagator are cleanly separated according to their total angular momentum. Thus, graphs which pass through a vector = intermediate state will not interfere with graphs that pass through a Higgs state. The Higgs must therefore interfere only with the Goldstone propagator which corresponds to the decay of longitudinal = into fermion pairs cs , udM , e 2 which are all suppressed by powers of the fermion masses. Furthermore, the C Goldstone propagator shows none of the resonance enhancement associated with the vector component. Thus, the cs is the most important contributor to the scalar component of the =-boson `bubblea. The and cs can be thought of as the compensating processes. So, in fact, the PRA goes as m m & A O 5 , O m m m R R 5 and is extremely small [63].
(2.50)
D. Atwood et al. / Physics Reports 347 (2001) 1}222
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However, as already mentioned before, just because PRA is vanishingly small does not, though, mean that there are no CP violation e!ects. Indeed, very important and large CPviolating asymmetries may arise in the decay tPb . First of all, one can construct an energy O asymmetry: E ! E O . (2.51) " O # E # E O O That is, compare, e.g., the average energy of the in tPb with that of the in tM PbM . Then will not su!er from the helicity suppression or constraints of CPT on and one expects # O m m #& R R , (2.52) m m O A O as explicit calculations con"rm. Indeed, a CP-violating asymmetry even bigger than the energy asymmetry, namely the transverse polarization asymmetry of the , resides in this =>}H> interference. In fact, the transverse polarization asymmetry is enhanced by another factor of m /m compared to the energy asymmetry R O as is shown in Section 5.1.3. 2.4. Resonant = ewects and CP violation in top decays The large mass of the top (m K174 GeV) means that it decays to a three-body "nal state R primarily through an on-shell =. This fact is of particular interest in the study of CP violation in such decays since there will be a large strong (i.e., CP-even) phase inherent in this =-propagator. In particular, since the =-width is substantial ( &2 GeV), the transverse modes of the = are 5 controlled by the Breit}Wigner propagator 1 , (2.53) G " 2 q !m #im 5 5 5 5 which will have a substantial strong phase. The enhancement of the imaginary part of G is evident by considering that, at q "m , 2 5 5 Im(G )"(m )\ . (2.54) 2 5 5 The real part swings through 0 at this point but in the vicinity of the resonance it will also be large. For instance, if q !m "m , then 5 5 5 5 Re(G )"(2m )\ . (2.55) 2 5 5 Since &O() this means that near qPm both the real and imaginary parts of the 5 5 amplitudes for decays such as tPbudM , bcbM ,2 behave as if they are O(1) in the gauge coupling constant. This phenomena is what we mean by `resonance enhancementa. The imaginary part here then provides the needed absorptive part (i.e., FSI phase) to lead to enhancement of CP-odd ¹ -even observables. Likewise, near q&m , the real part can magnify the e!ect of , 5 CP-odd ¹ -odd observables. ,
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The basic idea of FSI phase driven by particle widths in decays was discussed in [64,65]. Although originally the discussion [64,65] took place in the context of the SM in conjunction with the CKM phase, it should be completely clear that they can be equally well used with non-standard sources of CP violation. Indeed, as dealt in Section 5.1.3, the =-resonant e!ects provide a signi"cant enhancement of ¹ -even and ¹ -odd CP-violating e!ects in top decays in the context of an , , extended Higgs sector. 2.5. Ewective Lagrangians and observables One tool that is often used to catalog the e!ects of new physics at an energy scale, , much higher than the electroweak scale, is the e!ective Lagrangian (L ) method. If the underlying extended theory under consideration only becomes important at a scale , then it makes sense to expand the Lagrangian in powers of \ where the term is the SM Lagrangian and the other terms are the e!ective Lagrangian terms. Simple dimensional arguments tell us that the operator which multiplies \L must be of dimension n#4. This restriction together with symmetry considerations implies that at each order in \ there are only a "nite number of possible terms. Conversely, this implies that experimental tests for the existence of speci"c terms in L is a relatively model independent [71] way to search for new physics. Here we are interested in top quark physics which violates CP and so, in the e!ective Lagrangian approach, the operators of interest are further restricted. For example, in Section 4 we discuss the top electric dipole moment (and related e!ects) which can arise from a dimension 5 term in the e!ective Lagrangian: L JtM tFIJ . (2.56) IJ At dimension 6, photons can interact with the top quark via a CP-violating operator such as L J(tM t)(FIJF ) , (2.57) IJ which could arise, for instance, via a SUSY box diagram. Let us now consider as a speci"c example e!ective Lagrangian terms which would contribute to the process ggPttM . It is useful to recall that in such an expansion, operators that are proportional to the QCD equations of motion for the top or the gluon "elds may be eliminated by a "eld rede"nition and are therefore redundant [71]. There are a number of requirements that an operator has to satisfy for it to be relevant to CP violation in top production in hadronic collisions. These are: 1. It must violate CP. 2. Its Feynman rules must include couplings to two or fewer gluons. 3. It must not be proportional to q of one of the on-shell gluons in the initial state. For early references to resonance enhancement of CP violation in scattering processes, see [66,67]. For later references, see [68]. For subtelties resulting from non-gauge-invariance of Breit}Wigner form of the = propagator, see [69,70] and references therein.
D. Atwood et al. / Physics Reports 347 (2001) 1}222
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The need for the "rst requirement is clear. The second requirement is present since the events in question have two gluons in the initial state and no gluons in the "nal states. If one wanted to consider experiments where additional gluon jets were detected in the "nal state, clearly one would have to generalize this requirement. In constructing a basis of operators which satisfy the above conditions it can be shown that [72] one can eliminate, without loss of generality, any operator which is equal to 0 modulo the equations of motion. Equivalently, if the di!erence of two operators is 0 modulo equations of motion, then only one need to be included. Here is a set of operators that we choose which satisfy the requirements mentioned above and are of dimension six or less O "tM i[ f (!䊐)FIJ] ¹G t , G IJ ? ? O "tM i[ f (!䊐)FIJFG ] t , @ @ G IJ O "tM [ f (!䊐)F?@FAB]t , A ?@AB A G G O "tM i[ f (!䊐)FIJFI ] dGHI¹Gt , B B H IJ O "tM [ f (!䊐)F?@FABdGHI]¹G t , (2.58) C ?@AB A H I where F's are the gluon "eld strength tensor, 䊐"DID , the analytic functions f ,2, f are form I ? C factors and ¹G"G/2, G being the Gell}Mann color matrices. Note, that in general, higher order terms in 䊐 will imply the existence of couplings to additional numbers of gluons. As an illustration, let us consider further the operator O . This operator essentially corresponds ? to the chromo-electric dipole moment (CEDM) form factor. The experimental implications of the static analog of this quantity were considered in [35]. We can expand the operator to obtain the Feynman rule. The vertex for the one gluon interaction is i f (q)tM IJ ¹Htq H , (2.59) ? I J here is the polarization vector of the gluon. This is completely analogous to the EDM form I factor. However, O now also gives rise to a two gluon coupling given for on-shell gluons by ? h (q)!h (0) ? (q ) H q I !q ) I q H ) , (2.60) g tM IJ ¹GtFGHI h (q) H I # ? ? I J I J I J Q q
where q"q #q . Note that the second term that appears is needed to maintain gauge invariance. An important feature of f (as well as of the other form factors) is that the constant piece f (0) ? ? must be real while, at qO0, f may have an imaginary part due to the possibility of thresholds ? giving rise to absorptive pieces. Indeed these type of e!ects have also been considered in some particular extensions of the SM (see Section 4). The phenomenology of the static CEDM (i.e., for qK0) was considered extensively in [35]. It was shown that in a hadron collider with &10 ttM pairs, both of which decay leptonically, a precision of about 5;10\g cm for f (0) could be achieved. We can extend this consideration Q ? with a simplifying assumption that f is approximately constant above the ttM threshold. We can ? then introduce the quantity f "f (4m)!f (0). With this assumption and approximation we "nd ? R ?
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that, under ideal conditions, Re( f ) and Im( f ) can also be measured to a precision of about (2}3);10\g cm. Q The Weinberg model with an extended Higgs sector provides a speci"c example of non-standard physics where one can study this general feature of the operator analysis above. We recall that the source of CP violation now are the charged Higgs exchanges (see Section 3.2.4). Since O is the only ? operator which gives a one gluon Feynman rule, the electromagnetic form factor calculated in [31] is the quantity f except for the replacement of g with e, the electric charge. The result of that ? Q reference is that thus far QCD yields a limit around 5;10\g cm. In that work, it is also Q explicitly shown (see Section 4) that the q dependence is rather mild. Furthermore, above threshold the Im( f ) was also shown to have roughly the same ball park value. Thus, studies at ? a hadron collider could exhibit CP-violating signals although admittedly the experimental challenges are formidable. Another useful way to characterize the amplitude for ggPttM is to express it in terms of form factors. There are three possible color structures which such an amplitude can have. If A, B are the color indices of the gluon and i, j the indices of the t and tM , these color structures are " , D"d !¹! , F"f !¹! . GH GH GH
(2.61)
Let us de"ne P , P to be the momenta of the gluons and P , P to be the momenta of the t,tM E E R R quarks, respectively. Let us further de"ne the variables s"(P #P ), t"(P !P ), u"(P !P ) , E E E R E R z"(t!u)/(s!2m) . R
(2.62)
Let E and E be the polarizations of the gluons in a gauge where E ) P "E ) P "0. Here, we are interested in amplitudes which violate CP. These amplitudes must also be symmetric under the interchange of the two gluons. The helicity amplitudes which satisfy these conditions are aL "f L (s, z)(E ) E )(tM t)[D, , Fz] , aL "f L (s, z)( EI EJ PN PM )(tM t)[D, , Fz] , IJNM E E aL "zf L (s, z)( )EI EJ PN PM (tM t)[D, , Fz] , IJNM R R aL "zf L (s, z)EI EJ (tM t)[D, , Fz] , IJ aL "f L (s, z)( EI EJ (P #P )N)(tM Mt)[D, , Fz] , IJNM E E
(2.63)
where f is a function of s and z and the notation [D, , Fz] means that the term may be multiplied G by any of these color structures, the index n"1, 2, 3, respectively, depending on which of these color structures apply. As an explicit example, at tree-level, the CEDM operator discussed above will contribute to the amplitude a .
D. Atwood et al. / Physics Reports 347 (2001) 1}222
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2.6. Optimized observables Due to the exceedingly short lifetime of the top quark, measurement of its couplings requires considering top production and decay simultaneously. Consider, for example, the production and decay of ttM in e>e\ annihilation (see Fig. 3), e>(p )#e\(p )Pt(p )#tM (p M ) > \ R R
(2.64)
t(p )Pb(p )#=>(p > ) R @ 5
(2.65)
with
and tM (p M )PbM (p M )#=\(p \ ) . @ 5 R Indeed each of the =! also decays leptonically or hadronically (i.e., into jets). Thus,
(2.66)
=>(p > )Pl>(pl> )#l (p ) , 5 J =\(p \ )Pl\(pl\ )#l (p ) , J 5 =!Pj (p )#j (p ) . (2.67) H H This allows one to construct a multitude of observables, involving momenta of the initial beam and various decay products, to probe the presence of anomalous vertices in the top interactions. The case of the CP violating dipole moment interactions is especially interesting to this work. Such anomalous terms could occur at the tM t, tM tZ or tM tg vertices corresponding to electric, weak or chromo-electric dipole moment of the top quark. Of experimental interest are the values of the corresponding form factors at q"s, the square of the c.m. energy rather than the moments (q"0) themselves. It is possible, therefore, that the form factor is complex. The real and imaginary parts can thus lead to distinct experimental e!ects. Two examples of simple (or `naivea) observables that can be used to probe the presence of imaginary part of the dipole moment form factor are El> ! El\ , El> # El\
(2.68)
E ! E M @ @ . E # E M @ @
(2.69)
Fig. 3. Feynman diagram describing the process e>e\PttM followed by the ttM decays tPb=>Pbl>l and tM PbM =\PbM l\l .
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Examples of simple (or `naivea) observables that probe the presence of the real part of the dipole moment form factor are pI pJ pM pN IJMN l> l\ @ @M , (pl> ) pl\ p ) p M ) @ @ (pVl> pWl\ !pWl> pVl\ ) sgn (pXl> !pXl\ ) . (pl> ) pl\ )
(2.70) (2.71)
The ability to polarize the electron beams at a future e>e\ collider (e.g., the NLC) allows us to construct additional observables involving beam polarization. Clearly, while many observables can be constructed to probe the dipole moment, we may ask whether it is possible to construct an `optimal observablea i.e., one which will be the most sensitive or will have the largest `resolving powera. A general procedure for constructing an optimal observable was given in [29]. Here we will brie#y review the method. Let us write the di!erential cross-section as a sum of two terms ()" ()# () , (2.72) where is a parameter (e.g., dipole moment or magnetic moment form factor) and is some phase-space variable (including angular and polarization variables). For an ideal detector that accurately records the value of for each event that occurs, any method for determining the value of amounts to weighting the events with a phase-spacedependent function f () which we assume is CP-odd. Let us de"ne
f ()" f ()() d .
(2.73)
Thus, the change due to the contribution from the presence of is
f ()" f () () d, f ("1) .
(2.74)
f () then has to be compared to the error in its measurement. If n events are recorded, the error is
f"
f !( f ()) f + , n n
(2.75)
where
" d" d ,
(2.76)
is the total cross-section and
f " f d" f d .
(2.77)
Note that f ()J but f and do not depend on to "rst order; we will assume that is su$ciently small so that the above approximation is valid as well as f <( f ()). We now
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25
introduce the `resolving powera, R, which measures the e!ectiveness of an operator for determining . For the function f, R is de"ned as 1 ( f ()) ( f (1)) " . R" f n ( f )
(2.78)
The statistical signi"cance S, with which the presence of a non-zero value of may be ascertained, is given by f () S" "(nR . f
(2.79)
Thus, the larger R is the more e!ective f () is for measuring . Clearly, one would like to choose a function f () to determine for which R is maximal. A special case of such observables are asymmetries, which are observables where f ()"$1. For such observables f " so (see Eq. (2.78)), R"( f (1))/" f / .
(2.80)
In this case, it is conventional to denote for an asymmetry f f " , D
(2.81)
so that Eq. (2.79) becomes S" (n . D
(2.82)
Thus, the quantity (R is the natural generalization of for more general observables among D which we would like to "nd the optimal one, i.e., that which maximizes the value of R and therefore also the statistical signi"cance with which it can be measured. In order to de"ne such an optimal function (i.e., f ), it is useful to decompose an arbitrary function f as [29] f"A #fK ,
(2.83)
where A is de"ned as f d . A" ( / ) d
(2.84)
Since rescaling f does not change the value of R we can take A"1. If we now expand the de"nition of R to lowest order in and use the above decomposition we get f d ( / ) d R" " . f d ( / ) d# fK d
(2.85)
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Since the two terms in the denominator are each positive it is clear that R is maximal when fK "0. Thus, from Eq. (2.83), f"f " , (2.86) maximizes R. Another way to understand this derivation of f is to introduce the concept of a vector space as the set of all functions on which we de"ne the scalar product
g ) g " g ()g () () d ,
(2.87)
where g are functions of . If we denote f " / then in this notation Eq. (2.85) may be rewritten as R"( f ) f )/f ) f . (2.88) Thus R is maximized when f"f "f . Let us "rst illustrate these consideration with a toy example. Consider a 2P2 scattering process where the di!erential cross-section takes the form d "A#B#C , d
(2.89)
where "cos and B, C4A. According to the preceding discussion, the function most sensitive to B is f ". Thus explicitly we "nd "2A#C, f "B, f "A#C. Using Eq. (2.78), the resolving power for the optimum choice ( f ") is 1 1 1 [ B d] \ R( f )" " . (2.90) B Ad A d 3 A \ \ Suppose we consider measuring B via another function, which has the same symmetry properties as f and is de"ned as
g ()"
#1 if '0 , !1 if (0 .
(2.91)
Then one gets g"B, g"2A#C, so that R(g )" , which is not as good as R( f ). Generalizations. This simple method outlined above for an optimized observable has been widely used and a useful generalization has been proposed by Gunion et al. [73]. In general, we will assume that the di!erential cross-section can be expressed as d (2.92) (), " c f () , G G d G where c are model-dependent coe$cients and f () are known functions of the "nal state G G phase-space variable(s), . Thus, c may be couplings; some or all of which one is trying to extract G
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27
from experimental data. So, in analogy, with the discussion above, the task is to "nd an optimal choice for the functions f () such that, with a "xed set of data, c can be deduced with maximal G G statistical precision. The coe$cients c can be extracted by using the appropriate weighting functions w () such that G G w ()() d"c . As in the special example of the electric dipole moment discussed above, in G G general, there are an in"nite number of choices for w () that satisfy this condition which is G equivalent to
w () f () d" . G H GH
(2.93)
It can be shown [73] that minimizing the statistical error is equivalent to "nding the stationary point of w w () d. In other words, solving G H
w ()w ()() d"0 , G H
(2.94)
subject to the condition in Eq. (2.93). The solution to this condition is X f () w " H GH H , G () where X "M\ and GH GH f () f () H d . M , G GH ()
The desired coe$cients, c , are then given by G c " X I " M\I , G GI I GI I I I where
I , f () d . I I
(2.95)
(2.96)
(2.97)
(2.98)
It follows that with this method of determining c , the covariance matrix is G < , c c "M\ /N , (2.99) GH G H GH 2 where is the total cross-section and N"¸ is the total number of events, with ¸ being the 2 2 e!ective luminosity, i.e., luminosity times the e$ciencies. It is important to note that since the entire method has to be implemented numerically utilizing experimental data, experimental cuts and e$ciencies can be incorporated. Indeed the method has the additional virtue that, due to its generality, it can be applied to only a subset K of the kinematic variables that can be determined if some (say M ) among the total of cannot be determined. Situations such as this can occur due to experimental limitations. For example, in the case of the important reaction e>e\PttM H, H" neutral Higgs, to fully de"ne a point in phase-space one
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must be able to identify the momenta of the t and the tM and have no more than one invisible particle. This requires the use of those "nal states in which one top decays leptonically and the other hadronically so that the t and tM are reconstructible. If this can be done, then this case corresponds to all the kinematic variables being determined. On the other hand, for those events in which the t, tM both decay purely hadronically or both decay leptonically then the situation corresponds to only a part (K ) of the kinematic variables being determined. The technique for optimization can then be applied by using the subset of variables, K , that can be observed. The functions to be used can now be de"ned as
fK (K ), f () dM . G G
(2.100)
Then the entire procedure above can be used in conjunction with these functions fK . In fact, the method should work even if one or more of fK are zero. For example, for the reaction e>e\PttM H, if experimentally one is unable to distinguish t and tM then the f () that is a CP-odd function of the I variables reduces to fK "0. I The simple method, "rst suggested in [29], for extracting the dipole moment of the top quark has been extensively applied with appropriate modi"cations to deduce e.g., dipole moments of (from ZP>\, see e.g., [74,75]), for determination of the three gauge-boson couplings in e>e\P=>=\ [76], CP-violating phase(s) in b decays [77] as well as in ttM H production [78]. Many of these studies explicitly showed that optimized observables can be very e!ective, often yielding improvements over the `naivea operators by as much as an order of magnitude. The generalization discussed in [73] was originally used speci"cally to study the reaction e>e\PttM H. It demonstrated extraction of the ttM H and ZZH coupling a, b and c from the interaction terms: ttM H: tM (a#ib )t , (2.101) Z Z H: cg , (2.102) I J IJ respectively. An important consequence in this application is that the CP nature of a Higgs particle, i.e., whether it is CP-odd or CP-even, may well be deducible from studies of some momenta correlations in e>e\PttM H. We will discuss in greater detail the applications of this method, i.e., optimization of observables, to the top dipole moments (see Sections 6.1.1, 7.2.1 and 8.1.4) and to e>e\PttM H (see Section 6.2) in this review. 2.7. The naked top The large mass of the top quark makes it signi"cantly di!erent from all the other quarks in many important ways. Since m &174 GeV the top readily undergoes two-body weak decays R tPb#= , (2.103) with &1 GeV. The top lifetime, \&1 GeV\, is therefore much shorter than the `strong interaction time scalea &1/ &10 GeV\. As a result, the top quark, in complete contrast to /!"
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29
all the other quarks, cannot appear in traditional bound states [28]. Studies of this `naked topa are therefore not masked by the di$cult non-perturbative e!ects of the so-called `brown mucka. Thus, we should think of the top-quark as an elementary fermion; we can, for instance, try to study its anomalous magnetic moment, i.e., (g!2) and even more importantly its electric dipole moment [29}31]. The magnetic moment receives a large contribution from QCD and other SM interactions but the electric dipole moment is CP-violating and cannot arise at least to 2-loop order in the SM [79}81] and so is expected to be extremely small (;10\ e cm). Non-standard sources of CP violation can cause this to be signi"cantly bigger to the point that it may well be measurable. Now, generically, the electric dipole moment and its generalization to other gauge "elds (i.e., the corresponding weak and chromo-electric dipole moments) is an interaction of the type ) E, R i.e., the top spin with an external gauge "eld (photon, Z or gluon). Therefore, determination of the dipole moment understandably involves tracking the top spin. Fortunately this is possible for the top quark, just because it does not bind, even though it is notoriously di$cult to track the spin of b or the other quarks. As we shall see spin tracking provides an extremely important tool for studying all aspects of CP violation in the top system, not just for probing its electric dipole moment. 2.8. Elements of top polarimetry Decays of the top quark are very e!ective analyzers of its spin. It is indeed useful to introduce the notion of the `analyzing powera of a decay. The analyzing power ( ) measures the degree to which R the momentum of a decay product (A), in the reaction tPA#anything, is correlated with the top spin. For an angular distribution given by d "a(1# cos ) , R d cos
(2.104)
we can de"ne in terms of the expectation value of cos R "3 cos , R
(2.105)
where is the angle between p and the spin of the top (S ) in the top rest frame. Using the optimal R observable introduced in Section 2.6, it should be clear that with a distribution such as in Eq. (2.104), the best way to determine the polarization of the top is to experimentally measure cos . Using this method, it follows that the number of top decays that needs to be observed to determine the top polarization to within 1- is given by 3 , NN" R [ ] R R
(2.106)
where is the polarization of the top quark, R N(!)!N( ) , , R N(!)#N( )
(2.107)
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N(!) being the number of top quarks with spins along the positive z axis. Here are three concrete handles on the top spin: 1. Leptonic decays tPbl>l are the best analyzers, i.e., l"1 for the correlation between pl and R . Consider the decay chain tPb=>Pbe> . In the limit that the b quark is massless, the R C V}A nature of the weak interaction forces all the particles in the "nal state to be left-handed. Thus the amplitude for the top-quark decay is proportional to factors of fermion spins times [35]: [u (1! )u ][u I(1! )v ] . @ I R J C On Fierz transformation this becomes
(2.108)
[u (1# )uA ][v A (1! )u ] , (2.109) @ J C R where uA"Cu 2, C being the charge conjugate matrix. This implies that, in the rest frame of the top quark, the top spin is in the direction of the positron three-momentum. This statement is true in the same sense that in the decay \P\ the is polarized in the direction of the O momentum. Thus, for a top fully polarized in the z-direction, the angular distribution of the positron is J(1#cos ), hence C"1. C R 2. For hadronic decays of the =, if the momentum of the = can be determined, then it is easy to see that m!2m 5 K0.4 . (2.110) 5" R R m#2m R 5 This re#ects the fact that the =-momentum is an imprecise indicator of the dM type jet in the decay =>PdM u. Here the dM jet plays the same role as the e> in the previous case. 3. From the above two points, it should be clear that we can increase from 0.4 to 1 if we can identify the dM -type jet, which plays the role of the e>. Since this jet is one of the two that reconstructs to the = in method 2 we are approximating it with the =-momentum. In the limit that m is much greater than m , the two quark jets would coincide with the =-momentum and R 5 5P1; however, with the given top mass, the result is far from this limit. R In some cases, one could determine which jet was the dM -type one by detailed examination of the decay products. This may be feasible for the case of a b-jet. However, b-jets are highly suppressed in = decay so we will not consider this possibility further. Probably such methods will not o!er signi"cant improvements over the above. On the other hand, we have not exploited the full information available in the kinematics of the decay. It was suggested in [82] that the energy distribution of the two jets could give additional information concerning which is the desired dM -jet. In particular, in =>PdM u, the dM -jet is on the average less energetic than the u-jet: and so, if we take the polarization to be in the direction of the less energetic of these two jets, one would expect some improvement. Indeed, in In passing we wish to mention that while we "nd the dM -jet to be on average less energetic then the u-jet, Grza7 dkowski and Gunion [82] found it to be the other way around.
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this case we obtain U K0.5 . (2.111) R Further, if one uses the optimization methods discussed in Section 2.6 but restricts the experimentally available information to quantities which are symmetric under the interchange of the two jets, then one can improve this further K0.63 , (2.112) R which represents the best result that can be obtained without knowing the identities of the two =-jets.
3. Models of CP violation In this section we describe the key features of CP violation in the SM and in some popular models beyond the SM such as MHDM's (multi-Higgs doublet-models) and SUSY models. In doing so, we emphasize the relevance of the new CP-violating phases that appear in these models, to top quark physics. 3.1. CP violation and the standard model In the SM, CP violation emanates from a CP-odd phase in the CKM matrix [5,6] which in#uences directly only the quark sector (for some recent reviews on CP violation in the SM, see e.g., [83}86]). In this section we will brie#y describe the properties of this #avor mixing CKM matrix. 3.1.1. General remarks The electroweak (EW) Lagrangian of the SM can be symbolically written in the form (for notation, see e.g., [87]): L"L( f, G)#L( f, H)#L(G, H)#L(G)#<(H) ,
(3.1)
where f is the fermions (quarks, leptons), G the gauge-bosons (W and B), and H the Higgs doublet. The Lagrangian in Eq. (3.1) is constructed so that it is invariant under the local (space-time dependent) symmetry group SU(2) ;;(1) . * 7 The purpose of this section in not to review the structure of this Lagrangian but rather to explore the salient features of its CP-violating part. We will therefore present a self-contained discussion only of its CP-nonconserving pieces. All CP violation in the SM originates from the term L( f, H). The hadronic part of this term is given by
* > , q #>B (q, q ) q #h.c. , LF( f, H)" >S (q, q ) HI HI I0 H* !\ I0 H* H I
(3.2)
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where we introduced the multiplets of the quark weak eigenstates q H* , q , q , H0 H0 q
H*
j"1, 2,2, N ,
(3.3)
1# q, q " 0 2
1! q . q " * 2
(3.4)
and
Also, j, k are family indices, N denotes the number of families and >S , >B are the Yukawa HI HI couplings, which are arbitrary complex numbers. In our discussion we will consider N"3 corresponding to the SM with the three known families of fermions. This Lagrangian has no fermion mass term; fermion masses must therefore be induced by spontaneous symmetry breaking (SSB) of the SU(2);;(1) symmetry of the scalar potential term <(H). In the broken state, the scalar doublet assumes a vacuum expectation value (VEV) and thus one obtains the mass terms MS and MB of the charge 2/3 and charge !1/3 quarks, respectively, for the weak eigenstates of the SU(2) ;;(1) gauge theory (i.e., q MS q and q MB q ). These quark mass matrices are related * 7 G0 GH H* G0 GH H* to the Yukawa couplings via v v >S , MB " >B , MS " GH (2 GH GH (2 GH
(3.5)
where v is the VEV of the Higgs doublet. In general, the mass matrices MB S are not hermitian, and each one depends on 9 complex unknown parameters. Since an arbitrary matrix M can be written as M"Hu, with H hermitian and u unitary, there exists a "eld rede"nition such that MS and MB are hermitian, i.e., that MS"MSR and MB"MBR [88}90]. A unitary transformation on the u and d quark "elds gives the physical basis where the mass matrices are diagonal ;R MS; "diag(m , m , m ) , (3.6) 0 * S A R DR MBD "diag(m , m , m ) , (3.7) 0 * B Q @ where ; , ; , D and D are unitary matrices that relate the weak eigenstates to the physical 0 * 0 * eigenstates. It is worth mentioning already at this point that all CP violation in the SM emanates from the apparent mismatch between the gauge and mass (physical) eigenstates of the quark "elds. For the physical states thus de"ned, there is no longer an exact SU(2) identity between the left handed d and u quarks (since they are no longer the gauge eigenstates). To see this, write the Lagrangian in terms of the physical "elds and drop the numerical factors and the coupling constants. Thus one is left with the CP-violating charge current terms X "=>JI #h.c. , ! I ! where => is the charged, spin 1, SU(2) vector-boson and
(3.8)
d
JI "(u , c , tM ) I< s ! * b
*
.
(3.9)
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Here u, c, t, d, s and b are the quark mass eigenstates. The 3;3 unitary matrix < will therefore be the product of the unitary diagonalizing matrices since
< < < SB SQ S@ (3.10) <";R D , < < < . * * AB AQ A@ < < < RB RQ R@ Expressing the fermions in this basis, the term X is the one where all the CP violation in the SM ! resides. We will therefore consider the properties of < in some detail. CP conservation requires the matrix < to be real up to a trivial rephasing of the quark "elds. In general, for 3 families of quarks it can be speci"ed by 18 complex parameters of a general 3;3 unitary matrix. However, 9 of these 18 parameters are eliminated by the unitarity constraints
c c s c s e\ B < " !s c !c s s e B c c !s s s e B s c , (3.12) !)+ !c s !s c s e B c c s s !c c s e B where, as usual, c "cos , s "sin ; the indices 1, 2, 3 are `generationa labels and is the phase. GH GH GH GH With the advent of the b-quark lifetime measurements in 1983 [92,93], Wolfenstein [94] made the important observation that the magnitude of the CKM elements exhibit a speci"c hierarchical structure. The parameterization proposed by Wolfenstein uses the Cabibbo angle, s , as an expansion parameter making this hierarchy manifest by rewriting the matrix in terms of the 4 parameters, , A, and . These are de"ned as s ", s "A, s e\ B "A(!i ) . We then arrive at the Wolfenstein representation for the 3;3 matrix [95}98]:
1! ! A(!i ) 2 8 A 1 A < " !# (1!2)!iA 1! ! # A 5 2 2 2 8 A A 1! 1! (#i ) !A 1! [1#(#i )] 1! 2 2 2
#O() .
(3.13)
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For most purposes, a simpler form, with truncation to order su$ces [94]:
1! 2 !
1! < " 5 2 A(1!!i ) !A
A(!i ) A 1
#O() .
(3.14)
Notice that the matrix is diagonal to a good approximation. This is due to the fact that the couplings between quarks of the same family are close to unity and the o!-diagonal elements become smaller as the separation between the families gets larger. Note that all CP violation in the CKM matrix is proportional to since this parameter gives a complex phase to the CKM matrix, in particular to < and < , in the above parameterization. S@ RB 3.1.2. The Jarlskog invariant There is a unique invariant way to parameterize CP nonconservation which emerges from the mixing matrix < in Eq. (3.10). That is, to introduce one invariant quantity as considered in [88}90,99], which will be independent of any phase convention of the quark "elds and will enter in every CP-violating e!ect in the SM. It was shown in [99] that in order to obtain CP nonconservation in the SM, 14 conditions must be satis"ed. These conditions can be expressed as the equation det C"!2FSFBJO0 ,
(3.15)
where C is the commutator of the mass matrices of the up and down quarks de"ned by [MS, MB]"iC .
(3.16)
Also, FS and FB are given by FS,(m!m)(m!m)(m!m) , (3.17) R A R S A S FB,(m!m)(m!m)(m!m) , (3.18) @ Q @ B Q B and J, which is sometimes referred to as the Jarlskog invariant, is de"ned through [88}90,99]: (3.19) Im(< <
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One can use the invariant parameterization of the angles of the mixing matrix < to further examine the structure of CP violation in the SM by introducing the `unitarity trianglesa (sometimes called `CP violation trianglesa) [90,100}102]. To do so, de"ne a ,<
(3.22)
where (x, y, z)"[x!(y!z)][x!(y#z)]. Here x, y, z can have two sets of values I) x"< < , y"< < , z"< < , ?H ?I @H @I AH AI II) x"< < , y"< < , z"< < . (3.23) ?J @J ?H @H ?I @I For set I OO and jOk, while for set II O and jOkOl. Therefore, one can compute the angles of the unitarity triangles, up to an overall sign, in terms of only 4 moduli of the elements of < (note that in Eq. (3.23) only 4 independent moduli enter x, y, z for any of the allowed values of , , , i, j, l). In other words, the existence of CP violation or equivalently JO0, may be inferred in the SM with three generations if the three sides of a unitarity triangle can form a triangle with non-zero area. The moduli that enter into Eq. (3.23) may be obtained from purely CPconserving observations. However, an experimental measurement of CP violation is needed to "x the sign of J. An example of such a unitary triangle is the one commonly used in B physics studies [95}98], which is constructed out of the three vectors e , e and e in Eq. (3.21) corresponding to <
Pl>l and from hyperon decays, e.g., Ppe\ . In C addition, it is determined from a study of charm production via neutrino beams as well as from decays of charm to non-strange "nal states, albeit with much less precision. The average of the two determinations gives [95}97] < ""0.2205$0.0018 . (3.24) SQ The next best determined element is < and, through it, the Wolfenstein parameter A via A@ < "A . (3.25) A@
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< is deduced using semi-leptonic B decays to inclusive and exclusive "nal states, both at LEP A@ and at CLEO. The error in its determination is dominated by theory. In recent years, heavy quark symmetry and heavy quark e!ective theory [103,104], have had a signi"cant impact in reducing the model dependence. The average of various techniques now gives [95}97]: < "0.0397$0.0020 ; A@
(3.26)
A"0.81$0.04 .
(3.27)
thus,
The other two parameters, and , are poorly known. Considerable theoretical and experimental e!ort is being directed to improve their determinations. B physics experiments at e>e\ based B-factories as well as other facilities will surely improve our knowledge of , and . Indeed this will be the focus of intense theoretical and experimental activity in the near future in providing precision tests of the SM. In the context of decays of the b quark, it is very useful to consider the unitarity triangle which involves the b d elements
" 1! , 2
" 1! . 2
(3.29) (3.30) (3.31)
(3.32)
Factoring out the common quantity A, the three elements in Eqs. (3.29)}(3.31) can be given a geometrical representation in the (, ) plane of a triangle with apexes at A(, ), B(1, 0) and C(0, 0) (see Fig. 4).
Fig. 4. Unitarity triangle in the complex (, ) plane.
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Referring to that "gure we have 1! 2 < S@ "(# , AC" < A@ < (3.33) AB" RB "((1!)# . < A@ The angles , , of this triangle (see Fig. 4 for their de"nition) provide a basis for testing the SM especially with regard to its description of CP violation phenomena and CKM unitarity. In particular, unitarity implies that ##"1803. Let us now brie#y mention the key experimental and theoretical ingredients that enter to provide the current bounds on , or alternatively on , . The evidence for CP violation from the K!KM system through the indirect CP violation parameter, , plays a crucial role in constraining and via [13] ) G f m m S(x )# A(1!!(# )) S(x ) " $ ) ) 5 B A ! 1! ) A R ) 2 6(2m )
# S(x , x ) . A R
(3.34)
The represent QCD corrections, evaluated to next-to-leading order (NLO), the S's are functions G of x "m/m, f is the Kaon decay constant and B is the so called `baga parameter. Perhaps the O O R ) ) best determination of B comes from lattice calculations (for recent reviews see [105}108]) ) B K0.9$0.1 . (3.35) ) Using the experimental value of along with G , m , m , B , etc. in Eq. (3.34), leads to an ) $ 5 R ) allowed range for , as shown in Fig. 5. < and therefore , also enter intimately in controlling m , i.e., the mass di!erence between RB B the two mass eigenstates of the B!BM system. Thus, B B G m m " $ 5 A[(1!)# ]m B f B B B x F(x ) . (3.36) B R R 6 Here F(x ) is calculated perturbatively and is given to leading order in [109]. , f B and B B have R the same meaning as their values for the K system, as in Eq. (3.34). Once again f B , B B need to be calculated non-perturbatively and lattice QCD provides perhaps their best determination. The results from existing lattice calculations are best summarized as [95}97,105}108] f B "165$20$30 MeV ,
(3.37)
B B (2 GeV)"1.0$0.10$0.15 .
(3.38)
and
The value given here is the renormalization-group-invariant-B often denoted as BK . ) )
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Fig. 5. The allowed regions for the SM parameters and are shown. The solid curves show the 68% and 95% con"dence regions. The 1- bound originating from neutral K decays is shown by the dashed curves. The 1- bound * originating from the rate of bPc transitions is shown with the dot-dashed curve and the 1- bound originating from B!BM oscillations is shown with the dotted curve.
Using the experimental value of m in conjunction with m , G , f B (B B , etc., in Eq. (3.36), we can B R $ represent the allowed range on the , plane in the form of a hyperbola (see Fig. 5). A non-trivial test of the CKM paradigm is obtained by examining simultaneously the allowed range of , through a determination of < from semi-leptonic charmless B-decays. In this S@ regard, various techniques are used to deduce < /< from exclusive and inclusive semi-leptonic S@ A@ decays. For now, the uncertainties in the theoretical models of these transitions is quite substantial giving [21,22,95}97]
< S@ "0.08$0.02N(# "0.35$0.09 . (3.39) < A@ This compendium of experimental and theoretical information on , m and bPul is used to ) B obtain the best-"tted values [95}97]: "0.10> , "0.33> . (3.40) \ \ The corresponding 68% and 95% CL contours are shown in Fig. 5. In the future, the measurement of B !BM oscillations will provide an extremely important test Q Q of the SM. The point is that the ratio of the mass di!erences m f B < m B " B B B B RB , (3.41) m Q f Q B Q Q < m RQ Q will involve signi"cantly less uncertainty due to hadronic matrix elements (i.e., f B). Aside from the CKM elements, the most uncertain factor on the rhs of Eq. (3.41) is f B r , Q QB f B B
Q B
.
(3.42)
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However, theoretical uncertainties in extracting < from r are expected to become smaller RB QB [110,111] in comparison to the errors in extracting < from m alone. RB B The experiments at LEP have already made signi"cant progress in studying B !BM oscillations. Q Q A combined analysis of ALEPH, DELPHI and OPAL leads to [95}97,112]: m '8.0ps\ at 95% CL . (3.43) Q Incorporating this along with , m and bPul, into the , constraints one "nds [95}97]: ) B "0.11> , "0.33$0.06 . (3.44) \ Comparing this with Eq. (3.40) we see that the LEP bound on m is already reducing the negative Q error on appreciably [95}97]. Further slight changes (see the update of Buras in [14]) result from improvement, obtained by combining LEP/SLD/D0 data, of the limit on m to read: Q m '12.4ps\ at 95% CL , (3.45) Q and from the small increase of < /< to [113] S@ A@ < /< "0.091$0.016 . (3.46) S@ A@ Translated to and (see Fig. 4) yields sin 2"0.71$0.13,
sin "0.83$0.17 .
(3.47)
The above value of sin2 is consistent with the recent CDF result [23,114] sin 2"0.79> . \ 3.2. Multi-Higgs doublet models In the SM the interaction of the only neutral Higgs-boson with fermions is automatically P and C conserving as well as #avor conserving. This property is not valid in general in models beyond the SM. In this section we consider extensions to the SM involving the addition of extra Higgs doublets. CP-violating e!ects in such models can originate in the scalar sector and be manifested in the physics of fermions, particularly the top quark. For such an e!ect to occur, two or more complex SU(2) doublets of Higgs "elds are required; this was "rst pointed out by Lee [115,116]. However, the mere presence of more than one doublet does not guarantee CP violation in the Higgs sector. For instance, a CP-violating phase in the case of models with two Higgs doublets (2HDM) can be rotated out of the Higgs sector entirely if one imposes various discrete symmetries as will be discussed below. But if such phase(s) cannot be rotated away, this approach leads to CP violation from neutral Higgs-boson interactions, from charged Higgs-boson interactions, and perhaps in addition from the presence of a non-vanishing phase in the CKM matrix. In all cases, the various types of CP violation are presumably related at a fundamental level to CP violation in the Higgs potential, but because of our ignorance of the Higgs sector, in practice the parameters of each type of interaction are independent and should be separately measured. Another general feature of CP violation in an extended Higgs sector is that larger e!ects are expected in heavier quarks systems (compared to the usual SM approach), because Higgs-boson
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couplings to fermions are proportional to the fermions masses. This makes the top quark system an especially good testing ground for such phenomena. CP violation in the Higgs sector can arise in models where the Higgs potential may contain complex couplings. This might lead directly to a CP-violating interaction or to complex VEVs of the Higgs "elds which can induce CP-violating e!ects. In addition, as we shall see in some examples below, it is also possible that a real potential can lead to a ground state with a complex VEV, in which case CP is broken spontaneously. In any case, there are generally a large number of parameters in these models so that considerable experimental e!ort will eventually be required to determine them all. In particular, it is important to consider which predictions of such models di!er from the SM, so that might lead to early signs that extra scalar "elds are present. CP violation in top physics is especially useful since the SM contributions to CP violation in top quark reactions are negligible and the mass-dependent coupling of the Higgs means that top quark physics is very sensitive to such e!ects. It is convenient to classify CP symmetry-breaking in the scalar sector into three di!erent categories: hard (intrinsic), soft and spontaneous. Hard or intrinsic CP violation refers to symmetry-breaking terms with dimension four, for example, terms in the Lagrangian with complex Yukawa coupling constants, or with self-coupling of scalar "elds. Soft breaking is associated with terms in the Lagrangian with canonical dimension less than four. If the Lagrangian starting from the outset is CP invariant, CP violation can still be achieved by introducing complex phases from the VEVs of the scalar "elds (i.e., spontaneous CP violation). In the following, we will consider simple versions of 2HDM and three Higgs doublets model (3HDM) in which CP violation is manifested in the interactions of neutral and charged Higgs particles with fermions. 3.2.1. Two Higgs doublet models We start with a description of the most general 2HDM. The Higgs potential for such a 2HDM is given by [117,118] <( )"! R ! R !( R #h.c.) # ( R )# ( R )# ( R R )# ( R )( R ) # [ ( R )#h.c.]#[ R R # R R #h.c.] ,
(3.48)
where (i"1, 2) are Higgs "elds such that G
> " G . G G
(3.49)
The scalar spectrum of a 2HDM consists of three neutral and two charged Higgs-bosons which we will denote by HI (k"1, 2, 3) and H!, respectively. The important parameters in <( ) that may drive CP violation in the Higgs sector are: , , and . It is the di!erent choices of these parameters that will determine which type of mechanism is generating CP violation in the 2HDM.
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It should be noted that without the complex VEVs of the two doublets (or only one of them that can generate spontaneous CP violation), one needs at least two terms out of , , and to be non-zero in the Higgs potential (see Eq. (3.48)) in order to have CP non-conservation in the model. That is, if there is only one complex coe$cient in the Higgs potential prior to SSB, then SSB leads via the minimization condition, R
v " sin , (2
(3.50)
where v tan " v
and v"(v #v ,
(3.51)
then with non-zero and real, CP violation can arise from non-zero complex entries of one or more of , and . If , , and are all real then CP violation can occur spontan eously (i.e., through the relative phase ). If one of , and is complex then in addition to the complex VEVs above that appear after SSB, there is an explicit CP nonconservation in <( ). However, whether CP violation is spontaneous or explicit, the structure of the CP-violating sector of the model can be rede"ned to depend only on the relative phase . Let us consider, for example, the particular choice " "0 which follows, for example, from type II 2HDM, that we will describe later in this section. Here CP violation can occur only if Im[ /( )]O0 [119].
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If, on the other hand, arg( )"2arg( )", then the "eld rede"nitions Pexp(#i/4) and Pexp(!i/4) would eliminate this phase. A phase di!erence between and is therefore essential in order to get CP non-conservation in a 2HDM with no hard couplings of the type and and, in particular, cannot vanish. Let us remark at this point that in the minimal supersymmetric standard model (MSSM) with only two Higgs doublets, "0 is required by the supersymmetric nature of the Lagrangian. Therefore, no CP violation can arise from the pure Higgs sector in the MSSM. We will return to this point when we will describe CP violation in SUSY models (see Section 3.3). In models with extended Higgs sectors, an important experimental constraint is that the processes with #avor changing neutral currents (FCNC) are severely suppressed. To understand the extent to which such models will give rise to FCNC, consider the most general Yukawa interaction of quarks in a 2HDM [120}122]: (3.52) L " [(u , dM ) (; I #; I )(u ) #(u , dM ) (D #D )(d ) #h.c.] , G G * GH GH H 0 G G * GH GH H 0 7 G H where i and j are generation indices, u and d stand for up and down quarks, respectively, and I "i H while ;, ;, D and D are matrices in #avor space. In the general case presented above, where either both ; and ; or D and D are present in L , FCNC will appear in the 7 model. To avoid FCNC in a 2HDM, most models which have been considered impose an ad hoc discrete symmetry on the 2HDM Lagrangian [123]; the idea being that such a symmetry may originate from physics at a more fundamental level. We will return to this point later when we discuss CP violation in a 2HDM with no FCNC at tree-level. In the next two sections we examine two widely studied cases of the 2HDM where CP violation arises from the Higgs sector. In Section 3.2.2 we consider a model where FCNC are in fact present. In particular, we take '0 and real, and also , , are non-zero and real (corresponding to scenario 3 above). Since FCNC will be present, the parameters of the model will be constrained to keep them below the experimental limits. In Section 3.2.3 we consider a case which falls into scenario 2 and has #avor conservation built in at tree-level. In particular, " "0 and a discrete symmetry (which is only softly broken) is imposed on the model [30,124]. 3.2.2. 2HDM with CP nonconservation and FCNC In this model, no discrete symmetry on the Yukawa couplings of any kind is imposed, thus allowing for the presence of all the couplings which appear in <( ) and L . However, in view of the 7 low-energy data on FCNC, one has to require that the #avor changing (FC) parameters meet those experimental constraints. One systematic way that has been suggested to achieve this without "ne tuning the parameters is to impose an approximate global ;(1) symmetry which acts only on fermions (see [120}122,125] and references therein). This symmetry will be responsible for the smallness of the #avor changing couplings in this model and it leads to the Cheng}Sher ansatz [126,127] which imposes a hierarchy on the terms of L that can reasonably evade FCNC 7 constraints. We choose not to concern ourselves with the technical aspects of imposing such a ;(1) symmetry, but rather concentrate on the CP-violating consequences of a 2HDM with FCNC, the so called Model III. A more detailed discussion about the #avor changing parameters and their experimental constraints can be found, for example, in [128].
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We can rewrite L for this model in terms of the neutral and charged Higgs mass-eigenstates 7 HI and H!, respectively. We divide L into four terms [120}122]: 7
(3.53) L "((2G ) [L&#L& >$!#(2(L&!#L&!>$!)] , 7 7 7 $ 7 7 ! where L& >$!(L& >$! ) contains the FC e!ects for the neutral (charged) Higgs-bosons, and 7 7 L&(L&!) has no #avor changing e!ects other than the ones expected from the CKM matrix of the 7 7 SM part of the theory which also factors into the fermion charged Higgs coupling. These terms are written as follows: L&" (u G m G IG uG #dM G m G IG dG #h.c.)HI , (3.54) * B B 0 7 * S S 0 G I (3.55) L&!" (u G m H H < dH H>#dM G m H H $!" (u G (m G m H SIuH #dM G (m G m H BIdH #h.c.)HI , * B B GH 0 7 * S S GH 0 G$H I ! L& >$!" (u G < (m HY m H B dH H>#dM G
sin H 1 D (S$ ) , e ND F\BDH ! D K GH GH sin sin
(3.58) (3.59)
and (3.60) IG "f (R, G ) , D D (3.61) D I"D f (R) . GH GH Here f and f are functions of R, G and of R, respectively. S$ is an arbitrary o!-diagonal real D matrix. is the relative phase between the two VEVs as de"ned in Eq. (3.50) and G is the phase D
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associated with the mass m G and is de"ned through D (3.62) (2m G e ND BDG ,(g$G cos e ND F#g$G sin )v , D where g$G (g$G ) are diagonal elements of ;, D(;, D) de"ned in Eq. (3.52), and "# or ! for D up or down fermion, respectively. Evidently, L manifests four patterns of CP violation in the 2HDM being considered, all of 7 which are being driven by the relative phase (which appears after SSB) between the two VEVs and by some de"nite choice of the parameters , , and in <( ): 1. CP violation induced by the complex Yukawa couplings G which appear both in the neutral D and the charged Higgs sectors. 2. Scalar-pseudoscalar mixing in the couplings of a neutral Higgs species with fermions. That is, the mixing of the neutral imaginary and real parts of the two Higgs doublets which results from such a model generates neutral Higgs mass eigenstates that do not have a de"nite CP-property. Thus, the couplings of neutral Higgs with fermions will have the generic form (3.63) HI M "HI fM (aI #ibI ) f , DD D D where aI and bI are functions of R, tan and . D D D 3. The phases in D which yield CP violation in FCNC interactions both in the neutral and the GH charged Higgs sectors. 4. The usual CKM matrix < which gives CP violation in the charged Higgs interactions much like that of the charged =-boson interactions in the SM. 3.2.3. 2HDM with CP nonconservation and no FCNC There is a natural way suggested by Glashow and Weinberg [123] to have tree-level FCNC vanish. This idea that there may be a discrete symmetry present in the 2HDM Lagrangian, also implies the vanishing of CP violation if this discrete symmetry is exact (see discussion below). Depending on the discrete symmetry imposed, we can then obtain di!erent versions of a 2HDM. The two cases usually considered are the discrete symmetries D and D ' '' D : ; ; (d ) ; (u ) P! ; ;!(d ) ;!(u ) , (3.64) ' G0 G0 G0 G0 D : ; ; (d ) ; (u ) P! ; ;!(d ) ; (u ) . (3.65) '' G 0 G 0 G0 G0 The models with these symmetries are referred to as type I and II, respectively, depending on whether the ! and charge quarks are coupled to the same or to di!erent scalar doublets. Let us describe now the 2HDM of type II (sometimes also referred to just as Model II) following the notation in [124]. With the discrete symmetry D , we can build the Lagrangian for the 2HDM '' of type II as a special case of the general 2HDM Lagrangian described in the previous section, where: (a) In order that L in Eq. (3.52) be invariant under D , ; "D "0 is required. Thus, 7 '' GH GH L " [(u , dM ) ; I (u ) #(u , dM ) D (d ) #h.c.] , (3.66) 7 G G * GH H 0 G G * GH H 0 G H
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and hence only gives mass to charge # quarks and only is responsible for the mass generation of the charge ! quarks. (b) If all the terms that are non-invariant under the operation of D in the Higgs potential are zero '' and the symmetry is exact, then there is no CP violation in the theory. Therefore, allowing only for a one non-zero value of the soft breaking term O0, we have in the Higgs potential of Eq. (3.48), " "0. (c) Without loss of generality, is chosen as real (this can be done by `rephasinga ) and is chosen as complex, thus having explicit CP violation already at the Lagrangian, in addition to the relative phase between the two VEVs which arises after SSB. With the above three points and respecting the discrete symmetry (except for the soft breaking terms which do not introduce FCNC at tree-level), the 2HDM of type II can be extracted from the general 2HDM of the previous section by taking [129]: ", "0 ; B S
(3.67)
"tan , "!cot . B S
(3.68)
thus
Also because of point (a) above, we have D "0 in L ; thus, GH 7 ! L& >$!"L& >$!"0 , 7 7
(3.69)
and we are left with only two CP-violating mechanisms in this model: 1. The scalar-pseudoscalar mixing described in the previous section (i.e., Eq. (3.63)), where still aI and bI are functions of the 3;3 Higgs mixing mass matrix R, tan and . Note, however, D D D is now real and is given by Eq. (3.68) above. D 2. The usual CKM matrix < which gives CP violation in the charged Higgs interactions. For later use, let us introduce the following notation for the HI+M and HI<< (<"= or Z) parts of the Lagrangian in a general 2HDM g m LHI "! 5 D HIfM (aI #ibI ) f , DD D D (2 m5
(3.70)
LHI "g m C cIHIg
(3.71)
where C ,1; m /m . Note that in the SM the couplings in Eqs. (3.70) and (3.71) of the only 5_8 8 5 neutral Higgs present are a "( , b "0 and c"1, and there is no phase in the HI+M coupling. In D D Model II, for up quarks for example [124]: aI "R /sin , bI "R /tan , cI"R sin #R cos , S I S I I I
(3.72)
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where tan ,v /v and v (v ) is the VEV responsible for giving mass to the up(down) quark. R is S B S B the neutral Higgs rotation matrix which can be parameterized by three Euler angles as follows [124]:
c
s c s s , (3.73) c c c !s s c c s #s c R" !s c s s !c s c !c s !c s s #c c where s , sin and c , cos . G G G G A general feature of this class of 2HDMs, in which CP violation results from scalar}pseudoscalar mixing in the neutral Higgs interactions with fermions, is that only two out of the three neutral Higgs-bosons can simultaneously have a coupling to vector-bosons and a pseudoscalar-type of coupling to fermions. Denoting these two neutral Higgs-bosons by h and H with couplings aF , bF , cF and a&, b&, c& (corresponding to the light and heavy neutral Higgs, respectively), an D D D D important aspect of these 2HDMs, which has crucial phenomenological implications for CP violation, is that these couplings are subject to the constraint bF cF#b&c&"0. This is a general D D feature of CP violation induced by mass mixing and is due to the existence of a GIM-like cancellation, dictating that CP-odd e!ects must vanish when the two Higgs-bosons h and H are degenerate. To end this section let us brie#y comment on the existing experimental limits on the neutral and charged Higgs masses and on the couplings aI , bI , cI. There are very good reviews on this subject in D D literature and we only wish to point out the highlights of those investigations. The existing limits are usually given on tan and they depend on the mass of the charged Higgs-boson or the neutral Higgs-boson of the theory. They can be translated to bounds on aI , bI , cI using Eq. (3.72). In D D particular, the limits are obtained from the experimental constraints using low-energy data on B!BM , D!DM and K!KM mixing, , bPu, bPc and bPs transitions [130}134], on BP X ) O decays [135,136], on (g!2) [137], or from high-energy processes such as e>e\PZh [138], the I decay tPbH> [139}147] and from Z decays [148}150]. Typically, they "nd that tan (1 is allowed if the charged Higgs mass is above several hundred GeV (recall that a small tan enhances the Higgs coupling to the top quark). In order to have tan :0.5 the charged Higgs mass is required to be typically 9500 GeV. If tan is su$ciently large, i.e., tan &O(m /m ), then a light R @ charged Higgs-boson is possible, of the order of several tens of GeV. Note also that there are theoretical approximate lower and upper bounds on tan coming from perturbative considerations. That is, in order for a perturbative description to remain valid, tan has to roughly satisfy 0.1:tan :100 [131]. The lower (upper) limit corresponds to the a perturbative top (bottom) Yukawa coupling. The inclusive decay BPX , which is equivalent at the quark level to bPs, plays a unique role Q in constraining the parameter space of 2HDMs [132]. Recall "rst the CLEO observation [155] Br(BPX )"(2.32$0.57$0.35);10\ and the corresponding 95% CL bounds of 1;10\( Q Br(BPX )(4.2;10\. These were updated by CLEO to read Br(BPX )((3.11$0.8$ Q Q 0.72);10\ where, at 95% CL, 2.0;10\(Br(BPX )(4.5;10\ [156,157]. Furthermore, Q ALEPH presented the result [158] Br(BPX )((3.15$0.35 $0.32 $0.26 );10\, Q
consistent with CLEO. In Model I, the two contributions of the charged Higgs scale as cot causing signi"cant enhancements to the decay rate for small values of tan . An especially
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interesting aspect of Model I is that the contribution of the two charged Higgs interfere destructively for some values of the parameter space. Consequently, the lower and upper bounds from CLEO enable us to place stringent constraints on the mass (m ! )!tan plane [132]. Most notably, for & small value of tan (especially for tan :0.3), m ! :1 TeV gets excluded as can be seen from & Fig. 6 [151]. Br(bPs) also places important restriction on Model II [132], again, especially for small values of tan (see Fig. 7) [151}154]. An important di!erence with Model I is that Br(bPs) is now always larger than in the SM, independent of tan [132]. Thus, m ! :300 GeV is ruled out & practically for all values of tan . One can relax these bounds if MSSM contributions (chargino loops) are added to the charged Higgs in the loop [132]. 3.2.4. Three Higgs doublet model As was mentioned above, if one chooses to adopt natural #avor conservation (NFC), enforced by some discrete symmetry (see e.g., Eqs. (3.64) and (3.65)), then the minimum number of Higgs doublets which allows CP violation in the gauge model with NFC is three [159,160] (the other possibility is two Higgs doublets plus one singlet but we choose not to discuss it). In the Weinberg 3HDM [62,161,162], CP violation in the Higgs sector arises from the phase di!erences of the VEVs and from the complex quartic terms in the Higgs potential (as will be shown
Fig. 6. The excluded regions in the m ! !tan plane resulting from the present CLEO bounds in Model I and for & m "175 GeV. The excluded regions are (from left to right) to the left of the "rst curve and between the second and third R curves. Updated "gure from [132] (see [151]). Fig. 7. Constraints in the m ! !tan plane in Model II from the CLEO bound on Br(BPX ). The excluded region is & Q to the left and below the curves. The upper line is for m "181 GeV and the lower line is for m "169 GeV. We also R R display the restriction tan /m ! '0.52 GeV\ which arises from measurements of BPX as discussed in [152,153]. & Figure taken from [154].
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later). This model has many unknown parameters and it allows the standard CKM mechanism as one of the sources of CP non-conservation. Let us therefore present its spontaneous version which has a very attractive feature: spontaneous CP violation and NFC together with a real tree-level CKM matrix (for three generations) as a starting point, imply that CP violation is generated only after SSB via complex VEVs [163,164]. Thus, in such a scenario CP is a good symmetry of the Lagrangian before SSB and CP violation at tree-level comes solely from Higgs-boson exchanges once the VEVs are assigned a non-vanishing phase. The most general scalar potential with three Higgs doublets consistent with NFC is given by <( )" m R # a ( R )( R ) GH G G H H G G G G$H G
# b ( R )( R )# c ( R )( R )#h.c. . GH G H H G GH G H G H G$H G$H
(3.74)
Hermiticity of the scalar potential requires that the m terms, the a and the b be real G GH GH while c need only be hermitian. However, if CP is broken spontaneously, all the parameters of GH <( ) can be chosen to be real before gauge symmetry breaking and we choose to work in the latter scheme. The Higgs doublets can be written as
> > G , " G " G (v # #i ) ( G G G G
(3.75)
v "v e FG . G G
(3.76)
with
Assuming that the third Higgs doublet does not couple to quarks and that its VEV does participate in breaking SU(2);;(1), the general Yukawa interactions consistent with NFC read L ">DM G (>H;H #HDH )#>;M G ( ;H !>DH ) 7 GH 0 * * GH 0 * * # >EM G (>HNH #HEH )#h.c. , GH 0 * *
(3.77)
where ;"(u, c, t), D"(d, s, b), E"(e, , ) and N"( , , ). The Yukawa couplings are chosen C I O to be real as CP is assumed to be a good symmetry of L . 7 After SSB and after a phase rede"nition of the quark "elds in order to obtain real quark mass matrices, we can rewrite L as 7 > > > L>" ;M KM D ! ;M M KD # NM M E #h.c. , * " 0 0 3 * * # 0 7 v v v
(3.78)
Unlike Eq. (3.48) which describes the Higgs potential for a 2HDM, we use here the more generic form of the Higgs potential applicable also for arbitrary number of Higgs doublets.
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for charged Higgs-bosons, with " e FG and a real CKM matrix, denoted here by K. Also, G G L " DM M D#i DM M D# ;M M ;!i ;M M ; 7 v " " 3 3 v v v (3.79) # EM M E#i EM M E , # # v v for neutral Higgs-bosons, where M , M and M are diagonal mass matrices. The Yukawa 3 " # interactions of Eqs. (3.78) and (3.79) are still CP invariant, however, the scalar "elds , and are G G G not the physical Higgs states with de"nite masses. The mass matrix in the basis of >/v is G G X #X !X !i> !X #i> m" !X #i> X #X !X !i> , (3.80) !X #i> !X #i> X #X where
X "[b #c cos( ! )]v v , GH GH GH G H G H
(3.81)
and >"!c v v sin 2( ! )"!c v v sin 2( ! ) "c v v sin 2 . (3.82) Since the parameter > is, in general, non-zero, it is evident that CP violation in the charged Higgs-boson sector comes from the imaginary part of the o!-diagonal Higgs-boson mass matrix elements. The unitary matrix which relates the weak eigenstates > to the physical charged states H> is G G de"ned by
> G> (3.83) > ";> H> , > H> where G> is the charged Goldstone-boson which is absorbed into the =>. ;>, which has three arbitrary phases, of which two can be removed by a rede"nition of H> and H>, can be parameterized exactly in the same way as the CKM matrix [161,162]:
c
s c s s ;>, !s c (3.84) c c c #s s e B& c c s !s c e B& , !s s c s c !c s e B& c s s #c c e B& with s , sin I , c , cos I and are the charged Higgs mixing angles and phase, respectively. G G G G & From the gauge sector, it is straightforward to show that
(2(2G )\"v #v #v "v , $
(3.85)
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or that 1 G>" (v >#v >#v >) . v
(3.86)
It follows from Eqs. (3.83) and (3.84) that v "c v, v "!s c v, v "!s s v . (3.87) Therefore, the mixing angles I , I are determined by the VEVs v , v and v whereas I and depend on the parameters of the Higgs potential. & In terms of the Higgs mass eigenstates, the Yukawa interactions in Eq. (3.78) become L>"(2(2G ) ( ;M KM D # ;M M KD # NM M E )H>#h.c. , $ G * " 0 G 0 3 * G * # 0 G 7 G
(3.88)
with c c c #s s e B& c s c !c s e B& s c , " , " , " s c s s c s s c c s !s c e B& c s s #c c e B& " , " , " , c s c s s and we see that
(3.89)
Im( H)"!Im( H), Im( H)"!Im( H) , Im( H)"!Im( H) . (3.90) As in the charged Higgs case, we can write down an analogous 6;6 real mass matrix for the neutral scalar states. Then the neutral Higgs-boson Yukawa interactions in Eq. (3.79) become [160]: L "(2(2G ) (g DM M D#g DM M i D#g ;M M ; 7 $ G " G " G 3 G # g ;M M i ;#g EM M E#g EM M i E)H , (3.91) G 3 G # G # G where the couplings g are real. Since M and M i have opposite P, ¹ and CP transformation G properties, P and CP can be violated through the exchange of neutral Higgs-bosons. We will now brie#y mention some of the more notable constraints on this class of models [59,165}167]. One class of restrictions that are of some importance follow (as in the case of the 2HDM) if we further assume that the Higgs sector of the theory is perturbative [131]. These lead to :120, :6 . (3.92) G G Since these complex coupling constants arise from the diagonalization of the charged scalar mixing matrix, they obey the relation H"1 . G G G
(3.93)
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Thus [165], Im( H)"!Im( H) ,
(3.94)
and Im( H)4 :720 . (3.95) G G G G Assuming for simplicity that one of the H>, for instance H>, is very heavy, then B}BM mixing imposes an important constraint on [131,168]: :2 for m &m & 8 (3.96) :3 for m &2m . 8 & Using BPX, a constraint on is deduced [152,169]: (3.97) :2m /GeV . & A direct bound on Im(H) comes from the electric dipole moment of the neutron (d ) [170,171]: L Im(H):20 for m &m & 8 :100 for m &2m . (3.98) & 8 Interestingly enough, the strongest constraint so far actually comes from a CP-conserving process bPs [165,167]. The amplitude for bPs receives contributions from terms proportional to Im( H) that do not interfere with the other terms. G G Thus, these terms only enter quadratically in the expression for the rate for bPs. A conservative bound on Im(H) (where Im(H)"Im( H)"!Im( H)) is obtained by assuming that such a contribution saturates the measured rate for bPs. These constraints are displayed in Fig. 8 [132,151] as a function of the light charged Higgs mass (m ) for various values of the heavier & charged Higgs mass (m ), subject to the restriction m 4m . In the "gure, the bottom solid & & & curve corresponds to the case when m <m so that the contribution of the second charged & & Higgs is neglected. We note that the constraints depend strongly on m and they essentially & disappear when m Km due to a cancellation between the two contributions [132,151] & & resulting from a GIM-like mechanism. It is useful to note how stringent these constraints are. For example, Im(H):1.5 for m &m & 8 (3.99) :2.5 for m &2m , 8 & for m &500 GeV. A very important consequence of these tight bounds on Im(H) is that the & charged scalar exchange can only make a negligible contribution to the CP violation parameters in KP2, i.e., or . Therefore, CP violation in the 3HDM cannot be the sole source of the observed CP violation. We should note, though, that in the original Weinberg model for three Higgs doublets, CP is not assumed to be an a priori symmetry of the model. Thus CP violation arises from complex quartic terms in the Higgs potential as well as from the phase di!erences of the VEVs. In addition, one has the complex CKM phase as an independent source of CP violation which may be able to accommodate the CP violation in the Kaon system.
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Fig. 8. Constraints on Im(X>H) (,Im(H) in our notation) as a function of the lightest charged Higgs-boson mass M > ,m , with m "100, 250, 500 and 750 GeV corresponding to (from left to right) the dashed, dashed-dotted, solid & & & and dotted curves, respectively. The bottom dashed curve represents the case where the H! contributions have been neglected. The allowed region lies to the right and below the curves. m "175 GeV is used. Updated "gure from [132] R (see [151]).
3.3. Supersymmetric models Needless to say, the minimal SUSY extension of the SM is a very appealing theory (for reviews on supersymmetry see e.g., [172}177]). Among its compelling features are: it allows for radiative electroweak symmetry breaking (REWSB), it uni"es the gauge coupling constants, with masses of superpartners not much heavier then a TeV, it gives a well-grounded explanation to the hierarchy problem and it provides a good dark matter candidate } the lightest SUSY particle. New non-SM mechanisms of CP violation are introduced in each version of such SUSY models [44,172}179]. It is again the top quark sector in these models that may exhibit large CP-violating e!ects due to its very large mass. In particular, the supersymmetric partners of the top quark (these two scalar particles are often referred to as the stop and denoted by tI ), can be responsible for relatively large CP-violating phenomena. Such e!ects are enhanced by the possibility of having large mass splittings between the two stops which is in turn due to the relatively large top mass. This type of SUSY CP violation in top quark reactions has received considerable attention in the past few years. They are all strongly dependent on the magnitude of the low-energy phase of the soft trilinear breaking term A in the SUSY Lagrangian and we will describe some of these works in R the following sections. Although the top quark provides a good laboratory to investigate CP violation in SUSY models, 2-loop e!ects can also induce CP violation in the neutron electric dipole moment (NEDM) as well as in the electron electric dipole moment, and thus provide important constraints on such models as has been considered in [180,181].
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Another possible manifestation of the CP-violating phase arg(A ) is Baryogenesis in the early R universe. It was shown that with arg()P0, t squarks can mediate the charge transport mechanism needed to generate the observed baryon asymmetry, even with squark masses & hundred GeV, provided that arg(A ) is not much suppressed [43]. We will therefore emphasize here the phenomR enological importance of possible CP-violating e!ects which may reside in tI !tI mixing and are * 0 therefore proportional to arg(A ). Indeed, due to experimental constraints on the NEDM, the R possible phase in the Higgs mass term, i.e., arg(), is expected to be small (see below). Thus, arg(A ) R should be practically the only important SUSY CP-odd phase observable in high-energy reactions. Of course, the most natural place to look for such e!ects, driven by arg(A ), is high-energy processes R involving the top quark. Thus, CP-violating e!ects of the top quark observable in the laboratory may have direct bearing on Baryogenesis in the early Universe. It was stated in [182] that using the relations obtained from the renormalization group equations (RGE) of the imaginary parts in the SUSY Lagrangian, combined with the severe constraint on the low-energy phase of the Higgs mass parameter, , from the present experimental limit [183] on the NEDM, the phase in A at low-energy scales is likely to be very small provided R one imposes some de"nite boundary conditions for the SUSY soft breaking terms. As a consequence, at high energies, any CP-non-conserving e!ect that is driven by arg(A ) will then be R suppressed leaving top quark reactions almost insensitive to CP-violating e!ects of a SUSY origin in models with these assumptions. On the other hand, we will describe below the key phenomenological features of a general MSSM and a GUT-scale N"1 minimal SUperGRAvity (SUGRA) model. We will demonstrate that the prediction made in [182] depends on the GUT-scale boundary conditions, and therefore may be signi"cantly relaxed to yield a large CP-violating phase in the A term compatible with the R existing experimental limit on the NEDM. This should encourage SUSY CP violation studies in top quark systems as they may well be the only venue for constraining arg(A ) in high-energy R experiments at colliders in the foreseeable future. 3.3.1. General description and the SUSY Lagrangian The most general low-energy softly broken minimal SUSY Lagrangian which is invariant under SU(3);SU(2);;(1) consists of three generations of quarks and leptons, two Higgs doublets and the SU(3);SU(2);;(1) gauge "elds, along with their SUSY partners, can be written as [172}177, 184}186]:
L"kinetic terms# d=#L
.
(3.100)
Here = is the superpotential and is given by =" (g'(QK G HK H ;K #g'(QK G HK H DK A #g'(¸K G HK H RK A #HK G HK H ) . (3.101) GH 3 ' ( " ' ( # ' ( is the antisymmetric tensor with "1 and the usual convention was used for the super"elds GH QK , ;K , ¸K , RK and HK [184]. I, J"1, 2 or 3 are generation indices and i, j are SU(2) indices. We do not include R-parity violating terms in the SUSY Lagrangian below, since we do not discuss in this review any CP-violating e!ect which may be driven by such terms. We only brie#y mention in Section 11 the possible impact of R-parity violating SUSY interactions on CP violation studies in the top quark system.
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L consists of the soft breaking terms and can be divided into three pieces L ,L #L #L , (3.102) which are the soft supersymmetry breaking gaugino, scalar mass terms and the trilinear coupling terms. These are given by [185] L "(m #m ? ? #m @ @ ) , (3.103) 5 5 % % L "!m H!m H!m ¸G!m R!m QG!m D!m ; , & G & G * 0 / " 3 (3.104) L " (g A QGHH ;#g A QGHH D#g A ¸GHH R#BHG HH ) , (3.105) GH 3 3 " " # # where we have omitted the generation indices I and J in the soft breaking terms. The above scalar "elds correspond to the super"elds which were indicated in our notation by a `hata. , ? (with 5 a"1, 2 or 3) and @ (with b"1,2, 8) are the gauge superpartners of the ;(1), SU(2) and SU(3) % gauge-bosons, respectively. Also we remark that proportionality of the trilinear couplings to the Yukawa couplings (i.e., g , g and g ) is imposed in Eq. (3.105). 3 " # 3.3.2. CP violation in a general MSSM We now turn to a discussion of the CP-odd phases in the theory. In general, when no further assumptions are imposed on the pieces of the Lagrangian in Eqs. (3.101) and (3.103)}(3.105), there are several possible new sources (apart from the usual SM CKM and strong phases) of CP non-conservation at the scale } where the soft breaking terms are generated. These are 1 [44,172}178]: the trilinear couplings A (i.e., F";, D or E), the soft breaking Higgs coupling B, $ the gauginos mass parameters m (a"1, 2 or 3) and the Higgs mass parameter in the superpoten? tial. However, not all of them are physical and by a global phase change of one of the Higgs multiplets one can set arg(B)"0 ensuring real VEVs of the Higgs doublets and "xing the phase of to be arg()"!arg(B). Moreover, in the absence of the soft breaking Lagrangian, the MSSM possesses an additional ;(1) R-symmetry [187]. Thus, with an R-transformation one can remove an additional phase from the theory, say from one of the soft gaugino masses m . The remaining ? physical phases are: one phase for each arg(A ) (corresponding to a fermion f ), arg(B) and arg(m ), D ? say for a"1, 2. In the most general MSSM scenario, these remaining complex parameters at the -scale cannot simultaneously be made real by rede"ning the phases of "elds without introducing 1 phases in the other couplings. Of course, once the above phases are set to their -scale values, they feed into the SUSY 1 parameters of the theory at the EW-scale through the RGE. Instead of studying the RGE for the full theory, one needs to consider only a complete subset of the RGE of the complex parameters in the e!ective theory. Taking only the top and bottom Yukawa couplings and neglecting small e!ects from the other Yukawa couplings, such a complete subset was given in [182]: dm ? "2b m , ? ? ? dt
(3.106)
dA R "2c m #12 A #2 A , ? ? ? R R @ @ dt
(3.107)
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dA @ "2c m #12 A #2 A , ? ? ? @ @ R R dt
(3.108)
dA S A "2c m #6 A , ? ? ? R R dt
(3.109)
dA B Q "2c m #6 A , ? ? ? @ @ dt
(3.110)
dB "2c m #6 A #6 A , ? ? ? @ @ R R dt
(3.111)
d R "2 (!c #6 # ) , R ? ? R @ dt
(3.112)
d @ "2 (!c #6 # ) , @ ? ? @ R dt
(3.113)
d ? "2b , ? ? dt
(3.114)
where t,ln(Q/ )/4, a is summed from 1 to 3 and b "(, 1,!3), c "(, 3, ), c "( , 3, ), 1 ? ? ? c "(, 3, 0). Also, and are related to the corresponding quark masses via R @ ? 1 g m . (3.115) " R@ R@ 8 m sin (cos ) 5 We remark that in the general MSSM framework with arbitrary CP-violating phases at , the above RGE are of less importance and with such boundary conditions almost any 1 low-energy CP-violating scenario can be generated. In particular, large CP-violating phases at the EW-scale are not excluded in this unconstrained scenario. However, in a more constrained SUSY version, one can reduce the number of the physical CP-odd phases in the theory. In this case the RGE given above are crucial for determining the SUSY CP-violating phases at the EW-scale [182]. We will return to this point later when we discuss the N"1 minimal low-energy SUGRAGUT model. Let us consider now the phenomenological consequences of CP non-conservation in such a general SUSY model. First, we need to describe brie#y how these new CP-violating phases enter in reactions which are driven by supersymmetric particles. As it turns out, all CP violation in the low-energy SUSY vertices is driven by diagonalization of the complex mass matrices of the sfermions, charginos and neutralinos. For more detailed investigations of the diagonalization procedure and extraction of the mass spectrum and CP-violating phases from these mixing matrices we refer the reader to the existing literature, see e.g., [172}177] and [185,188}196]. Here we only wish to brie#y describe the key features of the formulation and the de"nitions. We denote by MI the mass squared matrix of the scalar partners of a fermion, and M and Q D M are the mass matrices of the charginos and neutralinos, respectively. Then, with the rotation Q
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matrices Z , Z , Z> and Z\, we can de"ne D , ZR MI Z "diag(mI , mI ) , D D D D D (Z\)RM Z>"diag(m , m ) , Q Q Q Z2 M Z "diag(m , m , m , m ) . Q Q Q Q , Q , MI is then given by D m !cos 2(¹ !Q sin )m #mI * D D 5 8 D MI " D D !m (R H#A ) D D D
(3.116) (3.117) (3.118) !m (R #AH) D D D m !cos 2Q sin m #mI 0 D D 5 8 D
, (3.119)
where m is the mass of the fermion f, Q its charge and ¹ the third component of the weak D D D isospin of a left-handed fermion f. mI * (mI 0 ) is the low-energy mass squared parameter for the left D D (right) sfermion fI ( fI ). R "cot (tan ) for ¹ "(!) where tan "v /v is the ratio between * 0 D D the two VEVs of the two Higgs doublets in the model. M and M are given by Q Q m (2m sin 5 M " , (3.120) Q (2m cos 5
m
0
M " Q !m cos sin 8 5 m sin sin 8 5
0 m
m cos cos 8 5 !m sin cos 8 5
!m cos sin 8 5 m cos cos 8 5 0 !
m sin sin 8 5 !m sin cos 8 5 , ! 0
(3.121)
where m (m ) is the mass parameter for the ;(1)(SU(2)) gaugino. Because of their relatively simple form, we will discuss below the way of parameterizing the CP-violating phases only of the sfermions and charginos diagonalizing matrices Z and Z>, Z\, D respectively. The diagonalization of the 4;4 neutralino mixing matrix with complex entries is more involved and may be estimated numerically, although in some limiting cases it may be approximated analytically (see e.g., [182]). If all elements of Z are real then the diagonalization , procedure can also be done analytically (see e.g., [191,193]). The mixing matrix of the sfermions is parameterized as
cos !e\ @D sin D D , (3.122) Z " D cos e @D sin D D where is the mixing angle and is the phase responsible for CP-violating phenomena in D D sfermions interactions with other particles in the theory, and is given by R sin !A sin D I D D , tan "! D R cos #A cos D I D D
(3.123)
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where we have used A "A e ?D and "e ?I . Recall that R "cot (tan ) for ¹ "(!) D D D D and tan "v /v . Also, the mixing angle is given by D !2m R #AH D D D . (3.124) tan " D cos 2(2Q sin !¹ )m #mI !mI D 5 D 8 D* D0 It is obvious from Eq. (3.123) and (3.124) that in the limit where all the quark masses are small except for m , only the phase of A leads to CP-violating e!ects (the limit m P0, fOt, is useful R R D when considering high-energy reactions). In particular, the other A-terms are multiplied by the light fermion masses (see also Eq. (3.119)) and, therefore, they have negligible e!ect on any physical quantity evaluated at high enough energies. That is, the o!-diagonal elements of Z are zero in this D limit (i.e., from Eq. (3.124) we see that sin P0 when m P0) and there is no mixing between the D D left and right components of the superpartners of light quarks. Of course, this is not the case for the NEDM which is particularly sensitive to the slight deviation from degeneracy of the supersymmetric partners of the u and the d quarks. We will return to a more detailed discussion on the `SUSY-CP problema of the NEDM in the following section. For a sfermion fI , it is useful to adopt a parameterization for its fI !fI mixing such that the * 0 sfermions of di!erent handedness are related to their mass eigenstates through the transformation fI "ZfI #ZfI , * D D fI "ZfI #ZfI , (3.125) 0 D D where fI are the two mass eigenstates. We note that in the case where all CP violation arises from fI !fI mixing, i.e., from the complex entries in the sfermion mixing matrix Z , it has to be * 0 D proportional to (!1)G\ sin 2 sin . Im(G ),Im(ZGHZG)" D D D D D 2
(3.126)
Clearly, G P0 if there is no mixing between the left and right sfermions such that they are nearly D degenerate. We will describe in the next sections CP-non-conserving e!ects in top quark systems which are driven by the possibly large mass splitting between the two stop mass eigenstates and are therefore proportional to Im(G ). R The charginos mixing matrices are given by Z!"P!O! ,
(3.127)
where
1
P>"
0
0 !e ?
, P\"
e ?
0
0
!1
,
(3.128)
and
cos !e\ @> sin > > , cos e @> sin > > e A cos !e\ @\ \A sin \ \ . O\" e A cos e @\ >A sin \ \ O>"
(3.129) (3.130)
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Here "arg(m ) and the CP-violating angles and above are given by ! sin tan ,! , (3.131) > cos #m cot / sin tan , , (3.132) \ cos #m tan / sin , (3.133) tan ,! cos #2m (m !)/gv v Q sin tan , , (3.134) cos #gm v v /2(m !m ) Q where , # and m G (i"1, 2) are the masses of the two charginos. and are also I Q > \ functions of m , , v and v (see [189,190]). 3.3.3. CP violation in a GUT-scale N"1 minimal SUGRA model Let us proceed by describing a more constrained supersymmetric model. In particular, we want to consider a spontaneously broken N"1 SUGRA, which apart from gravitational interactions, is essentially identical at low energies to a theory with softly broken supersymmetry with GUT motivated relations at the GUT mass scale. One of the most appealing consequences of such a constrained SUSY scenario is that it allows REWSB of SU(2);;(1) with the fewest number of free parameters. According to conventional wisdom, complete universality of the soft supersymmetric parameters at the GUT-scale (or at the scale where the SUSY soft breaking terms are generated) is assumed. More explicitly, a common scalar mass m and a common gaugino mass M at the GUT-scale m "m "m "m "m "m "m "m , & 0 * / 3 " & m "M , a"1, 2 or 3 , (3.135) ? and also universal boundary conditions for the soft breaking trilinear terms are assumed A "A "A ,A% . (3.136) # " 3 Of course, the above relations do not survive after renormalization e!ects from the GUT-scale (which is usually taken to be M &2;10 GeV) to the EW-scale are included. % It then follows that the universal parameters of the minimal SUGRA model at the GUT-scale are: m , M , A%, B%, % and tan . This is the most general set of independent parameters before REWSB. However, a bonus of this economical framework is that REWSB occurs and the parameters % and B% are no longer taken as independent but are set by m , M and tan (the magnitude of is adjusted to give the appropriate Z-boson mass but the sign of remains as an independent parameter). Thus, the number of independent parameters is reduced to "ve, namely m , M , A%, sign() and tan . In this GUT motivated SUGRA theory there are four possible new sources of CP nonconservation at the GUT-scale. These are [178]: the universal trilinear coupling A%, the Higgs mass
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parameter %, the gauge mass parameter M and the parameter B%. However, as was mentioned before, M can be made real by an R-transformation and by using one remaining phase freedom, a rede"nition of the Higgs "elds can set the product B%% to be real so that the VEVs of the two Higgs "elds in the theory are also made real. We therefore have arg(%)"!arg(B%). With this choice we are left with only two new SUSY CP-odd phases at the GUT-scale which are carried by A% and %, apart from the usual CKM phase that originates from the Yukawa couplings in the theory for three generations. One can proceed by choosing a more constrained CP-violating sector by setting one of these two phases to zero (we will address to this possibility in the next section), or even a `supera constrained CP-violating sector by taking arg(%)"arg(A%)"0, thus being left (at the GUT-scale) only with the usual CKM phase present as in the SM. Note that having a universal phase for all trilinear couplings of sfermions at the GUT-scale, does not necessarily mean that all sfermions will have the same phase (driven by their trilinear coupling A ) at the EW-scale. That is, the GUT-scale phases arg(A%) and arg(%) feed into the other D parameters of the theory through renormalization e!ects from the GUT-scale to the EW-scale. In particular, these GUT-scale phases can produce di!erent phases for di!erent squarks and sleptons of di!erent generations. However, irrespective of what those di!erent phases are at the EW-scale, they can all be expressed, in principle, with the two new CP-odd phases of A% and % through the RGE in Eqs. (3.106)}(3.114). Thus, it is evident from the very simple structure of the evolution equations of the gaugino masses (see Eq. (3.106)) that m , a"1}3, are left with no phase at any ? scale. Moreover, the di!erence between the three real low-energy gaugino mass parameters comes from the fact that they undergo a di!erent renormalization as they evolve from the GUT-scale to the EW-scale, due to the di!erent gauge structure of their interactions. In particular, they are related, at the EW-scale, by (see e.g., [191]) m m 3 cos m 5 "sin " , (3.137) 5 5 Q where is the weak mixing angle and m is the low-energy gluino mass. 5 Notice now that with arg(m )"0, the RGE (Eqs. (3.106)}(3.114)) simplify to a large extent, thus ? we have only to consider the evolution of the A%'s and B% (or equivalently %) to the EW-scale. This D constrained version of the MSSM has strong implications on the low-energy phase in A as was R suggested in [182]. In particular, it was shown that with and without universal trilinear couplings at the GUT-scale and with some de"nite boundary conditions for them (for example, arg(%)"arg(A%)"0 for fermion species f ), the low-energy CP-violating phase of A induces D R potentially large phases in A , A and B at the EW-scale (through relations obtained from the S B above set of the RGE). This in turn gives rise to a large NEDM which is ruled out by the present experimental limit. Therefore, severe constraints on the low-energy phases of A and where D obtained. We will return to this point in the next section. 3.3.4. A plausible low-energy MSSM framework and the `SUSY-CP problema of the NEDM The EDM of the neutron, d (for reviews on fermion electric dipole moments see e.g., [197}201]) L imposes important phenomenological constraints on SUSY models, see e.g., [178,182,189,190, 192,194}196,202], [203}214] and references therein. In particular, with a low-energy MSSM that originates from a GUT-scale SUGRA model (with complete universality of the soft breaking terms),
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keeping d within its allowed experimental value (i.e., d :10\ e cm [183]) requires the `"ne L L tuninga of the SUSY phases to be less than or of the order of 10\}10\ for SUSY particle masses of the order of the EW symmetry breaking scale. We remark, though, that it has recently been claimed in [194}196], that in some regions of the SUSY parameter space, cancellations among the di!erent components of the neutron EDM may occur and such a severe "ne tuning for either the SUSY masses or the SUSY CP-violating phases (i.e., at the order of 10\) may not be necessary. However, for such cancellations to occur SUSY parameters have to be suitably arranged and, also, several large SUSY phases have to be present, which renders this scenario less predictive and less attractive as well. The NEDM can be written schematically in any low-energy MSSM scenario in which the CP-phases originate from the and the A terms as [182] D A A d L "XI sin #X S S sin #X B B sin , (3.138) I S B M M M 10\ e cm 1 1 1 where M is the typical SUSY mass scale which may be used to describe the typical squark masses. 1 It was found in [182] that typical values of the XG's in almost every such low-energy SUSY realization are: X S B 'O(1) and XI'O(10) for M :500 GeV. Therefore, with A + 1 D +M , only moderate bounds can be put on sin and sin . In contrast, an unambiguous 1 S B severe constraint is obtained on the low-energy phase of the Higgs mass term , namely sin (O(10\) for M :500 GeV (see also [44,189,190,202] and references therein). I 1 The important "nding in [182] is that if a complete universality of the soft breaking terms is imposed at the GUT-scale and the two GUT phases are zero or very small, i.e., arg(A%)+ arg(B%):0.1, then this severe constraint on sin combined with the relations between Im(A ) and I R Im(), obtained from the RGE, leads to the comparable constraint sin (O(10\). With a univerR sal A term at the GUT-scale this will also imply (through relations obtained from the RGE) sin (O(10\). Moreover, the above strong constraints on the SUSY CP-violating phases hold S B even if the universality of the A terms is relaxed at the GUT-scale as long as the GUT-scale CP-violating phases are kept very small. This strong constraint on the low-energy phase of A would also eliminate any possible SUSY CP-non-conserving e!ect in top quark systems. R While this scenario with very small CP-violating phases at the GUT-scale provides an explanation of the smallness of the NEDM (in fact, the above constraints will drive the NEDM to a value of the order of 10\!10\ e cm), an unavoidable question then arises: why do the CP-violating phases happen to be so small wherever they appear? If so, then an underlying theory that screens the CP-violating phases is required. We therefore feel that a somewhat di!erent phenomenological approach is needed, namely, the implication that sin (O(10\) should be specially scrutinized in the top quark sector. The latter, R being very sensitive to sin at high energies, may serve as a unique probe for searching for R signi"cant deviations from the above upper bound, sin (O(10\) (see e.g., [215]). Indeed, if at R the GUT-scale the universality of the A terms is relaxed and no assumption is made on the magnitude of the CP-violating phases, then one can construct a plausible low-energy MSSM framework which incorporates an O(1) low-energy phase for A , while leaving the NEDM within its R experimental limit. The crucial di!erence in assuming non-universal boundary conditions for the Since M is the only SUSY mass scale associated with the squarks sector, it is natural to choose the soft breaking 1 terms to be of O(M ). 1
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soft breaking trilinear A terms is that, in this case, there is no a priori reason to believe that the low-energy phases associated with the di!erent A terms are related at the EW-scale. In particular, D the bounds on sin obtained from the experimental limit of the NEDM may not be used to S B constrain sin . In addition, when no further assumption is made on the magnitude of the R CP-violating phases at the GUT-scale, a value of sin P0 may be realized without contradicting I any existing relation from the RGE. As in [215] we therefore take the following phenomenological point of view in constructing a plausible low-energy set of the MSSM CP-violating phases and mass spectrum: 1. sin P0 as strongly implied from the analysis of the NEDM. I 2. sin , sin and sin are not correlated at the EW-scale, which implicitly assumes nonS B R universal boundary conditions at the GUT-scale. In particular, sin , sin may then be S B constrained only from the NEDM experimental limit, with no implications on the size of sin . R 3. Motivated by the theoretical prediction of the uni"cation of the SU(3), SU(2) and ;(1) gauge couplings when SUSY particles with a mass scale around 1 TeV are folded into the RGE, one may follow only the following traditional simplifying GUT assumption: there is an underlying grand uni"cation. As mentioned before, this leads us to have a common gaugino mass parameter de"ned at the GUT-scale which can be made real by an R-rotation. Thus, m , a"1}3 ? are left with no phase at any scale. Moreover, using the relation of Eq. (3.137), once the gluino mass is set at the EW-scale, the SU(2) and ;(1) gaugino masses m and m , respectively, are determined. 4. The typical SUSY-scale is M and all the squarks except for the light stop, are assumed to be 1 degenerate with mass M . 1 Note that in this low-energy framework one is only left with the phases of the various A terms at the EW-scale, out of which only A plays a signi"cant role in any high-energy D R reaction. For m P0 the superpartners of the light quarks are practically degenerate and D therefore the CP-violating e!ects from the phases of the other A terms, corresponding to the D light quarks, can be safely neglected. Note again that our approach, the EWPGUT approach, assumes a set of phases at the EW-scale, subject to existing experimental data, which implicitly assumes arbitrary phases at the scale in which the soft breaking terms are generated. As was mentioned above, this low-energy CP violation scenario can naturally arise from a GUT-scale SUGRA model if only the universality of the A terms is relaxed and no assumption is made on the magnitude of the GUT-scale phases. Also, the mass matrices of the neutralinos, M , and charginos, M , depend on the low-energy Q Q Higgs mass parameter , the two gaugino masses m and m (which are resolved by the gluino mass) and tan (see previous sections) and are therefore real in this scenario. Thus, once , m and tan are set to their low-energy values, the four physical neutralino species m L (n"1}4) Q and the two physical chargino species m K (m"1, 2) are extracted by diagonalizing the real Q matrices M and M . Q Q 5. Finally, the resulting SUSY mass spectrum may now be subject to the existing experimental limits, i.e., limits on the masses of squarks, gluino, neutralinos, charginos, etc. (for recent phenomenological reviews of supersymmetry and reported limits obtained and expected in existing and future experiments, see e.g., [216}223]).
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With these assumptions, when arg()P0, the leading contribution to a light quark EDM comes from gluino exchange, which with the approximation of degenerate u and dI squark masses (which we will denote by m ), can be written as [189,190] O A sin 2 O (rK(r) , d (G)" Q Q em O O O O m 3 O
(3.139)
where m (m ) is the quark(squark) mass and Q is its charge. Also, r,m /m (for the rest of this O % O O O section we denote the gluino mass by m ) and K(r) is given by %
1 r(2#r) K(r)" #r# ln r . (r!1) 1!r
(3.140)
Then, within the naive quark model, the NEDM can be obtained by relating it to the u and d quarks EDM's (i.e., d and d , respectively) as S B d "(4d !d )/3 . L B S
(3.141)
We now consider arg(A ) and arg(A ) to be free parameters of the model irrespective of arg(A ). In S B R Figs. 9(a) and (b) we have plotted the allowed regions in the sin !sin plane for d not to S B L exceed 1;10\ e cm (for the present experimental limit see [183]) and 3;10\ e cm. In calculating d , we assumed that the above naive quark model relation holds. Although there is no doubt L that it can serve as a good approximation for an order of magnitude estimate, it may still deviate from the true theoretical value which involves uncertainties in the calculation of the corresponding hadronic matrix elements. Note also that it was argued in [224] that the naive quark model overestimates the NEDM, as the strange quark may carry an appreciable fraction of the neutron spin which can partly screen the contributions to the NEDM coming from the u and the d quarks. To be on the safe side, we therefore slightly relax the theoretical limit on d in Fig. 9(b) to be L 3;10\ e cm. We have used, for these plots, m I "m "M "400 GeV, m "500 GeV and for simplicity we S 1 % B also took A "A "M . As remarked before, it is only natural to choose the mass scale of the S B 1 soft breaking terms according to our typical SUSY mass scale M . Also, we took the values for 1 current quark masses as m "10 MeV, m "5 MeV and (m )"0.118. B S Q 8 From Fig. 9(a) and in particular Fig. 9(b), it is evident that M "400 GeV and m "500 GeV 1 % can be safely assumed, leaving `enough rooma in the sin !sin plane for d not to exceed S B L 1}3;10\ e cm. We observe that while sin is basically not constrained, !0.35:sin :0.35 is S B needed for d (1;10\ e cm and !0.55:sin :0.55 is needed for d (3;10\ e cm. L B L Moreover, varying m between 250 and 650 GeV has almost no e!ect on the allowed areas in the % sin !sin plane that are shown in Figs. 9(a) and (b). That is, keeping M "400 GeV and S B 1 lowering m down to 250 GeV, very slightly shrinks the dark areas in Figs. 9(a) and (b), whereas, % Note that the function K(r) in Eq. (3.140) is slightly di!erent from that obtained in [182]. However, we "nd that numerically the di!erence is insigni"cant and does not change our predictions below. We will take M "400 GeV and vary m in this range in some of the CP-violating e!ects in collider experiments to 1 % be discussed in the following sections.
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Fig. 9. The allowed regions in the sin !sin plane for the NEDM not to exceed (a) 1;10\ e cm and S B (b) 3;10\ e cm. M "400 GeV and m "500 GeV is used. The shaded areas indicate the allowed regions. Figure 1 % taken from [215].
increasing m up to 650 GeV slightly widens them. Of course, d strongly depends on the scalar % L mass M } increasing M enlarges the allowed regions in Figs. 9(a) and (b) as expected from 1 1 Eq. (3.139). It is also very interesting to note that, in some instances, for a cancellation between the contributions of the u and d quarks to occur, sin , sin '0.1 is essential rather than being just S B possible. For example, with sin 90.75, sin 90.1 is required in order to keep d below its S B L experimental limit. We can therefore conclude that CP-odd phases in the A and A terms of the order of few;10\ S B can be accommodated without too much di$culty with the existing experimental constraint on the NEDM even for typical SUSY masses of :500 GeV. Therefore, we restate what is emphasized in [215]: somewhat in contrast to the commonly held viewpoint we do not "nd that a `"ne-tuninga at the level of 10\ is necessarily required for the SUSY CP-violating phases for squark masses of a few hundreds GeV or slightly heavier. 3.3.5. CP and the pure Higgs sector of the MSSM Since the superpotential is required to be a function of only left (or only right) chiral super"elds, it forbids the appearance of HK H and HK H in the superpotential = in Eq. (3.10). Therefore, since a QK G HK H ;K coupling in = is prohibited by gauge invariance, only H is responsible for giving mass ' ( to up quarks and H to down quarks [188]. As a consequence, the requirement that there will be no `harda breaking terms of the symmetry P! in the Higgs potential is automatically G G satis"ed in a minimal supersymmetric model. That is, " "0 in Eq. (3.48), for the Higgs
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potential in the MSSM. Moreover, no term of the form ( R )#h.c. , (3.142) appears in the Higgs potential of the MSSM [188], thus implying "0 in Eq. (3.48). It is then straightforward to observe that with the above constraints on the pure Higgs sector of the MSSM, any phase which may appear in a complex soft breaking parameter (i.e., proportional to in Eq. (3.48)) can be removed by a rede"nition of one of the Higgs doublet "elds, thus also setting the relative phase between the two VEVs to zero. Therefore, the pure Higgs sector in the MSSM possesses no CP violation [188]. Of course, CP violation may emerge in interactions of the Higgs "elds with the other "elds in the theory due to the CP-violating phases carried by these latter "elds.
4. Top dipole moments 4.1. Theoretical expectations A non-vanishing value for the EDM of a fermion is of special interest as it signi"es the presence of CP-violating interactions. We recall that the search for the EDM of the neutron and that of the electron have intensi"ed in recent years. Since the top is such an unusual fermion, in fact so heavy that it is very unlikely to exist as a bound state with another quark, it is clearly important to ask: What is its EDM? How can we measure it, if at all? This topic has been of interest to many for the past several years. In the SM quarks cannot have an EDM at least to three loops (for reviews on fermion electric dipole moments see [197}201]). For the electron the three loop contribution has been estimated to be dA (0)&10\ e cm. Simple-dimensional scaling then suggest for the top the value C dA(0)&10\}10\ e cm, much too small to be observable. R In contrast, in extensions of the SM, e.g., MHDMs and SUSY models, this situation changes sharply and the top dipole moment (TDM) can arise at the 1-loop level and as a result, the typical TDM is of the order of 10\!10\ e cm which is larger than the SM prediction by more than 10 orders of magnitude. The enhancement due to the large top mass is particularly evident in some models with an extended Higgs sector for which the dipole moments often scale as m . Since D at 1-loop light quarks (or neutron) in these models can get dipole moment of order 10\ e cm, the TDM could easily reach 10\ e cm or even more. It is at that level that measurable consequences can arise. Because of the unique importance of the top quark, it should be clear that measurements of the TDM will be extremely important. However, it should also be clear that due to the extraordinary short lifetime of the top quark (:10\ s) it will be extremely di$cult to actually measure the static
Strictly speaking, the term dipole moment refers to the static form factor (i.e., at q"0). Here we will mostly concern ourselves with the dipole moment form factor at qO0; for simplicity, we will still use the term TDM throughout. We will use the abbreviation TDM in general for dA 8 E denoting the top quark EDM, weak-EDM (ZEDM) and R Chromo-EDM (CEDM), unless we need to explicitly separate the type.
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(i.e., at q"0) TDM. Measurements of some of the e!ects driven by the presence of a dipole moment form factor may have a better chance. In fact, the TDM may be considered as a CP-odd form factor in the ttM , ZttM or gttM vertex that probes the interactions of a short-lived top quark with an o!-shell , Z or a gluon, respectively, and can be represented by A 8 E"idA 8 E(s)u (p ) pJv M (p M ) , I R R R IJ R R
(4.1)
where s"q, q"p #p M and color indices for E were suppressed. Therefore, depending on the I R R masses in the loops, the TDM form-factor can also develop an imaginary part (contrary to the static EDM of the electron or the quark) if the energy of the o!-shell , Z or gluon is su$cient for the particles in the loop to be on-shell, i.e., there is an absorptive cut. Since the dipole moment characterizes the e!ective coupling between the spin of the fermion and the external gauge "eld, to extract the dipole moment one needs information on the spin polarization of the top quark. Fortunately, the left-handed nature of the weak decays of the top allows us to determine its polarization quite readily. As discussed in Section 2.8, top quark decays can analyze the initial polarization of the top quark. For instance in the leptonic decay tPe> b, C in the rest frame of the top quark the top is 100% polarized in the direction of the e> momentum. This greatly simpli"es calculation of the top quark production followed by its subsequent decay. The problem is then essentially reduced to calculating the production of a polarized top quark. It is then straightforward to fold in the decay to the spin indices of the top quark. A serious limitation to be kept in mind about this procedure is that it is only valid when the decays are governed by the SM, since they assume the helicity structure of the SM. If non-standard interactions make large contributions to the decay, the decay distributions may be modi"ed to the point that the polarimetry we have discussed is only approximate. This point must be borne in mind when considering the e!ects of new physics. A detailed discussion of the feasibility of extracting the TDM in future collider experiments such as e>e\, ppPttM , will be given in subsequent sections. In this section we consider the contribution to the TDM which arises in extended Higgs sectors and SUSY models. 4.2. Arbitrary number of Higgs doublets and a CP-violating neutral Higgs sector It is instructive to calculate the TDM in models with an arbitrary number of Higgs doublets and singlets satisfying NFC (natural #avor conservation) constraints. CP violation arises as a result of scalar exchanges between quarks, being driven by the imaginary parts of the complex quantities (e.g., ZI ) de"ned as [119]: L (2G ZI $ L , , q!m L (v ) O & L
(4.2)
(2G Z 1 $ L , >H> , (v vH) O q!m L &Y L
(4.3)
1
where v and v are the VEVs of the neutral Higgs "elds , and the summation runs over all the mass eigenstates of neutral or charged scalars in the theory (H or H , respectively). Also, L L O
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Fig. 10. Feynman diagrams that contribute to the electric and weak top dipole moments in a two Higgs doublets model with CP-violating interaction of a neutral Higgs (h) with a top quark.
stands, for any pair of scalar "elds, and , for the momentum-dependent quantity
" dx 0¹[ (x) (0)]0 e\ OV . O
(4.4)
CP violation in the neutral Higgs sector then generates the dominant 1-loop contribution to the EDM of the top, i.e., dA, through the 1-loop graph with the external photon line in Fig. 10(a) (for this R discussion, h,H in Fig. 10(a)). This contribution is given by L
m 2(2 G m e Im ZI f &L dA(0)" $ R L R 3(4) m R L m m R Im ZI f &L "(1.4;10\ e cm) L m GeV R L where
,
(4.5)
2!r r r r!2r 1! ln r# arctan #arctan 2 (r(4!r) (r(4!r) (r(4!r)
f (r)" 3!4 ln 2
if r(4 , if r"4 ,
r (r!(r!4 r!2r 1! ln r! ln 2 2 (r(r!4)
(4.6)
if r'4
and r"m L /m for any value of n. For r<4, f (r) approaches 1/r(ln r!) asymptotically. The ZI 's & R L satisfy some important sum rules, for example [119]: Im ZI "0 , (4.7) L L so that dA will vanish if all the neutral Higgs-bosons were degenerate; no such degeneracy is of R course expected. For illustrative purposes let us assume that the lightest neutral Higgs-boson, with mass m , dominates the sum in Eq. (4.5). Taking m "100 GeV and m "2m (m "175 GeV) and F F F R R setting Im Z's to be of order unity, then dA is about 1.3;10\ e cm and 5.6;10\ e cm, R respectively. We note that for m 'm , dA(0) varies slowly with m [31]. F R R F
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For experimental purposes the top EDM at high q may be more relevant. This is given by [31]
m q 2(2 G m e Im ZI f &L , , dA(q)" L m m R 3(4) $ R R R L where
f (r, s)"
dx
\V
dy
x#y . (x#y)#(1!x!y)r!xys!i
(4.8)
(4.9)
For q'4m, dA(q) develops an imaginary part. In the following section, we will present explicit R R numerical results for the real and the imaginary parts of dA(q) and d8(q) in a 2HDM with CP R R violation in neutral Higgs exchanges. In this case, the top CEDM is immediately obtained by replacing the photon with a gluon in Fig. 10(a) and, therefore, e with g (the QCD coupling Q constant) in Eqs. (4.5) and (4.8). So, dE&1.5dA where dA is in e cm and dE in g cm. In theories with R R R R Q such a Higgs sector many CP violation e!ects are driven by the top CEDM. The Schmidt}Peskin energy asymmetry in ppPttM #X followed by top decay is one such interesting e!ect [33], and we will discuss it in detail in Section 7. 4.3. Expectations from 2HDMs with CP violation in the neutral Higgs sector The general analysis given in the previous section holds, of course, for any number of Higgs doublets. However, let us now focus on the simplest extension of the Higgs sector. That is, a 2HDM with CP violation in the neutral Higgs sector driven by a phase in the Higgs-fermion}fermion interaction [30,124]. The example that we will explicitly consider here is type II 2HDM; however, the analysis can also be applied to type I and III models with simple rede"nitions of the couplings (see also Section 3.2). In this model the dipole moment form factors for the top quark start to contribute at 1-loop order via the Feynman diagrams in Fig. 10. The required CP-odd phase is provided by the HIttM Lagrangian piece in Eq. (3.70), where HI, for k"1, 2, 3, stands for the three neutral Higgs particles in the model. The couplings aI, bI in Eq. (3.70) depend on the three Euler R R angles, i.e., , which parameterize the neutral Higgs mixing matrix and on tan which is the ratio between the two VEVs, v and v , corresponding to the two Higgs doublets of the model (for more details see Section 3.2). The top EDM and ZEDM within this class of 2HDMs was considered in [30,32]. They can be written as dA(s)" dA (s)g , I I R I 3gR 4 dA(s)# d 8 (s)g , d8(s)" R I I R 4 sin cos R 5 5 I where gR "1/2!4 sin /3 and in Model II 4 5 g "aIbI"R R cot /sin , I R R I I g "bIcI"R (R cos #R sin )cot . I R I I I
(4.10) (4.11)
(4.12) (4.13)
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The 3;3 neutral Higgs mixing mass matrix R is given in Eq. (3.73). cI in Eq. (4.13) is the coupling constant associated with the ZZHI vertex, see Eqs. (3.71) and (3.72). From Fig. 10(a)}(c), one can extract the functions dA and d 8 : I R I eQ (2G m C? $ R ; , dA (s)"! R I 4 2
(4.14)
egR C@ #CA !CA 4 , d 8 (s)" (2G m m ; (4.15) R I $ 8 R 8 sin cos 2 5 5 where Q "2/3. The three-point loop form factors CG , x311, 12 and i"a, b, c corresponding to R V diagrams (a)}(c) in Fig. 10, are given in our notation by C? "C (mHI , m, m, m, s, m) , R R R R V V (4.16) C@ "C (m, mHI , m , m, s, m) , 8 R R V V R CA "C (m, m , mHI , m, s, m) , R R V V R 8 and C (m , m , m , p , p , p ) is de"ned in Appendix A. Analytical expressions for the imaginary V parts of the three-point loop form factors, CG , may be derived through the Cutkosky rule [30]: V
(s!4m) C? R [Z\ ln(1#Z)!Z\] , Im G (s),Im " (4.17) I I I I 2 mHI (s
Im D (s),Im I
C@ #CA !CA " [s!(m #mHI )] 8 2
; A (s)# I
4m!m !mHI mHI !m R 8 8 B (s) , # I 2s 2(4m!s) R
(4.18)
where
and
Z "((s!4m)/mHI , I R
(4.19)
(w I A (s)" , I s(4m!s) R
(4.20)
b #(aw I !a\ ln I b !(aw I I B (s)" !2(w /b I I I
for a'0 , for a"0 ,
(4.21)
!2(!a)\ arctan((!aw /b ) for a(0 . I I
also
a"s(s!4m) , R
(4.22)
D. Atwood et al. / Physics Reports 347 (2001) 1}222
b "s[s!(m #mHI )] , I 8 w "[s!(m #mHI )][s!(m !mHI )] . I 8 8 The real parts are obtained from a dispersion relation
69
(4.23) (4.24)
1 Im G (s ) I Re G (s)" P ds
, (4.25) I s !s KR Im D (s ) 1 I ds
. (4.26) Re D (s)" P I s !s K8 >KHI Let us now assume again that the masses of the other two neutral Higgs particles are considerably larger then the lightest one and, therefore, the lightest neutral Higgs dominates the sums in Eqs. (4.10) and (4.11). Recall that the CP-violating e!ects would vanish if the Higgs were degenerate, i.e., dA"d8"0, due to the orthogonality properties of the neutral Higgs mixing R R matrix. With no loss of generality, we denote the lightest neutral Higgs by h with couplings g , g , corresponding to k"1 in Eqs. (4.12) and (4.13). Also, we scale out the couplings g , g and plot in Figs. 11(a) and (b) and 12(a) and (b) the real and imaginary parts of dA and d 8 (recall that d 8 is the R R R contribution to the ZEDM which arises through diagrams (b) and (c) in Fig. 10) for a variety of Higgs masses, m "100, 200, 300 GeV and m "175 GeV. F R We can see from Figs. 11(a) and (b) that Re(dA) and Im(dA) are typically &10\!10\ e cm R R for m "100}300 GeV. The peak in the threshold region, shown in Fig. 11(b), originates from F a Coulomb-like singularity present in diagram (a) of Fig. 10. This is more pronounced, of course,
Fig. 11. Imaginary (a) and real (b) parts of dA in units of 10\ e cm as a function of (s, for various masses of the lightest R neutral Higgs (h) and for m "175 GeV. R
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Fig. 12. Imaginary (a) and real (b) parts of d 8 in units of 10\ e cm as a function of (s, for various masses of the lightest R neutral Higgs (h) and for m "175 GeV. R
for light Higgs masses. Note that the contribution from the diagram in Fig. 10(a) to Re(d8) and R Im(d8) is smaller by a factor of 3gR /4 sin cos K0.35 than the contribution to Re(dA) and R 4 5 5 R Im(dA) as can be seen from Eq. (4.11). R As is evident from Figs. 11 and 12, Re(d 8) and Im(d 8) are typically about one order of R R magnitude smaller than Re(dA) and Im(dA), respectively. Clearly, disregarding the 1-loop form R R factors, this di!erence is in part due to the di!erent couplings in Eqs. (4.14) and (4.15). Thus, one "nds 2Q sin cos m dA 5 5 ; R K10.7 . R+ R gR m d 8 R 4 8
(4.27)
Moreover, the di!erence between the contributions from Fig. 10(a) and Figs. 10(b) and (c) becomes even more pronounced once the 2HDM couplings g and g are included. In particular, from Eqs. (4.12) and (4.13), one "nds that for tan (1, g J1/tan and g J1/tan . Thus, as we will show below, the ratio g /g may even become as large as &10 for tan :0.5. In Figs. 13(a) and (b), we show the dependence of the real and imaginary parts of dA and d 8 on the R R mass of the lightest Higgs-boson m . Evidently, the dependence of d 8 on m is rather insigni"cant, F R F while dA drops as m increases. R F The EDMs of the neutron and electron do not constrain the neutral Higgs mixing matrix in any signi"cant way. Thus, as already mentioned above, its matrix elements, which enters g and g in Eqs. (4.12) and (4.13) could be of O(1). Furthermore, for some versions of 2HDM, tan (1 is a viable alternative. In this case g , g can even become larger than one, further enhancing the
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Fig. 13. Imaginary and real parts of dA and d 8 in units of 10\ e cm as a function of m , for (a) (s"500 GeV and R R F (b) (s"1 TeV and for m "175 GeV. R
dipole form factors. Let us now include the factors g and g in calculating the top's EDM and ZEDM. For illustration, we choose three sets for the two Euler angles , and take tan "0.3. Set I : , "/4, /2, Set II : , "/4, 3/4 and Set III : , " /4, /4. Note that in Set I g is maximized, in Set II g '1 and in Set III g and g have opposite relative signs such that the contributions of d 8 and dA to d8 add. In particular, in terms of R R R g and g we have Set I: g K5.8, g K0.48 , (4.28) Set II: g K4.1, g K1.14 , (4.29) Set III: g K4.1, g K!0.46 . (4.30) In Table 2, we give the real and imaginary parts of dA and d8 (the total ZEDM including diagrams R R (a)}(c) in Fig. 10). For illustration, we present numbers for m "100, 200, 300 GeV and F (s"500, 1000 GeV. We see from Table 2 that dA ranges from a few ;10\ e cm to a few R ;10\ e cm. Also, as expected, d8 is typically smaller by about a factor of &3}4. The imaginary R parts tend to be bigger by factors ranging from 2 to 10 for (s"1000, 500 GeV, respectively. In passing, we also note that the weak dipole moment form factors obtained here are about an order Note that in the parameterization of Eq. (3.73) in Section 3.2.3, g and g are insensitive to and it is su$cient to consider only and .
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Table 2 Real and imaginary parts of dA and d8 in units of 10\ e cm, for m "100, 200 and 300 GeV and for (s"500 GeV and R R F (s"1 TeV (in parentheses). tan "0.3 and Sets I, II, III means , "/4, /2, /4, 3/4, /4, /4, respectively Type of moment (10\ e cm) P
The di!erent sets of , , tan "0.3
m F (GeV)P Set I
Set II
Set III
Re(dA) R
100 200 300
1.97(3.77) !3.36(2.26) !4.75(1.27)
1.40(2.66) !2.38(1.60) !3.36(0.90)
1.40(2.66) !2.38(1.60) !3.36(0.90)
Im(dA) R
100 200 300
!23.89(!5.44) !16.56(!4.91) !11.34(!4.33)
!16.88(!3.84) !11.70(!3.47) !8.02(!3.06)
!16.88(!3.84) !11.70(!3.47) !8.02(!3.06)
Re(d8) R
100 200 300
0.62(1.25) !1.17(0.74) !1.57(0.40)
0.36(0.83) !0.87(0.47) !1.04(0.24)
0.52(0.93) !0.78(0.57) !1.18(0.33)
Im(d8) R
100 200 300
!7.96(!1.81) !5.45(!1.62) !3.64(!1.42)
!5.41(!1.21) !3.58(!1.08) !2.22(!0.93)
!5.85(!1.34) !4.12(!1.22) !2.91(!1.08)
of magnitude bigger than that found in [31]. Note that, here also, the top CEDM is immediately obtained by replacing the photon with a gluon and, therefore, e with g . Q 4.4. Expectations from a CP-violating charged Higgs sector In models with three or more Higgs doublets [62,161,162], it is also possible to have CP violation in the charged Higgs sector. In this case, a top EDM, ZEDM and CEDM receives contributions from diagrams in Figs. 14(a) and (b) with q "d, s, b. In the case of the 3 Higgs doublet model as considered, for example, in Section 3.2.4 we may express the coupling of the lighter charged Higgs-bosons, say H>,H>, to the third generation of quarks as (see also Eq. (3.88) for our notation): g " 5 K (m tM b #m tM b )#h.c. , (4.31) R@ R 0 * @ * 0 R@& (2m 5 where K is the CKM matrix (we will assume that K +1); and are complex parameters of the R@ model (for a more complete description of these parameters see Section 3.2.4). Also, we neglect contributions from the H>td and H>ts couplings as those will yield a TDM smaller by a factor of &(m /m ) and&(m /m ), respectively, compared to the TDM coming from the H>tb Lagrangian B @ Q @ piece. The CP violation in this case is proportional to the quantity Im(<) where <,( H). We denote the coupling of a vector-boson to the b-quark by L
>
!iI(A@ #B@ ) , 4 4
(4.32)
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Fig. 14. Feynman diagrams that contribute to the electric, weak and chromo-electric top dipole moments due to CP-violating interactions of a charged Higgs with a top quark. <", Z (also <"gluon in diagram (a)) and q stands for d, s or b-quark.
and the coupling of a vector-boson to the pair of charged Higgs by iC&(p !p )I , (4.33) 4 &> &\ where p are the in-going momenta of the charged Higgs-bosons. &! In [225] we have calculated the contribution of the charged Higgs exchanges in Figs. 14(a) and (b) to the b-quark dipole moment. The top EDM, ZEDM or CEDM given by these diagrams may, therefore, be extracted from [225] by the replacement m m and are thus given by @ R m 1 Im(<) @ m A@ C? !C& C@ # C@ , (4.34) dA 8 E"! A 8 E 2 R m R A 8 E 4 sin 5 5 where CG , x30, 12 and i"a, b, are the three-point loop form factors corresponding to diagrams V (a) and (b) in Fig. 14 such that
C? "C (m > , m, m, m, s, m) , @ @ R R V V & (4.35) C@ "C (m, m > , m > , m, s, m) . & R R V V @ & Here s"(p #p M ) and C is de"ned in Appendix A. Also, V R R (4.36) A@ "!e, C&"e , A A e 1 2 A@ " ! # sin , C&"e cot 2 , (4.37) 8 2 sin cos 5 8 5 2 3 5 5 A "g , C&"0 , (4.38) E Q E where e is the electric charge, g is the strong coupling constant and is the weak mixing angle. Q 5 Note that in the case of the gluon, i.e., the CEDM, only Fig. 14(a) enters as the gH>H\ coupling is absent. Using m > "200 GeV (also m K175 GeV) and denoting dA 8 E,Im(<)A 8 E(s), we "nd that for R R R & 500 GeV((s(1000 GeV,
Im A 8 E, Re A 8 E:few;10\ e cm, g cm , (4.39) R R Q where, in fact, for (s"1000 GeV, Re A 8:10\ e cm. Moreover, as m > is increased the TDM R & drops rapidly and, for example, for m > "500 GeV, we "nd that, typically, Im A 8 E and Re A 8 E R R & are smaller than&10\ e cm or g cm. Such small TDM, residing in charged Higgs exchanges, is Q
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D. Atwood et al. / Physics Reports 347 (2001) 1}222
expected simply by comparing A 8 E above to the corresponding terms for neutral Higgs exchanges R which were given in the previous sections. In particular, one can immediately observe that the contribution to the TDM from charged Higgs exchanges is naively suppressed by a factor of &(m /m ) with respect to the neutral Higgs exchanges wherein the TDM was found to be typically @ R at the order of &10\ e cm. Let us brie#y consider what constraints can be placed on the parameter, Im(<), by use of the experimental results on the EDM of the neutron (NEDM) and on the decay bPs. For our discussion we will simply use the bound given in [183] on the NEDM dC :10\ e cm . (4.40) L Using our previous results given in Eq. (4.34), one can deduce an expression for the electric dipole moment of the light (u, d)-quarks in the charged Higgs model (see [225]). Making the simplifying assumption that the NEDM equals that of its valence quarks, one "nds (as has been noted before [62]) that there is signi"cant uncertainty from the numerical value of the mass of the light quark to be used in the above formula. For the purpose of obtaining an upper bound on Im(<) we can take the current mass to be &10 MeV. Then for m "180 GeV, m > in the range 200}500 GeV, we "nd R & from Eq. (4.40) and using Eq. (4.34) for the light u, d-quarks that Im(<):10 [225]. The experimental data on the decay bPs can also constrain this parameter. In particular, it was shown in [132,165] that, for m > "200}500 GeV and neglecting the e!ects of the second charged Higgs of & the model, a conservative upper bound of Im(<):3!9 ,
(4.41)
can be placed from the CLEO measurement of the decay rate of bPs (see also discussion in Section 3.2.4). It turns out, however, that even when using Im(<)"10, in conjunction with numerical values for A 8 E(q) as given before, the TDM is expected to be :10\ e cm in this class R of charged Higgs mediated CP violation. Thus, it is typically smaller by at least an order of magnitude than what one would expect from CP-violating neutral Higgs exchanges and from the MSSM (see the following section). Finally, it should be noted that in any given MHDM with new mechanisms of CP violation in the charged Higgs sector, i.e., three and more Higgs doublets, the neutral Higgs sector will also acquire new CP-violating phases which, in general, cannot be screened (see Section 3.2.4). Therefore, as was already mentioned before, in the top quark case, CP-violating neutral Higgs exchanges dominate the charged Higgs by a typical factor of&(m /m ). One power of (m /m ) originates from R @ R @ the ratio between the neutral and charged Higgs couplings to fermions and another power of (m /m ) comes in from the necessary mass insertion in the propagator of the fermion in the loop. R @ Thus, the charged Higgs contribution is negligible compared to one from a neutral Higgs. Note, however, that in light quark systems charged Higgs exchanges are expected to yield the dominant CP-violating e!ects since the above argument is basically reversed [225].
We note, however, that the limit on Im(<) gets weaker as the masses of the two charged Higgs approach the same value, due to a GIM-like cancellation of the CP-violating e!ect mediated by the two. For example, if m > "350 GeV and & the mass of the second charged Higgs is &500 GeV, then Im(<)910 does not contradict the upper bounds on bPs, and, of course, no such limit exist at all if the two charged Higgs are degenerate [132].
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4.5. Expectations from the MSSM Within the MSSM, the TDM can also arise already at 1-loop even without generation mixing. As was mentioned in Section 3.3, in general, the required CP-violating phases are provided by the chargino and neutralino mixing matrices as well as the squarks q !q mixing matrices. If one * 0 assumes GUT-scale uni"cation which leads to a common gaugino mass at the GUT-scale, then the phase in the gaugino mass term can be rotated away, leaving the gaugino masses phaseless at any scale. Thus, one is left with only three phases (neglecting generation mixing between the squarks) relevant for the TDM at the EW-scale; , and which arise from the Higgs mass term and the I R @ stop and sbottom mixing matrices, respectively (for the de"nitions of , and see Eq. (3.123) in I R @ Section 3.3.2). We recall (see Eqs. (3.122) and (3.125) in Section 3.3.2) that, for a sfermion fI , it is useful to adopt a parameterization for its fI !fI mixing such that the sfermions of di!erent handedness * 0 are related to their mass eigenstates through the transformation fI "cos fI !e\ @D sin fI , D * D (4.42) fI "e @D sin fI #cos fI , D D 0 where fI are the two mass eigenstates (i.e. physical states) and the phase is related to the phase D by Eq. (3.123). The contribution to the CP-violating TDM which arise from the above fI !fI D * 0 mixing matrix will always be proportional to the quantity (see also Eq. (3.126)) D ,2G "sin 2 sin . (4.43) !. D D D Clearly, from Eq. (4.42) we see that D P0 if the two sfermions are nearly degenerate. !. In the MSSM the TDM can therefore acquire a non-vanishing value through the Feynman diagrams depicted in Fig. 15. One can then distinguish between the following three contributions: 1. Gluino contribution, dA 8 , with tI tI Hg in the loop (see Fig. 15(b)). RE 2. Chargino contribution, dA 8 > , with > \bI (see Fig. 15(a)) and bI bI H > in the loop (see Fig. 15(b)). RQ 3. Neutralino contribution, dA 8 , with tI (see Fig. 15(a)) and tI tI H in the loop (see Fig. 15(b)). RQ The gluino contribution was considered in [226}229]. It is dA " Q Q R m ;(C#C !C!C ) , RE 3 R !. E
(4.44)
Fig. 15. Feynman diagrams contributing to dA and d8: (a) with two fermions and one scalar in the loop, (b) with two R R scalars and one fermion in the loop.
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Q d8 " R m RE 6 sin cos !. E 5 5 ;(cos ! sin )(C#C !C!C ) R 5 ;(sin ! sin )(C#C !C!C ) , (4.45) R 5 where the three-point 1-loop form factors, CGH, x30, 11 and i, j"1, 2, are given by V (4.46) CGH"C (m , mI G , mI H , m, s, m) , R V V E R R R and C is de"ned in Appendix A. Also, and R are de"ned in Eqs. (4.42) and (4.43), respectively. V R !. The chargino contribution was given in [230]: > R dA > "! RQ 4 sin 5 1 ; m >H CK HH!CK HH! (CH KK#CH KK) Q 6 K H ;(1!(!)K cos 2 )Im[; < ]# > (!)K sin 2 Im[; < e @@ ] , @ H H @ @ H H d8 > " ;( f>#f>#s>#s>) , RQ 16 sin cos 5 5 where
f>"!> m >H (CK HI!CK HI) R Q K H I ;( (1#2 cos 2 )#; !< ) HI 5 I I ;((1!(!)K cos 2 )Im[; < ]#> (!)K sin 2 Im[; < e @@ ]) @ H H @ @ H H ! > (1! )(!)K sin 2 Im[; < e @@ ] , @ HI @ H H f>"!m > sin 2 Im[;H ; e @@ ] R @ @ ; (!)K(CK #CK !2CK !2CK ) , K s>"!> m >I (CI KL#CI KL) R Q I K L 1 2 ; sin # (1!(!)L cos 2 ) 5 KL @ 2 3
(4.47) (4.48)
(4.49)
(4.50)
;(1!(!)K cos 2 )Im[; < ] @ I I # > (!)K sin 2 Im[; < e @@ ] , @ @ I I s>"!m > sin 2 (CI #CI !2CI !2CI )Im[;H ; e @@ ] , I I R @ @ I
(4.51) (4.52)
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where m, n are sbottom indices and j, k are the chargino indices. Thus, the above three-point 1-loop form factors Cl KL and CK HI for m, n"1, 2 and l"j or k"1, 2, x30, 11, 12, 21, 23, are given by V V l (4.53) C KL"C (m l> , m I K , mI L , m, s, m) , R @ @ R V V Q CK HI"C (mI K , m >H , m >I , m, s, m) , (4.54) V V @ R Q Q R and C is de"ned in Appendix A. Also, > and > are the top and bottom Yukawa couplings V R @ m m R @ >" , >" , (4.55) R (2m sin @ (2m cos 5 5 and, as usual, tan is the ratio between the VEVs of the two Higgs doublets in the theory. Furthermore, note that the phase , although not explicitly appearing in the above, is contained in I ; and < which are the 2;2 matrices that diagonalize the chargino mass matrix and, in the notation used in Section 3.3.2, we have ;H,(Z\)R and . The de"nitions of Z> and Z\ are given in Section 3.3.2 by Eqs. (3.127)}(3.134). The neutralino contribution was also given in [230]: m I (CI KK#CI KK) dA " Q RQ 12 sin 5 I K ;(!)K sin 2 Im[(h !f f H )e\ @R ] *I *I 0I R ! (1!(!)K cos 2 )Im[h f H ]! (1#(!)K cos 2 )Im[h f ] , R *I *I R *I 0I ;(2f #2f #s #s ) , d8 " RQ 16 sin cos 5 5 where 1 f " m H (CK HI!CK HI) Q 2 H I K ;(!)K sin 2 Im[O ( f f H !f f H )e\ @R ] *I 0H R HI *H 0I ! (1#(!)K cos 2 )Im[O (hH f H !hH f H )] R HI *H 0I *I 0H ! (1!(!)K cos 2 )Im[O (h f H !h f H )] , *I *H R HI *H *I 1 f " m (CK HI#CK HI!2CK HI!2CK HI) 2 R HI K ;2(!)K cos 2 Im[h O hH ] R *H HI *I ! 2(!)K sin 2 Im[( f H !f )O hH e @R ] *H 0H HI *I R # (1!(!)K cos 2 )Im[ f H O f ]# (1#(!)K cos 2 )Im[ f O f H ] , R *H HI *I R 0H HI 0I
(4.56) (4.57)
(4.58)
(4.59)
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1 s " m I (CI KL#CI KL) Q 2 I K L
; (!)K
8 sin !(1!(!)L cos 2 ) Im[(h !f f H )e\ @R ] *I *I 0I 5 KL R 3
!
8 sin (1#(!)K cos 2 )!(!)K>L sin 2 Im[h f ] 5 KL R R *I 0I 3
!
8 sin !(1!(!)L cos 2 ) (1!(!)K cos 2 ) 5 KL R R 3
;Im[h f H ] , *I *I
(4.60)
s "m sin 2 (CI #CI !2CI !2CI ) R R I ;Im[hH ( f H !f )e @R ] . *I *I 0I
(4.61)
For the neutralino contribution above, m, n are stop indices and j, k are the neutralino indices. Thus, the above three-point 1-loop form factors, Cl KL and CK HI for m, n"1, 2 and l"j or k"1, 2, V V x30, 11, 12, 21, 23, are given by Cl KL"C (m l , mI K , mI L , m, s, m) , R R R R V V Q
(4.62)
CK HI"C (mI K , m H , m I , m, s, m) , R V V R Q Q R
(4.63)
and C is again de"ned in Appendix A. Also, f , f are gaugino couplings and h are higgsino V *H 0I *H couplings that contain the large Yukawa coupling > . O contains elements of the neutralino R HI mixing matrices. The factors f , f , h , O are all given in [230]. *H 0I *H HI In order to be able to estimate the size of the TDM one has to choose a plausible set of the SUSY masses and parameters involved, i.e., m , , tan , m I I , m I I , cos , cos and the phases , and R @ R @ I R . A reference set of parameters was chosen in [230]: @ m "230, 360 GeV "250 GeV tan "2 4 " I 3
m I "150 GeV R m I "400 GeV R " R 9 " R 6
m I "270 GeV @ m I "280 GeV @ " @ 36 " @ 3
Note that with a common gaugino mass at the GUT-scale all the low-energy gaugino mass parameters are related and are proportional to m } the SU(2) gauginos mass term at the GUT-scale. For example, m +3m (for more details see Section 3.3). E
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Fig. 16. dA(s) and dI 8(s) (note that d8(s),(dI 8(s))/(2 cos sin )) in units of 10\ e cm, for the reference parameter set R R R R 5 5 with m "230 GeV. Note that in (a) Im dA(s) (full line), Im dI 8(s) (dashed line), and in (b) Re dA(s) (full line), Re dI 8(s) R R R R (dashed line). Figure taken from [230].
The real and imaginary parts of the top EDM and ZEDM, including all contributions in diagrams (a) and (b) in Fig. 15 (i.e., gluino, chargino and neutralino contributions), with the above set of the relevant SUSY parameters and as a function of the c.m. energy of the collider, (s, are given in Fig. 16. We see that typically Re dA 8(s), Im dA 8(s)&10\}10\ e cm . R R
(4.64)
Note that these results are about one order of magnitude smaller than what is expected in the 2HDM discussed in Section 4.3. Consider now the low-energy MSSM scenario described in Section 3.3.4. There we have taken P0, motivated by the experimental bound on the NEDM which strongly implies that I (10\}10\. Moreover, all squarks except from the light stop were assumed to be degenerate I with a mass M . It is instructive to evaluate the TDM in this limit in which the only relevant 1 CP-odd phase resides in tI !tI mixing and is proportional to sin (or equivalently to sin } the * 0 R R phase in the top trilinear soft breaking term A ). In this framework, dA 8 > "0 since the two sbottom R RQ particles are degenerate. Moreover, dA 8 gets its contribution only from terms proportional to RQ sin in Eqs. (4.56)}(4.61) where, in general, one "nds that dA 8 (dA 8 . Thus, in this scenario the RE R RQ TDM can be approximated by considering only the gluino exchange diagram in Fig. 15(b) for which only the masses m I , m I and m are relevant. R R E
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Fig. 17. Imaginary (a) and real (b) parts of dA and d8, due to only the gluino exchange diagram depicted in Fig. 15(b), in R R units of 10\ e cm and as a function of (s. We use: m I "50 GeV, m I "m "400 GeV and m "175 GeV. R R E R
In Figs. 17}19 we have plotted the imaginary and real parts of dA 8 in the above MSSM scenario RE with R "1 and with the approximation dA 8KdA 8 , as a function of (s, m and m I (the light stop E R !. R RE mass), respectively. Our reference set of masses for these "gures are m I "50 GeV and R m I "m "400 GeV. We again see that, typically, Re,Im(dA 8)&10\}10\ e cm. From R E R Fig. 18(a) we see that there is a small enhancement in the imaginary part of the TDM as the gluino mass gets smaller and, for example, we "nd Im(dA)K3.25;10\ e cm for m "200 GeV. R E Fig. 19(b) illustrates how the TDM vanishes when the two stop mass eigenstates are degenerate, i.e., m I "m I "400 GeV. R R It is important to note that within the MSSM, unlike in MHDM cases, the top CEDM cannot be calculated simply by replacing the o!-shell photon with an o!-shell gluon. The reason is that SUSY models give rise to an additional gg g coupling (i.e., gluon}gluino}gluino coupling). Thus, in addition to replacing the photon with the gluon in Fig. 15(b), a full calculation of dE has to include R the additional diagram with g g tI in the loop. This e!ect was considered in the context of CP violation in ppPttM #X by Schmidt [231], and we will return to it in Section 7. 4.6. Top dipole moments } summary In this section we have performed a detailed investigation of the top quark dipole moments in models beyond the SM in which new CP-violating phases appear rather naturally. The models that we have considered are MHDMs and the MSSM. In Table 3, we summarize our numerical results for the expected TDM within these class of models and for comparison we also write the expected
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Fig. 18. Imaginary (a) and real (b) parts of dA and d8, emanating from only the gluino exchange diagram depicted in R R Fig. 15(b), in units of 10\ e cm and as a function of the gluino mass m . We use: (s"500 GeV, m I "50 GeV, E R m I "400 GeV and m "175 GeV. R R
Fig. 19. Imaginary (a) and real (b) parts of dA and d8, emanating from only the gluino exchange diagram depicted in R R Fig. 15(b), in units of 10\ e cm and as a function of the light stop mass m I . We use: (s"500 GeV, R m I "m "400 GeV and m "175 GeV. R E R
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size of the TDM in the SM. For illustration, for each model the TDM is evaluated by setting the corresponding CP-violating quantities to plausible representative values, compatible with existing experimental limits if any. Let us also summarize below the salient features of the CP-violating mechanisms of these models which give rise to a non-vanishing TDM, and specify our choice of values used in Table 3 for the CP-violating quantities of each model. E MHDMs: In general, one can distinguish between two types of Higgs mediated CP-violating contributions to the TDM in MHDMs: 1. TDM from neutral Higgs exchanges: In any MHDM with or without NFC and with new CP-violating phases in the neutral Higgs sector, the neutral Higgs-fermion}fermion interaction Lagrangian may be generically expressed as (say for h } the lightest neutral Higgs) L
FDD
g m "! 5 D h fM (aF #ibF ) f . D D (2 m5
(4.65)
The hZZ vertex which is also needed for calculating the TDM in the case of 1-loop neutral Higgs exchanges is given by (see Eq. (3.71)) m 8 cFhg ZIZJ . (4.66) "g IJ 5 m 5 The CP-violating TDM then arises from the interference between the scalar, aF, and the R pseudoscalar, bF, couplings in Eq. (4.65) and the interference between the pseudoscalar coupling, R bF, and the hZZ coupling JcF in Eq. (4.66). R The numbers in the fourth column in Table 3 are given for masses of a neutral Higgs in the range 100}300 GeV (assuming that the masses of the other neutral Higgs particles in these models are much heavier) for L
F88
aF"bF"cF"1 , (4.67) R R and they hold for any MHDM, i.e., a 2HDM of type I and II with NFC, a 2HDM of type III with FCNC in the neutral Higgs sector or for three or more Higgs doublets which have the generic htt interaction Lagrangian in Eqs. (4.65) and (4.66). 2. TDM from charged Higgs exchanges: In models with three or more doublets the charged Higgs sector can acquire new CP-odd phases which can give rise to the CP-violating TDM. Again, one can parameterize a generic (assumed lightest) charged Higgs-up quark-down quark CP-violating interaction Lagrangian, which will appear in such models, as g " 5 K (m u d #m u d )#h.c. , (4.68) SB S 0 * B * 0 (2m 5 where K is the SM CKM matrix and u and d denote charge #2/3 and !1/3 quarks, respectively. The CP-violating TDM will then be proportional to Im(<),Im( H). As L
&>SB
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Table 3 The contribution to the top quark EDM (dA(s)) and ZEDM (d8(s)) form factors, in units of e cm, at s" R R (p #p M )"500,1000 GeV, for the SM (where it is a purely guess-estimate) and for some of its extensions. 4th column R R shows results for neutral Higgs exchanges in any MHDMs with a CP-violating htt coupling of the form (g /(2)(m /m )(aF#ibF ), an hZZ coupling g (m /m )cFg and with aF"bF"cF"1. 5th column is for charged R 5 R R 5 8 5 IJ R R 5 Higgs exchanges in any MHDMs of three or more doublets with a CP-violating H>tb coupling of the form (g /(2m )[m ; (1# )/2#m ; (1! )/2] and with Im(; ;H),Im(<)"5. Only the contribution from the 5 5 R R @ @ R @ lightest neutral or charged Higgs is retained. 6th column shows the results for the MSSM where only the dominant 1-loop gluino exchange diagram with gluino masses m "200}500 GeV is considered, in which CP violation arises from E tI !tI mixing and is proportional to R , sin(2 )sin( ), where and are the angle and phase that parameterize the * 0 !. R R R R tI !tI mixing matrix. The numbers are given for R "1 and for stop masses of 50 GeV (light stop) and 400 GeV (heavy * 0 !. stop) Type of moment (e cm) P
(s (GeV)P
Standard model
500 Im(dA) R
1000
1000
(4.1!2.0);10\
(29.1!2.1);10\
(3.3!0.9);10\
(0.9!0.8);10\
(15.7!1.0);10\
(1.2!0.8);10\
(0.3!0.8);10\
(33.4!1.5);10\
(0.3!0.9);10\
(0.7!0.2);10\
(0.3!2.7);10\
(1.1!0.3);10\
(1.1!0.2);10\
(15.8!2.5);10\
(1.1!0.3);10\
(0.2!0.2);10\
(9.2!1.2);10\
(0.4!0.3);10\
(1.6!0.2);10\
(22.9!0.8);10\
(0.1!0.3);10\
(0.2!1.4);10\
(0.6!1.9);10\
(0.4!0.1);10\
(10\ 1000 500
Re(d8) R
Supersymmetry m "200}500 E
(10\ 500
Im(d8) R
Charged Higgs m > "200}500 &
(10\ 500
Re(dA) R
Neutral Higgs m "100}300 F
(10\ 1000
mentioned before, the interaction Lagrangian in Eq. (4.68) is not the only source of CP violation in this class of models and one also has to take into account the CP-violating neutral Higgs contributions arising from the h+ coupling in Eq. (4.65). In fact, for the TDM we "nd that the CP-odd e!ect from a H>tb coupling in Eq. (4.68) is much smaller, i.e., typically by a factor of &(m /m ), than the one from the htt coupling in Eq. (4.65). @ R The numbers in the "fth column in Table 3 are given for masses of the charged Higgs in the range 200}500 GeV (again assuming that the masses of the other charged Higgs particles of these models are much heavier and therefore their contribution is negligible) and for Im(<)"5 ,
(4.69)
and they represent only the charged Higgs contribution to the TDM in any MHDM with three or more doublets, which have the generic H>tb interaction vertex in Eqs. (4.68). E MSSM: In the MSSM, as was shown in the previous section, if one neglects the phase in the Higgs mass parameter (as strongly implied from the existing limit on the NEDM) and the small mass
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splitting between the left and right superpartners of the light quarks, then the dominant contribution to the TDM arises from 1-loop gluino exchange. In that case the TDM emanates from tI !tI mixing and is proportional to the CP-violating quantity * 0 R , sin(2 )sin( ) , (4.70) !. R R where the angle represents the CP-phase in the soft trilinear breaking term associated with the R top, i.e. A . R The numbers in the sixth column in Table 3 are given for gluino masses in the range 200}500 GeV, for R "1 , !. and for
(4.71)
(4.72) m I "50 GeV , R m I "400 GeV , (4.73) R where m I are the masses of the two stop mass-eigenstates. They therefore represent only the R dominant gluino contribution to the TDM in any low-energy supersymmetric framework in which all squarks except from the stop are degenerate and the phase in the Higgs mass parameter, , is neglected. We observe from Table 3 that the expected magnitudes of the real and imaginary parts of the top EDM and ZEDM in these models for masses in the loops of, typically, several hundreds GeV and energy scales of 500}1000 GeV are Im dA 8&few;10\}10\ e cm , (4.74) R Re dA 8:10\ e cm , (4.75) R Charged Higgs: Im dA 8&few;10\}10\ e cm , (4.76) R Re dA 8&few;10\}10\ e cm , (4.77) R Supersymmetry: Im dA 8&few;10\}10\ e cm , (4.78) R Re dA 8:10\ e cm . (4.79) R The top CEDM dE, given in units of g cm, can be estimated within these models as follows: (i) In the R Q neutral Higgs exchange case, it is simply given by multiplying dA by 1/Q , where Q "2/3 is the top R R R quark charge. (ii) In the charged Higgs case, one cannot simply replace the o!-shell photon with an o!-shell gluon and dE has to be explicitly calculated from Eq. (4.34). Nonetheless, we "nd that R dE/(g cm) +dA/(e cm). (iii) In the MSSM case, it is also not possible to extract the top CEDM from R Q R the top EDM by the simple exchange of a photon with a gluon since there is an additional graph with two gluino propagators in the loop coming from a new gg g coupling. We have not estimated this additional contribution here. To conclude this section, we have shown that in MHDMs with CP violation in the neutral or the charged Higgs sector and in the MSSM, the TDM is always bounded to be smaller than about Neutral Higgs:
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&10\ e cm. This is rather discouraging since, as we will see in the next few sections, the attainable limits on the TDM that can be obtained in future e>e\ and hadronic colliders seem to fall short by about one order of magnitude compared to the above model-dependent expectations for these quantities. Basically, the strategy that we will describe in the following sections for such investigations of the various TDM in future colliders is to incorporate an e!ective Lagrangian approach which elaborates new e!ective interactions of a dipole moment type at the tt, Z, g vertices, with dimensions greater than 4 which can provide for a model independent investigation of new physics beyond the SM. The e!ects of such phenomenological vertices can then be studied in future e>e\ and hadron colliders. Of course, a hadron collider is appropriate for the study of the top CEDM and is not a very good environment for studying the top EDM and ZEDM couplings to a photon and a Z-boson; the EDM and ZEDM will obviously be masked by the gluon dynamics which will govern top quarks production in a hadronic collider. Therefore, a more natural place for such studies will be an e>e\ collider. Later we will discuss the feasibility of extracting information on the various TDMs in both hadron and e>e\ colliders through an investigation of CP-odd and even CP-even observables, e.g., cross-sections. It should be noted that the information that can be obtained on a CP-odd quantity by studying its e!ect on a CP-even observable is much less than what might be learned about the various EDMs of the top by measuring a non-vanishing CP-odd observable driven by these CP-odd e!ective couplings. In particular, folding into a given amplitude the various CP-violating EDM interaction terms, the corresponding di!erential cross-section will acquire a CP-odd piece driven by the interference of the tree-level process with the EDMs interactions (to leading order only one EDM e!ective coupling has to enter in each diagram). Then with an appropriate CP-odd observable, which linearly depends on the EDM of the top, one can, in principle, analyze directly and separately the possible CP-violating e!ects that can arise from each EDM interaction in collider experiments such as e>e\ or ppPttM . We will discuss CP-violating e!ects in these reactions in the following sections.
5. CP violation in top decays In this chapter we discuss CP violation in various top decays. In particular, we will consider two-body decays, i.e., tPd =, with k"1, 2, 3, the generation index, three-body decays as well as I radiative decays. The following CP-violating asymmetries will be reviewed (not for all decay modes): E E E E
Partial rate asymmetry (PRA). Partially integrated rate asymmetry (PIRA). Energy asymmetry. polarization asymmetry.
Although the polarization asymmetry tends to be the largest e!ect in models with CP violation phase(s) in the charged Higgs exchanges, for the sake of generality and completeness, we will "rst discuss the other e!ects. Asymmetries such as top polarization asymmetry, although intimately related to the top decays, but for which most, if not all discussions in the literature are speci"c to the
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production process, are discussed in Sections 6}8. Models included are: E E E E
Standard model (SM). Two Higgs doublet model (2HDM). Three Higgs doublet model (3HDM). Minimal supersymmetric standard model (MSSM).
Only PRAs are considered within the framework of all the models, while only the predictions of the 3HDM for tPb are presented for all the above asymmetries. O In addition, for the PRA, the form factor (FF) approach to the tb= vertex will be presented. In this approach, the FFs can assume complex values, thus emulating physical cuts in higher order Feynman diagrams, leading to non-vanishing CP-odd, ¹ -even observables such as PRA. This can , be contrasted [34,232,233], with the e!ective Lagrangian approach, assuming that all new particles lie above m , where the coe$cients are real. There, only ¹ -odd CP-violating asymmetries can R , emerge [234]. All CP-violating top decay asymmetries, within the SM, that have been studied so far are found to be too small to be measured. The same conclusion holds for CP-violating top production asymmetries. This results from severe GIM [36], or even double-GIM cancellations due to the fact that the masses of d, s, b are too small compared to the top quark mass. The obvious conclusion is that an observation of CP violation in top quark decays, will serve as a very strong indication for the existence of new physics beyond the SM. 5.1. Partial rate asymmetries In most models, the PRA, de"ned in Eq. (2.27) in Section 2.2 is found to be small. This can be readily understood, in a model-independent way, from the CP-CPT connection discussed in Section 2.3. Let us consider, for example, what seems to be the main decay of the top quark tPb=>. Due to CPT, to have a non-vanishing PRA, at least one additional decay channel should be available for the t. In other words, in the limit that tPb= becomes the only decay channel possible, then PRA has to vanish due to the fact that PRA then tends to become equal to the asymmetry in the total widths of t and tM which is constrained by CPT to vanish. In the SM, by virtue of < K1, there is very little competition to tPb=>, and the PRA turns out to be tiny R@ indeed. Larger asymmetries are obtained for other decay channels, but their rates are too small to result in an experimentally interesting signal. In models beyond the SM, the situation is slightly better since there is a possibility for new particles to be produced in top decays, leading to the absorptive part necessary for PRA. The largest credibly possible predicted PRA, is &0.3% for tPb=> in the MSSM with low tan which arises mainly since the top can have an appreciable decay rate into a tI (i.e. the stop and neutralino "nal state) in this scenario (see below). However, once the window which allows SUSY "nal states in t decays, such as tI is closed, then the PRA in top quark decays become vanishingly small in this model too, i.e., the MSSM. In the following, we elaborate on some of the issues mentioned above, and more. 5.1.1. PRA in the SM tPd =>: CP violation via PRA in the process tPd =>, where k"1, 2, 3 is the generation I I index, was discussed in [232,235,236]. The PRA results from interference of the two diagrams in
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Fig. 20. (a) Tree-level diagram for tPd = (k"1, 2, 3 for d, s, b). (b) Example of a 1-loop diagram for tPd =. I I
Fig. 20, i.e. from tree;loop interference, where the loop is the non-diagonal t!u self-energy. By H `interferencea, we actually mean `the di!erence between interferences for the process and its CP conjugated processa. The loop contributes the necessary imaginary part Im (m) through the cut G on d =. One has to sum over i, j, and obviously d Od and u Ot. Before continuing to discuss the G G I H above contribution of the absorptive part to the PRA, let us digress to show that, as stated in [235], the CP}CPT connection [55}58] forbids the self-energy of the = from contributing to the PRA. This will be shown within the SM; the proof holds for other models too, as can be easily generalized. Consider the case d "b (the generalization to k"1, 2 is trivial). The result of [235] is that the I PRA in tPb=> comes from the interference of diagram (a) with diagram (b) in Fig. 20, where the absorptive part is provided by the s= cut in diagram (b) (the d= cut is negligible here). Thus we can symbolically write that the result of [235] corresponds to (tPb=>)!(tM PbM =\)"Re a(b=);Im b(b=; s= cut) ,
(5.1)
where a, b denote the contribution of diagrams (a), (b), respectively, and the arguments of their real, Re, and imaginary, Im, parts denote the "nal state, with the additional information about the relevant cut in Im. Note that the CKM as well as numerical factors are suppressed in Eq. (5.1). Now, the rate di!erence in Eq. (5.1) can be written as (tPb=>)!(tM PbM =\)"Re 1(bu dM );Im 3(bu dM ; s= cut) I I I I #Re 1(bll );Im 3(bll ; s= cut) ,
(5.2)
where 1, 3 stand for diagrams (1) and (3), respectively, in Fig. 21. Also u dM denote summation over I I all quark pair states that the = can decay into, and ll means summation over lepton generations. Other q= cuts give negligible contributions to the PRA for tPb=>. To prove that the PRA contains no term from the interference of diagram (2) with (3), and of diagram (1) with (4), we have to show that, after summing over all "nal states Re 3;Im 2#Im 4;Re 1"0 .
(5.3)
But for a speci"c "nal state and cut (say, the bcs "nal state, with a cut) in the Im's, there is O a compensating contribution (say, the b "nal state, with a cs cut). Thus, summing, for each "nal O state, over all possible cuts on the = line (rescattering excluded), then summing over "nal states, it Note that near resonance diagrams (2) and (4) of Fig. 21 become O(g) and O(g), respectively.
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Fig. 21. The four diagrams considered in the proof that the = self-energy diagram does not contribute to CP violation for tPd =, within the SM. Dashed lines indicate cuts. Similar considerations hold for other models. I
is easy to show that Eq. (5.3) holds for tPb=. We are thus left with Re a(b=);Im b(b=;s= cut), which is, by the way, compensated by Re a(s=);Im b(s=; b= cut). Let us now recapitulate the calculation of [235], for the CP-violating PRA for tPb=, de"ned as (tPd =>)!(tM PdM =\) I I , A , I (tPd =>)#(tM PdM =\) I I
(5.4)
in the SM, with k"3. The two interfering amplitudes, Fig. 20(a) and (b), have a relative CP-odd phase and Fig. 20(b) has the required absorptive phase. They are given by A? "
(5.5)
m R A@ "!AK
(5.6)
where ig AK "! 5 u (p ) ¸u(p ) I . I R (2 I I
(5.7)
I is the =-boson polarization vector and ¸"(1! )/2. The PRA was then found to be Im(
mm S B , m 5
(5.9)
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where m,m!m and m,m!m. It was found in [235] that S R S B @ B mm @ A f (y) C , A " (5.10) I 8 sin m 5 I 5 5 independent of the Wolfenstein parameter A. f originates from the imaginary part of the 5 self-energy and is given by f (y)"!3 5
(1#y) , y
y,m /m R 5
(5.11)
and C "![(1!)# ]\ , (5.12) B C " , (5.13) Q m (5.14) C "! Q . @ m @ The largest asymmetry is obtained for d "d, A K10\, requiring at least 10 top quarks! I B Even smaller asymmetries are predicted for tPs= and tPb=. Thus, the discussion of PRAs for tPd = in the SM is of purely an academic value. Let us note, in passing, that in the I SM with four generations (a currently unfashionable approach) A may be substantially I enhanced [235]. tPc<: The branching ratios for the rare #avor changing processes tPc [37,237}239], tPcg [37,240] and tPcZ [37,241] in the SM are: 4.9;10\, 1.4;10\ and 4.4;10\, respectively. The CP-violating PRAs, resulting from the interference of two penguin diagrams, are largest for the even rarer (by more than an order of magnitude) decays tPu [239] and tPug [240] where they are K0.2%. A slightly larger asymmetry is obtained for tPugHPuu u [240]. Three-body t decays: In the above, we have discussed interferences of the type Im A ;Re A , where the loop is the o!-diagonal t!u self-energy (in contrast, in the MSSM, see below, `loopa H stands for `vertex correctionsa), and of the type Im A ;Re A
. The prime indicates that the imaginary and real parts of the penguins must have a di!erent weak phase. This is possible since three di!erent quark amplitudes are in the penguin graph. PRA in the SM for the six three-body top decays of the type tPcqq and tPuqq , q"d, s, b
(5.15)
was considered in Refs. [61,64,65]. There, PRA from the interference Im A ;Re A for each of the "nal states in Eq. (5.15) was calculated. This interference is present in top decays due to the fact that m 'm #m , thus endowing the tree diagram with a well-de"ned CP-even phase R 5 @ from the =-width [66}70] (see also footnote 3). Though the =-width turns a `treea diagram into a `loopa diagram, we will keep loosely calling it as a `treea. The above interference is there in addition to terms of the type: Re A ;Im A , which are analogous to the interference [60] that leads to the CP violation in b decays, such as bPsuu (see the discussion in Section 2.3). These interferences arise, since for each speci"c channel tPq qq , where q "c or u, there are two classes of
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possible paths: 1. The tree process tPq= followed by =Pq q . 2. The 1-loop penguin process tPq gH followed by gHPqq . The two Feynman amplitudes, tree and penguin, have di!erent weak and strong phases. Consequently, the interference of these two channels, provides the necessary condition mentioned in Section 2.2 for the observability of CP violation. Note that penguin;penguin terms are relatively small, except in the absence of tree terms. The PRA for the three-body decays tPq qq where q "c or u and q"d, s, b is [64,65,61]: 8g m A" Q 5 5 Im(KO [K m Re(¹ !¹ )!K Im(¹ !¹ )] H B H B , (5.16) ; 5 5 (q #q)(K# m ) 5 5 H where Z "< < (1!x)(1#x!2x) with x"m /m, K"m#m #2q ) q !m ; ¹ is 4 RO OYO 5 R O OY 5 H a lengthy expression for the resulting trace, where j indicates the virtual quark exchange in the loop (for which we have used CKM unitarity). Furthermore, K " !(=P q q), thus excluding 5 rescattering, i.e., q q ; q q [61]. The largest contribution to (¹ !¹ ) is from j"b, i.e., H B (¹ !¹ )<(¹ !¹ ). As a result, the asymmetry is completely negligible for the most abundant @ B Q B "nal state, tPbcbM as Im( . C
5.1.2. PRA in a 2HDM In a 2HDM, there are "ve physical Higgs particles: 3 neutral, HI, k"1, 2, 3 and 2 charged ones, H!. If one imposes discrete symmetry [123] to avoid FCNC at tree-level, then there is no CP violation in the charged Higgs sector (unlike the case in the 3HDM, discussed below) and CP violation then exists only in the neutral Higgs sector (see Section 3.2). This class of 2HDM is further subdivided into Models I and II (see Section 3.2.3). However, there is a version of 2HDM, the so-called Model III (see Section 3.2.2), where no discrete symmetry is imposed; the model then admits tree-level FCNC. In Model III, large tree-level FCNC may then be avoided for the light quarks by assuming [126] that the couplings hq q are proportional to (m m ) . Note that in the G H G H class of 2HDMs with NFC (i.e., Model I or II), CP violation resides in #avor-diagonal HI
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exchanges; consequently CPT forbids PRA in tPd = to arise. In model III, PRA in tPd = need I I not vanish, though it is expected to be very small due to the small HId d J(m G m H /m coupling G H B B 5 (iOj). tPd =>: The e!ect of a charged Higgs with the CKM phase was considered in [235] for I tPd =>. Assuming m > #m (m , then, since a charged Higgs exchange is added to the I & G R = exchange in Fig. 20(b), one has to make the following substitution in Eq. (5.10) (where a large 1/tan ,v /v is assumed) B S f (x, y) . (5.17) f (y)Pf (y)# & 5 5 tan f (y) is de"ned in Eq. (5.11) and 5 (1!x) , x,m /m . f (x, y)" R & & xy
(5.18)
Taking into account the experimental bounds on tan , there is no increase in the tiny PRA, A , I over what was obtained in the SM. This conclusion also holds for other observables governed by a charged Higgs exchange unless the CP-violating couplings to fermions are di!erent from the SM. 5.1.3. PRA in a 3HDM The 3HDM (Weinberg Model for CP violation [62]), as described in Section 3.2.4, can cause CP violation e!ects, through its CP-violating couplings of a charged Higgs-boson to fermion pairs. There are no tree-level FCNC within this model. The physical charged Higgs states are H! and H!, where, for simplicity, it is usually assumed that m <m , thus decoupling H from all & & predictions. It is also easy to show that any CP violation asymmetry vanishes for m "m & & through a GIM-like mechanism. In the 3HDM, the interesting new phenomenon is the existence of a CP-violating coupling in the leptonic term in the Yukawa part of the Lagrangian (see Eq. (3.88)) g " 5 (m tM b #m )H>#h.c. , (5.19) R 0 * O * 0 & (2m 5 where < "1 is assumed. Note the proportionality to the mass of the charged lepton, which R@ prompted studies of CP violation in the reaction tPb . The reaction tPbcs is not that useful in O view of the di$culty in identifying c, and especially s jets. In addition, the mode will enable, by following the decay products, measurements of spin related CP-violating observables. It is convenient [161,162] to parameterize CP violation in the Yukawa couplings with a CKMlike matrix (the SM CKM matrix itself is assumed to be real), then and in Eq. (5.19) are given by (see Eq. (3.89)) L
>
c c s !s c e B& , " s c c s s #c c e B& , " s s
(5.20) (5.21)
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Fig. 22. Tree-level diagrams and a representative set of box diagrams considered for PRA in tPb, within the 3HDM.
where s ,sin(I ) and c ,cos(I ), and I , are parameters of the model. CP violation will be G G G G G & proportional to combinations such as Im(;)"Im(H ) . (5.22) tPb> : Let us de"ne the PRA O (tPb> )!(tM PbM \ ) O O . (5.23) A" O (tPb> )#(tM PbM \ ) O O The lowest-order contribution to A arises due to the interference of the SM = mediated tree O diagram (see diagram (a) in Fig. 22) with the 3HDM H> mediated tree diagram (diagram (c) in Fig. 22). We assess the contribution of the second graph making two simplifying assumptions: (i) m > ,m ;m > , this allows us to neglect the e!ect of the heavier charged Higgs, H . Further& & & more, we also assume that (ii) m > 'm , thus the H> width becomes irrelevant. A will then be R O & proportional to Im(=!tree);Re(H!tree), where, in analogy to our previous discussion in Section 5.1.1, the CP-violating CKM-like angular function was factored out. It is easy to see that, because of the chirality miss match, only the longitudinal part of the =-propagator contributes to A . In other words, decomposing the =-propagator in the unitary gauge as O q q q q (5.24) D5 "i !g # I J G #i I J G , IJ 2 IJ q q *
only Im G will appear in A [242,243]. In fact A , obtained from the interference between * O O diagrams (a) and (c) in Fig. 22, is proportional to
KR \K@
f (q)Im GK , (5.25) * KO where GK indicates that is missing from G to respect CPT invariance and f (q) is a phase-space * * function. Now, while the transverse part of the =-propagator in Eq. (5.24) resonates, i.e., A JIm(;) O
1 , G K 2 q!m #i m 5 5 5
Note a typographical error in in [161,162].
(5.26)
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for qKm , there is no such enhancement for G . This is one of the reasons for an extremely small 5 * asymmetry, 10\ [243] or smaller [63], from tree;tree interference. The other reasons are the proportionality of both the Higgs coupling and Im GK to small fermion masses. * The next logical step is to capitalize on the resonance behavior of Im G and the fact that, unlike 2 Im G , it is not proportional to small masses, by considering interferences that are higher order in * the weak interaction coupling constant [59]. Thus, PRA from interferences of the type Im(a);Re(b) and Re(c);Im(d), is calculated; (a)}(d) denote the diagrams in Fig. 22, where (b) and (d) represent all box diagrams. Bremsstrahlung is included, but diagrams yielding A P0 for O G P0, see below, were neglected. The asymmetry is then given by * [Im(;)/512m]dq du Im GK Re(tree;box) R 2 , (5.27) AK O (tPb=Pb) where Im(;) is de"ned in Eq. (5.22) and u"(p #p ), q"(p #p ) , O @ J O m K 5 5 Im GK "! 2 (q!m )#( m ) 5 5 5
(5.28) (5.29)
with K " ! . (5.30) 5 5 5 OJ The maximal value of A turns out to be negative, and of order 10\. Subsequently, the following O contributions to A , explicitly neglected in [59], were calculated in [243]: O E Since the integration in Eq. (5.25) reaches up to (m !m ), it includes a region with q'm , for R @ 5 which a =! loop in G has to be taken into account. * E Imaginary parts can also appear in box and vertex diagrams corresponding to tPb=. Both new terms are non-resonant and do not su!er small mass suppression from fermion loops in G . They turn out to give a large correction, of about 50% and of the same sign, as compared to the * value of A calculated in [59] using only G . The minimal number of t-quarks required to observe O 2 CP violation in PRA within the 3HDM is } although many orders of magnitudes smaller than the best leptonic SM result (i.e., tPdl [64,65]) } of the order of 10}10 and thus not very promising. As we will see later, one can do much better in the 3HDM by considering PIRA rather than PRA. 5.1.4. PRA in the MSSM An extremely interesting possibility, investigated in [215,235,246,247], is that the CP-violating PRA in two-body modes (that was found to be extremely small in the SM, 2HDM and in the 3HDM) may receive appreciable contribution from new SUSY CP-odd phases. For example, While there is agreement in the literature as to the above facts, there is controversy } into which we do not enter here (due in part to the fact that it has no observational consequences) } regarding the form of Im G [63,242}245] to be * inserted in Eq. (5.25).
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Fig. 23. The SUSY-induced 1-loop Feynman diagrams that contribute to CP violation in the main top decay tPb=. is the chargino, is the neutralino, g is the gluino and tI , bI are the stop and sbottom particles, respectively.
consider the PRA A in Eq. (5.4) for the main top decay tPb=; the Feynman diagrams that can potentially contribute to A in the MSSM are depicted in Fig. 23. Recall that since A is ¹ -even, it requires an absorptive phase in the Feynman amplitude. This , necessitates radiative corrections to the tPb= to at least 1-loop order and, in particular, the SUSY particles exchanged in the loops have to be light enough such that absorptive cuts will arise. Of course, in addition to the strong phase from Final state interactions (FSI), a CP-odd phase is needed. We recall that, in the MSSM, with the most general boundary conditions for the soft breaking parameters at the scale where they are generated and ignoring generation mixing, only three places remain in the SUSY Lagrangian that can give rise to CP phases that cannot be rotated away: The superpotential contains a complex coe$cient in the term bilinear in the Higgs super"elds and the soft-supersymmetry breaking operators introduce two further complex terms, the gaugino masses m and the left- and right-handed squark mixing terms. The latter, being proportional to the trilinear soft breaking terms (i.e., the A terms) and to , may be complex O in general (for more details see Section 3.3). It is clear then that, in general, there are many sources of CP-violating phases. Therefore, reliable predictions cannot be made unless we make some simplifying assumptions. Let us "rst describe a convenient way to derive the PRA A . Following [215], the tPb=> and tM PbM =\ decay vertices can be parameterized as follows
D. pI g I R #D. I Pu , (5.31) JIR,i 5 u I R I m (2 .* 0 @ R D M . pI g I R #DM . I Pv , JIRM ,i 5 v (5.32) I I @ m (2 .* 0 R R where D* 0 and D* 0 de"ned in Eqs. (5.31) and (5.32), contain the CP-violating phases as well as I I the absorptive phases of the decay diagram (k) (k"a, b, c or d corresponding to diagrams (a)}(c) or (d) in Fig. 23). The important contributions to the D's above are likely to come from those diagrams in which one of the two on-shell superparticle is the lightest supersymmetric particle (LSP), e.g., the neutralino in our case. Such is the case for diagrams (b) and (d) in Fig. 23. Also, with a very light stop (i.e., &50 GeV) an absorptive cut can arise from diagram (a) in Fig. 23 if the gluino mass is below :130 GeV. The current experimental bounds on the superparticles involved in the loop of diagram (c) in Fig. 23 are already stringent enough that they are unlikely to have absorptive parts for m &175 GeV, see e.g., [216}223]. R
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Let us now write A in the most general case with no assumptions on the masses of the SUSY particles and taking into account all four diagrams in Fig. 23. In terms of the scalar (D ) and I vector (D ) form factors, the PRA A is given by I (x!1) A " Re(D0 #DM * )#Re(D* !DM * ) , (5.33) I I I I 2(x#2) I where x,m/m and the sum is carried out over all decay diagrams in Fig. 23 (i.e., k"a, b, c and R 5 d). It is easy to show that if one de"nes
I Q ;e BU , D0 &e BI (5.34) I I Q ;e BU , D* &e BI (5.35) I where I, I are the CP-even absorptive phases (i.e., FSI phases) and I, I are the CP-odd Q Q U U phases associated with diagrams (a)}(d) in Fig. 23, then I Q ;e\ BU , DM * &!e BI I I Q ;e\ BU . DM * &e BI I We then get for the scalar form factors in Eq. (5.33)
8 Re(D0 #DM * )"! Q m m OIm C? , ? ? 3 R E ? m [m OIm(C@ !C@ )#m L OIm C@ ] , Re(D0 #DM * )"! R R @ Q @ @ @ sin 5 m [m OIm(CA !CA )!m K OIm(CA !CA ) Re(D0 #DM * )" R R A Q A A A sin 5 #m L OIm(CA #CA )] , Q A Re(D0 #DM * )"Re(D0 #DM * )(m L P!m K , m K Pm L , Q Q Q B B A A Q O PO , Im CA PIm CB ) , A B GH GH while the vector form factors in Eq. (5.33) are given by
(5.36) (5.37)
(5.38) (5.39)
(5.40)
(5.41)
Re(D* !DM * )"0 , ? ?
(5.42)
OIm C@ , Re(D* !DM * )"! @ @ sin @ 5
(5.43)
In the discussion to follow we will evaluate A within a plausible set of the low-energy SUSY parameter space. We note that in [215] there is a misprint in one of the terms proportional to m in the form factor Re(D* !DM * ). 5 A A The correct form of this term is given in Eq. (5.44).
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1 Re(D* !DM * )" [O(mIm(CA !CA ) A A A R 2 sin 5 #m Im(CA #CA !CA !CA )#2Im CA ) 5 #m m K OIm(CA !CA ) R Q A !m m L OIm(CA #CA !CA ) R Q A !m K m L OIm CA ] , (5.44) Q Q A Re(D* !DM * )"Re(D* !DM * )(m L P!m K , m K Pm L , OG POG , Im CA PIm CB ) . B B A A Q Q Q Q A B GH GH (5.45) Here Im CI , x30, 11, 12, 21, 22, 23, 24 and k"a!d, are the imaginary parts, i.e., absorptive V parts, of the three-point form factors associated with the 1-loop integrals in diagrams (a)}(d) in Fig. 23. The CI are given by [215]: V (5.46) C? "C (mI H , mI G , m , m , m, m) , V V @ R % 5 R @ (5.47) C@ "C (mI H , mI G , m L , m , m, m) , V V @ R Q 5 R @ (5.48) CA "C (m L , m K , mI H , m , m, m) , @ 5 R @ V V Q Q CB "C (m K , m L , mI G , m , m, m) , (5.49) V V Q Q R 5 R @ and C (m , m , m , p , p , p ) is de"ned in Appendix A. The indices i, j"1, 2 stand for the two stop, V sbottom mass eigenstates, respectively, and m"1,2 and n"1}4 correspond to the two charginos and four neutralinos mass eigenstates, respectively; also, m is the gluino mass. % The OG 's in Eqs. (5.38)}(5.45) contain the SUSY CP-odd phases for the decay diagrams and they I were given in [215]. There, also the required Feynman rules for calculating the above PRA were given. For example, OG , for i"1}4, containing the SUSY CP-odd phases which appear in diagram B (d) are O"!Im(K\M) , B O"Im(K\M) , B O"Im(K>M) , B O"!Im(K>M) . B Here we have de"ned 1 ZLHZ\ , K>,ZLHZ\ # , K (2 , K As will be shown below, in our case this diagram will give rise to the leading contribution to A .
(5.50) (5.51) (5.52) (5.53)
(5.54)
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1 K\,ZLZ>H! ZLZ>H , , K (2 , K
97
(5.55)
m 1 R (Z> ¸>HGH!(2Z> ZLHG ) M, R K , R (2 M5 sin K 1 # (2
m R ZGZ> ZLH!(2ZGZ> ¸>H , R K , R K M sin 5
(5.56)
m 4 1 R tan ZGZ> ZL#(2ZGZ> ZL M, 5 R K , R K , (2 M5 sin 3
!
m 4 R Z> ZLGH# tan Z> ZLG , K , R 5 K , R M sin 3 (2 5 1
(5.57)
and G ,ZGHZG , R R R
(5.58)
1 ¸!, tan ZL$ZL . 5 , , 3
(5.59)
In Eqs. (5.54)}(5.59), Z , Z and Z\, Z> are the mixing matrices of the stops, neutralinos and R , charginos, respectively (i.e., with indices i, n and m), which are de"ned in Section 3.3.2. Obviously, to obtain an estimate of the numerical value of the asymmetry, one needs to know the de"nite form for the mixing matrices and various other parameters. Not knowing these makes it very di$cult to give a reliable quantitative prediction for the asymmetry. Therefore, one has to choose a reference set of the SUSY spectrum subject to theoretically motivated assumptions as well as experimental data. Such a reference set which constructs a plausible low-energy MSSM framework was described in [215] (and is also described in Section 3.3.4). The key assumptions made there are: E There is an underlying grand uni"cation which leads to the relation in Eq. (3.137) between ;(1) and SU(2) gaugino masses and the gluino mass. E All squarks except the lighter stop (with a mass denoted hereafter by m ) are degenerate with J a mass M ; in the analysis below we set M "400 GeV. 1 1 E The gluino mass is varied subject to m '250 GeV [216}223]. % E The parameters are chosen subject to the upper limit on the NEDM, d (1.1;10\ e cm [4]. L In particular, the Higgs parameter is chosen to be real as strongly implied from this upper bound on the NEDM when the squark masses are below &1 TeV. With the above criteria one is left with only one CP-odd phase arising from tI !tI mixing. That is, * 0 when is real all the elements in OG above except from G , de"ned in Eq. (5.58), are real. Recall that I R the stops of di!erent handedness are related to their mass eigenstates tI , tI through the following > \
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transformations (see Section 3.3.2): tI "cos tI !e\ @R sin tI , * R \ R > tI "e @R sin tI #cos tI . 0 R \ R > The asymmetry is thus proportional to the quantity (see also Eqs. (3.126) and (4.43))
(5.60)
R ,2G "sin(2 ) sin( ) , (5.61) !. R R R where G is de"ned in Eq. (5.58). R Although the CP-odd phases in the squarks sector generate the NEDM, the resulting restrictions on the CP-phases in the tI }tI mixing are rather weak. As we have demonstrated in Section 3.3.4, * 0 the main contribution to the NEDM (when is real) comes from the mixing of the superpartners of the lighter squarks. Therefore, if the trilinear soft breaking terms A , A and A are not correlated S B R at the EW-scale, as is the case in our low-energy MSSM framework described in Section 3.3.4, then it is not unreasonable to study the e!ects of maximal CP violation in the stop sector, i.e., R "1 !. without contradicting the current limit on the NEDM. With no further assumptions, the reference parameter set consists of M , m , m , , tan and R . 1 J % !. The neutralinos and charginos masses are extracted by diagonalizing the corresponding mass matrices which are functions of , m and tan (see Section 3.3.2). Note that the consequences of % such a low-energy MSSM scenarios on the various diagrams in Fig. 23 that can potentially contribute to the PRA, A , are: E For m 9250 GeV diagram (a) does not have the needed absorptive cut and thus does not % contribute to the PRA. E Diagram (c) does not have a CP-violating phase when arg()"0. This simpli"es our discussion to a great extent and we are therefore left with only two diagrams that can contribute to A . These are diagrams (b) and (d), where in fact we "nd that, by far, the leading contribution comes from diagram (d). In particular, we have calculated the PRA e!ect, A , arising from diagrams (b) and (d), for arg()"0, m "M "400 GeV and subject to 1 O m '50 GeV, m '250 GeV, the LSP (in our case the neutralino) mass to be above 20 GeV and J % the mass of the lighter chargino to be above 65 GeV. In Figs. 24 and 25, we plot A for two values of tan which correspond to a low (tan "1.5) and high (tan "35) tan scenarios, where the SUSY mass parameters are varied subject to all the above constraints and maximal CP violation is taken in the sense that R "1, thus presenting !. A in units of sin 2 sin . In particular, in Figs. 24(a) and (b) we plot the asymmetry as a function R R of for several values of m and for tan "1.5 and tan "35, respectively. In Figs. 25(a) and (b) % the asymmetry is plotted as a function of the gluino mass m for several values of and for % tan "1.5 and tan "35, respectively. In both "gures, we set M "400 GeV and m "50 GeV. 1 J Evidently, from Figs. 24 and 25 we see that a PRA in tPb= is very small over the whole range of our SUSY parameter space. In particular, we always "nd A (0.3% . (5.62) Of course, the asymmetry further drops as the mass of the lighter stop, m , is increased and vanishes J when m 9130 GeV since in that case there is no absorptive cut in the relevant contributing J
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Fig. 24. The SUSY-induced PRA A in the main top decay tPb=, as a function of , for several values of m and for (a) % tan "1.5 and (b) tan "35. M "400 GeV, m "50 GeV is used. Figure taken from [215]. 1 J
Fig. 25. The SUSY-induced PRA A in the main top decay tPb=, as a function of m , for several values of and for (a) % tan "1.5 and (b) tan "35. M "400 GeV, m "50 GeV is used. Figure taken from [215]. 1 J
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diagrams. Also, we "nd that the PRA is almost insensitive to tan in the range tan 910 and that A &0.3% become possible only for tan &O(1). The asymmetry we "nd is therefore somewhat small compared to the estimates of Grzadkowski and Keung (GK) [235] and of Christova and Fabbrichesi (CF) [246]. In the GK limit only the gluino exchange of diagram (a) was considered. They utilized the CP-violating, quark-squarkgluino interaction, occurring with coupling strength g (the QCD coupling) and the =tI bI Q interaction ig R tI =\I#h.c. . L "i(2g [tI H¹?(M ?t )#tI H ¹?(M ?t )]#(t b)! 5 < bI >@ Q * * 0 0 OOH (2 R@ * I *
(5.63)
As in our case, the most important source of CP violation is then the phase in the tI !tI mixing * 0 and, therefore, their e!ect is also proportional to R de"ned above. However, the GK limit is !. applicable only if m 'm #m I , so that an absorptive cut can occur in diagram (a). In the best case, R % R GK found a &1% asymmetry for m "m I "100 GeV. % @ On the other hand, in the CF limit, numerical results were given only for the neutralino exchange diagram (i.e., diagram (b)) wherein the CP-odd phase was chosen to be proportional to arg() and maximal CP violation with regard to arg() was taken. This can be parameterized by introducing a single CP-violating phase [246]: 1 f @ NH K sin , !. L L 2
(5.64)
where f @ and N appear in the q q Lagrangian I I L
1 (2m D NH (1! ) q ¸ " g q fD(1# )! 5 D L L D* OOQ 2m B L \D 2 5 D L D
1 (2m D NH # g q gD(1! )! (1# ) q 5 D L L D0 2 2m B L \D 5 D L D #h.c. ,
(5.65)
and are de"ned, together with gD and B , in [246]. For maximal CP violation, i.e., sin "1, and L D !. with m "m > "100 GeV and m "18 GeV, CF "nd A K2%. So an asymmetry in the main Q Q O two-body mode, tPb=, of a few percent can occur in their limit. However, these relatively large PRAs, reported by GK and CF in [235,246] su!er from the following drawbacks: E For the GK limit, m #m I (m is now essentially disallowed by the current experimental R % R bounds. E For the CF limit, arg()910\ is an unnatural choice in view of the stringent constraints on this phase coming from the experimental limits on the NEDM as discussed in Section 3.3.4. E For both the GK and CF limits, the large asymmetry arises once the masses of the superpartners of the light quarks are set to 100 GeV. Again, this is a rather unnatural choice as it is theoretically
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very hard, if at all possible, to meet the NEDM experimental limits when the masses of the squarks (except for the lighter stop) are of the order of 100 GeV. Besides, the current experimental limits disfavor down squarks lighter than about 200 GeV. We also remark that PRA in tPb= within the more constrained N"1 SUGRA model was investigated in [248], where similar numerical results (i.e. A (0.3%) for A were obtained. To conclude this section, although PRAs in the range of &10\}10\ in the main two-body decays of t,tM are appreciable, their measurements is likely to be very hard. Presumably an e>e\ collider (NLC) could be suitable due to its cleanliness. However, there may be about 10,000 to 50,000 ttM events a year. Therefore, bearing experimental e$ciency factors, under the best of circumstances only an asymmetry of the order a few percent could be measured in the NLC. The LHC, being able to produce 10}10 ttM pairs, might seem more appropriate for a measurement of such a small PRA. However, for a measurement of a&0.3% asymmetry, experimental systematics can pose serious limitations. 5.1.5. PRA within the form factor approach tPb=: The basic idea in the form factor approach is to write a model independent coupling, then investigate the dependence of various asymmetries on the form factors involved [34,232,233,249}251]. Thus one can write the amplitude for tPb=> as the sum M ,M #M , R@5 R@5 R@5 where M is the amplitude at the lowest order in the SM which is given by R@5 g M "! 5 < H(p > )u (p )I¸u (p ) . R R R@5 (2 R@ I 5 @ @
(5.66)
(5.67)
In this equation, (p > ) is the polarization vector of => with four momentum p > and p , p are 5 @ R I 5 the four momenta of b, t, respectively. M contains the new CP-violating interactions and can be R@5 written in general (for on-shell => and in the limit m "0) as follows: C f. g f . IP#i IpJ > P u (p ) , (5.68) M "! 5 < H(p > )u (p ) R@ I 5 @ @ J 5 R R R@5 m (2 5 .* 0
where P"¸ or R, ¸(R)"(1!(#) )/2 and the form factors f . and f . are complex, in general, they can both have an absorptive phase and a CP-violating phase. Note also that, in the SM, f * "1; f 0"f * "f 0"0, at tree level. Similarly, the non-standard part of the amplitude for tM PbM =\ is de"ned as
fM . fM . IP#i IpJ \ P v (p M ) . @ @ m J 5 5 .* 0 In general, the form factors f . and fM . can be further simpli"ed to the form G G f .,f . ;f . , G G !.! G !.4 fM .,fM . ;fM . , G G !.! G !.4 g M M "! 5
(5.69)
(5.70) (5.71)
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where the indices CPC and CPV stand for the CP-conserving and CP-violating parts in the above form factors. In particular, f . , fM . can be complex due to an absorptive phase (FSI phase), G !.! G !.! and f . , fM . are complex in the presence of a non-zero CP-violating phase. G !.4 G !.4 In terms of the CP-conserving and CP-violating parts of these form factors in Eqs. (5.70) and (5.71), it is useful to note that the following relations exist (between the form factors associated with tPb=> and those related to tM PbM =\) f* "fM * !.! !.!
and f 0 "fM 0 , !.! !.!
(5.72)
f* "( fM * )H and f 0 "( fM 0 )H , !.4 !.4 !.4 !.4
(5.73)
f* "fM 0 !.! !.!
(5.74)
and f 0 "fM * , !.! !.!
f* "( fM 0 )H and f 0 "( fM * )H . !.4 !.4 !.4 !.4
(5.75)
Using the relations above it is easy to show that any CP-violating observable must always be proportional to any one of the combinations: ( f * !fM * ), ( f 0 !fM 0), ( f * !fM 0) or ( f 0!fM * ). In particular, a CP-odd, ¹ -even quantity (like the PRA) will be proportional to the real parts of these , combinations, e.g., Re( f * !fM 0), but a CP-odd, ¹ -odd quantity will be proportional to their , imaginary parts, e.g., Im( f * !fM 0). Assuming these form factors to be purely CP-violating, i.e., Re( f . )"0 and Re( fM . )"0, G !.4 G !.4 the PRA de"ned in Eq. (5.4) for tPb= can be expressed as A " a.Re( f .) . G G G .* 0
(5.76)
In this context it was found that [232] a* K0.7, a0K!0.04, a* K0.04 and a0K!0.7. We thus see that the PRA is more sensitive to f * and f 0 than to f 0 and f * . 5.2. Partially integrated rate asymmetries In the rest of the section we will discuss CP asymmetries for the tPb decay within the O 3HDM, starting with PIRA. This asymmetry is de"ned as follows: A
.'0
(tPb> )! (tM PbM \ ) O .' O , , .' (tPb> )# (tM PbM \ ) .' O .' O
(5.77)
where stands for the partially integrated width, i.e., the width obtained by integrating over only .' a part of phase-space, rather than over the full kinematic range available to u and q, de"ned in Eq. (5.28). It is easy to see that, unlike the PRA, the PIRA is non-vanishing even for m P0 ( fO). D The reason is that in the calculation of the PRA, when the integration over the full range of u is
Note the slight di!erence between our de"nition of the form factors f ., fM . and the de"nition presented in [232]. G G
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performed, pI from the loop sandwiched between the = and the H>, necessarily gets replaced O O by qI. But then, the contribution of the transverse part of the =-propagator (i.e., G ) vanishes since 2 qIIJ"0 (IJ,!gIJ#qIqJ/q). On the other hand, when we calculate the PIRA, then the 2 2 relevant integration is over only a part of the full kinematic range of u which allows G to 2 contribute even to the =>-tree;H>-tree interference. Let us consider the integration over u for a "xed q in the rest frame of the =-boson, i.e., q"0. The integration over u is now equivalent to that over the angle between (!p ) and p . De"ne the O @ PIRA over positive values of cos to be A . Then, explicit calculation of the =>-tree;H>-tree > interference (i.e., diagrams (a) and (c) in Fig. 22) yields [59]: G mr Im(;) (2 $ O 5& , (5.78) A " > 4 (2#r )(1!r )B(=P ) 5R 5& O where Im(;) is de"ned in Eq. (5.22), r "m /m and r "m /m . For 200:m : 5R 5 R 5& 5 & & 300 GeV, A & a few ;10\ (see Table 4) so it is enhanced by two orders of magnitude over the > PRA, A , in Eq. (5.23). Even larger asymmetries are likely for m (m . O & R A related PIRA was investigated in [252]. There, a PRA for tPb was de"ned, speci"cally for O ggPtM t in a hadron collider. The imposition of experimental cuts (for details see [252]), turns the asymmetry into PIRA. Looking for the maximal CP-violating e!ect by choosing the most favorable values for the three CKM-like angles and "/2, subject to experimental constraints, & asymmetries of the order of a few;10\ were obtained from tree;tree interference with a resonant =. 5.3. Energy asymmetry Another explicit example of how an interesting CP-violating asymmetry can be sizable even when the PRA is vanishingly small is the energy asymmetry. Speci"cally, let us de"ne [59]: E > ! E \ O , (5.79) A " O # E > # E \ O O where E > is the average of the > energy in tPb>, etc. In the calculation of E > the O O integrand is of course equal to that for the PRA with just an additional factor of E > . Now G does 2 O Table 4 Results for the PIRA A , see Eq. (5.78), and (A B )\, in tPb within the Weinberg model for CP violation. > > > O s ,sin(I ), where I , "/2, are CKM-like angles chosen to maximize the charged Higgs coupling Im(;) (see G G G & Eq. (5.22)). Note (A B )\ is the number of ttM pairs required to observe the asymmetry to 1-. B K0.04 is the > > > appropriate branching fraction. Table taken from [59], updated to m "180 GeV R m > &
s
200 300
0.252 0.210
s (;10\)
s
8.29 9.99
0.707 0.707
A (;10\) >
(A B )\ > >
2.9 1.5
3.0;10 1.2;10
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contribute even when the integration is over the full range of u. Explicitly one "nds G mr (1!r )Im(;) (2 $ O 5& 5R A " . # 12 (1#3r #2r )(1!r )[B(=P )] 5R 5R 5& O The energy asymmetry is closely related to the PIRA and in fact numerically [59]:
(5.80)
A &A /3 . (5.81) # > Indeed, both are weighted CP-odd observables constructed from the outgoing momenta. Observables constructed in this way have the drawback that they are proportional to m . This factor of O m is in addition to a factor of m in the Yukawa coupling. The latter cannot be dispensed with, as O O long as we are dealing with tPb. However, the additional power of m entering these asymmetO ries can be overcome by examining the transverse polarization of the as we will discuss next. 5.4. -polarization asymmetry The advantage of using a polarization asymmetry over an energy or a rate asymmetry is that the latter asymmetries go as m/m , where one power of m comes from the Yukawa coupling at the O & O H vertex. The second power of m comes from the trace over the lepton loop in =>}H> O O interference, i.e., ¹r[I(p/ #m )(1! )p/ ]"4m pI . (5.82) O O J O J The only way to avoid this power of m is to avoid summing over the spin (s ) of in the preceding O O trace. Then the trace will take the form O 4i (, s , p , p )2 ¹r[I(p/ #m )(1# s. )(1! )p/ ]KP O O J O O O J
(5.83)
Thus the =>-tree;H>-tree interference will make a contribution to the transverse polarization of the , i.e., to s ) (p ;p ) without su!ering a suppression by an additional power of m (i.e. in O O J O addition to the Yukawa coupling) so this asymmetry will be enhanced over the PIRA and energy asymmetries by a factor of about m /m &100! R O We will consider the following CP asymmetries that involve the polarization [166]: >(!)!>( )#\(!)!\( ) , A , W >(!)#>( )#\(!)#\( )
(5.84)
>(!)!>( )!\(!)#\( ) A , , X >(!)#>( )#\(!)#\( )
(5.85)
where for A (A ) the arrows indicate the spin up or down in the direction y(z). The reference W X frame is de"ned to be the rest frame, such that the t momentum is in the !x direction (i.e. the x-axis is the boost axis from the top to the frame), the y-axis is de"ned to be in the decay plane with a positive y component for the b momentum. The z-axis is de"ned by the right-hand rule. Also, A (A ) is CP-odd, ¹ -even (CP-odd, ¹ -odd), and is therefore proportional to the absorptive W X , ,
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(dispersive) part of the bubble in the =-propagator. When integrated over the entire phase-space, A and A give W X 9 g Im(;)(x x 5 O 5 , A "! W 64 (1#2x )(x !x ) 5 & 5
(5.86)
9 g Im(;)(x 5 J A " f (x , y , x ) . X 64 (1!x )(1#2x )x 5 5 & 5 5 &
(5.87)
Recall that Im(;) is given by Eqs. (5.20)}(5.22) x ,m/m and y , /m. Also, f is de"ned as H H R 5 5 R the integral
f (x , y , x ), 5 5 &
(!x )x (1!)( 5 & d , [(!x )#x y ](x !) 5 5 5 &
(5.88)
and ,(p #p )/m. C J R A fully integrated polarization asymmetry as in Eq. (5.87), but weighting events di!erently for di!erent ranges of the invariant mass, was also given in [166]: O 9 g Im(;)(x 5 J A " f (x , y , x ) , X 64 (1!x )(1#2x )x 5 5 & 5 5 &
(5.89)
where f is the integral of Eq. (5.88) except that !x is replaced by !x . 5 5 The results for the above asymmetries, where experimental constraints were imposed on the relevant 3DHM parameters (for details see [166]), are a few percents for A and A , and W X a few tens of percents for A . As expected, the -polarization asymmetries are much larger and, X therefore, perhaps better suited to look for CP violation within the 3HDM, than any other asymmetry. 5.5. CP violation in top decays } summary As discussed in Section 2, CP violation can manifest in decays of particles. Such CPviolating signals may be driven by new physics containing new heavy particles. Thus, the large mass of the top may cause enhancements of CP violation in top decays as compared to the situation in light quarks decays. CP-odd signals in top decays, therefore, are attractive venues for such studies. In this section, we have discussed several types of CP-violating asymmetries in two- and three-body top decays. In particular, PRA, PIRA, energy asymmetry in the top decay products and -polarization asymmetries in the three-body decay tPb. In the SM, the CP violation in the top decays is found to be vanishingly small. This fact makes CP violation in top decays an extremely useful place for searching for new physics. We have, of course, also considered CP violation in top decays in extensions of the SM such as MHDMs and SUSY. We found that a sizable CP-violating PRA can arise in the main top decay, tPb=, in SUSY models. In particular, a stop-neutralino-chargino loop in the tb= vertex can give rise to a PRA of
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the order of 0.1% if the SUSY parameter space turns out to be favorable. Such a PRA is, in principle, within the reach of the LHC provided that detector systematics can be kept su$ciently under control. A much bigger CP-violating signal is expected in the three-body decay, tPb, in MHDMs with new CP-odd phases in the charged Higgs sector, e.g., a 3HDM. Indeed such CP violation may arise already at tree-level and is best observed through a CP-violating transverse -polarization asymmetry. In a favorable scenario this asymmetry may be as large as a few tens of percents, requiring &1000 top quarks for its detection. This is a particularly gratifying result since over 10 top quark pairs are expected to be produced in the future colliders. The decay tPb is therefore a very promising place to look for new signals of CP violation in top decays. 6. CP violation in e>e\ collider experiments The energy of circular e>e\ machines cannot be increased beyond the energy of LEP-II, due to heavy losses to the synchrotron radiation. Therefore, the next step in e>e\ physics will involve linear colliders only, the `existence proof a of which has been demonstrated at the 100 GeV scale by SLC. For recent reviews on linear colliders see [253}255]. The luminosity of future e>e\ colliders is projected to be L+10 cm\ s\, corresponding to a yearly integrated luminosity of L+100 fb\. The working assumption usually is to take it as tens of fb\ at the lower end of the scale of c.m. energies, and as hundreds of fb\ at its upper scale, corresponding to higher c.m. energies, to compensate for the decreasing cross-sections. In the "rst stage, the c.m. energy will cover the range approximately between LEP-II and 500 GeV, eventually reaching perhaps 1.6 TeV and hopefully even 2 TeV. Furthermore, beam polarization } which can help in clarifying many of the physics issues } is an interesting option. In this context recall that the SLC achieved polarization as high as 70}80%. 6.1. e>e\PttM In an high energy e>e\ collider running with c.m. energies of 500}2000 GeV and an integrated luminosity of L&O(100) fb\, 10}10 pairs of ttM will be produced mainly through the simple reaction e>e\P, ZPttM . This facility, especially due to its relatively clean environment, may therefore be thought of as a very e$cient `top factoryaand it is expected that many of the rare phenomena associated with top quark systems will be intensely studied there. Here we will focus on CP violation in the overall reaction P b=> e>e\Pt#tM . bM =\ P
(6.1)
Decays of the = also need to be included; the leptonic channels (=Pll , l"e, ) are perhaps the cleanest although experimental simulations suggest that ='s could be detected through jet topologies as well [256]. In what follows, we will not entertain the theoretical possibility that there is additional CP violation in =>, =\ decays and will focus only on e!ects directly related to the top quark.
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In general, in the limit m "0, the (or Z)ttM vertex can be modi"ed to include the top magnetic C and electric(or weak) dipole moments !i[I(A4#B4 )#IJq (ic4#d4 )] , (6.2) R R J R R where c4 and d4, for <" or Z, are the magnetic and electric dipole moment form factors of the R R top quark at q"s, assuming that they are deduced by the use of the reaction in Eq. (6.1). The tree-level SM values for these parameters are AA"e , R BA"0 , R
e 1 2 A8" ! sin , R 5 sin cos 4 3 5 5 e B8"! , R 4 sin cos 5 5 c4"d4"0 . R R Note also that, in the SM, the <e>e\ couplings in the notation of Eq. (6.2) are AA "!e , C BA "0 , C
(6.3)
1 e ! #sin , (6.4) A8" 5 C sin cos 4 5 5 e B8" . C 4 sin cos 5 5 The magnetic form factor, which is CP-conserving, has a signi"cant SM contribution at 1-loop due to QCD corrections and therefore is of lesser interest. Since in e>e\PttM we have q"s'4m, R these form factors are in general complex. In particular, with regard to the EDM form factors: Re dA 8(q) is ¹ -odd, and Im dA 8(q) is ¹ -even and, of course, all of these four quantities are R , R , CP-odd. Similar to the production vertex, the tb=> and tM bM =\ decay amplitudes may have CP-violating pieces. In order to take into account this possibility, for on-shell => and in the limit m "0, the C decay amplitudes for tPb=> may be decomposed with the most general form factors as (see also Section 5.1.5)
f. f . IP#i I pJ > P u (p ) , (6.5) J 5 R R m 5 .* 0 where H(p > ) is the polarization vector of => with four momentum p > and p , p are the four I 5 5 R @ momenta of the t, b, respectively. P"¸ or R, where ¸(R)"(1!(#) )/2 and the form factors f . and f . are complex in general. g M "! 5 < H(p > )u (p ) R@5 (2 R@ I 5 @ @
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Fig. 26. Tree-level (a) and CP-violating amplitudes (b)}(d) to leading order in the SM couplings and in CP-violating form factors.
Similarly, the amplitude for tM PbM =\ is de"ned as
fM . fM . IP#i IpJ \ P v (p M ) . (6.6) @ @ m J 5 5 .* 0 Furthermore, some useful relations exist (see Eqs. (5.70)}(5.75)) between pairs of ( f ., fM .) in G G terms of their CP-conserving and CP-violating parts. In particular, CP-violating observables associated with top decays must always be proportional to any one of the combinations: ( f * !fM * ), ( f 0!fM 0), ( f * !fM 0) or ( f 0!fM * ), such that a CP-odd, ¹ -even quantity will be propor , tional to the real parts of these combinations, but a CP-odd, ¹ -odd quantity will be proportional , to their imaginary parts (for details see Section 5.1.5). CP violation e!ects in e>e\PttM Pb=>bM =\ may thus enter in both the production and the decay vertices of the top and the anti-top. To leading order in the CP-violating form factors present in the production or the decay of the top, one has to include interferences of diagrams (b)}(d) with the SM diagram (a) in Fig. 26. In principle, in order to experimentally separate CP-non-conserving e!ects in the production vertex from the decay vertex, one has to construct appropriate observables with sensitivity to only one CP-violating vertex, i.e., either production or decay (see e.g., [258]), or alternatively some simplifying assumptions have to be made. It is important to note that the description of the e\PttM in terms of Eq. (6.2) is not necessarily su$cient. For example, in the MSSM, 1-loop box diagrams with exchanges of SUSY particles may contribute to CP violation in e>e\PttM . In these type of diagrams, a e\PttM , in the limit m "0. C g M M "! 5
6.1.1. Optimized observables As mentioned before, in the reaction e>e\PttM Pb=>bM =\, CP violation can arise from both the production (see Eq. (6.2)) and the decay (see Eqs. (6.5) and (6.6)) of the top. In this problem, For a comprehensive treatment of the helicity amplitudes for e>e\PttM and for the subsequent top decays tPbl J in the presence of the CP-violating couplings in Eqs. (6.2), (6.5) and (6.6), see e.g., [34,257].
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many momenta are available, so several ¹ -odd triple correlations can be constructed all of which, , in principle, can have non-vanishing expectation values that are proportional to Re dA 8(q). R Similarly several ¹ -even (CP-odd) observables can be constructed to measure Im dA 8(q). The , R sensitivity to Re dA 8(q) or Im dA 8(q) can vary considerably amongst the observables. It is, R R therefore, useful to devise a general procedure that represents a rough measure of the sensitivity of the observables in such situations. Thereby, one is led to consider the possibility of constructing `optimized observablesa, i.e., observables that have the maximum statistical sensitivity. Recall that for a given number of ttM events, the optimized observables will yield the smallest attainable limit on the real and the imaginary parts of dA 8(q). The basic idea of the optimized observables was "rst R outlined in [29]; the general recipe for construction of such observables is given in Section 2.6. Optimal observables have by now been used extensively in [29,260}264]. In [29,260,261] CP violation in the top decays was ignored and the CP-odd e!ect was attributed solely to the EDM (dA) R and ZEDM (d8) of the top in the ttM and ZttM production vertex. Indeed, in [227] it was shown that, R in model calculations such as 2HDM and MSSM, the dipole moment in the ttM production leads to larger CP-non-conserving e!ects than what might be expected in the top decays. In [262}264] the optimization technique was employed to the overall reaction e>e\PttM Pb=>bM =\, where CP violation from both the production and the decay vertices of the top were investigated. Using the general e\PttM may be expressed as () d" () d# (Re d4(s)R 4R ()#Im d4(s)I 4R ()) d . B R
B R 4A 8
(6.7)
As was explained in [29], the simplest optimized observables for the real and imaginary parts of dA 8(q) are R / , OA 8"I A 8 / . (6.8) OA 8"R A 8 BR
BR ' 0 These optimal observables are constructed simply from the available four momenta in e>e\PttM and the subsequent decays. It was found in [29] that, for example, with 10 ttM events in an NLC running at c.m. energies of (s"500 GeV, Re(dA 8) and Im(dA 8) of about &few;10\ e cm R R become accessible at the 1- signi"cance level. Recall that in model calculations, such as MHDMs and SUSY, the size of top EDM and ZEDM are typically at the level of :10\ e cm (see Section 4) if one pushes the CP-violating phases of these models to their largest allowed values. Thus, the 1- limit obtained in [29] is at least one order of magnitude above the theoretical expectation for these dipole form factors within extensions of the SM. Now, consider the reaction e>e\PttM Pb=>bM =\, where the =>, =\ further decay leptonically via =Pl or hadronically, i.e., to up and down quark jets. Of course, as is well known, the J top quark decay occurs in an extremely short time and one measures directly only the momenta of the decay products of the top. The optimization technique may be therefore improved to include all available 4-momenta in a given decay scenario of the ttM as, for example, was done in [260,261]. The basic idea there was to translate the CP-odd top spin correlations generated by the dipole moments in e>e\PttM to correlations among momenta of the decaying products of the t and tM . For this purpose, the most promising decay scenario is the single-leptonic decay channels, i.e., when one, say
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the t, decays leptonically and the other, i.e., tM , decays hadronically or vice versa ttM Pl>(q )#l #b#XM (q ) , (6.9) > 6M (6.10) ttM PX (q )#l\(q )#l #bM . 6 \ The decay scenarios in Eqs. (6.9) and (6.10) (each of which has a branching ratio, Bl &0.15, if l"e, ) allow for the reconstruction of the tM and t momentum, respectively, which, in turn, gives the rest frames of these quarks. The fact that the rest frames of t and tM may be accessible in these decay modes allows one to use CP-odd observables in terms of lepton unit momenta q( H in the ! corresponding top rest frames, instead of q( } de"ned in the e>e\ c.m. frame. ! The optimal observables that they used are, again, simply the ratio between the CP-odd and CP-even di!erential cross-sections and are given in [260]. However, in their optimal observables, the di!erential cross-section corresponds to the overall production and decay of the ttM and the leptonic momenta are taken in the corresponding t, tM rest frames. With the optimal observables they [261] calculated the best 1- sensitivity to the CP-violating dipole moments form factors Re(dA), Re(d8), Im(dA), Im(d8) assuming 100% tagging e$ciency of the single-leptonic decay modes R R R R of ttM in Eqs. (6.9) and (6.10); these are given in Table 5. We thus see that beam polarization (of the incoming electrons), denoted here by P , may increase the sensitivity to Re(dA) and Im(d8) by C R R almost an order of magnitude. Evidently, the results in Table 5 imply that a NLC running with a c.m. energy of (s"500 GeV and an integrated luminosity of 20 fb\, or (s"800 GeV and an integrated luminosity of 50 fb\, will be able to probe, at 1- and in the best cases, real and imaginary parts of the TDM, typically of the order of a few;10\}10\. As in their analysis, this may be achieved by investigating the single-leptonic decay mode of ttM . This improves the limits obtained in [29] by about one order of magnitude. However, at the 3- signi"cance level, the corresponding sensitivities are typically few;10\}10\ and so are still about an order of magnitude above the expectations from the models such as MHDMs and SUSY. It should be noted again that a CP-violating dipole moment at the ttM and ttM Z vertices may not, in general, account for the entire CP violation e!ect in ttM production. As mentioned before, this will, for example, be the case in the MSSM where 1-loop CP-violating box diagrams can cause an additional CP-odd e!ect [259]. We note, however, that speci"cally in the MSSM, these box contributions to CP violation in ttM cannot signi"cantly enhance the CP-violating signal. In fact, in some ranges of the relevant SUSY parameter space, the contribution of the box graphs comes with
Table 5 Attainable 1- sensitivities to the CP-violating dipole moment form factors in units of 10\ e cm, with (P "$1) and C without (P "0) beam polarization. m "180 GeV. Table taken from [261] C R P "0 C (Re dA) R (Re d8) R (Im dA) R (Im d8) R
4.6 1.6 1.3 7.3
20 fb\, (s"500 GeV P "#1 P "!1 C C 0.86 1.6 1.0 2.0
0.55 1.0 0.65 1.3
P "0 C 1.7 0.91 0.57 4.0
50 fb\, (s"800 GeV P "#1 P "!1 C C 0.35 0.85 0.49 0.89
0.23 0.55 0.32 0.58
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an opposite sign relative to the top dipole moments, such that cancellations may occur, thus decreasing the net CP violation e!ect in ttM production [259]. A more complete investigation was carried out in [262}264]. There the CP-violating form factors from both the production and the decay amplitudes of the t and tM were included. For this purpose, they used the single-leptonic energy spectrum 1 d! " c!f (x) , G G ! dx G where
(6.11)
c!"1, c!"G, c>"!Re( f 0), c\"!Re( fM * ) . (6.12) The functions f (x) are all given in [262] and f 0, fM * are de"ned in Eqs. (6.5) and (6.6). Also, G $ indicates the charge of the lepton and
El>(El\) (1!) , (6.13) m (1#) R El>(El\) being the energy of l>(l\) in the e>e\ c.m. frame and "(1!4/s. Speci"cally, for R m "180 GeV and (s"500 GeV R 1.76;10 K! ;(1.06Im(dA)#0.18Im(d8)) . (6.14) R R [e cm] x(x )"2
In [262], optimal observables that may be used to separately measure the CP-violating form factors in the production or the decay vertex were given. Their optimal observables utilize the single-leptonic energy spectrum in Eq. (6.11). With the optimization technique they showed that and Re( f 0}fM * ) may be extracted individually from the di!erence in the l> versus l\ energy spectra by convoluting the di!erential energy spectrum with approximately chosen kernel functions
1 d\ 1 d> 1 ! , " dx K \ dx > dx 2
(6.15)
1 d> 1 d\ ! , Re( f 0!fM * )" dx D > dx \ dx
(6.16)
where , are functions of f (x) de"ned in Eq. (6.11) and are given in [263,264]. The minimal D K G values of and Re( f 0!fM * ) that can be obtained with a statistical signi"cance NRR4 and NR@5, 1" 1" respectively, at the NLC with (s"500 GeV and m "180 GeV, can then be computed [262]: R NRR4 1" , (6.17) "11.3 ;pb * NR@5 1" Re( f 0!fM * )"13.1 , (6.18) ;pb * Notice the di!erence in our notation (Eq. (6.2)) and the one used in [262] for the top dipole moments. The translation is D "(4m sin /e);idA 8. A 8 R 5 R
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Here represents the square root of the e!ective luminosity for the single-leptonic ttM pairs at the * NLC. Thus, ,( L, where L is the integrated luminosity at the NLC and is the tagging RR * RR e$ciency for the single-leptonic mode. Note that, in the best case, "Bl +15%, if one assumes RR 100% e$ciency in measuring the single leptons from ttM . We see from Eqs. (6.17) and (6.18) that, for example, with L"100 fb\ and "0.15 we have RR +122.5 pb\ (note that with these values one has &9000 single-leptonic ttM events). Therefore, * with this number, a 3- detection of will be possible for +0.28. Using the relation between and Im(dA 8) in Eq. (6.14) we then get the following 3- equality: R 1.06Im(dA)#0.18Im(d8)K1.6;10\ e cm . (6.19) R R Eq. (6.19) implies that Im(dA 8) of the order of &10\ e cm may be detected at the 3- level at the R NLC, running with c.m. energy of (s"500 GeV and with an integrated luminosity of L"100 fb\, using the optimal observables suggested in [262]. This result is again about one order of magnitude better compared to the results obtained in [29] and it is comparable to the results shown in Table 5 which were obtained in [261]. As for the CP-violating form factors in the decay amplitude, we can use Eq. (6.18) to get the 3- limit on Re( f 0}fM * ). For +122.5 pb\ * Re( f 0!fM * )+0.32 . (6.20) In Section 5, we have discussed the theoretical expectations for couplings such as f 0 and fM * in the SM and its extensions. The SM prediction for such form factors, induced by the CKM matrix, is much too small to be observed. Moreover, even within MHDMs and the MSSM the resulting 3- limit in Eq. (6.20) falls short by at least one order of magnitude. Finally, in [263,264] the single-leptonic channel was compared to the double-leptonic mode, i.e., when both t and tM decay leptonically, using the optimization technique. It was found there that the single-leptonic mode comes out favorable by about a factor of 2. 6.1.2. Naive observables constructed from momenta of the top decay products Various types of `naivea observables to deduce the real and imaginary parts of the non-standard form factors in top production and decay were considered in [29,30,227,249,260,261,265}267]. These observables are constructed simply from correlations between momenta of the decaying products of the top quark. The basic idea again utilizes the fact that weak decays of the top quark act as very e$cient analyzer of the top spin. So the momenta of the decay products (via tPbl) can be used to construct the observables with the right transformation properties. In general, as expected, the `naivea operators are less e!ective than the optimal ones, sometimes by as much as an order of magnitude. However, it is important to bear in mind that this advantage pertains only with respect to statistical errors; in actual experimental considerations, systematic errors will also need to be taken into account and that could o!set some of the advantage of the optimized observables. In [29], amongst the various possible correlations, the best simple operators that they found are PIQJ H>NH\M for Re dA , IJNM @ 8 R PIQJ H>NH\M for Re d8 , IJNM C 8 R H\ ) Q for Im dA and Im d8 , 8 R R
(6.21)
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where the momenta above are given by P "p !p M , @ @ @ P "p>!p\ , C C C Q "p>#p\ , 8 C C H!"2(E> ) p )E> $2(E\ ) p )E\ , (6.22) 5 R 5 5 R 5 and E is the =-boson polarization for the reaction =(p )Pl(pl )l (p ). Because of the left5 5 J handed nature of the coupling of = to leptons its polarization can be constructed from the momenta of the decay products as ¹r[p/ . p/ l I(1! )] J . (6.23) EI " 5 4(p ) pl ) J Here is an arbitrary light-like vector that determines the phase convention for the polarization. The expression above requires the momenta. Recall that the "nal state consists of six particles bl>l bM l\l . Of these, only four are directly observable as the neutrinos escape detection. However, by imposing the following conditions, the momenta of the missing neutrinos may in fact be inferred. The conditions that need to be imposed are: (1) conservation of four-momentum together with the conditions that (2) the lepton and a neutrino reconstruct to the =! mass, (3) the b-quark together with a lepton and neutrino reconstruct the t, tM mass and (4) the neutrinos are massless. It was then found in [29] that the use of the = polarization in the operators of Eq. (6.21) can easily improve the sensitivity to the dipole moments by factors of 10}50 when compared to simple correlations which do not involve the = polarization vector. As compared to the optimal observables discussed in their work, the observables in Eq. (6.21) are less e!ective, typically, by about a factor of 2}10. In [30,249], the following CP-odd and ¹ -odd correlations were considered , (q( ;q( ) > H #(i j) , (6.24) ¹K "(q( !q( ) \ GH \ > G q( ;q( \ > (q( ;q( ) > , (6.25) AK "p( ) \ > q( ;q( \ > where p( is the unit momentum of the incoming positron and q( ! are the unit momenta of > a charged decay product from tPA, tM PBM in the overall c.m. system. Thus, q( are the directions of ! a charged lepton or a b jet. An interesting property of these correlations is that they are not sensitive to CP-violating e!ects in the t and tM decays and, therefore, they can be expressed in terms of only the real part of the EDM and ZEDM form factors, Re(dA 8). The mean values of ¹K and R GH AK plus the conjugate ones are given by [249] (s (c Re dA#c Re d8)s , (6.26) ¹K M # ¹K M "2 GH R 8 R GH GH e A (s (r Re dA#r Re d8) , AK M # AK M "2 R 8 R e A
(6.27)
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where, identifying the z-axis with the e> beam axis s "(p( p( !)"diag(!,!, ) . (6.28) GH >G >H The coe$cients c and r depend on the speci"c decay channel and were calculated as a function A 8 A 8 of s in [249] for correlations among AB"l>l\, bbM and l>b#l\bM . Possible CP-odd, ¹ -even correlations that use the momenta of the decay products of t and , tM were also examined in [249]: QK "(q( #q( ) (q( !q( ) #(i j) , (6.29) GH > \G \ >H AK "p( ) (q( #q( ) . (6.30) > > \ In contrast to the ¹ -odd observables in Eqs. (6.24) and (6.25), the ¹ -even observables QK and AK , , , GH which acquire absorptive phases, are sensitive also to the combinations Re( f . !fM .) (recall that G G P"¸ or R), e.g., Re( f0}fM * ), of the form factors in the decay amplitudes of Eqs. (6.5) and (6.6). In [249], CP-odd e!ects in the decay process were neglected when evaluating the two CP-odd, ¹ -even observables in Eqs. (6.29) and (6.30). In terms of the imaginary parts of the EDM and , ZEDM of the top they obtained (s QK M # QK M "2 (q Im dA#q Im d8)s , GH GH R 8 R GH e A
(6.31)
(s (p Im dA#p Im d8) . AK M # AK M "2 R 8 R e A
(6.32)
Here, again, the coe$cients q and p depend on the speci"c channel and were given as A 8 A 8 a function of s in [249] for correlations among AB"l>l\, bbM and l>b#l\bM . From the simultaneous measurement of the pairs ¹K , AK and QK , AK and for a given GH GH c.m. energy, Re(dA 8) and Im(dA 8) can be disentangled, respectively. Assuming 10 available ttM R R events at each c.m. energy, the 1- statistical sensitivity to Re dA and Re d8, using only the 33 R R component of ¹K , are GH ( (3r ¹K !c AK ) e 8 8 , (6.33) (Re dA)" R c ) r !c ) r (sN A 8 8 A ( (3r ¹K !c AK ) e A A , (6.34) (Re d8)" R c ) r !c ) r (sN 8 A A 8 where N "10;Br(tPA);Br(tM PBM ). Similar relations can be obtained for Im dA and Im d8 R R using QK and AK . Using this formalism, Ref. [249] presents the one standard deviation accuracies in measuring the real and imaginary parts of dA 8, again assuming 10 ttM events at c.m. energy (s"500 GeV and R with m "175 GeV R Re dA+1;10\, Im dA+1.2;10\ , (6.35) R R Re d8+5;10\, Im d8+7.5;10\ , (6.36) R R
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where the best sensitivity was obtained for correlations between the l>l\ momenta. We note that the sensitivity to Re dA 8 is slightly better than to Im dA 8. R R Cuypers and Rindani [268] have investigated the e!ect of polarized incoming electron beams. They found that the sensitivity of observables of the type A and A in Eqs. (6.25) and (6.30) to the real and imaginary parts of dA 8, respectively, can be enhanced if the incoming electron beam is R longitudinally polarized. Note also [260], the type of observables in Eqs. (6.24), (6.25), (6.29) and (6.30) may be improved (with respect to their sensitivity to the dipole moments) if one replaces the momentum of the charged lepton in the e>e\ c.m. frame by its momentum in the top (or anti-top) rest frame, in the single-leptonic ttM channel. As was shown in the previous section, the CP-violating e!ects in the top decay tPb=> and its conjugate may be isolated using the optimization procedure. This may also be achieved in some limiting cases using naive observables. In [227,249] a naive observable that projects onto CP violation in the top decay vertex was suggested
(q( lM ;q( ) (q( l ;q( M ) @ ! @ . O "p( ) > q( lM ;q( q( l ;q( M @ @
(6.37)
Although, in general, the expectation value of O above receives contributions both from the CP violation in ttM production and in the decay amplitude, it was shown in [249] that close to threshold, i.e., (sK2m , the contribution from the ttM production vertex vanishes. For example, with the R correlation O , for an appropriate e>e\ collider with m "150 GeV and c.m. energy (sK2m , R R they "nd O +0.15;Im( f 0!fM * ) ,
(6.38)
where f 0, fM * are form factors in the decay amplitudes de"ned in Eqs. (6.5) and (6.6). Assuming 3;10 ttM events in which tPbl>l and tM PbM l\l , they found that, to 1-, (Im( f 0!fM * ))+0.1 ,
(6.39)
can be determined from a measurement of O . Again, the limit in Eq. (6.39) falls short from model predictions for these form factors (see Section 5). This is easily understood from Eq. (6.38) which implies an asymmetry of the order of &10\ for Im( f 0!fM * )&0.1. Recall that the typical asymmetries in extensions of the SM that we have described in Section 5 are : a few times 10\ and in the SM they are a lot smaller than that. A related CP-odd and ¹ -odd asymmetry that projects only onto CP-violating e!ects , in the decay processes of the t and tM was suggested by Grzadkowski and Keung [251]. This asymmetry was de"ned by partially integrating over the azimuthal angle of l>(l\) in the =>(=\) rest frame and subtracting the integration in the range !, 0 from the integration in the range 0, and it is essentially proportional to the triple product pl ) (p ;p ). When evaluated within the MSSM, the asymmetry was found to be &10\}10\ @ 5 which again falls short by at least one order of magnitude from the limits that are anticipated to be obtainable, through the study of e>e\PttM Pl>l\l l bbM in a future e>e\ high-energy collider.
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6.1.3. Improved sensitivity using energy and angular distributions of top decay products and polarized electron beams Several di!erential leptonic asymmetries with respect to the charged lepton in tPbll and for longitudinal electron (positron) beam polarizations, P (P ), have been suggested in [29,262}264, C C 269}271]. Consider the CP-violating t!tM spin correlation p( ) (s !s M ). This spin correlation simply R R R translates to the asymmetry [N(t tM )!N(t tM )] * * 0 0 , N " *0 all ttM
(6.40)
suggested "rst by Schmidt and Peskin (SP) in [33] in the context of ttM production in hadron colliders (the SP e!ect will be discussed in more detail in Section 7). Now, N is related to the asymmetry in the energy spectrum de"ned as [32,269}271]: *0 d 1 d ! , (6.41) A (x)" # dx(l>) dx(l\)
through a simple multiplication by kinematic functions present in the lepton energy distribution functions (see [262]). The energy asymmetry in Eq. (6.41) is between distributions of l> and l\ at the same value of x"x(l>)"x(l\)"4 E(l!)/(s. Note that when CP violation is present in the top production and not in the top decay, then A (x)JN J, where is de"ned in Eq. (6.14). # *0 An up}down asymmetry, A , was also studied in [32,269,270]: SB > A () d cos , (6.42) A " SB SB \ where
1 d(l>, up) d(l>, down) d(l\, up) d(l\, down) A ()" ! # ! , SB 2 d cos d cos d cos d cos
(6.43)
and up/down refers to (p ! ) I0, (p ! ) being the y component of p ! with respect to a coordinate J W J W J system chosen in the e> e\ c.m. frame so that the z-axis is along p , and the y-axis is along p ;p . R C R The ttM production plane is thus the xz plane. refers to the angle between p and p in the c.m. R C frame. Note that the asymmetry A is related to spin components of the top transverse to the SB production plain and, therefore, it is a ¹ -odd quantity. , Three additional asymmetries were considered in [269,270]. The combined up}down and forward}backward asymmetry
A () d cos ! A () d cos , SB SB \ with A () given in Eq. (6.43). The left}right asymmetry SB > A " A () d cos , JP JP \ A " SB
(6.44)
(6.45)
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where
1 d (l>, left) d (l>, right) d (l\, left) d (l\, right) A ()" ! # ! , JP 2 d cos d cos d cos d cos
(6.46)
and left/right refers to (p ! ) I0. The combined left}right and forward}backward asymmetry J V A" A () d cos ! A () d cos (6.47) JP JP JP \ with A () given in Eq. (6.46). JP To leading order in the dipole form factors and on ignoring CP violation in the t and tM decays, all the above "ve CP-odd asymmetries are linear functions of dA and d8. The asymmetries A , A , A R R # JP JP are ¹ -even and are therefore proportional to Im dA 8, while A , A are ¹ -odd and are , R SB SB , proportional to Re dA 8. The ¹ -even asymmetries can be symbolically written as R , (6.48) A ,aA(P , P )Im dA#a8(P , P )Im d8 . R G C C R G G C C Similarly, for the ¹ -odd observables, one obtains , B ,bA(P , P )Re dA#b8(P , P )Re d8 , (6.49) G G C C R G C C R where A "A , A or A and B "A or A . The functions aA 8, bA 8 depend, among other G # JP JP G SB SB parameters, on the polarizations of the incoming electron and positron beams, P and P , C C respectively, and are explicitly given in [269,270]. It is evident from Eqs. (6.48) and (6.49) that, without beam polarization, by measuring only one asymmetry of the type A and/or B , one can extract information only on one combination of G G Im dA, Im d8 and/or Re dA, Re d8, respectively. However, any two asymmetries with the same R R R R ¹ property can be used to determine two independent combinations of the corresponding real or , imaginary parts of dA and d8, thus, giving Im dA and Im d8 or Re dA and Re d8 independently. Such R R R R R R an analysis was described in the previous section where the observable pairs ¹K , AK and QK , AK were used to "nd the sensitivity of a high-energy e>e\ collider to Re dA, Re d8 and Im dA, Im d8, R R R R respectively. However, it was suggested in [269,270] that if, in addition, beam polarization is included, then one ¹ -even(¹ -odd) asymmetry is su$cient to determine Im dA and Im d8(Re dA , , R R R and Re d8) independently by measuring this asymmetry for di!erent polarizations. Both apR proaches were adopted in [269,270]. Their best results are summarized in Table 6 where 90% con"dence level limits are given for (s"500 GeV and m "174 GeV. We can see from Table 6 R that the second approach, of incorporating beam polarization, increases the sensitivity to both the real and imaginary parts of the EDM and ZEDM of the top, in some cases, by about one order of magnitude. With 50% beam polarization, the 90% con"dence level limits for Im dA, Im d8, Re dA R R R and Re d8 are again at best around &few;10\ e cm. R An interesting di!erential asymmetry that combines information from both the production and decay vertices of the top was suggested in [263,264]:
( dx dx d/dx dx ! dx dx d/dx dx ) VV . All , VV dx dx d/dx dx
(6.50)
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Table 6 90% con"dence limits on the real and imaginary parts of the top dipole form factors dA and d8, in units of 10\ e cm, R R from di!erent asymmetries. In the unpolarized case, the asymmetries A , A are together used to get limits on Re dA 8 SB SB R and A , A to obtain the limits on Im dA 8. In the polarized case, the limits obtained from A and A are denoted by (a) JP JP R SB JP and the ones from A and A are denoted by (b). The numbers are for the single-leptonic ttM mode, for m "174 GeV, SB JP R (s"500 GeV and an integrated luminosity of L"10 fb\ Case
Re dA R
Re d8 R
Im dA R
Im d8 R
unpolarized (a) polarized(P "$0.5) C (b) polarized(P "$0.5) C
54.4 2.3 12.5
15.9 2.3 9.1
7.9 2.3 2.3
62.4 9.1 7.9
This asymmetry utilizes the double-leptonic energy distribution in e>e\PttM Pl>l\l l bbM 1 d " c f (x, x ) , G G dx dx G where x, x are de"ned in Eq. (6.13) and
(6.51)
(6.52) c "1, c ", c "!Re( f 0!fM * ) . f 0, fM * and are de"ned in Eqs. (6.5), (6.6) and (6.14), respectively. In terms of (or equivalently of Im dA 8) and Re( f 0!fM * ) and with (s"500 GeV, m "180 GeV and the SM parameters they R R obtained the simple relation (6.53) All "!0.34!0.31Re( f 0!fM * ) . Given a number of available double-leptonic ttM events in the NLC and using the relation in Eq. (6.53), one can now plot the 1-, 2- and 3- detectable regions in the Im dA 8!Re( f 0!fM * ) R plane. These regions are shown in Fig. 27 for &700 double-leptonic ttM events in a 500 GeV collider. We see from Fig. 27 that a 3- detection of CP violation through All is possible in a wide range of the parameters and f 0, fM * . However, if one parameter is very small, then it requires the other to be relatively large. Thus, for example, if Re( f 0!fM * )"0 then, at 3-, Im(dA 8)91.7;10\ e cm R (or, with the notation used in [263,264], Re(D )90.3, see also Fig. 27). This is again comparable A 8 to obtainable limits from other observables described in this section and, thus, falls short by about one order of magnitude when compared to model dependent predictions for the top dipole moment. Finally, interesting CP-violating asymmetries which involve correlations among b-quarks from ttM Pb=>bM =\ were suggested by Bartl et al. in [265}267]. They utilized the angular [265] and energy [266] distributions of b and bM with initial beam polarization, in e>e\PttM followed by tPb=> and tM PbM =\, to construct CP-violating asymmetries that can disentangle CP violation in the ttM production mechanism from CP violation in the top decay. Unfortunately, their asymmetries, when evaluated within the MSSM, range from 10\ to 10\ at best, and therefore also seem to be too small to be detectable at a future NLC. Also, in an interesting study of the CP-violating e!ects of the EDM, ZEDM and CEDM of the top in the ttM threshold region [272], it was suggested that at an e>e\ collider CP violating
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Fig. 27. One can verify the asymmetry All in Eq. (6.50) to be non-zero at 1-, 2- and 3- level when the parameters Re(D ) (horizontal axis) and Re( f 0!fM *) (vertical axis) are outside the two solid lines, dashed lines and dotted lines, A 8 respectively, given &700 double-leptonic ttM events in a 500 GeVe>e\ collider. Recall that D is related to dA 8 via: A 8 R D "(4m sin /e);idA 8, i.e., Re(D ) corresponds to Im dA 8. Figure taken from [263,264]. A 8 R 5 R A 8 R
correlations may be somewhat enhanced. Using CP-odd correlations of the top polarization projected onto the charged leptons from the top decays, they found that at the threshold region for ttM production, an e>e\ collider can be sensitive to an EDM (and ZEDM) at the level of &10\ e cm. Although this is comparable to the predicted sensitivity away from threshold, it has some advantage in that it may be achieved at the lowest possible energy needed for ttM production. 6.2. e>e\PttM h, ttM Z, examples of tree-level CP violation Let us now turn our attention to the reactions e>(p )#e\(p )Pt(p )#tM (p M )#h(p ) , F > \ R R
(6.54)
e>(p )#e\(p )Pt(p )#tM (p M )#Z(p ) , 8 > \ R R
(6.55)
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which may exhibit large CP violation asymmetries in a 2HDM [78,273}275]. Note that for reasons discussed below, for the CP-violating e!ects in e>e\PttM h as well as in e>e\PttM Z, only two out of the three neutral Higgs of the 2HDM are relevant. We denote these two neutral Higgs particles by h and H corresponding to the lighter and heavier Higgs-boson, respectively. In some instances, we denote a neutral Higgs by H, then H"h or H is to be understood. A novel feature of these reactions is that the e!ect arises at the tree-graph level. As a consequence, one can construct new type of asymmetries which are J (tree;tree)/(tree;tree) and are therefore a priori of O(1). This stands in contrast to loop-induced CP-violating e!ects in ttM production for which the CP asymmetries, in general, are J(tree;loop)/(tree;tree) and are therefore suppressed by additional small couplings to begin with, i.e., at 1-loop, typically, by } the "ne structure constant. Indeed, we will show below that CP violation at the level of tens of a percent is possible in the reactions in Eqs. (6.54) and (6.55). Basically, for the ttM h(ttM Z) "nal states, Higgs(Z) emission o! the t and tM interferes with the Higgs(Z) emission o! the s-channel Z-boson (see Fig. 28) [78,273,274]. We "nd that the processes e>e\PttM h and e>e\PttM Z provide two independent, but analogous, promising probes to search for the signatures of the same CP-odd phase, residing in the ttM -neutral Higgs coupling, if the value of tan (the ratio between the two VEVs in a 2HDM) is in the vicinity of 1. In particular, they serve as good examples for large CP-violating e!ects that could emanate from t systems due to the large mass of the top quark and, thus, they might illuminate the role of a neutral Higgs particle in CP violation. Although these reactions are not meant (necessarily) to lead to the discovery of a neutral Higgs, they will, no doubt, be intensely studied at the NLC since they stand out as very important channels for a variety of reasons. In particular, they could perhaps provide a unique opportunity to
Fig. 28. Tree-level Feynman diagrams contributing to e>e\PttM h (left-hand side) and e>e\PttM Z (right-hand side) in the unitary gauge, in a 2HDM. For e>e\PttM Z, diagram (a) on the right-hand side represents 8 diagrams in which either Z or are exchanged in the s-channel and the outgoing Z is emitted from e>, e\, t or tM .
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observe the top-Higgs Yukawa couplings directly [73,276}282]. In [73,279], using a generalization of the optimal observables technique outlined below (see also Section 2.6), Gunion et al. have extended the initial work [78] on CP violation in e>e\PttM h to include a detailed cross-section analysis such that all Higgs Yukawa couplings combinations are extracted (see below). A similar analysis which also uses the optimized observable technique for e>e\PttM Z is given in [275]. A detailed cross-section analysis of the reaction e>e\PttM Z in the SM was performed by Hagiwara et al. [280,281]. There, it was found that the Higgs exchange contribution of diagram (b) on the right-hand side of Fig. 28 will be almost invisible in a TeV e>e\ collider for neutral Higgs masses in the range m (2m . Interestingly, we will show here that, if the scalar sector is doubled, then the F R lightest neutral Higgs (h) may reveal itself through CP-violating interactions with the top quark even if m (2m . Obviously, a non-SM (e.g., larger) top-Higgs Yukawa coupling can cause an F R enhancement in the rates for both the ttM h and ttM Z "nal states. Thus, a `simplea cross-section study for these reactions may also come in handy for searching for new physics. However, one should keep in mind that, from the experimental point of view, asymmetries, i.e., ratios of cross-section, are easier to handle and, in particular, CP-violating signals are very distinctive evidence for new physics. This section will be divided to three parts. In the "rst part, we present a detailed analysis of the tree-level CP violation in the reactions e>e\PttM h and e>e\PttM Z which manifests itself as a ¹ -odd correlation of momenta. In the second part, we will consider the generalized optimization , technique developed by Gunion et al. and its application to the reaction e>e\PttM h. In the last part, we will discuss CP violation in the Higgs decay hPttM , where we take the Higgs to be produced through the Bjorken mechanism e>e\PZh. 6.2.1. Tree-level CP violation In the unitary gauge, the reactions in Eqs. (6.54) and (6.55) can proceed via the Feynman diagrams depicted in Fig. 28. We see that for e>e\PttM Z, diagram (b) on the right-hand side of Fig. 28, in which Z and H are produced (H"h or H is either a real or a virtual particle, i.e. mH '2m or mH (2m , respectively) followed by HPttM , is the only place where new R R CP-non-conserving dynamics from the Higgs sector can arise, being proportional to the CP-odd phase in the ttM H vertex. As mentioned above, in both the ttM h and the ttM Z "nal state cases, CP-violation arises due to interference of the diagrams where the neutral Higgs is coupled to a Z-boson with the diagrams where it is radiated o! the t or tM . We note that in the ttM Z case there is no CP-violating contribution coming from the interference between the diagrams with the ZZH coupling and the diagrams where the Z-boson is emitted from the incoming electron or positron lines (not shown in Fig. 28). The relevant pieces of the interaction Lagrangian involve the ttM HI Yukawa and the ZZHI couplings and are given in Eqs. (3.70) and (3.71). There, HI (k"1, 2 or 3) are the three neutral Higgs scalars in the theory. As usual the three couplings aI, bI and cI in Eqs. (3.70) and (3.71) are R R functions of tan ,v /v (the ratio of the two VEVs) and of the three mixing angles , , which characterize the Higgs mass matrix in Eq. (3.73) (for details see Section 3.2.3). As was also mentioned in Section 3.2.3, only two out of the three neutral Higgs can simultaneously have a coupling to vector-bosons and a pseudoscalar coupling to fermions. Therefore, only those two neutral Higgs particles are relevant for the present discussion and, without loss of generality, we denote them as H"h and H"H with couplings aF, bF, cF and a&, b&, c&, R R R R
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corresponding to the light, h, and heavy, H, neutral Higgs, respectively. This implies the existence of a `GIM-likea cancellation, namely, when both h and H contribute to CP violation, then all CP-non-conserving e!ects, being proportional to bFcF#b&c&, must vanish when the two Higgs R R states h and H are degenerate. In the following, we set the mass of the heavy Higgs, H, to be m "750 GeV or 1 TeV. & In the process e>e\PttM h, a Higgs particle is produced in the "nal state, therefore, the heavy Higgs-boson, H, is not important and this `GIM-likea mechanism is irrelevant. Note that there is an additional diagram contributing to e>e\PttM h, which involves the ZhH coupling and is not shown in Fig. 28. This diagram is, however, negligible compared to the others for the large m values used here. In contrast, in the process e>e\PttM Z, the Higgs is exchanged as a virtual or & a real particle and the e!ect of H is, although small compared to h, important in order to restore the `GIM-likea cancellation discussed above. For both the ttM h and ttM Z "nal states processes, we denote the tree-level polarized di!erential cross-section (DCS) by , where f"ttM h or f"ttM Z corresponding to the ttM h or ttM Z "nal states, HD respectively, and j"1(!1) for the left(right) polarized incoming electron beam. can be HD subdivided into its CP-even ( ) and CP-odd ( ) parts >HD \HD " # . (6.56) HD >HD \HD The CP-even and CP-odd DCS's can be further subdivided into di!erent terms which correspond to the various Higgs coupling combinations and which transform as even or odd (denoted by the letter n) under ¹ . For both "nal states, f"ttM h and f"ttM Z, we have , " gGL FGL , CP-even , >HD >D >HD G " gGL FGL , CP-odd , (6.57) \HD \D \HD G where gGL , gGL , n"#or!, are di!erent combinations of the Higgs couplings aH, bH, cH and R R >D \D FGL , FGL , again with n"# or !, are kinematical functions of phase space which transform >HD \HD like n under ¹ . , Let us "rst write the Higgs coupling combinations for the CP-even part. In the case of e>e\PttM h, neglecting the imaginary part in the s-channel Z-propagator, we have four relevant coupling combinations [73,78,274]: "(aF), g> "(bF), g> "(cF), g> "aFcF . (6.58) g> R >RRM F R >RRM F >RRM F R >RRM F In the case of e>e\PttM Z, apart from the SM contribution, which corresponds to interference terms among the four SM diagrams represented by diagram (a) on the right-hand side of Fig. 28, and keeping terms proportional to both the real and imaginary parts of the Higgs propagator, H , we get [273,274] "(aHcH)Re(H ), g\ "(aHcH)Im(H ) , g> >RRM 8 R R >RRM 8 H H "(a c )Re(H ), g> "(aHcH)Im(H ) , g> R R >RRM 8 >RRM 8 H H "(b c )Re(H ), g> "(bHcH)Im(H ) , g> R R >RRM 8 >RRM 8
(6.59)
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where H ,(s#m !mH !2p ) p #imH H )\ , 8 8 p,p #p and H is the width of H"h or H. \ > The CP-odd coupling combinations are
(6.60)
g\ "bFcF , \RRM F R for the ttM h "nal state and
(6.61)
(6.62) "bHcHRe(H ), g> "bHcHIm(H ) , g\ \RRM 8 R R \RRM 8 for ttM Z "nal state. The CP-even pieces, , yield the corresponding cross-sections (recall that f"ttM h or ttM Z) >HD
( ) d , " >HD HD
(6.63)
where stands for the phase-space variables. In Fig. 29(a) and (b), we plot the unpolarized cross-sections, M and M as a function of m and (s, for Model II (i.e., 2HDM of type II as RR8 F RRF de"ned in Section 3.2.3), with m "750 GeV and the set of values , , "/2, /4, 0 which & we denote as set II. Set II is also adopted later when discussing the CP-violating e!ect. Furthermore, for the ttM h "nal state we choose tan "0.5 while for ttM Z we choose tan "0.3. Afterwards,
Fig. 29. The cross-sections (in fb) for: (a) the reaction e>e\PttM h with tan "0.5 and (b) the reaction e>e\PttM Z with tan "0.3, assuming unpolarized electron and positron beams, for Model II with set II and as a function of m (solid and F dashed lines) and (s (dotted and dotted-dashed lines). Set II means , , ,/2, /4, 0. Figure taken from [274].
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we will discuss the dependence of the CP-violating e!ect on tan in the ttM h and ttM Z cases. One can observe the dissimilarities in the two cross-sections M and M : while M is at most &1.5 fb, RR8 RRF RRF M can reach &7 fb at around (s+750 GeV and m 92m . M drops with m while M grows in RR8 F RR8 F R RRF the range m :2m . M peaks at around m 92m and drops as m grows further. Moreover, F R F F R RR8 M peaks at around (s+1(1.5) TeV for m "100(360) GeV, while M peaks at around F RR8 RRF (s+750 GeV for both m "100 and 360 GeV. As we will see later, these di!erent features of the F two cross-sections are, in part, the cause for the di!erent behavior of the CP asymmetries discussed below. Let us now concentrate on the CP-odd, ¹ -odd e!ects in e>e\PttM h, ttM Z, emanating from the , , , respectively. From Eqs. (6.61) and (6.62), it is clear that the ¹ -odd pieces in , \HRRM F \HRRM 8 , have to be proportional to bFcF (in the ttM Z case there is an CP-violating pieces R \HRRM F \HRRM 8 additional similar piece corresponding to the heavy Higgs H). The corresponding CP-odd kinematic functions, F\ M , F\ M , being ¹ -odd, are pure tree-level quantities and are proportional , \HRRF \HRR8 to the only non-vanishing Levi}Civita tensor present, (p , p , p , p M ), when the spins of the top are \ > R R disregarded. The explicit expressions for F\ are (recall that j"1(!1) for left(right) polarized \HD incoming electron beam)
1 g m 5 R ¹c8 (p , p , p , p M ) F\ M "! \HRRF (2 c5 m8 8F 8 R H \ > R R ;j(F#FM )[(s!s !m)(3w\!w>)#m (w\!w>)] R R R F H H 8 H H # ¹c8 (F!FM ) f , (6.64) R H 8 R R 2g m 5 R ¹c8 (p , p , p , p M ) F\ M "!(2 \HRR8 c m 8 R H \ > R R 5 8 (6.65) ;j(8#8M )[m w\#(s !s)w>]# ¹c8 (8!8M ) f , 8 H R H R H 8 R R R R where s,2p ) p is the c.m. energy of the colliding electrons, s ,(p #p M ) and f, \ > R R R (p !p ) ) (p #p M ). Also, \ > R R (6.66) F M ,(2p M ) p #m)\, 8 M ,(2p M ) p #m )\ , RR RR F F RR RR 8 8 ,(s!m )\, ,s\, ,((p!p )!m )\ . (6.67) 8 8 A 8F F 8 and
w!,(s Q !¹)c8 $Q s c , (6.68) H 5 R R H 8 R 5 5 A where s (c ) is the sin(cos) of the weak mixing angle , Q and ¹ are the charge and the 5 5 5 D D z-component of the weak isospin of a fermion, respectively, and c8 "1/2!s , c8 "!s . \ 5 5 Since at tree-level there cannot be any absorptive phases, CP-violating asymmetries only of the ¹ -odd type are expected to occur in . Note that in the ttM Z case there is also a CP-odd, , \HD ¹ -even piece, bHcH Im(H );F> M (see Eq. (6.62)), in the DCS. However, being proportional to R \HRR8 , the absorptive part coming from the Higgs propagator, it is not a pure tree-level quantity.
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Simple examples of observables that can trace the tree-level CP-odd e!ect in e>e\PttM h; ttM Z are [78]: p ) ( p ;p M ) R R , O (ttM h)" \HRRM F ; O (ttM Z)" \HRRM 8 . (6.69) O" \ s M M >HRRF >HRR8 Here O (ttM h; ttM Z) are optimal observables in the sense that the statistical error in the measured is involved. As asymmetry is minimized [29]. Note also that only the ¹ -odd part of , \HRRM 8 mentioned before, since the "nal state consists of three particles, using only the available momenta, there is a unique antisymmetric combination of momenta that can be formed. Thus, both observables are proportional to ( p , p , p , p M ). Furthermore, O (ttM h; ttM Z) are related to \ > R R O through a multiplication by a CP-even function. In the following, we focus only on the CP-odd e!ects coming from the optimal observables. We remark, however, that the results for the simple observable O exhibit the same behavior, though slightly smaller then those for O . The theoretical statistical signi"cance, N , in which an asymmetry can be measured in an ideal 1" experiment is N "A(¸( (" M , M for the ttM h, ttM Z "nal states, respectively), where for the RRF RR8 1" observables O and O , the CP-odd quantity A, de"ned above, is A + O/( O, A +( O . (6.70) Also, ¸ is the e!ective luminosity for fully reconstructed ttM h or ttM Z events. In particular, we take ¸" L, where L is the total yearly integrated luminosity and is the overall e$ciency for reconstruction of the ttM h or ttM Z "nal states. In the following numerical analysis, we have used set II de"ned before for the angles , i.e., , , "/2, /4, 0. Figs. 30(a) and (b) show the expected asymmetry and statistical signi"cance in the unpolarized case, corresponding to O in Model II for the ttM h and ttM Z "nal states, respectively. The asymmetry is plotted as a function of the mass of the light Higgs (m ) where F again, m "750 GeV in the ttM Z case. We plot N /(¸, thus scaling out the luminosity factor from & 1" the theoretical prediction. We remark that set II corresponds to the largest CP-e!ect, though not unique since we are dealing with angles, i.e., , which may be rotated by or /2 leaving the relevant combinations of angles with the same value (e.g., bFJsin sin ). In the ttM h case tan "0.5 is favored, however, R the e!ect mildly depends on tan in the range 0.3:tan :1 [78,274]. In the ttM Z case, the e!ect is practically insensitive to and is roughly proportional to 1/tan , it therefore drops as tan is increased. Nonetheless, we "nd that N /(¸'0.1, even in the unpolarized case for tan :0.6 1" [273,274]; note that N /(¸ here is dimensionless if ¸ is in fb\. 1" From Fig. 30(a) we see that, in the ttM h case, as m grows the asymmetry increases while the F statistical signi"cance drops, in part because of the decrease in the cross-section. Evidently, the asymmetry can become quite large; it ranges from &15%, for m :100 GeV, to &35% for F m &600 GeV. Indeed, the CP-e!ect is more signi"cant for smaller masses of h, wherein A is F smaller. In contrast, Fig. 30(b) shows that, in the ttM Z case, A stays roughly "xed at around 7}8% Recall that for the ttM h "nal state we choose tan "0.5 while for the ttM Z "nal state we take tan "0.3. As a reference value, we note that for ¸"100 fb\, N /(¸"0.1 will correspond to a 1! e!ect. 1"
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Fig. 30. The asymmetry, A , and scaled statistical signi"cance, N /(¸, for the optimal observable O for: 1" (a) the reaction e>e\PttM h with tan "0.5 and (b) the reaction e>e\PttM Z with tan "0.3, as a function of the light Higgs mass m , for (s"1 TeV and 1.5 TeV. All graphs are with set II of the parameters, as in Fig. 29. Figure taken F from [274].
for m :2m , and then drops till it totally vanishes at m "m "750 GeV, due to the `GIMF R F & likea mechanism discussed above. The scaled statistical signi"cance N /(¸ behaves roughly as 1" A . That is, N /(¸+0.1}0.2 in the mass range 50 GeV:m :350 GeV, for both (s"1 and 1" F 1.5 TeV. Figs. 31(a) and (b) show the dependence of A and N /(¸ on the c.m. energy, (s, for the ttM h 1" and ttM Z cases, respectively. We see that, in the case of ttM h, the CP-e!ect peaks at (s+1.1(1.5) TeV for m "100(360) GeV and stays roughly the same as (s is further increased to 2 TeV. In the case F of ttM Z, the statistical signi"cance is maximal at around (s+1 TeV and then decreases slowly as (s grows for both m "100 and 360 GeV. Contrary to the ttM h case, where a light h is favored, in F the ttM Z case, the e!ect is best for m 92m . In that range, on-shell Z and h are produced followed by F R the h decay hPttM , thus, the Higgs exchange diagram becomes more dominant. In Tables 7 and 8 we present N for O , for the ttM h and ttM Z cases, respectively, in Model II with 1" set II, and we also compare the e!ect of beam polarization with the unpolarized case. As before, we take tan "0.5 and tan "0.3 for the ttM h and ttM Z cases, respectively, where for the ttM Z case we present numbers for both m "750 GeV (shown in the parentheses) and m "1 TeV, to demon& & strate the sensitivity of the CP-e!ect to the mass of the heavy Higgs. For illustrative purposes, we choose m "100, 160 and 360 GeV and show the numbers for (s"1 TeV with L"200 (fb)\ F
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Fig. 31. The asymmetry, A , and scaled statistical signi"cance, N /(¸, for the optimal observable O for: (a) the 1" reaction e>e\PttM h with tan "0.5 and (b) the reaction e>e\PttM Z with tan "0.3, as a function of the c.m. energy (s, for m "100 GeV and m "360 GeV. All graphs are with set II of the parameters, as in Fig. 29. Figure taken F F from [274].
and for (s"1.5 TeV with L"500 (fb)\ [283,284] (see also [253}255]). In both cases we take "0.5 assuming that there is no loss of luminosity when the electrons are polarized. Evidently, for both reactions, left polarized incoming electrons can probe CP violation slightly better than unpolarized ones. We see that in the ttM h case the CP-violating e!ect drops as the mass of the light Higgs (h) grows, while in the ttM Z case it grows with m . In particular, we "nd that with F (s"1.5 TeV and for m 92m the e!ect is comparable for both the ttM h and the ttM Z cases where it F R reaches above 3- for negatively polarized electrons. With a light Higgs mass in the range 100 GeV:m :160 GeV, the ttM h case is more sensitive to O and the CP-violating e!ect can reach F &4- for left polarized electrons. In that light Higgs mass range, the CP-violating e!ect reaches slightly below 2.5- for the ttM Z case. For a c.m. energy of (s"1 TeV and m "360 GeV, the ttM Z F case is much more sensitive to O and the e!ect can reach 2.2- for left polarized electron beam. However, with that c.m. energy, the ttM h mode gives a larger CP-odd e!ect in the range m &100}160 GeV. F Let us now summarize the above results and add some concluding remarks. We have shown that an extremely interesting CP-odd signal may arise at tree-level in the reactions e>e\PttM h and
Clearly if the e$ciency for ttM h and/or ttM Z reconstruction is "0.25, then our numbers would correspondingly require 2 yr of running.
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Table 7 The statistical signi"cance, N , in which the CP-nonconserving e!ects in e>e\PttM h can be detected in one year of 1" running of a future high-energy collider with either unpolarized or polarized incoming electron beam. We have used tan "0.5, a yearly integrated luminosity of L"200 and 500 (fb)\ for (s"1 and 1.5 TeV, respectively, and an e$ciency reconstruction factor of "0.5 for both energies. Recall that j"1(!1) stands for right(left) polarized electrons. Set II means , , ,/2, /4, 0. Table taken from [274] (s (TeV) P
1
1.5
j (GeV) N
m "100 F
!1 unpol 1 !1 unpol 1
2.2 2.0 1.8 4.0 3.6 3.2
e>e\PttM h (Model II with Set II) O m "160 F 2.0 1.9 1.7 3.9 3.5 3.1
m "360 F 1.1 1.0 0.9 3.2 2.9 2.6
Table 8 The same as Table 7 but for e>e\PttM Z, with tan "0.3. In this reaction, e!ects of the heavy Higgs, H, are included and N is given for both m "750 GeV (in parentheses) and m "1 TeV. Table taken from [274] 1" & & (s (TeV) P
1
1.5
j (GeV) N
m "100 F
!1 unpol 1 !1 unpol 1
(1.8) (1.6) (1.5) (2.3) (2.1) (1.8)
1.7 1.6 1.5 2.9 2.6 2.3
e>e\PttM Z (Model II with Set II) O m "160 m "360 F F (1.8) (1.7) (1.5) (2.4) (2.1) (1.8)
1.8 1.6 1.5 3.0 2.7 2.3
(2.2) (2.0) (1.8) (2.8) (2.5) (2.1)
2.2 2.0 1.8 3.3 3.0 2.6
e>e\PttM Z. The asymmetries that were found are &15}35% in the ttM h case and &5}10% for the ttM Z "nal state. These asymmetries may give rise in the best cases, i.e., for a favorable set of the relevant 2HDM parameters, to &3}4-, CP-odd, signals in a future e>e\ collider running with c.m. energies in the range 1 TeV:(s:2 TeV. Note, however, that the simple observable, O, as well as the optimal one, O , require the identi"cation of the t and tM and the knowledge of the transverse components of their momenta in each ttM h or ttM Z event. Thus, for the main top decay, tPb=, the most suitable scenario is when either the t or the tM decays leptonically and the other decays hadronically. Distinguishing between t and tM in the double-hadronic decay case will require more e!ort and still remains an experimental challenge. If, for example, the identi"cation of the charge of the b-jets coming from the t and the tM is possible, then the di$culty in reconstructing the transverse components of the t and tM momenta
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may be surmountable by using the momenta of the decay products in the processes e>e\PttM hPb=>bM =\h and e>e\PttM ZPb=>bM =\Z. For example, the observable (p , p , p , p M ) O " \ > @ @ @ s
(6.71)
may then be used. We have considered this observable for the reaction e>e\PttM hPb=>bM =\h in [78]. We found there that, close to threshold, this observable is not very e!ective. However, at higher energies, O is about as sensitive as the simple triple product correlation O de"ned in @ Eq. (6.69) and, therefore, only slightly less sensitive than O . Note also that for the light Higgs mass, m "100 GeV, the most suitable way to detect the Higgs F in e>e\PttM hPb=>bM =\h is via hPbbM with branching ratio &1. For m 92m , and speci"cally F R with set II used above, there are two competing Higgs decays, hPttM and hP=>=\, depending on the value of tan . For example, for tan "0.5, as was chosen above, one has Br(hPttM )+0.77 and Br(hP=>=\)+0.17, thus, the hPttM mode is more suitable. Of course, hPttM will dominate more for smaller values of tan and less if tan '0.5. In particular, for tan "0.3(1) one has Br(hPttM )+0.89(0.57) and Br(hP=>=\)+0.08(0.32). Finally, as emphasized before, the "nal states ttM h and ttM Z, in particular the ttM h, are expected to be the center of considerable attention at a linear collider. Extensive studies of these reactions are expected to teach us about the details of the couplings of the neutral Higgs to the top quark [285,286]. Thus, it is gratifying that the same "nal states promise to exhibit interesting e!ects of CP violation. It would be very instructive to examine the e!ects in other extended models. Numbers emerging from the 2HDM that was used, especially with the speci"c value of the parameters, should be viewed as illustrative examples. The important point is that the reactions e>e\PttM hP b=>bM =\h and e>e\PttM ZPb=>bM =\Z appear to be very powerful and very clean tools for extracting valuable information on the parameters of the underlying model for CP violation. 6.2.2. Generalized optimization technique and extraction of various Higgs couplings An optimization technique was employed by Gunion et al. in [73] for the process e>e\PttM h. It was shown that this reaction may provide a powerful tool for extracting the ttM h Yukawa couplings and the ZZh couplings. A similar analysis for the reaction e>e\PttM Z was likewise considered in [275]. This technique is outlined in Section 2.6. The basic idea is that the nature of the Higgs particle, i.e., whether it is CP-odd or CP-even, may well be distinguishable through studies of momentum correlations in e>e\PttM h. In particular, greater information on the detailed dependence of M ( ) on the variable is extracted to deduce limits that can be obtained on the di!erent HRRF Higgs couplings combinations in Eqs. (6.58) and (6.61). As described before, the di!erential e>e\ P ttM h cross-section contains "ve distinct terms which are explicitly written in Eqs. (6.58) and (6.61). The only CP-violating component is bFcF, while the others enter into the total cross-section R as in Eq. (6.63). Gunion et al. have investigated two issues. For a given c.m. energy and integrated luminosity at the NLC, they have examined: 1. The 1- error in the determination of the couplings aF, bF, cF, by "xing (ttM h)"1 for a given R R . input model with couplings g GL !RRM F 2. To what degree of statistical signi"cance can a model be ruled out, given a certain input model.
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Let us now elaborate more on how these two points were studied in [73]. With the optimal technique outlined in [73], Gunion et al. used unique weighting functions such that the statistical error in the determination of the various gGLM in Eqs. (6.58) and (6.61) is minimized. They write !RRF GL (6.72) g M " M\I , GI I !RRF I where I , M and the appropriate weighting functions are given in [73] (see also Section 2.6). Then, I GI , one can given an input model, for which the couplings are denoted with the superscript 0 as g GL !RRM F compute the con"dence level at which parameters of choice, di!erent from the input model, can be ruled out )(gHL !g HL )<\ , (ttM h)" (gGLM !g GL !RRM F !RRM F !RRM F GH !RRF G H where
(6.73)
M\ M (6.74) < , gGLM gHLM " GH RRF GH !RRF !RRF NM RRF is the covariance matrix. N M "¸ M is the total number of ttM h events, with ¸ the e!ective RRF RRF luminosity de"ned previously (after the e$ciency factor, , is included). Therefore, the sensitivity of (ttM h) to the couplings aF, bF, cF is determined by the covariance matrix directly. R R Three input models were considered in [73]. These correspond to di!erent choices of the set of parameters aF, bF and cF (see Eqs. (3.70) and (3.71) in Section 3.2.3), as follows: R R (I) A SM neutral Higgs, with aF"1/(2, bF"0, cF"1. R R (II) A pure CP-odd neutral Higgs, with aF"0, bF"1/(2, cF"0. R R (III) A CP-mixed neutral Higgs, with aF"bF"1/2, cF"1/(2. R R Given the couplings g HL in the above input models, they calculated (ttM h) as a function of the !RRM F location in aF, bF, cF parameter space, from which the 1- error in any one of these parameters was R R determined. Their results are shown in Table 9. We see that aF is well determined in all input models; in input models I and III, where aFO0, the R R 1- error is at the few percent level. In the same manner, bF is best determined in input model II, R where b+10}15%, at 1-. The error in cF is above the 50% level in all three input models. However, this can be improved by considering the reaction at hand, i.e., e>e\PttM h, combined with information extracted from the e>e\PZh cross-section as was done in [279]. Assuming real Higgs production and disregarding the subsequent h decay in the reaction e>e\PZh, the "(cF). Therecross-section (e>e\PZh) contains one useful Higgs coupling combination, g> >RRM F fore, the sensitivity to (cF) is increased when the above technique is also applied to (e>e\PZh). Let us continue with a discussion of the work of Ref. [73] in which only the ttM h "nal state was considered. The ability to distinguish between di!erent models was furthermore investigated by using the optimization technique. This is shown in Table 10. We see from this table, for example, that if the Higgs is the SM one, then the pure CP-odd case (input model II) and the equal CP-mixture case (input model III) are ruled out at the 9.5- and 4.8- level, respectively. Note also that negative beam polarization slightly improves the results.
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Table 9 The 1- errors in aF, bF and cF are given, for the three Higgs coupling cases I, II and III. Results are given for unpolarized R R beams and for 100% negative e\ polarization (P "!1) also (s"1 TeV, m "100 GeV, m "176 GeV and C F R ¸"50 fb\. Table taken from [73]
Case I
aF$aF R R
Unpolarized e\ bF$bF cF$cF R R
> \ 0> \ > \
0> \ > ( \ > \
(
II III
1> \ 0> \ > ( \
aF$aF R R
P "!1 C bF$bF R R
cF$cF
> \ 0> \ > \
0> \ > ( \ > \
1> \ 0> \ > ( \
(
Table 10 The number of standard deviations, ( , at which a given input model (I, II or III) can be distinguished from the other two models, are tabulated, for (s, m , m and ¸ as in Table 9. Table taken from [73] F R Unpolarized e\ Trial model
P "!1 C Trial model
Input model
I
II
III
I
II
III
I II III
* 34 6.3
9.5 * 6.3
4.8 17 *
* 40 7.3
11 * 7.3
5.5 20 *
Finally, the ability for determining a non-zero CP-violating component, bFcF, was also investiR gated in [73]. They found that with m "100 GeV, ¸"100 fb\, a non-zero bFcF coupling can F R be established at a level better then 1- in a 1 TeV e>e\ collider. 6.2.3. CP asymmetries in e>e\PZh and in the subsequent Higgs decay hPttM We now consider the process e>e\PZh followed by hPttM . As we have discussed above, in general, one cannot ignore the SM-like diagrams of class (a) on the right-hand side of Fig. 28 when analyzing CP violation in the reaction e>e\PttM Z. Moreover, inclusion of those diagrams and interfering them with diagram (b) on the right-hand side of Fig. 28, gives a bonus in the appearance of tree-level CP violation in e>e\PttM Z. However, let us assume that the Higgs has already been discovered, with a mass of m '2m , by the time a high-energy e>e\ collider starts its "rst run. In F R such a scenario, one should, in principle, be able to separate the contribution of the Higgs exchange graph in e>e\PttM Z from the rest of the SM-like diagrams which lead to the same "nal state, by imposing a suitable cut on the invariant ttM mass. Taking this viewpoint, we will consider Higgs production via e>e\PhZ and CP violation in the subsequent Higgs decay hPttM . This is the value considered by us in the previous Section 6.2.1 for which the results in Tables 7 and 8 are given.
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A general method for tracing CP-odd and CP-even t, tM spin-correlations, in hPttM , was introduced in [287]. There, it was assumed that an on-shell Higgs, with m '2m , is produced through, for F R example, e>e\PZh, l>l\h or even >\Ph, and that its rest system can be reconstructed. In [288], a helicity asymmetry in hPttM was suggested, where e>e\PZh was explicitly assumed as the Higgs production mechanism. We will describe below these two works. Other related works can be found in [82,289}295]. In the method suggested in [287], the decay hPttM stands out as an independent decay process, in which top spin-asymmetries can be formed. Consider, for example, the observables (6.75) S "kK ) (s !s M ) , R R R S "kK ) (s ;s M ) , (6.76) R R R S "s ) s M , (6.77) R R where s (s M ) is the spin operator of t(tM ) and k is the top quark 3-momentum in the ttM c.m. frame. R R R S and S are CP-odd, where S is ¹ -even and S is ¹ -odd. Therefore, a non-zero expectation , , value of S will also require absorptive parts, while S O0 can be generated already at the tree-level. S is CP-even and is also generated at the lowest order (i.e., tree-level). For a general ttM h Yukawa interaction Lagrangian as in Eq. (3.70), it was found in [287] that the spin-asymmetries, S in Eqs. (6.75)}(6.77), depend only on one combination of the couplings aF and bF R R bF R , (6.78) r" R aF#bF R R which takes values between 0 to 1, i.e., 04r 41, where the lower limit corresponds to bF"0 and R R the upper one to aF"0. R These observables can be translated to correlations between momenta of the t and tM decay products. To do so, one can de"ne decay scenarios of the t and tM , through which both the t and tM momenta and spins can be reconstructed in the most e$cient way [287]: A:
tP=>#bPl>#l #b ,
tM P=\#bM Pq#q #bM .
(6.79)
The sample A M is de"ned by the charge conjugate decay channels of the ttM pairs AM :
tP=>#bPq #q #b , tM P=\#bM Pl\#l #bM .
(6.80)
In these decay samples, either the t or tM decays leptonically while the other decays hadronically. Each of these samples has a branching fraction of about of all ttM pairs. With these decay scenarios Ref. [287] found the momentum correlations O which trace the spin correlations S , respectively, 2 (6.81) O " kK ) p( Hl> A # kK ) p( lH\ AM " S , R R 3
D. Atwood et al. / Physics Reports 347 (2001) 1}222
8 1!2x O " kK ) (p( lH> ;p( HM )A ! kK ) (p( lH\ ;p( H)AM " S , R @ R @ 9 1#2x
8 1!2x O " p( lH> ) p( HM A # p( Hl\ ) p( HAM " S , @ @ 9 1#2x
133
(6.82) (6.83)
where p( Hl> (p( lH\ ) is the #ight direction of l>(l\) in the t(tM ) quark rest system. Similarly, p( H(p( HM ) is the @ @ unit momentum of the b(bM ) in the t(tM ) rest system. Also, x,m /m and the factor 5 R (1!2x)/(1#2x)+0.41 measures the spin analyzing quality of the b(bM ) (see Section 2.8). of hPttM events that is required to establish a nonzero correlation O at The number N RRM the N (standard deviations) signi"cance level are given by 1" N 1" " , (6.84) N RRM Br(A);A where O A " , (6.85) ( O and Br(A)"Br(A M ) is the branching ratio of the decay samples A or AM . In particular, disregarding the leptons we have Br(A)" . The number of events needed for a 3- (i.e., N "3) observation 1" of the spin-correlations, S , of Eqs. (6.75)}(6.77) are given in Fig. 32, for m "400 GeV, F Br(A)" and as a function of the parameter r de"ned in Eq. (6.78). These can be simply obtained R in Eq. (6.84), by using the relations between O and S given in Eqs. (6.81)}(6.83). from N RRM It should be noted again that, while O are non-zero already at the tree-level, O , being ¹ -even, requires an absorptive phase and, therefore, its non-zero contribution "rst arises only at , the 1-loop level in perturbation theory. Thus, O is expected to be less e!ective (as can be seen from Fig. 32). In [287] the 1-loop QCD corrections to O were computed. For the operators, O , the QCD corrections were found to be of the order of a few percent compared to the leading tree-level contribution, and were therefore neglected. We see from Fig. 32 that, for example, values of r in the range, 0.18:r :0.52, would give rise to R R a 3- CP-odd e!ect in O with a data sample of NM +1500, i.e., &1500 (hPttM ) events. Note also RR that a simultaneous measurement of O and O , with these &1500 events, would have a 3- sensitivity to r from 0.18 to its maximal value of 1. Furthermore, production of a few thousands of R hPttM events through the Bjorken mechanism, e>e\PhZ, is indeed feasible. For example, if a CP-mixed neutral Higgs (with both scalar and pseudoscalar couplings aF and bF) with a mass R R m "400 GeV, has a ZZh coupling cF of a SM strength, then the cross-section for e>e\PhZ is of F the order of a few fb's for e>e\ c.m. energies in the range 500 GeV:(s:1000 GeV. Therefore, with a yearly integrated luminosity of L&10 fb\, hundreds of hZ pairs can be produced and, thus, a &2- limit on r may be achievable. R
The cross-section for e>e\PhZ is given, for example, in [91]. With m "400 GeV and for cF"1, i.e., the SM ZZh F coupling, (e>e\PhZ)&3.5(8.5) fb for (s"500(1000) GeV.
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Fig. 32. Number of events hPttM to establish a non-zero correlation S with 3 standard deviation signi"cance, as a function of r (see Eq. (6.78)) and for a "xed Higgs mass of m "400 GeV. The dashed line represents the result for NM , R F RR the solid line is the result for NM and the dotted line is the result for NM . m "175 GeV. Figure taken from [287]. RR RR R
An interesting CP-violating helicity asymmetry in hPttM was suggested in [288]: (##)!(!!) OF " , RR (##)#(##)
(6.86)
where (##) and (!!) are the decay widths of the lightest Higgs-boson h, into a pair of ttM with the indicated helicities. Since under CP: (##) (!!), non-zero OF would be a signal of CP RR violation. OF is CP-odd but ¹ -even, therefore, it requires a CP-odd as well as a CP-even absorptive phase RR , (i.e., FSI phase). As mentioned several times before, in a 2HDM with a CP-mixed neutral Higgs, the CP-odd phase is provided by the simultaneous presence of the scalar and pseudoscalar ttM h couplings in the ttM h interaction Lagrangian. The FSI absorptive phase is generated at the 1-loop level from the diagrams in Fig. 33. The expressions for the di!erent contributions to OF correRR sponding to the di!erent diagrams in Fig. 33 are given in [288]. There, it was found that with m "180 GeV and m 92m , OF of the order of 50% is possible. Explicitly, assuming the Higgs to R F R RR be produced through the Bjorken mechanism, e>e\PhZ, the statistical signi"cance, N , with 1" which this asymmetry can potentially be detected is (6.87) N "(¸((e>e\PhZ);OF . RR 1" In [288] an e!ective integrated luminosity of ¸"85 fb\ was assumed and a scan for maximal N was performed as a function of the 2HDM parameters tan and . It was found that with 1" tan "0.5, for example, and with m "180 GeV, m 92m , up to a 7- detection of a CP-violating R F R signal from OF is feasible, if the Higgs is produced via e>e\PhZ. It should be emphasized, RR however, that there is a potential background to the ttM pairs (coming from the subsequent Higgs decay in e>e\PhZ) from the SM-like diagrams included in Fig. 28. Therefore, as mentioned before, in order for such a study to be practical one has to know the mass of the Higgs prior to the
D. Atwood et al. / Physics Reports 347 (2001) 1}222
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Fig. 33. The 1-loop diagrams contributing to the asymmetry OF in a 2HDM. RR
actual experiment and, with a su$cient mass resolution, demand that the invariant ttM mass reconstructs the Higgs. 6.3. e>e\PttM g Given the importance of the top pair production at the NLC, it should be clear that the associated gluon emission will also receive considerable attention. Of course, the gluons will be radiated o! top quarks quite readily once the threshold for top pair production will be reached. One important advantage of the reaction e>e\PttM g is that it is rich in exhibiting several di!erent types of CP asymmetries which can be driven by 1-loop e!ects induced by extensions of the SM. For example, exchanges of neutral Higgs from MHDMs with CP violation in the scalar sector, or exchanges of SUSY particles which carry a CP-odd phase in their interaction vertices, could give rise to both ¹ -odd and ¹ -even type CP-violating dynamics. Therefore, both CP-odd, ¹ -odd , , , and CP-odd, ¹ -even type observables can be used to extract information on the real and , imaginary parts of the amplitude. With three particles in the "nal state there are enough linearly independent momenta available so that the construction of CP-odd, ¹ -odd observables is , straightforward; there is no need to involve the spins of the top. Following our work in [296], we give below the full analytical formulae of the tree-level di!erential cross-section (DCS) as well as a description for extracting the 1-loop CP-violating DCS that can be used for any given model. The SM tree-level diagrams, are depicted in Fig. 34. The incoming polarized electron}positron current can be written as JIH"v (p )IP u (p ) , C C > H C \
(6.88)
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Fig. 34. Tree-level Feynman diagrams contributing to e>e\PqqN g (for q"t).
where P "(1#j; ) and j"!1(1) for left (right) handed incoming electrons. p (p ) are the H > \ 4-momentum of the positron (electron) and p"(p #p ) is the 4-momentum of the s-channel > \ gauge-boson (the contribution from an s-channel Higgs vanishes for m P0). C We also de"ne the following constants: (4) ¹?g CC, C "(4)¹?g Q , C " Q 8 H A Q A O 8 2c s 5 5
(6.89)
where ¹? is the appropriate SU(3) generator, g is the strong coupling constant, Q is the charge of Q O the quark and c (s ) stands for cos (sin ), respectively. Also CC"CC (CC ) for j"!1(1) with 5 5 5 5 H * 0 CD "!2ID #2Q s and CD "2Q s . and are the gauge-boson propagators given by * D 5 0 D 5 8 A 1 1 " , " . A p 8 p!m 8
(6.90)
Then the tree-level matrix element is given by M,M #M #M #M , (6.91) ? ? @ @ where M , M , M and M are obtained from diagrams (a ), (a ), (b ) and (b ) in Fig. 34, ? ? @ @ respectively, all emanating from the SM. We thus get 1 ! ¹ ]#2C [ ¹ ! ¹ ]v(p ) . M" JIHu (p )C [ ¹ O 8 O ?I O ?I A O @I O @I O 2 C
(6.92)
Here p (p ) is the 4-momentum of the outgoing quark (anti-quark), p is the gluon's 4-momentum E O O and the quark and anti-quark propagators are given by 1 1 , " . " O 2p ) p O 2p ) p O E O E Furthermore, the hadronic vector elements in Eq. (6.92) are ¹ I " . (p/ #p/ #m ) C> , E O I *0 O ? ¹ I " C> (p/ #p/ !m ) . , ? E O I *0 O
(6.93)
(6.94) (6.95)
D. Atwood et al. / Physics Reports 347 (2001) 1}222
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Fig. 35. The cross-section for the reaction e>e\PttM g (in fb) as a function of the c.m. energy (s, for unpolarized (solid line), negatively polarized (dotted line) and positively polarized (dashed line) incoming electron beam. The cuts (p #p )5(m #m ) and (p #p M )5(m #m ), m "25 GeV, are imposed. Figure taken from [296]. E R R E R R
¹ "¹ I (C> P1) , @I ? *0
(6.96)
¹ "¹ I (C> P1) , @I ? *0
(6.97)
being the polarization vector of the gluon and C> "CO ¸#CO R, where ¸(R)"P (P ). ? *0 * 0 H\ H In Fig. 35, we have plotted the tree-level cross-section as a function of the c.m. energy in an e>e\ collider for polarized and unpolarized incoming electron beam. To facilitate experimental identi"cation as well as to avoid infrared singularities we have imposed a cut on the invariant mass of the jet pairs so that (p #p ) and (p #p M )5(m #m ) where we have taken m "25 GeV. This cut, R E R E R which e!ectively cuts the gluon energy, also removes soft gluon emission from the secondary b-quarks of the top decays. We see from Fig. 35 that with an integrated luminosity of L&200 fb\, a 1 TeV (500 GeV) e>e\ collider will be able to produce about &3;10 (&1;10) ttM g events. In a given model, the CP-violating corrections for the reaction e\(p )e>(p )Pq(p )q (p )g(p ) E \ > O O
(6.98)
requires the calculation of the corresponding 1-loop diagrams. Let us write the general form of the 1-loop matrix elements that violate CP as MM . For a given underlying model, the subscript N indicates the diagram and the superscript indicates which gauge particle is exchanged in the
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D. Atwood et al. / Physics Reports 347 (2001) 1}222
s-channel. Thus, MM "JIHu (p )HM v(p ) , (6.99) N C O NI O where HM is the `hadronic vectora corresponding to each diagram and exchanged quanta. NI Denoting the complete CP-violating 1-loop contribution by M4" MM , (6.100) N M N the M4 can be calculated within a given model, and the polarized CP-nonconserving DCS to 1-loop is then obtained from the interference terms between the 1-loop and the Born amplitudes (M4MH#MM4H) .
(6.101)
Here the sum is carried over the polarizations of e>, t, tM and g. 6.3.1. 2HDM and CP violation in e>e\PttM g In a 2HDM, CP-violating neutral Higgs exchanges, at 1-loop order, can give rise to the Feynman diagrams depicted in Fig. 36 [296]. We take the limit m P0, thus neglecting all the C diagrams that are proportional to the electron mass. This includes any diagram that involves electron coupling to the Goldstone modes, hence proportional to m . C The relevant Feynman rules for the diagrams in Fig. 36 can be extracted from parts of the Lagrangian involving the +M HI and ZZHI couplings aI , bI and cI de"ned in Eqs. (3.70) and (3.71), D D respectively. Again, for simplicity, we will consider only one light neutral Higgs, h, assuming that the remaining two are considerably heavier. Furthermore, whenever necessary we set the masses (m ) of the remaining two neutral Higgs particles to be 1 TeV, i.e., assume that they are degenerate. & All the CP-violating terms in the 1-loop amplitudes corresponding to the diagrams in Fig. 36 emerge through interference of the scalar coupling aF (for a quark q) with the pseudo-scalar O coupling bF in any exchange of a neutral Higgs. In the diagrams where the Higgs exchange is O generated at the ZZh vertex, the CP-violating terms will be proportional to bF ;cF. The CPO violating 1-loop amplitude can then be calculated (for details see [296]) and the DCS can be schematically written as (6.102) ()" ()#R()#I () . Here () is a CP-even piece and is a CP-odd piece which is further subdivided into two parts that depend on the real and imaginary components of the amplitude. Thus, CP-odd, ¹ -odd and , CP-odd, ¹ -even e!ects will emanate from R() and I (), respectively. , To estimate the CP-violating e!ects of both the ¹ -even and ¹ -odd types, the following two , , CP-odd observables were considered in [296] for the reaction e>e\PttM g p ) ( p #p M ) R R , ¹ -even: O , \ , G s
(6.103)
p ) ( p ;p M ) R R . ¹ -odd: O , \ , P s
(6.104)
Recall that HI stand for any one of the three, i.e., k"1, 2 or 3, neutral Higgs in a 2HDM.
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Fig. 36. CP violating Feynman diagrams contributing to e>e\Pqq g to 1-loop order in a 2HDM (h is a neutral Higgs-boson). Diagrams with permuted vertices (i.e., qPq ) are not shown.
A non-vanishing expectation value of any one of these would signal CP violation so that experimental searches for them can be performed without recourse to any model. However, as was mentioned in Section 2.6, within the context of any given model one can also construct optimal observables i.e., those observables which will be the most sensitive to CP violation e!ects in that model [29], (6.105) O ,I / , O ,R/ , P G The number of events needed in order to detect a CP-odd signal at the 1- level via each of the above four CP-violating observables, is shown in Figs. 37 and 38 for m "100 and 200 GeV, F respectively. In Fig. 39 we have magni"ed the range (s"400}600 GeV using the same Higgs masses. In these "gures, we have focused on the case of left polarized incoming electrons while in Table 11 we give a brief comparison of the left, right and unpolarized electron beam cases. Also, the following assumptions are made: (1) As mentioned before a cut on the invariant mass of the jet pairs was imposed, so that (p #p ) E R and (p #p M )5(m #m ), where we have taken m "25 GeV. R E R (2) The ¹ -odd observables, being proportional to the dispersive parts of the loop integrals, are , sensitive also to the mass of the two heavier neutral Higgs particles, m . For simplicity, we have & chosen these two Higgs particles to be degenerate with a mass of 1 TeV. (3) We set the relevant Higgs couplings to unity. That is, aF"bF"cF"1 (for q"t) which serves R R our purpose of "nding the order of magnitude of the CP-odd signal that can arise in this reaction. In fact, CP violation in e>e\PttM g is found to be dominated by the terms proportional to aF;bF. With regard to that, we note that, for low values of tan , i.e., tan :0.5, the R R product aF;bF can reach values above &5 and, therefore, our choice above of aF;bF"1 is R R R R rather conservative. Summarizing brie#y the numerical results presented in Figs. 37}39 and in Table 11, we can see that for the optimal observable O , for both a 500 GeV and a 1 TeV e>e\ collider, the number of G
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Fig. 37. Number of events (in units of 10) needed to detect CP violation via O , O , O and O to 1- G P G P level, as a function of the total beam energy, (s, for left-handed polarized incoming electron beam. m "100 GeV, F m "1 TeV and aF"bF"cF"1 are used. Also, the cuts (p #p )5(m #m ) and (p #p M )5(m #m ), R & R R E R R E R m "25 GeV, are imposed. Figure taken from [296].
Fig. 38. Same as Fig. 37 except m "200 GeV. Figure taken from [296]. F
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Fig. 39. Number of events (in units of 10) needed to detect CP violation via O and O to 1- level as a function P P of total beam energy in the range (s"400}600 GeV for m "1 TeV, m "100 and 200 GeV. The rest of the parameters & F are as in Figs. 37 and 38. Figure taken from [296].
Table 11 The unpolarized case is compared with left polarization ( j"!1) and right polarization ( j"#1) of the e\. The number of events in units of 10 needed for detection of asymmetries, to 1- level are given. The values of (s and m are F given in GeV. The results for the ¹ -odd observables are given for m "1 TeV, where m is the mass of the two heavy , & & Higgs (see also text). Table taken from [296] O (s
j
O
G m "100 F
O
P m "200 F
m "100 F
O P
G
m "100 F
m "200 F
m "200 F
m "100 F
m "200 F
!1 400 unpol. 1
1.8 22.5 2.3
11.5 134.8 17.1
0.07 0.05 0.05
0.05 0.05 0.04
1.0 6.5 1.3
6.0 37.0 8.4
0.05 0.05 0.04
0.05 0.04 0.03
!1 700 unpol. 1
3.4 48.6 4.5
20.0 263.9 30.8
2.2 2.2 1.9
2.1 1.9 1.8
1.7 12.2 2.0
5.1 38.6 5.9
1.9 1.8 1.5
1.7 1.4 1.2
!1 1000 unpol. 1
4.0 63.4 5.0
10.5 158.2 14.0
14.4 14.3 14.1
14.1 10.3 10.3
2.6 20.5 3.1
4.5 35.3 5.4
11.6 10.8 10.7
10.8 8.4 8.3
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needed ttM g events in order to detect a 1- CP-odd signal is comparable and is around few;10 with neutral Higgs masses in the range 100 GeV(m (200 GeV. With O the number of needed ttM g F P events at c.m. energies around 1 TeV is few;10. However, we see from Fig. 39 that, at a c.m. energy of 500 GeV and for 100 GeV(m (200 GeV, a 1- measurement of O will require F P few;10 ttM g events. From Table 11 we see that for the ¹ -even (i.e., O and O ) cases the , G G polarization makes a signi"cant di!erence and improves their e!ectiveness by about an order of magnitude or even more. For these it seems that the left-polarized case is marginally better than the right one. Bearing in mind that with an integrated luminosity of L&200 fb\, about &10 ttM g will be produced in a 500 GeV NLC, and few;10 in a 1 TeV NLC (see Fig. 35), the observability of a non-vanishing value for O , to the 1- level, in a NLC with c.m. energies of 500 GeV is marginal, P while O falls short by about an order of magnitude. Also, with a 1 TeV NLC that can produce up G to 3;10 ttM g's a year, the CP-odd signal from O falls short by almost two orders of magnitude, P while the number of events needed to detect a CP-odd e!ect through O is one order of G magnitude away from the expected number of available events in such a collider. Clearly, although the CP-violating e!ects driven by neutral Higgs exchanges that were found in [296] fall short by at least one order of magnitude for a 3- detection, this does not rule out the possibility of larger e!ects in other extensions of the SM (e.g., SUSY). Therefore, theoretical and experimental studies of CP violation in the process e>e\PttM g may still be worthwhile. 6.3.2. Model independent constraints on top dipole moments The e!ects of anomalous EDM (dA), ZEDM (d8) and CEDM (dE) couplings of the top quark to R R R a photon, Z-boson and a gluon, respectively, in e>e\PttM g were considered in [297,298]. Let us write again an e!ective top quark interaction with a neutral gauge-bosons <", Z or g, which involves the top magnetic and electric dipole moments (see also Section 2.5) ig (6.106) L " 4 tM qJ( !i )tFI . 4 4 4 4 2m IJ R Here g "g s "e, g "g /2c and g "g , where g (g ) is the weak(strong) coupling conA 5 5 8 5 5 E Q 5 Q stant, c ,cos , q is the gauge-boson 4-momentum, and F , <"A, Z or G, is the appropriate 5 5 4 gauge "eld (color index is suppressed). In Eq. (6.106) we have introduced the CP-conserving(violating) dimensionless e!ective anomalous couplings ( ) of the top quark to a gauge4 4 boson <", Z or g. Note that and are related to c4 (the magnetic-like dipole moments) and 4 4 R to d4 (the electric-like dipole moments), de"ned in Eq. (6.2), via R 2m , R ;c4 , (6.107) 4 R g 4 2m , R ;d4 , (6.108) 4 R g 4 and that and are, in general, complex. In particular, for the convenience of the reader we 4 4 note that &0.1(1) corresponds to a top electric-like dipole moment coupling of d4&0.55(5.5); 4 R 10\g cm. 4
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The implications of the e!ective top couplings in Eq. (6.106) can be studied by either considering CP-even or CP-odd observables in the reaction e>e\PttM g. Of course, it should be clear that an analysis of CP-even quantities like cross-sections and the shape of the gluon energy spectrum [297], or CP-even combination of polarization asymmetry [298], can place rather mild constraints on the absolute values of various EDMs as only enters into such quantities. Let us summarize 4 below the limits that can be obtained on the various EDM couplings of Eq. (6.106) by analyzing the e!ects of CP-even and CP-odd quantities on the reaction e>e\PttM g. 6.3.2.1. CP-even observables. In [297] it was suggested that the process e>e\PttM g can be used to obtain limits on the anomalous dipole-like couplings of the top to , g and Z through the analysis of the associated gluon energy spectrum. If the couplings of the top to the neutral gauge-bosons , Z and g are altered by the e!ective magnetic and electric-like interactions in Eq. (6.106), then by allowing one or more of the di!erent 's and 's to be non-zero, the shape of the gluon energy spectrum in the process e>e\PttM g can 4 4 be modi"ed. In [297], Monte Carlo data samples (assuming that the SM is correct) were generated and then a "t to the general expressions for the ! dependent spectrum were performed with 4 4 which a 95% CL allowed region in the ! was obtained. This procedure is done for each 4 4 gauge-boson separately. That is, in analyzing the limits that can be placed on the various dipole moment couplings, only one pair of , corresponding to one neutral gauge-boson, <", Z or 4 4 g, was allowed to have a non-zero value. As we have mentioned before the usefulness of the bound on the magnetic moment is rather limited as it receives signi"cant contribution from QCD. Summarizing now the limits obtained in [297], Fig. 40 shows the 95% CL allowed region in the ! plane for both a 500 GeV and a 1 TeV e>e\ NLC. We see that the CEDM coupling, , E E E can be bounded in a 500 GeV NLC to :0.6}0.8. For the CMDM coupling, for whatever it is E worth, the allowed values are &$few;10\, with integrated luminosities of 50}100 fb\ and E with a cut on the gluon energy of E '25 GeV. In a 1 TeV NLC, the limit on the CEDM coupling E is approximately twice as strong as what can be achieved in a 500 GeV NLC.
Fig. 40. 95% CL allowed region in the ! (recall that ,(2m /g );dE) plane obtained from "tting the gluon E E E R Q R spectrum. On the left above: E "25 GeV at a 500 GeV NLC assuming an integrated luminosity of 50(solid) or E 100(dotted) fb\. On the right above: for a 1 TeV collider with E "50 GeV and luminosities of 100(solid) and E 200(dotted) fb\. Figure taken from [297].
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Fig. 41. The 95% CL allowed regions obtained for the anomalous couplings (a) , and (b) , at a 500(1000) GeV A A 8 8 NLC, assuming a luminosity of 50(100) fb\, lie within the dashed(solid) curves (recall that ,(2m /e);dA and A R R ,(2m /g /2c );d8). The gluon energy range z,2E /(s50.1 was used in the "t. Only two anomalous couplings 8 R 5 5 R E are allowed to be non-zero at a time. Figure taken from [297].
It should be noted that in [298] two other CP-even quantities were considered: the cross-section itself and a CP-even combination of the top and anti-top polarizations. The limits obtained there for the CEDM of the top, , are somewhat weaker then those shown in Fig. 40. E In analyzing the anomalous EDMs of the top to a photon and a Z-boson, the `normalizeda gluon energy distribution was used in [297]: d(e>e\PttM g) 1 dR " . dz (e>e\PttM ) dz
(6.109)
It was then found that the EDM coupling of the top quark to a photon, , can be constrained at A the NLC by studying the reaction e>e\PttM g. From Fig. 41(a) we see that at a 500 GeV NLC with integrated luminosity of 50 fb\, only long narrow bands around &!1 or 0 are allowed which A then gives :0.4}0.6. In a 1 TeV NLC with integrated luminosity of 100 fb\, one circular A narrow band between !0.4: :0 is allowed giving 0: :0.2. A A The anomalous EDM coupling of the top quark to the Z, , is much less constrained. 8 In particular, from Fig. 41(b) we see that with a 500 GeV NLC and integrated luminosity of 50 fb\, 0: :0.5 is allowed if !0.5: :0.1. However, with a 1 TeV NLC and integrated 8 8 luminosity of 100 fb\, the allowed region in the ! plane is considerably reduced. Namely, 8 8 0: :0.1 can be achieved if !0.2: :0. Also, as was shown in [297], doubling the integrated 8 8 luminosity does not increase the sensitivity of the NLC to these anomalous couplings of the top quark. To conclude, note that while the process e>e\PttM will presumably be more appropriate for the exploration of CP-odd e!ects driven by the top dipole moment couplings to and Z (see Section 6.1), the reaction e>e\PttM g might be the only place for searching for the CEDM of the top quark at the NLC. In that sense, an investigation of the e!ects of on CP-odd quantities in the E ttM g "nal state at an e>e\ collider, is worthwhile. This was done in [298] by constructing a genuinely CP-odd observable out of the top and anti-top polarizations and is described below.
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6.3.2.2. CP-odd observables. An interesting CP-odd observable was suggested in [298]. This observable involves the top polarization and is de"ned as (6.110) \"[(!)!( )#(!)!( )] , where (!), (!) refer respectively to the cross-sections for top and anti-top with a positive spin component in its direction of #ight, and ( ), ( ) are the same quantities with a corresponding negative spin component. In [298] the sensitivity of \ to the CEDM of the top, , was studied. Note that \ is E CP-odd and ¹ -even and therefore depends on the imaginary parts of the combinations of , couplings Im( H ) and Im( ) in Eq. (6.106). The 90% CL limits on the values of Im( H ) and E E E E E Im( ) were obtained from [298] E (6.111) L \( , )!\"2.15(L (!)# (!) , E E 1+ 1+ 1+ where in the above expressions, the subscript `SMa denotes the value expected in the standard model, with " "0; is the top detection e$ciency and L is the integrated luminosity. E E Eq. (6.111) gives contours in the Im( H )!Im( ) plane which are shown in Figs. 42(a) and (b) E E E for c.m. energies of (s"0.5 and 1 TeV, respectively, for an integrated luminosity of L"50 fb\ and for "0.1. Also di!erent polarizations of the incoming electron beam, P , are considered in C Figs. 42(a) and (b). The allowed regions in Figs. 42(a) and (b) are the bands lying between the upper and lower straight lines. We see that the dependence of \ on the electron beam polarization is rather mild.
Fig. 42. \ contour plots in the Im( H ) (vertical axis)}Im( ) (horizontal axis) plane, with 90% con"dence level at E E E c.m. energies (s"500 GeV (left side) and (s"1000 GeV (right side), and for di!erent values of the electron beam polarization: P "#0.5, 0, !0.5, !1. Figure taken from [298]. C
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Fig. 43. Intersecting area in the Im( H )}Im( ) plane resulting from two independent \ measurements at E E E (s"500 GeV and (s"1000 GeV. P "!1 is used. See also caption to Fig. 42. Figure taken from [298]. C
It was also suggested in [298] that a measurement of \ at two di!erent c.m. energy may improve the limits on Im( H ) and Im( ). This is demonstrated in Fig. 43 where it is shown that E E E from measurements of \ at (s"0.5 and 1 TeV, the possible limits are !0.8(Im( H )(0.8, !11(Im( )(11 . (6.112) E E E Although dealing with a genuine CP-odd observable, the limits in Eq. (6.112) are still weaker by about an order of magnitude than those obtained through the study of the gluon jet energy distribution. Note however, that those limits are placed on the imaginary part of while the limits E obtained through the study of the gluon jet energy distribution are set on the absolute value of the top CEDM. 6.4. CP violation via == fusion in e>e\PttM C C At a NLC with a very high c.m. energy, above 1 TeV, the t-channel =>=\ fusion subprocesses =>=\PttM , where the two =-bosons are emitted from the initial e>e\ beams, starts to dominate over the simple s-channel production mechanism of a pair of ttM , i.e., e>e\P, ZPttM . As it turns out [299], the reaction e>e\P=>=\ PttM (6.113) C C C C can potentially exhibit large CP-violating phenomena, driven by CP-odd phases in the neutral Higgs sector in MHDMs. To lowest order there are four Feynman graphs, shown in Fig. 44, relevant to the reaction in Eq. (6.113). Indeed, at large c.m. energies, i.e., as s/m becomes very large, the cross-section for the 5
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Fig. 44. The Feynman diagrams that participate in the subprocess =>=\PttM . The blob in diagram (a) represents the width of the Higgs resonance and the cut across the blob is to indicate the imaginary part.
reaction in Eq. (6.113) is dominated by collisions of longitudinally polarized ='s and the subprocess =>=\PttM shown in Fig. 44, when calculated in the E!ective =-boson Approximation [300}305], serves as a good approximation to the reaction e>e\PttM . C C The key point here, as suggested in [299], is again to construct CP-odd observables utilizing the top polarization, which in turn can be traced through the top decays. Following [299], in the rest frame of the t one de"nes the basis vectors: !e J( p > #p \ ), e Jp > ;p \ and e "e ;e . 5 W 5 5 V W X X 5 For the anti-top one uses a similar set of the de"nitions in the tM rest frame related by charge conjugation: !e J( p \ #p > ), e Jp \ ;p > and e "e ;e . Now let P (for j"x, y or z) be 5 W 5 5 V W X H X 5 the polarization of t along e , e , e and similarly, PM the polarization of tM along e , e , e . One can V W X H V W X then combine information from the t and tM systems and de"ne the following asymmetries: A "(P #PM ), B "(P !PM ) , V V V V V V A "(P !PM ), B "(P #PM ) , W W W W W W (6.114) A "(P #PM ), B "(P !PM ) , X X X X X X where it is easy to verify that, within the above coordinate systems, the A's are CP-odd and the B's are CP-even. Moreover, A , B , A are CP¹ -odd whereas B , A , B are CP¹ -even. V W X , V W X , We note here that the CP-even spin observable B , being proportional to the imaginary part of W the Higgs propagator in Fig. 44(a), is also useful for experimentally measuring the Higgs width [299]. However, since here we are only interested in CP non-conservation e!ects in the reaction e>e\PttM , we will focus below on results obtained for the CP-odd observables, i.e., the A's in C C Eq. (6.114). Let us consider a 2HDM with the ttM HI and =>=\HI Lagrangian pieces of Eqs. (3.70) and (3.71), respectively. Here also, the simultaneous presence of the scalar, aI, and pseudoscalar R couplings, bI, in Eq. (3.70) is required for a non-zero expectation value of the CP-violating R asymmetries A , A , A . Therefore, since only two out of the three neutral Higgs particles, i.e., V W X k"1, 2 or 3, (say, h,H and H,H for the lighter and heavier ones, respectively) have a simultaneous scalar and pseudoscalar couplings to ttM (see Section 3.2.3), the third neutral Higgs need not be considered. A is expected to receive signi"cant contributions from loop corrections. X Therefore, we focus below on A and A only (see discussion in [299]). V W The two asymmetries A , A are shown in Fig. 45, for a NLC with a c.m. energy of (s"1.5 TeV, V W as a function of the lighter Higgs mass m . The heavier Higgs mass is "xed to m "1 TeV. Also, for F & illustration, we use tan "0.5 and choose , , "!/2, ,!/2, where , and
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Fig. 45. The asymmetries A (solid) and A (dashes) integrated over s( as a function of m for (s"1.5 TeV and V W F m "1 TeV. The coupling parameters are for tan "0.5 and , , "!/2, ,!/2 as described in the text. & Figure taken from [299].
are the three Euler angles that specify the 3;3 orthogonal mixing matrix of the three neutral Higgs-bosons (see Eq. (3.73)). With this set of parameters the ttM h, ttM H, =>=\h and =>=\H couplings are "xed according to Eqs. (3.70}3.73) in Section 3.2.3. We observe from Fig. 45 that for a wide range of the lighter Higgs mass the asymmetries are appreciable. In particular, A is about 10% for m &400}800 GeV whereas A is around 30% for V F W m &100}300 GeV. Although not shown in Fig. 45, the asymmetries vanish when m "m due to F F & a GIM-like cancellation as explained in Section 3.2.3. As mentioned above, in order to measure those top polarization asymmetries, one needs the momentum of the t and tM decay products in a given decay scenario. In [299] two such decay scenarios useful for top polarimetry were considered: 1. The decay tP=>b followed by =>Pl>, where l"e, ; in this case only the hadronic decays of tM are included. This case occurs with a branching ratio of B +()()" . 2. The decay tP=>b followed by =>Phadrons. Now the decay of tM to a \ is excluded. This case occurs with a branching ratio of B +()()". In the case of the leptonic decay of t (or equivalently tM ), the angular distribution of the lepton is J(1#R P cos l ), where P is its polarization, l is the angle between the polarization axis and the momentum of the lepton in the top rest frame and R "1 in the SM. Thus, the optimal method to obtain the value of P is to use P"3 cos l /R (see Section 2.8). Similarly, in the case of the hadronic t decay (or equivalently tM ), one uses the distribution of the = momentum in the top frame which is J(1#R P cos ) to extract the top polarization, where R "(m!2m )/(m#2m ). 5 R 5 R 5 Therefore, in this case P"3 cos /R . 5 Hence, bearing in mind that the leptonic decay of the top is self polarizing, the number of events needed to obtain a 3- signal in the t, tM decays of case 1 above is [299]: "()(R B #R B )\;a\ , NN RRM
(6.115)
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where a is the asymmetry in question (either A or A ). Therefore, given the above numbers for V W R , R , B and B , numerically NN M +52a\, thus requiring some 5200 events for an asymmetry of RR 10%. In fact, this can be further improved in the case of the hadronic decays of t (case 2 above) by observing that [82] the less energetic of the two jets from the decay of the = is more likely to be the dM -type quark as noted in Section 2.8. In particular, in case 2 one obtains NNM +32a\, thus RR reducing the requirement to 3200 events for an asymmetry of 10% [299]. For an asymmetry of about 30%, which was found possible in the case of A (see Fig. 45), only a few hundred events will W be needed. Indeed, the cross-section for the reaction in Eq. (6.113) was calculated in [299] for the case of a 2HDM with the couplings described above and it was found to be at the level of a few fb, reaching 910 fb for 350 GeV:m :550 GeV. Thus, given that at (s"1.5 TeV the projected luminosity F could be about 5;10 cm\ s\ [283,284] (see also [253}255]), a cross-section of 10 fb would yield about 5000 events rendering it feasible to detect asymmetries 910%. The conclusion is therefore that the top polarization asymmetries for the reaction e>e\PttM are accessible to the C C NLC and can serve as a powerful probe of CP violation driven by the neutral Higgs sector of a 2HDM. However, this last statement must be taken with some caution since, the two neutrinos in the "nal state, carry a substantial amount of missing energy and may therefore pose a problem in reconstructing the t and tM rest frames, as required for measuring the polarization asymmetries in question when the t or tM decays leptonically. No such problem arises if both the t and the tM decay purely hadronically but, in that case, it remains to be seen if it will be possible to distinguish t from tM which is also required for measuring A and A . V W An interesting generalization of this work [299] is to consider instead the reaction e>e\PttM e>e\. Now the fusion takes place via neutral gauge-bosons (, Z). Although there may be some loss of the cross-section, to compensate that, there is also the advantage that the di$culties in reconstructing the rest frames of t, tM may be far less formidable. 7. CP violation in pp collider experiments The LHC is a pp collider at CERN, with c.m. energy of 14 TeV, scheduled to start running around 2005 (For a recent review on machine parameters see [306].) Its design luminosity is L"10 cm\ s\, corresponding to a yearly integrated luminosity of 100 fb\. A low luminosity "rst stage of 10 fb\ is usually assumed in articles discussing physics at the LHC. The issues discussed in the following section, will be relevant for the future CMS and ATLAS experiments (for a review see [307]); heavy ions and LHC-B will not be discussed in the present work. For recent reviews on the physics at LHC, see [307,308]. 7.1. ppPttM #X: general comments In hadronic collisions ttM pairs are produced through the parton level subprocesses qq PttM and ggPttM . The latter, gg fusion process, dominates over the quark}anti-quark annihilation in a multi-TeV pp collider. For example, at the LHC, (ggPttM )&90% and (qq PttM )&10% are expected. It is therefore important to investigate the expected CP violation e!ects in ppPttM #X that can arise from CP non-conservation in the subprocess ggPttM .
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Note that the simple qq fusion process is the analog of the e>e\PttM production mechanism where an s-channel gauge-boson is exchanged. In the case of qq PgPttM , the CP-odd e!ect can therefore be attributed to the CEDM (dE) of the top present at the gttM vertex. In contrast, the gg R production process gives rise to a much richer possibility of CP-violating interactions and the resulting asymmetries in ggPttM need not be related merely to the CEDM of the top quark. This fact can be readily seen in model calculations (such as 2HDM and MSSM to be discussed below), where additional CP-violating 1-loop box diagrams as well as s( -channel resonant neutral Higgs exchange become relevant. We will "rst discuss an e!ective Lagrangian approach in which all CP-violating e!ects are assumed to originate only from the CEDM of the top. We will then present model dependent analysis of CP non-conservation in ggPttM where all possible CP-violating operators are taken into account. As will be shown, the typical size of the CP-violating asymmetries in ppPttM #X is &10\. Although, naively one may expect such asymmetries to be within the experimental reach of the LHC, which is expected to produce &10}10 ttM pairs, there are at least two types of hurdles that make this objective very di$cult to attain. First there is the detector-dependent systematics which are expected to present serious limitations for asymmetries at the &10\ level. Another serious di$culty is that the initial state (pp) is not an eigenstate of CP. Therefore, one expects fake asymmetries to arise at some level even though the underlying interactions do not violate CP. These backgrounds are process dependent and the fake asymmetries that they produce need to be much smaller in comparison to the CP-violating signal that is of interest. In some cases, e.g., an s( -channel resonant Higgs exchange within a 2HDM, as will be described in Section 7.3.2, by employing clever cuts on the ttM invariant mass one can obtain asymmetries at the percent level. In these cases, the CP signal is more robust and may be within the reach of the LHC if the 2HDM parameter space turns out favorable. 7.2. ppPttM #X: general form factor approach and the CEDM of the top As already mentioned in previous sections, in close analogy to the EDM and the weak(Z)}EDM of the top, one can generalize the top quark-gluon e!ective Lagrangian to include terms of dimension 5 which can give rise to a CEDM for the top quark (see Eq. (6.106) in Section 6.3.2). In general, the CEDM coupling, dE, may be considered as a form factor. Its momentum dependence is R generated by e!ective Lagrangian operators of dimension greater than 5. In model dependent calculations, this from factor may acquire momentum dependent imaginary parts as well as real parts. In momentum space, similar to the EDM and weak-EDM cases, the CEDM modi"es the ttg interaction to read (we will not concern ourselves here with the CP-conserving chromo-magnetic dipole moment of the top) !i¹ (g I#dEIJ k ) , (7.1) ? Q R J where k"p #p M is the gluon four-momentum and p (p M ) is the t(tM ) four momentum. R R R R The subprocess ggPttM then proceeds through diagrams (a)}(d) in Fig. 46 where the heavy dots indicate the vertices modi"ed by the CEDM of the top de"ned in Eq. (7.1). Diagram (d) involves an additional dimension 5 ttgg contact term and is needed to preserve gauge invariance (see Section 2.5). Assuming that dE is small enough such that one can expand the matrix element R
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Fig. 46. Feynman diagrams contributing to ggPttM in the presence of a top CEDM in the ttM g vertex which is denoted by the heavy dot.
squared to "rst order in dE, the di!erential cross-section for the subprocess ggPttM can be written, R similar to the e>e\PttM case, as [35] () d" () d#[Re dE(s( )R ()#Im dE(s( )I ()] d , (7.2) R R
where s( "x x s and x , x are the gluons momentum fractions. In Eq. (7.2) above, the gluon structure functions are included and thus represents the "nal state phase space including the gluon momentum fraction variables. Also, with no summation over the t and tM spins s and s M , R R respectively, the CP-odd di!erential cross-sections R () and I () are functions of s , s M .
R R Therefore, because of the correlation between the top spin and the momentum of the charged lepton from the top decay tPb=>Pbl>l , Eq. (7.2) with the s and s M dependence, gives in fact R R the di!erential cross-section for the complete process ggPttM including the subsequent leptonic decay chains of the tops. 7.2.1. Optimal observables With the e!ective Lagrangian in Eq. (6.106) in Section 6.3.2 and by ignoring operators of dimension greater than 5, only the e!ect of a constant real dE was investigated in [35]. Indeed, in R model calculations to be described below, the real part of the CEDM form factor is a constant to a good approximation. Similar to the e>e\PttM case, an optimal ¹ -odd, CP-violating observable , for ggPttM was de"ned in [35] as R (7.3) O" . In a realistic hadronic collider however, not all momenta which enter into the problem are immediately observable. For example, with leptonic decays of both t and tM , the momenta of the neutrinos and the longitudinal momenta of the initial gluons are not observed. As was shown in [35], this leads to a twofold or fourfold ambiguity (depending on the number of solutions to the kinematics which results in a quartic equation) in determining the neutrinos momenta. To bypass this di$culty an `improveda optimal observable, that averages over the reconstruction ambiguity, was suggested in [35]: R ( ) (7.4) O , G G , ( ) G G where the sum is over the di!erent possible reconstructions of the neutrino and anti-neutrino momenta from the leptonic t and tM decays, respectively.
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Using the optimal observables O and O , the attainable 1- limits on Re dE, assuming 10 dilepton R ttM decays, were given in [35]. Note that one can consider also leptonic}hadronic and purely hadronic decays of the ttM pairs. Due to the branching ratios of the =-boson, 10 leptonic ttM pairs implies a sample of &6;10 leptonic}hadronic ttM pairs and &9;10 hadronic ttM pairs. With m "175 GeV, for the `simplea optimal observable O, with dilepton ttM pairs, it was found that the R 1- limit is Re dE&2.8;10\g cm. For the `improveda optimal observable O , Re dE&3.0; R Q R 10\g cm with dilepton or hadronic ttM pairs and Re dE&2.0;10\g cm with leptonic}hadQ R Q ronic ttM pairs. Comparing the 1- limits on Re dE attainable with dileptonic ttM pairs and through the R use of the optimal observables O and O in Eqs. (7.3) and (7.4), respectively, we see that the reconstruction ambiguity does not cause any signi"cant changes. Evidently, with these optimal observables, Re dE may be measured to a precision of &10\g cm. O with the leptonic}hadronic R Q ttM channel seems to be the most sensitive to Re dE. Comparable results for Re dE were found in [309] R R by using the same type of optimal observables. Ref. [309] has also extended the analysis of [35] by including e!ects of the imaginary part of dE. They found that the attainable limit at the LHC for R Im dE is of the same order, i.e., Im dE&10\g cm, although slightly better than the one for Re dE. R R Q R This result is encouraging since, as we have discussed in Section 4 the CEDM of the top may be 910\g cm in some extensions of the SM, e.g., MHDMs and the MSSM. Q 7.2.2. Observable correlations between momenta of the top decay products It is also instructive to consider simple observables constructed exclusively out of momenta which are directly observed. With the decays tPbll and tM PbM lM l , the momenta pl , plM , p and @ p M will be directly observed and observables which involve correlations between those momenta are @ the most appropriate. Two such CP-odd, ¹ -odd correlations were considered in [35]: , M pI pJ pNp M f " IJNM C C @ @ , (7.5) (p ) p p ) p M ) C C @ @ (7.6) f "(pVpW !pW pV ) ) sgn(pX !pX )(p ) p ) , C C C C C C C C where sgn(X)"#1 for X50 and !1 for X(0. The attainable 1- limits on Re dE for the observables f and f , with m "175 GeV and R R assuming 10 dileptonic ttM pairs, were also given in [35]. The "ndings were for f : Re dE&5.3;10\g cm and for f : Re dE&3.0;10\g cm. We see that the limit that might be R Q R Q achieved with f is about an order of magnitude smaller than that from f . However, f depends only on the lepton momenta and is, therefore, easiest to determine experimentally. Also, the limit from f is about 2 times weaker then the one obtained from the `improveda optimal observable discussed previously. In [310] CP-odd ¹ -even observables which might be used to probe the imaginary part of the , CEDM, i.e., Im dE, were considered R (7.7) A "ElM !El , # (7.8) Q "2(pXlM #pXl )(pXlM !pXl )!( plM !pl ) . A is the energy asymmetry between l and lM and Q is an asymmetry originally suggested by # Bernreuther et al. in [249] (see Section 6.1.2). In a pp collider with (s"14 TeV, an integrated
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luminosity of 10 fb\ and an acceptance e$ciency of "10%, taking only leptonic (l"e, ) ttM pairs and assuming m "175 GeV the following 1- limits on Im dE were obtained through the R R observables A and Q # A : Im dE"8.58;10\g cm , (7.9) # R Q Q : Im dE"2.05;10\g cm . (7.10) R Q Thus, the limits on the imaginary part of the top CEDM are weaker by about an order of magnitude than those that might be obtained on the real part of the top CEDM, using the optimal observables discussed before. 7.2.3. Polarized proton beams A very interesting CP-violating polarization rate asymmetry was originally suggested by Gunion et al. in [311], for Higgs production through gg fusion in a pp collider. This asymmetry was applied to ppPttM #X in [310]. The basic idea is that, if the gluons in a polarized proton are polarized, then the initial CP-odd gluon}gluon con"guration allows to probe CP-violating e!ects without requiring full reconstruction of the ttM "nal state. The polarization rate asymmetry is de"ned as ! \ , (7.11) A , > NP # > \ where , in the subprocess ggPttM , is the cross-section for ttM production in collisions of an ! unpolarized proton with a proton of helicity $. Clearly, A is CP-odd and ¹ -even and therefore NP , can only probe the imaginary part of the top CEDM. Of course, a crucial point for such an analysis is the degree of polarization that can be achieved for gluons in the pp collider. The amount of gluon polarization in a positively polarized proton beam is de"ned by the structure functions di!erence g(x)"g (x)!g (x). The structure functions of polarized gluons, g , are not well known and > \ ! depend on the amount of the proton's spin carried by the gluons. In [310] the following, parameterization was adopted (g is the unpolarized gluon distribution)
g(x)
(x'x ) , A (7.12) (x/x )g(x) (x(x ) , A A where x &0.2 yields a value of g&2.5 at Q"10 GeV. The above distribution was actually A evaluated at Q"100 GeV disregarding any scale evolution. The 1- attainable limits on Im dE were calculated in [310] and are given in Table 12, for R various transverse-momentum cuts and for (s"14 TeV, m "175 GeV, L"10 fb\ and an R e$ciency acceptance of 10%. Also, in Table 12 N"N #N is the total number of ttM events, > \ NK "N !N and N (N ) is the number of ttM events predicted for positively(negatively) > \ > \ polarized proton. They have included all possible t decay modes so that the net branching ratio was taken as unity. We see that, even with high p -cuts, it is possible to put a 1- limit on Im(dE) up to 2 R the order of 10\g cm in the LHC with polarized incoming protons. This limit is more stringent Q than the ones obtained in Eqs. (7.9) and (7.10) through the leptonic correlations A and Q , # respectively, and is of the same order as that obtained on the real part of the top CEDM with the optimal observables discussed before. g(x)"
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Table 12 The number of ttM events N, the ratio NK /N (see text), and the attainable 1- limits on Im dE, for various p -cuts with R 2 (s"14 TeV, m "175 GeV and L"10 fb\. Table taken from [310] R p -cuts (GeV) 2
N (;10)
NK /N
Im dE (;10\g cm) R Q
0 20 40 60 80 100
2.62 2.55 2.36 2.08 1.74 1.41
1.44 1.42 1.37 1.30 1.22 1.14
0.766 0.788 0.847 0.951 1.107 1.313
Fig. 47. Feynman diagrams for the tree-level QCD and neutral Higgs exchanges (denoted by the dashed lines) which contribute to the production density matrix for ggPttM . Diagrams with crossed gluons are not shown.
7.3. 2HDM and CP violation in ppPttM #X In a 2HDM with the CP-violating ttM H couplings in Eq. (3.70), neutral Higgs exchanges can give rise to CP violation in ggPttM and qq PttM at the 1-loop order in perturbation theory. In Fig. 47(c)}(h) all possible 1-loop CP-violating Higgs exchanges in ggPttM are drawn and in Fig. 48(b) the only CP-violating 1-loop diagram for qq PttM is shown. Interference of diagrams (c)}(h) with the SM tree-level diagrams (a) and (b) in Fig. 47 and interference of diagram (b) with diagram (a) in Fig. 48 can then give rise to CP non-conservation e!ects in gg and qq fusion, respectively. One can then identify various CP-violating correlations to trace the resulting CP-odd quantities which appear in the corresponding di!erential cross-sections. Here also we assume that two out of the three neutral Higgs particles in the 2HDM model are very heavy or have very small CP-violating couplings, such that either way their e!ects decouple. Thus, only the couplings of the lightest neutral Higgs (denoted by h) are important and there will be only one dimensionless CP-odd quantity relevant for the study of CP violation in qq , ggPttM . Using the notation in [289,291,312] in conjunction with our parameterization in Eq. (3.70), this quantity is ,!2aFbF , !. R R
(7.13)
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Fig. 48. Born level QCD and relevant neutral Higgs exchange (denoted by the dashed line) Feynman diagrams for qq PttM .
where aF and bF are de"ned by the ttM h (say h"H) coupling in Eq. (3.70) and are functions of tan R R } the ratio between the two VEVs in the Higgs potential and of the three Euler angles which parameterize the Higgs mixing matrix (for details see Section 3.2.3). Below we present two very interesting approaches of probing CP violation in ppPttM #X. The "rst is the Schmidt and Peskin (SP) approach [33], which utilizes the distribution of the leptonic decay products of the top. The second is the Bernreuther and Brandenburg (BB) approach [289,291,312], which studies the CP-violating e!ect in the resonant s( -channel Higgs shown in Fig. 47(h). 7.3.1. Schmidt}Peskin signal Schmidt}Peskin (SP) proposed [33] a signature for CP violation in production and decays of ttM pairs for hadron colliders, namely via the reaction
P l\l P =\bM pp( pN )PttM #X . =>b P l>l P
(7.14)
Despite the complexity of the reaction and the hadronic environment, the signal for CP violation that they suggest, i.e., the lepton energy asymmetry El> ! El\ " # El> # El\
(7.15)
is very simple and robust. Such an asymmetry can only arise from non-SM sources such as an extended Higgs sector or supersymmetry. The size of the asymmetry is unfortunately rather small &10\. Since this asymmetry is CP-odd and ¹ -even, it requires an absorptive part to the Feynman , amplitude. Such an absorptive part is already present (see Figs. 47(c), (g) and (h) and 48(b)) as ttM pair production requires the kinematic threshold s( '4m , (7.16) R where s( is the square of the energy in the subprocess qq or gg c.m. frame. In particular, when a neutral Higgs exchange leads to the CP-violating phase (as in their study), then the absorptive part due to the threshold condition in Eq. (7.16) arises even if m's( . F
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Note again that for the subprocess qq PttM , the underlying cause of CP violation in extended Higgs models is the CEDM of the top quark. Of course given the extremely short lifetime of the top quark (&10\ s) the CEDM as such (i.e., at q"0) is extremely di$cult to be seen. Consider, however, the asymmetry [33]: [N(t tM )!N(t tM )] * * 0 0 , N " *0 all ttM
(7.17)
where N(t tM ) is the number of t tM pairs produced via qq (gg)PttM , etc. Clearly, N is CP-odd * * * * *0 and ¹ -even. The qq contribution to N arises from interference of Fig. 48(b) with the , *0 lowest-order graph for qq PttM depicted in Fig. 48(a). They found 2 Re(F ) , N " *0 3!
(7.18)
where "(1!4m/s( ) and, in their notation, F (s( ) is the CEDM form factor and Re(F ) R involves the absorptive part of the Feynman integral
m s( 1 m 4m R R 1! F ln 1# . (7.19) Re(F )" s( !. s( m 8 v F Here m is the mass of the lightest neutral Higgs and is de"ned in Eq. (7.13). It is easy to F !. understand [33] the e!ect intuitively: for s( <m, the gluon will predominantly couple to t tM or R * 0 t tM . However, when P0, t tM and t tM , which are related to each other via CP, are dominantly 0 * * * 0 0 produced, which may thus lead to N O0. The resulting asymmetry at the parton level, N , *0 *0 for the subprocess qq PttM for m "175 GeV, "( and for di!erent values of m and (s( , is F R !. found to be of order 10\. For ggPttM the calculation is more involved. In particular, in addition to the tK channel h exchange, now an s( -channel Higgs exchange graph is also present (see Fig. 47(h)). There is in fact constructive interference between these two channels for m (2m . The result for F R the asymmetry in the gg fusion case, but without the s( -channel Higgs exchange (see Fig. 48(b)), was also given in [33]. Near threshold, i.e., (s( 92m , the asymmetry in the gg fusion case is R about twice as big as that of the qq fusion case. However, although larger than the qq fusion subprocess, it is again at the level of 10\. Adding the qq and gg subprocesses, then N can reach *0 optimistically &10\, for low values of m and tan . In any case, the gg initial state gives rise to F a much richer possibility of CP-violating operators and, as was mentioned before, the resulting asymmetry cannot be attributed merely to the CEDM of the top quark. Indeed, as noted in [289,291,312], the s( -channel neutral Higgs exchange that was ignored in [33], can give rise to larger asymmetries in ggPttM and may be attainable at the LHC. We will return to this e!ect in the next section. As has been emphasized at several places in this review, the fact that top decays are a powerful spin analyzer comes in extremely handy here too in leading to a detectable signature. The CP Note that in this notation, tM means an anti-top quark with momentum, for instance, along #z and spin along !z. *
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violation in the production process causes the polarization asymmetry above, which leads to an asymmetry in the energies of the charged leptons emerging from t and tM decays. The distribution of the charged lepton in the t-rest frame is given by d 1#cos d " , 2 dEl dcos dEl
(7.20)
where is the angle between the top spin and the lepton momentum. When the top quark is boosted to the qq (gg) c.m. frame, Eq. (7.20) provides the correlation between the helicity of the top and the energy of the decay lepton. The resulting energy spectrum for t (tM ) and t (tM ) is * 0 0 * signi"cantly di!erent from each other as was shown in [33]. Clearly, their "ndings indicate that the energy spectrum of the leptons, serves as a useful spin analyzer. The asymmetry in pp collision is calculated by folding in as usual the parton distributions. For this purpose SP used the parton density functions proposed in [313]. The e!ects of the longitudinal boost of the parton}parton collision are eliminated by considering the transverse energy (E ) of the 2 leptons. The resulting asymmetry is [33]: d/dE l> !d/dE l\ 2 2 (7.21) N(E )" 2 d/dE l> #d/dE l\ 2 2 and it was calculated in [33] for m "100 GeV, m "150 GeV and "( . Unfortunately, F R !. numerically it is again only of order 10\. Let us brie#y discuss the background for these type of CP-violating asymmetries in ppPttM #X. As was mentioned at the beginning of this section, the initial state (pp) at hadron Supercolliders, such as the LHC, is not an eigenstate of CP. Consequently energy asymmetry in the decay lepton spectrum are not necessarily due to CP violation. The point is that the protons in the initial state produce more energetic quarks than anti-quarks. Also the reaction qq PttM has a small forward}backward asymmetry induced by corrections. Thus, the top quarks produced by this Q reaction tend to have a slightly higher energy than tM , leading to an asymmetry in the energy of the decay lepton. Such an e!ect, originating from higher order QCD corrections, causes an irreducible background. Fortunately this background is very small. First of all, qq annihilation is subdominant at such pp collider energies and the leading reaction ggPttM is free from such a forward}backward asymmetry. Also, as mentioned before, the background to the asymmetry arises from higher order (QCD) radiative corrections. Furthermore, since the forward}backward asymmetry mainly a!ects longitudinal variables, its e!ect on the transverse energy asymmetry in Eq. (7.21) would cancel if there were no lepton acceptance cuts. This background can be crudely estimated from the electromagnetic analog of the forward}backward asymmetry for e>e\P>\. The analogous asymmetry is crudely estimated by the replacement P[(d?@A)/32] "(5/12) . SP in [33] used the approximate formula in [314] Q Q which allows them to get an estimate for massless ttM pairs. This approximation tends to overestimate this background. For numerical estimates SP also impose a cuto! on the gluon energy of E/E"0.3. The resulting background was found to be of the order of 10\. Therefore, it is much smaller than the desired CP-violating e!ect and also it is essentially independent of the lepton energy.
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7.3.2. s( -channel resonance Higgs ewects } Bernreuther}Brandenberg approach For m '2m , as noted before, there is an interesting s( -channel Higgs contribution to ggPttM F R shown by diagram (h) in Fig. 47. This was explored in some detail by Bernreuther and Brandenberg [289,291], who recently improved their analysis in [312]. For the simple on-shell decay hPttM , a large ttM spin}spin correlation can be induced already at the tree-level if h is not a CP eigenstate, as happens in a class of 2HDMs. In a 2HDM, with the ttM h coupling of Eq. (3.70), this spin}spin correlation is given by [289] !. R , (7.22) kK ) (s ;s M )" R R R (bF)#(aF) R R R where "(1!4m/m) , s , s M are the spin operators of t and tM , respectively, kK is the unit vector R R R F R R of the momentum of the top quark and is de"ned in Eq. (7.13). It is remarkable that this !. CP-violating spin}spin correlation can, in principle, be as large as 0.5. In practice, though, this decay has to be coupled to some particular production process and the "nal asymmetry can vary signi"cantly between di!erent processes. Moreover, for pp collisions, there is an interference between the continuum and the resonant ttM production which tends to diminish the spin}spin correlation. For the gluon}gluon fusion, the CP-violating expectation value of kK ) (s ;s M ) was R R calculated in [291]. The resulting asymmetry was found to be at best only a few percent and falls signi"cantly short compared to the value of 0.5 mentioned above. In fact, when this is translated to an asymmetry that utilizes the t and tM decay products, as was done in [289,291], the signal-to-noise ratio for such an asymmetry was found to be at best &10\. The same non-vanishing spin}spin correlation of Eq. (7.22) can arise in qq PttM . The asymmetry for the qq fusion subprocess was also calculated in [291] for the same set of parameters as in the gg fusion case. As expected, in the case of qq fusion, the asymmetry is about one order of magnitude smaller than gg fusion, since in this channel the resonant Higgs graph is absent. Furthermore, the asymmetry gets smaller with growing Higgs-boson masses as opposed to the gg fusion case which we now discuss in some detail. Let us now focus on an improved analysis of the results mentioned above. This was recently suggested by Bernreuther, Brandenburg and Flesch (BBF) in [312]. In their analysis the basic idea was to include new cuts on the ttM invariant mass which signi"cantly improved their previous results in [291]. For the case when both t and tM decay leptonically, consider the CP-violating observables [312]: Q "kK ) q( !kK M ) q( R > R \
(7.23)
and Q "(kK !kK M ) ) (q( ;q( )/2 , (7.24) R R \ > where kK , kK M are the t, tM momentum directions in the parton c.m. system and q( , q( are the l>, l\ R R \ > momentum directions (from tPbl>l and tM PbM l\l ) in the t and tM rest frames, respectively. Note that the decay channels to l>, l\ (disregarding the lepton) may include all di!erent combinations A more detailed analysis of the possible spin}spin correlations in hPttM is given in Section 6.2.3. Recall that the lightest neutral Higgs is assumed to be h"H, i.e., k"1 in Eq. (3.70).
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of e and and, in [312], all possible combinations were summed over. It is useful to note that Q in Eq. (7.24) is, in fact, a transcription of the spin}spin correlation kK ) (s ;s M ) de"ned in Eq. (7.22) R R R above, and is a ¹ -odd quantity. Also, Q traces the spin}spin correlation kK ) (s !s M ) which , R R R corresponds to the CP asymmetry N de"ned in Eq. (7.17) and was suggested originally in [33]. *0 Clearly, it is ¹ -even, thus requiring an absorptive phase. , For the leptonic}hadronic decay channel of the ttM pairs, it is useful to consider the two possible decay scenarios: sample A in which the t decays leptonically and tM decays hadronically, and sample A M in which the t decays hadronically and tM decays leptonically (see Eqs. (6.79) and (6.80), respectively, in Section 6.2.3). One can then de"ne the following CP-violating quantities with respect to samples A and AM [312]: E " O A ! OM AM , E " O A # OM AM , where
(7.25) (7.26)
O ,kK ) q( , OM ,kK M ) q( , R > R \ O ,kK ) (q( ;q( M ), OM ,kK M ) (q( ;q( ) . (7.27) R > @ R \ @ Here q( and q( M denote the momentum direction of the b and bM jets in the t and tM rest frames, @ @ respectively. Again, E e!ectively corresponds to the spin}spin correlation kK ) (s !s M ) and is R R R a ¹ -even observable, and E traces the spin}spin correlation kK ) (s ;s M ) and is therefore a ¹ -odd , , R R R quantity. In [312] it was shown that, in the region m '2m , the magnitude of the asymmetry increases F R signi"cantly and the dominant contribution comes from the interference of the CP-violating terms in the amplitude of the neutral Higgs resonant diagram in Fig. 47(h) with the Born amplitude. They have evaluated the dependence of the di!erential expectation values of Q and Q on the ttM invariant mass, M M . An example of such a dependence for the ¹ -odd observable Q and in the , RR dilepton decay channels of the ttM pairs is shown in Fig. 49. In this "gure the resonant contribution in ggPhPttM is compared with the resonant#the remaining h contribution (i.e., all the nonresonant graphs), for di!erent values of m , for (s"14 TeV and setting the CP-violating quantity, F , from the ttM h vertex to be equal to 1. Also, since the CP-violating e!ect is sensitive to the neutral !. Higgs total width, it is therefore sensitive to the ZZh, ==h `reduceda coupling cF (de"ned in Eq. (3.71) for h"H, i.e., k"1), which determines the decay rates (hPZZ) and (hP==) [312]. In Fig. 49, cF"0 was chosen, in which case the above decay channels of a neutral Higgs to the massive gauge-bosons are forbidden at tree-level. Clearly, looking at Fig. 49, the non-resonant contributions are negligible with respect to the s( -channel h contribution which in turn gives rise to a CP asymmetry at the level of a percent when M M is in the vicinity of m . F RR The sharp peaks observed in the range M M &m give an extra handle in an attempt to enhance F RR the CP signal. Indeed, Figs. 49(b)}(d) show that, in the case of m '2m , Q changes sign as one F R goes from M M : m to M M 9 m , such that integrating over M M will diminish the CP-violating RR F RR F RR e!ect. Therefore, choosing appropriate M M mass bins below or above m , allows for a signi"cant F RR enhancement of the CP-odd signal. This is demonstrated in Table 13, where the three values cF"0, 0.2 and 0.4 were considered (see discussion above). Also, in Table 13, the left column gives
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Fig. 49. d Q /dM as a function of M (Q is de"ned in Eq. (7.24)) at (s"14 TeV, for reduced Yukawa couplings RR RR cF"0, aF"( , bF"!( , i.e., "1 (see text), and Higgs}boson masses: (a) m "320 GeV, (b) 350 GeV, (c) 400 GeV, R !. F R and (d) 500 GeV, in the dilepton channel. The dashed line represents the resonant and the solid line the sum of the resonant and non-resonant h contributions. Figure taken from [312].
the expectation value of Q in percent and the right column shows the statistical signi"cance in which this CP e!ect can be measured at the LHC with an integrated luminosity of 100 fb\. The M M intervals (i.e., mass bins) in Table 13 where chosen below m (see caption of Table 13). Also the F RR rows in Table 13 correspond in descending order to "1, 0.3, 0.09. !. The numbers for the statistical signi"cance of the CP-violating signal that were found in [312] and are shown in Table 13 are quite remarkable. In most cases, the CP-violating signal is well above the 3- level, perhaps even a lot better than 5- in the best case. As an example, note that with m "370 GeV and as low as 0.09, the observable Q can yield a 7- e!ect with an F !. appropriately chosen interval for M M . Recall that values as large as &4, corresponding to !. RR tan :0.5, are still allowed by present experimental data (see Section 3.2.3). A few additional remarks are in order regarding the analysis in [312]: 1. The expected statistical signi"cance for a CP-odd signal from the observable E in Eq. (7.26) corresponding to the leptonic}hadronic channel (i.e., lepton # jet from the ttM pairs) was found to be comparable to that of Q discussed above. Moreover, for the ¹ -even observables ,
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Table 13 The expectation value of Q and its sensitivity at the LHC with (s"14 TeV and an integrated luminosity of 100 fb\, for the dilepton ttM decay channels. The M M intervals are chosen below m such that: for m "370, 400, 500 GeV, Q was F F RR integrated over M M "15, 40, 80 GeV, in the M M ranges 355}370, 360}400 and 420}500 GeV, respectively. For each pair RR RR (m , cF) the "rst column is Q in percent and the second column is the sensitivity in standard deviations. The rows F correspond, in descending order, to (aF, bF)"(1,!1)/(2, (1,!0.3)/(2 and (0.3,!0.3)/(2, i.e., to "1, 0.3 and 0.09, R R !. respectively. Numbers for m are in GeV. The non-resonant h contributions have been neglected for these values of m . F F Table taken from [312]
m F
cF 0.2
0.0
0.4
370
4.4 3.9 1.2
29.8 23.4 6.6
4.1 2.9 0.75
27.4 16.7 4.1
3.3 1.6 0.39
20.9 9.0 2.1
400
2.3 1.3 0.49
24.4 13.4 4.9
2.1 1.1 0.35
22.8 11.1 3.5
1.8 0.75 0.21
18.7 7.5 2.1
500
0.65 0.31 0.14
8.6 4.1 1.9
0.59 0.26 0.10
7.9 3.5 1.4
0.46 0.18 0.06
6.0 2.4 0.77
Q (see Eq. (7.23)) and the corresponding one for the leptonic}hadronic channel E (see Eq. (7.25)) the CP-violating signal, although somewhat smaller, may yield more than a 3- e!ect as long as 90.3. !. 2. Apart from the cuts on the M M invariant mass, i.e., the chosen intervals/mass bins, Ref. [312] RR employed additional cuts on the rapidities of the t and tM and on the transverse momenta of the "nal state charged leptons and quarks in both the dilepton channel and leptonic}hadronic channel samples. 3. It is important to note that the ¹ -odd observables Q and E are insensitive to CP violation , from the subsequent t, tM decays to leading order in the CP-violating couplings. This is ensured by CPT invariance [312]. Moreover, the ¹ -even observables Q and E , although may acquire , contributions from CP-violating absorptive parts in the t, tM decays, at least in the 2HDM case, these absorptive parts are absent in the limit of vanishing b-quark mass (see also related discussion in Section 5.1.2). Therefore, for the 2HDM case, both the ¹ -even and the ¹ -odd , , quantities in Eqs. (7.23)}(7.26) are `cleana probes of CP violation in the production mechanism of ttM at the LHC. Refs. [289,291] considered possible contaminations to an asymmetry of the type Q (or equivalently the spin}spin correlation in Eq. (7.22)). Again, the key point is that the dominant gg fusion subprocess is free from undesired CP-conserving background to Q . Therefore, background considerations are relevant only for the case of qq PttM . Refs. [289,291] estimated the CPconserving background to Q to be of order 10\ which is about 3 orders of magnitude smaller than the actual asymmetry. The reason for that is that ¹ -odd observables such as Q do not ,
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receive contributions from CP-invariant interactions at the Born level but only from absorptive parts. Thus, in the case of qq PttM , the main background comes from order and Q Q absorptive parts [289,291]. Finally, let us note that the optimization technique (with no additional cuts), i.e., the use of optimal observables, employed in [309] for CP violation in ggPttM , yield roughly the same results as those obtained in [312]. That is, optimal observables can be sensitive to values down to &0.1 with no cuts on the ttM invariant mass. !. 7.4. SUSY and CP violation in ppPttM #X As we have discussed in previous sections, 1-loop exchanges of SUSY particles may give rise to CP-violating phenomena in top systems which are driven by SUSY CP-odd phases in the supersymmetric vertices. Such SUSY CP-violating 1-loop e!ects in ggPttM were investigated by Schmidt [231]. In the case of ggPttM , only exchanges of gluinos and stops are relevant and are shown in Fig. 50. The only CP-odd phase arises then from the o! diagonal elements of the tI !tI * 0 mixing matrix. Writing again (see also Sections 3.3.2 and 4.5) the top squarks of di!erent handedness in terms of their mass eigenstates, tI , tI , as > \ tI "cos tI !e\ @R sin tI , R > * R \ (7.28) tI "e @R sin tI # cos tI , R \ R > 0 the asymmetry will then be proportional to the quantity R "sin(2 ) sin( ) . (7.29) !. R R Schmidt neglected possible CP-non-conserving e!ects in the subdominant process qq PttM and considered the asymmetry N de"ned in Eq. (7.17) only for ggPttM . As mentioned before, N *0 *0 being CP-odd and ¹ -even, requires both a CP-odd phase and an absorptive phase. Such , absorptive phases are present in diagrams (a), (b) of Fig. 50 if the c.m. energy is large enough to produce on-shell gluino (g ) pairs and in diagrams (c)}(e) in Fig. 50 if the c.m. energy of the colliding gluons is su$cient to produce on-shell top squark (tI ) pairs. Obviously, if SUSY particles have masses of O(1 TeV), then this condition is satis"ed at the LHC in which the c.m. energy of the colliding protons is 14 TeV. Thus, N is a sum of the two contributions *0 d cos [AE (cos )#ARI (cos )] , (7.30) N , *0 (all ttM )
Fig. 50. 1-loop Feynman diagrams contributing to CP violation in ggPttM in supersymmetry. Diagrams with crossed gluons are not shown.
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where is the production angle of the top quark in the gg c.m. frame and AE (ARI ) is the contribution from on-shell gluino (stop) pairs. Schmidt found 9 Q R (!1)N ((s( !2m ) AE (cos )" E 64s( N 1 R (1! )K\!(1!)K\# sin (K\!K\) ; E R R E 1! cos R E 1 R (1! )K>!(1!)K># sin (K>!K>) # E R R E 1# cos E R
, (7.31)
and 9 Q R (!1)N ((s( !2m ) ARI (cos )" N 64s( N 1/81 10/81 R (1!)KM \! sin KM \ # ; N R N 1! cos 1# cos R N R 10/81 1/81 R (1! )KM >! sin KM > , # # (7.32) N R N 1# cos 1! cos R R N where "(1!4m/s( ) and the index refers to the two mass eigenstates of the stop. Also, here m m , E R R , (7.33) Q s( !. R and the form factors K! (i"0}3) are given in [231]. The asymmetry N was calculated for several values of m I \ , m I > (the masses of the two stop *0 R R eigenstates) and m , and for m "150 GeV, R "!1 (i.e., its maximal negative value). In general, E R !. the asymmetry was found to be dominated by the amplitudes which contain the intermediate gluinos (Fig. 50(a) and (b)) even if the intermediate stops in Fig. 50(c)}(e) can go on their mass shell. For example, with m "210 GeV, m I \ "100 GeV and m I > "500 GeV, Schmidt found that the R R E asymmetry at the parton (i.e. gluons) level can reach &2% if the c.m. energy of the colliding gluons is around 450 GeV. Note that, in this c.m. energy, and with the above stops masses, N receives *0 contributions only from diagrams (a) and (b) in Fig. 50 since there is no absorptive cut along the stops lines in diagrams (c)}(e) in Fig. 50. These results for N are about an order of magnitude *0 larger than what was found in the 2HDM case [33] and it is roughly comparable to the s( -channel Higgs resonant e!ect [289,291,312]. However, in a more realistic study, one will have to integrate over the structure functions of the incoming gluons and present asymmetries in the overall reaction ppPttM #X. Unfortunately, folding in the gluons structure functions, the CP-violating asymmetry drops again to the level of 10\. For this purpose Schmidt again considered the asymmetry in the transverse energy of the leptons de"ned in Eq. (7.21). He [231] found that, similar to the 2HDM case when no additional cuts are made (like the ones suggested in [312] and were described in the previous section), at the
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LHC with (s"14 TeV and with typical SUSY masses of a few hundreds GeV, the asymmetry N(E ) of Eq. (7.21) is again typically of the order of &few;10\. Schmidt also examined the 2 non-CP-violating background, and again found it to be negligible compared to the CP-violating e!ect. Clearly, the Schmidt SUSY CP-violating e!ect in ggPttM as it stands, is smaller than that of Ref. [312], i.e. the signal caused by the resonant Higgs contribution. This is mainly due to the appreciable improvement that can be achieved with appropriate cuts on the ttM invariant mass, as was discussed in the previous section. Note however, that by using optimal observables for extracting information on CP violation in ggPttM , it was shown in [309] that the signal to noise ratio for the CP-violating e!ect in ggPttM (driven by the diagrams in Fig. 50 that were considered by Schmidt) can reach the percent level after folding in the gluons structure functions. This allows a 1- detection of the CP-odd e!ect even for smaller values of R &0.1. !. 8. CP violation in pp collider experiments Shortly after the demise of the SSC, it was suggested to upgrade the energy of the Tevatron. More recently, substantial, two stage, upgrades in luminosity without a factor of two or so increase in energy remain as viable options. For a review see [315]. In the previous run at the Tevatron, at c.m. energy of 1.8 TeV, the D and CDF experiments accumulate more than 0.2 fb\ of integrated luminosity. In the "rst upgrade, called Run II, L will be increased from its current peak value to 2;10}10 cm\ s\ (or even to twice this value). In the second stage, so called Run III (or TeV-33), the luminosity will be further increased to L+10 cm\ s\. The working hypotheses are that in Runs II and III the integrated luminosity will be 2 and 30 fb\, respectively, with a modest increase of c.m. energy to 2 TeV. In addition, the D and CDF detectors are also being upgraded. 8.1. pp PttM #X Contrary to the LHC pp collider, where ttM pairs are produced predominantly through the gg fusion process ggPgPttM , in the Tevatron pp collider with c.m. energy of (sK2 TeV the main production mechanism of ttM pairs is the qq fusion, qq PgPttM . In particular, the qq fusion process is responsible for about 90% of the cross-section pp PttM #X. Being so, the processes pp PttM , ttM g#X, where g stands for an extra gluon jet in ttM production, will presumably be sensitive to the CEDM and CMDM of the top quark, which can be incorporated as e!ective interactions at the ttg vertex. As already mentioned in previous sections, being a CP-odd quantity, a nonvanishing CEDM coupling might give rise to observable CP violation in top systems in such a hadron collider. If so, this will be a clue for new physics, as in the SM the CP-non-conserving e!ects in the reactions pp PttM , ttM g#X are extremely small. In principle, CP-non-conserving e!ects due to the CEDM can be searched for through a study of either CP-even or CP-odd correlations in the reactions pp PttM , ttM g#X. Of course, CP-odd correlations are expected to be more sensitive to the CEDM than CP-even observables as the former are linear in CEDM whereas the latter are proportional to the square of the CEDM form factor.
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We will "rst present a study of the sensitivity of some CP-even observables to the CEDM, and then describe some interesting CP-odd correlations that may be applied to pp PttM #X. 8.1.1. CP-even observables in pp PttM #X and pp PttM #jet#X The e!ective Lagrangian for the interaction between the top quark and a gluon, that includes the CEDM and CMDM form factors of the top, is given in Eq. (6.106) ( for <"g). Recall that the e!ective interaction at the ttgg vertex, absent in the SM, is also required to ensure gauge invariance (see discussion in Section 2.5). The CMDM and CEDM dimensionless form factors and , which are de"ned in Eq. (6.106), E E can develop an imaginary part. However, these imaginary parts vanish at zero momentum transfer, and are only present if an on-shell intermediate state exists. Using form factors as a probe for new physics is most useful when the novel states can only be produced virtually and so here we consider the case where and are purely real. E E As mentioned above, the process pp PttM #X can proceed at the parton level via: (a) qq PttM , or (b) ggPttM . With the e!ective interactions in Eq. (6.106) and the additional ggtt e!ective vertex, the Feynman diagrams contributing to these two processes are shown in Fig. 51. The parton level cross-sections for pp PttM #X are then given by [35,316]
8 1 1 v d( M Q !v#z! # ( ! )# ( # ) , OO RR " E 4 E E E dtK 9s( 2 4z E d( EE RRM " Q 12s( dtK
4 !9 v
z 1 !v#2z 1! ! 1! E E v 2 2
(8.1)
7 1 1 5 # ( # ) (1! )# 1# E E z E 4 E 2v 2 #
1 1 1 4v ( # ) ! # # E 16 E z v z
,
(8.2)
where m (tK !m)(u( !m) R R . z" R , v" s( s(
(8.3)
The process pp PttM j#X, where j stands for a jet, can proceed via the following parton level subprocesses: (a) qq PttM g, (b) ggPttM g, or (c) q(q )gPttM q(q ) where an extra light quark jet is radiated. The number of diagrams for these subprocesses is large. A detailed description of the calculation of the cross-section for the reaction pp PttM j#X is given in [316]. A plot of the dependence of the total cross-section for ttM production on and was given in E E [316] and is shown in Fig. 52. Of course, as expected, the cross-section is symmetric about "0 E as only appears in the cross-section, being a CP-even observable. The SM point is given by E We will loosely refer here to and as the CMDM and CEDM form factors, respectively. They are related to the E E dimensionful CMDM and CEDM via Eqs. (6.107) and (6.108).
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Fig. 51. Feynman diagrams for the process (a) qq PQQM , and (b) ggPQQM with Q"t. The heavy dots represent the e!ective vertices involving and (see also text). E E
Fig. 52. Contours of the pp PttM #X cross-sections, in pb, in the (horizontal axis)} (vertical axis) plane. Figure taken E E from [316].
" "0 and the SM cross-section at this point is (ttM )&5 pb. Therefore, a measurement of E E (ttM )'5 pb will indicate the existence of a non-zero or . E E In [317] an `anomalous cross-sectiona was de"ned as ,( , )!(0, 0) , E E
(8.4)
and a plot of in the ( , ) plane was given. From the current experimentally allowed region of E E , not surprisingly, they found that a rather large CEDM coupling is permitted. For example, for
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Fig. 53. Contours of the ratio R"(ttM j)/(ttM ) in the (horizontal axis)} (vertical axis) plane, with p (j)'5, 10 and E E 2 20 GeV in (a), (b), and (c), respectively. Figure taken from [316].
certain values of , they found that 0.5: :1.5 is allowed, within 1. This corresponds to E E 2.8;10\g cm:Re dE:8.3;10\g cm, at 1!. Q R Q Considering again an associated extra jet, j, in top pair production at the 2 TeV Tevatron, it was shown in [316] that not much will be gained from a simple cross-section analysis of the ttM j "nal state compared to the ttM "nal state. However, a more interesting CP-conserving quantity was further suggested in [316]; it was shown that the ratio of the two cross-sections, R"(ttM j)/(ttM ), can be used to further constrain the CEDM and CMDM couplings. Fig. 53(a), (b) and (c) show the main behavior of the ratio R in the ( , ) plane for p (j)'5, 10, 20 GeV, respectively, where p (j) E E 2 2 is the transverse momentum of the jet j. The most interesting feature is the one depicted in Fig. 53(a); we see that for the cut p (j)'5 GeV, at some regions in the ( , ) plane, R'1. This 2 E E happens as a consequence of the new ttg and ttgg vertices (absent in the SM) in tt#jet production.
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A discussion on the possible limits that can be put on the CEDM and CMDM of the top by analyzing CP-even observables in single top production at the Tevatron was presented in [318]. There, single top production can occur either via the qq PtbM or the =-gluon fusion subprocess =gPtbM . An analysis of the cross-section for the reaction =gPtbM was performed [318], trying to further constrain the CEDM and CMDM of the top quark which may enter the ttg vertex present in this reaction. However, it was found that, for both the Tevatron and the LHC, the sensitivity of the =-gluon fusion subprocess leading to a single top quark for the CEDM and CMDM couplings is more than an order of magnitude smaller than the usual top pair production channel. 8.1.2. CP-odd observables in pp PttM #X As already mentioned before, if all non-standard e!ects reside in the ttM g coupling, then they can be parameterized by the e!ective ttg and ttgg interaction Lagrangian. However, upon neglecting the gg fusion process (which is a good approximation under the conditions of the Tevatron upgrade with c.m. energy of (s"2 TeV), the ttgg contact term plays no role and the only relevant e!ective interaction for the reaction qq PttM is the e!ective ttg CEDM interaction in Eq. (6.106). Similarly, the tb= vertex (with an on-shell =-boson) may give rise to additional non-standard couplings as in Eqs. (6.5) and (6.6), which may cause CP violation in the top and anti-top decays. Therefore, in the overall ttM production and decays, CP violation is parameterized by non-zero values of the CEDM, dE, in the production vertex, and/or by the quantities f * 0!fM 0 * (de"ned in Eqs. (6.5) and R (6.6)) emanating from the t, tM decays. In order to detect such CP-violating couplings one has to construct appropriate CP-odd observables. Following [319], let us again consider two decay scenarios for t and tM . In the "rst one, both t and tM decay leptonically pp PttM Pl>l\X ,
(8.5)
while in the second scenario only one of the t or tM decays leptonically (see also Eqs. (6.79) and (6.80) in Section 6.2.3) pp PttM Pl>X, pp PttM Pl\X .
(8.6)
The processes in Eq. (8.6) have better statistics than the one in Eq. (8.5) and give the best signature for the top quark identi"cation. Within these decay scenarios two possible CP-odd observables were considered in [319] which we will describe below. 8.1.3. Transverse energy asymmetry of charged leptons The transverse energy asymmetry of the charged leptons was originally suggested by Peskin and Schmidt in [33] for ttM production in a pp collider (see Eq. (7.21)) and was discussed in detail in Sections 7.3 and 7.4. Recall that in the case where both t and tM decay leptonically it can be de"ned as (using here the notation in [319]) (E\'E>)!(E>'E\) 2 2 2 . A " 2 2 (E\'E>)#(E>'E\) 2 2 2 2
(8.7)
Recall that this is not necessarily true in model calculations, e.g., in the MSSM } see discussion in Section 7.4.
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The result for the expected CP-violating asymmetry in the transverse energy of the muons (AI ) was 2 given in [319]. They considered AI as a function of the imaginary part of the top CEDM, Im dE 2 R (recall that A is ¹ -even thus requiring an absorptive phase), and of Re( f 0!fM * ). Also, the usual 2 , CDF cuts were applied. They found that CP violation from the production mechanism, i.e., JIm dE, is larger then that arising from the decay process, i.e., JRe( f 0!fM * ). For example, they R found that with Im dE&10\g cm and Re( f 0!fM * )&0.2 one can obtain an asymmetry around R Q the &10% level. In order to understand the feasibility of extracting such values for the CP-violating couplings in production and decays of the ttM , it is useful to decompose AI as follows [319]: 2 AI ,c A #c A , (8.8) 2 . . " " where the dimensionless couplings c and c are . " 1 m (8.9) c , R Im dE, c , Re( f 0!fM * ) . R " 2 . g Q Then, in terms of c and c , the statistical signi"cance for AI determination is given by . " 2 N2 ,c A #c A (Nll , (8.10) 1" . . " " where Nll is the number of dilepton events expected to be &80 and &1200 at an integrated luminosity ¸"2 and 30 fb\, respectively [320]. A and A can be calculated and, thus, it was . " found in [319] that a 3- e!ect will require the following relations to be satis"ed 2.5c #0.9c 51 for ¸"2 fb\ , (8.11) . " 9.8c #3.3c 51 for ¸"30 fb\ . (8.12) . " Clearly, AI is more sensitive to CP violation in the production mechanism than in the ttM decays. So, 2 for example, ¸"30 fb\ allows for an observation of c "c "0.08 at (s"2 TeV. Note that . " c "0.08 corresponds to Im dE"8.8;10\ which, again, is more than an order of magnitude . R larger than what is expected for the CEDM in beyond the SM scenarios such as MHDMs and SUSY (see Section 4). Similarly, the resulting 3- limit Re( f 0!fM * )"0.16 (corresponding to c "0.08) falls short by about one order of magnitude from model predictions for this quantity (see " discussion in Section 5). A related asymmetry which can be used in the case when only one top decays leptonically was also suggested in [319]: \(E\'E )!>(E>'E ) 2 2 2 2 . AI (E )" 2
(8.13)
The in Eq. (8.13) denotes the integrated cross-section with no cuts except for the standard experimental cuts. We note that, with the CP-violating coupling c and c of the order of 0.1, which . " tends to be somewhat optimistic, this asymmetry in Eq. (8.13) can also reach the &10% level. 8.1.4. Optimal observables It is useful to be able to experimentally separate CP violation in the production from that in the decay. The optimization method outlined in Section 2.6 can provide for such a detection and was also used in [319].
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Consider the transverse lepton energy spectrum in the single leptonic (say l>) and the dileptonic "nal states 1 d "f>(E>)#c f>(E>)#c f>(E>) , (8.14) 2 . . 2 " " 2 dE> 2 1 d "f!(E>, E\)#c f!(E>, E\)#c f!(E>, E\) , (8.15) 2 2 . . 2 2 " " 2 2 dE> dE\ 2 2 where in Eqs. (8.14) and (8.15) denotes the cross-section for the process pp Pl>#jets and pp Pl>l\#jets, respectively. Also, c , c are de"ned in Eq. (8.9) and f ! are known functions of . " E> and E\. 2 2 As can be seen from Eqs. (8.14) and (8.15), the transverse lepton energy spectrums, in both the single- and double-leptonic channels, are sensitive to c and c . Using the above transverse lepton . " energy spectrum, the optimal weighting functions can be obtained. This was done in [319] where in both cases the statistical signi"cances for the experimental determination of c and c , i.e., . " N. ",c /c , were calculated. For the single-leptonic events they obtained 1" . " . " c c (8.16) N. " . (Nl , N" " " (Nl , 1" 18.43 1" 2.37 and for the dileptonic events c c N. " . (Nll , N" " " (Nll , 1" 1.17 1" 5.76
(8.17)
where Nl and Nll are the expected number of single- and double-leptonic events, respectively. Nl &1300(20,000) and Nll &80(1200) for ¸"2(30) fb\, respectively [320]. The minimal values of c and c necessary to observe a 3- CP-violating e!ect with the optimization technique are . " listed in Tables 14 and 15 for the single- and double-leptonic channels, respectively. We see that the single-leptonic modes are more sensitive to the non-standard couplings. Thus, for example, the Tevatron upgrade Run III with ¸"30 fb\ will be able to probe Im dE down to values of R &5;10\g cm in the single-leptonic channel. Note that, as expected, this result is somewhat Q better than what can be achieved with `naivea observables such as AI in Eq. (8.13). We note in passing that comparable limits for Im dE but also for Re dE, i.e., Im, R R Re dE&few;10\g cm, were found also in [309] using optimal observables for the reaction R Q pp PttM #X and with the Tevatron upgrade parameters. Table 14 The minimal values of c and c necessary to observe CP violation in the single-lepton mode at the 3- level for . " ¸"2, 30 fb\. As a reference value, recall that c "1 corresponds to Im dE"1.1;10\g cm. Table taken from [319] . R Q ¸(fb\)
2
30
c . c "
0.20 1.50
0.05 0.40
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Table 15 The minimal values of c and c necessary to observe CP violation in the dilepton mode at the 3- level for . " ¸"2, 30 fb\. See also caption to Table 14. Table taken from [319] ¸(fb\)
2
30
c . c "
0.39 1.93
0.10 0.50
To summarize, as was noted in [319], it is not inconceivable that non-standard CP-violating couplings of the top quark to a gluon may be discovered at the Tevatron before precision measurements at the LHC are done. 8.2. pp PtbM #X In spite of the fact that at the Tevatron pp collider, top quarks are mainly produced as ttM pairs via an s-channel gluon exchange [321], the subleading electroweak production mechanism of a single top forms a signi"cant fraction of the ttM pair production. It will therefore be closely scrutinized in the next runs of the Tevatron [322}329] (see also [315]). The production rate of tbM through an s-channel o!-shell =-boson, pp P=HPtbM #X, (the corresponding partonic reaction, udM P=HPtbM , is depicted in Fig. 54) is expected to yield about 10% of the ttM production rate [322}329]. In this section, we examine CP violation asymmetries in top quark production and its subsequent decay via the basic quark level reactions [215,258,330]: udM PtbM Pb =>bM ,
u dPtM bPbM =\b .
(8.18)
Indeed this reaction is rich for CP violation studies as it exhibits many di!erent types of asymmetries. Some of these, which we consider below, involve the top spin. Therefore, the ability to track the top spin through its decays becomes important and top decays have to be examined as well (see e.g. [331]). The asymmetries in tbM production can be appreciably larger than those in ttM pair production wherein they tend to be about a few tenths of percent (see Sections 6 and 7). Moreover, while in the SM, CP-odd e!ects in pp P=HPtbM #X are expected to be extremely small since they are severely suppressed by the GIM mechanism (see e.g., [64,65,235]), it is shown below that, in extensions of the SM, CP asymmetries can be sizable } in some cases at the level of a few percent. Therefore, since the number of events needed to observe an asymmetry scales as (asymmetry)\, the enhanced CP-violating e!ects in tbM (tM b) may make up for the reduced production rates for tbM compared to ttM . In fact larger asymmetries could be essential as detector systematim cs can be a serious limitation for asymmetries less than about 1%. Let us discuss now the asymmetries in the udM (u d) subprocess. We consider four types of asymmetries that may be present [258]. First, is the CP-violating asymmetry in the cross-section (8.19) A "( M ! M )/( M # M ) , R@ R@ R@ R@ where M and M are the cross-sections for udM PtbM and u dPtM b, respectively, at s( "(p #p M ). R @ R@ R@
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Fig. 54. The tree-level Feynman diagram contributing to udM PtbM .
The spin of the top allows us to de"ne three additional types of CP-violating polarization asymmetries. For these it is convenient to introduce the coordinate system in the top quark (or anti-top) rest frame where the unit vectors are e J!p , e Jp ;p and e "e ;e . Here X @ W S @ V W X p and p are the 3-momenta of the bM -quark and the initial u-quark in that frame. We denote the @ S longitudinal polarization or helicity asymmetry as A(z( )"(N !N !NM #NM )/(N #N #NM #NM ) , (8.20) 0 * 0 * 0 * 0 * where N is the number of left-handed top quarks produced in udM PtbM and NM is the number of * 0 right-handed tM produced in u dPtM b, etc. Therefore, in the frame introduced above right-handed tops have spin up along the z-axis and left-handed ones spin down. We further de"ne the CP-violating spin asymmetries in the x and y directions as follows: !N #NM !NM )/(N #N #NM #NM ), V> V\ V> V\ V> V\ V> V\ (8.21) A(y( )"(N !N !NM #NM )/(N #N #NM #NM ) , W> W\ W> W\ W> W\ W> W\ where, for example, N (NM ) represent the number of t(tM ) with spin up with respect to jK -axis, for H> H> j"x, y, etc. While all these four asymmetries are manifestly CP-violating, A , A(z( ) and A(x( ) are even under naive time reversal (¹ ) whereas A(y( ) is ¹ -odd. So the "rst three, unlike A(y( ), require a complex , , amplitude, i.e., absorptive phases. These asymmetries are related to form factors arising from radiative corrections of the =HPtb production vertex due to non-standard physics. To see this, it is useful to parameterize the 1-loop tbM and tM b currents of the production amplitude as follows [215]: A(x( )"(N
P. pI g @ #P. I Pv , (8.22) JIR@M ,i 5 u @ (2 .* 0 R mR PM . pI g @ #PM . I Pv , (8.23) JIRM @,i 5 u R (2 .* 0 @ mR where P"¸ or R and ¸(R),(1!(#) )/2. P* 0 and P* 0, de"ned in Eqs. (8.22) and (8.23), contain the necessary absorptive phases as well as the CP-violating phases in a given model. It is easy to show that if one de"nes P* &e BQ ;e BU , P* &e BQ ;e BU ,
(8.24) (8.25)
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where , are the CP-even absorptive phases (i.e., FSI phases) and , are the CP-odd phases, Q Q U U then (8.26) PM 0&!e BQ ;e\ BU (8.27) PM * &e BQ ;e\ BU . In terms of these form factors, the two ¹ -even asymmetries A and A(z( ) are given by [258]: , (x!1) A " Re(P* #PM 0)!Re(P* !PM * ) , (8.28) 2(x#2) x!2 (x!1) Re(P* #PM 0)! Re(P* !PM * ) , A(z( )" x#2 2(x#2)
(8.29)
where x,m/s( . Notice that the three quantities A , A(z( ), A(x( ) are linear combinations of only R two form factors. Thus, one can show that A(x( )"!3x\ [(2#x)A #(2!x)A(z( )]/32 (8.30) which will therefore hold if the new CP violating physics takes place through such a vertex correction of =HPtb. Similarly, the asymmetry A(y( ) is proportional to the real parts of the 1-loop integrals and may therefore be obtained from the corresponding imaginary parts through the use of dispersion relations. In particular, since the ¹ -even asymmetries are proportional to the absorptive phases in , the above form factors, one can express A(y( ) in terms of A and A(z( ), 1!x 2#x 3 Re A(y( )[s( ]"! (!x)(!1#i ) 32 (2#x)(x
;[(2!x)A [s( ]#(2#x)A(z( )[s( ]] d .
(8.31)
Note that the integrand is 0 if s( is below the threshold to produce an imaginary part since then A and A(z( ) will vanish. Let us now evaluate the form factors de"ned in Eqs. (8.22) and (8.23) in two extensions of the SM: the 2HDM of type II and the MSSM. As was pointed above, once these form factors are calculated in a given model, the asymmetries A , A(z( ), A(x( ) and A(y( ) can be readily obtained. 8.2.1. 2HDM and the CP-violating asymmetries As emphasized throughout this article, in the 2HDM of type II, a CP-odd phase can reside in neutral Higgs exchanges and there is only one Feynman diagram that contributes to CP violation in udM P=HPtbM to 1-loop order. This diagram is shown in Fig. 55. The relevant Feynman rules for this diagram, required for calculating the asymmetries of interest, can be extracted from the parts of the 2HDM Lagrangian involving the ttM HI and ==HI couplings in Eqs. (3.70) and (3.71), with k"1, 2, 3 for the three neutral Higgs "elds. Recall again that the coupling constants, aI, bI and R R cI are functions of tan , which is the ratio between the two VEVs in this model, and of the three mixing angles which diagonalize the 3;3 Higgs mass matrix (see Section 3.2.3). As usual, for simplicity, we have assumed that two of the three neutral Higgs-bosons are much heavier
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Fig. 55. The CP-violating 1-loop graph in 2HDMs with CP violation from neutral Higgs exchanges; h denotes the lightest neutral Higgs.
compared to the third one which we denote by h. Thus, the CP-violating e!ect will be dominated by the lightest neutral Higgs, h, and is proportional to bFcF. Choosing the angles " "/2 and R "0 which give maximal e!ects [258], one obtains bFcFJ cos cot so the asymmetries are now R functions of tan and m only. F Using the Lagrangian in Eqs. (3.70) and (3.71), the form factors Re(P* #PM 0) and Re(P* !PM * ) can be readily calculated and, we get
bFcF m R R Im[2m (C !C ) Re(P* #PM 0)"! 5 (2 2 sin 5 m5 !m(C #C )!CI !CI ] , (8.32) R bFcF m R R Im[2m (C #C )!2C ] , Re(P* !PM * )" (8.33) 5 4 sin m (2 5 5 where the C above, x30, 11, 12, 21, 22, 23, 24, are the three-point form factor associated with the V 1-loop diagram in Fig. 55, and are given via [258,330]:
C "C (m, m , m, m, s( , m) , (8.34) V V R 5 F @ R where s( "(p #p M ) and C (m , m , m , p , p , p ) is de"ned in Appendix A. V R @ The quark level asymmetries of interest can be converted to the hadron (i.e., pp ) level by folding in the structure functions in the standard manner [91]. The results for the 2HDM case are shown in Fig. 56, for tan "0.3 and as a function of the lightest Higgs-boson mass, m . For the asymmetry F A(y( ) we apply a cut of s( '(m #m ). We can see that A and A(x( ) can reach above the percent F 5 level for m :200 GeV. The measurable consequences for such an asymmetry are discussed in F Section 8.2.3 below. It is interesting to note that in the 2HDM (to the 1-loop order) the ¹ -even asymmetries A , A(z( ) , and A(x( ) do not receive any contribution from the decay vertex in tPb=. The only diagram that can potentially drive CP violation in tPb= is the same one as shown in Fig. 55 with the momenta of the t and the = reversed. Thus, an important necessary condition for A , A(z( ), A(x( )O0, that there is an imaginary part in the decay amplitude, is not satis"ed. Moreover, as it turns out, the observed value of A(y( ) is not a!ected by CP violation in the decay process. The key point is that the measurement of A(y( ) through the decay chain u(p ) dM (p )PbM (p ) t(p ) followed by S B @ R t(p )Pb(p ) e>(p ) (p ) is equivalent to a measurement of a term proportional to (p , p , p , p ) R @ C J C B R @ ( being here the Levi-Civita tensor). On the other hand, CP violation arising from the decay process is proportional to (p , p , p , p ). It is easy to see that an observable related to the "rst of C B R @
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Fig. 56. The CP-violating asymmetries in pp PtbM #X for the 2HDM case and in the pp c.m. frame for (s"2 TeV, as a function of m (horizontal axis); A (solid), A(z( ) (dashed), A(x( ) (dotted) and A(y( ) (dot-dashed). Figure taken from [258]. F
these will be insensitive to the second. So, in this way asymmetries in the production can be separated from those in the decay. 8.2.2. SUSY and the CP-violating cross-section asymmetry The MSSM possesses several CP-odd phases that can give rise to CP violation in pp PtbM #X and in the subsequent top decay tPb= (see Section 5.1.4). In [215] we have constructed a plausible low-energy MSSM framework in which there are only "ve relevant free parameters needed to evaluate the cross-section asymmetry A (see also Sections 3.3.4 and 5.1.4). These are: M a typical SUSY mass scale that characterizes the mass of the heavy squarks, m the mass of the 1 % gluino, the Higgs mass parameter in the superpotential, m the mass of the lighter stop and tan J the ratio between the VEVs of the two Higgs "elds in the theory. In this framework there are two sources of CP violation that can potentially contribute to A [215]. The "rst may arise from the Higgs mass parameter which may be complex in general. The second CP-violating phase arises from tI !tI mixing (see e.g., Eq. (5.60)) and may be parameterized by a single quantity R de"ned * 0 !. in Eq. (5.61). In this scenario, when no further assumptions are made, there are 12 Feynman diagrams that can give rise to CP violation at the parton level process udM PtbM which are depicted in Fig. 57. However, if we assume that arg()"0 as implied from the existing experimental limit on the NEDM (see discussion in Section 3.3.4) and take m "m "m "0, then only diagrams (a), (b), (d), (e) and (g) S B @
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Fig. 57. CP-violating, SUSY induced, 1-loop diagrams for the processes udM PtbM . g is the gluino, is a chargino, is a neutralino and tI and bI are top and bottom squarks, respectively.
have the necessary CP-violating phase, being proportional to one quantity } R . In [215] we have !. shown that with M and m of about several hundred GeV, diagram (d) is, in fact, responsible for 1 % &90% of the total CP violation e!ect. Let us, therefore, present the results for the asymmetry A corresponding only to diagram (d) in Fig. 57. The Feynman rules needed to calculate A can be derived from the following parts of the SUSY Lagrangian [185]:
L K "u >dM S BQ
(2m (2m S ZGZ>H R# B ZGZ\ ¸ <SB A #h.c. , !g ZGZ>H# 5 S K S K S K K v v
L L "u \ S SQ L
g (2m S ZGHZL ¸ ! 5 ZGH¸>! S S , v (2
(8.35)
(2m 2(2 S ZGHZLH R u#h.c. , g tan ZGHZLH! (8.36) , , S S 5 5 3 v (8.37) L K L "g I(K\¸#K>R) =>#h.c. , L I 5Q Q 5 K where ¸(R)"(1!(#) ), u and u (dI and d) stand for up squark and up quark (down squark H and down quark), respectively, (m"1, 2) and (n"1!4) are the charginos and neutralinos, K L respectively. Also, in Eqs. (8.37) and (8.36) we have de"ned #
¸!, tan ZL$ZL , 5 , ,
(8.38)
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1 K>,ZLHZ\ # ZLHZ\ , , K (2 , K
(8.39)
1 K\,ZLZ>H! ZLZ>H , , K (2 , K
(8.40)
and the mixing matrices Z , Z , Z , Z\ and Z> were given in [215] (also given in Section 3.3.2). S B , Using the Lagrangian in Eqs. (8.35)}(8.37), the form factors Re(P* #PM 0) and Re(P* !PM * ) (corresponding to diagram (d) in Fig. 57) can be evaluated within the MSSM and are given by m [m O Im(CB #CB )#m L O Im CB Re(P* #PM 0)"! R R B Q B sin 5 (8.41) ! m K O Im(CB !CB )] , Q B 1 [O((s( !m) Im CB !s( Im CB Re(P* !PM * )" B R 2 sin 5 ! 2 Im CB !m Im CB ) R ! m m L O Im CB #m m K O Im(CB #CB ) R Q B R Q B (8.42) # m K m L O Im CB ] , Q Q B where s( "(p #p M ) and O contain the SUSY CP-weak phases associated with diagram (d) in R @ B Fig. 57. In fact, the same CP-violating phases occur also in the decay tPb= (see Section 5.1.4) and, therefore, the O above are the same as the ones given in Eqs. (5.50)}(5.53). The Im CB , B V x30, 11, 12, 21, 22, 23, 24, in Eqs. (8.41) and (8.42) are the imaginary parts of the three-point form factors associated with diagram (d) in Fig. 57. Thus, CB are given via [215]: V (8.43) CB "C (mI , m K , m L , m, s( , m) , Q @ R V V R Q and C (m , m , m , p , p , p ) is de"ned in Appendix A. V In [215], instead of calculating the cross-section asymmetry A , we considered a partially integrated cross-section asymmetry, A.'! , in which we have imposed a cut on the tb invariant mass, m (350 GeV. Such a cut on m may help to remove the ttM `backgrounda from a measureR@ R@ ment of a cross-section asymmetry in pp PtbM #X. The results in the SUSY case are shown in Figs. 58 and 59 as a function of and m , respectively, for M "400 GeV, m "50 GeV (m is the % 1 J J mass of the lighter stop particle) and for tan "1.5, 35. Maximal CP violation was chosen in the sense that R "1, thus the asymmetry plotted in Figs. 58 and 59 is, in fact, given in units of R . !. !. Evidently, for some values of around 100 GeV and with m &450 GeV the asymmetry can % almost reach the 3% level. The asymmetry is above the 1% level for several other choices of . It was also shown in [215] that, in general, in order for the asymmetry to be above the percent level the mass of the lighter stop is required to be below &75 GeV. Furthermore, the asymmetry tends to drop as tan is increased in the range 1:tan :10, and it is almost insensitive to tan for tan 910. In the MSSM, 1-loop radiative corrections to the amplitude of top decay tPb= that can violate CP are also present. In fact, disregarding the incoming u and d lines, diagrams (a)}(d) in Fig. 57 with
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Fig. 58. The SUSY-induced partially integrated cross-section asymmetry A.'! de"ned in the text, as a function of , for M "400 GeV, m "50 GeV and for (s"2 TeV. With (a) tan "1.5 and (b) tan "35. Figure taken from [215]. 1 J
the t and = momenta reversed, can give rise to a CP-violating tb= decay vertex. This was discussed in some detail in Section 5.1.4. To 1-loop order in perturbation theory, where the CP-violating virtual corrections enter only either the production or the decay vertices of the top in the overall reaction pp PtbM #XP=>bbM #X, and in the narrow width approximation for the decaying top, an overall CP asymmetry, A, can be broken into A"A #A , . "
(8.44)
In Eq. (8.44) A and A are the CP asymmetries emanating from the production and decay of the . " top, respectively, and are de"ned by (pp PtbM #X)!(pp PtM b#X) , A , . (pp PtbM #X)#(pp PtM b#X)
(8.45)
(tP=>b)!M (tM P=\bM ) A , . " (tP=>b)#M (tM P=\bM )
(8.46)
The PRA A , de"ned in Eq. (8.46), does not depend on the speci"c production mechanism of the " top and was calculated in Section 5.1.4. We have shown there that, with the low-energy MSSM parameters described above, one gets A (0.3% where, in most instances, it tends to be even "
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Fig. 59. The SUSY-induced partially integrated cross-section asymmetry A.'! de"ned in the text, as a function of m , % for several values of , M "400 GeV, m "50 GeV and for (s"2 TeV. With (a) tan "1.5 and (b) tan "35. Figure 1 J taken from [215].
smaller } A (0.1%. Therefore, it is about one order of magnitude smaller than the asymmetry in " the production of tb and its relevance to the overall asymmetry in pp PtbM #XP=>bbM #X is negligible. As a "nal remark here, let us recall that in the 2HDM case discussed before, the PRA A in " Eq. (8.46) is forbidden at 1-loop order because of CPT invariance, i.e., no rescattering of "nal states as shown in Section 2.3. This was also discussed in Section 5.1.2. 8.2.3. Feasibility of extraction from experiment To summarize the results of Sections 8.2.1 and 8.2.2, CP-violating asymmetries in single top production and decay at the Tevatron through pp PtbM #XP=>bbM #X may optimistically reach a few percent in extensions of the SM such as SUSY and 2HDMs. In future upgrades of the Tevatron to (s"2 TeV, the cross-section for pp PtbM #X is expected to be about 300 fb, if a cut of m (350 GeV is applied on the invariant mass of the tbM [215]. Therefore, with an integrated R@ luminosity of L"30 fb\ [315,322}329], an asymmetry of &3% can be naively detected with a statistical signi"cance of 3-. Therefore, a percent level CP-violating signal in the reaction pp PtbM #XP=>bbM #X is especially notable as it may become accessible at the near future 2 TeV pp collider. In particular, based on the results presented in this section, such a measurement
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may impose limits on the CP-violating parameters arg(A ) and bFcF of the MSSM and the 2HDM, R R respectively. However, it should be noted that such a detection at the Tevatron will require the identi"cation of all tbM pairs, which, in principle, can be achieved only if the top can be reconstructed even when the = decays hadronically. It will also be useful to explore SUSY or 2HDM mediated CP-violating e!ects that can originate from the =-gluon fusion subprocess which contributes to the same "nal state (i.e., =gPtbM d) and which has a comparable production rate to that of the simple udM PtbM in the 2 TeV Tevatron. While in the MSSM various 1-loop triangle and box corrections can give rise to CP nonconservation in the =-gluon fusion subprocess, in the 2HDM CP-violating radiative corrections to =g fusion, at the 1-loop order, do not yield absorptive parts in the limit m "0. Therefore, @ it will not contribute to CP asymmetries of the ¹ -even type in single top production. Note, , however, that the =-gluon fusion subprocess has its own characteristics, e.g., the extra light jet in the "nal state, which may be used in order to experimentally distinguish it from the `simplea ud fusion process (see e.g., [324]). 8.3. pp PtbM h#X, a case of tree-level CP violation Motivated by the large, tree-level, CP-violating e!ects found in the reaction e>e\PttM h (see Section 6.2), we were led to consider an analogous reaction in the Tevatron pp collider with a tbM h "nal state [330]. Thus, in this section we focus on CP violation, driven by 2HDM in the process pp PtbM h#X, where h is the lightest neutral Higgs in the 2HDM of type II. From the outset we remark that a statistically signi"cant CP study in the reaction pp PttM h#X in a future Tevatron upgrade with c.m. energy of 2 TeV and even 4 TeV, is unlikely due to the low tbM h event rate. As in the case of e>e\PttM h, a very interesting feature of the reaction udM PtbM h (at the parton level) is that it exhibits a CP asymmetry at the tree graph level. Such an e!ect arises from interference of the Higgs emission from the t (but not from the bM in the limit m P0) @ with the Higgs emission from the =-boson. Being a tree-level e!ect the resulting asymmetry is quite large. This asymmetry may be measurable, in principle, through a CP-odd, ¹ -odd , observable. Let us now discuss the tree-level cross-section and CP violation e!ects in our reactions, u(p )dM (p M )Pt(p )bM (p M )h(p ), u (p )d(p )PtM (p M )b(p )h(p ) . R @ F S B R @ F S B
(8.47)
In the limit m "0 (and also m "m "0), and to lowest order, the only two diagrams that can @ S B contribute to CP violation in the reactions of Eq. (8.47) are depicted in Fig. 60. The relevant Feynman rules needed to calculate the tree-level CP asymmetry are extracted from the Lagrangian in Eqs. (3.70) and (3.71). Here also, we assume that two of the three neutral Higgs particles are much heavier than the remaining one, i.e. h. We, therefore, omit the index k in Eqs. (3.70) and (3.71), and denote the couplings for the lightest neutral Higgs as: aF, bF and cF. The tree-level di!erential R R cross-section K at the parton level, is a sum of two terms: the CP-even and CP-odd terms K and > K , respectively, \ K ,K #K , > \
(8.48)
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Fig. 60. Tree-level Feynman diagrams contributing to udM PtbM h in the limit m "m "m "0; h stands for the lightest B S @ neutral Higgs in a 2HDM.
where K are calculated from the tree-level diagrams in Fig. 60. The expression for the CP-even ! part, i.e., K , can be parameterized as >
2 m 5 R ; (aF)A#(bF)B K " > R R R R sin 2 m 5 5
m # 2(cF) 5 C#2 aFcF D , R R 5F m 5F R
(8.49)
where the terms A, B, C and D are quite involved and were calculated in [330]. is the 5 =-boson propagator, and together with and are given by R 5F 1 , , 5 s( !m 5
1 , R 2p ) p #m R F F
1 , . 5F m!m #2p ) p M R 5 R @
(8.50)
Furthermore, p,p #p M is the s( -channel 4-momentum at the quark level, and s( is de"ned to be S B s( "p. The CP-violating piece of the tree-level di!erential cross-section is [330]:
2 m R bFcF; (p M , p M , p , p );( f!s #w) , K "2 \ @ B R S R sin m 5 5F R R 5 5
(8.51)
where s ,(p #p M ), f,(p !p M ) ) (p #p M ), w,(p #p M ) ) (p #p M ) and is the Levi-Civita S B R @ S B R @ R R @ tensor. For illustration, we adopt here also the value tan "0.3. We fold in the structure functions of the u and the dM inside the p and p , respectively, and plot in Fig. 61 the tree-level cross-section for pp PtbM h#X, with c.m. energies of (s"2 and (s"4 TeV. Four possible sets of the Higgs coupling constants aF, bF and cF were chosen. For illustrative purposes, the "rst two sets of R R parameters which are chosen for a 2 TeV collider are: set I with tan "0.3, , , " /4, /2, 3/4 and set II with tan "0.3, , , "/2, /2, 0. The other two, chosen for a 4 TeV collider are: set III with tan "0.3, , , "/4, /2, /2 and set IV with tan "0.3, , , "/2, 3/4, 3/4. The general feature of these sets is that sets I and III give rise to a large CP asymmetry but `smalla cross-section, while sets II and IV increase the event
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Fig. 61. The cross-section for the reaction pp PtbM h#X (in fb), for (a): (s"2 TeV and for sets I (solid line) and II (dashed line), (b): (s"4 TeV and for sets III (solid line) and IV (dashed line). For the de"nition of the sets I}IV, see text.
rate but decrease the asymmetry. Also, note that each set by itself is not unique, as there are other values of the angles , and for each set which lead to the same e!ect. As in the reaction e>e\PttM h discussed in Section 6.2, we are dealing here with a tree-level CP-violating e!ect. Thus, the CP-violating term K can probe only CP asymmetries of the ¹ -odd \ , type. In this case, the "nal state is not a CP eigenstate. Therefore, one has to construct a ¹ -odd, , triple correlation product (or equivalently, a Levi-Civita tensor) which takes into account the conjugate reaction as well (u dPtM bh), thus endowing the observable with de"nite CP properties. This led us to consider the following CP-odd, ¹ -odd observable: , O"( (p , p M , p , p )# (p , p , p M , p ))/s . S @ R F S @ R F
(8.52)
Fig. 62 shows the results for the signal-to-noise ratio, i.e., the asymmetry A , O/( O, for (s"2 TeV with sets I and II and for (s"4 TeV with sets III and IV. Evidently, the asymmetry A is of the order of 10}15% for a light Higgs particle in the mass range 50 GeV(m (100 GeV F and of the order of 20}30% for a heavy Higgs particle with mass in the range 200 GeV(m (250 GeV, for both set II (which corresponds to a 2 TeV collider) and set IV (which F we chose for the 4 TeV collider). Sets I and III give asymmetries of the order of a few percent. Although with these sets, i.e., sets I and III, the cross-section can be 10 times larger than that corresponding to sets II and IV, the statistical signi"cance of the CP-violating e!ect that can be achieved when the free parameters of the 2HDM are chosen according to sets I and III is much smaller than that with sets II and IV. This is simply due to the fact that the number of events needed to detect the CP-violating e!ect scale as (asymmetry)\. Therefore, the enhanced e!ect for sets II and IV makes up for the reduced production rate in those scenarios.
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Fig. 62. The asymmetry, A (see text) as a function of m for (a): (s"2 TeV with sets I (solid line) and II (dashed line), F (b): (s"4 TeV with sets III (solid line) and IV (dashed line). For the de"nition of the sets I}IV, see text.
Let us proceed by analyzing the two scenarios which give the large asymmetries. For the reaction at hand, the statistical signi"cance N of the CP-odd signal in the collider is 1" N "(L((pp PtbM h#X);A , 1" -
(8.53)
where L is the collider luminosity. From Fig. 61 we see that (pp PtbM h#X)&0.1}10 fb, depending on the parameters aF, bF and cF and the neutral Higgs mass. Thus, since A &0.1}0.3 in R R the best cases, it is evident from Eq. (8.53) that, typically, an integrated luminosity of about &100 fb\ will be required to be able to observe a statistically signi"cant e!ect in this reaction. Therefore, although the CP asymmetry in this process could reach the 10}30% level, it is, unfortunately, not likely to be able to detect a CP-violating signal in the next runs of the Tevatron with L"2 fb\ and even with 30 fb\.
9. CP violation in collider experiments Future electron}positron colliders include the attractive option of a linear collider, where each beam of photons is produced by Compton backscattering of laser light on an electron or positron beam. The peak energies and luminosities of the are expected to be slightly smaller than those of the corresponding e>e\ collider. The idea was originally suggested in [332}335]; for recent reviews see [336,337]. The attractive option of obtaining polarized photon beams is also being considered. Note also that collisions have been discussed in the context of heavy ion colliders (for a review see [338]) as well. Unfortunately, the invariant mass reach for the LHC (running in its heavy ion
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mode), will only be about 100 GeV, with lower values attainable at RHIC [338]. Therefore, this option will not be discussed here. 9.1. PX: general comments In a collider there are two distinguishable modes: unpolarized and polarized incoming photons. In the unpolarized case, in order to be able to detect CP violation in the reaction PttM , one needs information on the spins of the t and the tM , or equivalently, one needs to construct asymmetries involving the decay products of the top quark. In this case, including the subsequent decays of the t and tM , one can break the di!erential cross-section for the process (k , ) (k , )Pt(p , s )tM (p M , s M ) , R R R R to its CP-odd and CP-even parts as d,d#d .
(9.1)
(9.2)
In Eq. (9.1) , denote the helicities of the incoming photons , , respectively, and s , s M are R R the covariant spins of t, tM , respectively. In general, the CP-odd terms in Eq. (9.2) has the form (9.3) dJ q ) (s ;s M )# q ) (s !s M ) , \ R R > R R where q is a three momentum of any of the particles in the "nal or the initial state (i.e., q"k , k , p or p M ) and s (s M ) is the spin three vectors of the t(tM ). and are non-zero only if R R R R \ > there is CP violation in the underlying dynamics of the process PttM . Furthermore, is \ proportional to the dispersive, CP-odd, ¹ -odd contributions, while gets its contribution from , > absorptive, CP-odd, ¹ -even terms. Of course, as mentioned above, the CP-odd polarization , correlations of the top and the anti-top in Eq. (9.3) will lead to CP-odd correlations among the momenta of the decay products of the t and the tM . Asymmetries which involve the top decay products in the case of the unpolarized photons were investigated in [261,339]. If the incoming photons are polarized, then one can construct CP-odd correlations by using linearly polarized photons where no information on the momenta and polarization of the top quark decay products are needed. The amplitude squared for a general "nal state X, i.e., PX, in the case where the two photons are linearly polarized is given by [257] ![ cos (# )# cos(! )] Re( ) # [ sin(# )! sin(! )] Im( ) ! [ cos(# )! cos(! )] Re( ) # [ sin(# )# sin(! )] Im( ) # cos(2) Re( )# sin(2) Im( ) # cos(2 ) Re( )n# sin(2 ) Im( ) .
d( , ; , )"
(9.4)
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Here , are the degrees of linear polarization of the two initial photons, and are the azimuthal angle di!erence and sum, respectively, and the invariant functions are de"ned as [257] 1 " [M #M #M #M ] , 4 >> >\ \> \\ 6
(9.5)
1 " [M (MH #MH )#(M #M )MH ] , 2 >> >\ \> >\ \> \\ 6
(9.6)
1 " [M (MH !MH )!(M !M )MH ] , 2 >> >\ \> >\ \> \\ 6
(9.7)
1 " (M MH ) , >\ \> 2 6
(9.8)
1 " (M MH ) . 2 >> \\ 6
(9.9)
The subscripts 0 and 2 in Eqs. (9.6)}(9.9) represent the magnitude of the sum of the initial photon helicities and the notation for the helicity amplitudes for the reaction ( ) ( )PX using Eqs. (9.5)}(9.9) is M
H H
" XM .
(9.10)
Furthermore, the event rate of any "nal state production through fusion can be written, in general, as [340] dN"dL GH dGH , (9.11) AA G H where G(H) are the so called Stokes polarization parameters for ( ) with ""1. In particular, and are the mean helicities of and , respectively, and (()#() are their degrees of linear polarization. Also, dL is the luminosity of the two photons and GH are the AA corresponding cross-sections. There are only three CP-odd functions out of the nine invariant functions in Eqs. (9.5)}(9.9): Im( ), Im( ) and Re( ). While Im( ) and Im( ) are ¹ -odd, Re( ) is ¹ -even. , , A CP-odd asymmetry can be formed at a collider if, for example, the J"0 amplitudes of two photons in the CP-even and CP-odd states are both non-vanishing: M[PX(CP"#)]J ) , M[PX(CP"!)]J( ; ) ) k ,
(9.12)
where and are the polarizations of the two colliding photons and k is the momentum vector of one photon in the c.m. frame. Such an asymmetry can be constructed, for example, by taking
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Fig. 63. Lowest-order Feynman diagrams for the process I (k )J (k )Pt(p )tM (p M ). and are Lorenz indices. R R
the di!erence of distributions at "$/4 [257,341]:
d d ! d Im( ) d d " Q Q\ . A , (9.13) d d # d d d Q Q\ Alternatively, in terms of event rates which correspond to the (0, 0) and the unpolarized initial photon}photon states this reads N A " . N
(9.14)
9.2. PttM and the top EDM Recall that a top EDM, i.e., dA, modi"es the SM ttM coupling to read R "ie #idA (p #p M )J , (9.15) I I R IJ R R and CP-violation arising due to this EDM of the top can be studied in the reaction PttM of Eq. (9.1). The relevant lowest-order Feynman diagrams for PttM are shown in Fig. 63 wherein the top EDM can be folded into any of the ttM vertices in those two diagrams. We note however again that, in general, the CP-violating e!ects in PttM cannot necessarily be all attributed to the top quark EDM. For example, in a 2HDM (see next section) additional box diagrams can give rise to CP-non-conserving terms in the amplitude of the reaction PttM . With the notation M( , , , M ) for the amplitude, where , , and M correspond to the R R R R helicities of the two incoming photons, the top quark and the top anti-quark, respectively, the non-vanishing helicity amplitudes for the process in Eq. (9.1), obtained for combinations such as " " or "! " and M " or M "! , are given by [257,341,342] R R R A A R m M( , , , )"!4C R ( # ) A A R R R (s A R R
s ! idA 2m 2# ( ! ) sin R R R A R 4m R R R s 4m R # ( ! ) sin # (dA) A R 2 R R A R R s
,
(9.16)
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M( , , ,! )"!4m C A A R R R R ; sin cos [ idA!m (dA)] , R R R A R R R m M( ,! , , )"4C R A A R R R (s
(9.17)
s s ; #idA !(dA) sin , R R R 2m R R 2 R R R R
(9.18)
M( ,! , ,! )"2C sin ( # cos ) A A R R R R R A R R
s 4m R cos # (1! cos ) ! (dA) R A R R R R 2 s
,
(9.19)
where is the scattering angle in the c.m. frame and is the top quark velocity and R R C "eQ/(1! cos ). R R R R The CP-odd ¹ -odd distribution Im( ) de"ned in Eq. (9.9) depends linearly on dA and is given , R by [257] Im( )"24(1! cos ) Re(dA) , (9.20) R R R and the asymmetry A de"ned in Eq. (9.14) can then be calculated [257,341]. After extracting the top EDM from A and de"ning A ,Re( )AI , where ,2m dA/e is a dimensionless EDM A A R R form factor, as in Eq. (6.108), one obtains the allowed sensitivity (i.e., N " number of standard 1" deviations) to the dispersive part, Re( ), in the case that no asymmetry is found A (2N 1" Max(Re( ))" , (9.21) A AI ( N where is the detection e$ciency which was taken to be 10% in [257]. The kinematics of the Compton backscattering process at hand is characterized in part by the dimensionless parameter x,2p ) p /m. Larger x values are favored to produce highly energetic photons but the degree of C A C linear polarization is larger for smaller x values (for more details see [257]). In particular, the denominator on the r.h.s. of Eq. (9.21) depends on x, which for a given c.m. energy squared, s, is bounded by 2m R 4x42(1#(2) , (9.22) (s!2m R for the process PttM where the upper bound is required to prevent e>e\ pair production in the scattering of the photon while the lower bound is required to have photons energetic enough to produce top pairs. In [257] the x dependence of the Re( ) upper bound, i.e. Max(Re( )), was given which is A A shown in Fig. 64 for two c.m. energies (s"0.5 and 1 TeV. We see that Max(Re( )) can reach A below 0.1 where the optimal sensitivities are obtained with x"3.43 and x"0.85 for (s"0.5 and
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Fig. 64. The x dependence of the Re( ) upper bound, i.e., Max(Re( )), at (s"0.5 TeV (solid line) and 1 TeV (dashed A A line), from the asymmetry A . Figure taken from [257].
1 TeV, respectively. For these x values, and to 1-, the upper bounds that can be achieved are: Re( )"0.16 and Re( )"0.02 for (s"0.5 and 1 TeV, respectively. This corresponds to A A Re(dA)+0.9;10\ and 0.1;10\e cm for (s"0.5 and 1 TeV, respectively. R Di!erent type of asymmetries which involve the polarization of both the initial photons beams and the decay products of the t and tM (e.g., tPbll ) in the reaction PttM , were suggested in [342]. The "rst one is a charge asymmetry, A , which measures the di!erence between the number of leptons and anti-leptons produced as decay products of the top and anti-top, respectively. The second, A , is $ a sum of the forward}backward asymmetries of the leptons and anti-leptons and requires polarized laser beams. In terms of the di!erential cross-sections these asymmetries are given by \F dl (d>/dl !d\/dl ) A ( )" F . \F dl (d>/dl #d\/dl ) F
(9.23)
dl (d>/dl #d\/dl )! \F dl (d>/dl #d\/dl ) , A ( )" F $ \F dl (d>/dl #d\/dl ) F
(9.24)
and
where d>/dl and d\/dl refer to the l> and l\ distributions in the c.m. frame, respectively, and is a cuto! on the polar angle of the lepton. It is important to note that if there is no CP violation in the top decays, then the charge asymmetry is zero in the absence of the cuto! . Both A and A are ¹ -even asymmetries, thus they probe the imaginary part of the top EDM, $ , Im(dA). In [342], the case where one of the t or tM decays leptonically and the other decays R hadronically, was studied. Also, it was assumed that no CP-violation enters these top decays. The asymmetries A and A were then evaluated in the c.m. frame and 90% C.L. limits on the top $ Then the asymmetries correspond to samples A and A M de"ned in Eqs. (6.79) and (6.80) in Section 6.2.3.
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EDM, in the case that no asymmetry is found in the experiment, were obtained. The 90% C.L. limits were evaluated for di!erent electron and laser beam energies as well as for di!erent cuto! angles. Also, di!erent helicity combinations of the initial beam and di!erent values of the dimensionless parameter x de"ned before (see Eq. (9.22) and the discussion above), were analyzed. They found that for an electron beam energy of 250 GeV, and for a suitable choice of circular polarizations of the laser photons and longitudinal polarizations for the electron beams, and assuming a luminosity of 20 fb\ for the electron beam, in the best cases and in an ideal experiment, it is possible to obtain limits on the imaginary part of the top EDM, again, of the order of 10\e cm. However, an order of magnitude improvement may be possible if the beam energy is increased to 500 GeV [342]. To conclude this section, the reaction PttM can serve to limit both the real and the imaginary parts of the top EDM. However, it is worth mentioning that the limits that may be placed on the top EDM through a CP study in PttM are roughly comparable to those which might be obtained through a study of the reaction e>e\PttM at the NLC (for comparison see Section 6.1). Therefore, the motivation for going to a collider in order to study e!ects of the top EDM is somewhat arguable. On the other hand, model calculations of CP violation in PttM , such as the one described below, i.e., a 2HDM, show that CP-non-conserving signals in this reaction, which are not necessarily associated with the top EDM, may be sizable; i.e., at the detectable level in a future photon collider. 9.3. PttM and s-channel Higgs exchange in a 2HDM A collider can also provide an interesting possibility for producing an s-channel neutral Higgs-boson, via Ph, which can then decay to a pair of fermions, hP+M (recall that a related process was considered in the context of a pp collider in Section 7.3.2). Once again, h stands for the lightest neutral Higgs-boson in a MHDM and the other neutral Higgs particles are assumed to be much heavier, thus, neglecting their contribution in what follows. The decay of a neutral Higgs, for m '2m , to a pair of ttM will inevitably dominate the other F R fermionic decays of the Higgs due to the largeness of the top mass. CP violation in the reaction PttM was investigated within a MHDM in [339] for unpolarized incoming photons, where the e!ects of the s-channel Higgs were included. In [340] polarized laser beams were considered for the s-channel neutral Higgs production, Ph. In [339], the complete set of CP-non-conserving contributions to PttM , at the 1-loop order, were considered within a 2HDM of type II. Recall that, in the SM, CP violation cannot occur in this process at least to 2-loop order. This set of 1-loop Feynman diagrams is depicted in Fig. 65(b)}(h) and Fig. 65(a) represents the only tree-level diagram for this process; this tree-level diagram and its permuted one are also shown in Fig. 63. We de"ne M to be the amplitude for a diagram i in Fig. 65 (i.e., i"a,2, h) where we further G decompose M to its CP-odd part M and CP-even part M: M "M#M. Then, to G G G G G G leading order, the CP-odd (d) and the CP-even (d) parts of the di!erential cross-section are given by F d"2 Re M M R , ? G A G@
(9.25)
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Fig. 65. Feynman diagrams for PttM . In (a) the Born diagram is shown (see also Fig. 63), and in (b)}(h) the complete set of 1-loop diagrams that can violate CP are depicted. Diagrams with crossed lines are not shown.
F d" M MR# 2 Re M M R . ? ? ? G A GD
(9.26)
Note that the CP-even contribution from the interference of the s-channel Higgs graphs Fig. 65(f)}(h) with the Born amplitude of Fig. 65(a), which is explicitly included in Eq. (9.26), can become important and was taken into account in [339] because of the non-negligible width of the Higgs. Also, the CP-odd interference in Eq. (9.25) will give rise to the simple form of d in Eq. (9.3). Again, to e$ciently trace the CP-odd spin correlations in Eq. (9.3), one de"nes a ttM decay scenario where the t decays leptonically and the tM decays hadronically and vice versa. As in Eqs. (6.79) and (6.80) in Section 6.2.3, we denote by A the decay sample in the case that the top decays leptonically and the anti-top decays hadronically, and by A M the charged conjugate decay sample [261,339]. With these decay scenarios one can evaluate a few CP-odd asymmetries of both the ¹ -odd and , ¹ -even type which may acquire a non-vanishing value only if O0 and O0, respectively (see , \ > Eq. (9.3)). To do so, let us de"ne for sample A, i.e., tP=>bPl>l b and tM P=\bM Pqq bM , the following operators: O "(q( l> ;q( H \ ) ) p( M , 5 R
(9.27)
O "El> ,
(9.28)
O "q( l> ) p( M , R
(9.29)
where the asterisk denotes the t(tM ) rest frame. The corresponding ones for the sample A M are OM "(q( l\ ;q( H > ) ) p( , 5 R
(9.30)
OM "El\ ,
(9.31)
OM "q( l\ ) p( , R
(9.32)
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Thus, the CP-odd asymmetries are constructed as [261,339] ¹ !odd: " O # OM , (9.33) , ¹ !even: " O ! OM . (9.34) , To calculate the above asymmetries one has to fold in the distribution functions of the backscattered laser photons (for more details see [261,339]). Also, one has to choose a de"nite scheme for type II 2HDM couplings aF and bF of a neutral Higgs particles to a pair of ttM in Eq. (3.70). Recall R R that the scalar aF and pseudoscalar bF couplings are functions of the neutral Higgs mixing matrix R R R in Eq. (3.73) and of the ratio between the two VEVs, tan (see Section 3.2.3). In particular, as in HG [261,339], assuming that the other two neutral Higgs of the model are much heavier than h such that their mass lies above the c.m. energy, their contribution is neglected. Also, the mass of the charged Higgs-boson was taken as m ! "500 GeV and R "1/(3 for j"1, 2, 3 was assumed H & (see Eq. (3.73)). Let us present a sample of the results that were obtained in [261,339]. The ¹ -odd asymmetry, , i.e., the signal-to-noise ratio / , is shown in Fig. 66 and the ¹ -even asymmetry ratio / is , depicted in Fig. 67. The asymmetries are plotted for various values of tan . In Fig. 66 tan "0.5 (dashed line) and tan "1 (solid line) were used, while in Fig. 67 tan "0.3 (dashed line) and tan "0.627 (solid line) were chosen. In those "gures an e>e\ collider energy of (s"500 GeV was taken, and the asymmetries were plotted for the above values of the 2HDM free parameters and as a function of the lightest Higgs mass. We see from Figs. 66 and 67 that both the CP-odd ¹ -odd asymmetry, , and the CP-odd , ¹ -even asymmetry, , peak twice. First when the mass of the Higgs is close to the ttM threshold and , then when it is close to the maximal energy. With m +350 or 400 GeV these asymmetries can F
Fig. 66. The ratio / as a function of the lightest Higgs-boson mass, m , at (s"0.5 TeV. The dashed line F corresponds to tan "0.5 and the solid line to tan "1; m "175 GeV. Figure taken from [339]. R
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Fig. 67. The ratio / as a function of the lightest Higgs-boson mass, m , at (s"0.5 TeV. The dashed line F corresponds to tan "0.3 and the solid line to tan "0.627; m "180 GeV. Figure taken from [261]. R
reach above 10%. In particular, would lead to a somewhat higher CP-violating signal and we see from Fig. 67 that, for tan "0.3, / is above the 10% level in almost the entire mass range 100 GeV(m (500 GeV, and peaks around m +2m at 50%. Recall that the statistical F F R signi"cance NG of the CP-violating signal that can be measured with a CP-odd asymmetry is 1" given by (9.35) NG " (N , 1" where N "RA AM ;L is the number of expected events, with RA AM the branching ratios for the decay scenarios A, A M , respectively, and assuming a reconstruction e$ciency of 1. Furthermore, L is the collider integrated luminosity and is the cross-section which, in the leading order, is calculated from d in Eq. (9.26). With L"O(10) fb\ the expected number of PttM events is of the order of few;10 for collider c.m. energies of 500}700 GeV. Thus, for example, if we take RA AM "4/27 such that only leptonic top decays into electrons and muons are considered, then an asymmetry larger than 10% will correspond to a signal-to-noise ratio above the 3- level [261]. As previously discussed, if the polarization of the backscattered photons is adjustable then CP asymmetries involving these polarizations can be constructed and they, in turn, can serve as an e$cient tool for investigating the CP properties of the neutral Higgs-boson. Three such polarization asymmetries were suggested in [340]: M !M \\ , P , >> M #M >> \\
(9.36)
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2 Im(MH M ) \\ >> , P , M #M >> \\ 2 Re(MH M ) \\ >> . P , M #M >> \\
193
(9.37)
(9.38)
In the helicity basis, one can choose the polarization of (moving in the #z direction) as: !"G2\ (0, 1,$i, 0), and that of (moving in the !z direction) as: !"G2\ (0,!1,$i, 0). To understand how the above polarization asymmetries (P ) trace the CP properties of the neutral Higgs, note that a CP-even (CP-odd) scalar couples to two photons via F FIJ(F FI IJ) (see [340] and references therein). Therefore, as implied from Eq. (9.12), in the c.m. IJ IJ of the two photons this will yield a coupling proportional to ) (( ; ) ) for a CP-even X (CP-odd) neutral Higgs to a pair. For the above convention of the polarizations, one "nds i 1 ) "! (1# ), ( ; ) " (1# ) , X 2 2
(9.39)
where , "$1 are the helicities of , , respectively. Now, for a mixed CP state, the general amplitude to couple to will have both the CP-even and the CP-odd pieces in Eq. (9.12) and it can be written as M" ) # ( ; ) , X
(9.40)
where ( ) is the CP-even (CP-odd) coupling strength of the neutral Higgs to the two photons. Using Eq. (9.39), the squares of the helicity amplitudes which appear in P can be readily calculated [340]: M #M "2( # ) , >> \\ 2 Re(MH M )"2( ! ) , \\ >> M !M "!4 Im( H ) >> \\ 2 Im(MH M )"!4Re( H ) , \\ >>
(9.41) (9.42) (9.43) (9.44)
where and are given in [340] for a 2HDM with scalar and pseudoscalar couplings of a neutral Higgs to a pair of fermions. It is then evident that P , P O0 and P (1 only if both , O0. That is, only if both the CP-even and the CP-odd couplings are present. Using Eq. (9.11), for the Higgs-boson production of our interest, one gets [340] dN"dL d(M #M ) >> \\ AA ;[(1# )#( # )P # ( # )P # ( ! )P ] ,
(9.45)
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where d is the appropriate element of the "nal state phase space including the initial state #ux factor. Note that the properties of dL and of the various 's (appearing in Eq. (9.45)) as a function AA of the c.m. energy of the two photons are very important for this discussion as they depend strongly on the polarization of the incoming electrons and associated photons. Instead of presenting a detailed analysis of those parameters and the numerical results, we refer the reader to [340]. We will only give their summary for a general 2HDM in which the CP properties of a single neutral Higgs have to be determined. In particular, it was found in [340] that out of the three polarization asymmetries de"ned in Eqs. (9.36)}(9.38), P provides the best statistical signi"cance for the task at hand. A non-zero value for P requires that the h coupling has an imaginary part, as well as both CP-even and CP-odd contributions. For a mixed CP Higgs-boson with m :2m , a measurement F 5 of P will be easiest if tan is large since the b-quark loop, which makes the only large contribution to the imaginary part for such m values, will be enhanced. For m '2m , the required imaginary F F 5 part is dominated by the =-boson loop (or t-quark loop if m is also '2m ); large tan makes F R detection more di$cult since the dominant CP-odd contribution originates from the t-quark loop, which will be suppressed. To summarize, the production of a neutral Higgs-boson by fusion of backscattered laser beams can provide a systematic analysis of the CP properties of the Higgs particle. In particular, PhPttM would be a promising channel for exploring CP-violating e!ects that can arise from an extended Higgs sector, as for quite a large range of the 2HDM parameter space this reaction can exhibit statistically signi"cant CP-nonconserving signals in a high-energy collider running at c.m. energy of (sK500 GeV. Moreover, if the polarizations of the incoming photons are controlled, then detailed information on both the scalar and the pseudoscalar couplings of the neutral Higgs to a pair of fermions may be extracted by considering polarization asymmetries of the two colliding photons. If the neutral Higgs is a pure CP eigenstate, the polarization asymmetries P and P in Eqs. (9.36) and (9.37) will vanish, while, in Eq. (9.38) P "1 (!1) for a CP-even (odd) neutral Higgs. Therefore, a non-vanishing value for P and P and P (1 will imply the existence of an extended Higgs sector beyond the SM and of CP violation in the scalar potential.
10. CP violation in >\ collider experiments The idea to build a high-energy >\ collider is more than 25 years old [343,344]. It has recently gained interest in part due to the interesting possibility of using it as an `s-channel Higgs factorya. Of course, it may also be suitable for tackling some other physics issues, e.g., SUSY. For recent reviews see [345}350]. The c.m. energies considered range from 100 GeV to 4 TeV or even more, with luminosity comparable or higher than in linear e>e\ colliders. The subject is still in its infancy compared with the more established technologies of linear e>e\ colliders, and of hadronic colliders such as the Tevatron or the LHC. 10.1. >\PttM If there exists a Higgs-boson with mass of a few hundred GeV, a muon collider running at the Higgs resonance can provide the fascinating and unique possibility of an in-depth study of the
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Higgs particle in the s-channel. In particular, the CP-violating properties of its coupling to ttM may be studied via the reaction >\PHPttM ,
(10.1)
where we have generically denoted the neutral Higgs resonance under study by H. It is perhaps surprising that Higgs-bosons can be produced at an appreciable rate at a muon collider. Indeed, the H coupling is very small since it is proportional to the mass of the muon, m . On the other hand, if the c.m. energy of the accelerator can be tuned to be at s"mH , then the I cross-section receives appreciable enhancement due to the resonant production of the Higgs. To see how this works, consider a collider tuned precisely at the Higgs resonance, s"mH , then the cross-section H ,(>\PH) for neutral Higgs-boson production is given by 4 H " B , mH I
(10.2)
where B is the branching ratio of HP>\. It is useful to compare this with the point I cross-section "(>\PHPe>e\) .
(10.3)
Thus H 3 R(H)" " B , (10.4) I where is the "ne-structure constant. Therefore, H and R(H) are enhanced if the neutral Higgs has a narrow width, i.e., a relatively large B "(HP>\)/H . I One simple way to study CP-violation at a muon collider is via the decays HPttM . CP-violating correlations can be studied in the decays of the produced ttM pair. Again, this is possible due to the fact that the weak decays of the top quark are very e!ective in analyzing the top spin (see Section 2.8). 10.1.1. A general model for the Higgs couplings To keep the discussion completely general, we will assume that a single neutral Higgs-boson, H, is under study although the underlying model may contain several Higgs doublets. In practice, of course, the muon collider will only be tuned to one Higgs resonance at a time. In Section 3.2.3, we have written an example of a useful parameterization for the H+M ( f"fermion) interaction (see Eq. (3.70)), taking into account possible CP violation in this vertex due to an extended Higgs sector. It is, however, also instructive to introduce a di!erent notation } somewhat more compact } which is useful for the investigation at hand. Let us therefore parameterize the coupling of H to fermions with the Feynman rule [351]: (10.5) CH "iC e A HD , DD DD D where C "!(g /2)(m /m ) is the coupling in the SM and for each fermion f ( f"l, u, d DD 5 D 5 D where l"charged lepton) is a real constant which gives the magnitude of the coupling in relation to the standard model. The CP nature of the coupling is determined by the value of . In D particular, which is not a multiple of /2 is indicative of CP violation since the coupling will then D
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contain both scalar and pseudoscalar components. CP violation is thus essential in any scalar coupling which is not either pure scalar or pseudoscalar and so learning about is equivalent to D investigating CP violation in HP+M . Moreover, in models with an extended Higgs sector the coupling of H to the boson sector of the theory may be characterized as either scalar, H, or pseudoscalar, A. If H"A then it cannot couple to gauge-bosons while if H"H we can parameterize its coupling to two vector-bosons as (10.6) CH "C cos , 44 4 where, again, C is the coupling in the SM, <"Z or =, given by (see also Eq. (3.71)) 4 C "m g , C "m g /cos , (10.7) 5 5 5 8 8 5 5 and is the angle between the observed Higgs-boson H and the orientation of the vacuum in the Higgs space. The mass eigenstate could also be a mixture of H and A which again would violate CP. This aspect of CP violation in the Higgs sector can lead to enhanced CP violation if the two masses are close together as discussed in Section 10.1.3. Such mixing could also lead to a CP-violating coupling to fermions (Eq. (10.5)) whose implications will be discussed in the following sections. In the following sections, we will consider a number of methods to investigate CP-violating couplings of the s-channel neutral Higgs-boson to top quarks at a muon collider. We "rst consider the reaction >\PttM . Note that in this reaction both initial and "nal states are CP eigenstates. Furthermore, there is no CP-odd observable that one can construct out of the total cross-section (such as a partial rate asymmetry); if one considers angular distributions of the "nal t-quark, such distributions depend only on the angle , i.e., the angle between the \ and the t-quark momenta IR in the c.m. frame. Since cos is a C-even P-even quantity we clearly need more information if we IR are to observe CP violation. Indeed, if the dominant amplitude is mediated by scalar exchange the angular distribution will be isotropic in any case. To construct CP-violating observables we therefore need information about the polarization of the fermions: either the "nal state ttM or the initial state >\. In Section 10.1.2, we consider the use of correlations in the top polarization in >\PHPttM to measure the CP-violating parameter of HttM coupling, i.e., sin 2 , where is the angle in Eq. (10.5). R R In Section 10.1.3, we consider measurement of CP violation in the same reaction (>\P HPttM ) except this time the asymmetry we construct is based on polarized muons. Clearly, to perform such experiments it is necessary to have a muon collider capable of producing muons with a signi"cant polarization. Finally in Section 10.2 we consider the possibility of #avor changing neutral Higgs couplings which could give rise to CP violation in the reaction >\Ptc versus >\PtM c. Large couplings of this sort may be expected in 2HDM of type III which is described in some detail in Section 3.2.2. Here again the use of top and/or muon polarization is essential to obtain CPviolating signals.
We note that in the language of the interaction Lagrangian in Eq. (3.70), O/2 corresponds to having aH, bHO0, D R R where aH(bH) is the scalar (pseudoscalar) H coupling to a ttM pair. R R
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10.1.2. Decay correlation asymmetry Let us now consider the case of a muon collider where unpolarized muons produce ttM through >\PHPttM . Thus, in order to learn about the coupling in Eq. (10.5), we can observe the polarization of the top quarks through their decays [351], which is discussed extensively in Section 2.8. Here we will just consider the determination of top polarization by its correlation with the momentum of a particle in various decay modes. Thus, if X is a decay product of a top decay, we de"ne the `analyzing powera: R ,3 cos , (10.8) 6 6 where is the angle between p and the spin of the top in the top rest frame. 6 6 Let us now extend this idea to study the correlations of the polarizations of the top quarks where the polarizations are indicated by the momenta of speci"c decay particles. As discussed in Section 2.8, for the case of a single polarized top, some further optimization may follow from going beyond this which we do not consider here. Following [351], we work in the rest frame of the Higgs-boson with the t momentum along the z-axis. Let each of the t-quarks undergo a decay which can analyze the top polarization. Let x and G x be the outgoing particles which we wish to correlate with the t and tM polarization respectively; for H instance, the lepton or =!. Also, let y and y denote the rest of the decay products. Thus the two G H decays are tPx y and tM Px y . We can then de"ne the azimuthal angle between p G and G G H H V p H projected on to the x, y plane as V ( p ;p H ) ) p V R . (10.9) sin( )" VG GH p G p H p V V R The azimuthal di!erential distribution of t and tM events is then given by d "1! R R cos 2 cos # R R sin 2 sin . (10.10) R GH 16 G H R R GH d 16 G H R GH The coupling is de"ned in Eq. (10.5) for f"t and R , R are the analyzing power of the decays R G H (de"ned by Eq. (10.8)). Also, , are phase space factors which approach 1 as mH <2m . They are R R R given by 1!!(1#)cos 2 R R R , "! R cos 2 [1#!(1!)cos 2 ] R R R R R , " R 1!(1!)cos R R
(10.11) (10.12)
where "(1!4m/mH . R R We may now de"ne the following CP-violating, ¹ -odd, P-odd, azimuthal asymmetry by , (sin '0)!(sin (0) GH GH AR " . (10.13) GH (sin '0)#(sin (0) GH GH
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From the distribution above, the value for this observable will be AR " R R sin 2 . R GH 8 G H R
(10.14)
For each pair i, j of top decays that can be used to analyze the polarization of the t and tM , one will obtain an experimental value of AR from which one can infer the value of the quantity sin 2 . GH R Clearly, we would like to combine the information from all of the modes together in order to obtain better statistical signi"cance for the determination of sin 2 . To combine the asymmetries from R di!erent pairs of modes i, j in an optimal fashion we form a weighted average of the asymmetries de"ned by a set of weights w with the normalization de"ned by GH w B B "B B , (10.15) GH G H G H where the summation is over the modes under observation and B is the branching ratio of mode i. G The total weighted asymmetry is thus de"ned as AR"w AR B B , (10.16) GH GH G H which is maximized by taking w JAR [29]. With these weights, then, the maximal asymmetry is GH GH (10.17) AR" ( R) sin 2 , R R 8 where R"[B ( R )] . (10.18) G G Now, for the top quark, the branching ratio into electron or muon, B "B & and the analyzing C I power is R & R "1. For decays (of =) into hadrons (i.e., jets), B " with R "0.39. The decays F C I F into have, B " and the analyzing power is taken to be the same as into the jet modes, R "0.39. O O Using these we can then deduce, via Eq. (10.18) RK0.58 .
(10.19)
In order to quantify how well such experiments might detect CP violation, let us de"ne y( N to be H the number of years needed to accumulate a 3 signal for the CP asymmetry, AH, in the "nal state j. Then R#RH B(HPj) H y( N"9 , (10.20) H (AH)RH B(HPj) L where L is the integrated luminosity per year, is the cross-section for >\PHPe>e\ and for a "nal state X, R "(>\PX)/ . In the above R is R from SM processes only, which 6 H H needs to be included as it contributes to the background. Note that the interference between the SM and the Higgs exchange will be negligible; from helicity considerations, such an interference term will be suppressed by m /mH . The SM does I however contribute as a background, hence the term R in the numerator of Eq. (10.20). We must H
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Fig. 68. The values of y( N (i.e., the number of years required for a 3 e!ect) for the three scenarios discussed in the text obtained in [351]. The solid line is y( N for scenario (1) using top polarization correlation. The upper dash-dot line is RRM y( N for scenario (2) using top polarization correlation while the lower dash-dot line (also in scenario (2)) is for the case M RR where the initial muon beams have a longitudinal polarization P"0.9. The short dashed line is y( N obtained in scenario G (2) using transverse polarization of the initial muon beams. The long dash line is y( N for scenario (3) using top RRM polarization correlation. Here we take L"10 cm\ s\.
also consider the e!ect of all of the other decay modes of the Higgs taken together since RH is proportional to B and hence inversely proportional to the total width (see Eq. (10.20)). To get an I idea of how large the CP-violating e!ects can be, we consider y( N as a function of mH ,(s in H Fig. 68 in a number of di!erent scenarios: (1) H"H with "1 and "453 for all fermions and "453. D D (2) H"A with "1 and "453 for all fermions. D D (3) H"A with " "5 and "1/5, "453 for all fermions. J B S D We assume that " " and " " . The signi"cance of H"H or A, as discussed S A R B Q @ above, is that only if H"H does HP==, ZZ contribute to the total width. Thus, in particular, the value of is not relevant to cases (2) and (3), since if H"A, then no boson pairs are produced by the resonance at tree-level. In Fig. 68 which shows the results from [351], we take a nominal luminosity of 10 cm\ s\ and a year of 10 s. (i.e., with running e$ciency of ). The solid line gives the result in the case of scenario (1) while the upper dot-dash line is scenario (2). In both of these cases, y( NM starts at about RR 5 yr near threshold and increases thereafter. The result for scenario (3) is shown with the long dashed line and is considerably smaller, 0.01}0.1 years, due to the narrow width of the neutral Higgs H in this case.
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One can enhance the signal with respect to the SM through the use of longitudinally polarized beams. If both of the > and \ beams are left polarized with polarization P, then the Higgs production is multiplied by (1#P) thus enhanced while the SM backgrounds are multiplied by (1!P) and thus reduced. More generally, if the > has polarization P> and the \ beam has polarization P\ then the Higgs cross-section gets multiplied by (1#P>P\) while the SM background gets multiplied by (1!P>P\). In the lower dash-dot curve of Fig. 68, we consider the results for scenario (2) where we have taken P"0.9 which gives a reduction of nearly an order of magnitude in the number of years required to observe a 3- CP violating signal. 10.1.3. Production asymmetry in >\PttM via polarized muons As discussed above, some knowledge about fermion polarization is required if information about CP violation in >\PttM is to be obtained. Above we considered the case where we used the polarization of the top quarks to learn about sin 2 . Here we consider the case where the muons R are polarized. The initial production of muons results in a substantial longitudinal polarization since the weak decay P produces predominantly left-handed \ (and right-handed >). If one constructed a single pass colliding beam machine, it should not be too di$cult to preserve this polarization. On the other hand, in the case of muon storage rings the polarization would have to be manipulated in some way since a longitudinally polarized beam will precess at a rate proportional to g!2. In a recent paper Grza7 dkowski et al. [352] discuss the details of how the polarization of muons in a storage ring may be used to make measurements on the Higgs resonance of the type considered in the following sections. Here we will assume that it is possible to prepare muons in a given initial state of transverse or longitudinal polarization. Let us "rst consider an experiment where the muon beams are polarized transversely to the beam axis. The cross-section is then measured as a function of the angle between the polarizations. We can take the z-axis in the c.m. frame to be in the direction of the \ beam and the x-axis to be its polarization while the > beam is polarized at an angle of to the x-axis, that is in the direction I (cos , sin , 0). I I If the ! beams have polarization P then the cross-section (e.g., for >\PttM ) as a function of ! is [351] I ( )"(1!P P cos 2 cos #P P sin 2 sin ) , (10.21) I > \ I I > \ I I where is the corresponding unpolarized cross-section. We could therefore look for the presence of CP-violation by comparing ( "#903) with ( "!903). Thus, we de"ne the CP-odd I I (C-even and P-odd), ¹ -odd asymmetry , (#903)!(!903) "P P sin 2 . (10.22) A " > \ I I (#903)#(!903) Clearly if appreciable polarizations are available and sin 2 +1 these e!ects are dramatic. I In this experiment, we are simply observing a change in Higgs production as a function of , so I in the approximation that the Higgs resonance is dominant, it would not matter in fact what the "nal state is. In practice, the SM e!ects will also produce the same "nal states. Again using this asymmetry we can quantify the amount of run time required to see a signal through Eq. (10.20). In Fig. 68, we
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show with the short dashed line the value from [351] of y( N, which is the number of years required G to obtain a 3 signal using the initial polarizations for this asymmetry in scenario 2 taking P "P "1. > \ It is also useful in some cases to consider asymmetries which make use of longitudinally polarized muons in the initial state. Such an asymmetry which is ¹ -even, C-even and P-odd was , considered in [353]: (\>PttM )!(\>PttM ) 0 0 . (10.23) A " * * !. (\>PttM )#(\>PttM ) * * 0 0 This asymmetry in the pair production cross-section clearly requires longitudinally polarized muon beams. In principle, a similar asymmetry is possible for other fermions as well. Since this asymmetry is ¹ -even, some absorptive phase is required. If the collider is running near the Higgs , resonance, this will be naturally provided by the complex phase in the Higgs propagator. Thus, a mechanism for generating this asymmetry is the CP violation originating from the mixing between H and Z and/or between the scalar (H) and pseudoscalar (A) Higgs that can occur in extended models. In particular, the CP invariance of the Higgs sector may be broken by the presence of heavy Majorana fermions. Such a scenario can occur in the minimal SUSY model, in which heavy neutralinos are Majorana fermions. E inspired theoretical scenarios o!er another possibility for heavy Majorana neutrinos at the TeV mass scale. The most interesting situation occurs when a CP-even H mixes with a CP-odd Higgs scalar, A, and, as is natural in SUSY models, for M <M , the two states are roughly 8 degenerate, M KM . In particular, if M '2M the broadening of the H due to the two vector & & 8 decay channel can allow signi"cant mixing between the two states. Consequently, Pilaftsis [353] "nds !2K &[ImK &&(m )!ImK (m )] & & , (10.24) A & !. (m !m )#[ImK (m )]#[ImK &&(m )] & & & where GH are coupled channel propagators derived in that paper. Note, in particular, the proportionality to the imaginary parts as expected since the asymmetry is CP¹ -even. , Fig. 69 shows the results from [353] in a model where & is generated by heavy Majorana neutrinos with masses M "0.5, 1.0 and 1.5 TeV. It is assumed that (s"m . Two scenarios for , & the masses and couplings of the Higgs-bosons are considered: (a) M "170 GeV and cos "1, "2"1/ and the asymmetry is observed with a bbM "nal B S state. (b) M "400 GeV and cos "0.1, "2"1/ and the asymmetry is observed with a ttM "nal B S state. The cross-section is shown with solid curves while the asymmetry is shown with dotted curves. Scenario (a) is shown with the curves in the region around (s"170 GeV where the "nal state is bbM while scenario (b) corresponds to the curves in the region (s"400 GeV, with a ttM "nal state. The enhancement of the asymmetry from the imaginary part of the scalar propagators is apparent in the case where the A and H masses are close together, within about 10% of each other.
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Fig. 69. This "gure shows the cross-section (solid curves) and A in Eq. (10.24) (dotted curves) as a function of (s"m !. & in a model with A}H mixing induced by heavy Majorana neutrinos with masses M "0.5, 1.0 and 1.5 TeV and with , "2"1/ . The two curves at the left are for case (a) where M "170 GeV and A is observed in the bbM channel (see B S !. text). The curves at the right are for case (b) where M "400 GeV and A is observed in the ttM channel (see text). Note !. that in both cases the curves are shown in the vicinity of M &M where the mixing e!ects are likely to be most & prominent. Figure taken from [353].
10.2. CP-violation in the yavor changing reaction >\Ptc As mentioned before, one of the unique properties of a muon collider is that, under favorable conditions, it may produce neutral Higgs states in the s-channel. If the Higgs sector contains two or more doublets, then the Higgs couplings may be #avor changing (FC), see e.g., [128,354,355] (see also Section 3.2.2). This can lead to a dramatic tree-level signature of >\Ptc (or ctM ) due to the neutral Higgs resonance [356]. At the same time FC processes do occur at the loop level in the SM and in practically all of its extensions, even if they are forbidden at the tree-level. Thus, a continuum of FCNC reactions of the form >\PZH, HPtc , tM c are expected. Indeed such couplings with CP-violating phases may also naturally arise in R-parity violating SUSY models [357]. Of course such processes are GIM suppressed in the SM but for the purpose of this discussion we are assuming that there is a FC Higgs sector as in Section 3.2.2, thus for the reactions ZH, HPtc , tM c, rates appreciably larger than the SM may be expected [128,358]. Since many such extensions of the SM contain a large number of unconstrained Yukawa couplings, they will, in general, also contain CP-violating phases. Therefore, the interference between the resonant and the continuum processes can lead to CP-odd observables; it is this possibility which we wish to study in this section. Consider now the two processes shown in Fig. 70(a) and (b). Since the Higgs #ips the helicity of the while the Z does not, for unpolarized or longitudinally polarized beams the interference will be proportional to the mass of the muon and consequently exceedingly small and uninteresting. Such a suppression will not occur if the beams are transversely polarized whence a large interference signal may be produced, especially if the resonant (Fig. 70(a)) and the continuum (Fig. 70(b)) processes are of similar strength.
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Fig. 70. (a) Feynman diagram for >\Ptc through s-channel neutral Higgs exchange. (b) Feynman diagram for >\Ptc through virtual Z and exchanges, where the circle indicates a vertex correction. (c) An example of a vertex correction contributing to >\Ptc, where H is a neutral Higgs with #avor changing interactions to fermions and H! is a charged Higgs.
Bearing all this in mind, we will thus proceed as follows: "rst, we will consider the general case of the resonance production of tc interfering with the continuum and then we will consider, more speci"cally, what signals are produced in models similar to the ones discussed by Atwood et al. [128]. The process which produces tc via s-channel Higgs exchange is controlled by the terms in the Lagrangian (for more details see Section 3.2.2) L "[ #tM c#2#h.c.]H , & I RA where with the parameterization de"ned in Eq. (10.5),
(10.25)
"C e A HI , # . (10.26) I II I I I Similarly, for we can write RA , # . (10.27) RA RA RA Note that unitarity implies that is real while is imaginary. I I CP-violation will then occur if there exists another mechanism for producing tc which the Higgs may interfere with. Here we take this process to be >\PHPtc and/or >\PZHPtc with the amplitude M "e ( ) ) (tM M c) , 8 A 8 A M I RA where , , tM and c above are Dirac spinors and
(10.28)
"s[(s!m )s c ]\, "1 , 8 8 5 5 A ,A8 A#B8 A , ,A8 A#B8 A . (10.29) I I I RA RA RA Here s "sin , c "cos , where is the weak mixing angle. A8 A, B8 A are real and they, of 5 5 5 5 5 I I course, occur in the tree-level SM Lagrangian. A8 A, B8 A are form factors which are induced at the RA RA loop level. As mentioned previously, although small in the SM, they may be generated at reasonable levels in some extensions of the SM, for instance, in multi Higgs scenarios which give HPtc , see e.g. [354}356]. Since we are interferring a vector continuum with a scalar resonance, this interference is naturally suppressed by m . This suppression, however, does not apply if at least one of the beams I
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is transversely polarized. To construct a quantity which is CP odd, we consider the case where the \ beam is polarized in the #x direction and add it to the result where the > is also polarized in the #x direction. We will also consider the case where the continuum is dominated by the Z exchange. Combining the transversely polarized > and transversely polarized \ data as suggested above, we now consider some angular distributions of this combined data which have speci"c properties under CP and ¹ . Let us de"ne the polar coordinates (, ) of p ; in particular, is the azimuthal , A separation between the beam polarization and p . For each event of the form >\Ptc or tM c, let us A also de"ne, ¸ to be #1 for the tc "nal state and !1 for the tM c "nal state. It is natural, therefore, to R consider the following possible asymmetries x " (cos ), x " ¸ (cos) , R x " (sin), x " ¸ (sin) , (10.30) R where (x)"#1 if x50 and !1 if x(0. These expectation values can be characterized in terms of their symmetry properties. Thus x , x are CP-odd; x is ¹ -even and x is ¹ -odd. x and x are CP-even; x is ¹ -even and x is , , , ¹ -odd, indicating that x and x require complex Feynman amplitudes, i.e., FSI phase(s). In the , process at hand, one source of this is the Higgs propagator (see Fig. 70(a)). In fact, since the Higgs is close to resonance, in the experiments being envisioned here, these (CP¹ -odd) observables (i.e., , x and x ) are likely to be the most prominent of the observables since they are enhanced by the resonant phase of the Higgs propagator. In order to observe the signals suggested above one "rst requires a muon collider which is able to deliver beams with a large transverse polarization as well as an energy spread for the beam which is comparable to or smaller than the Higgs peak. Clearly, theories which produce detectable signals should, of course, have fairly large FC tc couplings. As a speci"c example let us consider 2HDM of types III (see e.g., [128,354,355]), also discussed in Section 3.2.2. In these scenarios a 2HDM is considered where the second Higgs doublet has arbitrary Yukawa couplings. The popular Cheng-Sher Ansatz [126]: (m m R A , (10.31) Kg RA 5 m 5 for in Eq. (10.25) is then imposed where is a parameter that needs to be extracted from RA experiment. It is perhaps natural to expect to be of O(1). It is clear that the "rst obstacle to a large signal is having the ZH-exchange continuum in Fig. 70(c), generated by loop corrections, to be sizable. Let us de"ne R "(>\PHPtc , tM c)/(>\PHPe>e\) , (10.32) & R "(>\PZHPtc , tM c)/(>\PHPe>e\) . (10.33) 8 Clearly then, a necessary condition for there to be large O(1) asymmetries is that R +R . For & 8 m &150}350 GeV, typically R &10\}10\ [358]. Such a signal would have a marginal 8 &
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chance at a L"10 cm\ s\ luminosity machine. The Higgs signal in this case can also be of order R &10\ } 10\ in the scenario where the Higgs decay to two vector-bosons is allowed. In & that case, &O(1 GeV), so the spread in beam energy needed to be in the resonance region of the & Higgs should be achievable at a muon collider. This could allow asymmetries of a few tens of percents provided that the CP-odd weak phase di!erence were large. Since R &R &10\, & 8 observing the asymmetry would still require a 10 cm\ s\ collider. The situation, of course, would improve considerably if the continuum were larger; for example, this happens if '1 in Eq. (10.31). Since the continuum scales as the integrated luminosity required to observe these asymmetries scales as \.
11. Summary and outlook There are only two known systems which have been shown to violate the CP symmetry: the neutral kaon through the parameters and and the entire universe through the dominance of matter over antimatter. It would be of great signi"cance to understand the relation between these two e!ects or trace them to a common origin. Although experiments in the near term are likely to clarify the source of CP violation in the kaon system, the mechanisms of baryogenesis remain in the realm of theoretical speculation and may not be directly tested in the lab for some time. The top quark, however, o!ers a unique system where new CP-violating e!ects could be discovered which could, in time, shed light on the processes which were important in the early universe. The immediate source of CP violation in the kaon is thought to be the CKM phase in the SM. This will be tested in detail in the next few years through the study of the B meson. Ironically, although the exchange of virtual top quarks generates the large CP violation in the B and K mesons, the CKM phase will not produce any signal in top quark systems that is large enough to be of experimental interest. Instead, if a CP-violating signal is seen in the top quark, it must be due to some inherently large, non-standard, CP-odd phase which becomes manifest only at high energy scales. Since the e!ect of the CKM phase is also thought to be too weak to explain baryogenesis, the required phase for this process is likely to show up in top quark physics as well. Thus, the observation of CP violation in top quark reactions is an unambiguous signal of physics beyond the SM which may well shed light on baryogenesis. In this review, we have considered a number of laboratory tests of CP violation in the top quark in the context of various non-standard models for physics beyond the SM. In particular, we focus on two classes of models which are described in some detail in Section 3: 1. Multi Higgs models containing at least two Higgs doublets with phases in the Yukawa couplings. Here CP violation manifests either in the neutral or in charged Higgs sectors. 2. SUSY models wherein we consider in detail the MSSM with the Yukawa couplings given by N"1 minimal SUGRA models. One manifestation of SUSY CP violation is through mixing in the sfermion sector. We chose to focus on these models because they seem to be representative of models for physics beyond the SM which could give rise to CP violation and are most often considered in the literature. It is likely that the ability of a particular signal to detect CP violation in one of these models is a good indication of its general utility.
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In this review we highlight some notable CP-violating phenomena which follow from these models: E The transverse polarization of the in tPb which follow from CP violation in the charged Higgs sector (see Section 5). E CP-violating correlations in e>e\PttM H, ttM Z and e>e\PttM at high energy e>e\ colliders C C generated by CP violation in the neutral Higgs sector. Since these e!ects arise through the interference of two tree-level graphs the resulting correlations can be very large indeed (see Section 6). E CP-violating correlations in hadronic top pair and single top production which can originate from CP-odd phase(s) in the neutral Higgs sector or in the squark sector of SUSY models (see Sections 7 and 8). E CP-violating top polarizations may arise in top production at muon and/or photon colliders. In particular at such colliders the neutral Higgs(es) can be produced in the s-channel, giving rise to a distinct resonant enhancement which in turn may magnify the CP-violating e!ect in reactions such as >\, PttM or even in the #avor changing channels >\, Ptc #tM c (Sections 9 and 10). E CP-violating moments of the top analogous to the electric dipole moment which may be observed at an NLC from top polarimetry in the reaction e>e\PttM . Such moments can be generated in SUSY models as well as models with an extended neutral Higgs sector (Section 4). E CP-violation in the main top decay tPb=. In this case a CP-odd phase in the stop sector of the MSSM can cause a partial rate asymmetry in tPb=> at the level of a few;0.1% (Section 5). A common feature of both the SM and the models mentioned throughout this review is that CP violation is driven directly or indirectly by Yukawa couplings in the scalar sector of the theory. In the SM the CKM matrix which contains the CP-violating parameter results from the Higgsfermion coupling while with multi Higgs models additional CP violation may result from the couplings between the various Higgs "elds, either from explicit CP violation in the Higgs potential (e.g., Model II) and/or in the Yukawa interaction terms (Model III), or CP can be violated spontaneously if there are more than two Higgs doublets. In SUSY models, phases may be associated with the scalar Lagrangian as well, for instance, from squarks and sleptons mixing. As in the SM, the amount of CP violation is proportional to the non-degeneracy of the relevant mass spectrum. For instance, in MHDMs the non-degeneracy of the Higgs particles is required while in SUSY it is the non-degeneracy of squarks or sleptons of di!erent helicities. It is important to emphasize that, in many ways, the phenomena of CP violation in top quark systems strongly relies on the large mass of the top which, therefore, becomes the key property of the top as far as CP violation is concerned: E The large mass of the top quark allows its polarization to be determined by its weak decays because, unlike the other 5 quark #avors, it decays before it hadronizes and so the information carried by its spin is not diluted. As discussed in Section 2, this allows experiments to consider CP-odd observables involving polarization (i.e., top spin correlations) which is crucial since in many settings no CP-violating observables could be constructed without this information. For example, the CP-violating transverse top polarization asymmetry, suggested in Section 8, may
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E
E
E
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be used to probe tree-level CP violation in pp PtbM which otherwise (i.e., without the use of top spins) cannot be observed. In MHDMs, it is the enhancement in the Yukawa coupling of a neutral Higgs to the top quark that is responsible for the enhanced CP-violating e!ect. As discussed in Section 6, this clearly manifests in e.g., e>e\PttM Z, where the only CP-violating diagram present, i.e., the one with a virtual neutral Higgs exchange, is comparable in size to the CP-even SM diagrams that contribute to the same "nal state, due to the fact that the CP-odd, HttM Yukawa coupling may be as large as the gauge coupling. As mentioned above, in the CP-non-conserving e!ects associated with SUSY particles exchanges, it is the large mass splitting between the two stop mass eigenstates of the theory that may be the cause for an enhanced CP-violating e!ect, again, due the corresponding large mass of their SM partner } the top quark. As discussed in Sections 7 and 8, this is the case for example in ppPttM and pp PtbM where the e!ect arises from CP-violating loop exchanges of stop particles. Large m enables the study of CP violation in cases where the CP-odd e!ect is driven by new R thresholds (i.e., absorptive cuts across heavy particles of the underlying theory). As discussed in Section 5, this is the case in e.g., PRA in tPb= within supersymmetry, where it is only viable if m 'm I #m } still allowed by present experimental data, basically, because of the heaviness of Q R R the top. The cases where the CP-odd e!ect is enhanced to the detectable level due only to an intermediate resonance are also a clear manifestation of the important role played by the large m in CP R violation studies. Such is the case in e.g., CP violation in the decay tPb , as discussed in O Section 5, where the intermediate =-boson resonance provides the necessary enhancement, of course, since m 'm . R 5
It is therefore evident that, due to its large mass, the top is very sensitive to new e!ects from possible new short distance theories. This sensitivity of the top quark to short distance e!ects from many models leads one to consider a more general approach for such studies. For instance, by parameterizing CP violation in a model independent way using CP-violating form factors. Such form factors which contain the information of the dynamics of some new physics scenarios at higher energy scales are expected to be more pronounced in top quark interactions. This technique is a useful prescription for extracting limits on various CP-violating couplings that may arise in new physics. Examples of such e!ective form factors are the top dipole moments and the CP-violating form factors in the top decays which were separately discussed in Sections 4 and 5, respectively. In Section 4, we "nd that in models with extra Higgs doublets as well as in SUSY models, one can expect an EDM and ZEDM of the top on the order of &10\ e cm and, likewise, a CEDM of &10\g cm. These values are many orders of magnitude larger than the SM prediction for these Q quantities. Thus, a discovery of such an e!ect in ttM production in leptonic or hadronic colliders and perhaps also in photon and muon colliders, would be a clear signal of beyond the SM dynamics. In Section 6 we discuss the sensitivity of an e>e\ NLC collider to these EDM and ZEDM. We "nd that optimal observable techniques seem to indicate that high-energy e>e\ colliders will be sensitive, at best, to a top dipole moment at the level of &10\}10\ e cm, about one to two orders of magnitude larger than what is expected in the models mentioned above.
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In Section 9, we "nd that a similar statement is true at colliders based on the backscattered laser light from an NLC. On the other hand, hadron colliders are expected to be more sensitive to the CEDM, in particular, in Section 7, we "nd that a CEDM at the level of 10\}10\g cm Q might be observable at the LHC. Although this sensitivity of a NLC to the top electric dipole moment may seem a little discouraging, there is still very strong motivation to look for this e!ect; the observation of a top dipole moment with this magnitude (i.e., &10\ e cm), will clearly be a surprise, since such a large dipole moment cannot be accounted for in the popular models such as SUSY and MHDMs. Thus, in spite of the very large enhancements expected in such beyond the SM scenarios for the top dipole moments, it is evident that this type of signal is useful only if the dipole moments are on the very large side of the theoretical range. That being the case, one would like to search for other alternatives for the observation of CP violation in top quark systems. It may, for example, be more promising to look for speci"c signals of CP violation in the production or decay of top quarks which are not related to the dipole moments. In Section 5, we consider the CP-violating e!ects which might be present in the decay of top quarks. The simplest kind of signal is a PRA in the decay tPb= (mentioned above) which in SUSY can have an asymmetry of &10\ and thus may be detectable at the LHC. Another promising signal which is particularly applicable to 3HDM or other models with charged scalars is polarization asymmetries in the decay tPb which arises from the interference of the W pole with the charged Higgs propagator and could result in asymmetries on the order of a few tens of percents. While CP-violating e!ects in the decay of top quarks may be searched for at any experiment where top quarks are produced, there are a broader range of signals where the CP violation occurs in the production of the top quark. In this case, one must consider each kind of top quark production mechanism separately. CP violation in the production mechanism of the top was discussed in the context of an e>e\ collider (in Section 6), hadronic colliders (in Sections 7 and 8), photon collider (in Section 9) and muon collider (in Section 10). Each of these machines has its own characteristics and so special attention needs to be given in constructing appropriate CP-violating observables. For example, lepton and photon colliders have the advantage of their relative cleanliness as far as background is concerned; it should be easier to reconstruct the top quark in such colliders even when it decays via purely hadronic modes. On the other hand, it may be quite challenging for such colliders, e.g., the NLC, to posses the necessary luminosity for studying rare phenomena in top physics such as loop induced CP-violating e!ects. Hadron colliders such as the LHC may have an advantage in this context, since top quarks will be more readily produced there. However, the hadronic environment requires more e!ort in disentangling the CP-violating signal both from the experimental and the theoretical points of view. In addition for hadron colliders, it should be noted that one would prefer, in principle, to always use a pp collider (such as the Tevatron) for CP studies since then the initial state is a CP eigenstate. Unfortunately, the LHC which is expected to produce a very large number of top pairs is a pp collider. It turns out, however, that the initial state at the LHC may not e!ect the CP studies considered here greatly, primarily because the dominating ttM production mechanism there is in fact through a CP eigenstate, i.e., gluon}gluon fusion. Nonetheless, in such colliders it is important to use clean CP-violating observables that can reduce the backgrounds. One such useful observable for the LHC, that was suggested by Schmidt
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and Peskin was discussed in Section 7. Their CP-odd signal uses the di!erence between the energy of the positrons from tPbe> and electrons from tM PbM e\ in the overall rather complicated C C reaction, ggPttM Pb=>bM =\Pbe> bM e\ . Unfortunately, the expected asymmetry is unlikely C C to be larger than a few times 10\. However, Bernreuther, Brandenburg and Flesch have shown that a considerable improvement may be achieved in this reaction by employing clever cuts on the ttM invariant mass. By doing that they were able to isolate the possible CP-violating contribution from an s-channel Higgs exchange in ggPttM . Thus, in their analysis, asymmetries at the level of a few percent may arise leading to a CP-odd signal well above the 3! level in ppPttM #X at the LHC. Another useful CP-violating signal designed for the Tevatron setting was discussed in Section 8. Speci"cally, it uses an apparent advantage of pp colliders: that there should be a high rate of virtual = production via udM annihilation. In this case a number of asymmetries involving the transverse and longitudinal components of the top spin may be constructed. In both MHDMs and SUSY models we "nd that asymmetries around 1% may thus occur in single top production at the Tevatron. Loop induced CP-violating e!ects such as that of Schmidt and Peskin as well as dipole moments tend to give asymmetries at the level of &0.1!1%. Thus experimental detection of rare CP-violation e!ects in top physics, both in hadronic and leptonic colliders, leads to at least two important challenges. (1) Can detector systematics be controlled to the point that a CP asymmetry of O (0.1%) can be observed? (2) Can CP violation be studied with purely hadronic decay modes of the ttM pair. That is, to what extent will the experimentalists be able to distinguish between the top and the anti-top via purely hadronic modes; if that can be done to a signi"cant level, then the increased statistics will improve the prospects for the observability of such rare CP-odd signals. The small CP-asymmetries which arise from phenomena that occur at 1-loop may make most of those signals too small to be of great use in putting bounds on models of new physics. On the other hand, signals which arise from the interference of tree graphs only are likely to give rise to larger asymmetries. In Section 6, we discuss some candidate signals of this type such as e>e\PttM H, ttM Z, and ttM . In addition, the decay discussed in Section 5, tPb C C O falls into this category. In these cases one "nds that CP asymmetries at the level of tens of percents are possible in models with CP-odd phase(s) in the Higgs sector. This makes that type of CP-violating mechanism quite robust, requiring about a few thousands ttM events per year in order to be detected; such a number may well be within the reach of the future colliders presently under consideration. The two other exotic technologies which may be used in the future in this context (i.e., tree-level CP violation) are colliders and muon storage rings. In Section 9 we discuss reactions which can take place at a collider constructed from an e>e\ NLC collider by backscattering laser light from the e! beams. As mentioned above, these machines can be used to produce ttM pairs through an intermediate Higgs state and interfere it with the born cross-section for top pair production. In this case, observables constructed by considering the top polarization can give asymmetries of up to &10% in 2HDM. In Section 10 we further discuss experiments at muon colliders. Clearly any experiment which can be performed at an e>e\ collider may also be performed at a muon collider. In addition, however, the larger mass of the muon allows us to contemplate the production of Higgs bosons in the s-channel. In such scenarios one can analyze the scalar versus pseudoscalar
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couplings of the Higgs to ttM by studying the polarization correlations of the tops produced which, again, could give &10% asymmetries. Now, since a large portion of the experimental e!ort in these future colliders will be devoted to the search of supersymmetry, it is particularly gratifying that for such studies of tree-level CP violation, supersymmetry may play an important role in our understanding of the underlying mechanism for CP violation. In particular, once SUSY is discovered and SUSY particles are readily produced in high energy collider experiments, the next step would clearly be to start scrutinizing the basic ingredients of the SUSY Lagrangian, e.g., its CP-violating sector. Indeed, due to the potential richness of CP-odd phases in SUSY theories, tree-level CP violation can easily occur in production and decay of SM#SUSY particles. A promising venue to investigate such tree-level SUSY CP violation may be to search for reactions involving associated top production in "nal states which contain additional SUSY particles and to probe the CPviolating e!ect through top polarimetry, i.e., bypassing the missing energy limitation (typical to SUSY signatures) by using the top spins. A simple example may be CP violation in e>e\Pt#X#missing energy versus e>e\PtM #XM #missing energy, where X is some nonSUSY hadronic "nal state. Another interesting related venue in the context of tree-level CP violation within SUSY models is to search for CP-odd signals in reactions where, although involving SUSY particles, only SM particles are produced in the "nal state. Indeed, if SUSY theories posses R-parity violating interactions, CP may be violated at tree-level even in 2P2 processes in which the initial and "nal states consist of SM particles only. In particular, through SUSY scalar exchanges in which the CP-odd phases are carried by the R-parity violating couplings in the interaction vertices of a pair of SM particles to squarks and/or sleptons. Again, such tree-level CP violation may be probed even in a 2P2 process if one uses top spin asymmetries. Consider, for example, single top production at the Tevatron, pp PtbM #tM b. As was discussed in Section 8, a transverse top polarization asymmetry can potentially probe tree-level CP violation in this process. Indeed, since s-channel exchanges of charged sleptons can mediate pp PtbM #tM b in R-parity violating SUSY, this transverse top polarization asymmetry can potentially lead to large tree-level SUSY CP-violating signal in this reaction. These types of tree-level CP violation in SUSY models were not discussed in this review or anywhere else in the literature to date and could be useful to examine in the future, especially once SUSY is directly observed. More generally, the subject of tree-level CP violation seems promising and requires additional e!ort from the theoretical point of view. In parting, the study of CP violation in top quark physics deserves to be one of the main issues on the agenda of the future high-energy colliders. The expected high production rate of top quarks in these colliders turns these machines into practically top factories enabling the examination of what is presently considered rare phenomena in top physics. In particular, these colliders provide a unique opportunity for the study of CP violation } a phenomenon that till now seems to be essentially con"ned only to the kaon system } and its relation to top quark dynamics. The manifestation of CP violation in heavy particles systems in general and in the top quark system in particular, can shed light on new aspects of this phenomena due to the high-energy scales involved, possibly on new physics related to the dynamics of our universe in its very early stages. One, of course, should not forget the importance of the up coming CP measurements in the B system. On the other hand, it is also important to note that there is a very interesting interplay
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Table 16 The underlying source of CP-odd phase and the mechanism responsible for CP violation in the processes indicated. / means CP / from tI !tI mixing, HCP / means CP / from scalar}pseudo-scalar mixing in the Htt vertex Note that: tI CP * 0 / means a CP / phase in the H>tb and/or H> vertices. See also Section 3 and H>CP O Process
CP source
Mechanism
tPb=
/) MSSM (tI CP
1-loop
tPb O
/) MHDM (H>CP
=>-resonance in tree-level =>!H> interference
pp PtbM
/) MSSM (tI CP & /) MHDM (HCP
1-loop
pp PttM
/) MSSM (tI CP & /) MHDM (HCP
Top } CEDM (1-loop)
ppPttM
/) N MSSM (tI CP & /) N MHDM (HCP
1-loop s-channel H & 1-loop
e>e\PttM
/) MSSM (tI CP & /) MHDM (HCP
Top } EDM, ZEDM (1-loop)
e>e\PttM H, ttM Z
/) MHDM (HCP
Tree-level interference
e>e\PttM C C
/) MHDM (HCP
s-channel H in Tree-level interference
>\PttM
/) MHDM (HCP
H-resonance
PttM
/) MHDM (HCP
s-channel H & 1-loop
between CP violation in b physics and in t physics. In b physics, one expects large CP-violating signals due to the CKM phase alone. Therefore, non-observation of CP violation in B decays would, in fact, stand out as a signal of new physics. This is, of course, in complete contrast to the situation in the top system in which one does not expect any CP-odd signal with the CKM phase of the SM. Therefore, any signal of CP violation in top reactions will unambiguously prove the existence of new physics. Moreover, in order to disentangle e!ects of new physics in the B system, one will need precision measurements and cross-checking of the di!erent available CP-violating B decay channels. In top systems the advantage is that no signi"cant e!ort is needed in order to establish the existence of new physics phenomena in CP-odd top correlations } any measured CP-non-conserving e!ect in top systems will su$ce. Finally, in Tables 16 and 17, we summarize the main features of some of the most interesting CP-violating signals that were discussed in this review.
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Table 17 The asymmetries that can probe CP violation in the processes indicated and their expected size. Also indicated is the place in which each asymmetry was discussed in the review, i.e., its equation number. The size of the asymmetry given tends to be optimistic, i.e., on the large side of its theoretical range Process
Type of asymmetry
tPb=
PRA } A (Eq. (5.4)) pol. } e.g., (transverse) A (Eq. (5.85)) X Cross-section } A (Eq. (8.19))
tPb O pp PtbM pp PttM ppPttM
e>e\PttM
Size (%) 0.1 10
1 Top pol. } e.g., (transverse) A(y( ) (Eq. (8.21)) Lepton energy } e.g., (transverse) A (Eq. (8.7)) 2 Optimal observable } e.g., O (Eq. (7.4)) Top pol./lepton momenta } e.g., N (Eq. (7.17)) *0 Lepton energy } e.g., (transverse) N(E ) (Eq. (7.21)) 2 Optimal observable } e.g., O (Eq. (6.8)) 0 Top pol./lepton momenta } e.g., ¹K (Eq. (6.24)) GH
0.1 0.1}1
0.1
Angular distributions } e.g., A () (Eq. (6.43)) SB Energy distributions } e.g., Al l (Eq. (6.50)) e>e\PttM H, ttM Z e>e\PttM C C
Top momenta, optimal observable } O, O (Eq. (6.69)) Top pol./lepton momenta } e.g., A (Eq. (6.114)) W Top pol./lepton momenta } AR (Eq. (10.13))
>\PttM
PttM
10 10 10
Muon beam pol. } e.g., A (Eq. (10.22)) I Top pol./lepton momenta } e.g., (Eq. (9.33))
10
Photon pol. } e.g., P (Eq. (9.36))
12. Note on literature survey The literature survey for this review was primarily completed in Dec. 1999.
Acknowledgements Two of us (G.E. and A.S.) are most grateful to the US}Israel Binational Science Foundation for its support that proved very valuable during the long period that took to write this review. GE would also like to thank the Israel Science Foundation and the Fund for Promotion of Research at the Technion for partial support. This work was also supported in part by US DOE Contract Nos. DE-FG02-94ER40817 (ISU), DE-AC02-98CH10886 (BNL) and DE-FG03-94ER40837 (UCR).
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Appendix A. One-loop C functions In this appendix, we de"ne the coe$cients C , (x"0, 11, 12, 21, 22, 23, 24, see below) correspondV ing to 1-loop integrals with three internal propagators in the loop, i.e. `triangle-likea 1-loop diagrams. In the review they appear in Section 4.3 (Eqs. (4.16) and (4.17)), in Section 4.4 (Eqs. (4.35) and (4.36)), in Section 4.5 (Eqs. (4.46), (4.53), (4.54), (4.62) and (4.63)), in Section 5.1.4 (Eqs. (5.38)}(5.45)), in Section 8.2.1 (Eqs. (8.32) and (8.33)) and in Section 8.2.2 (Eqs. (8.41) and (8.42)). These three-point loop from factors which are functions of masses and momenta are de"ned by the 1-loop momentum integrals as follows [359,360]:
dk 1; k ; kk ; k k I I I J , C ; C ; CI ; C (m , m , m , p , p , p ), I I IJ i D D D where
(A.1)
D ,k!m , (A.2) D ,(k#p )!m , (A.3) D ,(k!p )!m , (A.4) and p "0, i"1!3, is to be understood above. G G The three-point loop from factors are then given through the following relations [361]: C "p C #p C , (A.5) I I I CI "p CI #p CI , (A.6) I I I C "p p C #p p C #p p C #g C , (A.7) IJ I J I J IJ IJ where ab ,a b #a b . The numerical evaluation of the above form factors can be performed IJ I J J I using the algorithm developed in [359,360].
Appendix B. Abbreviations We list in this appendix all the abbreviations used throughout this review: SM CKM GIM NLC LHC PRA FSI PIRA MHDM
standard model Cabibbo}Kobayashi}Maskawa Glashow}Iliopoulos}Maiani next linear collider large hadron collider partial rate asymmetry "nal state interactions partially integrated rate asymmetry multi Higgs doublet models
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SUSY SSB VEV NLO 2HDM 3HDM MSSM FCNC FC NFC REWSB NEDM EDM RGE SUGRA EW TDM ZEDM CEDM FF LSP DCS
supersymmetry or supersymmetric spontaneous symmetry breaking vacuum expectation value next-to-leading order two Higgs doublet model three Higgs doublet model minimal supersymmetric standard model #avor changing neutral currents #avor changing natural #avor conservation radiative electroweak symmetry breaking neutron electric dipole moment electric dipole moment renormalization group equations supergravity electroweak top dipole moment weak(Z)-dipole moment chromo-electric dipole moment form factor lightest supersymmetric particle di!erential cross-section
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
G.C. Branco, L. Lavoura, J.P. Silva, CP Violation, Oxford Science Publications, Oxford, 1999. I.I. Bigi, A.I. Sanda, CP Violation, Cambridge University Press, Cambridge, 2000. J.H. Christenson, J.W. Cronin, V.L. Fitch, R. Turlay, Phys. Rev. Lett. 13 (1964) 138. C. Caso et al., Review of Particle Physics, Eur. Phys. J. C 3 (1998) 1. N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531. M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. M. Ciuchini, E. Franco, G. Martinelli, L. Reina, Phys. Lett. B 301 (1993) 263. M. Ciuchini, E. Franco, G. Martinelli, L. Reina, 2nd DAPHNE Physics Handbook, 27, hep-ph/9503277. A.J. Buras, M. Jamin, M.E. Lautenbacher, Phys. Lett. B 389 (1996) 749. S. Bertolini, J.O. Eeg, M. Fabbrichesi, E.I. Lashin, Nucl. Phys. B 514 (1998) 93. Y.-Y. Keum, U. Nierste, A.I. Sanda, Phys. Lett. B 457 (1999) 157. S. Bosch et al., Nucl. Phys. B 565 (2000) 3. For a recent review see A.J. Buras, in: F. David, R. Gupta (Eds.), Probing the Standard Model of Particle Interactions, Elsevier Science, 1998, hep-ph/9806471. Numerical results of Ref. [13] are updated in hep-ph/9901409, to appear in: P. Breitenlohner, D. Maison, J. Wess (Eds.), Recent Developments in Quantum Field Theory, Springer, Berlin, 2000 B. Winstein, L. Wolfenstein, Rev. of Mod. Phys. 65 (1993) 1113. G.D. Barr et al., (NA31 Collaboration), Phys. Lett. B 317 (1993) 233. L.K. Gibbons et al., (E731 Collaboration), Phys. Rev. Lett. 70 (1993) 1203. For the combined result of NA31 and E731, see Ref. [4]; for the latest world average, see D.E. Groom et al., Review of Particle Physics, Eur. Phys. J. C 15 (2000) 1. A. Alavi-Harati et al., (KTeV Collaboration), Phys. Rev. Lett. 83 (1999) 22.
D. Atwood et al. / Physics Reports 347 (2001) 1}222
215
[20] L. Wolfenstien, Phys. Rev. Lett. 13 (1964) 562. [21] See e.g., F.J. Gilman, K. Kleinknecht, B. Renk, in Ref. [4], p. 103. [22] P.F. Harrison, H.R. Quinn (Eds.), The BaBar Physics Book: Physics at an Asymmetric B Factory, SLAC-Report SLAC-R-504, 1998. [23] T. A!older et al., (CDF Collaboration), Phys. Rev. D 61 (2000) 072005. [24] F. Abe et al., (CDF Collaboration), Phys. Rev. Lett. 74 (1995) 2626. [25] S. Abachi et al., (D0 Collaboration), Phys. Rev. Lett. 74 (1995) 2632. [26] For the most recent D0 published values of the top quark mass see B. Abbott et al., (D0 Collaboration), Phys. Rev. D 60 (1999) 0521001. [27] For the most recent CDF published values of the top quark mass see F. Abe et al., (CDF Collaboration), Phys. Rev. Lett. 82 (1999) 271. [28] I.I. Bigi et al., Phys. Lett. B 181 (1986) 157. [29] D. Atwood, A. Soni, Phys. Rev. D 45 (1992) 2405. [30] W. Bernreuther, T. SchroK der, T.N. Pham, Phys. Lett. B 279 (1992) 389. [31] A. Soni, R.M. Xu, Phys. Rev. Lett. 69 (1992) 33. [32] D. Chang, W.-Y. Keung, I. Phillips, Nucl. Phys. B 408 (1993) 286; Erratum-ibid. B 429 (1994) 255. [33] C.R. Schmidt, M. Peskin, Phys. Rev. Lett. 69 (1992) 410. [34] G.L. Kane, G.A. Ladinsky, C.-P. Yuan, Phys. Rev. D 45 (1992) 124. [35] D. Atwood, A. Aeppli, A. Soni, Phys. Rev. Lett. 69 (1992) 2754. [36] S.L. Glashow, J. Iliopoulos, L. Maiani, Phys. Rev. D 2 (1970) 1285. [37] G. Eilam, J.L. Hewett, A. Soni, Phys. Rev. D 44 (1991) 1473; Erratum-ibid. D 59 (1999) 039901. [38] B. Grza7 dkowski, J.F. Gunion, P. Krawczyk, Phys. Lett. B 268 (1991) 106. [39] A.D. Sakharov, JETP Lett. 5 (1967) 24. [40] See e.g., A.G. Cohen, D.B. Kaplan, A.E. Nelson, Ann. Rev. Nucl. Part. Sci. 43 (1993) 27. [41] For a recent review on baryogenesis see A. Riotto, M. Trodden, Ann. Rev. Nucl. Part. Sci. 49 (1999) 35. [42] See e.g., A.E. Nelson, D.B. Kaplan, A.G. Cohen, Nucl. Phys. B 373 (1992) 453. [43] M. Aoki, A. Sugamoto, N. Oshimo, Prog. Theor. Phys. 98 (1997) 1325. [44] Y. Grossman, Y. Nir, R. Rattazzi, in: A.J. Buras, M. Lindner (Eds.), Heavy Flavors II, World Scienti"c, Singapore, 1997, p. 755. [45] W. Bernreuther, in: M. Jamin et al., (Eds.), Heidelberg 1996, Non-Perturbative Particle Theory and Experimental Tests, World Scienti"c, Singapore, 1997, p. 141, hep-ph/9701357 and references therein. [46] J. Schwinger, Phys. Rev. 82 (1951) 914. [47] J. Schwinger, Phys. Rev. 91 (1953) 713. [48] G. LuK ders, Kgl. Dansk. Vidensk. Selsk. Mat.-Fys. Medd. 28 C5 (1954). [49] W. Pauli, in: Niels Bohr and the Development of Physics, McGraw-Hill, New York, Pergamon, London, 1955. [50] W. Bernreuther, O. Nachtmann, P. Overmann, T. Schroder, Nucl. Phys. B 388 (1992) 53; Erratum-ibid. B 406 (1993) 516. [51] C.-P. Yuan, Mod. Phys. Lett. A 10 (1995) 627. [52] J. BernabeH u, M.B. Gavela, in: C. Jarlskog (Ed.), CP Violation, World Scienti"c, Singapore, 1989, p. 269. [53] C. Itzykson, J.-B. Zuber, Quantum Field Theory, McGraw-Hill, New York 1980, in particular Chapter 5. [54] S. Weinberg, The Quantum Theory of Fields, Vol. 1, Cambridge University Press, New York, 1995 (Chapter 3). [55] H. Simma, G. Eilam, D. Wyler, Nucl. Phys. B 352 (1991) 367. [56] J.-M. GeH rard, W.-S. Hou, Phys. Rev. Lett. 62 (1989) 855. [57] J.-M. GeH rard, W.-S. Hou, Phys. Rev. D 43 (1991) 2909. [58] L. Wolfenstein, Phys. Rev. D 43 (1991) 151. [59] D. Atwood et al., Phys. Rev. Lett. 70 (1993) 1364. [60] M. Bander, D. Silverman, A. Soni, Phys. Rev. Lett. 43 (1979) 242. [61] J.M. Soares, Phys. Rev. Lett. 68 (1992) 2102. [62] S. Weinberg, Phys. Rev. Lett. 37 (1976) 657. [63] T. Arens, L.M. Sehgal, Phys. Rev. D 51 (1995) 3525.
216 [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106]
D. Atwood et al. / Physics Reports 347 (2001) 1}222 G. Eilam, J.L. Hewett, A. Soni, Phys. Rev. Lett. 67 (1991) 1979. G. Eilam, J.L. Hewett, A. Soni, Phys. Rev. Lett. 68 (1992) 2103. M. Nowakowski, A. Pilaftsis, Phys. Lett. B 245 (1990) 185. M. Nowakowski, A. Pilaftsis, Mod. Phys. Lett. 6A (1991) 1933. A. Pilaftsis, Nucl. Phys. B 504 (1997) 61. S. Dittmaier, Proceedings of the Fourth International Symposium on Radiative Corrections, Barcelona, Spain, 1998, hep-ph/9811434. J. Papavassiliou, A. Pilaftsis, Phys. Rev. D 58 (1998) 053002. C.P. Burgess, J.A. Robinson, in: S. Dawson, A. Soni (Eds.), BNL Summer Study on CP Violation, World Scienti"c, Singapore, 1990, p. 205. D. Atwood et al., Phys. Lett. B 256 (1991) 471. J.F. Gunion, B. Grza7 dkowski, X.-G. He, Phys. Rev. Lett. 77 (1996) 5172. M. Davier et al., Phys. Lett. B 306 (1993) 411. K. Ackersta! et al., (OPAL Collaboration), Zeit. f. Phys. C 74 (1997) 403. M. Diehl, O. Nachtmann, Zeit. f. Phys. C 62 (1994) 397. D. Atwood, A. Soni, Phys. Rev. Lett. 74 (1995) 220. D. Atwood et al., Phys. Rev. D 53 (1996) 1162. J.F. Donoghue, Phys. Rev. D 18 (1978) 1632. E.P. Shabalin, Sov. J. Nucl. Phys. 28 (1978) 75. A. Czarnecki, B. Krause, Phys. Rev. Lett. 78 (1997) 4339. B. Grza7 dkowski, J.F. Gunion, Phys. Lett. B 350 (1995) 218. X.-G. He, hep-ph/9710551. H.R. Quinn, in Ref. [4], p. 558. Y. Nir, in: Proceedings of the 1998 European School of High-Energy Physics, St. Andrews, Scotland, August 1998, hep-ph/9810520. I.I. Bigi, hep-ph/9901366. C. Jarlskog, in: C. Jarlskog (Ed.), CP Violation, World Scienti"c, Singapore, 1989, p. 3. C. Jarlskog, Zeit. f. Phys. C 29 (1985) 491. C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039. L.-L. Chau, W.-Y. Keung, Phys. Rev. Lett. 53 (1984) 1802. V.D. Barger, R.J.N. Phillips, Collider Physics, Addison-Wesley, New York, 1987. E. Fernandez et al., Phys. Rev. Lett. 51 (1983) 1022. N.S. Lockyer et al., Phys. Rev. Lett. 51 (1983) 1316. L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945. P. Paganini, F. Parodi, P. Roudeau, A. Stocchi, Phys. Scripta. 58 (1998) 556. A. Stocchi, Proceedings of the Fourth Conference on Heavy Quarks at Fixed Target, Batavia, IL, October 1998, hep-ex/9902004. M. Artuso, Proceedings of the International Europhysics Conference On High-Energy Physics, Tampere, Finland, July 1999, hep-ph/9911347 and references therein. A.J. Buras, M.E. Lautenbacher, G. Ostermaier, Phys. Rev. D 50 (1994) 3433. C. Jarlskog, R. Stora, Phys. Lett. B 208 (1988) 268. J.D. Bjorken, Phys. Rev. D 39 (1989) 1396. C. Hamzaoui, J.L. Rosner, A.I. Sanda, in: A.J. Slaughter, N. Lockyer, M. Schmidt (Eds.), Proceedings of the Workshop on High Sensitivity Beauty Physics at Fermilab, Fermilab, Batavia, 1988, p. 215. J.L. Rosner, A.I. Sanda, M.P. Schmidt, in: A.J. Slaughter, N. Lockyer, M. Schmidt (Eds.), Proceedings of the Workshop on High Sensitivity Beauty Physics at Fermilab, Fermilab, Batavia, 1988, p. 165. N. Isgur, M. Wise, Phys. Rev. D 42 (1990) 2388. M. Neubert, Phys. Rep. 245 (1994) 259. Y. Kuramashi, Proceedings of the 17th International Symposium on Lattice Field Theory, Pisa, Italy, July 1999, hep-lat/9910032. G. Martinelli, Nucl. Phys. Proc. Suppl. 73 (1999) 58.
D. Atwood et al. / Physics Reports 347 (2001) 1}222
217
[107] A. Soni, Proceedings of the Third International Conference on B Physics and CP Violation, Taipei, Taiwan, December 1999, hep-ph/0003092. [108] S. Hashimoto, Proceedings of the 17th International Symposium on Lattice Field Theory, Pisa, Italy, July 1999, hep-lat/9909136. [109] T. Inami, C.S. Lim, Prog. Theor. Phys. 65 (1981) 297. [110] C. Bernard, T. Blum, A. Soni, Phys. Rev. D 58 (1998) 014501. [111] L. Lellouch et al., Proceedings of the Eighth International Symposium on Heavy Flavor Physics, Southampton, England, July 1999, hep-ph/9912322. [112] M. Feindt, Proceedings of the International Europhysics Conference on High-Energy Physics, Jerusalem, Israel, August 1997, hep-ph/9802380. [113] See the contributions by J. Alexander, D. Karlen, P. Rosnet, Proceedings of the International Conference on High Energy Physics, Vancouver, Canada, July 1998. [114] http://www-cdf.fnal.gov/physics/new/bottom/cdf4855/cdf4855.html. [115] T.D. Lee, Phys. Rev. D 8 (1973) 1226. [116] T.D. Lee, Phys. Rep. 9 (1974) 143. [117] J.F. Gunion, H.E. Haber, G.L. Kane, S. Dawson, The Higgs Hunter's Guide, Addison-Wesley, New York, 1990. [118] H.E. Haber, in: G.L. Kane (Ed.), Perspectives on Higgs Physics II, World Scienti"c, Singapore, 1997, p. 23. [119] S. Weinberg, Phys. Rev. D 42 (1990) 860. [120] Y.-L. Wu, hep-ph/9404241 (unpublished). [121] Y.-L. Wu, hep-ph/9404271 (unpublished). [122] Y.-L. Wu, L. Wolfenstein, Phys. Rev. Lett. 73 (1994) 1762. [123] S. Glashow, S. Weinberg, Phys. Rev. D 15 (1977) 1958. [124] C.D. Froggatt, R.G. Moorhouse, I.G. Knowles, Nucl. Phys. B 386 (1992) 63. [125] W. Bernreuther, Proceedings of the 37th International University School of Nuclear and Particle Physics: Broken Symmetries, Schladming, Austria, March 1998, hep-ph/9808453. [126] T.P. Cheng, M. Sher, Phys. Rev. D 35 (1987) 3484. [127] M. Sher, Y. Yuan, Phys. Rev. D 44 (1991) 1461. [128] D. Atwood, L. Reina, A. Soni, Phys. Rev. D 55 (1997) 3156. [129] L. Wolfenstein, Y.-L. Wu, Phys. Rev. Lett. 73 (1994) 2809. [130] J.F. Gunion, B. Grza7 dkowski, Phys. Lett. B 243 (1990) 301. [131] V. Barger, J.L. Hewett, R.J.N. Phillips, Phys. Rev. D 41 (1990) 3421. [132] For a review see J. L. Hewett, Proceedings of the SLAC Summer Inst. on Particle Physics: Spin Structure in High Energy Processes, Stanford, CA, 1993, hep-ph/9406302. [133] G. Cvetic, Proceedings of the Workshop on Electroweak Symmetry Breaking, World Scienti"c, Singapore, 1995, Budapest, Hungary, 1994, hep-ph/9411436. [134] D. Chakraverty, A. Kundu, Mod. Phys. Lett. A 11 (1996) 675. [135] A. Coarasa, R.A. Jimenez, J. Sola, Phys. Lett. B 406 (1997) 337. [136] K. Kiers, A. Soni, Phys. Rev. D 56 (1997) 5786. [137] M. Krawczyk, J. Zochowski, Phys. Rev. D 55 (1997) 6968. [138] J.F. Gunion, B. Grza7 dkowski, H.E. Haber, J. Kalinowski, Phys. Rev. Lett. 79 (1997) 982. [139] J.L. Hewett, T. Takeuchi, S. Thomas, in: T. Barklow et al., (Eds.), Electroweak Symmetry Breaking and New Physics at TeV Scale, World Scienti"c, Singapore, 1996, p. 548, hep-ph/9603391. [140] D.A. Dicus, D.J. Muller, S. Nandi, Phys. Rev. D 59 (1999) 055007. [141] F. Abe et al., (CDF Collaboration), Phys. Rev. Lett. 79 (1997) 357. [142] P. Abreu et al., (DELPHI Collaboration), Phys. Lett. B 420 (1998) 140. [143] M. Acciarri et al., (L3 Collaboration), Phys. Lett. B 446 (1999) 368. [144] G. Abbiendi et al., (OPAL Collaboration), Eur. Phys. J. C 7 (1999) 407. [145] R. Barate et al., (ALEPH Collaboration), hep-ex/9902031. [146] B. Abbott et al., (D0 Collaboration), Phys. Rev. Lett. 82 (1999) 4975. [147] K. Desch, Proceedings of the International Conference on High Energy Physics, Vancouver, Canada, July 1998. [148] A. Sopczak, Mod. Phys. Lett. A 10 (1995) 1057.
218
D. Atwood et al. / Physics Reports 347 (2001) 1}222
[149] M. Krawczyk, in: Tran Thanh Van (Ed.), Les Arcs 1996, Electroweak Interactions and Uni"ed Theories, page 221, Edition FrontieH rs, Dreux, 1996, hep-ph/9607268. [150] M. Krawczyk, J. Zochowski, P. Mattig, Eur. Phys. J. C 8 (1999) 495. [151] We thank JoAnne Hewett for providing us with updates of "gures from Ref. [132]. [152] P. Krawczyk, S. Pokorski, Phys. Rev. Lett. 60 (1988) 182. [153] Y. Grossman, H. Haber, Y. Nir, Phys. Lett. B 357 (1995) 630. [154] J.L. Hewett, J.D. Wells, Phys. Rev. D 55 (1997) 5549. [155] M.S. Alam et al., (CLEO Collaboration), Phys. Rev. Lett. 74 (1995) 2885. [156] S. Glenn et al., (CLEO Collaboration), ICHEP-98-1011, Proceedings of the International Conference on High Energy Physics, Vancouver, Canada, July 1998. [157] G. Eigen, in: Proceedings of the Fourth International Symposium on Radiative Corrections: Applications of Quantum Field Theory to Phenomenology, Barcelona, Catalonia, Spain, September 1998, hep-ex/9901005. [158] R. Barate et al., (ALEPH Collaboration), Phys. Lett. B 429 (1998) 169. [159] G.C. Branco, A.J. Buras, J.M. GeH rard, Nucl. Phys. B 259 (1985) 306. [160] H.-Y. Cheng, Int. J. Mod. Phys. A 7 (1992) 1059. [161] C.H. Albright, J. Smith, S.-H.H. Tye, Phys. Rev. D 21 (1980) 711. [162] K. Shizuya, S.-H.H. Tye, Phys. Rev. D 23 (1981) 1613. [163] G.C. Branco, Phys. Rev. Lett. 44 (1980) 504. [164] G.C. Branco, Phys. Rev. D 22 (1980) 2901. [165] Y. Grossman, Y. Nir, Phys. Lett. B 313 (1993) 126. [166] D. Atwood, G. Eilam, A. Soni, Phys. Rev. Lett. 71 (1993) 492. [167] Y. Grossman, Nucl. Phys. B 426 (1994) 355. [168] A.J. Buras, P. Krawczyk, M.E. Lautenbacher, C. Salazar, Nucl. Phys. B 337 (1990) 284. [169] G. Isidori, Phys. Lett. B 298 (1993) 409. [170] P. Krawczyk, S. Pokorski, Nucl. Phys. B 364 (1991) 10. [171] Y. Nir, in: L. Vassilian (Ed.), Proceedings of the SLAC Summer Inst. on Particle Physics: The Third Family and the Physics of Flavor, SLAC-Report-412, 1993, p. 81. [172] H.P. Nilles, Phys. Rep. 110 (1984) 1. [173] H.E. Haber, G.L. Kane, Phys. Rep. 117 (1985) 75. [174] Articles, in: G.L. Kane (Ed.), Perspectives on Supersymmetry, World Scienti"c, Singapore, 1998. [175] H.E. Haber in Ref. [4], p. 743. [176] S. Pokorski, Acta Phys. Polon. B 30 (1999) 1759. [177] H. Dreiner, Proceedings of the Zuoz Summer School on Hidden Symmetries and Higgs Phenomena, Zuoz, Switzerland, August 1998, hep-ph/9902347. [178] M. Dugan, B. Grinstein, L. Hall, Nucl. Phys. B 255 (1985) 413. [179] K.S. Babu, C. Kolda, J. March-Russel, F. Wilczek, Phys. Rev. D 59 (1999) 016004. [180] D. Chang, W. Keung, A. Pilaftsis, Phys. Rev. Lett. 82 (1999) 900; Erratum-ibid. 83 (1999) 3972. [181] A. Pilaftsis, Phys. Lett. B 471 (1999) 174. [182] R. Garisto, J.D. Wells, Phys. Rev. D 55 (1997) 1611. [183] The most recent Review of Particle Physics [4] gives d (0.97;10\ e cm (see P.G. Harris et al., Phys. Rev. L Lett. 82 (1999) 904). Both results are at 90% C.L.. [184] L. Alvarez-Gaume, J. Polchinski, M.B. Wise, Nucl. Phys. B 221 (1983) 495. [185] J. Rosiek, Phys. Rev. D 41 (1990) 3464 and hep-ph/9511250 (unpublished erratum). [186] M. Kuroda, hep-ph/9902340. [187] See, e.g., S. Dimopoulos, S. Thomas, Nucl. Phys. B 465 (1996) 23. [188] J.F. Gunion, H.E. Haber, Nucl. Phys. B 272 (1986) 1. [189] Y. Kizukuri, N. Oshimo, Phys. Rev. D 46 (1992) 3025. [190] Y. Kizukuri, N. Oshimo, Phys. Rev. D 45 (1992) 1806. [191] V. Barger, M.S. Berger, P. Ohmann, Phys. Rev. D 49 (1994) 4908. [192] P. Nath, Phys. Rev. Lett. 66 (1991) 2565. [193] M.M. El Kheishen, A.A. Sha"k, A.A. Aboshousha, Phys. Rev. D 45 (1992) 4345.
D. Atwood et al. / Physics Reports 347 (2001) 1}222 [194] [195] [196] [197] [198] [199] [200] [201] [202] [203] [204] [205] [206] [207] [208] [209] [210] [211] [212] [213] [214] [215] [216] [217]
[218] [219] [220] [221] [222] [223] [224] [225] [226] [227] [228] [229] [230] [231] [232] [233] [234] [235] [236] [237]
219
T. Ibrahim, P. Nath, Phys. Rev. D 57 (1998) 478, Erratum-ibid. D 58 (1998) 019901. T. Ibrahim, P. Nath, Phys. Rev. D 58 (1998) 111301. M. Brhlik, G.J. Good, G.L. Kane, Phys. Rev. D 59 (1999) 115004. W. Marciano, in: S. Dawson, A. Soni (Eds.), Proceedings of the Summer Study on CP Violation, World Scienti"c, Singapore, 1990, Upton, New York, 1990. S. Barr, W. Marciano, in: C. Jarlskog (Ed.), CP Violation, World Scienti"c, Singapore, 1989, p. 455. W. Bernreuther, M. Suzuki, Rev. Mod. Phys. 63 (1991) 313. N.F. Ramsey, Annu. Rev. Nucl. Part. Sci. 40 (1991) 1. E. Shabalin, in: Tran Thanh Van (Ed.), Electroweak Interactions and Uni"ed Theories, Edition FrontieH rs, 1997, p. 483. T. Falk, Nucl. Phys. Proc. Suppl. A 52 (1997) 78. J. Ellis, S. Ferrara, D.V. Nanopoulos, Phys. Lett. B 114 (1982) 231. W. BuchmuK ller, D. Wyler, Phys. Lett. B 121 (1983) 321. J. Polchinski, M.B. Wise, Phys. Lett. B 125 (1983) 393. F. del Aguila, M.B. Gavela, J.A. Grifols, A. Mendez, Phys. Lett. B 126 (1983) 71. J.M. GeH rard, W. Grimus, A. Raychaudhauri, Phys. Lett. B 145 (1984). E. Franco, M. Mangano, Phys. Lett. B 135 (1984) 445. J.M. GeH rard et al., Nucl. Phys. B 253 (1985) 93. A.I. Sanda, Phys. Rev. D 32 (1985) 2992. T. Kurimoto, Prog. Theor. Phys. 73 (1985) 209. W. Fischler, S. Paban, S. Thomas, Phys. Lett. B 289 (1992) 373. Y. Nir, R. Rattazzi, Phys. Lett. B 382 (1996) 363. T. Moroi, Phys. Lett. B 447 (1999) 75. S. Bar-Shalom, D. Atwood, A. Soni, Phys. Rev. D 57 (1998) 1495. S. Dawson, TASI 97 lectures, hep-ph/9712464. X. Tata, in: J.C.A. Barata, A.P.C. Malbouisson, S.F. Novaes (Eds.), Proceedings of the 9th Jorge Andre Swieca Summer School: Particles and Fields, World Scienti"c, Singapore, 1998, Sao Paulo, Brazil, 1997, p. 404, hep-ph/9706307. M. Carena et al., hep-ex/9712022. S. Katsanevas, J. Phys. G 24 (1998) 337. D.P. Roy, Proceedings of the 13th Topical Conference on Hadron Collider Physics, Mumbai, India, January 1999, hep-ph/9902298. M. Dittmar, Proceedings of the Zuoz Summer School on Hidden Symmetries and Higgs Phenomena, Zuoz, Switzerland, August 1998, hep-ex/9901004. E.L. Berger, B. Harris, M. Klasen, T. Tait, summary given at Physics at Run II: Workshop on Supersymmetry/Higgs: Summary Meeting, Batavia, IL, November 1998, hep-ph/9903237. J. Erler, D.M. Pierce, Nucl. Phys. B 526 (1998) 53 updated in J. Erler, hep-ph/9903449. J. Ellis, R.A. Flores, Phys. Lett. B 377 (1996) 83. D. Atwood, S. Bar-Shalom, A. Soni, Phys. Rev. D 51 (1995) 1034. E. Christova, M. Fabbrichesi, Phys. Lett. B 315 (1993) 338. W. Bernreuther, P. Overmann, Z. Phys. C 61 (1994) 599. B. Grza7 dkowski, Phys. Lett. B 305 (1993) 384. A. Bartl, E. Christova, W. Majerotto, Nucl. Phys. B 460 (1996) 235; Erratum-ibid. Nucl. Phys. B 465 (1996) 365. A. Bartl, E. Christova, T. Gajdosik, W. Majerotto, Nucl. Phys. B 507 (1997) 35; Erratum-ibid. B 531 (1998) 653. C.R. Schmidt, Phys. Lett. B 293 (1992) 111. J.P. Ma, A. Brandenburg, Z. Phys. C 56 (1992) 97. C. Nelson, B.T. Kress, M. Lopez, T.P. McCauley, Phys. Rev. D 56 (1997) 5928. J.M. Yang, B.-L. Young, Phys. Rev. D 56 (1997) 5907. B. Grza7 dkowski, W.-Y. Keung, Phys. Lett. B 319 (1993) 526. T. Hasuike, T. Hattori, S. Wakaizumi, Phys. Rev. D 58 (1998) 095008. B. Dutta-Roy et al., Phys. Rev. Lett. 65 (1990) 827.
220 [238] [239] [240] [241] [242] [243] [244] [245] [246] [247] [248] [249] [250] [251] [252] [253] [254] [255] [256] [257] [258] [259] [260] [261] [262] [263] [264] [265] [266] [267] [268] [269] [270] [271] [272] [273] [274]
[275] [276] [277] [278] [279]
D. Atwood et al. / Physics Reports 347 (2001) 1}222 B. Dutta-Roy et al., Phys. Rev. D 43 (1991) 268. N.G. Deshpande et al., Phys. Rev. D 43 (1991) 3591. N.G. Deshpande, B. Margolis, H.D. Trottier, Phys. Rev. D 45 (1992) 178. B. Grza7 dkowski, J.F. Gunion, P. Krawczyk, Phys. Lett. B 286 (1991) 106. D. Atwood, G. Eilam, A. Soni, Phys. Rev. D 49 (1994) 289. J. Papavassiliou, in: J. Gunion, T. Han, J. Ohnemus (Eds.), Proceedings of Beyond the Standard Model IV, World Scienti"c 1995, Lake Tahoe, December 1994, p. 509. J. Liu, Phys. Rev. D 47 (1993) 1741, in: D.I. Britton, D.B. MacFarlane, P.M. Patel (Eds.), Proceedings of the 5th Workshop on Heavy Flavors Physics, Editions Frontiers, Dreux, 1994, p. 639, Montreal, July 1993. G. LoH pez Castro, J.L. Lucio, J. Pestieau, Int. J. Mod. Phys. A 11 (1996) 563, and references therein. E. Christova, M. Fabbrichesi, Phys. Lett. B 320 (1994) 299. X. Bi, Y. Dai, Eur. Phys. J. C 12 (2000) 125. M. Aoki, N. Oshimo, Nucl. Phys. B 531 (1998) 49. W. Bernreuther, O. Nachtmann, P. Overmann, T. Schroder, Nucl. Phys. B 388 (1992) 53; Erratum-ibid. B 406 (1993) 516. B. Grza7 dkowski, J.F. Gunion, Phys. Lett. B 287 (1992) 237. B. Grza7 dkowski, W.-Y. Keung, Phys. Lett. B 316 (1993) 137. R. Cruz, B. Grza7 dkowski, J.F. Gunion, Phys. Lett. B 289 (1992) 440. E. Accomando et al., (ECFA/DESY LC Physics Working Group), Phys. Rep. 299 (1998) 1. S. Kuhlman et al., (NLC ZDR Design Group and NLC Physics Working Group), hep-ex/9605011. H. Murayama, M.E. Peskin, Ann. Rev. Nucl. Part. Sci. 49 (1996) 513. R. Frey, in: A. Miyamoto, Y. Fujii, T. Matsui, S. Iwata (Eds.), Proceedings of the 3rd Workshop on Physics and Experiments with e>e\ Linear Colliders, World Scienti"c, Singapore, 1996, p. 144, hep-ph/9606201. M.S. Baek, S.Y. Choi, C.S. Kim, Phys. Rev. D 56 (1997) 6835. D. Atwood, S. Bar-Shalom, G. Eilam, A. Soni, Phys. Rev. D 54 (1996) 5412. W. Hollik et al., Nucl. Phys. B 551 (1999) 3. W. Bernreuther, P. Overmann, Z. Phys. C 72 (1996) 461. W. Bernreuther, A. Brandenburg, P. Overmann, in: P.M. Zerwas (Ed.), e>e\ Linear Collisions: The Physics Potential, 1995, p. 49, hep-ph/9602273. B. Grza7 dkowski, Z. Hioki, Nucl. Phys. B 484 (1997) 17. B. Grza7 dkowski, Z. Hioki, Phys. Lett. B 391 (1997) 172. L. Brzezinski, B. Grza7 dkowski, Z. Hioki, Int. J. Mod. Phys. A 14 (1999) 1261. A. Bartl, E. Christova, T. Gajdosik, W. Majerotto, Phys. Rev. D 58 (1998) 074007. A. Bartl, E. Christova, T. Gajdosik, W. Majerotto, hep-ph/9803426. A. Bartl, E. Christova, T. Gajdosik, W. Majerotto, Int. J. Mod. Phys. A 14 (1999) 1. F. Cuypers, S.D. Rindani, Phys. Lett. B 343 (1995) 333. P. Poulose, S.D. Rindani, Phys. Lett. B 349 (1995) 379. P. Poulose, S.D. Rindani, Phys. Rev. D 54 (1996) 4326. T. Arens, L.M. Sehgal, Phys. Rev. D 50 (1994) 4372. M. Jezabek, T. Nagano, Y. Sumino, Phys. Rev. D 62 (2000) 014034. S. Bar-Shalom, D. Atwood, A. Soni, Phys. Lett. B 419 (1998) 340. S. Bar-Shalom, (expanded version of a talk given at International Europhysics Conference on High-Energy Physics (HEP 97), Jerusalem, Israel, 1997), in: D. Lellouch, G. Mikenberg, E. Rabinovici (Eds.), Proceedings of the International Europhysics Conference on High-Energy Physics, Springer, Berlin, Germany, 1999, p. 1182, hep-ph/9710355. B. Grzadkowski, J. Pliszka, Phys. Rev. D 60 (1999) 115018. A. Djouadi, J. Kalinowski, P.M. Zerwas, Mod. Phys. Lett. A 7 (1992) 1765. A. Djouadi, J. Kalinowski, P.M. Zerwas, Z. Phys. C 54 (1992) 255. X. Zhang, S.K. Lee, K. Whisnant, B.L. Young, Phys. Rev. D 50 (1994) 7042. J.F. Gunion, X.-G. He, in: Proceedings of the 1996 DPF/DPB Summer Study for New Directions in High Energy Physics, Snowmass, Colorado, June 1996, hep-ph/9609453.
D. Atwood et al. / Physics Reports 347 (2001) 1}222 [280] [281] [282] [283] [284] [285] [286] [287] [288] [289] [290] [291] [292] [293] [294] [295] [296] [297] [298] [299] [300] [301] [302] [303] [304] [305] [306] [307] [308] [309] [310] [311] [312] [313] [314] [315] [316] [317] [318] [319] [320] [321]
[322]
221
K. Hagiwara, H. Murayama, I. Watanabe, Nucl. Phys. B 367 (1991) 257. K. Fujii, Proceedings of the 4th KEK Topical Conference on Flavor Physics, 1996 and references therein. T. Han, T. Huang, Z.H. Lin, J.X. Wang, X. Zhang, Phys. Rev. D 61 (2000) 015006. F. Harris et al. (Eds.), Proceedings of the Workshop on Physics and Experiments with Linear e>e\ Colliders, World Scienti"c, Singapore, 1993. A. Miyamoto, Y. Fujii (Eds.), Proceedings of the Workshop on Physics and Experiments with Linear e>e\ Colliders, World Scienti"c, Singapore, 1996. H. Baer, S. Dawson, L. Reina, Phys. Rev. D 61 (2000) 013002. S. Dawson, L. Reina, in: Proceedings of the Fourth International Workshop on Linear Colliders, Barcelona, Spain, April 1999, hep-ph/9910228. W. Bernreuther, A. Brandenburg, M. Flesch, Phys. Rev. D 56 (1997) 90. B. Grza7 dkowski, Phys. Lett. B 338 (1994) 71. W. Bernreuther, A. Brandenburg, Phys. Lett. B 314 (1993) 104. D. Chang, W.-Y. Keung, I. Phillips, Phys. Rev. D 48 (1993) 3225. W. Bernreuther, A. Brandenburg, Phys. Rev. D 49 (1994) 4481. X.-G. He, J.P. Ma, B.H.J. McKellar, Phys. Rev. D 49 (1994) 4548. X.-G. He, J.P. Ma, B.H.J. McKellar, Mod. Phys. Lett. A 9 (1994) 205. M. KraK mer, J. Kuhn, M.L. Strong, P.M. Zerwas, Z. Phys. C 64 (1994) 21. T. Arens, U. Gieseler, L.M. Sehgal, Phys. Lett. B 339 (1994) 127. S. Bar-Shalom, D. Atwood, G. Eilam, A. Soni, Z. Phys. C 72 (1996) 79. T. Rizzo, talk given at the 1996 DPF/DPB Summer Study on New Directions for High-energy Physics, Snowmass, CO, 1996, hep-ph/9610373. S.D. Rindani, M.M. Tung, Phys. Lett. B 424 (1998) 125. D. Atwood, A. Soni, hep-ph/9607481. R. Cahn, S. Dawson, Phys. Lett. B 136 (1984) 196; Erratum-ibid. B 136 (1984) 464. S. Dawson, Nucl. Phys. B 249 (1985) 42. M. Chanowitz, M.K. Gaillard, Phys. Lett. B 142 (1984) 85. G.L. Kane, W.W. Repko, W.B. Rolnick, Phys. Lett. B 148 (1984) 367. R. Kau!man, Phys. Rev. D 41 (1989) 3343. P.W. Johnson, F.I. Olness, W.-K. Tung, Phys. Rev. D 36 (1987) 291. L.R. Evans, CERN LHC Project Report 101, talk given at the Particle Accelerator Conference (PAC 97), Vancouver, May 1997. N.V. Krasnikov, V.A. Matveev, Phys. Part. Nucl. 28 (1997) 441. I. Hinchli!e, in: Z. Parsa (Ed.), Precision Physics at the LHC and Future High Energy Colliders, American Inst. of Phys. pub., 1996, p. 129. H.-Y. Zhou, Phys. Rev. D 58 (1998) 114002. S.Y. Choi, C.S. Kim, J. Lee, Phys. Lett. B 415 (1997) 67. J.F. Gunion, T.C. Yuan, B. Grza7 dkowski, Phys. Rev. Lett. 71 (1993) 488; Erratum-ibid. 71 (1997) 2681. W. Bernreuther, A. Brandenburg, M. Flesch, hep-ph/9812387. M. Diemoz et al., Zeit. f. Phys. C 39 (1988) 21. R. Brown et al., Phys. Lett. B 43 (1973) 403. D. Amidei et al., (TeV-2000 Study Group), FERMILAB-PUB-96/082, April 1996. K. Cheung, Phys. Rev. D 53 (1996) 3604. P. Harbel, O. Nachtmann, A. Wilch, Phys. Rev. D 53 (1996) 4875. T. Rizzo, Phys. Rev. D 53 (1996) 6218. B. Grza7 dkowski, B. Lampe, K.J. Abraham, Phys. Lett. B 415 (1997) 193. R. Frey et al., hep-ph/9704243. J.-M. Yang et al., Physics at the "rst muon collider, talk given at Workshop on Physics at the First Muon Collider and at the Front End of the Muon Collider, Batavia, IL, November 1997, p. 783, hep-ph/9802305 and references therein. T. Stelzer, S. Willenbrock, Phys. Lett. B 357 (1995) 125.
222
D. Atwood et al. / Physics Reports 347 (2001) 1}222
[323] D.O. Carlson, C.-P. Yuan, Proceedings of the Workshop on Physics of the Top Quark, Ames, IA, May 1995, hep-ph/9509208. [324] A.P. Heinson, A.S. Belyaev, E.E. Boos, Phys. Rev. D 56 (1996) 3114. [325] T. Stelzer, Z. Sullivan, S. Willenbrock, Phys. Rev. D 56 (1997) 5919. [326] T. Tait, C.-P. Yuan, hep-ph/9710372. [327] A. Belyaev, E.E. Boos, L. Dudko, Proceedings of the 32nd Rencontres de Moriond: Electroweak Interactions and Uni"ed Theories, Les Arcs, France, March 1997, hep-ph/9804328. [328] A. Belyaev, E.E. Boos, L. Dudko, Phys. Rev. D 59 (1999) 075001. [329] T. Stelzer, Z. Sullivan, S. Willenbrock, Phys. Rev. D 58 (1998) 094021. [330] See Shaouly Bar-Shalom, Heavy Quark Physics and CP-Violation, Ph.D Thesis, Technion-IIT, Israel, 1996. [331] G. Mahlon, hep-ph/9811219. [332] I. Ginzburg et al., Pizma ZhETF 34 (1981) 514. [333] I. Ginzburg et al., JETP Lett. 34 (1982) 491. [334] I. Ginzburg et al., Nucl. Instr. and Meth. 205 (1983) 47. [335] I. Ginzburg et al., Nucl. Instr. and Meth. 219 (1984) 5. [336] V. Telnov, Turk. J. Phys. 22 (1998) 541. [337] R. Brinkman et al., Nucl. Instrum. Meth. A 406 (1998) 13. [338] G. Baur, K. Hencken, Trautmann, J. Phys. G 24 (1998) 1657. [339] H. Anlauf, W. Bernreuther, A. Brandenburg, Phys. Rev. D 52 (1995) 3803; Erratum-ibid. D 53 (1996) 1725. [340] B. Grza7 dkowski, J.F. Gunion, Phys. Lett. B 294 (1992) 361. [341] S.Y. Choi, K. Hagiwara, Phys. Lett. B 359 (1995) 369. [342] P. Poulose, S.D. Rindani, Phys. Rev. D 57 (1998) 5444. [343] D. Neu!er, FERMILAB-FN (1979) 319. [344] D. Neu!er, Particle Accelerators, 14 (1983) 75, in: F.T. Cole, R. Donaldson (Eds.), Proceedings of the 12th International Conference on High Energy Accelerators, Batavia, August 1983, p. 481. [345] D.B. Cline (Ed.), Proceedings of the 3rd Symposium on Physics Potential and Development of > \ Colliders, San Francisco, CA, December 1995 (Nucl. Phys. Proc. Supp. A 51 (1996)). [346] V. Barger et al., Proceedings of the ITP Conference on Future High-energy Colliders, Santa Barbara, CA, October 1996, hep-ph/9704290. [347] V. Barger et al., Phys. Rep. 286 (1997) 1. [348] D.B. Cline, in: J. Gunion, T. Han, J. Ohnemus (Eds.), Proceedings of the Workshop on Beyond the Standard Model IV, 1994, p. 164. [349] C.M. Ankenbrandt et al., Phys. Rev. ST Accel. Beams 2 (1999) 081001. [350] B. Austin, A. Blondel, J. Ellis (Eds.), Prospective Study of Muon Storage Rings at CERN, CERN 99-02, 1999. [351] D. Atwood, A. Soni, Phys. Rev. D 52 (1995) 6271. [352] B. Grzadkowski, J.F. Gunion, J. Pliszka, hep-ph/0003091. [353] A. Pilaftsis, Phys. Rev. Lett. 77 (1996) 4996. [354] W.-S. Hou, Phys. Lett. B 296 (1992) 179. [355] M. Luke, M.J. Savage, Phys. Lett. B 307 (1993) 387. [356] D. Atwood, L. Reina, A. Soni, Phys. Rev. Lett. 75 (1995) 3800. [357] M. Chemtob, G. Moreau, hep-ph/9910543. [358] D. Atwood, L. Reina, A. Soni, Phys. Rev. D 53 (1996) 1199. [359] For the FF-package that was used for numerical evaluation of loop integrals see G.J. van Oldenborgh, Comput. Phys. Commun. 66 (1991) 1. [360] For the algorithms used in the FF-package see G.J. van Oldenborgh, J.A.M. Vermaseren, Z. Phys. C 46 (1990) 425. [361] G. Passarino, M. Veltman, Nucl. Phys. B 160 (1979) 151.
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EMBEDDED RANDOM MATRIX ENSEMBLES FOR COMPLEXITY AND CHAOS IN FINITE INTERACTING PARTICLE SYSTEMS
V.K.B. KOTAa,b a
b
Physical Research Laboratory, Ahmedabad 380 009, India Max-Planck-Institut fuK r Kernphysik, Postfach 10 39 80, D-69029 Heidelberg, Germany
AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO
Physics Reports 347 (2001) 223}288
Embedded random matrix ensembles for complexity and chaos in "nite interacting particle systems V.K.B. Kota * Physical Research Laboratory, Ahmedabad 380 009, India Max-Planck-Institut fu( r Kernphysik, Postfach 10 39 80, D-69029 Heidelberg, Germany Received September 2000; editor: W. Weise
Contents 1. Introduction 2. Embedded ensembles: EGOE(k) 2.1. De"nition 2.2. Basic results 2.3. Statistical spectroscopy 3. Deformed embedded ensembles 3.1. Onset of chaos in "nite interacting many-particle systems 3.2. Strength functions: transition from Breit}Wigner to Gaussian form 3.3. Statistical mechanics for "nite systems of interacting particles via smoothed strength functions 3.4. Other deformed EGOE 4. Interaction-driven thermalization and Fockspace localization 4.1. Chaos and interaction-driven thermalization
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4.2. Embedded ensembles and Fock space localization 5. Conclusions Acknowledgements Appendix A. Some basic results for classical random matrix ensembles Appendix B. EGOE(2) for Boson systems Appendix C. Edgeworth expansions Appendix D. EGOE results for NPC and S Appendix E. Unitary decomposition of the Hamiltonian and trace propagation E.1. Unitary decomposition of operators E.2. Trace propagation Appendix F. Convolution forms in statistical spectroscopy References
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Abstract Universal properties of simple quantum systems whose classical counter parts are chaotic, are modeled by the classical random matrix ensembles and their interpolations/deformations. However for "nite interacting
* Corresponding author. Physical Research Laboratory, Ahmedabad 380 009, India. Tel.: 91-79-6302129; fax: 91-796301502. E-mail address: [email protected] (V.K.B. Kota). 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 1 3 - 7
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many-particle systems such as atoms, molecules, nuclei and mesoscopic systems (atomic clusters, helium droplets, quantum dots, etc.) for wider range of phenomena, it is essential to include information such as particle number, number of single-particle orbits, lower particle rank of the interaction, etc. These considerations led to resurgence of interest in investigating in detail the so-called embedded random matrix ensembles and their various deformed versions. Besides giving a overview of the basic results of embedded ensembles for the smoothed state densities and transition matrix elements, recent progress in investigating these ensembles with various deformations, for deriving a statistical mechanics (with relationships between quantum chaos, thermalization, phase transitions and Fock space localization, etc.) for isolated "nite systems with few particles is brie#y discussed. These results constitute new progress in deriving a basis for statistical spectroscopy (introduced and applied in nuclear structure physics and more recently in atomic physics) and its domains of applicability. 2001 Elsevier Science B.V. All rights reserved. PACS: 02.50.Ey; 05.45.Mt; 21.10.!k; 21.60.Cs; 24.60.Lz Keywords: Chaos; Shell model; Random matrix ensembles; GOE; EGOE; Information entropy; Bivariate strength distributions; Strength functions; Statistical spectroscopy; Statistical mechanics; Finite interacting many-particle systems; Fock space localization
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1. Introduction The slow neutron resonances mark a region of chaos in heavy nuclei [1,2]. The level and strength #uctuations in this region are described by the Gaussian orthogonal ensemble of random matrices which are "rst introduced and studied by Wigner in the 1950s. The classical (Gaussian orthogonal (GOE), unitary (GUE) and sympletic (GSE)) ensembles are classi"ed by Dyson and studied in detail by Wigner, Dyson, Mehta, Gaudin, Porter, Rosenzweig, French, Pandey, Bohigas and others in the 1960s and 1970s. Signi"cant results of GOE are: (i) the nearest-neighbor spacing (S) distribution P(S) dS (of unfolded spectra) well represented by the Wigner's surmise P(S) dS&Se\1 dS (but not by the Poisson law e\1 dS); (ii) the Dyson}Mehta spectral statistic showing spectral rigidity; (iii) locally renormalized transition strengths (x) obey the Porter}Thomas law P(x) dx&x\ e\V dx (Appendix A). These results are well found in the slow neutron resonance data produced by the Columbia group in the 1960s and early 1970s, however a decisive positive test of the GOE model via these data came from the much improved data analysis by Bohigas et al. [3]. This and the seminal paper by Bohigas et al. in 1984 [4] on the analysis of level #uctuations of the quantum Sinai's billiard whose classical counter part is known to be completely chaotic has established that the #uctuation properties of classical random matrices are generic and therefore applicable for local spectral statistics of a wide variety of quantal systems. In fact, as Berry states [5] if the system is classically integrable corresponds to that of Poisson systems, if the system is classically chaotic and has no symmetry corresponds to that of GUE and if the system is chaotic and has time reversal symmetry corresponds to that of GOE. Study of various billiard systems, kicked rotor, kicked top, quartic oscillator, quantum maps, polynomial potentials in two dimensions, experiments with hydrogen atom in a strong magnetic "eld, microwaves in metal cavities (superconducting microwave billiards) and acoustic resonances in aluminum blocks, etc. by large number of authors in post Bohigas era (i.e. from 1984 onwards) established random matrix physics to be one of the central themes of quantum chaotic systems. In the study of these systems and applications to real physical systems (atoms, molecules, nuclei, mesoscopic systems, etc.), it became clear that one has to consider various interpolations and deformations of the Gaussian ensembles with or without the regular Poisson and uniform spectra. To this end studied are: (i) GOE}GUE, GOE}GSE, GSE}GUE interpolations, with, for example GOE}GUE giving bounds on the time reversal non-invariant (TRNI) part of the nucleon}nucleon interaction, statistics of levels in a metallic ring when pierced by a magnetic "eld, etc.; (ii) Poisson to GOE, GUE and GSE and uniform to GOE, GUE and GSE, with, for example Poisson to GOE being important for order}chaos transitions in the low-lying and near-yrast levels in nuclei, Poisson to GUE for metal}insulator transitions, etc.; (iii) partitioned random matrix ensembles with the 2;2
The word chaos (or quantum chaos) refers to quantum systems which in the classical limit show chaotic dynamics. However, the usage of this phrase even for systems like atomic nuclei, which do not possess a classical limit, is now quite wide spread. At present, it should be understood that complex quantum states are referred as chaotic. In fact, most of the studies presented in this article attempt to characterize and quantify complexity and/or chaos in interacting many particle systems. The dynamics of a classical Hamiltonian system is called regular if the orbits of the system are stable, to in"nitesimal variations of initial conditions. It is called chaotic if the orbits are unstable to in"nitesimal variations of initial conditions. Useful quantities to calculate this behavior are the Lyapunov exponents.
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ensembles giving bounds on isospin breaking in Al, the 5;5 ensembles being important for two coupled quartic oscillators, mixing of several GOE's being important for ¸S to JJ coupling transitions in atoms, etc.; (iv) Dyson's circular ensembles which do not need unfolding of energy levels; (v) banded random matrices which are, for example, used to describe one-dimensional disordered models in solid-state physics (for example quasi-1D disordered wires); (vi) sparse random matrices and quasi-random matrices for localization; (vii) random matrices related to orthogonal polynomials and non-Gaussian random matrix ensembles; (viii) Verbaarschot's chiral Gaussian ensembles (chGOE, chGUE, chGSE) in the study of the Dirac operator of QCD; (ix) parametric random matrix theory (Gaussian process corresponding to GOE, GUE, GSE) for universal statistical correlations; (x) the C, D, CI, DIII class of random matrices introduced by Zirnbauer with applications in mesoscopic systems; (xi) Ginibre's non-Hermitian random matrices with applications for example in dissipative quantum chaos; (xii) embedded random matrix ensembles (they are de"ned and described in detail ahead) for interacting many-particle systems, etc. It is useful to mention that many diversi"ed methods are used to derive results for the ensembles mentioned above and they are: (i) in some limited situations via 2;2 matrices; (ii) Monte-Carlo methods using powerful computers; (iii) exact matrix methods (largely due to Mehta); (iv) Dyson's Brownian motion method which is, for example, used recently in deriving analytical results for banded random matrices; (v) supersymmetry method (nonlinear model) of Efetov which was later extended by Verbaarschot, WeidenmuK ller and Zirnbauer (using this they developed, for example, the theory for cross-section #uctuations in the chaotic compound-nucleus resonances); (vi) the so-called binary correlation approximation due to Wigner which was later used extensively by French for studying embedded ensembles and also the classical ensembles; (vii) in some situations perturbation theory, etc. Finally, we mention that, in the nuclear context nuclear models such as the shell model, cranked shell model for high-spin states, quasipaticlephonon nuclear model (QPNM) of Soloviev, particle-rotor model, etc. on the one hand and group theoretical models such as the interacting boson model (IBM), interacting boson}fermion and fermion}fermion models (IBFM, IBFFM), Ginocchio's SO(8) model, Feng's FDSM model and the simple Lipkin}Meshkov}Glick SU(3) model, etc. on the other are used to study relationships between level and strength #uctuations in quantum systems, random matrix predictions and classical chaos, onset of chaos, order}chaos transitions, etc. Group theoretical models for interacting particle systems in "nite-dimensional Hilbert spaces allow one to obtain their classical analogues via coherent states. Other advantage of these models is that they are semi-realistic and in these models the Hamiltonian matrix dimensions are usually not very large. As the literature for all these is quite vast, we refer the readers to some representative reviews and papers [1}41]; large bibliography is given in the latest review article by Guhr et al. [26]. Besides the analysis of the slow neutron resonance data for testing the GOE results for level and strength #uctuations [3], low-energy data (complete level schemes with respect to JL) of nuclei with 244A4244 by Von Egidy, Mitchell and Shriner, energy levels up to &4 MeV in Sn by Raman et al., near vibrational nuclei by Abul-Magd and rotational bands in deformed nuclei with Z"62}75, A"155}185 by Garrett et al. are analyzed for identifying the region of onset of chaos in nuclei and for order-chaos transitions interpreted mainly in terms of the Brody parameter There are several phenomena in nuclear spectra which can be termed as precusors to chaos and some examples are backbending, signature splitting in B(M1)'s, ¸"4 staggering in superdeformed bands, etc. [42].
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(Appendix A). Chaotic nature of the slow neutron resonance domain is exploited: (i) to derive a bound on the TRNI part of the nucleon}nucleon interaction by French and collaborators via level and strength #uctuations and statistical spectroscopy; (ii) bounds on parity breaking meason exchange coupling constants in the nucleon}nucleon interaction via experiments with polarized neutrons by the TRIPLE collaboration. Similarly, French and collaborators and WeidenmuK ller and collaborators derived bounds on the TRNI part of the nucleon}nucleon interaction from cross-section #uctuations in detailed balance pair of reactions proceeding through compound nucleus, the major example till to date is Mg# Al#p with intermediate Si compound nucleus. Finally, Mitchell, Shriner and collaborators carried out analysis of the ¹"0 and 1 levels in 0}8 MeV excitation (and the electromagnetic transition strengths among them) in Al for order}chaos transitions due to possible isospin breaking and more recently for the same in P and Cl. See Refs. [3,19,23,43}45] for details of all these data analysis. In addition to all these, the relationship between quantum chaos and random matrix results for #uctuations is being applied to the problem of decay from superdeformed band to normaldeformed (low-lying) states [46] via chaos assisted tunneling [17] and to problems in dissipative collective dynamics as occurs in the decay of giant resonances, "ssion and in collisions between two heavy nuclei, etc. [47]. At this stage, it is appropriate to recall the purpose, as stated by the organizers Altshuler, Bohigas and WeidenmuK ller, of a recent workshop (held at ECTH, Trento in February 1997) on chaotic dynamics of many-body systems: &The study of quantum manifestations of classical chaos has known important developments, particularly for systems with few degrees of freedom. Now, we understand much better how the universal and system-specixc properties of &simple chaotic systems' are connected with the underlying classical dynamics. The time has come to extend, from this perspective, our understanding to objects with many degrees of freedom, such as interacting many-body systems. Problems of nuclear, atomic, and molecular theory as well as the theory of mesoscopic systems will be discussed at the workshop'. Working in this direction, several research groups recently recognized the importance of investigating the embedded ensembles (EE) in detail. With the development of nuclear shell model codes in late 1960s French recognized very early that the statistical properties of nuclear levels and strengths (produced by various transition operators) have their origin in EE (with a shell model code one can construct EE) which primarily take into account the fact that the interaction rank of the nuclear force is much smaller compared to the number of valence nucleons and the many particle states, in the shell model, are direct products of single-particle states. French and collaborators have, in fact, investigated in considerable detail both spectral averages and #uctuations (the later coinciding with the results of the classical ensembles). With this one has statistical spectroscopy (SS) [48}67]. These developments are brie#y reviewed in Section 2. However in the last three years, shell model results for Si [22], spectroscopic calculations for the Ce atom [68}70] and the description of shell model results for the measures, number of principal components (NPC) and information entropy (S ) which depend both on averages and #uctuations, of complexity and chaos [66] substantiating many results of EE and SS, new interest is generated in EE with deeper connections to chaos giving simplicities and possible applications to many other interacting particle systems such as atomic clusters, isolated quantum dots, etc. Nuclear shell model codes, the binary correlation approximation and Monte-Carlo methods, etc. are employed recently in investigating: (i) critical energy and temperature for onset of chaos in "nite interacting many-particle systems; (ii) change in the shape of strength functions as the strength of
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the interaction is varied, (iii) statistical mechanics for "nite system of interacting particles via smoothed strength functions or partial densities with examples of occupancies, Gamow} Teller (GT) strength sums and transition matrix elements of one-body operators; (iv) 2;2 partitioned EE for mixing between distant con"gurations and EE with additional symmetries generating order out of chaos; (v) chaos and interaction driven thermalization in isolated "nite interacting many-particle systems. Topics (i)}(iv) are discussed brie#y in Sections 3.1}3.4, respectively, with special emphasis on their relevance in future nuclear spectroscopy. Topic (v) and a brief discussion of EE and Fock-space localization form Section 4. These results give a new basis, in terms of chaos and complexity, for SS and enlarge its scope. Finally Section 5 gives concluding remarks.
2. Embedded ensembles: EGOE(k) 2.1. Dexnition Embedded ensembles, in particular, the embedded Gaussian orthogonal enemble of random matrices with k-body interactions (EGOE(k)), are introduced by French and Wong [71] and Bohigas and Flores [72]. Early studies used, for analyzing EGOE(k), the nuclear shell model codes along with Monte-Carlo methods. However, good insight into EGOE(k) is obtained by using the binary correlation approximation [49,57]. The EGOE(k) for many fermion (boson) systems assumes at the outset that the many particle spaces are direct product spaces, of single-particle states, as in the nuclear shell model [73] (in the interacting boson model [74]). Before going further let us de"ne EGOE(k) for m (m'k) particle systems (bosons or fermions) with the particles distributed say in N single-particle states. The EGOE(k) is generated by dexning the Hamiltonian H, which is k-body, to be GOE in the k-particle spaces and then propagating it to the m-particle spaces by using the geometry (direct product structure) of the m-particle spaces. To make clear this de"nition, let us consider EGOE(2) for fermions which is appropriate for atomic nuclei when studied using the shell model (note that most of the discussion in this article is restricted to fermion systems). Given the single-particle states , i"1, 2,2, N, the two-particle G Hamiltonian H(2) is de"ned by H aRJ aRI a G a H , H(2)" I J G H ? J J J J JG JH JI JJ
(1)
where aRJ creates a fermion in the state and similarly a J destroys a fermion in the state J J J . The symmetries for the antisymmetrized two-body matrix elements (TBME) H J I J G H ? being, H "! H , I J H G ? I J G H ? H " H . I J G H ? G H I J ?
(2)
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The Hamiltonian H(m) in m-particle spaces is de"ned in terms of the TBME via the direct product structure. The non-zero matrix elements of H(m) are of three types, 2 H 2 " H , K K ? G H G H ? JG JH XJK JK 2 H 2 " H , N K K ? N G G ? JG J 2 H 2 " H , N O K K ? N O ?
(3)
all other 2 H 2 "0 due to the two-body selection rules. ? The EGOE(2) is de"ned by (1)}(3) with GOE representation for H in the two-particle spaces, i.e. H I J G H ?
are independent Gaussian random variables ,
H "0 , I J G H ? H "v(1# ). I J G H ? GH IJ
(4)
In (4) bar denotes ensemble average and v is a constant. Note that the H(m) matrix dimension d is d(N, m)"(,) and the number of independent matrix elements ime are ime(N)"d (d #1)/2 K where the two-particle space dimension d "N(N!1)/2. For example, d(11, 4)"330, d(12, 5)"792, d(12, 6)"924, d(14, 6)"3003, d(14, 7)"3432, d(40, 6)"3838380, d(80, 4)"15815 80, etc. and similarly ime(11)"1540, ime(12)"2211, ime(14)"4186. It should be mentioned that the EGOE(2) is also called two-body random ensemble (TBRE). Using (1)}(4), construction of EGOE(2) on a machine is straightforward. In general for fermions, in the dilute limit (de"ned ahead) EGOE(k) is more tractable and this limit is considered in detail in Section 2.2. With EGOE(k) operating in the m-particle spaces, as the corresponding Hamiltonian matrix in the k-particle spaces is represented by a GOE, the m-particle Fock space can be referred as strongly interacting (in the situations that the interaction is not strong, as discussed ahead in Sections 3 and 4, modi"cations of the GOE structure of H in the k-particle spaces are called for). Let us mention that extension of (1)}(4) for interacting boson systems is straightforward; see Appendix B and Refs. [65,75,76]. Similarly de"ning EGOE for mixed particle rank Hamiltonians (in nuclear case H is (1#2)-body) and for Hamiltonians with other extra information is direct and they are considered in Section 3. 2.2. Basic results Investigating EGOE(k) numerically (using shell model with realistic interactions and shell model#Monte-Carlo) and analytically (using the so-called binary correlation approximation; see for example [7]), some generic results that are essentially valid in the dilute limit, which corresponds to (N, m, k)PR, m/NP0 and k/mP0, are derived. These basic results form Sections 2.2.1}2.2.3.
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2.2.1. State density The state density (eigenvalue density) I(E) or its normalized version (E) takes Gaussian form and it is de"ned by [49] I(E)"[(H!E)\"d(H!E)"d(E) ,
1 1 E! #%-# . exp! (E) P (E)"G (E)" 2 (2
(5)
In (5) [2\ denotes trace (similarly 2 denotes average), the , and d are centroid, width ( is variance) and dimensionality, respectively, of the space over which I(E) is de"ned. Note that "H, "(H! ), &G' stands for Gaussian and the bar over (E) indicates ensemble average (smoothing) with respect to EGOE. The binary correlation approximation, originally used by Wigner for deriving the semi-circle state density for GOE (k"m in EGOE(k)) was employed by Mon and French [49] to derive (5) via the m-particle space moments (H(k))NK of I(E). Keeping technical details to minimum, the dilute limit results are brie#y described here. Firstly, it is seen that by de"nition all the odd moments of I(E) will vanish. Given the k-body Hamiltonian H(k)" = PR(k)Q(k) where PR(k) creates the . / . / k-particle state (k) and Q(k) destroys (k), the m-particle trace of H is generated by the trace . / equivalent HH" = = PR(k)Q(k)QR(k)P(k). Under EGOE(k) ensemble average and in the . / . / / . dilute limit, using the normalization HK"1, the binary correlated pair HHP(L); n is the I number operator. Then HK"(K). More generally, under ensemble average and in the dilute I limit,
H(k) O(t) H(k)N
n!t k
O(t) .
(6)
In (6) O(t) is a t-body operator and the binary link denotes averaging over the pair of H's below which the link is drawn. Using (6) and that in the trace HNK binary associations dominate, formulas for the moments are derived by writing down all the possible binary associations in HN. Denoting the binary linked pairs as A, B, 2 (A, B, etc. are independent), it is easily seen that H contains three patterns (diagrams), (H(k))K"AABB ABBA ABABKN2 AABB ABABK
"2
m m!k # k k
m k
.
(7)
The values of the irreducible diagrams AABB and ABBA are simple while that of ABAB follows from (6). Then the 4th reduced moment and cumulant k are,
k " !3" HK \HK!3"
m!k k
; m \ !1IPK !k/m . k
(8)
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Similarly for the 6th moment (H(k)) there are 15 binary association diagrams and they are ACCBAB ACBCAB ACBACB ACBABC ABCCAB ABACCB ABACBC ABABCC ACCABB AACCBB AABCCB AACBCB ACCBBA ABCCBA ACBCBA N5 AABBCC 6 ABABCC ABCABC 3 ACBCAB .
(9)
Here the second row gives the irreducible diagrams. Evaluating them (as in (15) ahead) and taking the dilute limit gives for the 6th cumulant and similarly for the 8th cumulant [57,59] k(6k!1) !4k(23k!9) k " #O(1/m), k " #O(1/m) . m m
(10)
Further details of (6}10) can be found in [49,77] (let us mention that the special case of EGOE(2) was also discussed by Gervois [78]). From (8) and (10) one recovers, in the dilute limit, the Gaussian form for the state densities (note that we need in fact not k/mP0 but k/mP0). Thus, for a two-body interaction m&12 gives a good Gaussian. Note that for m"4 one has k "!1 implying semi-circle shape as seen in many numerical calculations. In practice, one has to apply Edgeworth (mostly 3rd and 4th moment/cumulant) corrections [79] to the Gaussian form; Appendix C gives the Edgeworth form. The EGOE derives its signi"cance from the important result that local #uctuations in energy levels are of GOE type [7]. In fact, it is seen that there is a natural separation of information into long and short wavelengths with damping of intermediate wavelengths. The long wavelength parts correspond to the smoothed Gaussian form (5) 䉴 Fig. 1. (a) State density I(EK ) vs. EK "(E! )/ for a EGOE(1#2) ensemble H "h(1)# <(2) , "0.1 in the 924-dimensional N"12, m"6 space. The one-body Hamiltonian h(1) is de"ned by the single-particle energies "(i)#(1/i) for i"1, 2,2,12 just as in [81,82]. Similarly <(2) is a EGOE(2) ensemble. In the calculations a 25 G member GOE is constructed in the two-particle spaces (matrix dimension is (,)) and then <(2) in the m-particle spaces, in the basis de"ned by occupation numbers, is generated by applying (1)}(4) with v"1 in (4). Note that h(1) adds only to the diagonal matrix elements of H and for a m-particle state 2 , its contribution is simply H . It should be K H J pointed out that the EGOE(1#2) ensemble is in fact introduced ahead in Section 3 (see the discussion following Eq. (23)). In the calculations spectra of all the ensemble members are "rst zero centered ( is centroid) and scaled to unit width ( is width) and then the ensemble average is carried out. It is seen that the EGOE(1#2) state density (histogram) is well represented by the Edgeworth corrected Gaussian (ED in the "gure); for the ensemble "0 and "!0.342. In the plots the integral of the densities is 924. (b) For the ensemble in (a): (i) NNSD compared with the Poisson and Wigner (GOE) distributions; (ii) statistic (¸) vs. ¸. The spectrum for each member of the ensemble is unfolded using Edgeworth corrected Gaussian for the state density and then NNSD and are calculated. More sophisticated unfolding procedures are described in [7,80] but they are not employed in the present calculations. It is seen from the "gures that for the EGOE(1#2) ensemble considered, the level #uctuations follow GOE. (c) Level density I(EK ) vs. EK "(E! )/ for the JL"4> levels in Ca. The histogram is for the shell model eigenvalues in (1 f 2p) space (matrix dimension is 1755). The Hamiltonian is H"h(1)#<(2) de"ned by the so called KB3 two-body matrix elements (<(2)) [83] with singleparticle energies "0 MeV, "2 MeV, "4 MeV and "6.5 MeV; hereafter this is referred as KB3 D N N D interaction and the (1f 2p) shell as ( fp) shell. The shell model eigenvalues are supplied by A.P. Zuker and they are obtained using the large-scale Strasbourg}Madrid group ( fp)-shell code [83]. The shell model density is well represented by ED; here "!0.01, "!0.19. Note that the ground state appears at EK "!3.219. (d) Same as (b) but for the Ca shell model eigenvalues in (c). The #uctuation analysis are carried out for the middle 1355 levels. Just as in the case with EGOE(1#2), the shell model energy level #uctuations also follow GOE.
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(with Edgeworth corrections) and the short wavelength parts giving the GOE #uctuations. There are numerical examples [7,80] and some analytical understanding [49,7] of this result. Finally, Fig. 1 gives EGOE and shell model examples for the smoothed Gaussian form of the state densities and GOE #uctuations in the energy levels.
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2.2.2. Transition matrix elements Similar to (5), the bivariate strength densities take bivariate Gaussian form [57]. The strength R(E , E ) generated by a transition operator O in the H-diagonal basis is R(E , E )"E OE . G D G D D G Correspondingly, the bivariate strength density I O (E , E ) or O (E , E ) which is positive _ G D _ G D de"nite and normalized to unity is de"ned by I O (E , E )"[OR(H!E )O(H!E )\ _ G D D G "ID(E )E OE IG(E )"[ORO\ O (E , E ) , D D G G _ G D P O (E , E )" (E , E ; , , , , ) O (E , E )#%-# _ G D \G_O G D G D G D _ G D
E ! 1 1 G G exp ! " 2(1!) 2 (1! G G D E ! E ! E ! G D D # D D . (11) !2 G G D D In (11) and are the centroids and and are the variances of the marginal densities O (E ) G D G D G_ G and O (E ) respectively of the bivariate density O . The bivariate reduced central moments of _ D_ D O are "OR((H! )/ )OO((H! )/ )N/ORO and " is the bivariate correlation NO D D G G _ coe$cient. Let us consider a brief discussion of the evaluation of as given in [57]. Firstly, H is NO represented by EGOE(k) and the transition operator O by EGOE(t); it is also assumed that the H and O ensembles are uncorrelated. Now the correlations in arise due to the non-commutabilNO ity of H and O operators. Firstly, it is seen that all with p#q odd will vanish on ensemble NO average and also " . Moreover "OROHK/O-OK"(K) and ". Thus, the "rst I D G NO ON G non-trivial moment is and evaluating it using (6) gives, " "OR(t)H(k)O(t)H(k)K/ OR(t)O(t)KH(k)H(k)K PABABK/AAKBBK
m!t
"
k
m \ kt k(k!1)t(t!1) #O(1/m) . "1! # 2m m k
(12)
In the cases with p#q"4, the moments to be evaluated are " , " and . The diagrams for these follow by putting OR and O at appropriate places in the H diagrams in (7). The "nal results are,
"[ORAABBOK ORABBAOK ORABABOK]/[OROK(HK)] m!k m \ "2# , k k
"[ORAABOBK ORABBOAK ORABAOBK]/[OROK(HK)]" ,
"[ORAAOBBK ORABOBAK ORABOABK]/[OROK(HK)] m m!t m!k!t m!t m \ # # . " k k k k k
(13)
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Using these, formulas for the fourth order cumulants are obtained:
k "k " !3"
m!k k
m \ k k(k!1) !1"! # #O(1/m) , m 2m k
k k[(k!1)#2kt] k "k " !3 "k "! # #O(1/m) , m 2m
k " !2 !1"
m!k!t k
m!t \ !1 k
k k[(k!1)#4kt!2t] "! # #O(1/m) . m 2m
(14)
In order to establish the structure of the bivariate cumulants, the cumulants to order p#q"6 are also derived in [57] starting with the 15 diagrams in (9). The binary association diagrams for
" , " , " and follow directly from (9) and evaluating them give
"[5ORAABBCCOK 6ORABABCCOK 3ORACBCABOK . ORABCABCOK]/[OROK(HK)]
m!k
"5#6
k
m \ m!k m \ #3 k k k
m!2k
m!k
k
k
#
m \ "\ , k
(15)
"[ ORAAOCCBB ORAAOBCCB ORACOCABB ORACOCBBA ORABOCCBA ORABOABCC ORABOCCAB ORABOACCB ORACOBCBA ORACOCBAB ORACOBABC ORABOACBC ORAAOCBCB ORACOBACB ORACOBCABK]/[OROK(HK)]
" 2
#2
#
#
m m!t m m!k!t #3 #3 k k k k m!t m!k m!t #2 k k k
m m!k m!t # k k k m!t I m! k k JI
m!t
m
k
k
m!t!k
m!k
k
k
m!t!k
m!2k
k
k
m!t!k
k
!k
2k!
m \ , k
(16)
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"[ ORACCOABB ORAACOCBB ORAABOCCB ORACCOBBA ORABAOBCC ORAACOBCB ORACCOBAB ORABAOCCB ORABCOCBA ORACBOCAB ORACBOABC ORABAOCBC ORABCOCAB ORACBOCBA ORACBOACBK]/[OROK(HK)]
" 4
#
m!t k
m m!t #4 k k
m!k
m
k
k
m!t m!t!k m!t #2 k k k m!t
#
k
m!k m!t m!t!k #2 k k k
m!t
m!t!k
m!t!2k
k
k
k
#
m \ . k
(17)
Note that only the last term in (16) is non-trivial [49]. Using (15}17) formulas for the sixth-order cumulants are obtained: k "k " !15 #30 k(6k!1) k(k!1)(7k!1) " ! #O(1/m) , m m k(6k!1) k[(k!1)(7k!1)#kt(6k!1)] ! #O(1/m) , k "k "k " m m k " ! !8 !6 #24#6 k(6k!1) k[(k!1)(7k!1)#t(12k!6k!1)] " ! #O(1/m) , m m k " !6 !9 #12#18 k(6k!1) k[(k!1)(7k!1)#t(16k!13k#2)] " ! #O(1/m) . m m
(18)
Using (12)}(18), in the dilute limit (just as in the case of state densities, here also one needs k/mP0), one recovers the bivariate Gaussian form for the strength densities. However, it should be noted that the bivariate cumulants are not symmetric in the E #E and E !E variables. G D G D Discussion on this result and for further details of (12)}(18) see [57]. Finally, in practice one has to
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Table 1 Shell model r.m.s. matrix elements of a two-body transition operator O in the 307 dimensional (ds)K ( 2 space. The operator O is de"ned in [57] (it is a part of the Hamiltonian) and the Hamiltonian is Kuo interaction de"ned in the caption to Fig. 2. The 307;307 symmetric matrix E OE is divided into 9 blocks by 9 blocks and the r.m.s. D G transition matrix element in each 34;34 block is calculated; the 307th eigenvalue is dropped. In each block, the "rst row gives exact shell model r.m.s. matrix element and second row is the bivariate Gaussian (11) prediction (correlation coe$cient "0.58). Further details are given in [34,57] 0.469 0.489
0.353 0.362
0.319 0.297
0.291 0.250
0.247 0.210
0.174 0.173
0.128 0.137
0.090 0.101
0.047 0.057
0.319 0.339
0.284 0.310
0.271 0.283
0.279 0.256
0.254 0.226
0.206 0.193
0.145 0.156
0.083 0.101
0.297 0.302
0.279 0.290
0.252 0.274
0.252 0.253
0.234 0.228
0.195 0.194
0.132 0.138
0.283 0.288
0.290 0.282
0.273 0.270
0.259 0.253
0.225 0.226
0.163 0.173
0.292 0.284
0.289 0.282
0.280 0.274
0.256 0.257
0.205 0.210
0.294 0.290
0.295 0.293
0.291 0.287
0.247 0.255
0.291 0.308
0.306 0.318
0.310 0.308
0.314 0.350
0.378 0.380 0.559 0.520
apply the bivariate Edgeworth corrections (given in Appendix C) to the bivariate Gaussian strength density in (11). Just as in the case of energy levels, EGOE gives local strength #uctuations to be of GOE type, i.e. strength #uctuations are Porter}Thomas (P}T) ((1)) in nature (see (A.8)). The EGOE smoothed Gaussian forms for the state densities and bivariate strength densities together with the result that the local strength #uctuations are P}T are used [66] to derive the EGOE formulas for information entropy (S ) and number of principal components (NPC) or the inverse participation ratio (IPR), which are measures of complexity and chaos, in the transition strength distributions; Appendix D gives some details. Finally, Table 1 and Fig. 2 give shell model examples for the smoothed bivariate Gaussian form of the strength densities and P}T GOE #uctuations in the transition strengths. There are further examples in [7,57].
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Fig. 2. (a) Distribution of renormalized shell model transition strengths R(E , E )/R(E , E ), R(E , E )"E OE , G D G D G D D G shown as a Garrison plot [7,84] and compared with the GOE Porter}Thomas (P}T) form (A.8). Shell model (SM) calculations for R(E , E ) are in (2s1d)K ( 2 space with the Hamiltonian de"ned by the Wildenthal interaction G D [85] and the transition operator is the two-body part of the Hamiltonian obtained after substracting the so-called con"guration isospin centroid part [86]. The locally averaged strengths R(E , E ) are calculated using the EGOE G D bivariate Gaussian form (11). The SM#EGOE result is in good agreement with P}T. (b) Number of principal components NPC and information entropy S (denoted by S in the "gure) vs. energy (E) for a strength distribution in 307-dimensional (2s1d)K ( 2 space. The (2s1d) shell Hamiltonian is H"h(1)#<(2) de"ned by Kuo's [87] two-body matrix elements (<(2)) and O single-particle energies (h(1) 0 "!4.15 MeV, "0.93 MeV, B B "!3.28 MeV); hereafter this is referred as Kuo interaction and the (2s1d) shell as (ds) shell. The transition operator Q is same as in (a). Shown in the "gure is also the ratio exp(S)/NPC vs. E. The exact shell model results (SM) are compared with the GOE and EGOE predictions; the EGOE predictions are given by (D.3). (c) Same as (b) but for wavefunctions (see Appendix D). All the shell model calculations are carried out using the Rochester}Oak Ridge shell model code [73]. Results in the "gures are "rst reported in [88] and Fig. 2b is taken from [66].
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2.2.3. Transition strength sums Given an operator K"ORO, expectation values K# are the diagonal elements of K in the Hamiltonian (H) diagonal basis (a more precise de"nition is given ahead in Eq. (19)); they give the strength sums generated by the transition operator O acting on the eigenstates with energy E. In nuclear structure, signi"cance of the transition strength sums is well known [89]. Two major examples are: (i) single-particle transfer strength sums, which are expectation values of the number operators, giving occupancies of the single-particle states (they determine thermodynamic behaviour); (ii) Gamow}Teller (GT) strength sums, as a function of the excitation energy, which are relevant in astrophysics (presupernova evolution and stellar collapse). With Eqs. (5) and (11) it is easily seen that the transition strength sums take the form of a ratio of two Gaussians [58,59]. The expectation value density (E)"K#(E) is the marginal of the bivariate density in (11) and ) hence it is a Gaussian. Thus,
K#"[d(E)]\ EKE "I (E)/I(E)" (E)/(E) , ) ) ?Z# #%-# P (E)/(E)" G (E)/G (E) , ) )
(19)
(E)"K#(E)"d\I (E)"d\[K(H!E)\, K"ORO . ) ) Note that (E) can be treated as a probability density function and the smoothed form for K# ) given by (19) takes into account (K, H) and (K, H) correlations which de"ne the centroid and width of (E). Recently, it is recognized that the smoothed K# vs. E will give information about ) order}chaos transitions just as energies, wavefunction amplitudes and transition strengths and this topic is discussed in detail ahead; see [88,90]. In deriving (19), it is assumed that the smoothed form of (E)/(E) reduces to the ratio ) of the smoothed forms of (E) and (E). This result ignores the #uctuations in both (E) and ) ) (E). The mean square deviation (E) at energy E in the smoothed transition strength sum M(E)"ORO# follows by using the P}T assumption for strength #uctuations. The "nal result [52,65], in terms of the bivariate strength density O (E, E ) and the state density D(E ), is D D _ \ 2 ( O (E, E )) D dE O (E, E ) dE _ (E)/(M(E))" ; D _ D D dD D(E ) D 2 2 " " , (20) 3(NPC) d (E) # where dD is the dimensionality of the E space. The EGOE formula for (NPC) or the e!ective D # dimension d (E) in (20), is given by (D.3) and they are essentially determined by the bivariate correlation coe$cient de"ning O . Figs. 3 and 4 give numerical shell model examples for _ the EGOE results (19) and (20), respectively.
2.3. Statistical spectroscopy To the extent that the EGOE results (5,11,19) apply, spectroscopy of many-particle systems will be simple and then we have a statistical theory (statistical spectroscopy) for "nite
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Fig. 3. (a) Occupation numbers for d and d orbits vs. excitation energy (E) for the 325-dimensional eight particle (ds)-shell space with J"0, ¹"0. The exact shell model results for the Kuo interaction are compared with EGOE smoothed form given by (19). (b) Gamow}Teller (GT) strength sums vs. excitation energy (E) for the 814-dimensional six particle ( fp)-shell space with J"0, ¹"0. The exact shell-model results for the KB3 interaction are compared with EGOE. In the (ds) shell example the K-density centroid , width , skewness and excess for the d and ) ) ) ) d densities are (!55.30, 12.31 MeV, 0.01, !0.06) and (!48.70, 13.42 MeV, 0.07, 0.05) respectively. Similarly for the state density the parameters are "!52.59 MeV, "13.15 MeV, "0.10 and "0.03. In the ( fp) shell example, "11.12 MeV, "8.65 MeV, "0.09, "!0.18, "9.51 MeV, "8.62 MeV, "0.10 and "!0.19. ) ) ) ) It is seen that the EGOE form (19) describes very well the shell model results except at the edges of the spectra. The reason for the deviation at the edges is well known; here the states are not chaotic (su$ciently complex). See [88,90] for further examples and details. The fp-shell "gure is from [90].
Fig. 4. Smoothed transition strength sums as given by EGOE (ratio of Gaussians (19)) and the r.m.s. deviation (E) estimated using the EGOE # P}T formula (20,D.3), compared with exact shell model transition strength sums. The transition operator is the two-body part of the Hamiltonian as de"ned in [57]. The calculations (same as in Table 1) are in the 307-dimensional six particle (ds)-shell space with J"2, ¹"0 and the Hamiltonian is the Kuo interaction. In the example shown in the "gure the bivariate correlation coe$cient "0.58. The e!ective dimensions d (E) which determine (E) are 49, 89, 144, 206, 260, 290, 286, 250, 193, 132, 79, 42 and 20 at the energies !60 MeV, !55MeV,2,0 MeV, respectively. Results for the strength sums (shell model and EGOE smoothed form) in the "gure are "rst reported in [57] and for the (E) in [65].
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systems of interacting particles. To this end, one needs some more results as we shall discuss now. Given the (,)-dimensional m-particle Hilbert space, it can be decomposed into subspaces in K a variety of ways. Often more important and interesting decompositions are de"ned by the irreducible representations (irreps) of a group structure in the Hilbert space; see [51] and Appendix E. In the context of the nuclear shell model, decompositions according to spherical orbit con"gurations, Elliott's SU(3), Wigner's spin}isospin S;(4)}S¹, Talmi's generalized seniority, etc. are considered in the past [48,50,51,56]; the case of spherical orbit con"gurations is discussed in detail in Appendix E. Now, the important result is that, under mP, the state and expectation value densities decompose into the corresponding densities de"ned over the subspaces , I(E)"[(H!E)\" [(H!E)\" I(E) ,
I(E)PI(E)"IG (E) , I (E)"[K(H!E)\" [K(H!E)\" I (E) , ) ) (21) I (E)PI (E)"I G (E) . ) ) ) Note that the IPI and I PI decompositions in (21) are exact. For interacting particle systems ) ) with a mean-"eld (due to h(1)) and a chaos generating two-body interaction (<(2) } representable by EGOE(2)), i.e. for H h(1)# <(2) , a decomposition of the bivariate strength density I O into partial strength densities IG OD (in this case, both the initial and "nal spaces are _ _ the subspaces are de"ned by h(1). Then [34,59,62,63], decomposed) is possible (via F.3) and here D (E , E ) , I O (E , E )PI O (E , E )" O IG }G _ G D _ G D D G _O G D G D
O "[d( )d( )]\ O , D G G D D G ?ZG @ZD D (E , E )"d( )d( ) G D (E , E ; , , , , ) , IG }G G D }G_O G D G D G D _O G D
OR
(22)
Note that in (22), the bivariate correlation coe$cient arises out of the non-commutability of < and the transition operator O; in principle, it is possible to add the Edgeworth corrections to . Assuming that the smoothed forms in (5) and (19) apply, as indicated in (21), to the partial \G_O state and strength sum densities I(E) and I (E) respectively (Refs. [48,68,91] and the results in ) Section 3.2 give numerical evidence for this result) together with the more restrictive smoothed result (22) for the transition strength densities and incorporating the exact/approximate values for the various parameters de"ning the partial densities in (21), one has a statisical theory, i.e. SS or a method (called in nuclear physics literature as spectral distribution method or spectral averaging theory), for calculating spectroscopic observables such as (i) occupation numbers, (ii) spin-cuto!
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factors, (iii) level densities, (iv) GT strength sums and distributions for nuclear structure, presupernova evolution calculations and neutrino detection, (v) giant dipole resonance (GDR) distributions and widths, (vi) estimates for Wigner's spin}isospin S;(4)}S¹ symmetry breaking, etc. (within the errors determined by GOE #uctuations). This approach derives its power from the important result that the traces that de"ne the parameters of the various densities in (5,11,19,21,22) propagate and hence can be evaluated without recourse to exact matrix construction (Appendix E). Further progress in statistical spectroscopy (SS) is in the recognition that the sums in (21) and (22) can be simpli"ed by converting them into convolution forms or otherwise by paying attention to the fact that H is (1#2)-body and decomposing it into parts (tensors) with respect to the underlying ;(N) group and its various direct sum sub-groups; see Appendix F and Section 3.3.2. For shell model tests, details of the formulations and applications of SS see [1,7,23,48}67] and references therein. It should be mentioned that there is recent interest in these methods in atoms [12,68}70,92] and molecules and solids [93]; also there are now good reasons to believe that SS applies to mesoscopic systems (example of quantum dots is discussed in Section 4). However there are many questions: (i) in what domain of the interacting particle system (nuclear) spectrum SS or EGOE operates } it is expected that SS (EGOE) operates in the domain of chaos (or perhaps from the region of onset of chaos and in some special situations even below this reference energy) [94]; (ii) how to determine the region of onset of chaos; (iii) is there new physics in the tails of the distributions (the tails are not well represented by a Gaussian) in (5), (11), (19), (21) and (22) and is it related to exponential localization; (iv) are there more general derivations for the smoothed forms for the partial densities in (21,22), how they will change from the region of onset of chaos to the domain of chaos, what changes may be there due to the fact that the Hamiltonian is (1#2)-body; (v) what is the special role played by NPC and S in connecting SS to chaos and thermalization; (vi) role and applications of various deformed EE in the study of interacting many-particle systems. Various investigations presented in Sections 3 and 4 go long ways in providing answers to these questions. With this, i.e. EGOE operating in the chaotic domain, one has for example a basis for using SS for deriving physics from experiments involving compound nucleus levels such as the parity breaking experiments [95], as the compound nucleus domain is known to be a region of chaos [2,23].
3. Deformed embedded ensembles Hamiltonians for interacting particle systems contain a mean-"eld part (one-body part h(1)) and a two-body residual interaction <(2) mixing the con"gurations built out of the distribution of particles in the mean-"eld single-particle orbits; h(1) is de"ned by the single-particle energies (SPE) , i"1!N and <(2) is de"ned by the TBME (see (1,2)). Then it is more realistic to use EE(1#2), G the embedded ensemble of (1#2)-body Hamiltonians, EE(1#2) : H "[h(1)]# <(2) .
(23)
In (23) [h(1)] and <(2) are independent and <(2) is EE(2), i.e. it is EGOE(2) with v"1 in (4) or an ensemble with TBME being independent random variables with a distribution di!erent from Gaussian (for example uniform distribution). Similarly [h(1)] represents a "xed Hamiltonian h(1)
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(i.e. the SPE that de"ne h(1) are "xed) or an ensemble with the SPE chosen random but G G following some distribution. In the "rst case, H "h(1)# <(2) and in the second situation H " h(1) # <(2) . Let us mention that in the literature EE(1#2) is also called TBRIM (two-body random interaction model) [81,82]. Hereafter the EE(1#2) ensemble (23) with <(2) being EGOE(2) is called EGOE(1#2). In most of the studies presented in the remainder of this article, the EGOE(1#2) with a "xed h(1) is employed, i.e. H "h(1)# <(2) . The EE(1#2) (or EGOE(1#2)) can be studied using Monte-Carlo methods, i.e. ensemble (23) can be constructed on a machine (for moderate values of m and N) using (1}3) and study its properties as a function of the parameter . Alternatively, assuming that the Hamiltonian H"h(1)#<(2), with the shell model giving h(1) and <(2), is a typical member of EE(1#2) (or EGOE(1#2)), one can infer some generic features of EE(1#2) (or EGOE(1#2)) from the shell model calculations with h#<. In general, one may use other models (say group theoretical models) of many-body systems but, unlike the shell model, they may not be realistic (for nuclei) over a wider energy range. The third approach is to use an analytical method such as the binary correlation method. It is to be expected that the generic features of EGOE(1#2) (or EE(1#2) in general) approach those of EGOE(2) for su$ciently large values of (this statement will be made more precise later) and signi"cant results emerge as is varied starting from "0 as discussed in Sections 3.1}3.3 ("rst study with variation, using EE(1#2) for observables in rotating nuclei, is due to A berg [96]). A second class of EE are the partitioned embedded ensembles (p-EE) where the Hamiltonian in the de"ning space is a partitioned random matrix. For example, for p-EGOE(2) the two-particle space is block structured with H represented by partitioned GOE, i.e. the two-particle space divides into subspaces with variance of the matrix elements in each GOE block being v for the block Y connecting and subspaces. The p-EE are appropriate in situations where the mixing between distant con"gurations is weak; for example, a 2;2 p-EE is needed for treating the mixing between 0 and 2 spaces in the shell model [55]. Third class of EE are EE-sym where v "0 for O Y and these ensembles are important for the Hamiltonians carrying symmetries (for example J or J¹ in nuclei and J or ¸S¹ in atoms). Recently, EGOE(2)-J¹ (or TBRE-J¹) with the choice of v determined by particle}hole symmetry is considered [97]. Finally, there are also the modi"ed K#EE ensembles of the type H "K# H where K is a "xed operator (for example pairing plus quadrupole}quadrupole) and H is EE(1#2) [98]. Some results of these deformed EE are discussed in Section 3.4. 3.1. Onset of chaos in xnite interacting many-particle systems The border for the emergence of quantum chaos and thermalization in "nite interacting many-particle systems depends on number of particles m, number of single-particle states N, the interaction strength (for ensemble (23) with EGOE(1#2) this implies v"1 in (4)) and the mean single-particle level spacing . This border can be determined by assuming that the GOE result (Wigner's nearest-neighbor spacing distribution (A.5)) for P(S) implies chaos (or thermalization with Fermi}Dirac statistics } see Section 4.1) and this is because Wigner's distribution implies excitation of many unperturbed modes and mixing. Jacquod and Shepelyansky [99] studied this problem recently using EE(1#2). For the unperturbed m-particle spectrum (i.e. for "0 in (23)), for N<m<1, the ground state energy E "(m/2), the maximum excitation energy E "mN and the Fermi energy is "m. Given an unperturbed m-particle state, it is
$
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coupled to maximum number of states K where K follows directly from (3), K" 1#m(N!m)#m(m!1)(N!m)(N!m!1)/4&mN/4. Similarly, with "0, the two-particle spectrum span B"(2N!4). Then a state at energy E of the m-particle system will be strongly F mixed if the TBME strength is of the order of the spacing B/K of the directly coupled states. This perturbation theory argument which is good at su$ciently high energies E , where all the K states F are coupled, gives the critical above which chaos sets in, B ; B"(2N!4), K&mN/4 (24) "C K
or
B 2C "C + . K m In (24) is the two-particle density and C is a constant; "(B/d(2))\&N/4. To test (24), in [99] a EE(1#2) ensemble is constructed using SPE randomly distributed over the interval [0, N] with the average spacing "1 and the TBME are chosen to be uniform random variables in the interval [!1,1]. Calculations are carried out for 24m48 and 44N480. Using the middle 25% levels for di!erent values of , P(S) is constructed for each ensemble and the parameter and thereby are determined, Q Q (S)] dS [P (S)!P (S)] dS . " [P(S)!P %-# . %-# ( )" "0.3 . (25) In (25) S "0.4729 and at this value the Poisson and GOE P(S)'s intersect. Note that "1 for Poisson and "0 for GOE. Thus in this analysis, the Bohigas}Berry result for chaos and order vs. GOE and Poisson is used. The deduced values of are in good agreement with Eq. (24) (Fig. 5); C is found to be 0.58. In the place of of (25), Berkovits and Avishai [100] employed "[P(S) dS!e\]/[e\!e\]. However, instead of using the spacing distributions in cri terion (25), one can use the transition curve de"ned by (A.14) which is much simpler [101]. In the low-energy part of the spectrum, for "xed , it is important to know the crtical energy (E ) and/or temperature (¹ ) above which there will be chaos and thermalization. For this purpose (24) is modi"ed by introducing e!ective two-particle density and e!ective number of interacting particles m near the Fermi sea. For a given temperature ¹, it can be seen that &¹/ and m &¹/. Putting these values in (24) gives, ¹ "C (/) , E "C (/) . (26) In (26) C is a constant. The second equation in (26) follows from the "rst and the relation E&¹m . In the calculations using the EE(1#2) de"ned earlier, results in (26) are well veri"ed (Fig. 5); for m"6, N"12, /"0.147 gives C "1.08. Experiments with metallic quantum dots appear to con"rm (26) for the onset of chaos/thermalization [99].
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Fig. 5. Figure shows: (i) rescaled critical interaction strength /B vs. number of directly coupled states K for 24m48 and 44N480; (ii) vs. E/ for m"6, N"12 and /"0.147. In the /B vs. K plot, the line shows theory (24) with C"0.58 and the calculations are with EE(1#2) embedded ensemble de"ned in the text. Similarly in the vs. E/ plot, the straight line marks " "0.3 and circles are for EE(1#2). The inset in this "gure gives the numerically found E / vs. /; the line shows theory (26) with C "1.08 and black squares are from the EE(1#2) calculation. Plots are taken with permission from [99].
Results in (24,26) are well corroborated in [100,102}104] and they are applied recently [105] to spin glass clusters (`shardsa) in a random transverse magnetic "eld and determined where quantum chaos emerge from the integrable limits of weak and strong "elds. Note that the spin glass shards are described by the Hamiltonian H" J VV# X where the are the Pauli matrices GH GH G H G G G G for the spin i, represent the local random magnetic "eld and J represent the exchange G GH interaction. In the model study, are distributed uniformly in the interval [0, ] and J in the G GH interval [!J/(n, J/(n] where n is total number of spins. In the limit J/P0, one obtains a paramagnetic phase and in the limit J/PR a spin glass phase. Although (24) and (26) work well, there are important questions regarding the nature of the transition from order to chaos in interacting many particle systems [106,107] and some of them are related to Fock space localization [108]. We will return to these questions in Section 4.
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In the context of nuclear physics, determination of for realistic nuclear Hamiltonians will be relevant for the discussion in Sections 3.2 and 3.3 ahead and then one will also understand why shell model calculations with realistic interactions give GOE statistics. There are estimates [94] from the analysis of low-energy spectra in various parts of the periodic table that E &6}8 MeV for (ds) and ( fp) shell nuclei, &4 MeV in A&100 region and &2 MeV for rare-earths and actinides; can one derive these from "rst principles, say using (24) and (26)? In fact, it is useful to determine the J and J¹ dependence of ¹ and E and they will give information about onset of chaos near the yrast line at high-spins [39] and near the drip-lines (it is possible that the answers to this question come by analyzing p-EE and EE-sym ensembles). Finally, it is worth mentioning that in SS one uses a reference energy [48,64] to get rid of the low-lying regular levels and this reference energy is nothing but the E given by (26). 3.2. Strength functions: transition from Breit}Wigner to Gaussian form Strength functions (also called local density of states (LDOS) in literature) are basic ingredients of a many-particle system. Given a compound state , the probability of its decay into stationary I states (generated by H) is given by . Taking into account the degeneracies in the # I # spectrum, the strength function F (E) is, I " C# , I I # # (27) F (E)" C#Y (E!E)"C#I(E)"(H!E)I . I I I #Y In practice, to derive a form for F (E), one can choose the state to be one of the shell model basis I I states (say a mean-"eld basis state). Instead of considering a single state , it is possible to consider I a set of states that belong to say a label (in SS is always taken to be an irrep of a group structure in the Hilbert space) and construct the average strength function F (E). The partial densities I(E) introduced in (21) are nothing but F (E) (sometimes denoted as (E)), I(E)" F (E), F (E)"(E)"I(E)/d() . (28) I IZ A quite di!erent and useful way to look at strength functions is to think of as a compound state I generated by the action of a transition operator O on a state I (for example ground state), then it # is seen immediately that the resulting strength function F (E) is nothing but a conditional density of I the bivariate strength density introduced in (11), O I # I " , I I "1 , # # # [ I ORO I ] # # [OR(H!E)O(H!E )\ I . (29) I " C#I 0 F (E)"F O (E)" # # I I_ # [ORO(H!E )\ I # Starting with the EGOE(1#2) ensemble H "h(1)# <(2) , it is to be expected that for su$ciently large values of , the EGOE(2) description should be valid and therefore applying (11)
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gives the shape of the strength function, for large , to be Gaussian (conditional density of a bivariate Gaussian is a Gaussian) and hence its width is independent of the energy E or k. I I These results are indeed seen in the numerical calculations and we will return to them shortly. Yet another useful form for F (E) is "rst to identify the states by their energies (rather than by an I I index) E "kHk and say there is a H that gives the E 's, E 0 H . Then I I I I I [(H !E )(H!E)\ I I . (30) F I (E)" # [(H !E )\ I I Eq. (30) once again shows that F (E) is a conditional density of the bivariate density in the I numerator of (30). The representation given by (30) is more appropriate (and convenient) for ensemble studies of the strength functions; see Fig. 7 ahead. The standard form, normally employed in many applications, for the strength functions is the Breit}Wigner (BW) form [109] characterized by a spreading width , I 1 I . (31) F (E)" I 5 2 (E!E )#/4 I I The centroid E and the spreading width of the BW are de"ned by the median and the upper I I and lower quartiles, i.e. by E in # F (E) dE"p with p", and , respectively. In practice \ I instead of using these, one can just "t the BW form to the calculated F (E) and get the best "t E I I and . For the EGOE(1#2) ensemble with h(1) de"ning the states, it is easily seen that the I I assumptions that give F will break down when the mixing is strong, i.e. for large [22]. I 5 Therefore, it is expected that the BW form and the Gaussian form should appear as is varied starting from "0. Nature of the strength functions, as a many-particle system makes order}chaos transition, is being investigated only recently. Some of the studies carried out so far are: (i) using EGOE(1#2) de"ned by (23) and constructing F (E) for various values of with the I being the mean-"eld (h(1)) basis states ("nal results mentioned in [107]; see also [110]); I (ii) starting with the nuclear (ds) } shell Wildenthal's SPE (h) plus TBME (<) and carrying out shell model calculations with H "h#< in the 839-dimensional (ds)K ( 2 space as H a function of with the chosen to be the mean-"eld basis states [111]; I (iii) just as in the shell model calculations but using the three-orbital Lipkin}Meshkov}Glick model [91]; (iv) using a symmetrized coupled two-rotor model [112]; (v) just as in (i) but in the 924-dimensional (N"12, m"6) space with 25 members for <(2) and also in a N"14, m"7 example [113]. One advantage of the model studies (iii) and (iv) being that here the classical counter parts can be constructed and therefore the corresponding classical motion can be studied at di!erent energies as a function of . The basic conclusion of all these studies is that the form of the strength function F (E) changes gradually from BW to Gaussian with exponential tails as increases (in the model I studies (iii) and (iv) this result is seen to be be valid within some system speci"c e!ects), orderPchaos 0 F
I 5
(E)PF G (E) . I
(32)
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For illustration shell model results from (ii) and the EGOE(1#2) results for the six-particle case from (v) are shown in Figs. 6 and 7, respectively. From Fig. 6, it is seen that the BW form is good for 40.3 and the Gaussian form is good for 90.6. Comparing with the corresponding study of P(S) dS and for level #uctuations [21,22], it is seen that the value of " I around which the $ BW to Gaussian transition occurs in F (E) is much greater than " "0.3 that generates I order}chaos transition in the energy levels. Thus the BW form survives for some distance in after . This observation is further con"rmed by the EGOE(1#2) results presented in Fig. 7. In these calculations, it is known that +0.08 (see Figs. 10, 11 ahead and [88]) and Fig. 7 shows that I +0.2. Thus, the BW form for F (E), which begins some what before approaches (for ; I $ the strength functions are delta functions with perturbative corrections), extends much into the chaotic domain (de"ned by ' ) and the transition to the Gaussian shape takes place in the second layer de"ned by I ( I < ); in the EGOE(1#2) calculations presented in [102], this $ $ transition is not seen as the values considered in this work are not large enough. Therefore in the region of chaos (de"ned by ' I ), we have basically Gaussians and SS in action. This conclusion $ is well supported by the nuclear shell model calculations in [7,111] as here for realistic interactions the "1 value is much larger than both and I . However, it appears that in the Ce [68,70] and $ Au> [92] atom calculations the of the e!ective interaction appears to be less than I . For the $ EE(1#2), given (N, m, ) with being the mean single particle level spacing, a theory for determining is given in Section 3.1 but there is not yet a theory for determining I . In general, it $ is essential to derive a form for F (E) in the transition domain between BW and Gaussian and this I topic is discussed in some recent papers [91,107,110]. Similarly there are attempts to write down empirical formulas for the tails of the Gaussian and there are arguments that the exponential form of the tails (as seen in shell model calculations for nuclei and atoms) is related to exponential localization [68,91,107,111]. Another question is regarding the spreading width . It is easily seen that for small we have the I BW [22,109], and for 91, "2 , 5 I I " "2< /DM for small I 5 IH "2 for 91 , I I (33) " H l "(H!HI)I , I I l $I where < is the mean squared matrix element of the two-body interaction and DM is the mean IH spacing of the "nal j-states (proper meaning of the index j is given detail in [22,109]). The interpretation that F (E) is a conditional density (see (29)) gives, as stated before, that (or ) is I I I independent of k in the chaotic domain (constancy of is responsible for the information entropy S becoming identical to the thermodynamic entropy (de"ned by state density) S\ in the chaotic domain [22,107], this follows from the relations S &ln NPC, NPC&/DM ). Constancy of is well known (numerically) in SS and it is called &constant width approximation' [51,55,56]. I In addition, depends quadratically on for small and becomes linear in in the chaotic domain. A formal (analytical) study of using EE(1#2) is not yet available; see however [81,107,110,111]. In particular in [110] it is argued that as increases, in (31) starts becoming I strongly energy dependent and acquires a Gaussian shape leading to the BW to Gaussian transition for the strength functions. Let us recall that in all the numerical calculations F (E) is I
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Fig. 6. Breit}Wigner and Gaussian "ts (solid lines) to the strength functions F (E) obtained from shell model calculations I with H "h(1)# <(2) in the 839-dimensional (ds)K ( 2 space. The are chosen to be mean-"eld basis states. H I In the "gures shell model results are averaged over middle 400 states after F (E) are zero centered in each case. Results I I in the panels (a), (b) and (c) are for "0.1, 0.2 and 0.3 with BW "t and similarly (d), (e), (f ) and (g) are for " 0.6, 0.8, 1.0 and 1.2 with Gaussian "t. Calculations use Wildenthal's h(1) and <(2) [85]. The "gure is constructed by combining, with permission, two "gures in [111].
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Fig. 7. Strength function F (E) vs. E for the 25 member EGOE(1#2) ensemble, de"ned in Fig. 1, in the 924-dimensional I N"12, m"6 space for various values of the interpolation parameter . The states are the mean-"eld states de"ned I by the distribution of m particles in the N single-particle states. The energies E are the diagonal matrix elements of H in I the m-particle basis states, E "kh(1)# <(2)k. The centroids ( ) of the E spectra are same as that of the eigenvalue I I (E) spectra but their widths are di!erent (for "0.3 their ratio is 0.7, for "0.5 it is 0.51). In the calculations E and E are I zero centered (for each member) and scaled by the spectrum (E's) width . Then the new energies are EK "(E ! )/ and I I EK "(E! )/. For each member C# are summed over the basis states in the energy window EK $ and then I I I ensemble averaged F K I (EK ) vs. EK is constructed as a histogram; the value of is chosen to be 0.025 for (0.1 and beyond # this "0.1. Results are shown for "0.01, 0.05, 0.1, 0.2, 0.3 and 0.5. For each there are two plots and they are for, (a) EK "!0.5 and (b) EK "0. In the plots F (EK ) dEK "1. The histograms are EGOE(1#2) results, continuous curves are I I I BW "t and dotted curves are Edgeworth corrected Gaussians (ED). The BW "t for "0.01, 0.05 and 0.1, for 0.2 both BW and ED and for 0.3 and 0.5 only ED are shown. For example, for EK "0 the BW width is 0.03, 0.1, 0.3 and 0.75 for I "0.01, 0.05, 0.1 and 0.2 respectively. Similarly the ED parameters (, , ) for "0.2, 0.3 and 0.5 are (0.571, 0.01, 0.92), (0.721, 0.01, 0.02) and (0.863, 0.01,!0.44) respectively. In the calculations I +0.2 and clearly there is $ BW to Gaussian transition. The calculations are due to Kota and Sahu [113].
constructed for the mean-"eld (h) states. Had we used some other (say, in the nuclear shell I I model, the states with good S;(3) [S;(4)}S¹]), do we get the BW to Gaussian transition?. It is plausible that (except for some singular 's) in the chaotic domain Gaussian form may be generic I and this may have to do with Percival's conjucture [114]; results in Figs. 6}11 support this
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conclusion. In SS this result is implicit (I(E)'s, independent of , are always taken to be Edgeworth corrected Gaussians). 3.3. Statistical mechanics for xnite systems of interacting particles via smoothed strength functions Let us begin with the introduction in a recent paper by Flambaum and Izrailev [107]: `As is known, quantum statistical laws have been derived for systems with in"nite number of particles, or for systems in a heat bath; therefore, their applicability to isolated "nite systems of a few particles is, at least, questionable. However, density of many-particle energy levels increases extremely fast } for this reason, even a weak interaction between particles can lead to a strong mixing } resulting in the so-called chaotic eigenstates. If the components of such eigenstates can be treated as random variables (the onset of quantum chaos), statistical methods are expected to be valid even for isolated dynamical systems. One should stress that a statistical description of such isolated systems can be quite di!erent from that based on standard canonical distribution; therefore, application of the famous Fermi}Dirac or Bose}Einstein formulas may give incorrect results. Moreover, for isolated few-particle systems, a serious problem arises in the de"nition of temperature or other thermodynamic variables like entropy and speci"c heat (this contrasts with in"nite systems for which di!erent de"nitions give the same result.) 2a. Following this, Flambaum and Izrailev [107] argued that a proper statistical description of isolated few-particle systems with interactions is to de"ne expectation values of operators in terms of the (smoothed) strength functions F (E), just as in SS, I instead of using the standard canonical distribution (e\#I 2#). More generally, there is a newly emerging understanding that in the chaotic domain of isolated "nite interacting many-particle systems smoothed densities (they include strength functions) de"ne the statistical description of these systems and these densities follow from embedded random matrix ensembles. This conjucture is based on the calculations for transition strength sums and transition matrix elements with Hamiltonans that generate order}chaos transitions. 3.3.1. Occupancies and GT strength sums Let us begin with the exact forms, written in terms of densities via (19), (21) and (28) for the expectation values of an operator K (here we restrict ourselves to operators that generate transition strength sums, K"ORO where O is a transition operator), [K(H!E)\ I (E) [K(H!E)\ I (E) ) " ) " " K#" [(H!E)\ I(E) I(E) [(H!E)\ [K(H!E)\IP P F P (E) " IP " I )_ I . (34) P [(H!E)\IP P F P (E) I I I The second and third forms in (34) further simply if we choose the subspaces or k to be the P eigenspaces of the operator K. Some examples are mean-"eld basis states for the single state number operators in EE(1#2), spherical con"gurations for spherical orbit number operators in nuclei, S;(4)}S¹ subspaces in the case of GT strength sums, etc. Then, KI(E) K#" I(E)
or
P KIP F P (E) I I . P F P (E) I I
(35)
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Using (34) and (35) the behavior of occupancies, GT and electromagnetic (EM) transition strength sums, etc. are studied recently, by several research groups, for a variety of interacting particle systems using Hamiltonians that generate order}chaos transitions. Some of them are: (i) occupancies using a 20 member EGOE(1#2) in the 330-dimensional N"11, m"4 space with h(1) de"ned by the SPE "i#(1/i); i"1, 2,2, 11 [107]; G (ii) occupancies and transition strength sums for one-body operators using the EGOE(1#2) in item (i) but for the (N"12, m"6) system [88,113]; (iii) occupancies in Ce and Au> atoms [68,70,92]; (iv) occupancies in a symmetrized coupled two-rotor model [115]; (v) occupancies and GT strength sums in Mg using two di!erent order}chaos generating Hamiltonians [88,90]; (vi) occupancies, GT and EM strength sums for several ( fp) shell nuclei using the modern shell model code NATHAN [116]. Most signi"cant conclusion of these studies is that the transition strength sums show quite di!erent behavior in regular and chaotic domains of the spectrum and the agreement shown in Fig. 3 between EGOE and shell model, for the strength sums, is a consequence of the chaoticity of the shell model spectrum. Let us "rst consider the shell model results shown in Fig. 8 for GT strength sums in Mg. They are studied using two di!erent interpolating Hamiltonians. First set of calculations use the spherical shell model mean-"eld (MF) Hamiltonian h(1) as the unperturbed Hamiltonian H and in this case the occupation number operators commute with H , H (MF)"h(1)#<(2)"H #(H !H ) . (36) H 1+ Note that H "h(1)#<(2); the h(1) is de"ned by O single-particle energies and <(2) by Kuo's 1+ two-body matrix elements as in Figs. 2}4 (ds)-shell examples. In Figs. 8 and 9 these calculations are denoted by MF. It is easily seen that the spherical con"gurations (generated by distributing the nucleons in the three (ds) shell orbits) are eigenstates for H . Therefore for "0 in (36), the spectrum will have degeneracies. In the second set of calculations, the S;(4)}S¹ scalar part of H is used as the unperturbed Hamiltonian, H 1+ 13}12 #(H !H )"H #(H !H ) . (37) H (S;(4))"H 1+ 13}12 QA?J?P 1+ H 13}12 QA?J?P In Figs. 8 and 9 calculations with (37) are denoted by SU(4). Note that the GT operator commutes with the SU(4) Hamiltonian H . For H"H , the eigenvalues and eigenvectors 13}12 13}12 are given easily by the S;(4)}S¹ algebra. The eigenstates are labelled, for a given number m of valence nucleons, by ¸, S, J, ¹ and the ;(4) irreducible representations (irreps) f " f f f f or the S;(4) irreps F " F F F where f #f #f #f "m, f 5f 5f 5f 50, F " ( f #f !f !f )/2, F "( f !f #f !f )/2, F "( f !f !f #f )/2 with additional restrictions on f 's coming from the spatial part. In addition, as the GT operator is a generator of G the S;(4) group, the S;(4) algebra gives directly K(GT)# in terms of the S;(4)}S¹ quantum part of H are given in [117]. It is clear from the numbers. Methods for constructing the H 13}12 results in Figs. 8 and 9 that the EGOE smoothed form (19) cannot give a proper description of the exact results in the case of regular motion. For &0 there are several (approximately) good quantum numbers with the nearby levels carrying di!erent sets of quantum numbers and therefore
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Fig. 8. GT strength sum vs. excitation energy (E) in the 325-dimensional (ds)K ( 2 shell model space. All the calculations use the interpolating Hamiltonian H "H # (H !H ) where H "h(1)#<(2) is the Kuo interacH 1+ 1+ tion. Results for the unperturbed Hamiltonian H ("0), for the full Hamiltonian ("1) and for the interpolating cases with "0.1, 0.3, 0.5 and 0.8 are compared with the EGOE predictions given by (19,38). For "1, the ED parameters ( , , , ) for the GT strength sum density are (!48.37, 12.64 MeV, 0.06, !0.1); the corresponding parameters for the state density are given in Fig. 3. First set of calculations use the spherical shell model mean-"eld (MF) Hamiltonian h(1) as the unperturbed Hamiltonian H and in this case the GT operator do not commute with H . In the "gures these calculations are denoted by MF. In the second set of calculations the S;(4)}S¹ scalar part H of H is used 13\12 1+ as the unperturbed Hamiltonian. In the "gures these calculations are denoted by SU(4). Some of the SU(4) results in the "gure are "rst reported in [90].
expectation values show large #uctuations as a function of excitation energy. The order}chaos transition as increases is clearly illustrated by the spectral rigidity in Figs. 9a and b (also by the distribution of nearest-neighbor level spacings as shown in [90]). As concluded in [90], the GT strength sums behave like the statistic. This similarity is probably due to the fact that both statistics are related to long-range correlations between the energy levels or wavefunctions. Thus, in the quantum chaotic domain transition strength sums, independent of the Hamiltonian, follow EGOE forms and we have statistical spectroscopy or a statistical mechanics in the chaotic domain. Another important observation that follows from Figs. 8 and 9 is: as the interacting particle system becomes chaotic, expectation values take smoothed forms (within P}T #uctuations as given by (20)) and hence described by the smoothed I (E) and I(E) densities; see (38)}(40) ahead. For < )
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Fig. 9. (a) statistic (¸) vs. ¸ for the eigenvalues from the MF calculations in Fig. 8. (b) Same as (a) but for the SU(4) interpolating Hamiltonian. Error bars give the standard deviation of (¸) over overlapping intervals of length ¸. Note that the dashed curves are for Poisson and the continuous curves are for GOE. Some of the SU(4) results in the "gure are "rst reported in [90].
( corresponds to order}chaos border and is di!erent for the MF and S;(4) Hamiltonians as it is seen from Figs. 8 and 9; &0.3 for MF and &0.5 for SU(4)) they take the EGOE form given by (19). This generic result, which is of central interest in quantum chaos studies of "nite interacting particle systems, is further substantiated by the EGOE(1#2) results in Figs. 10 and 11 and now we will turn to them. Fig. 10 shows the results for occupancies and Fig. 11 for transition strength sums for the one-body operator aR a calculated for various values of the interpolating parameter in the EGOE(1#2) Hamiltonian. In the system considered in Figs. 10 and 11 there are 12 singleparticle states and 6 particles, and aR a destroys a particle in state 9 and creates one in state 2. In these calculations, the estimate for for the order}chaos border can be obtained from (24). It gives B"20, K"262, "C (0.08) and then with C&0.6, &0.05 in agreement with the results in the "gure. It is clearly seen that below the region of chaos strength sums show strong #uctuations (in the regular ground state domain perturbation theory applies). In this region there is no equilibrium distribution for F (E) (or I(E), I (E), I(E)). However in the chaotic domain (NPC<1 ) I and the root mean square admixing two-particle matrix element (v) is larger than the mean spacing D of the states directly coupled by <(2), i.e. 'D with v"1) F (E) and similarly the I other densities in (34) and (35) can be replaced by their smoothed forms. Then for example, K#"I (E)/I(E) )
(38)
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Fig. 10. Occupation numbers for the 25 member 924-dimensional EGOE(1#2) ensemble de"ned in Fig. 1. Results are shown for the lowest 5 single-particle states and for six values of the interpolating parameter . The estimate (24) gives &0.05 for order}chaos border in the present EGOE(1#2) example. It is clearly seen that once chaos sets in, the occupation numbers take stable smoothed forms. For "0.08 and 0.1, the EGOE(1#2) results are compared with the EGOE smoothed form (19) which is a ratio of Gaussians (smoothed curves in the "gure } here Edgeworth corrections are added). For example, for "0.1 the occupancy density centroid shifts (from the state density centroids and scaled by state density width) are !0.42,!0.37,!0.3,!0.21 and !0.14, the widths (relative to the state density width) are 0.91, 0.93, 0.95, 0.98 and 0.99 and similarly ( , ) are (0.03,!0.35), (0.03,!0.37), (0.02,!0.38), (0.01,!0.36) and (0.01,!0.35) for the single-particle states 1, 2, 3, 4 and 5 respectively. Some of the results in the "gure are "rst reported in [88,113]. Fig. 11. Same as Fig. 10 but for the transition strength sums generated by the one-body transition operator aR a . Again it is seen that in the chaotic domain strength sums are well represented by EGOE; for "0.08 and 0.1 the EGOE(1#2) numerical results are compared with the EGOE smoothed form (19) (continuous curves in the "gure) as well as the r.m.s. deviation from EGOE smoothed form as given by (20) (dashed curves in the "gure, the upper and lower curves give $(E) from the middle smoothed curve). Just as in Fig. 10, for example for "0.1 the strength sum density centroid shift is 0.534, width is 0.92 and ( , )"(!0.04,!0.38). Some of the results in the "gure are "rst reported in [113].
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or K#" I (E)/ I(E) )
(39)
or K#" KIP F P (E)/ F P (E) . (40) I I IP IP Thus, as can be seen from Figs. 10 and 11 transition strength sums (expectation values of operators) take smoothed forms in the chaotic domain. Note that (38) and (39) are same as (19) and (21) of SS and (40) is valid only when k are eigen subspaces of the operator K. Thus, the results in the "gures P show not only that the strength sums take smoothed forms in the chaotic domain but also that the smoothed forms follow from EGOE with r.m.s. deviation given by (20) via P}T for strength #uctuations, i.e. EGOE operates in the chaotic domain of the spectrum. In fact, Flambaum et al. [107] (see also [115,118]), via EGOE(1#2) studies, argued that in the spectrum four regions are possible giving di!erent forms for the occupancies and other transition strength sums. Region I corresponds to the situation when NPC&1, then the eigenstates are strongly localized (`collective statesa [52]) which is the case with low-lying levels (or near yrast levels) in nuclei and here conventional perturbation theory should work. Region II corresponds to the case with NPC<1 but (D ; D is same as B/K in (24). This is the region of `initial chaotizationa and in this situation there is no equilibrium distribution for F (E) (or I(E), I (E), ) I I(E)) and it is sparse, i.e. NPC cannot be given as /DM ; DM is the mean spacing between energy levels. In this region occupation numbers n show strong #uctuations. Region III correspond to the case with NPC<1 and 'D . In this situation the system is chaotic and then F (E) and similarly I I(E), I(E), I (E), I (E) and F P (E) in (34) and (35) can be replaced by their smoothed forms. For ) )_I ) small particle numbers clear departures from the Fermi}Dirac (F}D) form, for occupation numbers, n( : E)"[1#eC\I# 2#]\ are seen in the calculations. Transition to the F}D form follows via (38)}(40) if the local mean-"eld (at a given energy E) approximation is valid, i.e when ;m and the particle number m is large (however here the chemical potentials and temperatures take shifted values, i.e. shifted from those de"ned via n and E); this is region IV. Classi"cation of the four regions I}IV give the criteria for the onset of chaos, equilibrium and thermalization. The criterion for onset of chaos here is basically same as the prescription given in Section 3.1. We will return to region IV with interaction driven thermalization, in Section 4. Finally, the investigations in [107,115,118] and in Figs. 6}11 appear to give the "rst con"rmation that SS de"ned by (5), (11), (19), (21) and (22) operates from the region (II) of onset of chaos (NPC <1) in the spectrum, it becomes accurate (to within GOE #uctuations) in the chaotic region (III) and in favourable situations reduces to canonical distribution treatment (with e!ective chemical potentials and temperatures). 3.3.2. Matrix elements of one-body transition operators Going beyond the transition strength sums (and expectation values in general), the transition matrix elements E OE themselves are expected to exhibit simple structure in the quantum D G chaotic domain (i.e. E and E are chaotic many-body states) of interacting particle systems. G D
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Towards this end let us "rst consider the recent Flambaum et al. [68}70] theory for the matrix elements of one-body transition operators and then re-examine (22). General form of a one-body transition operator written in the occupation number representation is O" aR a with and denoting single-particle states. The Flambaume et al. formula ? @ ?@ ? @ for the transition matrix elements E OE in the chaotic domain is derived as follows: D G (i) Using the expansion E "C#GG k , the transition matrix elements are written in terms of the I G G matrix elements in the mean-"eld basis states k , G E OE " C#GG C#DD k Ok I I D G D G IG ID C#GG C#GG C#DD C#DD k Ok k Ok "diag#o! diag . (41) # I I I I D G D G IG $IG ID $ID (ii) Assuming that the transitions between di!erent pairs of mean-"eld basis states are uncorrelated, the o! diag term in (41) is neglected. (iii) Using Eq. (12) of [62], evaluation of k Ok in the diag term is simple and then, D G E OE PE OE D G D G
(42) " n (1!n )EG C#EGG C#EDD EG @ ? , @ ? \C >C ?@ E G ? @ where are the single-particle energies and E are the mean-"eld basis states energies. ? G (iv) Assuming that k n (1!n )k do not vary much over the basis states that contribute to the G @ ? G given initial (E ) or "nal (E ) state in the chaotic domain, i.e. over the number of principal G D components (say N is NPC for the mean-"eld basis states), it can be replaced by its corresponding average giving n (1!n )EG Pn (1!n )#G . This result is well veri"ed @ ? @ ? using EE(1#2) calculations in [81]. (v) Applying (iv), one is left with a term involving summation over the C's and it can be changed to an integral with the additional assumption that the strength functions are a function of (E!E )/s where s is a scale parameter (spreading width for BW and the spectral width G G G G for Gaussian). Then we have the results (with D(E) denoting mean spacing), G F (E)"F (E; E , s )"(s )\f ((E!E )/s ), I I I I I I I
F (E) dE"1, I
C#"D(E)F (E)N(N f (0))\f ((E!E )/s ), D(E)"[N f (0)]\s I I I dE I " F (E; E , s ) dE . (43) N C#P C# I D(E) I I I I I (vi) For the last step in (43) is to be valid, constancy, in the chaotic domain, of the scale parameters s is needed (s Ps as indicated in (43)) and this is well supported by the results discussed in I I the last part of Section 3.2. (vii) Applying (v) and (vi), summation over E in Eq. (42) is converted into an integral (with s G G denoting average scale parameter de"ning strength functions for all the initial mean-"eld
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states k and similarly s for the "nal states), G D E OE " n (1!n )#G D(E )F(, s , s ) ; D G ?@ @ ? D G D ? @ "E !E # ! , D G @ ?
(44)
F(, s , s )" F G (E ; E , s )F D (E ; E "E ! # , s ) dE . G D I G G G I D D G @ ? D G (viii) Flambaum et al. assumed that the strength functions are BW in the chaotic domain. Substituting the BW form for the F's in the integral in (44) gives again a BW form for F, # 1 G D F(, , ) " . G D 5 2 #( # )/4 G D
(45)
In (45) and are the average BW spreading widths for the basis states over the initial and G D "nal many-particle states, respectively. The two questionable assumptions in (i)}(viii) are the neglect of the o! diag term in (41) and using the BW form for the F's. From the results in Section 3.2 and Figs. 6 and 7, it is clear that the appropriate form for the strength functions in the chaotic domain (i.e. for ' I in Fig. 7) is $ a Gaussian. Then the form of F is 1 " exp! . F(, , ) G D % (2( # ) 2( # ) G D G D Rewritting (22) in a slightly di!erent form [62,63],
(46)
IGG (E )IGD (E ) G D E OE " D G G D (E )IK (E ) IK G D G G G D G DG O (E , E ; , , , , ) ; O \ _ G D G D G D , (47) D G GG (E )GD (E ) G D it is easily seen that (with Pk ), for "0 (47) reduces to the theory given by (44) and (46). G G Therefore, the French et al. formulation (22) for one-body transition operators is same as Flambaum et al. theory but, more importantly, it takes into account e!ects due to the bivariate correlation coe$cient "OR
(48)
It is seen from (48), for "0 and P1, there is enhancement by the factor &1/((1!) ( for & ) in the matrix elements compared to the diag approximation (44) and (46). In addition, the G D
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Fig. 12. Transition strengths E OE vs. (E , E ): (a) for the EGOE(1#2) example in Fig. 1 with "0.3 and the D G G D transition operator O"aR a as in Fig. 11; (b) theory (48) with the correlation coe$cient "2/3 as given by the binary correlation approximation (12). In the "gures M.E. denotes E OE , Ei"EK "(E ! )/ and Ef"EK "(E ! )/ D G G G D D where ( , ) are state density centroid and width. The value "0.3 is chosen so that the form of the strength functions is close to a Gaussian (see Fig. 7) and therefore the bivariate Gaussian form is a good approximation for the strength densities. In the calculations, the strengths in the window EK $/2 and EK $/2 are summed and plotted at (EK , EK ); G D G D is chosen to be 0.1. For "2/3 the agreement between theory (48) and calculations is close and without (i.e. for "0) theory and calculations do not show any agreement. Results in the "gure are from [119].
enhancement grows with m as grows with m, a result that follows directly from (12). These results, appear to be `peculiara, are seen in the EE(1#2) tests [81] of the theory given by (44) and (45); in [69,70] (44) and (45) are used in the analysis of dipole (E1) transitions in Ce atom. Thus takes into the account e!ects due to the o!-diag term in (41). Comparision in Fig. 12 clearly con"rms the role of the bivariate correlation coe$cient and without it is not possible to get a meaningful description of the transition matrix elements [119]. Starting with the EGOE(1#2) Hamiltonian H "h(1)# <(2) , it can be concluded that (22) or the more simpler (48) provides a theory for the matrix elements of one-body transition operators in the chaotic domain (' I ) of isolated "nite interacting particle systems. In the domain $ (( I ) where the BW form is good for F (E), extensions of the Flambaum et al. theory (44) and I $ (45) to include the e!ects due to need to be worked out. Finally, further investigations of the various assumptions that led to (22), (44) and (48) are desirable.
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3.4. Other deformed EGOE 3.4.1. A p-EGOE model for mixing between distant conxgurations In inde"nitely large spectroscopic spaces a single EGOE is not appropriate as the space divides into distant subspaces that interact weakly (the interaction within a given subspace is usually strong). This is infact the case whenever there is shell structure (as in atoms, nuclei, atomic clusters, 2). In these situations it is more appropriate to consider p-EE, the partitioned embedded ensembles. For example, in many nuclear structure studies such as level densities, Gamow}Teller strength distributions, giant dipole resonance strengths and widths etc., it is important to include multi- excitations in the shell model spaces. In such nuclear structure calculations it is important to remember that shell model stability ensures that the mixing between the distant multi- con"gurations is weak. Therefore, the H matrix divides into a block structured matrix with each diagonal block denoting the con"gurations that are far apart (for light nuclei, they correspond to 0 , 2 , 4 ,2 for states with parity same as the ground state) and the o!-diagonal blocks giving the mixing between these con"gurations. Then we have p-EGOE(2) (or p-EGOE(1#2)) de"ned by a partitioned GOE in the two particle spaces [55,120,121]. Here, unlike in (4), the variances of the TBME in each diagonal as well as o!-diagonal blocks are di!erent and also the centriod of the diagonal blocks are di!erent (one-point function for the partitioned GOE is solved in [122]). In m (many)-particle spaces, it is straightforward to construct p-EGOE(2) H matrices on a machine by applying (1)}(3). For insight into the physics of one-point functions (state densities) generated by p-EGOE, a simple two-spike model was introduced in [55] where only the S"0 2 subspace of the m-particle space is considered and in addition the diagonal GOE's in two-particle space are put to zero. Then the m-particle Hamiltonian H(m) is a 2;2 block matrix,
H(m)"
A
XI
X
P
B
0
X
XI I
.
(49)
In (49), dimensions of the matrix A and B are, say, d and d with d 4d . With and denoting the quantum numbers of the basis states de"ning the matrices A and B, respectively, the eigenstates #B C . The state density I (E) and its decompositake the form E"B C ? ? # @ @ # tion into the partial densities I (E) and I (E) are given by I (E)"[(H!E)\"I (E)#I (E) , B , "1, 2, 3,2, d . I (E)"[(H!E)\ " C ? # ? The moments of
(50)
1 I I (E) are M (I )"HN " (E)EN . N d # )"1, M (I )"0 and M (I )" where the partial variance is de"ned For example, M (I by " d \ X . The shape of the densities in (50) depends on the mixing parameter G H GH
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" /. The solution to I (E) is obtained by writing it "rst in terms of I (x), E! I ((E(E!)), E40, E5 I (E)" E
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(51)
and then solving for I (x) [55,120]. The I (x) is generated by the matrix X which corresponds to S"0PS"2 excitations produced by the de"ning two-body interaction (which is an o!-diagonal GOE for m"2). The binary correlation approximation gives M (I )"!( )J and then, J I (E)"d E( )\exp(!E/ ), !R4E4#R. (52) N\#%-# Thus, the I (E) which is a spike for "0, with the mixing of the space A states with the faro! B states, becomes bimodal as increases. For completness let us mention that in the case the m-particle matrix X in (49) is represented by an o!-diagonal GOE (with v giving the variance of the matrix elements), the form of I (E) is I (E)"(2vE)\([(R !E)(E!R )] , N\%-# > \ R 4E4R and R "v(d $(d . (53) \ > ! Recently [121], for testing the p-EGOE(2) result (51) and (52) a shell model example is constructed in (ds) ( 2_1 [(ds);( f
)]( 2_1 space. Using (50), the shell model histograms for I (E) are constructed and compared with the p-EGOE (51) and (52) and p-GOE (51) and (53) forms in Fig. 13 for "0.1, 0.5, 1.0 and 10.0. These shell model results clearly con"rm the form (51) and (52) for the one-point function I (E) as given by the binary correlation approximation theory. Thus mixing with distant con"gurations give bi-modal and in general multi-modal densities. In practical applications it is expected that 40.3, and it is seen that in this situation only a fraction of the partial variance goes into the variance of the lower mode (most of it goes into separating the two modes); for example, for "0.3 only 8% will contribute. By switching on the diagonal A and B matrices in (49), it is expected that in the "rst approximation (by convolution argument) the variance of the lower mode of I (E), i.e : /10, will add to the variance of the unperturbed I (E). Although these basic results are important, for practical applications (as mentioned in the beginning of this section) the general case of mixing of S"0 2 42 spaces should be solved, i.e. go beyond the 2;2 block matrix case. There are attempts to solve this in some special cases as discussed ahead in Section 4.2. 3.4.2. EGOE(2)-JT ensemble with particle}hole symmetry Besides the real symmetric nature, the EE do not have any other symmetries. Symmetries such as angular momentum and/or isospin are essential for nuclei, atoms, etc. and hence the need for EE-sym embedded ensembles. Till now only EGOE(2)-J¹ which is a special case of EE-sym has received attention. Let us consider proton-neutron systems with nucleons in single-particle j-orbits and denote the two-particle J¹ values as J ¹ . EGOE(2)-J¹ follows by de"ning each of the two-particle J ¹ matrix to be an independent GOE (but the variances v of the matrix (2 elements in each J ¹ space are chosen to be independent of J¹; v "v) and then constructing (2 the many particle J¹ matrices. Note that in the literature EGOE(2)-J¹ is also referred as
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Fig. 13. I (E) partial density vs. E for various values of the mixing parameter . Shell model results (histograms) are compared with the two-spike p-EGOE results (continuous curves) given by (51) and (52) and p-GOE results (dashed curves) given by (51) and (53). The shell model calculations are in (ds)_1 [(ds);( f )]1 space with J"0, ¹"0. The dimensions of the S"0 (A block) and S"2 (B block) spaces are d "71 and d "253. The matrix dimensionality of H is d"d #d "324. In order to generate a typical member of the two spike p-EGOE matrix in (49), the A and B matrices are modi"ed such that A "0 and B "() ; the X matrix is generated by sdPf two-body Kuo matrix GH GH GH elements (see the second reference in [48] for these matrix elements). Some of the results in the "gure are "rst reported in [121].
TBRE-J¹. Numerical calculations showed that TBRE-J¹ gives the same forms for the various densities as in (5), (11), (19) and to some extent because of this analytical studies of this ensemble are not undertaken but, as we shall see, they are needed in future. Recently, there is new interest in TBRE-J¹ as further constraints on the variances v of each two-particle J¹ matrix are found to (2 give some important new physics [97]. If the ensemble is invariant under particle}hole (ph) conjugation (i.e. under Pandya transform [123]), it implies that v "v/(2J #1)(2¹ #1) (for (2 identical particle systems there is no ¹ and v "v/(2J #1)). With this constraint on the ( two-particle GOE matrices, EGOE(2)-J¹ : ph ensemble (called random quasiparticle ensemble (RQE) in [97]) is constructed, in the m-particle spaces, for several choices of the single particle j-orbits, m and more importantly J¹ values (by putting the single-particle energies to zero). The following remarkable results are found [97]: (i) most members (&70%) of EGOE(2)-J¹:ph give J"0> as ground state and they are separated by a gap from the excited states typical of a BCS system. For the states with JO0 the gap is much smaller; (ii) there is evidence for low-lying vibrational states (analyzed via the spacing between the lowest J"0>, 2>, 4> states) and noncollective rotational states (i.e. energies E of the yrast states have overall quadratic increase with (
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respect to J but with large #uctuations from one J to the next) when averaged over large number of yrast states; (iii) there is no evidence for rotational collectivity. The source of these regularities generated by EGOE(2)-J¹ : ph is not yet known (see however [124]); recently it is demonstrated [125], employing EGUE(2), that time-reversal invariance cannot be the origin of the 0> dominance. It is clear that EE with extra symmetries should be explored further as we may understand how regular patterns (rotations, vibrations, etc.) suggestive of group symmetries evolve out of complex two-particle interactions in many-body systems such as molecules, nuclei, etc. Recent interacting boson model calculations using EGOE(1#2)-¸ (here the bosons carry angular momenta l"0> (s) and l"2> (d) and ¸ is the angular momentum of a N boson system) gave strong evidence [126] for a predominance of ¸"0> ground states and the occurrence of both vibrational and rotational band structures in nuclei. An alternative to EGOE-J¹ : ph is to use K# EE where EE may be EGOE(2), EGOE(1#2) or EGOE-J¹:ph, etc. with K being a "xed Hamiltonian such as the pairing (P) plus quadrupole}quadrupole (QQ) interaction which generates regular features seen in the low-lying states in nuclei. Preliminary studies of K# EE are given in [98]. It is useful to point out that the recently proposed Hamiltonian decompositions by Zuker et al. [127] give better forms for the "xed operator K in nuclei.
4. Interaction-driven thermalization and Fock-space localization With quantum chaos and random matrices operating in interacting particle systems, questions related to chaos and interaction-driven thermalization in isolated systems and also chaos and Fock-space localization have started receiving attention recently. Results in these two topics, with EE playing a role, are described brie#y in this section. 4.1. Chaos and interaction-driven thermalization For isolated "nite interacting many-particle systems, in Section 3.1 a border for order}chaos transition is derived and in Section 3.3 it is argued that in the chaotic domain there is a statistical mechanics with various densities taking smoothed forms (with GOE #uctuations) as given by EGOE and observables like occupancies, GT strength sums, etc. follow from them. But a natural question is: in the chaotic domain will there be a point from where thermalization occurs, i.e. will there be a domain where the smoothed forms reproduce the conventional Fermi}Dirac or Bose}Einstein statistics? Then there is quantum ergodicity for isolated quntum many-body systems. The preliminary investigations carried out so far are: (i) Fermi}Dirac form for the occupancies studied using the EE(1#2) ensemble [107]; (ii) di!erent de"nitions of entropy studied using nuclear shell model with realistic interactions [128]; (iii) di!erent de"nitions of temperature studied using symmetrized coupled two-rotor model [115]. As already stated in Section 3.3, Flambaum and Izrailev [107] found, using numerical EE(1#2) calculations, the criterion for the occupancies to follow Fermi}Dirac (FD) distribution, i.e.
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NPC<1 for the mean-"eld basis states expanded in terms of the eigenstates, particle number m is large and the spreading widths ;m (this is region IV mentioned in Section 3.3). This result is deduced by comparing occupancies calculated via strength functions using (40) and the FD form. In region IV, Flambaum et al. argued [70] that occupation numbers can be calculated using the canonical distribution with temperature ¹, n I exp(!E /¹)(E ) dE I I I , (54) n (¹)" G G exp(!E /¹)(E ) dE I I I where k are the mean-"eld basis states, E "kHk are the k-basis states energies and (E ) is I I the density of states for E . The temparature ¹ and energy E are related by I E exp(!E /¹)(E ) dE I I I . E" I (55) exp(!E /¹)(E ) dE I I I Note that the expression for E in terms of the SPE and TBME will follow from the results given I in Appendix E. Horoi et al. [128] compared in a (ds)K calculation, by varying in H"h(1)# <(2) (here h(1) and <(2) correspond to the Wildenthal's interaction [85]), the information entropy S for shell model eigenstates expanded in terms of the mean-"eld basis states with the thermodynamic entropy S\ calculated using the state density and the single-particle entropy S calculated using the occupation numbers f # "n #/(2j#1): JH JH S\"ln(I(E)), S"! (2j#1)[ f # ln f # #(1!f # ) ln(1!f # )] . (56) JH JH JH JH JH They found that the S coincides (basically over the entire energy range) with S\ and S for "1 while this is not the case for ;1 or <1. For small there is basically equilibrium picture of non-interacting particles and S is weakly correlated with S\. Similarly for <1 there is strong mean-"eld admixing TBME (in real calculations one has to ensure that the h and < parts of H are orthogonal in some well-de"ned sense; see Appendix E for a criterion) and S is same as S and they are at their maximum but S\ is same as the case with "1. Therefore, in the absence of the mean-"eld S is not useful for characterizing thermalization. Thus, with a mean"eld and su$cient amount of chaoticity, all the di!erent de"nitions of entropy coincide. Borgonovi et al. [115] studied in a symmetrized coupled two-rotor model the occupancy numbers and compared di!erent de"nitions of temperature. The di!erent de"nitions are: (i) ¹ via canonical expression (55) for the energy; (ii) ¹ de"ned by the state density of the total Hamiltonian, ¹ " d ln((E))/dE \; (iii) ¹ de"ned by the Bose}Einstein distribution for # occupancies, n "(e@ # CG \I!1)\ where "(¹ )\, is the energy of the ith single-particle # # G G state and is chemical potential. The three di!erent de"nitions are found to coincide (just as in the case of entropy in shell model calculations) in the region of thermalization and the occupancies are described by Bose}Einstein distribution. Basic conclusion of all the studies is that strong enough interaction plays the role of a heat bath, thus leading to thermalization. As an aside it is important to mention that one of the "rst
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calculations of obtaining e!ective chemical potentials and temperatures via occupancies for interacting particle systems is given in [129]. 4.2. Embedded ensembles and Fock space localization Altshuler, Gefen, Kamenev and Levitov (AGKL) in a pioneering contribution [108] addressed the basic question, in general terms, of how complexity (and chaos) develops in interacting particle systems with a mean "eld. Starting with (23), i.e. with the (1#2)-body Hamiltonian H"h(1)#<(2) for a many fermion system, the Fock-space is de"ned by the eigenstates (they are Slater determinants) of the mean-"eld producing h(1) part of the Hamiltonian. With N singleparticle states and m fermions, number of Fock states are (,). The distance in Fock space between K two Fock states is de"ned as twice the minimum number of fermions to be moved from one single-particle state to the other in order to get from one state to other. From (1)}(3) it should be obvious that the matrix elements of H will be zero unless the Fock distance 44. Now, the basic question is how are the interacting eigenstates composed out of the non-interacting Fock space eigenvectors (to some extent this question is answered in the investigations presented in Sections 3.1}3.3, 4.1). For small values of the interaction strength and excitation energy, it is expected that the interacting eigenstates are composed of small number of non-interacting eigenvectors while for large interaction strength they will be composed of many non-interacting eigenvectors. For describing the transition between these two extremes AGKL introduced the term `localization in Fock spacea, for localized states the energy of the constituent quasiparticles is a good quantum number, whereas for the extended delocalized states only total energy is conserved. General answers are expected to come from the study of the EE(1#2) ensemble (23). AGKL analyzed analytically EE(1#2) beyond perturbation theory, by mapping the Fock space problem onto a tight-binding model on the in"nite Bethe lattice using a hierarchy of approximations (important among them is the restriction to couplings with Fock distance only 2). Then they could use the phenomenon of Anderson localization on a Cayley tree. In terms of the parameter g, g" .
(57)
AGKL identi"ed localized and delocalized regimes in the spectrum. In (57) is the average single-particle level spacing and is interaction strength as de"ned by (23). In mesoscopic physics g has a special signi"cance, it is related to the Thouless energy E . For a disordered system 2 E " D/¸, D being the di!usion constant of the electrons and ¸ is the relevant sample length. 2 Then the Thouless number E / ( being the single particle level spacing near the Fermi energy) is the dimensionless conductance (i.e. conductance in the units of e/ ) and it is nothing but g. For g<1 AGKL showed the existence of three regimes separated by two characteristic energies. The transition from localized to extented states occurs at E &(g/ln g and above E &(g the many-particle states are completely mixed; between E and E the states are non-ergodic. These results were further studied using supersymmetry method in [104], with few interacting particles in a random potential in [103] and by introducing the concept of "nite energy scaling and diagonalizing a microscopic Hamiltonian in [100]. More importantly, in [104] it was shown that a proper treatment of "nite size e!ects changes E to g exactly as in (26). One of the main motivations
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behind the AGKL study is the Sivan et al experiment. Sivan et al. in 1994 [130] measured the quasiparticle spectrum of a di!usive quantum dot via its tunneling conductance. The data showed few narrow peaks (of the order of g of the dot) near the Fermi energy followed by a quasicontinuum. Does this data display localized to extended states transition predicted by AGKL theory? In order to avoid the various approximations in the analytical treatment, MejiaMonasterio, Richert, Rupp and WeidenmuK ller (MRRW) [131] analyzed numerically the EE(1#2) adopted to the problem at hand. In the MRRW model [131] EGOE(1#2) ensemble de"ned by (23) is constructed with [h(1)] representing a random potential. To this end the SPE de"ning [h(1)] are drawn from the center of a large GOE after unfolding the eigenvalues with a semi-circle. The many-particle states are divided into classes, with class M of Fock states de"ned by M-particle } (M!1) hole excitation over the ground (lowest energy) state given by h(1). Then the states are labelled by M, i, M"1, 2,2 and i is an index for each of the states in the class M. The diagonal part of <(2) is removed by assuming that it is included in h(1) by the Hartree}Fock method. The matrix elements of <(2) between two di!erent Fock states M, i and N, j vanish unless M!N"0 or 1 and unless the Fock distance is 44 (the assumption that couplings with M!N"2 do not change the results is veri"ed numerically). Denoting the energies, due to h(1) of the M, i states by EM , all Fock states with energies EM 918 are dropped (this cuto! is used as the interest is in the G G vicinity of the Fermi surface) and with this the EE(1#2) ensemble (23) is constructed for various values and diagonalized. Note that "/g and for Sivan et al. data g"5!15. Finally, <(2) with 100 members and only a single realization of [h(1)] are used in the discussion of the "nal results. The strength functions for the 1p}0h Fock states 1, i and their BW spreading widths and NPC are studied in detail. The quasiparticle spectrum of the 1p}0h states is shown in Fig. 14 for
Fig. 14. Quasiparticle line shapes F (E) for the 1p}0h states 1, 2,1, 6 and 1, 11 (full curves). To reduce the density of G the data set, the numbers are averaged over small energy windows and they yield the dots shown for these states. The dotted curve is the quasiparticle spectrum of all 1p}0h states. Results are for g"10. The inset shows the BW spreading widths vs. E for g"5, 10 and 15 together with the GR prediction (full curve). The "gure is taken with permission from [131].
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sensible values of g and it reproduces the essential features of Sivan et al. data. The spreading widths and NPC are found to change smoothly with excitation energy and do not display the transition predicted by AGKL. The spreading widths follow the Golden rule (GR) formula (E)"2(/g) (E) as in (33) with (E) being the density of the 2p}1h states; this is consistent %0 with the "nding in [102] that one should include only the density of directly coupled states in the spreading width expression (33). Similarly at su$ciently high energies E, (NPC) is found to follow # the GR formula (NPC) " (E) (E) where is the total density of states. # %0 R R Leyronas, Tworzydlo and Beenakker (LTB) [132] addressed the same problem but using the layer model of Georgeot and Shepelyansky [102]. The layer model is a particular form of p-EGOE(1#2) and with this model it is possible to handle systems with very large number of single-particle states. The model is based on the observation that in the situation ;, only the eigenstates of h(1) within the energy layer of width are strongly coupled by <(2). This is implemented by constructing a p-EGOE(1#2) with the TBME k, l<(2)i, j"0 unless i, j, k, l are distinct indices and satisfy the relation i#j"k#l. As the thermodynamic limit g<1 is expected to be essential for observing the localized to delocalized phase transition, LTB employed the layer model and carried out calculations with &50 single-particle states and g&500. In such large systems, localization transition in Fock space predicted by AGKL is indeed seen. However, it is not clear whether the layer model results in [132] are appropriate for analyzing Sivan et al. data. Independent of this, the AGKL, MRRW, LTB and other related studies [99,102}108,131,132]
Fig. 15. (a) Various levels of chaotization of single or many-particle excited states in a "nite interacting many particle system according to Silvestrov [106]. (b) Classi"cation of the spectrum of a "nite interacting particle system into four regions I}IV according to Flambaum and Izrailev [107]. Shown also is (c) typical spectrum of a heavy nucleus (constructed out of a picture drawn by French and Kota [133]). See text for de"nitions of the various symbols appearing in the "gure.
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inspired by the Sivan et al. experiment certainly opened up investigations in characterizing chaotization of the many-particle excited states. For example, by examining in considerable detail (NPC) for EGOE(1#2) eigenstates in the mean-"eld basis, Silvestrov [106] proposed that there # are several layers of chaotization as shown in Fig. 15. Shown in this "gure are also the classi"cation, discussed in Section 3.3, due to Flambaum and Izrailev [107] and the division of a typical spectrum of a heavy nucleus [133].
5. Conclusions In the last few years, the study of quantum chaos in isolated "nite interacting particle systems has turned from spectral statistics to properties of eigenfuctions and transition strengths. With this, the nature of occupancies (of single-particle states), strength functions, information entropy, inverse participation ratio or the number of principal components, transition strength sums (for example Gamow}Teller strength sums in nuclei) and transition matrix elements of one-body operators, interaction-driven thermalization, Fock-space localization and transition to delocalized phase etc. in the chaotic domain of interacting particle systems is being studied in several systems in an attempt to characterize and quantify quantum chaos in many-particle systems. In this article, an attempt is made to review all the results derived in this subject, for many-particle systems, using embedded random matrix ensembles. Let us mention that there are also recent attempts to study some of these questions using chaotic interacting particle systems with few particles (say 2 or 3) whose classical analogues can be constructed [112,115,134,135], banded random matrices [136], billiard models of N-body systems [137], etc. (discussion of these results is beyond the scope of the present article). The various results presented in Sections 3 and 4 are persuasive enough to claim that EGOE operates in the quantum chaotic domain of isolated "nite interacting particle systems (this gives a new basis for statistical spectroscopy). Certainly much remains to be done in exploring and applying EGOE and its various deformed versions. Many questions in chaotization of many-particle systems are only touched upon and they deserve further investigations. Finally, though many of the earlier embedded ensemble results are derived in nuclear physics context (Section 2), it is clear from Sections 3 and 4 that, as pointed out in [97], embedded ensembles are relevant for generic many-body systems such as atoms, molecules, atomic clusters, quantum dots, etc. and therefore the results in Sections 2}4 should be considered in as broad an arena as possible.
Acknowledgements The author is deeply indebted to O. Bohigas and H.A. WeidenmuK ller for reading a draft of the article and for many fruitful discussions. Thanks are due to J.M.G. GoH mez, R.U. Haq, K. Kar, After the submission of the present article two important preprints have appeared: (i) Benet and WeidenmuK ller (cond-mat/0005103) introduced an analytical approach based on the supersymmetry method [8] and showed that, within some bounds, EGOE(EGUE) level #uctuations are same as that of GOE(GUE) #uctuations; (ii) Shepelyansky (quantph/0006073) has brought out the links and importance to quantum computers, the border for emergence of quantum chaos in energy level statistics in interacting many particle systems (de"ned by in Section 3.1).
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D. Majumdar, V. Potbhare, J. Retamosa, R. Sahu and S. Sumedha for collaborations and for many stimulating discussions. The author is thankful to J.B. French for many years of research collaboration on statistical spectroscopy. The author thanks V.V. Flambaum for bringing to his attention Ref. [110]. Thanks are due to A.P. Zuker for supplying Ca eigenvalues. Finally, thanks are due to D.L. Shepelyansky, V. Zelevinsky and H.A. WeidenmuK ller for permisson to reproduce, from their papers, Figs. 5, 6 and 14, respectively.
Appendix A. Some basic results for classical random matrix ensembles Following Dyson [6], Hamiltonians for quantal systems that preserve angular momentum and time reversal are representable by GOE, an ensemble of real symmetric matrices H(S) with the matrix elements H (S) chosen to be independent zero centered Gaussian random variables with GH variance H (S)"v(1# ). Similarly, systems for which TRI is completely broken, i.e. for time GH GH reversal non-invariant (TRNI) systems, the Hamiltonian is representable by GUE, an ensemble of complex Hermitian matrices H(C) " H(S) #i H(A) where H(A) is a real antisymmetric matrix; H (S) is already de"ned and H (A) are zero centered Gaussian random variables with GH GH variance H (A)"v(1! ). In addition to GOE and GUE there is also the symplectic ensemble GH GH GSE appropriate for systems with good TRI but with half integral spin and broken rotational symmetry. The other extreme to the GOE, GUE and GSE spectra are the picket fence (or uniform) spectrum and the Poisson spectrum. The latter can be realized for example, by random superposition of good symmetry (say GOE) spectra. Recent applications of GSE are given in [38,138] and references therein and we will not refer to GSE any further in this article. Though the nearestneighbor spacing distribution (NNSD) gives information about #uctuations (in fact about level repulsion), more satisfying are the measures introduced by Dyson which derive from correlation functions of various orders (and types). The lowest-order function which su$ces for most purposes [7] is the two-point correlation function SM(x, y), SM(x, y)"(x)(y)!(x) (y) ,
(A.1)
where (x)((y)) is the normalized eigenvalue density and (x)((y)) is a semi-circle for GOE; the bar in (A.1) denotes ensemble average. The integral version S$(x, y) is also useful as we shall see ahead,
S$(x, y)"
V
dx
W
dy SM(x, y)"F(x)F(y)!F(x) (y) ,
(A.2)
\ \ where F(x)"!V (x) dx which counts number of eigenvalues below x when multiplied by d the dimensionality. The number variance (n ) which gives the root mean square deviation of actual number (n) of levels from the average number of levels (n ) in the interval n "(y!x)/DM , where DM is the mean spacing and (y!x) is the energy interval, is given by (n )"d[S$(x, x)#S$(x#n DM , x#n DM )!2S$(x, x#n DM ) ,
(A.3)
which is thus an exact two-point measure. The (n ) in many situations is related to the variance (k) of the kth nearest-neighbor spacing distribution by the simple formula [7] (k)" (k#1)!. This relationship is used recently in solving a novel problem for non-interacting
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particle systems by Bohigas et al. [139]. The Dyson}Mehta statistic which gives the root mean square deviation of the staircase function F(x) in the interval x$n DM /2 from the best-"t straight line is also related to (n ),
2 L (n !2n r#r)(r) dr . M (n )" n
(A.4)
For obvious reasons, the statistic is a measure of spectral rigidity. Expressions for the NNSD P(S) dS, (n ) and (n ) for Poisson, GOE and GUE are [6],
exp !S dS
Poisson ,
P(S) dS" (S/2) exp !S/4 dS
GOE ,
(A.5)
(32S/) exp !4S/ dS GUE ,
n 1 2 ln(2n )##1! #O (n )" n 8
1 1 (ln(2n )##1)#O n
n /15 9 1 (n )! M (n )" 2 %-# 4 1 9 (n )! 2 %3# 8
Poisson , GOE , (A.6) GUE ,
Poisson , GOE ,
(A.7)
GUE .
In (A.5) S is in DM units. Finally, for GOE the transition strengths are distributed according to the Porter}Thomas (P}T) law. Given the locally renormalized transition strengths x, the P}T form is 1 x\ exp !x/2 dx . P } (x) dx" .2 (2
(A.8)
The changes in the nature of level #uctuations as a symmetry is gradually broken is studied by interpolating the Gaussian ensembles [35]. For GOE}GUE transition one starts with the interpolating ensemble H (v) " H(S; v) # i H(A; v) which gives GOE for "0 and GUE for ? "1; note that the variance v is now shown explicitly in denoting the ensembles. The signi"cant parameter for the transition however, is not the `globala but rather d . This is because the spectral width is proportional to d and therefore the local average level spacing DM is proportional to d\ . Then by simple second-order peturbation theory one can see that the transition parameter is d or more precisely which is given by, ()"H (¹RNI)/DM "v/DM . For small , the GH spacing variance for kth nearest spacings is easily given by (k; )"(k; 0)!4#2 . This
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equation gives the slope of the transition curve. The complete transition curve can indeed be constructed [35] and to this end we give the "nal formula for (n ),
n 1 ln 1# . (n ; )" (n )# } %3# %-# %3# 4(#2) 2
(A.9)
The cut o! parameter "0.615 and this value follows from the GOE}GUE di!erence for n "1. The GOE}GUE transition curve given by (A.9) is used to derive an upper bound on the amount of the TRNI part of the nucleon}nucleon interaction [57]. Poisson to GOE and GUE transitions received attention of large number of research groups as they represent order to chaos transitions. There are several di!erent formulas, given by Brody [140], Berry and Robnik [33], Hasegawa et al. [141], Izrailev [142], Abul-Magd [143], etc., for the NNSD P (S) dS interpolating Poisson .\%-# (S) dS interpolating Poisson and GUE [142,144,145]. For and GOE and similarly for P } . %3# example the well-known Brody (Br) distribution for Poisson to GOE transition, with the Brody parameter , based on the Wigner's surmise, has no physical basis but "ts data embarrassingly well, is [140]
#2 (S) dS"a SS exp !bSS> ; a"(#1)b, b" P } . %-# #1
S>
.
(A.10)
The Br-distribution (A.10) reduces to Poisson for "0 and Wigner (GOE) form for "1. For 0((1, the distribution (A.10) vanishes as SP0 but has an in"nite derivative at that point, an unrealistic feature. The one parameter () Berry}Robnik (BR) formulas for Poisson to GOE and GUE are [33,144], P }0 (S) dS"(1!) exp !(1!)S erfc((S/2) . %-# # (2(1!)#S/2) exp !(1!)S!S/4 ,
(A.11)
2 P 0 (S) dS"(2(1!)!(1!)S) exp !(1!)S erfc S .\%3# ( #
8 4 32 S# (1!)S#(1!) exp !(1!)S! S , (A.12)
where is fractional volume, in phase space, of the chaotic region and 1! is fractional volume of all regular regions put together. The BR forms are good when there is only one dominant chaotic region coexisting with regular regions. Note that "0 in (A.11) and (A.12) gives Poisson and "1 Wigner (GOE or GUE). The limits at "0 and "1 are correct but for 0((1, the distributions have non-zero value for S"0 which is unrealistic. Extending Wigner's 2;2 matrix formalism, recently [146] transition curves are constructed, for the variance of the NNSD for Poisson to GOE and GUE, in terms of a transition parameter ( is mean squared admixing GOE or GUE matrix element devided by times square of the mean spacing D of the Poisson spectrum, "1 for GOE and "2 for GUE). The NNSD for Poisson to GOE is
SK SK (SK ) dSK "dSK exp !! exp !SK /8 I d, SK "S/D . P" } . %-# 4 8 4
(A.13)
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For "0, (A.13) gives Poisson and for PRthe Wigner (GOE) form. Eq. (A.13) is also derived by Haake et al. [147] and [28] gives examples of chaotic systems where it is applied. It is easily proved that P"(S) goes to zero as S goes to zero for non-zero values of (i.e. there is level repulsion as soon as GOE is switched on). More signi"cantly the variance of the NNSD (0 : )"(S/SM )!1 for Poisson to GOE transition, which de"nes a transition curve, is given by 8#2 } (0 : )" !1 . . %-# [(!1/2, 0, 2)]
(A.14)
In (A.14) is Kummer's function [148]. It is instructive to consider small expansion, ; } (0 : ) P 1#4(ln()#1#!ln 2) , . %-#
(A.15)
where is Euler's constant. Note the ln() term also appears in the perturbation theory result for the number variance [35],
; n (n , ) P n !2 ln #!1#ln 4 . .\%-# 2
(A.16)
The NNSD for Poisson to GUE from 2;2 matrix formalism given in [146] is
SK SK P" (SK ) dSK "dSK exp !SK /8 \ exp !! Sinh d . .}%3# 8 4 (2 Then the expression for (0 : ) is
(A.17)
12#2 (0 : )" !1 , .\%3# [X()]
1/2, 1 ; 2 X()"2 !Ei(2)#4(2/ F 3/2, 3/2 # (8/#e[1!((2)] .
(A.18)
In (A.18) Ei is exponential integral, is error function and F is a generalized hypergeometric function [148]. Once again it is instructive to consider small expansion, ; (0 : ) P 1#8(ln ()###ln 2) . .\%3#
(A.19)
Just as in the case of Poisson to GOE, here also there is the ln() term. Finally the perturbation theory result for the number variance in this case is [35],
; n (n , ) P n !4 ln # . .\%3# 2
(A.20)
The results in (A.13), (A.14), (A.17) and (A.18) are in fact applicable to general N;N matrices (or for any interacting many particle system) through the transition parameter and this is indeed veri"ed by the results in Fig. 4 of the second paper in Ref. [147]. Eqs. (A.14) and (A.18) de"ne Poisson to
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GOE and GUE transition curves for (0) statistic and it is seen that the Poisson to GOE and Poisson to GUE transitions are nearly complete for &0.3. Moreover, using the BR formula for (0 : ) it is seen that K/20(1!) for 90.05; the BR parameter represents fractional } . %-# volume, in phase space, of the chaotic region of a complex dynamical system. However for :0.01 results of Eq. (A.14) and the corresponding BR formula di!er signi"cantly. Experiments with several coupled microwave billiards [27,28] will be able to test the Poisson to GOE transition curve given by (A.14). Construction of transition curves for partitioned GOE (p-GOE) are considered in [35,37] and the signi"cance of the simple 2;2 p-GOE in the study of realistic systems is recognized in the analysis of Al data [43,45]. For this nucleus there is complete spectroscopy for positive parity levels upto 8 MeV excitation containing JL"1> to 5> (15}25 levels for each J) with 75 ¹"0 and 25 ¹"1 levels; the ¹"0 and ¹"1 levels coexist. Thus there are 2 GOE's for this system (one for ¹"1 and other for ¹"0) with possible mixing between the two due to, say, coulomb (c) force. Then the appropriate random matrix model is a 2;2 p-GOE with 2GOE to 1GOE transition. The ensemble H then is de"ned by H #< where H is a 2;2 block matrix with dimension d"d #d (d is dimension of the upper block and d of the lower block) and < is a GOE with variance v. The o!-diagonal block of H is zero, upper block H with dimension d is _ a GOE(v ) where v "v(d #d )/d and similarly the lower block H with d is a GOE(v ) _ with v "v(d #d )/d (a slightly di!erent 2;2 p-GOE was considered in [149]). Thus "0 corresponds to a superposition of two GOE's and PRgives GOE. With the transition parameter "v/DM , the number variance in the so-called binary correlation approximation is [35],
r 1 . (r, )"(r,R)# ln 1# " 4(#)
(A.21)
The cut-o! parameter is determined using the result (r, 0)" ([d /d]r)# ([d /d]r) " %-# %-# and formula (A.21) is good for r'2. Note that (r,R) is the GOE value. As discussed before, (r) formula gives the expression for M (r) (Leitner in [37]), 2 1 1 1 ! ! ln(1#Xr) M (r, )"M (r,R)# " 2 Xr 2Xr
#
4 1 9 tan\(Xr)# ! ; X" . Xr 2Xr 4 2(#)
(A.22)
A direct and good test of (A.21) came recently from experiments with two-coupled #at superconducting microwave billiards [150]. It is not out of place to mention that in [150] Eq. (A.21) is ascribed to Leitner [37] although the equation is derived much earlier in [35]. Appendix B. EGOE(2) for Boson systems Let us consider a system of interacting bosons occupying N single-particle states , i"1, 2,2, N and the Hamiltonian is say two-body. Then H(2) is G H I J G H Q bR bR b b H(2)" (B.1) ((1# )(1# ) JI JJ JG JH JG XJH JI XJJ GH IJ
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with the symmetries for the symmetrized TBME H being, I J G H Q H " H , I J H G Q I J G H Q H " H . (B.2) I J G H Q G H I J Q The Hamiltonian H(m) in m-particle spaces is de"ned in terms of the TBME via the direct product structure of the m-particle states in occupation number representation. The non-zero matrix elements are of three types,
n (n ! ) GH H , ( )LP H ( )LP " G H P P G H G H Q (1# ) 2 2 Q GYH GH PG H PG H
( )LG \( )LH > ( )LPY H ( )LP G H PY P Q PYI J 2 PG H 2 n (n #1)(n ! ) IY IYG " G H H , IY H IY G Q (1# )(1# ) IYG IYH IY
( )LG >( )LH >( )LI \( )LJ \ ( )LP H ( )LP G H I J P P Q PK L 2 PG H 2 n (n ! )(n #1)(n #1# ) IJ G H GH " I J H . (B.3) G H I J Q (1# )(1# ) GH IJ In the second equation in (B.3) iOj and in the third equation four combinations are possible: (i) k"l, i"j, kOi; (ii) k"l, iOj, kOi, kOj; (iii) kOl, i"j, iOk, iOl; (iv) iOjOkOl. EGOE(2) for bosons is de"ned by (B.1), (B.2) and (B.3) with the two-particle H being GOE. Note that the H(m) matrix dimension d is
d(N, m)"
N#m!1 m
and the number of independent matrix elements ime are ime(N)"d (d #1)/2 where the two particle space dimension d "N(N#1)/2. For example, d(4, 11)"364, d(5, 10)"1001, d(6, 12)"6188. Similarly, ime(4)"55, ime(5)"120 and ime(6)"231. For interacting bosons, in general, the dense limit (mPR, NPRand m/NPR) is more interesting as this limit does not exist for fermion systems. Nature of energy level #uctuations in dense boson systems is being studied using EGOE(2) [76]. In this article we will not consider any further the EGOE(2) for boson systems.
Appendix C. Edgeworth expansions Given the standardized variable x( "(x! )/ and the corresponding Gaussian density 1 x( exp! , G (x( )" 2 (2
(C.1)
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the Edgeworth expansion to fourth-order is [79]
#"
(x( )"G (x( ) 1# He (x( ) # He (x( )# He (x( ) 6 24 72
,
He (x( )"x( !3x( , He (x( )"x( !6x( #3 , He (x( )"x( !15x( #45x( !15 ,
(C.2)
where He (x( ) are Hermite polynomials. It should be noted that the centroid and the width of (x) P and G (x) that correspond to (x( ) and G (x( ), respectively, are identical in the above Edgeworth #" expansion. It is worth noting that several alternatives to Edgeworth expansion (C.2) are suggested in the literature [151]. Given the bivariate Gaussian
1 x( !2x( x( #x( G (x( , x( )" exp ! 2(1!) 2((1!)
,
(C.3)
the bivariate Edgeworth expansion including bivariate cumulants k corrections with r#s44 is PQ [62,79]
(x( , x( )" 1# \#"
k k He (x( , x( )# He (x( , x( ) 2 6
k k # He (x( , x( )# He (x( , x( ) # 6 2
k k He (x( , x( )# He (x( , x( ) 6 24
k k k # He (x( , x( )# He (x( , x( )# He (x( , x( ) 6 24 4
#
k k k k k k He (x( , x( )# He (x( , x( )# # He (x( , x( ) 12 12 72 8
#
k k k k k k k # He (x( , x( )# # He (x( , x( ) 4 12 36 8
k k k # He (x( , x( )# He (x( , x( ) 12 72
G (x( , x( ) .
The bivariate Hermite polynomials He (x( , x( ) in (C.4) are generated by KK RK RK He (x( , x( )G (x( , x( )"(!1)K >K G (x( , x( ) KK Rx( K Rx( K
(C.4)
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and they satisfy the recursion relation (1!)He (x( , x( )"(x( !x( )He (x( , x( ) K > K K K ! m He (x( , x( )#m He (x( , x( ) , K \ K K K \ He (x( , x( )"1 , He (x( , x( )"(x( !x( )/(1!) , He (x( , x( )"(x( !x( )/(1!) , (x( !x( )(x( !x( ) # He (x( , x( )" . (1!) 1!
(C.5)
Appendix D. EGOE results for NPC and Sinfo Information entropy (S ) and number of principal components (NPC) are measures of complexity and chaos in many-body systems. GOE gives d/3 for NPC and ln(0.48d) for S where d is matrix dimension. Recently, the corresponding EGOE formulas are derived [66] and these results are brie#y described here. Let us introduce normalized strength R, average (smoothed) normalized strength RM and locally renormalized strength RK (for a transition operator O), where R(E, E )" EOROE \E OE , D D R(E, E )" EOROE \E OE , D D RK (E, E )" E OE \E OE . D D D
(D.1)
Then the measures NPC and S for strength distributions are
(NPC) " R(E, E ) D # #D
\
,
(S ) "! R(E, E ) ln R(E, E ) . D D # #D
(D.2)
The EGOE expression for NPC is derived by "rst writing (NPC) in terms of (RK ) and RM # and then using the bivariate Gaussian form (11) for the smoothed transition strength densities plus the fact that RK (E, E ) is Porter}Thomas; i.e. the locally renormalized amplitudes D E OE/ E OE are Gaussian distributed with zero center and unit variance. The "nal D D
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formulas are [66],
(NPC) #%-# P RK (E, E ) R(E, E ) # D D #D
"(dD) ( (1!X exp!
\ \ " 3 R(E, E ) D #D
( EK #K d (E) " , X 3
(S ) #%-# P ! R(E, E ) RK (E, E ) ln RK (E, E ) # D D D #D ! (RK (E, E )) R(E, E ) ln R(E, E ) D D D #D
"ln 0.48dD ( (1! exp
1!( (1!) (( EK #K ) exp! 2 2
;
( " /, K "( ! )/, EK "(E! )/ , X"[2!(( )(1!)] . D D G G
(D.3)
In (D.3) and are the centroid and width of ID(E ) and (11) de"nes the other parameters. The D EGOE formulas for NPC and S in shell model transition strength distributions are tested by performing shell model calculations in (ds)K ( 2 space using a two-body transition operator. Results shown in Fig. 2b con"rm that the EGOE (but not GOE) describes shell model results; see [66] for details. The NPC and S in shell model wavefunctions are de"ned by expanding the eigenfunctions
in terms of the shell model (mean-"eld) basis states (de"ned basically by h) , # I " C# , # I I I
(NPC) " C# I # I
\
,
(S ) "! C# ln C# . # I I I
(D.4)
The EGOE formulas (D.3) are applicable to the NPC and S in wavefunctions de"ned by (D.4). Here ( "1, K "0, ( , ) are same as the centroid and width of the eigenvalue distriG G bution and "(1!( / ). Note that is the average width of the basis states I I # ("d\ O H ) and " . Shell model wavefunctions for the same Hamiltonian I G H G H # G used in Fig. 2b are analyzed for (NPC) and (S ) and the results are shown in Fig. 2c. For this # # example, "10.24 MeV and "0.68. Just as in Fig. 2c, the EGOE formulas (D.3) also explain # the shell model results for NPC and S in wavefunctions given in [22]. See [152] for a very recent discussion on NPC in EGOE(2) wavefunctions.
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Appendix E. Unitary decomposition of the Hamiltonian and trace propagation Let us consider m identical fermions distributed over r spherical orbits each carrying angular momentum j . Then there are N"P (2j #1) single-particle states and one can recognize the ? ? ? appearance of the ;(N) group which is generated by the N operators aR? ?Ya @ @Y ; , "1, 2,2, r, HK HK !j 4m 4j , !j 4m 4j . The (,) antisymmetric states of the m-fermions form an irrep of ? ?Y ? @ @Y @ K the group ;(N), usually denoted by the Young shape 1K . With this the single-particle creation operators aR? ? belong to 1 and the destruction operators a @ @ belong to 1,\ . The only scalar HK HK operator in the m-particle spaces is the number operator n as it remains invariant under the transformations produced by the generators of the ;(N) group. Partitioning of the shell model ;(N) space can be carried out in many di!erent ways and the partitioning de"ned by irreps of groups that can be realized in shell model spaces are in fact the most signi"cant ones. The signi"cance derives from the fact that the moments de"ned over the irreps in fact propagate from the few particle spaces to the many particle spaces as discussed ahead. Simplest and signi"cant (from shell model mean-"eld point of view) partitioning of m-particle spaces is according to spherical and unitary con"gurations [62,64,94]. The spherical con"gurations m"(m , m 2) ? @ where m is the number of particles in the orbit and m" m ; mP m. Note that the ? ? ? dimensionality d(m) of the con"guration m is
N ? . d(m)" m ? ? For example, for identical particles, denoting 1d , 2s and 1d orbits as C1, C2 and C3 orbits, the (ds) spherical con"gurations are m"(m , m , m )"(3,0,0) (2,1,0) (2,0,1) (1,2,0) (1,1,1) (1,0,2) (0,2,1) (0,1,2) (0,0,3). The unitary group ;(N ) acting in each spherical orbit ? generates m of the spherical con"gurations m; i.e. m behaves as 1K? 1K@ 2 with respect ? to the direct sum group ;(N ) ;(N ) 2 . Thus in the spherical con"gurations space, the ? @ scalar operators are n 's. A unitary orbit is de"ned as a set of spherical orbits. With this, ? decomposition of the m-particle spaces into unitary con"gurations [m] is possible; mP [m]. As above, a unitary con"guration [m]"(m , m ,2); m" m . Using the convention that the spherical orbits belong to unitary orbit and similarly the orbits belong to etc., the number of single-particle states in a unitary orbit is N " N . The dimensionality of the con"guration ?Z ? [m] is d([m])" (, ). For example in the above ds-shell case one can choose (1d , 2s ) K and (1d ) to be two unitary orbits (C1, C2 orbits) and then (ds)P[m]"(m , m )" (3,0) (2,1) (1,2) (0,3). It is important to recognize that [m]P m and the set of spherical con"gurations m that belong to a given unitary con"guration is easy to enumerate. For example in the (ds) case, (3,0)P(3,0,0) (2,1,0) (1,2,0); (2,1)P(2,0,1) (1,1,1) (0,2,1); (12)P(1,0,2) (0,1,2); (03)P(003). The unitary group ;(N ) acting in each unitary orbit generates m of a unitary con"guration [m]; i.e. [m] behaves as 1K 1K 2 with respect to the direct sum group ;(N ) ;(N ) 2 . Thus in the unitary con"gurations space, the scalar operators are n 's. With spherical and unitary con"gurations, decomposition of the m-particle space is mP [m], [m]P m .
(E.1)
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It is worth remarking that this decomposition automatically produces "xed-parity subspaces by choosing the unitary orbits to be set of spherical orbits with all of them having same parity. Before turning to unitary decomposition and trace propagation, it is useful to introduce some symbols. With m particles in N single-particle states, the number of holes m""N!m. Similarly for m particles ? in N states and m particles in N states, m""N !m and m" "N !m . Correspondingly, ? ? ? ? n""N !n and n" "N !n . Finally, the symbols [X] , X 2 and X2 are P ?@ ? ? ? [X] "X(X!1)(X!2)2(X!r#1) , P X 2 "X (X ! )(X ! ! )2 , ?@A ? @ @? A A? A@ X2 "X (X ! )(X ! ! )2 , X"N, m, m", n, n" .
(E.2)
E.1. Unitary decomposition of operators One can in general seek a decomposition of a given operator into tensor operators with respect to the ;(N) group similar to what one does with respect to O(3) in angular momentum algebra. Let us consider the tensor decomposition of shell model Hamiltonian H"h(1)#<(2) and restrict ourselves to identical particle systems (extension of the results in this appendix to proton-neutron or nucleon systems is straightforward). The one-body Hamiltonian h" n ; are SPE and as ? ? ? stated before n is number operator for the spherical orbit . Similarly the two-body interaction ? < is de"ned by the TBME <( "()J<()J and for many purposes it is easy to work out ?@AB results in terms of the average two-particle matrix elements < " N /(1# ) \ [J]<( ; ( ?@?@ ?@ ?@ ?@ [J]"(2J#1). The scalar part h of h with respect to the ;(N) group should be a "rst-order polynomial in the number operator n"n and therefore h"a #a n. Now taking traces on ? both sides over zero- and one-particle spaces gives the unitary decomposition of h, h" n "h#h , ? ? ? h" n, "N\ N , h" (1)n , (1)" ! . (E.3) ? ? ? ? ? ? ? ? The scalar part < of < with respect to the ;(N) group should be a second-order polynomial in n. Then <"a #a n#a n and with this < can be written simply in terms of < . Just as ?@ h has h part, < should have a one-body part < with respect to the ;(N) group and this part by de"nition vanishes in zero- and one-particle spaces. Therefore, the form of < is <"(n!1)F"(n!1) (2)n . Then,
<"
n 2
<M "[n] [N] \ N /(1# ) < , ?@ ?@ ?@ ?Y@
<"(n!1) (2)n ; (2)"(N!2)\ (N ! )(< !<) , ? ? ? A ?A ?A A <"
(E.4)
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Thus "nally, the ;(N) decomposition of a one plus two-body Hamiltonian H"h(1)#<(2) into irreducible "0, 1 and 2 parts (for a given m) is, H"h(1)#<(2)"[h#<]#[h#<]#<"H#H#H ,
m
<M , H" (m)n , (m)" (1)#(m!1) (2) , ? ? ? ? ? ? H"<"H!H!H .
H"m #
2
(E.5)
One real signi"cance of the ;(N) tensor decomposition is that it is orthogonal with respect to the ;(N) trace, HJHJY" (HJ). Thus there is a ;(N) geometry [48]. JJY With respect to the spherical con"guration group ;(N ) ;(N ) 2, the non-interacting ? @ particle Hamiltonian h is a scalar and therefore, h" n "h . (E.6) ? ? In (E.6) h "h 2 denotes that it is a scalar with respect to each spherical orbit. The scalar part < "< 2 of < with respect to spherical con"gurations follows by recognizing that it must be a second-order polynomial in n 's. Therefore < "a # b n # C n and then taking ? ? ? ?Y@ ?@ ?@ ? traces on both sides in zero-, one- and two-particle spaces, one has the result < " < n /(1# ) . (E.7) ?@ ?@ ?@ ?Y@ As a one-body Hamiltonian will be a scalar with respect to spherical con"gurations group (see [48] for exceptions), there cannot be an e!ective one-body part of < with respect to spherical con"gurations group. Thus V"
and A"A 0 #A 1 #A 2 ; A 0 "B 0 #C 0 , A 1 "B 1 #C 1 and A 2 "C 2 . Explicit expressions for B , C and A are derived using the methods that gave (E.3)}(E.8) and the "nal results are [62],
, B 0 " [E (A)]n , E (A)" E N N\ ? ? ?Z B 1 "[E 1 (A)]n , E 1 (A)"E ![E (A)] , ? ? ? ?
C 0 0 C (A)"
N C [N ]\ , ?@ ?@ ?Z @Z
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C 1 0 E 1 _(A)" ?
281
(N ! )C !(N ! )C ; N !2 \ , @ ?@ ?@ @Z C 2 0 C 2 (A)"C ![C (A)]! [E 1 _(A)]#[E 1 _(A)] , ? @ ?@ ?@ A 0 " [E (A)]n # [C (A)] n /(1# ) , Y
A 1 " [E 1 (A)]# (n ! )[E 1 _(A)] n , ? ? ? ? P m ([m]:A)n , ? ? ? ([m]: A)"E 1 (A)# 1 ([m]: A) , ? ? ? 1 ([m]: A)" (m ! )E 1 _(A) , ? ? @ A 2 0 C 2 (A) . (E.9) ?@ Just as the case with (E.4) and (E.5), ([m]) are the induced SPE and 1 ([m]) are the renormalized ? ? SPE (for a "xed unitary con"guration [m]). E.2. Trace propagation Given the (,)-dimensional m-particle space generated by distributing the particles in N singleK particle states, the ;(N) (or scalar) average (or trace)
N \ mOm m ?ZK of an operator O (de"ned, for fermions, by the antisymmetric irrep 1K of ;(N)) propagates from the corresponding averages in few particle spaces, i.e. the ;(N) average is a polynomial in the particle number m (the only ;(N) scalar) with the expansion coe$cients determined by the basic few particle (or input) traces. For example, for a k-body operator F(k) one has the elementary result F(k)K"(K)F(k)I and hence h(1)K"m and <(2)K"(K)<M . Similarly, the scalar I average of a (0#1#2#2#k)-body operator F(0!k) is a kth degree polynomial in m, F(0!k)K"I a mP. The a 's can be written in terms of the input traces F(0!k)KY for P P any (k#1) values ofP m. By exploiting the properties of the operator under question it is possible to reduce the input traces to a minimal set of easily calculable traces. There are further simpli"cations, from particle}hole symmetries, in trace propagation if operators with de"nite unitary tensor rank are used. For illustration some examples are, O"
mm" ( )N , (h)K" ? ? [N] ? mm"(m"!m) ( )N (h)K" ? ? [N] ?
,
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mm"N (h)K" [N(N#1)!6mm"] N\ ( )N ? ? [N] ? # 3(m!1)(m"!1)N N\ ( )N , ? ? ? [m] [m"] (<J)K" [J](<( ("2)) , ?@AB [N] ?Y@ AYB ( HI K"[m] [N!m] [N!2]\ (N!3)(m!2) (m!2) 1 ; HI ! HI # HI ,\ (N!1)(m!1) (N!1)(N!m!1) 2
mm" " ( #(m!1) (2))N ? ? ? [N] ? [m] [m"] # [J](<( ("2)) , ?@AB [N] ?Y@ AYB ( [m] [m"] (m"!m) 1 h(<J)K" [J] (<( ("2))(1# )(1# ) . (E.10) P PRST PR ST [N] 2 PRST In fact, the principles that led to (E.10) can be used in much more complicated situations such as the UVWK trace (U, V and W are three di!erent two-body interactions; note that as stated below (E.7) U, V and W are automatically "2 with respect to the ;(N) group) by putting the operator in normal order form which follows by applying the anticommutation properties of single fermion state operators; [aR aR ] "0, [a a ] "0, [a aR ] " . In the "nal simpli"cations only the > > > traces of basic one, two and three-body operators are needed and they are: [aR a \" ; [aR aR a a \" ! ; [aR aR aR a a a \" ! ! ! " ! " ! " ! " # $ $ "! # $ " #! # "! $ # # ! , where A, B, C etc. are single particle state indices. The # " $! " #! $ " # $! formula for UVWK can be read o! from (E.14) given ahead. The principles used in deriving scalar trace propagation equations (E.10) apply directly for con"guration traces Fm and F m of an operator F but in many cases these principles prove to be too cumbersome to apply [48,56]. However there is a simpler method, as recognized in [94], when F is a product of operators with each one having "xed unitary rank (with respect to the spherical or unitary con"gurations group as the case may be). To derive spherical con"guration propagation equation for A J B J 2m one starts with the formula for AJ BJ 2K, attach appropriate spherical orbit indices to particle number m and degeneracy N in the corresponding scalar propagation formulas and carry out the summation over all the spherical orbit indices after multiplying the propagators with the inputs having spherical orbit indices (the input matrix elements of AJ , BJ ,2, etc. should be replaced by the matrix elements of A J , B J ,2, respectively). In the case of unitary con"guration traces F m (after carrying out tensorial decomposition of the operators involved with respect to the unitary con"guration group) the unitary orbit indices are to be attached with m, N etc. of the propagators of the corresponding scalar trace propagation formulas and the summation is now over all unitary orbit indices. For the inputs, the indices are spherical orbit indices that belong to those particular unitary orbits and then the summations over
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these spherical orbit indices are to be carried out. The only complication in implementing this method being that the right correspondence between the indices of inputs and those of propagators is to be decided and this is best done with a computer program. Simple examples are the formulas for UVm and UV m and they follow from the formula for (<J)K given in (E.10),
1 UVm" m m" [N ]\ [J]U( V( (1# )(1# ) , ?@ AB ?@AB ?@AB ?@AB ?@ AB 4 ? @ A B ( UV m " [mrs m" tu ](Nrstu )\ rstu 1 ; [J]U(PQRS V(PQRS (1#PQ )(1#RS ) . 4 ( PZrt QZsu RZ SZ Similarly, the trace propagation equations for (h) 1 (V) m and UVW m are
(E.11)
(E.12)
1 (h) 1 (V) m " [N\ rtuvr mrt m" uvr !N\ uvrrt muvr m" tr ] 2rtuv (E.13) ; [J] 1 V( V( (1# )(1# ) , P PRST STPR PR ST ( PZr RZt SZu TZv UVW m " mrsy m" tux N\ rsytux Ar s t u x y # mrs m" tuxy N\ rstuxy #mxytu m" rs N\ xyturs Br s t u x y , rst u x y (!1)P\R\[]\ (U) (V) (W) , PRQS VWQS VWRP PZur QZsx RZt y SZ VZ WZ
Ar s t u x y "
1 Br s t u x y " 8
[J]U( V( W( (1# )(1# )(1# ) , PQRS RSVW VWPQ PQ RS VW ( PZur QZsx RZt y SZ VZ WZ r s J (X)" (!1)Q>R>(> [] [J] PQRS u t (
;[(1# )(1# )] X( ; X"U, V or W . PQ RS PQRS
(E.14)
Eqs. (E.13) and (E.14) are derived starting from the scalar and spherical con"guration traces given in [56] and using a computer search for indices in the propagators and the corresponding inputs (V.K.B. Kota and R.U. Haq, unpublished). The variance V m of the density VG m (x)"(V!x) m and similarly the centroids and variances of the spin-cut o! density V m "J (V!x) m /J m follow from (E.12)}(E.14). They are used recently in calculating 8 8 (8 realistic nuclear level densities [64].
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Appendix F. Convolution forms in statistical spectroscopy In strongly interacting shell model subspaces, (i) using a signi"cant unitary group decomposition (see Appendix E) under which a (1#2)-body nuclear Hamiltonian decomposes into orthogonal e!ective one-body part h and irreducible two-body part V (HPh#V), (ii) decomposing the spectroscopic space into spherical con"gurations (generated by the mean-"eld h) and (iii) using the well veri"ed result that the spreading functions (partial densities I(E) de"ned in (21)) have essentially constant variance (they are produced by V), the state densities take a convolution form [55], I&(E)"[(H!E)\"Ih VG [E] ,
(F.1)
where Ih is renormalized non-interacting particle state density, G is a normalized spreading Gaussian due to V. Convolution forms similar to (F.1) are derived for spin-cut o! and occupancy densities [58,64]; h V I& O (E)"[O(H!E)\"IO O G [E] , V O (y)"O(V!y)/O; O"J or n . (F.2) 8 ? It is expected that the convolution form in (F.2) applies to the GT strength sum and other operators [63]. Similarly, under some plausible conditions, bivariate strength density I& O (E , E ) for G D a transition operator O takes a convolution form [57,62], h V [E , E ] , (F.3) I& O (E , E )"IO G D \G_O G D where IhO is the bivariate NIP strength density produced by h and V G O is the spreading bivariate \ _ Gaussian due to V. Convolution forms (F.1)}(F.3) are applied in calculating nuclear level densities, -decay rates and GDR strength distributions in [64,94], [63] and [67], respectively. In these applications, the trace propagation equations (E.12)}(E.14) given in Appendix E and those in [62] are used.
References [1] J.B. French, in: E.C.G. Sudarshan (Ed.), A Gift of Prophecy } Essays in Celebration of the Life of R.E. Marshak, World Scienti"c, Singapore, 1995, p. 156. [2] G.E. Mitchell, J.D. Bowman, H.A. WeidenmuK ller, Rev. Mod. Phys. 71 (1999) 445. ** [3] O. Bohigas, R.U. Haq, A. Pandey, in: K.H. Bockho! (Ed.), Nuclear Data for Science and Technology, Reidel, Dordrecht, 1983, p. 809; R.U. Haq, A. Pandey, O. Bohigas, Phys. Rev. Lett. 48 (1982) 1086. [4] O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev. Lett. 52 (1984) 1. *** [5] M.V. Berry, Proc. Roy. Soc. (London) A 400 (1985) 229. [6] C.E. Porter, Statistical Theories of Spectra: Fluctuations, Academic Press, New York, 1965. [7] T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey, S.S.M. Wong, Rev. Mod. Phys. 53 (1981) 385. ** [8] K.B. Efetov, Adv. Phys. 32 (1983) 53; J.J.M. Verbaarschot, H.A. Weidenmuller, M.R. Zirnbauer, Phys. Rep. 129 (1985) 367; K.B. Efetov, Supersymmetry in Disorder and Chaos, Cambridge University Press, Cambridge, 1997. [9] O. Bohigas, M.J. Giannoni, in: J.S. Dehesa, J.M.G. GoH mez, A. Polls (Eds.), Mathematical and Computational Methods in Nuclear Physics, Lecture Notes in Physics, Vol. 209, Springer, New York, 1984, p. 1. [10] O. Bohigas, H. WeidenmuK ller, Ann. Rev. Nucl. Part. Sci. 38 (1988) 421.
V.K.B. Kota / Physics Reports 347 (2001) 223}288
285
[11] O. Bohigas, in: M.J. Giannoni, A. Voros, J. Zinn-Justin (Eds.), Chaos and Quantum Physics, North-Holland, Amsterdam, 1991, p. 89. [12] J. Bauche, C. Bauche-Arnoult, Phys. Rep. 12 (1990) 1. [13] M.S. Gutzwiller, Chaos in Quantum and Classical Mechanics, Springer, New York, 1990. [14] F.M. Izrailev, Phys. Rep. 196 (1990) 299. [15] M.L. Mehta, Random Matrices, Academic Press, Boston, 1991. [16] F. Haake, Quantum Signatures of Chaos, Springer, New York, 1991. [17] O. Bohigas, S. Tomsovic, D. Ullmo, Phys. Rep. 223 (1993) 45. [18] K. Ikeda (Ed.), Quantum and Chaos: How Incompatible, Prog. Theor. Phys. 116 (Suppl.) (1994). [19] M.T. Lopez-Arias, V.R. Manfredi, L. Salasnich, Riv. Nuovo Cimento 17 (5) (1994) 1. [20] W.M. Zhang, D.H. Feng, Phys. Rep. 252 (1995) 1. [21] V. Zelevinsky, Annu. Rev. Nucl. Part. Sci. 46 (1996) 237. [22] V. Zelevinsky, B.A. Brown, N. Frazier, M. Horoi, Phys. Rep. 276 (1996) 85. *** [23] N. Auerbach, J.D. Bowman (Eds.), Parity and Time Reversal Violation in Compound Nuclear States and Related Topics, World Scienti"c, Singapore, 1996. [24] M.R. Zirnbauer, J. Math. Phys. 37 (1996) 4986. [25] C.W.J. Beenakker, Rev. Mod. Phys. 69 (1997) 731; A.D. Mirlin, Phys. Rep. 326 (2000) 259. [26] T. Guhr, A. MuK ller-Groeling, H.A. WeidenmuK ller, Phys. Rep. 299 (1998) 189. * [27] A. Richter, in: D.A. Hejhal, J. Friedman, M.C. Gutzwiller, A.M. Odlyzko (Eds.), Emerging Applications of Number Theory, IMA Vol. 109, Springer, Heidelberg, 1998, p. 479; K. Schaadt, A. Kudrolli, Phys. Rev. E 60 (1999) R3479. [28] H.-J. StoK ckmann, Quantum Chaos: An Introduction, Cambridge University Press, Cambridge, 1999. [29] I.V. Lerner, J.P. Keating, D.E. Khmelnitskii (Eds.), in: Supersymmetry and Trace formulae: Chaos and Disorder, NATO Asi Series, Series B, Physics, Vol. 370, Kluwer Academic, Dordrecht, 1999. [30] J. Bellissard, O. Bohigas, G. Casati, D.L. Shepelyansky (Eds.), Classical Chaos and its Quantum Manifestations, Physica D 131 (1999). [31] J. Ginibre, J. Math. Phys. 6 (1965) 440; R. Grobe, F. Haake, H.-J. Sommers, Phys. Rev. Lett. 61 (1988) 1899; K.B. Efetov, Phys. Rev. Lett. 79 (1997) 491; B. Mehlig, J.T. Chalker, Ann. Phys. 7 (1998) 427; M. Markum, R. Pullirsch, T. Wettig, Phys. Rev. Lett. 83 (1999) 484. [32] M.V. Berry, M. Tabor, Proc. Roy. Soc. (London) A 349 (1976) 101. [33] M.V. Berry, M. Robnik, J. Phys. A 17 (1984) 2413; A 19 (1986) 649. [34] S. Tomsovic, Ph.D. Thesis, University of Rochester, 1986, unpublished. * [35] J.B. French, V.K.B. Kota, A. Pandey, S. Tomsovic, Ann. Phys. (N.Y.) 181 (1988) 198. [36] Y. Alhassid, N. Whelan, Phys. Rev. Lett. 67 (1991) 816; N. Whelan, Y. Alhassid, Nucl. Phys. A 556 (1993) 42; D. Kusnezov, Phys. Rev. Lett. 79 (1997), 537; P. Cejnar, J. Jolie, Phys. Lett. B 420 (1998) 241. [37] D.M. Leitner, Phys. Rev. E 48 (1993) 2536; A. Pandey, Chaos Solitons Fractals 5 (1995) 1275; T. Guhr, Ann. Phys. (N.Y.) 250 (1996) 145; K.M. Frahm, T. Guhr, A. Muller-Groeling, Ann. Phys. (N.Y.) 270 (1998) 292. [38] J.J.M. Verbaarschot, Phys. Rev. Lett. 72 (1994) 2531; M.E. Berbenni-Bitsch, S. Meyer, A. SchaK fer, J.J.M. Verbaarschot, T. Wettig, Phys. Rev. Lett. 80 (1998) 1146; M.E. Berbenni-Bitsch, S. Meyer, T. Wettig, Phys. Rev. D 58 (1998) 071502; R.G. Edwards, Urs M. Heller, J. Kiskis, R. Narayanan, Phys. Rev. Lett. 82 (1999) 4188. [39] M. Matsuo, T. Dossing, E. Vigezzi, S. Aberg, Nucl. Phys. A 620 (1997) 296; V.G. Soloviev, Nucl. Phys. A 586 (1995) 265. [40] V.E. Kravstov, K.A. Muttalib, Phys. Rev. Lett. 79 (1997) 1913; T. Nagao, P.J. Forrester, Nucl. Phys. B 530 (1998) 742; E. Kanzieper, Phys. Rev. Lett. 82 (1999) 3030. [41] V. Plerou, P. Gopikrishnan, B. Rosenow, L.A.N. Amaral, H.E. Stanley, Phys. Rev. Lett. 83 (1999) 1471; R.N. Mantegna, H.E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, Cambridge, 2000. [42] Z. Szymanski, Fast Nuclear Rotation, Clarendon Press, Oxford, 1983; I.M. Pavlichenkov, Phys. Rep. 226 (1993) 173; I. Hamamoto, B. Mottelson, Phys. Lett. B 333 (1994) 294; I.M. Pavlichenkov, S. Flibotte, Phys. Rev. C 51 (1995) R460; V.K.B. Kota, Phys. Rev. C 53 (1996) 2550; D.S. Haslip et al., Phys. Rev. Lett. 78 (1997) 3447; M. Dudeja, S.S. Malik, A.K. Jain, Phys. Lett. B 412 (1997) 14.
286
V.K.B. Kota / Physics Reports 347 (2001) 223}288
[43] J.F. Shriner Jr., G.E. Mitchell, E.G. Bilpuch, Phys. Rev. Lett. 59 (1987) 435; N.R. Roberson, C.R. Gould, J.D. Bowman (Eds.), Tests of Time Reversal Invariance in Neutron Physics, World Scienti"c, Singapore, 1987; G.E. Mitchell, E.G. Bilpuch, P.B. Endt, J.F. Shriner Jr., Phys. Rev. Lett. 61 (1988) 1473; G.E. Mitchell, J.F. Shriner, in: D. Koltun, A. Das (Eds.), Spectroscopy to Chaos, World Scienti"c, Singapore, 1995, p. 77; Triangle Universities Nuclear Laboratory, North Carolina, Report XXXVII 1997}98, p. 9. [44] A.Y. Abul-Magd, H.A. WeidenmuK ller, Phys. Lett. B 162 (1985) 223; S. Raman et al., Phys. Rev. C 43 (1991) 521; J.F. Shriner Jr., G.E. Mitchell, T. von Egidy, Z. Phys. A 338 (1991) 309; A.Y. Abul-Magd, Phys. Rev. C 54 (1996) 1675; J.D. Garrett, J.Q. Robinson, A.J. Foglia, H.-Q. Jin, Phys. Lett. B 392 (1997) 24. [45] A.A. Adams, G.E. Mitchell, J.F. Shriner Jr., Phys. Lett. B 422 (1998) 13; C.I. Barbosa, T. Guhr, H.L. Harney, Phys. Rev. E 62 (2000) 1936. [46] H.A. WeidenmuK ller, P. von Brentano, B.R. Barrett, Phys. Rev. Lett. 81 (1998) 3603; S. Aberg, Phys. Rev. Lett. 82 (1999) 299; C.A. Sta!ord, B.R. Barrett, Phys. Rev. C 60 (1999) 051305; J.-z. Gu, H.A. WeidenmuK ller, Nucl. Phys. A 660 (1999) 197. [47] J. Blocki, J.J. Shi, W.J. Swiatcki, Nucl. Phys. A 554 (1993) 387; Ph. Chomaz, Nucl. Phys. A 569 (1994) 203c; J. Blocki, J. Skalski, W.J. Swiatcki, Nucl. Phys. A 594 (1995) 137; S. Pal, Pramana- J. Phys. 48 (1997) 425; S.R. Jain, A.K. Pati, Phys. Rev. Lett. 80 (1998) 650. [48] J.B. French, K.F. Ratcli!, Phys. Rev. C 3 (1971) 94; K.F. Ratcli!, Phys. Rev. C 3 (1971) 117; F.S. Chang, J.B. French, T.H. Thio, Ann. Phys. (N.Y.) 66 (1971) 137. ** [49] K.K. Mon, J.B. French, Ann. Phys. (N.Y.) 95 (1975) 90. *** [50] C. Quesne, J. Math. Phys. 16 (1975) 2427. [51] J.C. Parikh, Group Symmetries in Nuclear Structure, Plenum, New York, 1978. [52] J.P. Draayer, J.B. French, S.S.M. Wong, Ann. Phys. (N.Y.) 106 (1977) 472, 503. [53] B.J. Dalton, S.M. Grimes, J.P. Vary, S.A. Williams (Eds.), Moment Methods in Many Fermion Systems, Plenum, New York, 1980; K. Kar, Nucl. Phys. A 368 (1981) 285; J.J.M. Verbaarschot, Ph.D. Thesis, Fysisch Laboratorium, Utrecht, The Netherlands, 1982, unpublished; J.J.M. Verbaarschot, P.J. Brussaard, Z. Phys. A 321 (1985) 125. [54] J.B. French, V.K.B. Kota, Annu. Rev. Nucl. Part. Sci. 32 (1982) 35. [55] J.B. French, V.K.B. Kota, Phys. Rev. Lett. 51 (1983) 2183. ** [56] S.S.M. Wong, Nuclear Statistical Spectroscopy, Oxford University Press, New York, 1986. [57] J.B. French, V.K.B. Kota, A. Pandey, S. Tomsovic, Phys. Rev. Lett. 58 (1987) 2400; Ann. Phys. (N.Y.) 181 (1988) 235. *** [58] J.B. French, V.K.B. Kota, J.F. Smith, University of Rochester Report, UR-1122 (ER40425-245), 1989. * [59] V.K.B. Kota, K. Kar, Pramana } J. Phys. 32 (1989) 647. [60] V.K.B. Kota, in: R.W. Ho! (Ed.), Capture Gamma-Ray Spectroscopy and Related Topics, AIP Conference Proceedings, Vol. 238, New York, 1991, p. 685; V.K.B. Kota, in: N.G. Puttaswamy (Ed.), Direct Nuclear Reactions, Indian Academy of Sciences, Bangalore, India, 1991, p. 321; R.U. Haq, in: D. Koltun, A. Das (Eds.), Spectroscopy to Chaos, World Scienti"c, Singapore, 1995, p. 89; V.K.B. Kota, in: G. Shanmugam, N. Arunachalam (Eds.), Recent Trends in Nuclear Structure Physics, M.S. University, Tirunelveli, India, 1997, p. I2-1. [61] V. Potbhare, N. Tresseler, Nucl. Phys. A 530 (1991) 171. [62] V.K.B. Kota, D. Majumdar, Z. Phys. A 351 (1995) 365. * [63] V.K.B. Kota, D. Majumdar, Z. Phys. A 351 (1995) 377; K. Kar, S. Sarkar, A. Ray, Ap. J. 434 (1994) 662. [64] V.K.B. Kota, D. Majumdar, Nucl. Phys. A 604 (1996) 129. [65] V.K.B. Kota, Proceedings of Symposium on Nuclear Physics, Vol. 39 A, Library and Information Services, B.A.R.C., Bombay, India, 1997, p. 208. [66] V.K.B. Kota, R. Sahu, Phys. Lett. B 429 (1998) 1. ** [67] D. Majumdar, K. Kar, A. Ansari, J. Phys. G 24 (1998) 2103. [68] V.V. Flambaum, A.A. Gribakina, G.F. Gribakin, M.G. Kozlov, Phys. Rev. A 50 (1994) 267. ** [69] V.V. Flambaum, A.A. Gribakina, G.F. Gribakin, Phys. Rev. A 58 (1998) 230. [70] V.V. Flambaum, A.A. Gribakina, G.F. Gribakin, I.V. Ponomarev, Physica D 131 (1999) 205. * [71] J.B. French, S.S.M. Wong, Phys. Lett. B 33 (1970) 447; B 35 (1971) 5. [72] O. Bohigas, J. Flores, Phys. Lett. B 34 (1971) 261; B 35 (1971) 383. [73] J.B. French, E.C. Halbert, J.B. McGrory, S.S.M. Wong, Adv. Nucl. Phys. 3 (1969) 193. [74] F. Iachello, A. Arima, The Interacting Boson Model, Cambridge University Press, Cambridge, 1987.
V.K.B. Kota / Physics Reports 347 (2001) 223}288
287
[75] M. Nomura, Ann. Phys. (N.Y.) 91 (1975) 83; V.K.B. Kota, V. Potbhare, Phys. Rev. C 21 (1980) 2637; V.K.B. Kota, Ann. Phys. (N.Y.) 134 (1981) 221; V.R. Manfredi, Rev. Nuovo Cimento 9(9) (1986) 1. [76] K. Patel, M.S. Desai, V. Potbhare, V.K.B. Kota, Phys. Lett. A 275 (2000) 329. [77] K.K. Mon, B.A. Dissertation, Princeton University, 1973, unpublished. [78] A. Gervois, Nucl. Phys. A 184 (1972) 507. [79] M.G. Kendall, A. Stuart, Advanced Theory of Statistics, Vol. 1, 3rd Edition, Hafner Publishing Company, New York, 1969. [80] G.J.H. Laberge, R.U. Haq, Can. J. Phys. 68 (1990) 301. [81] V.V. Flambaum, G.F. Gribakin, F.M. Izrailev, Phys. Rev. E 53 (1996) 5729. * [82] V.V. Flambaum, F.M. Izrailev, G. Casati, Phys. Rev. E 54 (1996) 2136; V.V. Flambaum, F.M. Izrailev, Phys. Rev. E 55 (1997) R13. [83] G. Martinez-Pinedo, A.P. Zuker, A. Poves, E. Caurier, Phys. Rev. C 55 (1997) 187; E. Caurier, G. MartinezPinedo, F. Nowacki, J. Retamosa, A.P. Zuker, Phys. Rev. C 59 (1999) 2033. [84] J.D. Garrison, Ann. Phys. (N.Y.) 30 (1964) 269. [85] B.A. Brown, B.H. Wildenthal, Annu. Rev. Nucl. Part. Sci. 38 (1988) 29. [86] J.B. French, in: D.H.Wilkinson (Ed.), Isospin in Nuclear Physics, North-Holland, Amsterdam, 1969, p. 259. [87] T.T.S. Kuo, Nucl. Phys. A 103 (1967) 71. [88] V.K.B. Kota, R. Sahu, K. Kar, J.M.G. GoH mez, J. Retamosa, in: C.R. Praharaj (Ed.), Perspectives in Nuclear Physics, Narosa publishers, New Delhi, India, 2000, to appear (nucl-th/0005066). [89] J.B. French, in: C. Bloch (Ed.), Many-Body Description of Nuclei and Reactions, Proceedings of the International School of Physics, Enrico Fermi, Course 36, Academic Press, New York, 1966, p. 278. [90] V.K.B. Kota, R. Sahu, K. Kar, J.M.G. GoH mez, J. Retamosa, Phys. Rev. C 60 (1999) 051306. *** [91] W. Wang, F.M. Izrailev, G. Casati, Phys. Rev. E 57 (1998) 323. [92] G.F. Gribakin, A.A. Gribakina, V.V. Flambaum, Aust. J. Phys. 52 (1999) 443. [93] J. Karwowski, Int. J. Quantum Chem. 51 (1994) 425; D.B. Waz, J. Karwowski, Phys. Rev. A 52 (1995) 1067; J. Planelles, F. Rajadell, J. Karwowski, J. Phys. A 30 (1997) 2181. [94] J.B. French, R.U. Haq, V.K.B. Kota, S. Rab, J.F. Smith, 1999, unpublished. [95] S. Tomsovic, private communication, 1999. [96] S. A berg, Phys. Rev. Lett. 64 (1990) 3119. [97] C.W. Johnson, G.F. Bertsch, D.J. Dean, Phys. Rev. Lett. 30 (1998) 2749; C.W. Johnson, G.F. Bertsch, D.J. Dean, I. Talmi, Phys. Rev. C 61 (1999) 014311. ** [98] A. Pandey, J.B. French, J. Phys. A 12 (1979) L83; A. Cortes, R.U. Haq, A.P. Zuker, Phys. Lett. B 115 (1982) 1. [99] Ph. Jacquod, D.L. Shepelyansky, Phys. Rev. Lett. 79 (1997) 1837. *** [100] R. Berkovits, Y. Avishai, Phys. Rev. Lett. 80 (1998) 568. [101] V.K.B. Kota, in: A.K. Jain, R.K. Bhowmik (Eds.), Nuclear Structure and Dynamics, Phoenix publishing, New Delhi, India, 2000, p. 179. [102] B. Georgeot, D.L. Shepelyansky, Phys. Rev. Lett. 79 (1997) 4365. [103] D. Weinmann, J.-L. Pichard, I. Imry, J. Phys. I (France) 7 (1997) 1559. [104] A.D. Mirlin, Y.V. Fyodorov, Phys. Rev. B 56 (1997) 13393. [105] B. Georgeot, D.L. Shepelyansky, Phys. Rev. Lett. 81 (1998) 5129. [106] P.G. Silvestrov, Phys. Rev. E 58 (1998) 5629. * [107] V.V. Flambaum, F.M. Izrailev, Phys. Rev. E 56 (1997) 5144. *** [108] B.L. Altshuler, Y. Gefen, A. Kamenev, L.S. Levitov, Phys. Rev. Lett. 78 (1997) 2803. *** [109] A. Bohr, B. Mottelson, Nuclear Structure, Vol. 1, Benjamin, New York, 1969. [110] V.V. Flambaum, F.M. Izrailev, Phys. Rev. E 61 (2000) 2539. [111] N. Frazier, B.A. Brown, V. Zelevinsky, Phys. Rev. C 54 (1996) 1665. [112] F. Borgonovi, I. Guarneri, F.M. Izrailev, Phys. Rev. E 57 (1998) 5291. [113] V.K.B. Kota, R. Sahu, in: V. Srinivasan, P.K. Panigrahi (Eds.), Proceedings of Conference on Dynamical Systems: Recent Developments, Allied publishers, Hyderabad, India, 2000, to appear (nlin.CD/0006003); nucl-th/0006079. [114] I.C. Percival, J. Phys. B 6 (1973) L229. [115] F. Borgonovi, I. Guarneri, F.M. Izrailev, G. Casati, Phys. Lett. A 247 (1998) 140.
288
V.K.B. Kota / Physics Reports 347 (2001) 223}288
[116] J.M.G. GoH mez, K. Kar, V.R. Manfredi, R.A. Molina, J. Retamosa, Phys. Lett. B 480 (2000) 245; J.M.G. GoH mez, K. Kar, V.K.B. Kota, J. Retamosa, R. Sahu, 2000, to be published. [117] T.R. Halemane, K. Kar, J.P. Draayer, Nucl. Phys. A 311 (1978) 301. [118] V.V. Flambaum, A.A. Gribakina, G.F. Gribakin, I.V. Ponomarev, Phys. Rev. E 57 (1998) 4933. [119] V.K.B. Kota, R. Sahu, Phys. Rev. E 62 (2000) 3568. [120] J.B. French, in: J.S. Dehesa, J.M.G. GoH mez, A. Polls (Eds.), Mathematical and Computational Methods in Nuclear Physics, Lecture Notes in Physics, Vol. 209, Springer, New York, 1984, p. 100. [121] V.K.B. Kota, D. Majumdar, R.U. Haq, R.J. Leclair, Can. J. Phys. 77 (1999) 893; R.J. Leclair, M.Sc. Thesis, Laurentian University, Canada, 1994, unpublished. [122] Z. Pluhar, H.A. WeidenmuK ller, Phys. Rev. C 38 (1988) 1046; A. Pandey, Ann. Phys. (N.Y.) 134 (1981) 110. [123] S.P. Pandya, Phys. Rev. 103 (1956) 956. [124] D. Mulhall, A. Volya, V. Zelevinsky, Phys. Rev. Lett. 85 (2000) 4016; S. Droz dz , M. WoH jcik, nucl-th/0007045. [125] R. Bijker, A. Frank, S. Pittel, Phys. Rev. C 60 (1999) 021302. [126] R. Bijker, A. Frank, Phys. Rev. Lett. 84 (2000) 420; Phys. Rev. C 62 (2000) 014303. [127] M. Dufour, A.P. Zuker, Phys. Rev. C 54 (1996) 1641; J. Du#o, A.P. Zuker, Phys. Rev. C 59 (1999) R2347. [128] M. Horoi, V. Zelevinsky, B.A. Brown, Phys. Rev. Lett. 74 (1995) 5194. ** [129] J.F. Smith, Ph. D. Thesis, University of Rochester, 1987, unpublished. [130] U. Sivan, F.P. Milliken, K. Milkove, S. Rishton, Y. Lee, J.M. Hong, V. Boegli, D. Kern, M. DeFranza, Europhys. Lett. 25 (1994) 605; U. Sivan, Y. Imry, A.G. Aronov, Europhys. Lett. 28 (1994) 115. [131] C. Mejia-Monasterio, J. Richert, T. Rupp, H.A. WeidenmuK ller, Phys. Rev. Lett. 81 (1998) 5189. * [132] X. Leyronars, J. Tworzydlo, C.W.J. Beenakker, Phys. Rev. Lett. 82 (1999) 4894. [133] J.B. French, V.K.B. Kota, Pramana- J. Phys. 32 (1989) 459. [134] T. Papenbrock, T. Prosen, Phys. Rev. Lett. 84 (2000) 262. [135] L. Benet, F.M. Izrailev, T.H. Seligman, A. Suarez-Moreno, Phys. Lett. A 277 (2000) 87. [136] D. Cohen, F.M. Izrailev, T. Kottos, Phys. Rev. Lett. 84 (2000) 2052. [137] T. Papenbrock, Phys. Rev. C 61 (2000) 034602. [138] T. Guhr, J.-Z. Ma, S. Meyer, T. Wilke, Phys. Rev. D 59 (1999) 054501. [139] O. Bohigas, P. Leboeuf, M.J. SaH nchez, Physica D 131 (1999) 186. [140] T.A. Brody, Lett. Nuovo Cimento 7 (1973) 482. [141] H. Hasegawa, H.J. Mikeska, H. Frahm, Phys. Rev. A 38 (1988) 395. [142] F.M. Izrailev, J. Phys. A 22 (1989) 865; G. Casati, F.M. Izrailev, L. Molinari, J. Phys. A 24 (1991) 4755. [143] A.Y. Abul-Magd, J. Phys. A 29 (1996) 1. [144] M. Robnik, J. Dobnikar, T. Prosen, J. Phys. A 32 (1999) 1427. [145] M. Moshe, H. Neuberger, B. Shapiro, Phys. Rev. Lett. 73 (1994) 1497. [146] V.K.B. Kota, S. Sumedha, Phys. Rev. E 60 (1999) 3405. [147] G. Lenz, F. Haake, Phys. Rev. Lett. 67 (1991) 1; G. Lenz, K. Zyczkowski, D. Saher, Phys. Rev. A 44 (1991) 8043. [148] M. Abramowtiz, I.A. Stegun (Eds.), Handbook of Mathmatical functions, NBS Applied Mathematics Series, Vol. 55, U.S. Govt. Printing O$ce, Washington, DC, 1964. [149] T. Guhr, H.A. WeidenmuK ller, Ann. Phys. (N.Y.) 199 (1990) 412. [150] H. Alt, C.I. Barbosa, H.D. Graf, T. Guhr, H.L. Harney, R. Ho!erbert, H. Rehfeld, A. Richter, Phys. Rev. Lett. 81 (1998) 4847. [151] V.K.B. Kota, V. Potbhare, P. Shenoy, Phys. Rev. C 34 (1986) 2330; S.M. Grimes, T.N. Massey, Phys. Rev. C 51 (1995) 606; A.P. Zuker, nucl-th/9910002. [152] L. Kaplan, T. Papenbrock, Phys. Rev. Lett. 84 (2000) 4553.
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DENSE NUCLEAR MATTER: LANDAU FERMI-LIQUID THEORY AND CHIRAL LAGRANGIAN WITH SCALING
C. SONG Department of Physics & Astronomy, State University of New York, Stony Brook, New York 11794-3800, USA
AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO
Physics Reports 347 (2001) 289}371
Dense nuclear matter: Landau Fermi-liquid theory and chiral Lagrangian with scaling夽 Chaejun Song Department of Physics & Astronomy, State University of New York, Stony Brook, New York 11794-3800, USA Received September 2000; editor: G.E. Brown Contents 1. Introduction 2. Landau Fermi-liquid theory 2.1. A mini-primer 2.2. Renormalization group approach 2.3. Relations for relativistic Fermiliquid 3. Chiral e!ective Lagrangian for nuclei 3.1. QCD: basis of strong interactions 3.2. E!ective "eld theories 3.3. E!ective Lagrangian in medium and Landau theory 3.4. A chiral e!ective model: FTS1 model 3.5. Anomalous dimension in FTS1 model 4. Brown}Rho scaling 4.1. Freund}Nambu model 4.2. Brown}Rho scaling in in-medium e!ective Lagrangian 4.3. Duality
292 295 295 298 302 305 305 307 308 310 312 318 318 319 323
5. E!ective Lagrangian with BR scaling 5.1. A hybrid BR-scaled model 5.2. Model with BR scaling 5.3. Thermodynamic consistency in medium 5.4. Results 5.5. Landau Fermi-liquid properties of the BR-scaled model 5.6. Mesons in medium 6. Fermi-liquid theory vs. chiral Lagrangian 6.1. Electromagnetic current 6.2. Axial charge transition 7. Summary 8. Open issues Acknowledgements Appendix A. E!ect of many-body correlations on EOS Appendix B. Relativistic calculation of FL Appendix C. Relativistic calculation of F () and J S References
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Abstract The relation between the e!ective chiral Lagrangian whose parameters scale according to Brown and Rho scaling (`BR scalinga) and Landau Fermi-liquid theory for hadronic matter is discussed in order to make
夽
In part based on the Ph.D. thesis (February 99) of Seoul National University. E-mail address: [email protected] (C. Song). 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 0 8 - 3
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a basis to describe the #uctuations under the extreme condition relevant to neutron stars. It is suggested that BR scaling gives the background around which the #uctuations are weak. A simple model with BR-scaled parameters is constructed and reproduces the properties of the nuclear ground state at normal nuclear matter density successfully. It shows that the tree level in the model Lagrangian is enough to describe the #uctuations around BR-scaled background. The model Lagrangian is consistent thermodynamically and reproduces relativistic Landau Fermi-liquid properties. Such points are important for dealing with hadronic matter under extreme condition. On the other hand, it is shown that the vector current obtained from the chiral Lagrangian is the same as that obtained from Landau}Migdal approach. We can determine the Landau parameter in terms of BR-scaled parameter. However, these two approaches provide di!erent results, when applied to the axial charge. The numerical di!erence is small. It shows that the axial response is not included properly in the Landau}Migdal approach. 2001 Elsevier Science B.V. All rights reserved. PACS: 12.39.Fe; 21.30.Fe; 21.65.#f; 71.10.Ay Keywords: E!ective chiral Lagrangian; Landau Fermi-liquid theory; Brown}Rho scaling
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1. Introduction Although QCD which deals with quarks and gluons is believed to be the fundamental theory for strong interactions, it is generally accepted that the appropriate theory at very low energy is the e!ective quantum "eld theory which incorporates the observed degrees of freedom in low-energy nuclear process, i.e., pions, nucleons and other low-mass hadrons [1}7]. The e!ective Lagrangian in matter-free or dilute space is governed by QCD symmetries with its parameters to be determined from experiments in free space. The energy scale of the experiments in which one is interested determines which hadrons play an important role in the theory. For example, it is shown [7] that one can integrate out even the pions for the two nucleon systems at very low energy. According to the results of [7], the deuteron and low-energy nucleon}nucleon scattering properties can be described very accurately by an e!ective theory given in terms of the nuclonic degrees of freedom only with a cuto! around the natural scale of the theory which for low energy is the pion mass. Since there is currently growing interplay between the physics of hadrons and the physics of compact objects in astrophysics through the properties of hot and dense environments, we want to extend such successful strategy of e!ective "eld theories to a dense medium. First of all, our main goal is to understand the properties of dense hadronic medium which can be tested in various heavy ion collisions. Recent dilepton experiments (CERES and HELIOS-3) gave us very important information on the properties of hadrons in dense medium, which we will detail later. Understanding the properties of dense hadronic medium through heavy ion collision experiments is essential for understanding the properties of neutron stars which are believed to be formed in the center of supernovae at the time of explosions. Especially the determination of the maximum neutron star mass is one of the most important issues in astrophysics in explaining controversies between the observations and theoretical estimations, and it will give some hints for the detectability of the gravitational wave detectors. A dense matter makes the probed energy scale larger than that in free space. Therefore, we have to introduce more massive degrees of freedom which are usually vector meson and/or higher-order operators in the nucleon "elds. We must also consider a new energy scale, Fermi energy of nucleons in bound system. The standard strategy to attack the dense hadronic matter is to obtain the ground state of matter and compute the excitations around it, based on an e!ective Lagrangian whose parameters are obtained in free space. Although such an approach can give satisfactory results with a su$cient number of parameters, it is not obvious whether we can extend the results for the di!erent density. When the extension does not work, we usually face with very complicated loop diagrams which may lead to the impasse. Another strategy is to start from the in-medium Lagrangian which is built on the reasonable assumptions, instead of deriving the hadronic matter properties from the matter-free e!ective Lagrangian de"ned in free space. In this approach, one regards its mean "eld solution as a solution of the Wilsonian e!ective action in which the high-energy modes are integrated out into the coe$cient. We can compare it with the well-known Landau Fermi-liquid theory, which works at low-energy excitation in strongly correlated Fermi system. Landau Fermi-liquid theory is described by quasiparticles, which are the low-energy excitations in Fermi liquid, and their interactions under the assumption of one-to-one correspondence between the quasiparticle in the liquid and the particle in a non-interacting gas. It is a "xed point theory [8,9] under Wilsonian renormalization to the Fermi surface [10] with /k P0 if Bardeen}Cooper}Schrie!er (BCS) $
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instability does not exist, where is the cuto! of the theory relative to the Fermi surface and k is $ Fermi momentum. Since the result after the repeated renormalization does not depend on k , the $ argument for Fermi liquid holds as long as there is no phase transition. A famous example of such a Lagrangian is Walecka model. Its extension and justi"cation were studied recently by Furnstahl et al. [11}13]. Their Lagrangian is constrainted by QCD symmetry and by Georgi's naturalness condition and the coupling constants in it are tuned in order to describe nuclear ground state. Bulk properties of nuclei are described by it very successfully. But it is somewhat unclear to know how to approach the #uctuation on the ground state. In this review, we will approach the nuclear matter in a di!erent way. We wish to apply the strategy of the e!ective "eld theory to the in-medium theory. We assume that the in-medium e!ective Lagrangian has the same structure as in free space according to the symmetry constraint of the fundamental theory QCD but that its parameters are modi"ed in medium. It means that the e!ect of embedding a hadron in matter appears mainly in the change of the `vacuuma, i.e., quark and gluon condensate in QCD variables and the parameters in an e!ective theory. In this strategy, the density-dependent parameters include many-body correlations. Brown}Rho (BR) scaling [14] is one speci"c way to de"ne such in-medium parameters. Brown}Rho scaling is the scaling of the dynamically generated masses of hadrons which consist of chiral quarks, i.e., u and d quarks. Brown and Rho phrased the scaling with the large N Lagrangian, i.e., Skyrmion, under the assumption that the chiral symmetry and the scale symmetry of QCD are relevant. Their Lagrangian is implemented with scale anomaly of QCD, too. The masses and pion decay constant of this QCD e!ective theory scale universally: f 夹() m夹() m夹() M夹() . ()+ L + + N + M f m m L N
(1)
The star represents in-medium quantities here. v is vector meson degree of freedom and s an isoscalar scalar meson which has a mass &500 MeV in nuclear matter. M represents a free nucleon mass and M夹 a scaled nucleon mass which is somewhat di!erent from the Landau e!ective mass discussed in Section 6.1. It is known that BR scaling describes that the light-quark vector meson property in the extreme condition very successfully. One can make the extreme condition through relativistic heavy ion collisions. Specially, the dileptons provide a good probe of the earlier dense and hot stage of relativistic heavy ion collision since the interaction of leptons is not subject to the strong interactions of the "nal state. Cherenkov ring electron spectrometer (CERES) collaboration observed in heavy ion collision (S#Au) that the dilepton production with invariant masses from 250 MeV to about 500 MeV is enhanced much more than the predicted from the superposition of pp collision [15]. And HELIOS-3(CERN Super Proton Synchrotron detector) [16] also observed the dilepton enhancement in S#W. It is shown by Li et al. [17] that a chiral Lagrangian with BR-scaled meson masses describes most economically and beautifully the enhancement, which is found to come from the dropping vector meson masses in dense matter. Furthermore, the excitation into the kaonic direction above the given ground state seems to describe the properties of kaons in medium [18,19] with scaled parameters [20]. Since BR scaling gives the universal scaling mass relation among hadrons, it must work for the baryon properties in medium. In recent works [21}25], it has been discussed how the BR scaling,
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which is applied to meson properties successfully, can be applied to the baryon property in dense medium and how it can be extrapolated to a hadronic matter under extreme conditions from the known normal nuclear matter. The construction of such bridges will be necessary to understand the various phenomena in relativistic heavy ion collisions and in compact stars in the universe. For those purposes, we develop arguments for mapping the e!ective chiral Lagrangian whose parameters are governed by BR scaling to Landau Fermi-liquid theory. We will relate the meson mass scaling with baryon mass scaling and also relate matter properties with BR scaling parameter, which implies the vacuum structure characterized by quark condensate, via Landau parameter. Fermi liquid theory may work up to chiral phase transition, though the relevant degrees of freedom are changed to quasiquarks from quasihadrons. In this review, we approach the aim in two ways. The "rst is to obtain the ground state with BR scaling. How the Fermi surface is obtained in BR scaling framework is not yet understood clearly. So people usually assume that the ground state of hadronic matter is determined by the conventional matter from a standard many-body theory and use density-dependent e!ective chiral Lagrangians to compute mesonic #uctuations above the ground state. Though various #uctuation phenomena can be described successfully in this way, such a treatment gives no constraints for consistency between the excitations and the ground state. Though BR scaling is applied very successfully to describe meson properties in medium [17], the way to deal with the matter properties is disconnected from BR scaling. Such a procedure is not satisfactory in going to the higher density region from the normal nuclear matter density. We can take the kaon condensation in neutron star as an example. In dealing with it, we come across a change of the ground state from that of a non-strange matter to a strange matter. The works up to date [26,27] treated KN interaction and the ground state separately. This is not satisfactory. Ground state properties might e!ect the condensation. For example, the e!ect of the four-Fermi interactions which play an important role in determining ground state, suppresses a pion condensation [28]. So we must deal with the whole bulk involving the ground state and excitations on top of it on the same footing. We assume that the ground state is given by the same e!ective Lagrangian that is supposed to include higher-order corrections as the mean "eld of the BR-scaled chiral Lagrangian. We want to make an initial step for dealing with the ground state and the #uctuation on top of it on the same basis. We bridge these two properties by constructing a simple model whose parameters scale in the manner of BR and which describes nuclear matter properties well. It is shown that the model can be mapped to Landau Fermi-liquid theory. The next step to achieve our aim is to identify the parameters of the BR-scaled e!ective Lagrangian with the "xed point quantities in Landau Fermi-liquid theory, given a hadronic matter with a Fermi surface. With this identi"cation, certain mean "eld quantities of heavy meson (e.g. , ) can be related to BR scaling through the Landau parameters. We show how such arguments work for the electromagnetic currents in nuclear matter. We can link a set of BR-scaled parameters at nuclear matter density with the orbital gyromagnetic ratio in terms of the Landau parameter FS which comes from integrated-out isoscalar vector degree of freedom in the e!ective Lagrangian. Then we will try to derive the corresponding formulas for the axial current in a similar way. This review is organized as follows. In Section 2, Landau Fermi-liquid theory and its interpretation in terms of renormalization group language are summarized brie#y. And we show how thermodynamic observables are related to the relativistic Landau parameters. In Section 3, the strategy of an e!ective "eld theory and how it is applied to a dense matter are explained. We
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proceed to discuss how nuclear matter described by an in-medium e!ective Lagrangian can be identi"ed with Landau Fermi liquid. The model of Furnstahl et al. [11] (referred to FTS1) which imposes the symmetries of QCD is examined as an example of the application of a general strategy of an e!ective chiral theory to a medium. In Section 4, Brown}Rho scaling is derived with QCD-oriented e!ective Lagrangian. A model where BR scaling governs the parameters of a chiral Lagrangian and determines the background at a given density is constructed in Section 5 in order to describe in weak coupling the same physics as FTS1 which has strong coupling in the form of a large anomalous dimension of a dilaton. It describes well normal nuclear matter properties and has thermodynamic consistency and Fermi-liquid structure needed for the extrapolation to higher density region. In Section 6, vector current and axial charge transition matrix elements for a nucleon above the given Fermi sea in Landau}Migdal theory and in chiral e!ective Lagrangian are calculated and compared. A summary and comments on some unsolved problems are given in Sections 7 and 8, respectively. Appendix A shows how sensitive the equation of state is to the many-body correlation parameters for ' . Appendix B shows how to compute relativistically the pionic contribution to Landau parameter F by means of Fierz transformation. And the vector-mesonic contribution to F and to electromagnetic convection current is calculated in Appendix C relativistically with random phase approximation.
2. Landau Fermi-liquid theory We discuss brie#y Landau Fermi-liquid theory in this section, before presenting the relation between the chiral e!ective theory for nuclear matter and Landau Fermi-liquid theory. A miniprimer on Landau Fermi-liquid theory is given in the "rst subsection to de"ne the quantities involved. In the second subsection, it is discussed that Landau Fermi-liquid theory is considered as an e!ective theory and is shown to be a "xed point theory in renormalization group (RG) language. And we will show how the thermodynamic quantities are related to relativistic Landau parameters in Section 2.3. 2.1. A mini-primer Landau's Fermi-liquid theory is a semi-phenomenological approach to strongly interacting normal Fermi systems at small excitation energies. The elementary excitations of the Fermi-liquid, which correspond to single-particle degrees of freedom of the Fermi gas, are called quasiparticles in Landau Fermi-liquid theory. It is assumed that a one-to-one correspondence exists between the low-energy excitations of the Fermi liquid near Fermi surface, i.e., quasiparticles, and those of a non-interacting Fermi gas. A quasiparticle state of the interacting liquid is obtained by turning on the interaction adiabatically at the corresponding state of non-interacting Fermi gas. The quasiparticle properties, e.g. the mass, in general di!er from those of free particles due to interaction e!ects. In addition there is a residual quasiparticle interaction, which is parameterized in terms of the so-called Landau parameters. The adiabatic process described above is possible in the vicinity of the Fermi surface only. Let us see the Fermi liquid at ¹"0. Since Pauli exclusion principle makes the states below the Fermi surface "lled, quasiparticle with energy loses energy less than ! when colliding the $
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background particles. It means that the quasiparticles which can interact with the quasiparticle are those with an energy within ! of the Fermi surface. And the "nal state momenta are $ also restricted by (. Pauli exclusion principle and the corresponding rarity of "nal states make the quasiparticle life time proportional to ! \ at ¹"0 case. $ Fermi-liquid theory is a prototype e!ective theory, which works because there is a separation of scales. The theory is applicable to low-energy phenomena, while the parameters of the theory are determined by interactions at higher energies. The separation of scales is due to the Pauli principle and the "nite range of the interaction. Pauli principle makes the low-energy quasiparticle physics possible near the Fermi surface and the "nite range of interaction makes a few quasiparticles around Fermi surface, who appear by the small change of the energy in low-energy physics, form a gas. Fermi-liquid theory has proven very useful [29] for describing the properties of e.g. liquid He and provides a theoretical foundation for the nuclear shell model [30] as well as nuclear dynamics of low-energy excitations [31,32]. The interaction between two quasiparticles p and p at the Fermi surface of symmetric nuclear matter can be written in terms of a few spin and isospin invariants [33] fp
1 " F(cos )#F(cos ) ) #G(cos ) ) N O N O N(0) p
q # G(cos ) ) ) # H(cos )S (q( ) k $ q # H(cos )S (q( ) ) , (2) k $ where is the angle between p and p and N(0)"k /(2)(dp/d) is the density of states at the $ $ Fermi surface. In this review, natural units where "1 are used. The spin and isospin degeneracy factor is equal to 4 in symmetric nuclear matter. Furthermore, q"p !p and S (q( )"3 ) q( ) q( ! ) , (3) where q( "q/q. The tensor interactions H and H turn out to be important for the axial charge [33]. The functions F, F,2 are expanded in Legendre polynomials,
F(cos )" Fl Pl (cos ) , l
(4)
with analogous expansion for the spin- and isospin-dependent interactions. The energy of a quasiparticle with momentum p"p, spin and isospin is denoted by and the N N O corresponding quasiparticle number distribution by n . From now on, the spin and isospin N N O indices and will be omitted from the formulas to avoid overcrowding, except where needed to avoid ambiguities. n (r, t)"n()# n (r, t) , N N N . dp (r, t)"# f n (r, t) , N N (2) NNY NY NY OY
(5) (6)
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where n (r, t) is the long wavelength excitations from the ground state n() in the vicinity of N N Fermi surface. The space and time dependence of the quantities will also be omitted, e.g., ,(r, t). The Landau e!ective mass and velocity of a quasiparticle on the Fermi surface is de"ned by
k " $夹 ,v夹 . $ m NI$ * We must note that the total current is not d N dp
(7)
dp 夹 " * n (2) $ N N O in Landau Fermi-liquid theory. The quasiparticle distribution n (r, t) obeys N Rn dr dp N # ) n # ) n "0 . P N Rt dt dt N N J
/.
(8)
(9)
The key assumption of Landau Fermi-liquid kinetic theory is that (r, t) plays the role of the N quasiparticle Hamiltonian; dp "! , P N dt
(10)
dr " . N N dt
(11)
The kinetic equation is Rn N # n ) ! n ) "I[n ] , P N N N N N P N NY Rt
(12)
where I[n ] is an internal collision integral which represents the sudden change of quasiparticle NY momenta. We consider the system without external forces. Integrating over p, that cancels the e!ect of the internal collision under the assumption that the quasiparticle energy, momentum and number are locally conserved, (12) becomes (to order n (r, t)), by integration by part, N dp R n dp R N # ) (n )" # )
n # n P N N N P N N N (2) Rt (2) N N N Rt N O N O "0 (13)
where is the total particle (or quasiparticle) density #uctuation. We can easily see that the conserved current is J" N O " N O
dp n (2) N N N dp 夹 * n (2) $ N
(14)
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with the excitation from the local equilibrium n((r, t)); N n (r, t)"n((r, t))# n (r, t) . N N N Comparing (15) with (5) and (6), we obtain
(15)
dp Rn f n . (16)
n " n # N N N (2) NNY N R NY OY The modi"cation J !J can be interpreted as the e!ect of the return #ow of the surrounding /. matter to the localized wave packet carrying J and is called back-#ow current. 2.2. Renormalization group approach Landau Fermi-liquid theory was based on Landau's reasonable intuition. After his work, his theory was derived and justi"ed microscopically [34]. Recent development of Wilsonian RG method in medium [10] provides us a new understanding of Fermi-liquid theory. Landau Fermi-liquid theory is a "xed point theory described by marginal coupling, i.e. F, m夹 [8]. In this * section, we will review the RG arguments for Fermi-liquid theory. To do this, we consider a nonrelativistic system of spinless fermions whose Fermi surface is spherical characterized by k for simplicity. Then non-interacting one-particle Hamiltonian near $ Fermi surface is k k K H" ! $ + k ,v k $ 2m 2m m $
(17)
with k"K!k . The free fermion "eld action $
S "
(k) (i!vk) (k)
(18)
in momentum space where
dk d . (19) (2) (2) \ \ and are Grassmannian eigenvalue with fermion operator K ; K " and M K R" . Note that a shell of thickness on either side of the Fermi surface is taken for low-energy physics as seen in Fig. 1. Pauli exclusion principle lets only small deviations from the Fermi surface, not from the origin, be important in low-energy physics of fermion matter. This de"nes the starting point of an in-medium renormalization group procedure. The "rst step for the renormalization is decimation; to integrate out the high-energy mode whose momentum is larger than /s and to reduce the cuto! from to /s. For example, free action (18) becomes " :
d (2)
d S" (2)
Q
dk d (k) (i!vk) (k) . (2) (2) \ Q \
(20)
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Fig. 1. Our interest lies in the spherical shell which is of the thickness of on either side of Fermi sphere. In RG transformation we eliminate the mode within the two thick-lined shells whose thickness is d.
Next, we rescale the momenta in order to compare the old and the new; (, k)P(s, sk) .
(21)
The last step is to absorb the uninteresting multiplicative constant by Ps\ .
(22)
The RG transformation consists of these three steps. After such RG transformation, free action (18) returns to the old. When a coupling is turned on, the coupling is called relevant if it increases after RG transformation. If it decreases, it is called irrelevant and if it remains "xed like the free action, it is called marginal. Now let us turn on four-Fermi interaction. The appropriate action is
S"
[i!v夹k] # 夹 $
1 # 2!2!
u(4, 3, 2, 1) (4) (3) (2) (1) ,
(23)
where (i) represents ( , k , ) and G G G
" : G
d (2)
dk d (!k ) . (2) (2) \ \
(24)
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Fig. 2. Momentum conservation and the physics near Fermi surface force the angle between K and K to be the same as that between K and K unless K "!K .
with a cuto! function for k , (!k ), which is needed to make all the momenta lie within the band of width 2 around the Fermi surface. Here v夹"k /m夹 and $ $ 1 m夹" Z(1#(m/k ) (R/Rk)) $
(25)
is the e!ective mass of the nucleon which will be equal to the Landau mass m夹 as will be elaborated * on later. (, k) is self-energy and 1/Z"1#iR/R. The e!ective mass arises because the eliminated mode contributes to i and k di!erently. By de"ning "s\ Z\ , we "x the coe$cient of i and de"ne the e!ective mass. The term with 夹 is a counter term added to assure that the Fermi momentum is "xed (that is, the density is "xed). What this term does is to cancel loop contributions involving the four-Fermi interaction to the nucleon self-energy (i.e., the tadpole) which contributes marginally so that the v夹 is at the "xed point. Since other contributions $ which can be written as ! are irrelevant, m moves to m夹 in earlier stage of renormalization but becomes a "xed point characterized by some m夹. This means that the counter term essentially assures that the e!ective mass m夹 be at the "xed point. Without this procedure, the term quadratic in the fermion "eld would be `relevanta and hence would be unnatural [8]. Let us see the quartic coupling u at tree level. The cuto! function (!k ) in (24) makes the coupling depend on angles on the Fermi surface. Since all the momenta are on the thin spherical shell near k , momentum conservation makes the new quartic coupling u(, k, ) decay after the $ RG transformation at tree level except for the two cases. One is that K and K are the rotation of K and K around the K #K as seen in Fig. 2. In that case, the opening angle cos\(u( ) u( ) is
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Fig. 3. One-loop diagrams for the renormalization of the marginal quartic couplings F and V.
"xed where u( is a unit vector in the direction of K . The other, so-called BCS coupling, is that G G u( "!u( and u( "!u( . These cases are represented by functions; u( " )"F( , ) , u( " )"V( ) , (26) where "u( ) u( and is the angle between the planes containing (K , K ) and (K , K ), respec GH G H tively. F and V are marginal at tree level. Then the next question is how F and V evolve at one-loop level. Fig. 3 shows the one-loop diagrams for the evolution. Integrating out the momentum shell of thickness d at k"$ and sending /k P0, all the diagrams in Fig. 3 do not contribute to F. So F is left marginal to the $ one-loop order. In the case of V, the third diagram in Fig. 3 makes a #ow. When we expand V in terms of angular momentum eigenfunction, V becomes irrelevant only if all the V 's are repulsive. J However, if any V is attractive, it becomes relevant and causes BCS instability. We call it BCS J channel. Since Landau theory assures that there is no phase transition for one-to-one correspondence between particles and quasiparticles, BCS channel destabilizes in Landau Fermi-liquid theory. If we divide the angular part of the shell integration in action (18) into the cells of size &/k , $ the shell is split into &k / cells. By analogy with large N theory each cell corresponds to one $ species, i.e., N&k /. 1/N expansion tells that all higher-loop corrections except for the bubbles of $ the "rst diagram in Fig. 3 are down by powers of 1/N"/k . And the contributions of bubbles of $ the "rst diagram survive but are irrelevant to the function. Only one-loop contributions to function survive in the large N limit. So the four-Fermi interactions in the phonon channel F are also at the "xed points in addition to the Fermi surface "xed point with the e!ective mass m夹. Note that only forward scattering F("0) is important for responses to soft probes, since non-forward amplitudes in loop calculations are also suppressed by 1/N. Six- and higher-Fermi interactions are irrelevant and contribute at most screening of the "xed-point constants. The Landau parameter F can be identi"ed with forward scattering (m夹/2k )F("0) easily [8]. We arrive at the $ Fermi-liquid "xed point theory in the absence of BCS interactions. Chen et al. [9] obtain the same result in the 1/N expansion where their N is taken to be N& with 1/ being the width of the e!ective wave vector space around the Fermi sea which can be considered as the ratio of the microscopic scale to the mesoscopic scale. More speci"cally, if one rescales the four-Fermi interaction such that one de"nes the dimensionless constant g&u /k $ where u is the leading term (i.e., constant term) in the Taylor series of the quantity u in (23), then
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the fermion wave function renormalization Z, the Fermi velocity v and the constant g are found $ not to #ow up to order O(g/N). Thus in the large N limit, the system #ows to Landau "xed point theory to all orders of loop corrections. This result is correct provided there are no long-range interactions and if the BCS channel is turned o!. 2.3. Relations for relativistic Fermi-liquid In this section, we brie#y summarize the relations of physical properties of the relativistic Landau Fermi-liquid. The extension of Landau Fermi-liquid theory to relativistic region for high-density matter is found in [35]. It should be pointed out that one can use all the standard Landau Fermi-liquid relations established below in our calculations once "xed point quantities are identi"ed in the chiral Lagrangian. 2.3.1. Compression modulus and F The density of states at Fermi surface is
k m夹 " $ * . (27) 2 I$ Note that m夹"(m #k for relativistic non-interacting Fermi gas. m夹 de"ned by (7) includes the * , $ * kinetic energy for relativistic Fermi liquid. The chemical potential is de"ned by N(0)"
R R
RE , " , $ R
(28)
where E represents the energy per volume. Using (27) and (28), one can derive the relativistic relation between compression modulus which represents the change of volume with pressure and F in the same way as the non-relativistic one: R R # d f n K"9 "9 NNY NY R N R
3k " 夹$ (1#F ) . (29) m * Here is the baryon number density and n is the Fermi distribution function for the state of N momentum p. 2.3.2. Landau ewective mass In deriving the Landau mass formula, we shall compare the rest frame and the boosted frame with very small velocity u and check Lorentz symmetry. Since u,u is small we neglect the order of u in the process of the derivation. (Note that "(1!u)\ +1 in that order.) S Let us "rst derive the relativistic relation between the Landau e!ective mass and the velocity dependence of the quasiparticle interaction. When we add a particle (or equivalently quasiparticle) of momentum p in the rest frame to the system, the energy and the momentum of the system increases by p and (0), respectively. In a moving frame with velocity !u, the momentum and N
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energy increase by p"p!u( (u( ) p)(1! )# (0)u S N S +p# (0)u , N (u)"( (0)#p ) u) NY N S + (0)#p ) u . N From (u). (30) and (31), we have N (u)" N (0)#p ) u" (0)! (0)u ) (0)#p ) u . N N N N N N\C S and
(30)
(31)
(32)
(u)" (0)# d f (n (u)!n (0)) . N N NN N N
(33)
Since n (u)"n (0), we can obtain NY N n (u)+n N (0)"n (0)!e (0)u ) n (0) N N\C S N N N N using (30) and (31). Then (33) becomes
(34)
(u)" (0)! d f (0)u ) n (0) . NN N N N N N
(35)
Comparing it with (32)
p" (0) (0)! d f (0) n (0) . N N N NN N N N
(36)
In the ground state R n (0)"! ( !) N p( N N N Rp
(37)
(38)
gives p"
R R N # d ( !) f p( ) p( N . N N NN N Rp Rp
The chemical potential is $ . Thus (36) becomes on the Fermi surface N R p f "p ! $ . N $ Rp $ 2 3 N Using (27), one obtains
F m夹 * "1# . 3
(39)
(40)
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This is the relativistically extended relation of the famous Landau mass formula F m夹 * "1# M 3
(41)
that follows from Galilean invariance in the same way [29]. 2.3.3. First sound velocity The "rst sound is a density oscillation mode under the circumstances where there are many quasiparticle collisions during the time of interest. The su$ciently often collisions produce the required local equilibrium in the time scale of the period of motion. When "rst sound wave gives small change in density of a static homogeneous relativistic #uid in a comoving frame without changing entropy, the continuity equation requires R
"! ) * . Rt
(42)
Under the condition of the "xed entropy 1 1 !(p#E) "0 k ds" (p v# )" ¹ ¹
(43)
with the entropy per particle ks, the volume per particle v"1/, and the energy per particle "E/, the relativistic equation of motion becomes R* 1
p "! p"! . Rt p#E E
(44)
Applying (44) to (42), we obtain
Rp R
" . RE Rt
(45)
Thus the "rst sound velocity of the relativistic Landau Fermi-liquid is Rp c , RE
R R RE/ " RE R R
K " 9 k 1#F " $ 3 1#F /3 from the above results.
(46)
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3. Chiral e4ective Lagrangian for nuclei In this section, we come to hadron/nuclear phenomena. We simply explain the basic theory of strong interaction, QCD, and how successfully low-energy phenomena can be described by its e!ective theory. Then it is discussed how the e!ective theory can be applied to the nuclear/hadronic matter ground state. FTS1 model will be studied as an example. 3.1. QCD: basis of strong interactions Quantum Chromodynamics is believed to be the fundamental theory for the strong interaction. It is a non-abelian gauge description of the strong interaction and its building blocks are quarks and gluons. The QCD Lagrangian is L
/!"
"!Tr G GIJ#q (iR. !gG. !m )q IJ O
(47)
with q"(u, d,2)2 and m "diag[m , m ,2] in #avor space. In this section Tr represents the trace O S B over #avor indices. The gauge "eld strength tensor is G "R G !R G !ig[G , G ] , IJ I J J I I J
(48)
where G "G? ?/2. ? is the well-known Gell-Mann's 3;3 traceless hermitian matrix with color I I indices a"1, 2,2, 8. Quantum Chromodynamics conserves its relevant symmetries; Lorentz symmetry, parity, charge conjugation, time reversal and color SU(3) gauge symmetry. In addition, it has approximate symmetries. Isospin symmetry is a good one because of small di!erence between u and d masses. There is another well-known and important QCD approximate symmetry, chiral symmetry. It is based on the fact that the mass of light quarks u and d are much smaller than chiral symmetry breaking scale &4f &1 GeV. We can consider the massless limit, i.e. m P0 when dealing Q O with light quark physics. It means that chiral symmetry becomes relevant. With projection operator 1G , P " ! 2
(49)
we can decompose the spinor into the eigenstates of helicity ) kK , ) kK q"$q !
(50)
with q ,P q. Without quark mass term, q and q are decoupled each other ! ! > \ L
/!"
P!Tr G GIJ#q (iR. !gG. )q !q (iR. !gG. )q . IJ > > \ \
(51)
So the separate transformations q Pe\ C! q ! !
(52)
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and q Pe\ ! tq ! !
(53)
with real parameters and m leave the massless QCD Lagrangian invariant, where t is the ! N general SU(2) generators. We can construct SU(2) and U(1) vector transformations as qPe\ 4 tq, qPe\ C4 q
(54)
and axial ones as qPe\ tA q, qPe\ C A q
(55)
with " $ and " $ . The corresponding currents are ! 4 ! 4 V I"q Itq,
AI"q I tq, AI"q I q . Q
(56) (57)
Although massless QCD Lagrangian is invariant under these transformations, its U(1) axial current of QCD is not conserved due to anomaly even if the quark mass m P0; O N g R AI" $ Tr G GI IJ#2iq m q I Q IJ O 16
(58)
with GI IJ"IJ?@G . So the massless QCD has chiral symmetry (SU(2) ;SU(2) ) and fermion ?@ 0 * number symmetry (U(1) ). Chiral symmetry does not appear in hadron spectrum. So it is generally 4 assumed that the chiral symmetry is dynamically broken into SU(2) in hadron physics and the 4 light pseudoscalar mesons are regarded as Goldstone bosons. Since the massless QCD Lagrangian contains no dimensional parameters, massless QCD action is invariant under scale transformation; x"ax, q(x)"a\ q(x), G (x)"a\G (x) . I I
(59)
Thus, massless QCD seems to have another approximate symmetry, scale symmetry. However, the renormalization prescriptions which have the running coupling constant break the scale symmetry. It means that we need to introduce a dimensional scale in order to specify the value of running coupling constant. Such speci"cation of a scale breaks the scale invariance of QCD; R DI"I " Tr G GIJ#q (1# )m q IJ O O I I g
(60)
with the anomalous dimension of quarks "diag[ , ,2] even if quark masses are zero. DI is O S B dilatation current, I is the improved energy}momentum tensor [36] and J ,
2 g Rg " N !11 #O(g) 3 $ 16 R
(61)
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with the scale of renormalization . Contrary to the axial anomaly which includes only one-loop contributions, scale anomaly includes multi-loop contributions in . Scale anomaly which plays a major role in breaking scale invariance of light quark QCD gives a basis of BR scaling argument which will appear in Section 4.2. 3.2. Ewective xeld theories We know that the particles which appear in nuclear physics, e.g. pions, nucleons, etc., are not elementary particles. In the framework of QCD, they are known to consist of quarks and gluons which give non-perturbative contributions in low-energy processes. In addition, QCD is a part of the Standard Model which is also an e!ective theory, not a fundamental theory. In order to deal with low-energy hadrons, one can construct an e!ective "eld theory that is appropriate to the probed energy scale Q. The relevant degrees of freedom in such e!ective "eld theory are the low-energy particles which appear at the energy scale of the observed experiments. The particles with energies higher than the relevant energy scale are integrated out and absorbed into the couplings among the relevant degrees of freedom;
VL O O E E E H
[Dq][Dq ][Dg]e VL M + E E H
" [DB][DBM ][DM]e ,
(62)
where , , and j are the external sources of elementary fermions q(quarks), anti-fermions q (antiquarks), and bosons g(gluons) respectively in fundamental theory(QCD) and B, BM , and M represent fermions(baryons), anti-fermions(anti-baryons), and bosons(mesons), respectively, in e!ective "eld theory. Although one can in principle calculate the couplings in e!ective theory for strong interactions from the fundamental QCD from Eq. (62), it seems an impossible mission because we do not know how to perform such a highly non-perturbative calculation. However, we do not have to deal with, nor to know exactly, the fundamental theory. The strategy to build an e!ective theory is simple. It consists of writing down the most general Lagrangian which conserves all the relevant symmetries that "gure at a given energy scale, and satisfy the basic principles (e.g. quantum mechanics, cluster decomposition) of the theory. Then any theory under the same constraints looks like his/hers at su$ciently low-energy scale though he/she cannot insist that the right theory necessarily leads only to his/hers. (This is called `folk theorema by Weinberg [37] though he used it to explain the usefulness of quantum "eld theory.) By imposing the relevant symmetries of the QCD or of a more fundamental theory, e.g., chiral symmetry, and the basic principles we can build an e!ective "eld theory for low-energy nuclear processes where the composite hadrons are taken as elementary. The number of terms in the e!ective Lagrangian which are consistent with fundamental or assumed symmetries may be in"nite, but one can manage to describe the probed physical processes by expanding the terms in Q/, where is s suitable cut-o! scale and Q the scale probed (Q;). So only a few leading terms are relevant for low-energy processes. And the couplings can be determined from available experiments.
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Such e!ective "eld theories with chiral symmetry so designed work very well in matter-free or dilute space. They are used to calculate soft-pion processes "rstly [1], and recently extended to the process in light nuclei though nucleons are not soft [2}7]. Park et al.'s work for two-nucleon systems at very low energies [7] is one of the best and newest examples of the success. Setting the cuto! near one pion mass, they integrate out all mesonic degrees of freedom, even the pions. Since the deuteron bound state is dilute, the parameters of the theory can be determined by free space experiments. The results in [7] con"rm that the strategy of the e!ective "eld theory works remarkably well. When the pion "eld is included in addition, it provides a new degree of freedom and improves the theory even further allowing one to go higher in energy scale [7]. But in heavier nuclei, the energy scales of the system will be higher since the interactions between nucleons in such systems sample all length scales and hence other degrees of freedom than nucleonic and pionic need be introduced. It means that the irrelevant and neglected terms in the computation for the light nuclei become more relevant. We need to consider more and more terms as the density of the system becomes higher and higher. The power of e!ective "eld theory whose parameters are determined from the matter-free experiments gets weakened in dense system. How can we proceed as density increases beyond the ordinary matter density for which there are practically no experimental data? 3.3. Ewective Lagrangian in medium and Landau theory Recently, Lynn made a progress to give a good hint to answer the above questions [38]. He proposed that the ground-state matter is `chiral liquida which arises as a non-topological soliton. Fluctuation around this ground state should give an accurate description of the observables that we are dealing with. We shall here extend this argument further and make contact with Landau's Fermi-liquid theory of nuclear matter. This will allow us to understand the nuclear/hadron matter description in terms of chiral Lagrangians and Fermi-liquid "xed point theory thereby giving a uni"ed picture of ordinary nuclear matter and extreme state of matter probed in heavy-ion collisions. This is the attempt to connect the physics of the two vastly di!erent regimes. The basic assumption we start with is that the chiral liquid arises from a quantum e!ective action resulting from integrating out the degrees of freedom lying above the chiral scale &4f & Q 1 GeV.
[d ]e 1Q ( " [d ][d ]e 1( (
(63)
where the subscript ((') represents the sector ( (' ) of the given set of "elds . Q Q As explained in Section 3.2, S " g OK (64) Q G G G is the sum of all possible terms consistent with symmetries of QCD. This corresponds to the "rst stage of `decimationa [9] in our scheme. The mean "eld solution of this action is then supposed to yield the ground state of nuclear matter with the Fermi surface characterized by the Fermi momentum k . The e!ective Lagrangian was given in terms of the baryon, pion, quarkonium scalar $
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and vector "elds. The gluonium scalars are integrated out. Instead of treating the scalar and vector "elds explicitly, we will consider here integrating them out further from the e!ective Lagrangian. This will lead to four-Fermi, six-Fermi, etc., interactions in the Lagrangian with various powers of derivatives acting on the Fermi "eld. The resulting e!ective Lagrangian will then consist of the baryons and pions coupled bilinearly in the baryon "eld and four- and higher-Fermi interactions with various powers of derivatives, all consistent with chiral symmetry. A minimum version of such Lagrangian in mean "eld can be shown to lead to the original (naive) Walecka model [5]. The next step is to decimate successively the degrees of freedom present in the excitations with the scale E( [9]. To do this, we consider excitations near the Fermi surface, which we take to Q be spherical for convenience characterized by k . First, we integrate out the excitations with $ momentum p5$ (where p" p and ( ) measured relative to the Fermi surface (correQ sponding to the particle}hole excitations with momentum greater than 2). We are thus restricting ourselves to the physics of excitations whose momenta lie below 2 as in Section 2.2. Leaving out the pion for the moment and formulated non-relativistically, the appropriate action to consider can be written in a simpli"ed and schematic form as Eq. (23). In nuclear matter, the spin and isospin degrees of freedom need to be taken into account into the four-Fermi interaction. All these can be written symbolically in the action (23). The function u in the four-Fermi interaction term therefore contains spin and isospin factors as well as space dependence that takes into account non-locality and derivatives. For simplicity, we will consider it to be a constant depending in general on spin and isospin factors. Non-constant terms will be `irrelevanta. In our discussions, the BCS channel that corresponds to a particle}particle channel does not "gure and hence will not be considered explicitly. The successive mode elimination
夹
[dJ ]e 1 (J " [dJ ][dF ]e 1Q (J (F ,
(65)
which satis"es RG equation explained in Section 2.2, will give S夹" g夹OK 夹 . (66) G G G The starred coupling constant g夹 and operators OK 夹 depend on density through s and have the same G G structure of (64). l(h) represents the components p(/s (p'/s). The upshot of the analysis in Section 2.2 is that the resulting theory is the Fermi-liquid "xed point theory with the limit /k P0. In sum, we arrive at the picture where the chiral liquid solution of the quantum e!ective $ chiral action gives the Fermi-liquid "xed point theory. The parameters of the four-Fermi interactions in the phonon channel are then identi"ed with the "xed-point Landau parameters. There are two steps to apply such scheme to nuclear/hadron phenomenology. The "rst is to derive the in-medium e!ective Lagrangian directly from QCD or from the matter-free e!ective Lagrangian and the second is to solve the built in-medium e!ective Lagranian. The "rst is very
The pion will be introduced in the Section 6.1 in terms of a non-local four-Fermi interaction that enters in the ground state property and gives the nucleon Landau mass formula in terms of BR scaling and pionic Fock term.
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di$cult. So we usually go to the second step, after building the in-medium e!ective Lagrangian by reasonable guesses. We assume that the e!ective Lagrangian satis"es
S" dxL
(67)
where S is a Wilsonian e!ective action arrived at after integrating out high-frequency modes of the nucleon and other heavy degrees of freedom. This action is then given in terms of sum of terms organized in chiral order in the sense of e!ective theories at low energy. One way to build the chiral e!ective Lagrangian for nuclear matter has been studied by Furnstahl et al. [11,12]. They formulated their theory in terms of a chiral Lagrangian constructed by using the terms which are governed by QCD symmetry and applying the `naturalnessa condition for all relevant "elds. In doing this, they introduced in the FTS1 a quarkonium "eld that is associated with the trace anomaly with its potential constrained by Vainshtein et al.'s low-energy theorem [39]. And Georgi's `naive-dimensional analysisa [40] was used in the FTS2 instead of the trace anomaly. It was argued in [12] that a Lagrangian so constructed contains, in principle, arbitrarily higher-order many-body e!ects including those loop corrections that can be expressed as counter terms involving matter "elds (e.g., baryons). This is essentially equivalent to Lynn's chiral e!ective action [38] that purports to include all orders of quantum loops in chiral expansion supplemented with counter terms consistent with the order to which loops are calculated. Though it is a little hard to de"ne the #uctuation on its ground state, their models are very successful in describing the bulk properties of nuclei. Another way is to apply the strategy of the e!ective theory to the in-medium Lagrangian. The parameters of the e!ective Lagrangian are related to the vacuum state at a given density so depend on the density. One famous example is BR scaling [14]. Brown and Rho point [41] that the mean "eld solution of the chiral e!ective Lagrangian given by BR scaling approximates
S"0 .
(68)
The detail of BR scaling is reviewed in the next section. The aim of this review is to cast BR scaling in a suitable form starting with a chiral Lagrangian description of the ground state as speci"ed above around which #uctuations in a various #avor sectors are to be made. To do this, we study phenomenologically successful FTS1 model here. 3.4. A chiral ewective model: FTS1 model Furnstahl et al. [11] constructed an e!ective non-linear chiral model that will be referred to as FTS1 model in this review. The FTS1 model incorporates the scale anomaly of QCD in terms of a light (`quarkoniuma) scalar "eld S and a heavy (`gluoniuma) scalar "eld and gives a good description of basic nuclear properties in mean "eld. One can also build a model appealing to The model in [11] is referred to as FTS1 and the one in [12] as FTS2. Since the strong interactions have two relevant scales f and and is much larger than f , we can apply Georgi's Q Q naive dimensional analysis to the low-energy hadron physics.
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general notions of e!ective "eld theories such as `naturalness conditiona as in [12]. This avoids the use of the scale anomaly of QCD. The FTS2 model is also an e!ective mean "eld theory which gives an equally satisfactory phenomenology as the FTS1. However, the FTS1 is found to be more convenient for studying the role of the light scalar "eld in the scale anomaly we are interested in since the FTS1 Lagrangian includes the scale anomaly term explicitly. The underlying assumption in the FTS1 is that the light scalar "eld transforms under scale transformation as S(a\x)"aBS(x)
(69)
with a parameter d that can be di!erent from its canonical scale dimension, i.e. unity, while the scale dimension of the heavy gluonium, which is integrated out in the e!ective Lagrangian for normal nuclear matter, is taken to be unity. This assumption imposes that quantum #uctuations in the scalar channel be incorporated into an anomalous dimension d "d!1O0. An RG #ow argument in Section 3.3 justi"es this assumption heuristically. One further assumption of the FTS1 is that there is no mixing between the light scalar S(x) and the heavy scalar in the trace anomaly. The FTS1 Lagrangian has the form
S B 1 1 S 1 L"L !H ln ! !H ln ! , (70) Q E O S 2d S 4 4 where L is the chiral- and scale-invariant Lagrangian containing , S, N, , , etc. Here and Q S are the vacuum expectation values (VEV) with the vacuum 0 de"ned in the matter-free space: ,0 0 , S ,0S0 . (71) The trace of the improved energy}momentum tensor [36], i.e. the divergence of the dilatation current DI, from the Lagrangian is
S B . (72) R DI"I "!H !H O S I I E The mass scale associated with the gluonium degree of freedom is higher than that of chiral symmetry, &1 GeV. For example, the mass of the scalar glueball calculated by Weingarten Q [42] is 1.6&1.8 GeV. This invites us to integrate out the gluonium. The resulting FTS1 e!ective Lagrangian takes the form L"NM [i (RI#ivI#ig I#g aI)!M#g ]N I Q 1 1 ! F FIJ# g( I) IJ 4! I 4
f 1 tr(R ;RI;R)#m I 1# I I 2 S 2 1 m B 1 1 1 # R RI! Q S d 1! ln 1! ! # , d 2 I 4 S S 4 4 #
(73)
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where S"S !, and are unknown parameters to be "xed and ";"e D , i v "! (RR #R R) , I I I 2 i a "! (RR !R R) . I I I 2 Note that a given VEV of the "eld scales down the pion decay constant and the mass in the same way at the lowest chiral order as in [14]. The static mean "eld equations of motion for FTS1 are
\B B g NM N " !m 1! ln 1! ! m , (74) Q G G Q 2S S S G (75) g NM RN " #m 1# # g 6 G G S G with the static mean "eld solutions and . Eq. (75) is a constraint because is not a dynamical degree of freedom. It is important to note that the FTS1 Lagrangian is an e!ective Lagrangian in the sense explained in Section 3.2. The e!ect of high-frequency modes of the nucleon "eld and other massive degrees of freedom appears in the parameters of the Lagrangian and in the counter terms that render the expansion meaningful. It presumably includes also vacuum #uctuations in the Dirac sea of the nucleons [11,13]. In general, it could be much more complicated. Indeed, one does not yet know how to implement this strategy in full rigor given that one does not know what the matching conditions are. In [11,12], the major work is, however, done by choosing the relevant parameters of the FTS1 Lagrangian to "t the empirical informations. The energy density for uniform nuclear matter obtained from (73) is
I$ m dk(k#(M!g )! 1# #g ! g " Q 4! 2 S (2) B 1 1 1 m ln 1! ! # . # Q S d 1! d S S 4 4 4 Here is the degeneracy factor.
(76)
3.5. Anomalous dimension in FTS1 model The best "t to the properties of nuclear matter and "nite nuclei is obtained with the parameter set T1 when the scale dimension of the scalar S is near d"2.7 in the FTS1 [11]. The large anomalous dimension means that one is #uctuating around a wrong ground state. Brown}Rho scaling is Explicitly the T1 parameters are: g"99.3, g"154.5, "!0.496, and "0.0402. Q
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meant to avoid this. In this section, we analyze how this comes out and present what we understand on the role of the large anomalous dimension d "d!1+1.7 in nuclear dynamics. In what follows, the parameter T1 with this anomalous dimension will be taken as a canonical parameter set. 3.5.1. Scale anomaly Following Coleman and Jackiw [36], the scale anomaly can be discussed in terms of an anomalous Ward identity. De"ne (p, q) and (p, q) by IJ
G(p) (p, q)G(p#q)" dx dye O Ve N W0¹H (x) (y) (0)0 IJ IJ
(77)
G(p)G(p#q)" dx dye O Ve N W0¹H[RID ](x) (y) (0)0 I with the renormalized propagator G(p) and the renormalized "elds (x). Here ¹H represents the covariant ¹-product, D (x) the dilatation current, and the improved energy momentum tensor I IJ with D "xJ . A naive consideration on Ward identities would give I IJ g IJ(p, q)"(p, q)!idG\(p)!idG\(p#q) (78) IJ with d the scale dimension of (x). However, is ill-de"ned due to singularity and so has to be regularized. With the regularization, the Ward identity reads g IJ(p, q)"(p, q)!idG\(p)!idG\(p#q)#A(p, q) IJ
(79)
A(p, q), lim (p, q, )!(p, q) , (80) where the additional term, A, is the anomaly. This anomaly corresponds to a shift in the dimension of the "eld involved at the lowest loop order but at higher orders there are vertex corrections. One obtains however a simple result when the beta functions vanish at zero momentum transfer [36]. Indeed in this case, the only e!ect of the anomaly will appear as an anomalous dimension. In general, this simpli"cation does not occur. However, it can take place when there are nontrivial "xed points in the low-energy theory. Under the reasoning developed in condensed matter physics [8], it is argued later that nuclear matter is given, in the absence of BCS channel, by a Landau Fermi-liquid "xed point theory with vanishing beta functions of the four-Fermi interactions and that all quantum #uctuation e!ects would therefore appear in the anomalous dimension of the scalar "eld S. That nuclear matter is a Fermi-liquid "xed point seems to be well veri"ed at least phenomenologically as seen in Section 5. However that #uctuations into the scalar channel can be summarized into an anomalous dimension is a conjecture that remains to be proved. We conjecture here that this is one way we can understand the success of the FTS1 model. 3.5.2. Nuclear matter properties at d +5/3 The FTS1 theory has some remarkable features associated with the large anomalous dimension. Particularly striking is the dependence on the anomalous dimension of the compression modulus K and many-body forces.
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Table 1 Equilibrium Fermi momentum k and binding energy B"M!E/A as a function of d for Fig. 4 d
K(MeV)
k (MeV)
B(MeV)
2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1
1960 1275 687 309 196 184 180 175 169
313 308 297 279 257 241 231 223 217
50.4 37.0 27.1 20.4 16.4 14.0 12.4 11.2 10.3
Fig. 4. Compression modulus vs. anomalous dimension. The parameter set used here is the T1 in FTS1. This shows the sensitivity of the compression modulus to the anomalous dimension.
In Table 1 are listed the compression modulus K and the equilibrium Fermi momentum k vs. the scale dimension d of the scalar "eld . As the d increases, the K drops very rapidly and stabilizes at K&200 MeV consistent with experiments for d+2.6 and stays nearly constant for d'2.6. This can be seen in Fig. 4. The equilibrium Fermi momentum on the other hand slowly decreases uniformly as the d increases.
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Unfortunately, we have no simple understanding of the mechanism that makes the compression modulus K stabilize at the particular value d +. We believe there is a non-trivial correlation between this behavior of K and the observation made below that the scalar logarithmic interaction brought in by the scale anomaly is entirely given at the saturation point by the quadratic term at the same d with the higher polynomial terms (i.e. many-body interactions) contributing more repulsion for increasing anomalous dimension. At present, our understanding is purely numerical and hence incomplete. In mean "eld, the logarithmic potential in Eqs. (73) and (76) contains n-body-force (for n52) contributions to the energy density. For the FTS1 parameters, these n-body terms are strongly suppressed for d92.6. This is shown in Fig. 5 where it is seen that the entire potential term is accurately reproduced by the quadratic term m for d &. Furthermore, a close examination Q of the results reveals that each of the n-body terms are separately suppressed. This phenomenon is in some sense consistent with chiral symmetry [2] and is observed in the spectroscopy of light nuclei [43]. Since there are additional polynomial terms in vector "elds (i.e. terms like ), the near complete suppression of the scalar polynomials does not mean the same for the total many-body forces. In fact, we should not expect it. To explain why this is so, we plot in Fig. 6 the three-body contributions of the and forms. We also compare the FTS1 results with the FTS2 results that are based on the naturalness condition. In FTS1 the term that turns to repulsion from attraction for d' contributes little, so the main repulsion arises from the -type term. This, together with an attraction from a term, is needed for the saturation of nuclear matter at the right density. This raises the question as to how one can understand the result obtained by Brown et al. [44] where it is argued that the chiral phase transition in dense medium is of mean "eld with the bag constant given entirely by the quadratic term &m. The answer to this question is as Q follows. Firstly, we expect that the anomalous dimension will stay locked at d "d!1& near the phase transition (this is because the anomalous dimension associated with the trace anomaly } a consequence of ultraviolet regularization } is not expected to depend upon density), so the L terms for n'2 will continue to be suppressed as density approaches the critical value. Secondly, near the chiral phase transition, the gauge coupling of the vector meson will go to zero in accordance with the Georgi vector limit [45], where chiral symmetry is restored by m P0, and M vector meson decoupling takes place as argued in [46]. So the many-body forces associated with the vector mesons will also be suppressed as density increases to the critical density. 3.5.3. Anomalous dimension and the scalar meson mass We would like to understand how the large anomalous dimension needed here could arise in the theory and its role in the scalar sector. As suggested in [21] and elaborated more in Section 3.3, one appealing way of understanding the FTS1 mean "eld theory is to consider all channels to be at Fermi-liquid "xed points except that because of scale anomaly, the scalar "eld develops an anomalous dimension, thereby a!ecting four-Fermi interaction in the scalar channel resulting when the scalar "eld is integrated out. If the anomalous dimension were su$ciently negative so that marginal terms became marginally relevant, then the system would become unstable as in the case of the NJL model or superconductivity, with the resulting spontaneous symmetry breaking. However, if the anomalous dimension is positive, then the resulting e!ect will instead be a screening. The positive anomalous dimension we need here belongs to the latter case. We can see this as
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Fig. 5. Comparison between the interaction and the logarithmic self-interaction of the scalar "eld with the FTS1 parameters. The dashed lines represent (m/2) and the solid lines (m/4)S d[(1!/S ) B[(1/d)ln(1!/S )!]#] Q Q for (from top to bottom) d"1, 2, 2.7, 3.5, respectively.
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Fig. 6. The 3-body contributions to the energy per nucleon vs. Fermi momentum in the FTS models. The short-dashed line represents the contribution of the term in the FTS2 with the Q1 parameters. The long-dashed and the solid lines represent the contributions of the cubic terms ( and ) in the FTS1 with the T1 parameters for d"2.7.
follows. Consider the potential given with the low-lying scalar S (with the gluonium component integrated out):
S B 1 S 1 1 ln ! #2 , (81) <(S,2)" m dS S d S 4 4 1 where m is the light-quarkonium mass in the free space (&700 MeV) and the ellipses stand for 1 other contributions such as pions, quark masses, etc. that we are not interested in. The scalar excitation on a given background S夹 is given by the double derivative of < with respect to S at S"S夹
S夹 B\ 4 S夹 m夹"m 1# !1 ln . 1 1 S S d We may identify the ratio S夹/S with the BR scaling factor [21]: S夹 f 夹 m夹 "" L " . S f m L Then we have m夹 1 "() () B m 1
(82)
(83)
(84)
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with
4 . ()" B\ 1# !1 ln B d
(85)
One can see that for d"1 which would correspond to the canonical dimension of a scalar "eld the scalar mass falls much faster, for a () that decreases as a function of density, than what would be given by BR scaling. Increasing the d (and hence the anomalous dimension) makes the scalar mass fall less rapidly. With d+2, +1 and we recover the BR scaling. Since the dropping scalar mass B is associated with an increasing attraction, we see that the anomalous dimension plays the role of bringing in an e!ective repulsion. One may therefore interpret this as a screening e!ect of the scalar attraction. In particular, that d!2+0.7'0 means that in FTS1, an additional screening of the BR-scaled scalar exchange (or an e!ective repulsion) is present.
4. Brown}Rho scaling Brown and Rho develop a strategy of density-dependent e!ective "eld theory for an in-medium e!ective theory in [14]. They assume that the e!ective Lagrangian even for the hadrons in matter also keeps the symmetries of QCD (e.g., chiral symmetry) and that the parameters of the e!ective theory are determined at a given density. The change of vacuum in the presence of medium is assumed to be expressible by the change of parameters of the theory. Brown}Rho scaling is the relation among the density-dependent parameters in medium. The quasiparticle picture associated with BR scaling is a successful way to describe hadrons in medium. In this section BR scaling is "rst derived from a QCD-oriented e!ective Lagrangian and then discussed. 4.1. Freund}Nambu model Before discussing the e!ective theory for dense matter and BR scaling, we study in this section the Freund}Nambu (FN) model, in order to describe the idea of BR scaling. The FN model is a simple model where scale symmetry is realized in Goldstone mode. The FN model Lagrangian is L$,"L$, #L$, , L$, "RIR #RI R ! f , I I c L$,"! ( !v) 8
(86)
with matter "eld and dilaton "eld both of which have scale dimension 1:
"(1#x ) R),
"(1#x ) R) .
(87)
Eq. (87) comes from (ax)"a\(x) with a"1# where represents "elds of scale dimension 1. The equations of motion for the FN model R!f "0 ,
(88)
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c R !f ! ( !v)"0 2
319
(89)
have non-trivial constant solutions; "0, "$v .
(90)
Let "v and shift the dilation "eld as " #v .
(91)
The scale transformation of the newly de"ned does not show a de"nite scale dimension, but show a symptom of Goldstone mode;
"(1#x ) R) #v .
(92)
With the "eld rede"nition
v !1 . , 2 v
(93)
Eq. (86) becomes 1 m m 1 1 L$, " RIR ! ( ! ( # RIR 2 I I 1#2 /v 2 2 v m L$,"! N 2
(94)
with (95) m "f , m "(c . ( N Scale symmetry in L becomes invisible. We can see some interesting features in the FN model. Both the matter and dilaton "eld masses depend on . It means that both masses move in the same way if the vacuum expectation value of is changed. In addition, the mass can be left massive even if cP0. We also note that the explicit scale symmetry breaking term is necessary for the FN model. It renders possible the spontaneous scale symmetry breaking of L . 4.2. Brown}Rho scaling in in-medium ewective Lagrangian We introduce the elementary scalars that QCD contains. Following Schechter [47], scalar "elds represented by the trace anomaly (60) can be introduced. We divide the trace anomaly into a hard part, which is associated with the gluonium , and a soft part, which comes from the E quarkonium . I "(I ) #(I ) . I I I
(96)
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The gluon condensate in (I ) determines the gluonium mass m E &1.6}1.8 GeV [42] and does I Q not vanish [48]. Since the gluonium mass is much larger than the chiral scale &1 GeV, the Q gluonium "elds can be integrated out in low-energy regime. Adami and Brown [49] show by QCD sum rule that the gluon condensate is less important for the masses of light-quark hadrons. So we focus on the quarkonium scalar which has mostly to do with (I ) . I In matter-free space, there are no scalars whose mass is small enough, i.e.,; . However, Q according to Weinberg's mended symmetry [50] there must be a scalar to form a quartet with pions near the chiral phase transition. In addition, lattice simulations [51] near the chiral phase transition show that two light #avor QCD transition reproduces a scaling relation with O(4) exponents as argued "rst by Pisarski and Wilczek [52]. Beane and van Kolck [53] suggest that the Goldstone boson from spontaneous scale symmetry breaking plays the role of the chiral partner of the pion, i.e., the chiral singlet scalar in the scale anomaly approaches the pions and makes up the quartet of O(4) symmetry in medium as density increases to the critical density of the chiral restoration. We note that the quark condensate, which contributes to the quarkonium mass, goes to zero as chiral symmetry is restored. So we assume that the quarkonim mass in the scale anomaly becomes m夹; as density increases, though m & in free space. Q Q Q Q Integrating out the hard part of the scalar "eld , we introduce into a Skyrme Lagrangian E } which represents the low-energy QCD for the in"nite number of colors } the soft part of scalar, , in order to make our e!ective theory consistent with QCD scale property: L"L #L 1 1 f 1 Tr[;RR ;, ;RR ;]#2 L "L Tr(R ;RI;R)# R RI # I J I 32e 4 2 I L "!<( )#pion mass term#2 1
(97) (98) (99)
with ;"e DL ,
(100)
"0 0 . (101) We make L scale correctly by multiplying the proper powers of . Since L do not contribute to I , we add the scale breaking potential term <( ) due to scale anomaly. <( ) includes radiative I corrections of high order and gives the trace anomaly of QCD in terms of . Let us break the scale symmetry spontaneously following the strategy of the FN model. If we de"ne 夹 as the mean "eld value of in dense matter 夹 , , M we can expand as " 夹 # .
(102)
(103)
Stars will be a$xed to all the quantities that appear in nuclear matter from now on. L becomes in terms of , L
f 夹 1 1 " L Tr(R ;RI;R)# R RI # Tr[;RR ;, ;RR ;]#2 I I J 4 2 I 32e
(104)
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with the e!ective pion decay constant de"ned at mean "eld level as 夹 . (105) f 夹"f L L 夹 夹 Note that ;夹"e DL with 夹, 夹 / is the same as ; in (100) by de"nition (105). In our e!ective "eld theory both scale symmetry and chiral symmetry are realized in the Goldstone mode and their Goldstone bosons are and 's, respectively. According to Gell}Mann}Oaks}Renner (GMOR) relation m #m B 0u u#dM d0 , f m"! S L L 2
(106)
the pion mass squared is proportional to the quark mass. Since the quark masses come from explicit chiral symmetry breaking which has something to do with the electroweak symmetry breaking scale < , the pion mass problem is out of our interesting range. In the prediction #5 Q of chiral perturbation theory in medium [54], the pole mass of the pion does not decrease up to nuclear matter density. In fact, a recent analysis of deeply bound pionic states in heavy nuclei [55] shows that the pole mass of the pion could be even a few per cents higher than the free space value at nuclear matter density. The m夹 in our in-medium e!ective chiral Lagrangian is not necessarily L the pole mass. Thus, it is not clear how to incorporate this empirical information into our theory. We will assume here that m夹 does not scale. This assumption may be justi"ed by using the L fact that the pion is an almost perfect Goldstone boson. Under the assumption m夹"m , Eq. (106) L L may give
f夹 q q 夹 L& . (107) q q f L It is an example to show the relation between the BR-scaling factor and the change of the vacuum in medium. Since the hedgehog solution of the Skyrme Lagrangian (98) gives the Skyrmion mass proportional to (g f , the nucleon mass must scale in the matter as L
m夹 g夹 f 夹 ," L . (108) M g f L 夹 Here M represents the nucleon mass in free space and m is the in-medium e!ective mass of , nucleon (later identi"ed with the Landau e!ective mass). In the simple Skyrme model g is inversely proportional to e which does not scale since the quartic Skyrme term is classically scale-invariant. So g does not change in our simple model and the nucleon mass scales in the same way as pion decay constant in medium. The successful low-energy results of the tree level in the chiral e!ective theory implemented with hidden local symmetry, i.e., Kawarabayashi}Suzuki}Riazuddin}Fayyazudin (KSRF) relation, coupling universality, -meson dominance, etc., are shown to remain valid with the loop e!ects at low energy [56]. Though it is proven in free space, it might hold in medium. If we assume KSRF relation m"2ef L
(109)
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is satis"ed in nuclear matter, we obtain m夹 f 夹 " L (110) m f L with the subscript v standing for light-quark vector mesons and . The #uctuating component in the soft is expected to represent multi-pion excitations in scalar channel and to give a scalar e!ective "eld in dense medium by interpolation. It is known that the correlated 2 exchange can be approximated by a scalar "eld with a broad mass distribution [57]. Durso, Kim, and Wambach's recent calculation of NNM P helicity amplitude in the scalar channel f ( for & with BR scaling of vector meson mass shows that the > resonance of the scalar mass becomes very sharper and that its value shifts downward +500 MeV [58]. represents the normal nuclear matter density (0.16/fm). The light (; ) and decreasing Q mass of the scalar "eld suggests that it is the expected Beane and van Kolck's dilaton [53]. Since the main } rescattering comes from -meson exchange [57], the shift of the scalar mass in medium is a!ected by the density-dependence of the mass. It is clear for low densities, where the linear approximation works well, that m夹 m夹 N+ . (111) m m N Now, we "nd that the hadron masses and pion decay constant decrease in the similar way. f 夹() m夹() m夹() m夹() ()+ L + N + + . (112) f m m M L N For the moment, we are ignoring the scaling of g to which we will return later. M夹 represents the e!ective nucleon mass obtained with a "xed g夹 . The universal factor () can be determined by experiments and/or QCD sum rules. Note that the scalar "eld that governs BR scaling is the quarkonium component of the scale anomaly, not the hard gluonic component. The latter gives dominant contribution to the scale anomaly in QCD but is integrated out in the e!ective Lagrangian. This structure imposes the hadron scaling relation (112) by a Nambu}Jona}Lasinio (NJL) mechanism as described in [44]. Recently, Liu et al.'s detailed lattice analysis [59] for the source of the mass of a constituent quark supports this structure. They show that dynamical symmetry breaking contribution gives most of the mass of the chiral constituent quark. It means that the change of the vacuum in (112) can be related only in a subtle way to the light quark hadron mass. When considering BR scaling, one must note that the BR-scaled masses and decay constants are the parameters in an e!ective theory. An e!ective parameter de"ned in one theory can be di!erent from the parameter de"ned in another theory, even though two theories have the same physics in common. So the connection between the BR-scaled parameters and the parameters in the other theories needs be established via observed quantities, i.e., experiments. One must also note that BR scaling is the result of mean "eld approximation. So when one considers the cases where higher-order modi"cations are important, that is, the mean-"eld In some recent papers, deviations from BR scaling are obtained in higher order. For example, see [60].
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Fig. 7. The comparison of CERES dilepton experiments and the theoretical predictions with the free-mass mesons and with scaled mass mesons. The "gure comes from [61].
approximation is not reliable, BR scaling cannot be applied without further corrections. Let us take Goldberger}Treiman relation g
f "g M (113) L,, L as an example. In the real world, g decreases from &1.26 in free space to &1 at normal nuclear matter density since it is a!ected by the short-range interaction between baryons. So the naive BR scaling, which would have g remain constant, cannot explain Goldberger}Treiman relation in L,, the low-density region. It means higher-order modi"cations spoil the tree-order BR scaling at low densities. At high densities above the normal nuclear matter density, however, g夹 remains at 1 while f 夹 continues to drop and hence the coupling constant ratio increases. L Although Brown and Rho's arguments about the scaling relation of e!ective parameters may seem a bit drastic, the experiments (e.g., CERES [15], HELIOS-3 [16]) provide support for this scheme. The explanation of CERES data is one of the good examples. As seen in Fig. 7, the scaling of e!ective masses of hadrons in medium reproduces the CERES results very simply at the mean "eld level [17]. 4.3. Duality Brown}Rho scaling describes the low-mass dilepton enhancement on the basis of the quasiquark picture. But the increase of dilepton yields can also be described by a hadronic theory with free In Ref. [17], BR scaling is invoked at the constituent quark level [62]: Li, Ko, and Brown used Walecka theory to implement the density dependence of the constituent quarks.
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masses, introducing the appropriate variables, e.g. nucleonic excitations. Rapp et al. [63] described the CERES results successfully using conventional many-body theory. To evaluate the in-medium rho meson propagator they renormalized the intermediate two-pion states, which dress rho mesons and interact with the surrounding nucleons and deltas, and considered the direct interaction of rho mesons with surrounding hadrons, especially -like baryon-hole excitations (`rhosobara). They found that such medium e!ects broaden the spectral function of rho meson and the dilepton production at &m /2 is enhanced. M Rapp et al.'s hadronic rescattering approach and Brown}Rho's quasiparticle approach have merits and demerits, compared with each other. Since BR scaling is approximate at mean-"eld level, Rapp et al.'s theory is more reliable at low densities than BR scaling. On the other hand at higher densities many diagrams have to be considered in the hadronic rescattering approach. BR scaling gives a medium-modi"ed vacuum around which the weak #uctuations can be dealt with. If both descriptions are correct, the e!ective variables are expected to change smoothly from hadrons to quasiquarks subject to BR scaling and both descriptions must show duality around the hadron-quark changeover densities. Such duality was suggested by Brown et al. [65] and Y. Kim et al. [66] studied it more precisely. For rho mesons in medium they studied a two-level model, which consists of the collective rhosobar [NH(1520)N\] and the &elementary' . The in-medium -meson propagator is D "[q !q!(m)! ! ]\ , M M MLL M ,
(114)
where the indicates self-energies and m is the bare mass of . Taking the free rho meson mass M m "(m )#Re , we obtain the dispersion relation in the q"0 limit, M M MLL q "m#Re H . M M, ,
(115)
The phenomeological Lagrangian for the s-wave interaction between the elementary and the -like baryon-hole excitation is L
M ,
f " M , BR(q s ) !s ) q)t N#h.c. ? ? ? m M
(116)
with appropriate spin operator s and isospin operators t [68]. The self-energy from interaction ? (116) for B"NH(1520) is
8 q 2E (q )+ f H L , H M, , 3 M, , m 4 (q #i /2)!(E) M R
(117)
Rapp and Wambach suggested [64] to interpret the strong broadening of the -meson resonance as a manifestation of hadron-quark continuity. The NH(1520) resonance gives the most important contribution to the photoabsorption cross-sections in the dileption analysis [67]. For p-wave coupling, the Lagrangian is f " M , BR(s;) ) t N#h.c. ? ? m M
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Fig. 8. The in-medium -meson mass and Z factor obtained in [66] for "0.
where is the nuclear density and the total width includes the free width of M H (1520) and its L R , modi"cation in medium. Neglecting nuclear Fermi motion (q"0 limit), E"M H !M . , , Kim et al. [66] showed that dispersion relation (115) gives the two states, which have -meson quantum numbers, with the spectral weight
R \ Re H Z" 1! M, , Rq
(118)
and that one of them can be interpreted as an in-medium vector meson whose mass decreases. Fig. 8 shows the results with "0. R/= indicates that m in (117) is the free mass and B/R M indicates that q /m in (117) is replaced by 1, i.e., replacing m by m夹. Note that q i.e., the M M M in-medium -meson mass, cannot go to zero at any density as seen in R/= of Fig. 8 if m in the M denominator of (117) is "xed. It matches neither BR scaling [14] nor the prediction that m夹P0 at M the chiral transition point [44]. Brown et al. [65] suggested the replacement of m in (117) by m夹 in M M order to go from Rapp et al.'s hadronic theory to BR scaling which predicts zero vector meson mass at some high density. B/R of Fig. 8 shows that in-medium -meson mass goes to zero near &2.75 as predicted in [44]. The study of the schematic model [65,66] provides how BR scaling can be interpolated from a hadronic rescattering description.
5. E4ective Lagrangian with BR scaling If the large anomalous dimension of the scalar "eld in FTS1 is a symptom of a strong-coupling regime, it suggests that one should rede"ne the vacuum in such a way that the #uctuations around the new vacuum become weak-coupling. This is the basis of the BR scaling introduced in [14]. The basic idea is to #uctuate around the `vacuuma de"ned at + characterized by the quark
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condensate q q ,q q 夹. This theory was developed with a chiral Lagrangian implemented with M the trace anomaly of QCD as seen in the last section. The Lagrangian used was the one valid in the large N -limit of QCD and hence given entirely in terms of boson "elds from which baryons arise as solitons (skyrmions): Baryon properties are therefore dictated by the structure of the bosonic Lagrangian, thereby leading to a sort of universal scaling between mesons and baryons. One can see, using a dilated chiral quark model, that BR scaling is a generic feature also at high temperature in the large N limit [69]. In this description, one is approximating the complicated strong interaction process at a given nuclear matter density in terms of `quasiparticlea excitations for both baryons and bosons in medium. This means that the properties of fermions and bosons in medium at + are given in terms of tree diagrams with the properties de"ned in terms of the masses and coupling constants universally determined by the quark condensates at that density. The question then is: How can one marry the FTS1 Lagrangian with the BR-scaled Lagrangian? The next question is how to identify BR-scaled parameters with the Landau parameters. In this section, we will provide some answers to these two questions. 5.1. A hybrid BR-scaled model As a "rst attempt to answer these questions, we consider the hybrid model in which the ground state is given by the mean "eld of the FTS1 Lagrangian L and the #uctuation $21 around the ground state is described by the tree diagrams of the BR-scaled Lagrangian L, L"L #L . (119) $21 Let us see how the particles behave in the background of the FTS1 ground state given by L . $21 The nucleon of course scales a% la BR as mentioned above. We can say nothing on the pion and the meson with the FTS1 theory. However there is nothing which would preclude the scaling a% la BR and the pion non-scaling within the scheme. What is encoded in the FTS1 theory is the behavior of the and the scalar S which "gure importantly in Walecka mean "elds. Let us therefore focus on these two "elds in medium near normal nuclear matter density. We have already shown in Section 3.5.3 that the mass of the scalar "eld S drops less rapidly than BR scaling for d'2. One can think of this as a screening of the four-Fermi interaction in the scalar channel that arises when the scalar meson with the BR-scaled mass is integrated out. This feature and the property of the "eld can be seen by the toy model of the FTS1 Lagrangian (that includes terms corresponding up to three-body forces) L
m m , "L # S (2#) ! Q $21 0 S 3S 2
(120)
If one calculates f 夹(¹) and m夹 (¹) at zero density in chiral perturbation theory, the temperature-dependence deviates L , from BR scaling at low ¹ [70]. In [69], Y. Kim, H.K. Lee and M. Rho argued that BR scaling in ¹ should instead hold at high enough density as one approaches chiral restoration.
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where
2 m 2 m ! Q 1! . L "NM (i (RI#ig I)!M#g )N# S 1! 0 I Q S 2 3S 2 We have written L such that the BR scaling is incorporated at mean xeld level as 0 夹 m夹 m夹 M " Q " S +1! ()" M m m S Q S with
(121)
(122)
S "0S0 "M/g . (123) Q Here we are ignoring the deviation of the scaling of the e!ective nucleon mass (denoted later as m夹 ) , from the universal scaling () [21]. This will be incorporated in the next subsection. We can see from (120) that the FTS1 theory brings in an additional repulsive three-body force coming from a cubic scalar "eld term while if one takes "!2, the "eld will have a BR scaling mass in nuclear matter. Fit to experiments favors +! instead of !2, thus indicating that the FTS1 theory requires a many-body suppression of the repulsion due to the exchange two-body force. (In the simple model with BR scaling that we will construct below, we shall use this feature by introducing a `runninga vector coupling g夹 that drops as a function of density.) The e!ective mass may be written as
m夹+ 1# m . (124) S S S For (0, we have a falling mass corresponding to BR scaling (modulo, of course, the numerical value of ). In FTS1, there is a quartic term & which is attractive and hence increases the mass. In fact, because of the attractive quartic term, we have m夹 S +1.12 (125) m S at the saturation density with T1 parameter set. This would seem to suggest that due to higher polynomial (many-body) e!ects, the mass does not follow BR scaling in medium. Furthermore, the e!ective mass increases slowly around this equilibrium value. On the other hand, Klingl, Waas, and Weise's recent sum rule analysis on current correlation function [71,72] follows BR scaling. It shows that e!ective meson mass scales down by about 15% at normal nuclear matter density from its free value. We will discuss the shift of vector meson mass in medium in detail in Section 5.6. 5.2. Model with BR scaling The above hybrid model suggests how to construct an e!ective Lagrangian model that implements BR scaling and contains the same physics as the FTS1 theory. The crucial point is that such
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a Lagrangian is to give in the mean "eld the chiral liquid soliton solution. This can be done by making the following replacements in (121): M!g PM夹 , Q
2 m 1! Pm夹 , S S S 2 m 1! Pm夹 Q Q S
(126)
and write 1 1 m夹 m夹 L "NM (i (RI#ig I)!M夹#h)N! F # (R )# S ! Q 0 I 2 2 4 IJ 2 I
(127)
M夹 m夹 m夹 " S " Q "() . M m m S Q
(128)
with
The additional term NM hN is put in to account for the di!erence between the Landau mass m夹 to * be given later and the BR scaling mass M夹. In the chiral Lagrangian approach with BR scaling, the di!erence comes through the Fock term involving non-local pion exchange. This will be discussed further in the next section. For simplicity, we will take the scaling in the form 1 ()" 1#y/
(129)
with y"0.28 so as to give ( )"0.78 (corresponding to k "260 MeV) found in QCD sum rule $ calculations [73], as well as from the in-medium GMOR relation [46]. Note that the Lagrangian (127) treated at the mean "eld level would give a Walecka-type model with the meson masses replaced by the BR scaling mass. Now in order to describe nuclear matter in the spirit of the FTS1 theory, we introduce terms cubic and higher in and "elds to be treated as perturbations around the BR background as L
LU
"a#b#c#d#e#2 ,
(130)
where a } e are `naturala (possibly density-dependent) constants to be determined. By inserting for the and "elds the solutions of the static mean "eld equations given by L , 0 m夹M "h NM N , Q G G G
(131)
m夹 "g NRN , S G G G
(132)
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we see that at the mean-"eld level, L generates three- and higher-body forces with the LU exchanged masses density-dependent a% la BR. Note that at this point, the scaling factor and the mean "eld value (131) are not necessarily locked to each other by self-consistency. As the "rst trial, we will consider the drastically simpli"ed model by dropping the n-body term (130) and minimally modifying the BR Lagrangian (127). We shall do this by letting as mentioned above the vector coupling `runa as a function of density. For this, we use the observation made in [74] that the baryon #ow probing higher density requires that g夹/m夹 be independent of density at S low densities and decrease at high densities. We shall therefore take, to simulate this particular many-body correlation e!ect, the vector coupling to scale as g夹 1 " (133) g 1#z/ with z equal to or slightly greater than y. The h is assumed not to scale although it is easy to take into account the density dependence if necessary. Scaling (133) seems at odds with the prediction made with the Skyrme model [75] where using the Skyrme model with the quartic Skyrme term inversely proportional to the coupling e, it was found that e g夹 . & e夹 g It is tempting to identify (via SU(6) symmetry) e with g that we are discussing here since the Skyrme quartic term can formally be obtained from a hidden gauge-symmetric Lagrangian by integrating out the meson "eld. If this were correct, one would predict that the vector coupling increases } and not decreases } as density increases since we know that g夹 is quenched in dense matter. This identi"cation could be too naive and incomplete in two respects, however. First of all, this skyrmion formula is a large-N relation and secondly the Skyrme quartic term embodies all short-distance physics in one dimension}four term in a derivative expansion. Thus, the constant 1/e must represent a lot more than just the vector-meson () degree of freedom. Furthermore, we are concerned with the degree of freedom which in a naive derivative expansion would give a six-derivative term. The BR-scaled model we are constructing should involve not only shortdistance physics presumably represented by the 1/e term (consistent with the understanding that the quenching of g is a short-distance phenomena) but also longer-range correlations. Therefore, the qualitative di!erence should surprise no one. The truncated Lagrangian that we shall consider then is
1 1 m夹 m夹 L "NM (i (RI#ig夹I)!M夹#h)N! F # (R )# S ! Q . 0 I 4 IJ 2 I 2 2
(134)
As suggested in [76,41,5], chiral in-medium Lagrangians can be brought to a form equivalent to a Walecka-type model. The scalar "eld appearing here transforms as a singlet, not as the fourth component of O(4) of the linear sigma model. As it stands, the Lagrangian (134) does not look chirally invariant. This is because, we have dropped pion "elds which play no role in the ground state of nuclear matter. In considering #uctuations around the ground state, they (and other pseudo-Goldstone "elds such as kaons) should be reinstated. The chiral singlet "eld and "eld
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can be considered as auxiliary "elds brought in from a Lagrangian consisting of multi-Fermion "eld operators [5] via a Hubbard}Stratonovich transformation. Since we treat density-dependent parameters, we must be careful in thermodynamic consistency. After showing the way to treat density-dependent parameters in the next section, we display the results of our model for nuclear matter properties. We will argue in the next subsection that the energy density from (151) is independent of the way how the parameters move as density increases. 5.3. Thermodynamic consistency in medium In this subsection, we address the issue of thermodynamic consistency of the Lagrangian (134) treated in the mean "eld approximation. For instance, it is not obvious that the presence of the density-dependent parameters in the Lagrangian does not spoil the self-consistency of the model, in particular, energy}momentum conservation in the medium and also certain relations of Fermiliquid structure of the matter. The purpose of this section is to show that there is no inconsistency in doing a "eld theory with BR scaling masses and other parameters. This point has not been fully appreciated by workers in the "eld. The Euler}Lagrange equations of motion are in the same as the ones that arise from the "eld for the Lagrangian wherein the masses and constants are not BR scaling. While this procedure gives correct energy density, pressure and compression modulus, the energy}momentum conservation is not automatically assured. In fact, if one were to compute the pressure from the energy-density E, one would "nd that it does not give ¹ (where ¹ is the IJ GG conserved energy}momentum tensor and the bracket means the quantity evaluated in the mean"eld approximation as de"ned before) unless one drops certain terms without justi"cation. This suggests that it is incorrect to take the masses and coupling constants independent of "elds in deriving, by Noether theorem, the energy}momentum tensor. So the question is: how do we treat the "eld dependence of the BR scaling masses and constants? One possible solution to this problem is as follows. In Section 4.2, the density dependence of the Lagrangian arose as the vacuum expectation value of the scalar "eld that "gures in the QCD trace anomaly. By vacuum, we mean the state of baryon number zero modi"ed from that of true vacuum by the strong in#uence of the baryons in the system. See later for more on this point. It corresponded to the condensate of a quarkonium component of the scalar with the gluonium component } which lies higher than the chiral scale } integrated out. It was assumed to scale in dense medium in a Skyrmion-type Lagrangian subject to chiral symmetry. Now in the language of a chiral Lagrangian consisting of the nucleonic matter "eld N with other massive "elds integrated out, this scalar condensate would be some function of the vacuum expectation value of NM N or NM N coming from multi-Fermion "eld operators mentioned above. How these four- and higher Fermi "eld terms can lead to BR scaling in the framework of chiral perturbation theory was discussed in [5]. We shall follow this strategy in this paper leaving other possibilities (such as dependence on the mean "elds of the massive mesons) for later investigation. For this, it is convenient to de"ne \ uI,NM IN
(135)
with unit #uid 4-velocity 1 1 (1, *)" (, j) uI" (!j (1!*
(136)
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with the baryon current density j"NM N
(137)
and the baryon number density "NRN " n . (138) G G We will take n to be given by the Fermi distribution function, n "(k !k ) at ¹"0. We G G $ G should replace in (134) by \ for consistency of the model. The de"nition of \ makes our Lagrangian Lorentz-invariant which will later turn out to be useful in deriving relativistic Landau formulas. With this, the Euler}Lagrange equation of motion for the nucleon "eld is
L RL RL R\ " #
NM RNM R\ RNM "[iI(R #ig夹 !iu [ )!M夹#h]N I I I "0
(139)
with RL [ " R\ Rm夹 Rm夹 Rg夹 RM夹 "m夹 S !m夹 Q !NM I N !NM N . S Q I R\ R\ R\ R\
(140)
This additional term which may be related to what is referred to in many-body theory as `rearrangement termsa plays a crucial role in what follows. The equations of motion for the bosonic "elds are (141) (RIR #m夹)"hNM N , Q I R FJI#m夹I"g夹NM IN . (142) J S S We start with the conserved canonical energy}momentum tensor constructed a la Noether from the Lagrangian (134): ¹IJ"iNM IRJN#RIRJ!RI RJH H 夹 ! [(R)!m !(R)#m夹!2[ NM u. N]gIJ . Q S
(143)
For a recent discussion on rearrangement terms as well as thermodynamic consistency in the context of standard many-body theory, see [77]. The notion of density-dependent parameters in many-body problems is of course an old one [78].
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We shall compute thermodynamic quantities from (143) using the mean "eld approximation which amounts to taking 1 N" a (< G G
E G #m夹 G , exp(i ) x!i(g夹 !u #E )t) ) G G G 夹 2E G G E G #m G ,
m夹 , h"CNM N "C n F F G (#m夹 G G ,
G g夹"C(, j)"C n 1, , 夹 G ( #m G , G where a is the annihilation operator of the nucleon i, with G n "aRa , G G G and
(144)
(145)
"[ ,
(146)
,k !Cj#u , G G
(147)
and E G "(#m夹 . G G , is the spinor and is the Pauli matrix. We have de"ned g夹(\ ) C (\ ), 夹 m (\ ) S
(148)
(149)
and 1 h . C (\ ), 夹 , F m (\ ) " (\ ) Q F In this approximation, the energy density is
(150)
E"¹
1 1 " iNM RN# m夹! m夹#[ NM u. N Q 2 2 S
1 1 " C(#j)# CI (m夹 !M夹)# n (#m夹!u ) j . (151) J J , 2 2 F , J Note that the -dependent terms cancel out in the comoving frame ("0), so that the resulting energy-density is identical to what one would obtain from the Lagrangian in the mean "eld with the density-dependent parameters taken as "eld-independent quantities.
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Given the energy density (151), the pressure can be calculated by (at ¹"0) RE/ RE "!E p"! " R R< 1 1 " C()! ! CI ()(m夹 !M夹()) , 2 F 2 !
m夹 1 m夹 , k ! k ! , ln[(k #E )/m夹 ] , E $ $ , 8 2 $ 8 $ 12 $
(152)
where is the chemical potential } the "rst derivative of the energy density with respect to in the comoving frame ("0): R , E* "C#E ! $ R
(153)
with E "(k #m夹 and "[ * . To check that this is consistent, we calculate the pressure , $ $ from the energy}momentum tensor (143) in the mean "eld at ¹"0: 1 p , ¹ * 3 GG
1 1 " iNM R N! (m夹!m夹!2[ NRN)g G G Q GG 3 2 S
*
1 1 " C()! CI ()(m夹 !M夹())! , 2 F 2 !
m夹 1 m夹 , k ! k ! , ln[(k #E )/m夹 ] . E $ $ , 8 2 $ 8 $ 12 $
(154)
This agrees with (152). Thus our equation of state conserves energy and momentum. We showed that a simple e!ective chiral Lagrangian with BR scaling parameters is thermodynamically consistent, a point which is important for studying nuclear matter under extreme conditions. It is clear however that this does not require that the masses appearing in the Lagrangian scale according to BR scaling only. What is shown in this subsection is that masses and coupling constants could depend on density without getting into inconsistency with general constraints of chiral Lagrangian "eld theory. This point is important for applying (134) to the density regime &3 appropriate for the CERES dilepton experiments and also kaon produc tion at GSI (Gesellschaft fuK r Schwerionenforschung) where deviation from the simple BR scaling of [14] might occur. 5.4. Results Based on the thermodynamic consistency of density-dependent e!ective theories proven at the mean "eld level, we check whether the model Lagrangian (134) can describe the in"nite nuclear
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Table 2 Parameters for the Lagrangian (134) with y"0.28, m "700 MeV, m "783 MeV, M"939 MeV Q S SET
h
g
z
S1 S2 S3
6.62 5.62 5.30
15.8 15.3 15.2
0.28 0.30 0.31
Table 3 Nuclear matter properties predicted with the parameters of Table 2. The e!ective nucleon mass (later identi"ed with the Landau mass) is m夹 "M夹!h , SET
E/A!M(MeV)
k (MeV)
K(MeV)
m夹 /M ,
(k )
S1 S2 S3
!16.0 !16.2 !16.1
257.3 256.9 258.2
296 263 259
0.619 0.666 0.675
0.79 0.79 0.78
matter properties successfully. The characteristic properties we try to reproduce are compression modulus, m夹 , and binding energy at normal nuclear matter density, and the saturation density , itself. In Table 2, three sets of parameters are listed. We take the measured free-space masses for the and the nucleon and for the scalar for which the free-space mass cannot be precisely given, we take m "700 MeV (consistent with what is argued in [46]) so that at nuclear matter density, it Q comes close to what enters in the FTS1. The resulting "ts to the properties of nuclear matter are given in Table 3 for the parameters given in Table 2. These results are encouraging. Considering the simplicity of the model, the model } in particular with the S2 and S3 set } is remarkably close in nuclear matter to the full FTS1. The compression modulus comes down toward the low value that is currently favored. In fact, the somewhat higher value obtained here can be easily brought down to about 200 MeV without modifying other quantities if one admits a small admixture of the residual many-body terms (130), as we shall shortly show. The e!ective nucleon Landau mass m夹 /M+0.67 is in good agreement with what was obtained in QCD sum rule calculations [79] , and also in the next section (i.e., 0.69) by mapping BR scaling to Landau}Migdal Fermiliquid theory. We shall see below that this has strong support from low-energy nuclear properties. What is also noteworthy is that the ratio R,(g夹/m夹 ) forced upon us } though S not predicted } is independent of the density (set S1) or slightly decreasing with density (sets S2 and S3), as required in the nucleon #ow data as found by Sahu et al. [74]. In FTS1 theory, it is the higher polynomial terms in and de"ning the mean "elds that are responsible for the reduction in R needed in [74]. In Dirac}Brueckner}Hartree}Fock theory, it is found [80] that while R takes the free-space value R for + , it decreases to R+0.64R at +3 due to rescattering terms which in our language would correspond to the many-body correlations.
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Table 4 E!ect of many-body correlations on nuclear matter properties using the Lagrangian (134) # (155). We have "xed the free-space masses m "700 MeV, m "783 MeV, M"939 MeV and set "0 for simplicity. The equilibrium density Q S k , the compression modulus K, and the binding energy B"M!E/A are all given in units of MeV SET
h
g
y
z
S3 B1 B2 B3 B4 C1 C2
5.30 5.7 5.7 5.6 5.6 5.7 5.8
15.2 15.3 15.3 15.27 15.3 15.3 15.3
0.28 0.28 0.28 0.28 0.28 0.28 0.28
0.31 0.30 0.30 0.30 0.31 0.30 0.30
0.5
!0.055 0.18 0.31 0.9 !0.05 !0.11
0.155 0.35
k
258.2 !4.9 256.0 257.3 !4.1 259.1 !8.1 256.4 256.3 256.1
mH /M ,
K
B
0.675 0.666 0.661 0.659 0.669 0.665 0.662
259 209 201 185 191 218 161
16.1 16.2 16.1 16.1 16.1 16.2 16.2
The assumption that the many-body correlation terms in (130) can be entirely subsumed in the dropping vector coupling may seem too drastic. Let us see what small residual three-body and four-body terms in (130) as many-body correlations (over and above what is included in the running vector coupling constant) can do to nuclear matter properties. For convenience, we rewrite (130) by inserting dimensional factors as L
LU
" m ! m # g! m # m 3! Q f 4! 4! Q f 2 Sf 2 Sf
(155)
and demand that the coe$cients , and so de"ned be natural. The results of this analysis are given in Table 4 and Fig. 9 for various values of the residual many-body terms and compared with those of the truncated model (134) with S3 parameter set. The coe$cients are chosen somewhat arbitrarily to bring our points home, with no attempt made for a systematic "t. (It would be easy to "ne-tune the parameters to make the model as close as one wishes to FTS1 theory.) It should be noted that while the equilibrium density or Fermi momentum k , the e!ective nucleon mass m夹 and the binding energy B stay more or less unchanged, within the range of the parameters , chosen, from what is given by the BR-scaled model (134) with the S3 parameters, the compression modulus K can be substantially decreased by the residual many-body terms. Fig. 9 shows that as expected, lowering of the compression modulus is accompanied by softening of the equation of state at ' . While the equilibrium property other than the compression modulus is insensitive to the many-body correlation terms, the equation of state at larger density can be quite sensitive to them. This is because for the generic parameters chosen, the m夹 can vanish at a given density above , at which the approximation is expected to break down and hence the resulting value cannot be trusted. The B2 and B4 models do show this instability at 91.5 (see Fig. 16). It is quite encouraging that simple minimal model (134) with BR scaling captures so much of the physics of nuclear matter. Of course, by itself, there is no big deal in what is obtained by the truncated model: It is not a prediction. What is not trivial, however, is that once we have a Lagrangian of the form (134) which de"nes the mean "elds, then we are able to control with some con"dence the background around which we can calculate #uctuations, which was the principal
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Fig. 9. E/A!M vs. for FTS1 theory (`T1a parameter), the `S3a, `B1a and `B3a models de"ned in Table 4.
objective we set at the beginning of this approach. The power of the simple Lagrangian is that we can now treat #uctuations at higher densities as one encounters in heavy-ion collisions, not just at an equilibrium point. The description of such #uctuations does not su!er from the sensitivity with which the equation of state depends at ' on the many-body correlation terms (130). Simple Lagrangian (134) embodies the e!ective "eld theory of QCD discussed by Furnstahl et al. [11] anchored on general considerations of chiral symmetry. This Lagrangian should be viewed as an e!ective Lagrangian that results from two successive renormalization group `decimationsa, one leading to a chiral liquid structure [38] at the chiral symmetry scale and the other with respect to the Fermi surface [8,9]. The advantage of (134) is that it can, on the one hand, be connected to Landau Fermi-liquid "xed point theory of nuclear matter and, on the other hand, be extrapolated to the regime of hadronic matter produced under extreme conditions as encountered in relativistic heavy ion processes. It would, for instance, allow one, starting from the ground state of nuclear matter, to treat on the same footing the dilepton processes observed in CERES experiments as explained in [17] and kaon production at SIS (Schwerionen-Synchrotron) energy and kaon condensation in dense matter relevant to the formation of compact stars as discussed in [81]. 5.5. Landau Fermi-liquid properties of the BR-scaled model The next issue we address is the connection between the mean-"eld theory of the chiral Lagrangian (5.2) and Landau's Fermi-liquid "xed point theory as formulated in Section 3.3. As far
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as we know, this connection is the only means available to implement chiral symmetry of QCD in dense matter based on e!ective "eld theory. For this, we shall follow closely Matsui's analysis of Walecka model in mean "eld [82] exploiting the similarity of our model to the latter. 5.5.1. Quasiparticle interactions The quasiparticle energy and quasiparticle interaction f are, respectively, given by "rst and G GH second derivatives with respect to n : G RE , (156) " G Rn G R (157) f " G . GH Rn H A straightforward calculation gives RC RC "C#(#m夹#C !Cj G G , Rn Rn G G 夹 RC I RM Rj # CI (m夹 !M夹) F !CI (m夹 !M夹) !u ) F , F , Rn Rn Rn G G G
(158)
and
R f " G GH Rn j * H
RC RC RC m夹 Rm夹 , # #C "C#4C # , R E Rn R R G H R RC I RCI RCI F #CI F (m夹 !M夹) F #2CI # (m夹 !M夹) (m夹 !M夹) , , F R , F R Rn R H RCI RM夹 RM夹 R RM夹 F (m夹 !M夹) ! 2CI !CI (m夹 !M夹)!CI (m夹 !M夹) F R F R Rn F , , , R R H k Rj ! C! G ) (159) E Rn G H with E "(k#m夹,. Note that C , CI , and M夹 are functions of G G F \ "u !u ) j (160) in the mean "eld approximation. In arriving at (159), we have used the observation that in the limit jP0, we have
Ru P0 , Rn G Ru 1 Rj Rj P ) , Rn Rn Rn Rn G H G H
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Ru 1 Rj P , Rn Rn G G R\ P1 , Rn G 1 Rj Rj R\ P! ) Rn Rn Rn Rn H G G H and that if f is taken to be a function of the expectation value of \ , then as jP0, we have Rf R\ Rf " R\ Rn Rn G G Rf P R
(161)
Rf R\ R\ Rf R\ Rf " ) # R\ Rn Rn R\ Rn Rn Rn Rn H G G H G H Rf 1 Rf Rj Rj P ! ) . R R Rn Rn G H In the absence of the baryon current, j"0, the quantities Rm夹 /Rn and Rj/Rn simplify to , H H 夹 夹 夹 RmH RM /R!2C (RC /R) n m /E !Cm /E F , H F F J J , J ," 1#C n k/E Rn F J J J J H and Rj k /E H H " . Rn 1#(C! /) n (k#m夹)/E , J H J J J Writing in a standard way
(162)
(163)
(164)
d k )k f "(2l#1) P G H f (k "k "k ) , (165) J J GH G H $ 4 k $ we see that the last term in (159) contributes to f and the sum of the rest at the Fermi surface (i.e. k "k ) to f . So H $ k E 3E F , $ $ f " $ f 2 k $ RC m夹 Rm夹 RC RC 3E #C , # (166) " $ C#4C # , R R R k E Rn $ $ H RC I RCI RC I R F #2CI F (m夹 !M夹) F #CI # (m夹 !M夹) (m夹 !M夹) F F , , , R R R Rn H RM夹 RM夹 R 夹 RM夹 RCI F (m夹 !M夹) !CI (m !M夹)!CI (m夹 !M夹) ! 2CI , F R Rn , F , F R R R H
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and k E F , $ $f 2 3(C! /) "! . E #(C! /) $
(167)
5.5.2. Some relations for relativistic Fermi-liquid Here we bridge model (134) to relativistic Fermi-liquid theory. For it, we will show that the thermodynamic properties of any model like (134), which has density-dependent parameters and is Walecka-type, can be described in terms of relativistic Landau parameters derived from the mean-"eld approximation of the model in the same way as in Section 2.3. First let us calculate the compression modulus K de"ned by K,9
RE( j"0) . R
(168)
It comes out to be
3k RC RC RC m夹 Rm夹 #C , # K" $ #9 C#4C # , R R E R E R $ $ RCI RCI RCI R F #CI F #2CI F (m夹 !M夹) (m夹 !M夹) #(m夹 !M夹) , , F R F R R R ,
(169)
RM夹 RM夹 RM夹 R RCI F (m夹 !M夹) !CI (m夹 !M夹)!CI (m夹 !M夹) . ! 2CI , F R R , F , F R R R Comparing (166) and (169), we obtain Eq. (29): 3k K" $ (1#F ) . E $ In our model
(170)
Rk G (171) "E ,(k#m夹 . , $ R $ $ * G II$ It is veri"ed that our model satis"es the relativistic Landau Fermi-liquid formula for the compression modulus (29). As shown in Section 2.3, the relativistic Landau liquid satis"es the mass relation (40): k
Rk G "(1#F /3) . $ R G II$ * One can see from Eqs. (153), (167), and (171) that (40) is satis"ed exactly in our model. k
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And lastly the relativistic relation for the "rst sound velocity (46) is satis"ed automatically since (29) and (40) are satis"ed in our model. So all the relativistic Landau Fermi-liquid relations in Section 2.3 are satis"ed with density-dependent Walecka-type models like (134). 5.5.3. Discussions The crucial question is really how to understand the scaling masses and constants as one varies temperature and density as considered in [14]. If one takes the basic assumption that the chiral Lagrangian in the mean "eld with BR scaling parameters corresponds to Landau's Fermi-liquid "xed point theory, then one should consider "rst "xing the Fermi momentum k and let $ renormalization group #ow come to the "xed points of the e!ective mass m夹 for the nucleon and * Landau parameters F [8]. In this case, the scaling quantities would seem to be dependent upon /k , not on the "elds entering into the e!ective Lagrangian. This paper however shows that if one $ wants to approach the Fermi-liquid "xed point theory starting from an e!ective chiral Lagrangian of QCD, it is necessary to take into account the fact that the scaling arises from the e!ect of multi-Fermi interactions "guring in chiral Lagrangians as implied by chiral perturbation theory described in [5]. This is probably due to the fact that we are dealing with two-stage `decimationsa in the present problem } with the Fermi surface formed from a chiral Lagrangian as a nontopological soliton (i.e., `chiral liquida [38]) } in contrast to condensed matter situations where one starts ab initio with the Fermi surface without worrying about how the Fermi surface is formed. Our result suggests that there will be a duality in describing processes manifesting the scaling behavior. In other words, the change of `vacuuma by density exploited in [14] could equally be represented by a certain (possibly in"nite) set of interactions among hadrons } e.g., four- and higher-Fermi terms in chiral Lagrangians } canonically taken into account in many-body theories starting from the usual matter-free vacuum. A notable evidence may be found in the two plausible explanations of the low mass enhancement in CERES dilepton yields in terms of scaling vector meson masses [17] and in terms of hadronic interactions giving rise to increased widths [63,83]. Recently, the relation between two descriptions are discussed by Brown et al. [65] and also by Kim et al. [66]. It is argued that the description based on the reaction dynamics and on the meson spectral function should be reliable at low density where the e!ective degrees of freedom are hadrons and have no contradiction with the description on BR scaling. There should however be a `crossovera region at higher density at which BR scaling will become more e$cient or we should go over to constituent quarks. How to relate the two description in the `crossovera density regime remains an open problem. In discussing the properties of dense matter, such as the BR scaling of masses and coupling constants, e.g., f 夹 , we have been using a Lagrangian which preserves Lorentz invariance. This L seems to be at odds with the fact that the medium breaks Lorentz symmetry. One would expect, for instance, that the space and time components of a current would be characterized by di!erent constants. Speci"cally such quantities as g , f , etc. would be di!erent if they were associated with L the space component or time component of the axial current. So a possible question is: How is the medium-induced symmetry breaking accommodated in the formalism which will be discussed in the next section? Section 5.3 and this section provide the answer to this question. Here the argument is given in an exact parallel to Walecka mean "eld theory of nuclear matter. One writes an e!ective Lagrangian with all symmetries of QCD which in the mean-"eld de"nes the parameters relevant to the state of
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matter with density. The parameters that become constants (masses, coupling constants, etc.) at given density are actually functionals of chiral invariant bilinears in the nucleon "elds. When the scalar "eld and the bilinear R , where is the nucleon "eld, develop a non-vanishing expectation value Lorentz invariance is broken and the time and space components of a nuclear current pick up di!erent constants. This is how, for instance, the Gamow}Teller constant g measured in the space component of the axial current is quenched in medium while the axial charge measured in the axial charge transitions is enhanced as described in the next chapter. If one were to calculate the pion decay constant in medium, one would also "nd that the quantity measured in the space component is di!erent from the time component. The way Lorentz-invariant Lagrangians "gure in nuclear physics is in some sense similar to what happens in condensed matter physics. 5.6. Mesons in medium It should be recalled that we extracted the scaling parameter from the in-medium property of the vector mesons. Here we will present evidences for the predicted scaling in the meson masses. There are some preliminary experimental indications for decrease in matter of the meson mass in recent nuclear experiments [84,85] but we expect more de"nitive results from future experiments at GSI and other laboratories. In fact, this is currently a hot issue in connection with the recent dilepton data coming from relativistic heavy ion experiments at CERN (European Organization for Nuclear Research). When heavy mesons such as the vector mesons , and the scalar are reinstated in the chiral Lagrangian, then the mass parameters of those particles in the Lagrangian, when written in a chirally invariant way, are supposed to appear with star and are assumed to scale according to Eq. (112). The question is: What is the physical role of these mass parameters? If we assume that the mesons behave also like quasiparticles, that is, like weakly interacting particles with the `dropping masses,a then physical observables will be principally dictated by the tree diagrams of those particles endowed with the scaling masses. In this case, the masses "guring in the Lagrangian could be identi"ed in some sense as `e!ectivea masses of the particles in the matter. This line of reasoning was used in the work of Li et al. [17] to interpret the low mass enhancement of the CERES data [15]. As discussed in Section 5.2, this treatment is consistent with an e!ective Lagrangian which in the mean "eld approximation yields the nuclear matter ground state as well as #uctuations around the ground state. The parameters of the theory, as well as their density dependence are determined by the properties of the ground state. The work of this section shows that this scheme is internally consistent. However, we emphasize that the scaling we have established is for the mesons that are highly o!-shell and it may not be applied to mesons that are near on-shell without further corrections (e.g., widths, etc.). Suppose one probes the propagation of an meson in nuclear medium as in HADES (High Acceptance Di-Electron Spectrometer) or TJNAF (Thomas Je!erson National Accelerator Facility) experiments, say through dilepton production. The 's will decay primarily outside of the nuclear medium, but let us suppose that experimental conditions are chosen so that the leptons from the decaying inside dense matter can be detected. See [86] for discussions on this issue. The question is whether the dileptons will probe the BR-scaled mass or the quantity given by (125). The behavior of the mass would di!er drastically in the two scenarios. A straightforward application
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of FTS1 theory would suggest that at a density : , the mass as `seena by the dileptons will increase slightly instead of decrease. Since in FTS1 theory, the vector coupling g does not scale, this means that (g夹/m夹 ) will e!ectively decrease. On the other hand, if the vector coupling constant S drops together with the mass at increasing density as in the BR scaling model, the situation could be quite di!erent, particularly if dileptons are produced at a density &3 as in the CERES experiments: The will then be expected to BR-scale up to the phase transition. It has recently been suggested [87] that at some high density, Lorentz symmetry can be spontaneously broken giving rise to light mesons as `almost Goldstonea bosons when a small explicit Lorentz symmetry breaking term via chemical potential is introduced. By introducing the term, they assume a state which is chirally symmetric (q q "0) but breaks Lorentz symmetry (qRq O0). The assumed state is metastable at ( but becomes a global minimum at ' . At ' , A A A would become light but not massless due to the explicit breaking. Such mesons could be a source of copious dileptons at some density higher than normal matter density. Thus, measuring the mass shift could be a key test of the BR scaling idea as opposed to the FTS1-type interpretations. This interesting issue will be studied in forthcoming experiments at GSI and TJNAF. It is interesting that the dropping mass is also found in a recent QCD sum rule calculation based on current correlation functions by Klingl et al. [71] who, however, do not see the dropping of the mass because of the large broadening of peak. If we can describe the meson in medium as a quasiparticle an -nuclear bound state is feasible even in light nuclei [72]. The process like Li(d,He) He is expected to be seen in GSI [88] if such a bound state exists. S 6. Fermi-liquid theory vs. chiral Lagrangian 6.1. Electromagnetic current We will here give a brief derivation of the Landau}Migdal formula for the convection current for a particle of momentum k sitting on top of the Fermi sea responding to a slowly varying electromagnetic (EM) "eld. We will then analyze it in terms of the speci"c degrees of freedom that contribute to the current. This will be followed by a description in terms of a chiral Lagrangian as discussed in [21]. This procedure will provide the link between the two approaches. 6.1.1. Landau}Migdal formula for the convection current Following Landau's original reasoning adapted by Migdal to nuclear systems, we start with the convection current given by
dp 1 ( )n (1# ) , (172) J" N N N (2) 2 N O where the sum goes over the spin and isospin which in spin- and isospin-saturated systems may be written as a trace over the and operators. More precisely, this is a matrix element of the current operator corresponding to the response to the EM "eld of a nucleon (proton or neutron) sitting on top of the Fermi sea. The sum over spin and isospin and the momentum integral go over all occupied states up to the valence particle. What we want is a current operator and it is deduced
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after the calculation is completed. One can of course work directly with the operator but the result is the same. We consider a variation of the distribution function from that of an equilibrium state n "n# n , (173) N N N where the superscript 0 refers to equilibrium. The variation of the distribution function induces a variation of the quasiparticle energy "# . N N N In the equilibrium state, the current is zero by symmetry, so we have
(174)
dp 1 (( ) n #( )n) (1# ), J" N N N N N N (2 ) 2 N O dp 1 (( ) n !( n ) )) (1# ) (175) " N N N 2 (2 ) N N N N O to linear order in the variation. We consider a nucleon added at the Fermi surface of a system in its ground state. Then 1 1$
n " ( p!k) N < 2
(176)
and p n "! (p!k ) (177) N N $ k $ where k with k"k is the momentum of the quasiparticle. The modi"cation of the quasiparticle $ energies due to the additional particle is given by
dp
" f n . (178) N (2 ) N NY NYNYOY NY OY Combining (2), (175), (176) and (178), one "nds that the "rst term of (175) gives the operator k 1# , J" 夹 2 m * where k is taken to be at the Fermi surface. The second term yields
k FI #FI
J" J # J " Q M 6
,
(179)
(180)
where k 1F ,
J " 夹 Q m 2 3 * k F k F !F k F .
J " 夹 " 夹 # 夹 m 2 3 3 m 2 3 m 2 * * *
(181) (182)
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For convenience, let us de"ne FI ,(M/m夹)F * J J with analogous de"nitions of FI , etc. It gives another representation of (41) J M FI "1! . m夹 3 * Putting everything together we recover the well-known result of Migdal [30,32]
k 1# 1 k # (FI !FI ) , J" g " J 2 M 6 M
(183)
(184)
(185)
where 1# # g g" J J 2
(186)
is the orbital gyromagnetic ratio and
g "(FI !FI ) . (187) J Thus, the renormalization of g is purely isovector. This is due to Galilean invariance, which implies J a cancellation in the isoscalar channel. We have derived Migdal's result using standard Fermi-liquid theory arguments. This result can also be obtained [89] by using the Ward identity, which follows from gauge invariance of the electro-magnetic interaction. This is of course physically equivalent to the above formulation. We shall now identify speci"c hadronic contributions to the current (185) in two ways: the Fermi-liquid theory approach and the chiral Lagrangian approach. 6.1.2. Pionic contribution In Fermi-liquid theory approach, all we need to do is to compute the Landau parameter F from the pion exchange. The one-pion-exchange contribution to the quasiparticle interaction is
q 3! ) 1 1 f S (q( )# (3! ) ) , (188) " fpLU NO pYNYOY 3 m q#m 2 2 L L where q"p!p and f"g (m /2M)+1. In a relativistic formulation sketched in Appendix B, L,, L we can Fierz the one-pion exchange. Done in this way, the Fierzed scalar channel is canceled by a part of the vector channel and the remaining vector channel makes a natural contribution to the pionic piece of F . The one-pion-exchange contribution to the Landau parameter relevant for the convection current is 3f m夹 F ( ) *I , "!F ( )"! 8k 3 $
The de"nition of m夹 is in (7). *
(189)
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where
m 4k x "!2# 1# L ln 1# $ . (190) 2k m 1!x#m/2k \ $ L L $ Note that F ( ) satis"es 3m夹 d (p) L (191) F ( )"! * dp k $ NI $ with one-pion-exchange Fock contribution to the self-energy (p) and includes the higher-order contribution in m夹. Thus, from Eq. (187), the one-pion-exchange contribution to the gyromagnetic * ratio is I "
dx
M f I . (192)
gL" J k 4 $ In the next subsubsection, we include contributions also from other degrees of freedom. Let us obtain the convection current from a chiral Lagrangian and compare it with the results given above. In absence of other meson degrees of freedom, we can simply calculate Feynman diagrams given by a chiral Lagrangian de"ned in matter-free space. Non-perturbative e!ects due to the presence of heavy mesons introduce a subtlety that will be treated below. In the leading chiral order, there is the single-particle contribution (Fig. 10a) which for a particle on the Fermi surface with the momentum k is given by J
U
k 1# . " M 2
(193)
Note that the nucleon mass appearing in (193) is the free-space mass M as it appears in the Lagrangian, not the e!ective mass m夹 that enters in the Fermi-liquid approach, (179). To the * next-to-leading order, we have two soft-pion terms as discussed in [5,90,91,6]. We should recall a well-known caveat here discussed already in [91]. If one were to blindly calculate the convection current coming from Fig. 10b, there would be a gauge non-invariant term that is present because the hole line is o!-shell. Fig. 10c contains also a gauge non-invariant term which is exactly the same as in Fig. 10b but with an opposite sign, so in the sum of the two graphs, the two cancel exactly so that only the gauge-invariant term survives. Of course, we now know that the o!-shell dependence is not physical and could be removed by "eld rede"nition ab initio. To the convection current we need, only Fig. 10b contributes k 1 k f I " (FI ( )!FI ( )) . (194) " M6 k 4 $ We should emphasize that the Landau parameters FI and FI are entirely "xed by a chiral e!ective Lagrangian for any density. The sum of (193) and (194) agrees precisely with the Fermi-liquid theory result (185) and (189). This formula "rst derived in [92] in connection with the Landau}Migdal parameter is of course the J
U
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Fig. 10. Feynman diagrams contributing to the EM convection current in e!ective chiral Lagrangian "eld theory. (a) Single-particle term and (b, c) the next-to-leading chiral order pion-exchange current term. (c) Does not contribute to the convection current; it renormalizes the spin gyromagnetic ratio.
same as the Miyazawa formula [93] derived nearly half a century ago. Note the remarkable simplicity in the derivation starting from a chiral Lagrangian. However, we should caution that there are some non-trivial assumptions to go with the validity of the formula. As we will see shortly, we will not have this luxury of simplicity when other degrees of freedom enter. 6.1.3. Vector meson contributions and BR scaling So far we have computed only the pion contribution to g . In nuclear physics, more massive J degrees of freedom such as the vector mesons and of mass 700&800 MeV and the scalar meson of mass 600&700 MeV play an important role. When integrated out from the chiral Lagrangian, they give rise to e!ective four-Fermion interactions: C C C L " ( (NM N)! S (NM N)! M (NM N)#2 , I I 2 2 2
(195)
where the coe$cients C's can be identi"ed with g C " + with M", , . (196) + m + For the moment, we make no distinction as to whether one is taking into account BR scaling or not. For the Fermi-liquid approach, this is not relevant since the parameters are not calculated. However with chiral Lagrangians, we will specify the scaling which is essential. Such interaction terms are `irrelevanta in the renormalization group #ow sense but can make crucial contributions by becoming `marginala in some particular kinematic situation. A detailed discussion of this point can be found in [8]. The e!ective four-Fermion interactions play a key role in stabilizing the Fermi-liquid state and leads to the "xed points for the Landau parameters. (The other "xed-point quantity, i.e. the e!ective mass, is put in by "at to keep the density "xed.) In the two-nucleon systems studied in [7], they enter into the next-to-leading order term of the potential, which is crucial in providing the cut-o! independence found for cut-o! masses 9m . L
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347
Again, it su$ces to compute the Landau parameters coming from the velocity-dependent part of heavy meson exchanges in the Fermi-liquid theory approach. We treat the e!ective four-Fermion interaction (195) in the Hartree approximation. Then the only velocity-dependent contributions are due to the current couplings mediated by and exchanges. The corresponding contributions to the Landau parameters are 2k $ F ()"!C S M
(197)
2k F ()"!C $ . M M
(198)
and
The derivations of relativistic F () and F () are shown in Appendix C. Now, the calculation of the convection current and the nucleon e!ective mass with interaction (195) in the Landau method goes through the same way as in the case of the pion. The net result is just Eq. (185) including the contribution of the contact interactions (197,198), i.e., FI "FI ( )#FI () ,
(199)
FI "FI ( )#FI () .
(200)
Similarly, the nucleon e!ective mass is determined by (41) with F "F ( )#F () .
(201)
In chiral Lagrangian approach, the most e$cient way to bring in the vector mesons into the chiral Lagrangian is to implement BR scaling in the parameters of the Lagrangian. We shall take the masses of the relevant degrees of freedom to scale in the manner of BR as (128). Note again that M夹 is a BR scaling nucleon mass which will turn out to be di!erent from the Landau e!ective mass m夹 [21]. For our purpose, it is more convenient to integrate out the vector and scalar "elds and * employ the resulting four-Fermi interactions (195). The coupling coe$cients are modi"ed compared to Eq. (196), because the meson masses are replaced by e!ective ones: g C " 夹+ + m +
with M", , .
(202)
The coupling constants may also scale [22] but we omit their density dependence for the moment. The "rst thing we need is the relation between the BR scaling factor which was proposed in [14] to re#ect the quark condensate in the presence of matter and the contribution to the Landau parameter F from the isoscalar vector () meson. For this we "rst calculate the Landau e!ective mass m夹 in the presence of the pion and "elds [21] ,
m夹 \ 1 1 * "1# (F ()#F ( ))" 1! (FI ()#FI ( )) . M 3 3
(203)
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Next, we compute the nucleon self-energy using the chiral Lagrangian. Given the single quasiparticle energy p " 夹 #C NRN # (p) , N M S L
(204)
we get the e!ective mass as in [21]
m夹 d \ \ k 1 *" $ " \! FI ( ) (205) M m dp N 3 $ NI , from Eqs. (128), (183), and (191). Comparing (203) and (205), we obtain the important result FI ()"3(1!\) . (206) This is an intriguing relation. It shows that the BR factor, which was originally proposed as a precursor manifestation of the chiral phase transition characterized by the vanishing of the quark condensate at the critical point [14], is intimately related (at least up to + ) to the Landau parameter F , which describes the quasiparticle interaction in a particular channel. We believe that the BR factor can be computed by QCD sum rule methods or obtained from current algebra relations such as the GMOR relation evaluated in medium. As was shown in [21], Eq. (206) implies that the BR factor governs in some, perhaps, intricate way low-energy nuclear dynamics. The equivalence discussed above between the physics of the vacuum property and that of the quasiparticle interaction F due to the massive vector-meson degree of freedom suggests that the `bottom-upa approach } going up in density with a Lagrangian whose parameters are "xed at zero density } and the `top-downa approach } extrapolating with a Lagrangian whose parameters are "xed at some high density } can be made equivalent at some intermediate point. If this is so in the hot and dense regime probed by relativistic heavy ion collisions, then the CERES data should also be understandable in terms of hadronic interactions without making reference to QCD variables. Because of the complexity of hadronic descriptions, it will be di$cult to relate the two directly but the recent alternative explanation of the CERES data in terms of `melting of the vector mesonsa inside nuclear matter manifested in the increased width of the mesons due to hadronic interactions [83] may be an indication for a possible `duala description at low density between what is given in QCD variables (e.g., quark condensates) and what is given in hadronic variables (e.g., the Landau parameter), somewhat reminiscent of the quark-hadron duality in heavy}light-quark systems [94]. A possible mechanism that could make the link between the two descriptions was suggested recently by Brown et al. [65] and by Kim et al. [66]. In the presence of the BR scaling, a non-interacting nucleon in the chiral Lagrangian propagates with a mass M夹, not the free-space mass M. Thus, the single-particle current Fig. 10a is not given by (193) but instead by J
k 1# . " 夹 U M 2
(207)
Now, current (207) on its own does not carry conserved charge as long as M夹OM. This means that two-body currents are indispensable to restore charge conservation. Note that the situation is quite di!erent from the case of Fermi-liquid theory. In the latter case, the quasiparticle propagates
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Fig. 11. (a) Feynman diagram contributing to the EM convection current from four-Fermi interactions corresponding to the and channel (contact interaction indicated by the blob) in e!ective chiral Lagrangian-"eld theory. The NM denotes the anti-nucleon state that is given in the chiral Lagrangian as a 1/M correction and the one without arrow is a Pauli-blocked or occupied state. (b) The equivalent graph in heavy-fermion formalism with the anti-nucleon line shrunk to a point.
with the Landau e!ective mass m夹 and it is the gauge invariance that restores m夹 to M. In * * condensed matter physics, this is related to a phenomenon that the cyclotron frequency depends on the bare mass, not on Landau e!ective mass. It may be referred to as Kohn e!ect [95]. The bare mass in Kohn e!ect is restored due to the quasiparticle interactions with Galilean invariance in the same way as for the convection current in Section 6.1.1 [96]. This clearly indicates that gauge invariance is more intricate when BR scaling is implemented. Indeed if the notion of BR scaling and the associated chiral Lagrangian are to make sense, we have to recover the charge conservation from higher-order terms in the chiral Lagrangian. This constitutes a strong constraint on the theory. Let us now calculate the contributions from the pion and heavy meson degrees of freedom. The pion contributes in the same way as before, so we can carry over the previous result of Fig. 10b, k 1 (FI ( )!FI ( )) . JL " U M 6
(208)
This is of the same form as (194) obtained in the absence of BR scaling. It is in fact identical to (194) if we assume that one-pion-exchange graph does not scale in medium at least up to nuclear matter density. This assumption is supported by observations in pion-induced processes in heavy nuclei. This means that the observation that the one-pion-exchange potential does not scale implies that the constant g夹 /f 夹 remains unscaling at least up to normal nuclear matter density with non-scaling L pion mass. In what follows, we will make this assumption implicitly. The contributions from the vector meson degrees of freedom are a bit trickier. They are given by Fig. 11. Both the (isoscalar) and (isovector) channels contribute through the antiparticle intermediate state as shown in Fig. 11a. The antiparticle is explicitly indicated in the "gure. However in the heavy-fermion formalism, the backward-going anti-nucleon line should be shrunk to a point as Fig. 11b, leaving behind an explicit 1/M dependence folded with a factor of nuclear density signaling the 1/M correction in the chiral expansion. One can interpret Fig. 11a as saturating the corresponding counter term although this has to be yet veri"ed by writing the full set
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Fig. 12. Particle}hole contributions to the convection current. Here backward-going nucleon line N\ denotes a hole. These graphs vanish in the q/P0 limit.
of counter terms at the same order. These terms with Fig. 11a k 1 JS " FI () , U M 6
(209)
k 1 FI () , JM " U M 6
(210)
where FI () and FI () are given by Eqs. (197), (198) with M replaced by M夹. Their relativistic forms are given in Appendix C. The total current given by the sum of (207)}(210) precisely agrees with the Fermi-liquid theory result (185) when we take FI "FI ()#FI ( ) , (211) FI "FI ()#FI ( ) . (212) The way in which this precise agreement comes about is non-trivial. What happens is that part of the channel restores the BR-scaled mass M夹 back to the free-space mass M in the isoscalar current. (It has been known since sometime that something similar happens in the standard Walecka model (without pions and BR scaling) [97].) Thus, the leading single-particle operator combines with the sub-leading four-Fermi interaction to restore the charge conservation as required by the Ward identity. This is essentially the `back-#ow mechanisma which is an important ingredient in Fermi-liquid theory. We describe below the standard back-#ow mechanism as given in Section 2.1, adapted to nuclear systems with isospin degrees of freedom, and elucidate the connection to the results obtained with the chiral Lagrangian in this subsection. The current so constructed is valid for a process occurring very near the Fermi surface corresponding to the limit (, q)P(0, 0) where q is the spatial momentum transfer and is the energy transfer. In the diagrams considered so far (Figs. 10 and 11) the order of the limiting processes does not matter. However, the particle}hole contribution, which we illustrate in Fig. 12 with the pion contribution, does depend on the order in which q"q and approach zero. Thus, in the limit q/P0, the particle}hole contributions vanish whereas in the opposite case /qP0, they do not. This can be seen by examining the particle}hole propagator n (1!n ) n (1!n ) I>O I I I>O ! , q # ! #i q # ! !i
I I>O I I>O
(213)
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where (q , q) is the four-momentum of the external (EM) "eld. This vanishes if we set qP0 with q non-zero but its real part is non-zero if we interchange the limiting process since for q "0 we have q ) kK
(k !k) . $ !q ) k/M
(214)
Fig. 12 was computed by several authors (e.g., see [89]) and is given in the limit /qP0 by
dp 1# (2) J "! (1) ) p( (k !p) f L , (215) $ Q (2) 2 OY where f L is the spatial and spin part of the quasiparticle interaction which is (188) and (268) without Q isospin part (3! ) )/2. The isospin factor is given by the Fierz transformation:
(1) ) OY
1# 3 1 3 1 (2) " ! ) # tr[ ]! ) tr[ ] 2 4 4 4 4 OY 3 1 " ! . 2 2
(216)
Note that the factor comes from f and from f . In the limit that we are concerned with L L (i.e. ¹"0 and /qP0), the particle}hole contribution to the current is 1 J "! kK k ( f #f ) 3 $
k FI ( )#FI ( ) "! 6 M
.
(217)
This holds, in general, regardless of what is being exchanged as long as the exchanged particle has the right quantum numbers. Contributions from heavy-meson exchanges are calculated in a similar way. Adding the particle}hole contribution (217) to the Fermi-liquid result (185) we obtain the current of a dressed or localized quasiparticle
k 1# . " 夹 (218) m 2 * Note that J precisely cancels J, Eq. (180). The current J is the total current carried by the /. wave packet of a localized quasiparticle with group velocity * "k/m夹. However, the physical * $ situation corresponds to homogeneous (plane wave) quasiparticle excitations. The current carried by a localized quasiparticle equals that of a homogeneous quasiparticle excitation modi"ed by the so called back-#ow current [98]. The back-#ow contribution (J !J ) is just the particle}hole /. *+ polarization current in the /qP0 limit, Eq. (217). J
/.
6.1.4. Phenomenological test It is not obvious that the e!ective nucleon mass computed in the chiral Lagrangian approach is directly connected to a measurable quantity although quasielastic electron scattering from nuclei
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does probe some kind of e!ective nucleon mass and Walecka model describes such a process in terms of an e!ective mass. To the extent that the bulk of m夹 is related to the condensate through , BR scaling as we can see in (205), the e!ective mass in the chiral Lagrangian can be related to the quantity calculated in the QCD sum rule approach for in-medium hadron masses. In BR scaling, the parameter is related to the scaling of the vector meson () mass. There are several QCD sum rule calculations for the meson in-medium mass starting with [99]. The most recent one which closely agrees with the GMOR formula in medium for the pion decay constant f夹 (see Eq. (112)) is L the one by Jin and Leinweber [73]: ( )"0.78$0.08 . (219) We shall take this value in what follows but one should be aware of the possibility that this value is not quite "rm. A caveat to this result was recently discussed by Klingl et al. [71], who show that the QCD sum rule can be saturated without the mass shifting downward by increasing the vector meson width in medium. For a discussion of the empirical constraints on the in-medium widths of vector mesons, see Friman [100]. Given this, we can compute m夹 using (205) for nuclear matter density since the pionic contribu, tion FI ( ) is known. One "nds [21] m夹 , (" )+0.70 . (220) M This can be tested in an indirect way by looking at certain magnetic response functions of nuclei as described below. An additional evidence comes from QCD sum rule calculations. Again there are caveats in the QCD sum rule calculation for the nucleon mass even in free-space and certainly more so in medium. Nevertheless, the most recent result by Furnstahl et al. [79] is rather close to the prediction (220):
m夹 ( ) , M
"0.69> . \
/!" If one writes the gyromagnetic ratio g as J 1# # g , g" J J 2
(221)
(222)
then the chiral Lagrangian prediction is
g "(FI !FI ) "[\!1!FI ( )] . (223) J In writing the second equality we have used (189), (206) and the nonet relation FI ()"FI ()/9. At nuclear matter density, we get, using (219),
g ( )+0.23 . J This agrees with the value extracted from the dipole sum rule in Bi [101],
g"0.23$0.03 J
(224)
(225)
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and agrees roughly with magnetic moment data in heavy nuclei. Nuclear magnetic moments are complicated due to conventional nuclear e!ects. To make a meaningful comparison, one would have to extract all `triviala nuclear e!ects and this operation brings in inestimable uncertainties. It should be stressed that the gyromagnetic ratio provides a test for the scaling nucleon mass at + . It also gives a check of the relation between the baryon property on the left-hand side of Eq. (206) and the meson property on the right-hand side. Instead of using (219) as an input to calculate g , we could take the experimental value (225) to determine, using (223), the BR scaling J factor at + . We would of course get (219), a value which is consistent with what one obtains in the QCD sum rule calculation and also in the in-medium GMOR relation. Recall that because of the pions which provide (perturbative) non-local interactions to the Landau interaction, the Landau mass for the nucleon scales di!erently from that of the vector mesons (see (112) and (205)). This di!erence is manifested in the skyrmion description by the fact that the coe$cient of the Skyrme quartic term must also scale. In the original discussion of the scaling based on the quark condensates using the trace anomaly [14], the Skyrme quartic term was scale-invariant and hence the corresponding g夹 was non-scaling. So the scaling implied by (205) indicates that the scaling of g夹 is associated with the pionic degrees of freedom. This is consistent with the description based on the Landau}Migdal g interaction between a nucleon and a reson ance [102}104] and also with the QCD sum rule description of Drukarev and Levin [105], who attribute about 80% of the quenching to the !N e!ect. If we equate the skyrmion relation [21,75]
g夹 m夹 ," M g to (205), we get
(226)
\ g夹 1 1 " 1# F ( ) " 1! FI ( ) . g 3 3 At nuclear matter density, this predicts g夹 ( )+1
(227)
(228)
and g夹 f夹 + L " . (229) g f L We will use this relation in deriving (254). It should be emphasized that this relation, being unrelated to the vacuum property, cannot hold beyond + . Indeed as suggested by the scaling given in [14], g夹 ()+ g with a constant independent of density, for 9 . It would be a good approximation to set g夹 equal to 1 for 9 . Since one expects that when chiral symmetry is restored, g will approach 1, it may be thought that the evidence for g夹 +1 in nuclei is directly connected with chiral restoration. This is not really the case. Neither in the skyrmion picture nor in QCD sum rules is the quenching of g simply related to a precursor behavior of chiral restoration. This does not however mean that the
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Fig. 13. Examples of the second-order core polarization contribution to Gamow}Teller transition. Downward nucleon line denote a hole. (a) 2p1h (b) 3p2h. Fig. 14. The resonance-exchange graph in [104] for four-Fermi contact interaction contribution. Its Hartree contribution decreases axial vector coupling constant in medium.
quenching of g carries no information on the chiral symmetry restoration. As suggested recently by Chanfray et al. [106], if one were to compute all pion-exchange-current graphs at one-loop order that contribute to the in-medium g , the e!ect of medium-induced change in the quark condesate would be largely accounted for. In a way, this argument is akin to that for the Cheshire}Cat (or dual) phenomenon we are advocating in the description of the quark condensate in terms of quasiparticles. Another issue that has generated lots of debate in the past and yet remains confusing is the interpretation of an e!ective constant g+1 actually observed in medium and heavy nuclei. The debate has been whether the observed `quenchinga is due to `core polarizationsa or ` -hole e!ecta (or other non-standard mechanisms). Our view is that in the presence of BR scaling, both are involved. In light nuclei in which the Gamow}Teller transition takes place in low density, the tensor force is mainly operative and the core polarization (i.e., multiparticle}multihole con"gurations) mediated by this tensor force is expected to do most of the quenching, while the -hole e!ect directly proportional to density is largely suppressed. The typical diagrammatic representations for the second order core polarization is shown in Fig. 13. In heavy nuclei, on the other hand, the tensor force is quenched due to BR scaling, rendering the core polarization mechanism ine!ective while the increased density makes the -hole e!ect dominant. Recently, Park et al. [104] applied chiral perturbation theory to calculate g夹 at normal nuclear matter density . The resonance-exchange graphs that contribute are shown in Figs. 14 and 15 and the Landau}Migdal g e!ect contains both. The Hartree contribution from Fig. 14, i.e. -hole e!ect, makes g夹 quenched and the Fock contribution from Fig. 15 enhances g夹 . The magnitude of quenching is two or three times larger than the enhancement. What is seen in nature, in our view, is the interplay between these two. The second form of (227) shows that the quenching of g in matter is quite complex, both the pionic e!ect and the vacuum condensate e!ect being confounded together. Again for the reason
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Fig. 15. The resonance-exchange graphs in [104] for one-pion-exchange contribution. Their Fock contributions enhance the axial vector coupling constant with incorporating the short-range correlation between nucleons.
given above, this relation cannot be extended beyond the regime with + . We have no understanding of how this formula and the -hole mechanism of [102,103] are related. Our e!ort thus far has met with no success. Understanding the connection would presumably require the short-distance physics implied by both the Landau}Migdal g interaction and the Skyrme quartic term (which is known to be more than just what results when the meson is integrated out of the chiral Lagrangian). 6.2. Axial charge transition No one has yet derived the analogue to (185) for the axial current. Attempts using axial Ward identities in analogy to the EM case have not met with success [107]. The di$culty has presumably to do with the role of the Gold-stone bosons in nuclear matter which is not well understood. In this subsection, we analyze the expression for the axial charge operator obtained by a straightforward application of the Fermi-liquid theory arguments of Landau and Migdal and compare this expression with that obtained directly from the chiral Lagrangian using current algebra. For the vector current, we found precise agreement between the two approaches. 6.2.1. Applying Landau quasiparticle argument The obvious thing to do is to simply mimic the steps used for the vector current to deduce a Landau}Migdal expression for the axial charge operator. We use both methods developed above and "nd that they give the same result. In free space, the axial charge operator nonrelativistically is & ) * where *"k/M is the velocity. In the in"nite momentum frame, it is the relativistic invariant helicity ) ( . It is thus tempting to assume that near the Fermi surface, the axial charge operator for a local quasiparticle in a wave packet moving with the group velocity * "k/m夹 is simply & ) * . * $ $ This suggests that we take the axial charge operator for a localized quasiparticle to have
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the form ) k G AG "g . (230) /. m夹 2 * As in the vector current case, we take (230) to be the /qP0 limit of the axial charge operator. The next step is to compute the particle}hole contribution to Fig. 12 (with the vector current replaced by the axial current) in the /qP0 limit. A simple calculation gives ) k G AG "!g 夹 m 2 *
(231)
with f k m夹
" $ * (I !I ) 4m L where I was de"ned in (190) and 4k 1 I " "ln 1# $ . dx m 1!x#m/2k \ L L $ In an exact parallel to the procedure used for the vector current, we take the di!erence
AG !AG /. and identify it with the corresponding `Landau axial chargea (LAC):
(232)
(233)
(234)
) k G (1# ) . (235) AG "g * ! m夹 2 * Let us now rederive (235) with an argument analogous to that proven to be powerful for the convection current. We shall do the calculation using the pion exchange only but the argument goes through when the contact interaction (195) is included. We begin by assuming that the axial charge } in analogy to (172) for the convection current } takes the form
dp G ) ( )n , AG "g N N N2 (2) NO where n and are 2;2 matrices with matrix elements N N [n (r, t)] "n (r, t) #s (r, t) ) , N ??Y N ??Y N ??Y and
(236)
(237)
[ (r, t)] " (r, t) # (r, t) ) N ??Y N ??Y N ??Y
(238)
1 s (r, t)" [n (r, t)] . N ??Y N ?Y? 2 ??Y
(239)
with
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In general, n"4 in the spin}isospin space. But without loss of generality, we could con"ne ourselves to n"2 in the spin space with the isospin operator explicited as in Eq. (236). Then upon linearizing, we obtain from (236) AG "g NO where
n
dp G ( ) ( ) n ! ) ( n ) ) #2 , N N NNO N N NNO 2 (2)
1# G 1 " (p!k) NNO < 2 2
(240)
(241)
and
dp
" f
n (242) NNO (2) NNO NYNYOY NYNYOY NY OY in analogy with (178). Eq. (240) is justi"ed if the density of polarized spins is much less than the total density of particles (assumed to hold here). The "rst term of (240) with (241) yields the quasiparticle charge operator ) k G AG "g 夹 , (243) /. m 2 * while the second term represents the polarization of the medium, due to the pion}exchange interaction (188) ) k G (244)
AG "g 夹 . m 2 * The sum of (243) and (244) agrees precisely with the Landau charge (235). It is not di$cult to take into account the full Landau}Migdal interactions (2) which includes the one-pion-exchange interaction as well as other contributions to the quasiparticle interaction. Thus, the general expression is obtained by making the replacement 10 4 2
P G ! H # H ! H 3 3 15
(245)
in (244). This combination of Fermi-liquid parameters corresponds to a l"l"1, J"0 distortion of the Fermi sea [33]. We will see later that the result obtained with the naive Landau argument may not be the whole story, since the one-pion-exchange contribution disagrees, though by a small amount, with the chiral Lagrangian prediction derived below. 6.2.2. Chiral Lagrangian prediction We now calculate the axial charge using our chiral Lagrangian that reproduced the Landau}Migdal formula for the convection current. Consider "rst only the pion-exchange contribution. In this case, we can take the unperturbed nucleon propagator to carry the free space mass M.
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The single-particle transition operator corresponding to Fig. 10a is given by ) k G AG "g . U M 2
(246)
There is no contribution of the type of Fig. 10b because of the (G-)parity conservation. The only contribution to the two-body current comes from Fig. 10c and is of the form [108] ) k G AG U "g
M 2
(247)
with
m f k M $ I !I ! I . (248)
" 2k 2g m $ The factor (1/g ) in (248) arose from replacing 1/f by g /g M using the Goldberger}Treiman ,, relation. Now consider what happens when the vector degrees of freedom are taken into account. Within the approximation adopted, the only thing that needs be done is to implement the BR scaling. The irect intervention of the vector mesons and in the axial-charge operator is suppressed by the chiral counting, so they will be ignored here. This means that in the single-particle charge operator, all that one has to do is to replace M by M夹"M in (246): ) k G AG U "g M 2
(249)
and that in the two-body charge operator (247), f should be replaced by f and M by M: ) k G AG U "g
. (250) M 2 In the two-body operator, there is a factor (g /f ) coming from the NN vertex which as mentioned before, is assumed to be non-scaling at least up to nuclear matter density [109], in consistency with the observation that the pion-exchange current does not scale in medium. The total predicted by the chiral Lagrangian (modulo higher-order corrections) is then g
) k G (1# ) , M 2
(251)
which di!ers from the charge operator obtained by Landau method (235). 6.2.3. Comparison between vector and axial current An immediate question (to which we have no convincing answer) is whether or not the di!erence between the two approaches } the Fermi-liquid vs. the chiral Lagrangian } is genuine or a defect in either or both of the approaches. One possible cause of the di!erence could be that both the assumed localized quasiparticle charge, Eq. (230), and the e!ective axial charge, Eq. (236), are
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incomplete. We have looked for possible additional terms that could contribute but we have been unable to "nd them. So while not ruling out this possibility, we turn to the possibility that the di!erence is genuine. It is a well-known fact that the conservation of the vector current assures that the EM charge or the weak vector charge is g "1 but the conservation of the axial vector charge does not constrain 4 the value of the axial charge g , that is, g can be anything. This is because the axial symmetry is spontaneously broken. In the Wigner phase in which the axial symmetry would be restored, one would expect that g "1. It therefore seems that the Goldstone structure of the `vacuuma of the nuclear matter is at the origin of the di!erence. To see whether there can be basic di!erences, let us look at the e!ect of the pion "eld. The cancellation between the two-body current J (208) and J (217) leaving only a term that U 夹 夹 changes M to m in the one-body operator with a BR scaling mass, Eq. (207), in the EM case can * be understood as follows. Both terms involve the two-body interaction mediated by a pionexchange. It is obvious how this is so in the latter. To see it in the former, we note that it involves the insertion of an EM current in the propagator of the pion. Thus, the sum of the two terms corresponds to the insertion of an EM current in all internal hadronic lines of the one-pion exchange self-energy graph of the nucleon. The two-body pionic current } that together with the single-particle current preserves gauge invariance } is in turn related to the one-pion-exchange potential < . Therefore, what is calculated is essentially an e!ect of a nuclear force. Now, the point is that the density-dependent part of the sum (that is, the ones containing one hole line) } apart from a term that changes M夹 to m夹 in (207) } vanishes in the /qP0 limit. In contrast, the * cancellation between (244) and (231) in the case of the axial charge, has no corresponding interpretation. While the one-pion-exchange interaction is involved in the particle}hole term (231) and (244) cannot be interpreted as an insertion of the axial vector current into the pion propagator since such an insertion is forbidden by parity. In other words, Eq. (244) does not have a corresponding Feynman graph which can be linked to a potential. We interpret this as indicating that there is no corresponding Landau formula for the axial charge in the same sense as in the vector current case. In a chiral Lagrangian formalism, each term is associated with a Feynman diagram. As mentioned, there is no contribution to the convection current from a diagram of the type Fig. 10c (apart from a gauge non-invariant o!-shell term which cancels the counter part in Fig. 10b). Instead this diagram renormalizes the spin gyromagnetic ratio. In contrast, the corresponding diagram for the axial current does contribute to the axial charge (247). As "rst shown in [90], the contribution from Fig. 10c for both the vector current and the axial-vector current is current algebra in origin and constrained by chiral symmetry. Furthermore, it does not have a simple connection to nuclear force. While the convection current is completely constrained by gauge invariance of the EM "eld, and hence chiral invariance has little to say, both the EM spin current and the axial charge are principally dictated by the chiral symmetry. This again suggests that the Landau approach to the axial charge cannot give the complete answer even at the level of quasiparticle description. There is however a caveat here: in the Landau approach, the non-local pionic and local four-Fermion interactions (195) enter together in an intricate way as we saw in the EM case. Perhaps this is also the case in the axial charge, with an added subtlety due to the presence of Goldstone pions. It is possible that the di!erence is due to the contribution of the four-Fermion interaction term to (245) which cancels out in the limit /qP0 but contributes in the q/P0 limit. This term cannot be
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given a simple interpretation in terms of chiral Lagrangians. Amusingly, the di!erence between the results (see below) turns out to be small. 6.2.4. Numerical comparison To compare the two results, we rewrite the sum of (243) and (244), i.e., `Landau axial chargea (LAC), using (41) and (189) ) k G (1# I ) , AG "g * ! M 2
(252)
where
3m f k M I !I # I \
I " $ 2k 4m $
(253)
and the sum of (249) and (250), i.e., the `current-algebra axial chargea (CAAC), as ) k G AG "g (1# ) , ! ! M 2
(254)
where
m f k M $ I !I ! I . (255)
" 2k 2g m $ We shall compare I and for two densities " (k "1.50 m ) and " (k "1.89 m ) $ $ where is the normal nuclear matter density 0.16/fm. For numerical estimates, we take
\ (256) which gives ( )"0.78 found in QCD sum rule calculations [22]. Somewhat surprisingly, the resulting values for I and are close; they agree within 10%. For instance, at + /2, I +0.48 while +0.43 and at + , I +0.56 while +0.61. Whether this close agreement is coinciden tal or has a deep origin is not known. ()" 1#0.28
6.2.5. Test: axial charge transition in heavy nuclei The axial charge transition in heavy nuclei A(J>) B(J\)
(257)
with change of one unit of isospin ¹"1 provides a test of the axial charge operator (254) or (252). To check this, consider the Warburton ratio [110] +#! "M /M , (258) +#!
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where M is the measured matrix element for the axial charge transition and M is the theoretical single-particle matrix element. There are theoretical uncertainties in de"ning the latter, so the ratio is not an unambiguous object but what is signi"cant is Warburton's observation that in heavy nuclei, can be as large as 2: +#! & , "1.9&2.0 . (259) +#! More recent measurements } and their analyses } in di!erent nuclei [111] quantitatively con"rm this result of Warburton. To compare our theoretical prediction with the Warburton analysis, we calculate the same ratio using (254) ! !"\(1# ) . (260) +#! Formula (254) di!ers from what was obtained in [112] in that here the non-scaling in medium of the pion mass and the ratio g /f is taken into account. We believe that the scaling used in [112] (which amounted to having / in place of in (260)) is not correct. The enhancement corresponding to the `Landau formulaa (252) is obtained by replacing by
I in (260). Using the value for and at nuclear matter density, we "nd +2.1 (2.0) (261) +#! in good agreement with the experimental results of [110,111]. Here the value in parenthesis is obtained with the Landau formula (252). The di!erence between the two formulas (i.e., current algebra vs. Landau) is indeed small. This is a check of the scaling of f in combination with the scaling of the Gamow}Teller constant g in medium. 7. Summary An attempt is made and some success is obtained in this review to relate an e!ective chiral Lagrangian to an e!ective "eld theory for nuclear matter. The aim is to bridge between what we know at normal nuclear density and what can be expected under the extreme condition, relevant in neutron stars and in relativistic heavy ion collisions. Furnstahl, Serot and Tang's e!ective chiral model Lagrangian FTS1 [11], which describes successfully the phenomenology of "nite nuclei and in"nite nuclear matter, is taken to imply that an e!ective chiral Lagrangian calculated in high chiral orders corresponds to Lynn's chiral soliton with the chiral liquid structure [38] in mean "eld. This provides the ground state around which quantum #uctuations can be calculated. Note that FTS1 is simply one of the available theories that are consistent with the symmetries of QCD and successful phenomenologically. We do not imply that FTS1 is the best one can construct as an e!ective theory of nuclear matter. The scalar sector in FTS1 develops a large anomalous dimension, which is interpreted as a signal of a strong coupling situation. It is suggested that the strong coupling theory can be transformed into a weak coupling theory if the chiral Lagrangian is rewritten in terms of the parameters given by BR scaling. A simple model, whose mass parameters are BR-scaled, is constructed and is shown to describe ground state properties of nuclear matter very well with "ts comparable to the full
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FTS1 theory. The simple BR-scaled Lagrangian gives the background at any arbitrary density around which #uctuations can be calculated. Tree diagrams yield the dominant contributions. It is shown that we can make simple Walecka-type models, including our simple model. The models are thermodynamically consistent and the dependence of parameters on density are represented by the interactions of hadrons. One can also map the density-dependent model Lagrangian into relativistic Landau Fermi-liquid theory. Thus a quasiparticle picture of a strongly correlated system at densities away from the normal nuclear matter density is obtained. The BR scaling parameter has been identi"ed with a Landau Fermi-liquid parameter by means of nuclear responses to the EM convection current. The Landau e!ective mass of the nucleon m夹 is given in terms of and pion cloud, i.e. the Goldstone boson of the broken chiral * symmetry, through the Landau parameter FI . The relation between the exchange current correc tion to the orbital gyromagnetic ratio g and m夹 provides the crucial link between and FS which * J comes from the massive degree of freedom in the isoscalar vector channel dominated by the meson. The axial charge transition in heavy nuclei provides a relation between and the in-medium pion decay constant f 夹/f . These relations are found to be satis"ed very accurately and to connect physics of relativistic heavy ion collision data, e.g., dilepton data of CERES and nucleon and kaon #ow data of FOPI (4 pi multiparticle detector) and KaoS (Kaon Spectrometer), etc. to low energy spectroscopic properties, e.g., m夹, g , etc. in heavy nuclei via BR scaling. * J 8. Open issues While as an exploration our results are satisfying, there are several crucial links that remain conjectural in the work and require a lot more work. We mention some issues for future studies. E We have not yet established in a convincing way that a non-topological soliton coming from a high-order e!ective chiral Lagrangian accurately describes nuclear matter that we know of. The "rst obstacle here is that a realistic e!ective Lagrangian that contains su$ciently high-order loop corrections including non-analytic terms has not yet been constructed. Lynn's argument for the existence of such a soliton solution and identi"cation with a drop of nuclear matter is based on a highly-truncated Lagrangian (ignoring non-analytic terms). We are simply assuming that the FTS1 Lagrangian is a su$ciently realistic version (in terms of explicit vector and scalar degrees of freedom that are integrated out by Lynn) of Lynn's e!ective Lagrangian. To prove that this assumption is valid is an open problem. E We do not understand clearly the role and the origin of the anomalous dimension d +5/3 for the quarkonium scalar "eld in FTS1. It is an interesting problem how the scalar in FTS1 comes to include higher-order interactions in its anomalous scaling dimension through decimations. And our argument for interpreting the FTS1 with such large anomalous dimension as a strongcoupling theory which can be reinterpreted in terms of a weak-coupling theory expressed with BR scaling is heuristic at best and needs to be sharpened, although our results strongly indicate that it is correct. E There is also the practical question as to how far in density the predictive power of the BR-scaled e!ective Lagrangian can be pushed. In our simple numerical calculation, we used a parameterization for the scaling function () of the simple geometric form which can be valid, if at all, up to
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the normal matter density as seems to be supported by QCD sum rule and dynamical model calculations. At higher densities, the form used has no reason to be accurate. By using the empirical information coming from nucleon and kaon #ows, one could infer its structure up to, say, &3 and if our argument for kaon condensation is correct } and hence kaon condensa tion takes place at :3 , then this will be good enough to make a prediction for the critical density for kaon condensation. In calculating compact-star properties in supernovae explosions, however, the equation of state for densities considerably higher than the normal matter density, say, 95 is required. It is unlikely that this high density can be accessed within the presently employed approximations. Not only will the structure of the scaling function be more complicated but also the correlation terms that are small perturbations at normal density may no longer be so at higher densities, as pointed out by Pandharipande et al. [113], who approach the e!ect of correlations from the high-density limit. In particular, the notion of the scaling function will have to be modi"ed in such a way that it will become a non-linear function of the "elds that "gure in the process. This would alter the structure of the Lagrangian "eld theory. Furthermore, there may be a phase transition (such as spontaneously broken Lorentz symmetry, Georgi vector limit, chiral phase transition or meson condensation) lurking nearby in which case the present theory would have already broken down. These caveats will have to be carefully examined before one can extrapolate the notion of BR scaling to a high-density regime as required for a reliable calculation of the compact-star structure. How the scaling parameters extrapolate beyond normal nuclear matter density is not predicted by theory and should be deduced from lattice measurements and heavy-ion experiments that are to come. Corrections to BR scaling as massive mesons approach on-shell need be taken into account. The "t to the available CERES data indicates however that the extrapolation to higher density } perhaps up to the chiral phase transition } is at least approximately correct under the conditions that prevail in nucleus-nucleus collisions at SPS energies. How this could come about was discussed in [65]. E In addition, the behavior of g夹 at ' also deviates from our simple form. It is expected to T drop more rapidly [74]. Indeed a recent calculation [114] of kaon attraction to O(Q) in chiral perturbation theory that is highly constrained by the ensemble of on-shell kaon-nucleon data and that includes both Pauli and short-range correlations for many-body e!ects is found to give at most about 120 MeV attraction at nuclear matter density. Thus, the crucial input here is the strength of the K\-nuclear interaction in dense medium. The attraction decreases from the analysis of the K-mesic atom by Friedman et al. [115] indicating the 200 MeV attraction. If the attraction came down to 100&120 MeV as found in [114], this would give a strong constraint on the constants that enter in the four-Fermi interactions in the chiral Lagrangian. This would presumably account for the need for a dropping vector coupling g夹 required for 9 . T Moreover, Kim and Lee [116] found recently by renormalization group analysis that the coupling constant g drops as density increases. This crucial information is also expected to M,, come from on-going heavy-ion experiments. E Although we show that BR scaling parameters can be written in terms of Landau parameter via non-relativistic EM current, it remains to formulate the relativistic mapping along the line developed in Section 5.5 where thermodynamic properties of a simple BR-scaled chiral model Lagrangian in the mean "eld were shown to be consistent with relativistic Landau formula derived in Section 2.3. This work is needed to go to higher-density region. Such a work is in progress. Furthermore, quasiquarks must become relevant degrees of freedom at high density.
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Thus quasihadron liquid is shifted to quasiquark liquid as density increases. The investigation of the change in the shift may give a good way to connect the low-density physics to the higher one. E As seen in Section 6.2, it is not clear how we deal with Goldstone bosons in the scheme of Landau}Migdal approach. The Landau axial charge and current algebra axial charge are not the same, though they give similar numerical values. We do not know even whether it is possible or not that the Landau}Migdal approach can treat Goldstone boson properly. The study of it will show the way to treat the Landau Fermi-liquid theory and its scope. E Finally, one could ask more theoretical questions as to in what way our e!ective Lagrangian approach is connected to standard chiral e!ective theory, which does not concern scale symmetry and its anomaly, proper and if the theory is to be fully predictive, how one can proceed to calculate the corrections to the tree-diagram results we have obtained. The "rst issue, a rigorous derivation of BR scaling starting from an e!ective chiral action via multiple scale decimations required for the problem is yet to be formulated but the main ingredients, both theoretical and phenomenological, seem to be available. The second issue is of course closely tied with what the appropriate expansion parameter is in the theory. These matters are addressed in the paper but they are somewhat scattered all over the place and it might be helpful to summarize them here. The answers to these questions are not straightforward since there are two stages of `decimationa in the construction of our e!ective Lagrangian: the "rst is the elimination of high-energy degrees of freedom for the e!ective Lagrangian that gives rise to a soliton (i.e., chiral liquid) and here the relevant scale is the chiral symmetry breaking scale &1 GeV and the second is that given a chiral liquid which we argued can be identi"ed as the Fermi-liquid "xed point, the decimation involved here is for the excitations of scale above (and below) the Fermi surface for which the expansion is made in 1/N. As discussed in Section 2.2, 1/N&/k where $ is the cut-o! in the Fermi system. In bringing in a BR-scaled chiral Lagrangian, we are relying on chiral symmetry considerations applied to a system with a density de"ned by nuclear matter. Thus, the link to QCD proper of the e!ective theory we use for describing #uctuations around the nuclear matter ground state must be tenuous at best. As recently re-emphasized by Weinberg [37], low-energy e!ective theories need not be in one-to-one correspondence with a fundamental theory meaning that one low-energy e!ective theory could arise through decimation from several di!erent `fundamentala theories. This applies not only to theories with global symmetry but also to those with local gauge symmetry. In the present case, this aspect is more relevant since there is a change in degrees of freedom between the non-perturbative regime in which we are working and the perturbative regime in which QCD proper is operative.
Acknowledgements I am very grateful to Professor D.-P. Min and Professor M. Rho for their guidance and encouragement throughout my graduate years. Collaboration with Professors G.E. Brown and B. Friman has been a great pleasure to me. I would like to thank C.-H. Lee and R. Rapp for useful comments and discussions. This work was supported in part by the U.S. Department of Energy under DE-FG02-88ER40388, by the Korea Science and Engineering Foundation through Center for Theoretical Physics of Seoul National University, and by the Korea Ministry of Education under BSRI-98-2418.
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Appendix A. E4ect of many-body correlations on EOS In this appendix, we brie#y discuss the sensitivity of the EOS to the correlation parameters of (155) at a density ' . This is shown in Fig. 16. While the parameter sets B1, B2, B3 and B4 give more or less the same equilibrium density and binding energy (see Table 4), the parameter set B2 has an instability and B4 a local minimum at &2 times the normal matter density whereas the sets B1 and B3 give a stable state at all density, possibly up to meson condensations and/or chiral phase transition. It is not clear what this means for describing #uctuations at a density above but it indicates that given data at ordinary nuclear matter density, it will not be feasible to extrapolate in a unique way to higher densities unless one has constraints from experimental data at the corresponding density. In our discussion, we relied on the data from KaoS and FOPI collaborations to avoid the "ne-tuning of the parameters. Appendix B. Relativistic calculation of F 1 In the text, the Landau parameter F (or f ) was calculated nonrelativistically via the Fock term of Fig. 17. Here we calculate it relativistically by Fierz-transforming the one-pion-exchange graph and taking the Hartree term. This procedure is important for implementing relativity in the connection between Fermi-liquid theory and chiral Lagrangian theory along the line discussed by Baym and Chin [35].
Fig. 16. E/A!M vs. for the B1, B2, B3 and B4 models given in Table 4 compared with FTS1 theory.
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Fig. 17. The-one-pion-exchange diagram that gives rise to FL .
The one-pion-exchange potential in Fig. 17 is u u u u < "!g ( ) ) . ,, (p !p )!m The Dirac spinors are normalized by
(B.1)
uR(p, s)u(p, s)"
(B.2)
) "(3 ! ) )
(B.3)
. QQY By a Fierz transformation, we have
and u u u u "[u u u u !u Iu u u I # u IJu u u #u Iu u u #u u u u ] . (B.4) IJ I Remembering a minus sign for the fermion exchange, we obtain the corresponding pionic contribution to the quasiparticle interaction at the Fermi surface, f "!< (p "p "p, p "p " p, p"p"k ) (see (2)). Decomposing f as $ 3! ) f " ( f #f #f #f #f ) , (B.5) 1 4 2 . 2 where S, <, ¹, A and P represent scalar, vector, tensor, axial vector and pseudoscalar channel, respectively, we "nd Mf 1 f "! , 1 E m q#m L $ L Mf 1 q f " 1# , 4 E m q#m 2M $ L L
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1 Mf q 2 ) p ) p! ) p ) p! ) p ) p f "! ) 1# # , 2 E m q#m 2M 2M L $ L 1 Mf 2 ) p ) p! ) p ) p! ) p ) p f " ) ! , E m q#m 2M L $ L f "0 . . with E "(k #M and q"p!p. Thus, we obtain $ $ f M 1 q(1! ) ) 3! ) f L" ) q ) q! m E q#m 2 2 L $ L 3! ) 1 f M q ) q ) q 1 " 3 ! ) # (3! ) ) . 2 3 m E q#m q 2 L $ L
(B.6)
(B.7)
In the non-relativistic limit, E &M and we recover (188). The factor M/E comes since there is $ $ one particle in the unit volume which decreases relativistically as the speed increases. Note that only f and f in (267) are spin-independent and contribute to FL . The f is completely canceled by 1 4 1 the leading term of f with the remainder giving FL . In this way of deriving the Landau parameter 4 F , it is the vector channel that plays the essential role. Appendix C. Relativistic calculation of F1 ( ) and J 2 -body Here we compute the contribution of vector meson channel to Landau parameter F and EM current relativistically. The way to compute the contribution of -meson channel is almost the same as channel. So we treat here -meson channel only. The blob in Fig. 11a corresponding to four-Fermi interactions can be expanded as Fig. 18 in random phase approximation. One bubble is represented by
dp tr[IS(p)JS(p#q)] , (C.1) IJ" lim lim 2 (2) SO O where S(p) is the Fermion propagator. The factor 2 come from isospin contribution. In the presence of Fermi sea, we can divide S(p) into the four parts;
1!n 1 n #p ) !M夹 N N S(p)" ( !p ) #M夹) # # N N p # !i
p ! #i p ! !i
2 N N N N
(C.2)
with "(M夹#p and n "(k !p) at ¹"0. The "rst term is the free particle propagator N $ N in vacuum. The second is the particle propagator for Pauli-blocked state in medium. The third is for hole and the fourth is for antiparticle. Since vacuum contribution, i.e. antiparticle}particle in vacuum contribution, is canceled by counter terms and particle}hole contribution vanishes in our limit q/ P0, the antiparticle}particle in Pauli-blocked state contribution remains. O
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Fig. 18. Quasiparticle interactions in vector meson channel represented by four-Fermi interaction. The large blob corresponds to the blob in Fig. 11.
Then (269) becomes
dp n N tr[I( !p ) #M夹)J( #p ) !M夹) N N (2) 4 N (C.3) # I( #p ) !M夹)J( !p ) #M夹)] . N N Because of rotational invariance and zero energy}momentum transfer, only GG does not vanish. IJ"
4 I$ dp 3M夹#2p " GG" E 3 (2) N $ 夹 with E "(M #k . $ $ The quasiparticle interaction in -meson channel in Fig. 18 gives
(C.4)
G u (p)Gu(p) f S "C u (p)Iu(p)u (p) u(p)!u (p) u(p) !C I G SE NNY S $ G C /E S $ u (p)Gu(p) "C u (p)Iu(p)u (p) u(p)#u (p) u(p) S I G 1#C /E S $ p ) p "C !C S S E $ with chemical potential "E #C . Thus, $ S 2k F ()"!C $ . S
(C.5)
(C.6)
And the EM current in Fig. 11a is C /E S $ JS "u (k)u(k) U 1#C /E S $ k FI () " . 6
(C.7)
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References [1] [2] [3] [4]
[5] [6] [7] [8]
[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
[38] [39]
S. Weinberg, Phys. Rev. Lett. 18 (1967) 188; Phys. Rev. 166 (1968) 1568; Physica A 96 (1979) 327. S. Weinberg, Nucl. Phys. B 363 (1991); Phys. Lett. B 251 (1990) 288; Phys. Lett. B 295 (1992) 114. T.-S. Park, D.-P. Min, M. Rho, Phys. Rept. 233 (1993) 341. C. OrdoH nez, U. van Kolck, Phys. Lett. B 291 (1992) 459; C. OrdoH nez, L. Ray, U. van Kolck, Phys. Rev. Lett. 72 (1994) 1982; U. van Kolck, Phys. Rev. C 49 (1994) 2932; S.R. Beane, C.Y. Lee, U. van Kolck, Phys. Rev. C 52 (1995) 2914; C. OrdoH nez, L. Ray, U. van Kolck, Phys. Rev. C 53 (1996) 2086. T.-S. Park, D.-P. Min, M. Rho, Nucl. Phys. A 596 (1996) 515. T.-S. Park, D.-P. Min, M. Rho, Phys. Rev. Lett. 74 (1995) 4153. T.-S. Park, K. Kubodera, D.-P. Min, M. Rho, Phys. Rev. C 58 (1998) R637; Nucl. Phys. 646 (1999) 83. J. Polchinski, E!ective "eld theory and the Fermi surface, in: J. Harvey, J. Polchinski (Eds.), Recent Directions in Particle Theory: From Superstrings and Black Holes to the Standard Model, World Scienti"c, Singapore, 1994, pp. 235}274; R. Shankar, Rev. Mod. Phys. 66 (1994) 129. T. Chen, J. FroK hlich, M. Seifert, Renormalization group methods: Landau Fermi liquid and BCS superconductors, lectures at Les Houches summer school, August 1994, e-print cond-mat/9508063. G. Benfatto, G. Gallavotti, J. Stat. Phys. 59 (1990) 541; Phys. Rev. B 42 (1990) 9967. R.J. Furnstahl, H.-B. Tang, S.D. Serot, Phys. Rev. C 52 (1995) 1368. R.J. Furnstahl, S.D. Serot, H.-B. Tang, Nucl. Phys. A 615 (1997) 441; Erratum-ibid. A 640 (1998) 505. R.J. Furnstahl, S.D. Serot, H.-B. Tang, Nucl. Phys. A 618 (1997) 446. G.E. Brown, M. Rho, Phys. Rev. Lett. 66 (1991) 2720. G. Agakichiev et al., Phys. Rev. Lett. 75 (1995) 1272. M. Masera, Nucl. Phys. A 590 (1995) 93c. G.Q. Li, C.M. Ko, G.E. Brown, Phys. Rev. Lett. 75 (1995) 4007; Nucl. Phys. A 606 (1996) 568; G.Q. Li, C.M. Ko, G.E. Brown, H. Sorge, Nucl. Phys. A 611 (1996) 539. P. Senger et al., Nucl. Phys. A 553 (1993) 757c; D.M. Minskowiec et al., Phys. Rev. Lett. 72 (1994) 3650. J.L. Ritman et al., Z. Phys. A 352 (1995) 355. G.Q. Li, G.E. Brown, C.-H. Lee, C.M. Ko, Strangeness production and #ow in heavy-ion collisions, in: W. Bauer, A. Mignerey (Eds.), Marathon 1997, Advances in Nuclear Dynamics 3, Plenum, New York, 1997, pp. 107}114. B. Friman, M. Rho, Nucl. Phys. A 606 (1996) 303. C. Song, G.E. Brown, D.-P. Min, M. Rho, Phys. Rev. C 56 (1997) 2244. C. Song, D.-P. Min, M. Rho, Phys. Lett. B 424 (1998) 226. B. Friman, M. Rho, C. Song, Phys. Rev. C 59 (1999) 3357. R. Rapp, R. Machleidt, J.W. Durso, G.E. Brown, Phys. Rev. Lett. 82 (1999) 1827. D.B. Kaplan, A.E. Nelson, Phys. Lett. B 175 (1986) 57. C.-H. Lee, G.E. Brown, D.-P. Min, M. Rho, Nucl. Phys. B 585 (1995) 401. D. Montano, H.D. Politzer, M.B. Wise, Nucl. Phys. B 375 (1992) 507. G. Baym, C. Pethick, Landau Fermi-Liquid Theory: Concepts and Applications, Wiley, New York, 1992. A.B. Migdal, Nuclear Theory: The Quasiparticle Method, W.A. Benjamin, New York, 1968. S. Krewald, K. Nakayama, J. Speth, Phys. Rep. 161 (1988) 103. G. Baym, W. Weise, G.E. Brown, J. Speth, Comments Nucl. Part. Phys. 17 (1987) 39. S.-O. BaK ckman, O. SjoK berg, A.D. Jackson, Nucl. Phys. A 321 (1979) 10. A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover Publications, New York, 1975. G. Baym, S. Chin, Nucl. Phys. A 262 (1976) 527. S. Coleman, R. Jackiw, Ann. Phys. 67 (1971) 552. S. Weinberg, What is quantum "eld theory, and what did we think it is?, talk given at the Conference on Historical and Philosophical Re#ections on the Foundations of Quantum Field Theory, March 1996, Boston, United States, e-print hep-th/9702027. B.W. Lynn, Nucl. Phys. B 402 (1993) 281. A.I. Vainshtein, V.I. Zakharov, V.A. Novikov, M.A. Shifman, Sov. J. Part. Nucl. 13 (1982) 224.
370
C. Song / Physics Reports 347 (2001) 289}371
[40] H. Georgi, Phys. Lett. B 298 (1993) 187. [41] G.E. Brown, M. Rho, Nucl. Phys. A 596 (1996) 503. [42] D. Weingarten, Nucl. Phys. B (Proc. Suppl.) 34 (1994) 29; J. Sexton, A. Vaccarino, D. Weingarten, Phys. Rev. Lett. 75 (1995) 4563. [43] J.L. Friar, Chiral symmetry in nuclei, in: D.-P. Min, M. Rho (Eds.), Chiral Dynamics in Hadrons and Nuclei, Seoul National University Press, 1995, pp. 187}206; Few Body Syst. 99 (1996) 1. [44] G.E. Brown, M. Buballa, M. Rho, Nucl. Phys. A 609 (1996) 519. [45] H. Georgi, Phys. Rev. Lett. 63 (1989) 1917; Nucl. Phys. B 331 (1990) 311. [46] G.E. Brown, M. Rho, Phys. Rep. 269 (1996) 333. [47] J. Schechter, Phys. Rev. D 21 (1980) 3393. [48] S.-H. Lee, Phys. Rev. D 40 (1989) 2484; V. Koch, G.E. Brown, Nucl. Phys. A 560 (1993) 345. [49] C. Adami, G.E. Brown, Phys. Rev. D 46 (1992) 478. [50] S. Weinberg, Phys. Rev. Lett. 65 (1990) 1177. [51] Y. Iwasaki, K. Kanaya, S. Kaya, T. YoshieH , Phys. Rev. Lett. 78 (1997) 179. [52] R.D. Pisarski, F. Wilczek, Phys. Rev. D 29 (1984) 338. [53] S. Beane, U. van Kolck, Phys. Lett. B 328 (1994) 137. [54] V. Thorsson, A. Wirzba, Nucl. Phys. A 589 (1995) 633. [55] T. Waas, R. Brockmann, W. Weise, Phys. Lett. B 405 (1997) 215. [56] M. Harada, K. Yamawaki, Phys. Lett. B 297 (1992) 151; M. Harada, T. Kugo, K. Yamawaki, Phys. Rev. Lett. 71 (1993) 1299. [57] J.W. Durso, A.D. Jackson, B.J. Verwest, Nucl. Phys. A 345 (1980) 471. [58] J.W. Durso, H.-C. Kim, J. Wambach, Phys. Lett. B 298 (1993) 267. [59] K.F. Liu, S.J. Dong, T. Draper, D. Leinweber, J. Sloan, W. Wilcox, R.M. Woloshyn, Phys. Rev. D 59 (1999) 112001. [60] M. Harada, F. Sannino, J. Schechter, H. Weigel, Phys. Lett. B 384 (1996) 5. [61] C.M. Ko, G.Q. Li, J. Phys. G 22 (1996) 1673. [62] K. Saito, A.W. Thomas, Phys. Lett. B 327 (1994) 9. [63] R. Rapp, G. Chanfray, J. Wambach, Nucl. Phys. A 617 (1997) 472. [64] R. Rapp, J. Wambach, Eur. Phys. J. A 6 (1999) 415. [65] G.E. Brown, C.Q. Li, R. Rapp, M. Rho, J. Wambach, Acta Phys. Polon. B 29 (1998) 2309. [66] Y. Kim, R. Rapp, G.E. Brown, M. Rho, A schematic model for density dependent vector meson masses, talk given at the AIP/KKG Memorial Meeting, October 1999, Upton, U.S.A., e-print nucl-th/9902009; Phys. Rev. C 62 (2000) 015202. [67] R. Rapp, M. Urban, M. Buballa, J. Wambach, Phys. Lett. B 417 (1998) 1. [68] W. Peters, M. Post, H. Lenske, S. Leupold, U. Mosel, Nucl. Phys. A 632 (1997) 197. [69] Y. Kim, H.K. Lee, Phys. Rev. C 55 (1997) 3100; Y. Kim, H.K. Lee, M. Rho, Phys. Rev. C 52 (1995) 1184. [70] J. Gasser, H. Leutwyler, Phys. Lett. B 184 (1987) 83; H. Leutwyler, A.V. Smilga, Nucl. Phys. 342 (1990) 302. [71] F. Klingl, N. Kaiser, W. Weise, Nucl. Phys. A 624 (1997) 527. [72] F. Klingl, N. Kaiser, W. Weise, Nucl. Phys. A 650 (1999) 299. [73] X. Jin, D.B. Leinweber, Phys. Rev. C 52 (1995) 3344. [74] P.K. Sahu, A. Hombach, W. Cassing, M. E!enberger, U. Mosel, Nucl. Phys. A 640 (1998) 493; P.K. Sahu, W. Cassing, U. Mosel, A. Ohnishi, Nucl. Phys. A 672 (2000) 376. [75] M. Rho, Phys. Rev. Lett. 54 (1985) 767. [76] G. Gelmini, B. Ritzi, Phys. Lett. B 357 (1995) 431. [77] C. Fuchs, H. Lenske, H.H. Wolter, Phys. Rev. C 52 (1995) 3043. [78] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [79] R.J. Furnstahl, X. Jin, D.B. Leinweber, Phys. Lett. B 387 (1996) 253. [80] R. Brockmann, H. Toki, Phys. Rev. Lett. 68 (1992) 3048. [81] G.Q. Li, C.-H. Lee, G.E. Brown, Phys. Rev. Lett. 79 (1997) 5214. [82] T. Matsui, Nucl. Phys. A 370 (1981) 365. [83] W. Cassing, E.L. Bratkovskaya, R. Rapp, J. Wambach, Phys. Rev. C 57 (1998) 916.
C. Song / Physics Reports 347 (2001) 289}371 [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116]
371
G.J. Lolos et al., Phys. Rev. Lett. 80 (1998) 241. E.J. Stephenson et al., Phys. Rev. Lett. 78 (1997) 1636. B. Friman, M. Soyeur, Nucl. Phys. A 600 (1996) 477. K. Langfeld, H. Reinhardt, M. Rho, Nucl. Phys. A 622 (1997) 620. R.S. Hayano, S. Hirenzaki, A. Gillitzer, Eur. Phys. J. A 6 (1999) 99. W. Bentz, A. Arima, H. Hyuga, K. Shimizu, K. Yazaki, Nucl. Phys. A 436 (1985) 593. K. Kubodera, J. Delorme, M. Rho, Phys. Rev. Lett. 40 (1978) 755. M. Chemtob, M. Rho, Nucl. Phys. A 163 (1971) 1. G.E. Brown, M. Rho, Nucl. Phys. A 339 (1980) 269. H. Miyazawa, Prog. Theor. Phys. 6 (1951) 801. B. Grinstein, R.F. Lebed, Phys. Rev. D 57 (1998) 1366. W. Kohn, Phys. Rev. 123 (1961) 1242. A. Houghton, H.-J. Kwon, J.B. Marston, Adv. Phys. 49 (2000) 141. H. Kurasawa, T. Suzuki, Phys. Lett. B 165 (1985) 234. D. Pines, P. Nozie`res, The Theory of Quantum Liquids, Vol. I, Benjamin, New York, 1966. T. Hatsuda, S.H. Lee, Phys. Rev. C 46 (1992) R34. B. Friman, Vector meson propagation in dense matter, talk given at the APCTP Workshop on Astro-Hadron Physics, October 1997, Seoul, Korea, e-print nucl-th/9801053. R. Nolte, A. Baumann, K.W. Rose, M. Schumacher, Phys. Lett. B 173 (1986) 388. M. Rho, Nucl. Phys. A 231 (1974) 493. K. Ohta, M. Wakamatsu, Nucl. Phys. A 234 (1974) 445. T.-S. Park, H. Jung, D.-P. Min, Phys. Lett. B 409 (1997) 26. E.G. Drukarev, E.M. Levin, Nucl. Phys. A 532 (1991) 695. G. Chanfray, J. Delorme, M. Ericson, Nucl. Phys. A 637 (1998) 421. W. Bentz, A. Arima, H. Baier, Ann. Phys. (NY) 200 (1990) 127. J. Delorme, Nucl. Phys. A 374 (1982) 541c. G.E. Brown, Chiral symmetry and changes of properties in nuclei, in: W.C. Haxton, E.M. Henley (Eds.), Symmetries and Fundamental Interactions in Nuclei, World Scienti"c, Singapore, 1995, pp. 169}180. E.K. Warburton, Phys. Rev. Lett. 66 (1991) 1823; Phys. Rev. C 44 (1991) 233; E.K. Warburton, I.S. Towner, Phys. Lett. B 294 (1992) 1. P. Baumann et al., Phys. Rev. C 58 (1998) 1970. K. Kubodera, M. Rho, Phys. Rev. Lett. 67 (1991) 3479. V. Pandharipande, C.J. Pethick, V. Thorsson, Phys. Rev. Lett. 75 (1995) 4567. T. Waas, M. Rho, W. Weise, Nucl. Phys. A 617 (1997) 449. E. Friedman, A. Gal, C.J. Batty, Phys. Lett. B 308 (1993) 6; Nucl. Phys. A 579 (1994) 518. Y. Kim, H.K. Lee, Phys. Rev. C 62 (2000) 037901.
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THE THREE-BODY PROBLEM WITH SHORT-RANGE INTERACTIONS
E. NIELSENa,b, D.V. FEDOROVc, A.S. JENSENc, E. GARRIDOd a
Institute of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark b Department of Physics, Kansas State University, Manhatten, KA 66506-2601, USA c Institute of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark d Instituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain
AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO
Physics Reports 347 (2001) 373}459
The three-body problem with short-range interactions E. Nielsen , D.V. Fedorov, A.S. Jensen, E. Garrido * Institute of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark Department of Physics, Kansas State University, Manhatten, KA 66506-2601, USA Institute of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark Instituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain Received August 2000; editor: G.E. Brown
Contents 1. Introduction 1.1. The three-body problem in various branches of physics 1.2. Theoretical few-body methods 1.3. The philosophy and structure of the report 2. Hyperspherical description of three-body systems 2.1. Hyperspherical coordinates 2.2. Hyperspherical adiabatic expansion 2.3. Faddeev equations 3. The hyperspherical adiabatic potentials 3.1. The free angular solutions 3.2. Small distance solutions 3.3. Intermediate distance solutions 3.4. Finite spins of the particles 4. Large-distance asymptotic behaviour 4.1. Expansion of eigenvalue equation 4.2. Three-body continuum states 4.3. Two-cluster continuum states 4.4. Asymptotic behaviour of the nondiagonal coupling terms 5. The E"mov and Thomas e!ects in d dimensions 5.1. Three identical bosons in 3 dimensions 5.2. Occurrence conditions for the Thomas and E"mov e!ects
376 376 377 378 379 380 382 384 386 387 389 394 399 399 399 401 405 407 409 409 413
5.3. Conclusion 6. Three-body systems in two dimensions 7. Helium trimers: accurate numerical calculations 7.1. The two-body phenomenological potentials 7.2. The adiabatic potentials 7.3. Bound states: energy and structure 7.4. Helium trimers in external "elds 8. Nuclear three-body halos 8.1. Hypertriton: the simplest strange halo 8.2. Borromean two-neutron halo nuclei: He and Li 9. Continuum structure and scattering 9.1. Scattering in the hyperspherical adiabatic approach 9.2. Continuum structure of nuclear halos 9.3. Atomic recombination reactions 10. Summary and conclusions Appendix A. Jacobi functions Appendix B. Spherical coordinates in d dimensions Appendix C. Basic properties of two-body systems in d dimensions References
* Corresponding author. E-mail address: [email protected] (E. Garrido). 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 0 7 - 1
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Abstract The quantum mechanical three-body problem is studied for general short-range interactions. We work in coordinate space to facilitate accurate computations of weakly bound and spatially extended systems. Hyperspherical coordinates are used in both the interpretation and as an integral part of the numerical method. Universal properties and model independence are discussed throughout the report. We present an overview of the hyperspherical adiabatic Faddeev equations. The wave function is expanded on hyperspherical angular eigenfunctions which in turn are found numerically using the Faddeev equations. We generalize the formalism to any dimension of space d greater or equal to two. We present two numerical techniques for solving the Faddeev equations on the hypersphere. These techniques are e!ective for short and intermediate/large distances including use for hard core repulsive potentials. We study the asymptotic limit of large hyperradius and derive the analytic behaviour of the angular eigenvalues and eigenfunctions. We discuss four applications of the general method. We "rst analyze the E"mov and Thomas e!ects for arbitrary angular momenta and for arbitrary dimensions d. Second we apply the method to extract the general behaviour of weakly bound three-body systems in two dimensions. Third we illustrate the method in three dimensions by structure computations of Borromean halo nuclei, the hypertriton and helium molecules. Fourth we investigate in three dimensions three-body continuum properties of Borromean halo nuclei and recombination reactions of helium atoms as an example of direct relevance for the stability of Bose}Einstein condensates. 2001 Elsevier Science B.V. All rights reserved. PACS: 21.45.#v; 31.15.!p
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1. Introduction The classical three-body problem, Moon}Earth}Sun, still contains unanswered questions [1,2]. Also the quantum mechanical three-body problem has been around almost as long as quantum mechanics [3] and still in general considered `unsolveda. Both structure and dynamical behaviour of this problem arises in all sub"elds of physics including chemical physics [4}19]. Precise and rigorous solutions are highly desirable both for use directly as well as in high accuracy studies of three-body correlations within various many-body systems. New and e$cient methods providing both insight and high precision are most welcome as that developed recently [20,21]. This method is applicable for short-range interactions and may prove useful in the future also when long-range interactions are involved [22]. In this report we shall describe the method, generalize to d dimensions, extract a number of analytical conclusions and investigate numerically several topical physical systems. Further details and discussions are available in [23]. 1.1. The three-body problem in various branches of physics The two-body problem is classi"ed as `solveda due to angular momentum conservation, which e!ectively reduces the problem to that of one-dimensional radial motion. The energy is then still conserved and in both classical and quantum mechanics the task is to solve a second-order ordinary di!erential equation corresponding to constant energy. For the three-body problem the con"guration space is six-dimensional in the centre of mass system. Again spherically symmetric interactions between each pair of particles imply total angular momentum conservation, but this only provides three constants of motion e!ectively leaving a three-dimensional problem. For classical mechanics we have three coupled second order non-linear di!erential equations. In quantum mechanics we have instead a three-dimensional partial second-order di!erential equation and no e$cient general numerical procedure exists. In atomic physics the constituent particles are to a very good approximation point particles and the long-range Coulomb interaction is known exactly. The stability of Coulomb interacting three-body systems is, apart from borderline cases and possible additional forces, completely understood [24]. The equation of motion can immediately be written down and the (sometimes very di$cult) problem is to "nd the solutions [8,24,25]. These long-range interacting systems are abundant, but beyond the scope of the present report. In molecular or chemical physics three-body systems e!ectively arise when the Born}Oppenheimer approximation is used to separate out the electron motion [17,26}30]. Mixed molecular and atomic systems also exist [18,19,31], e.g. muonic molecules, dt\, are of practical interest for muon catalyzed fusion [32]. In nuclear physics there are the three-nucleon systems [7] and the more complicated nuclear clusters approximately described by three-body models [5,22,33}37]. Finally we have the nucleons (three-quarks) themselves [4]. In both nuclear and molecular physics the decoupling into clusters is much less e$cient than for atomic systems. The particles are composed of other particles like quarks, other nuclei or electrons. The interactions are not known exactly and may e!ectively depend on the relative position and motion of all three particles. In nuclear physics the experimental data is often used to determine the e!ective two-body interaction [7], which in molecular physics most often is calculated in the
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Born}Oppenheimer approximation as the ground state energy of the electrons with frozen nuclear positions [17,26}30]. The dependence of observables on the potentials has been discussed both in two [38,39] and three dimensions [13,35}37,40}43]. This question of model independence is interesting in nuclear physics aiming at e!ective nucleon}nucleon potentials and consistent models with predictive power. In molecular physics complicated models might be approximated by use of simpler potentials providing better physical insight and easier to handle numerically. Model independence relates di!erent physical systems within di!erent sub"elds of physics. Borromean systems [5] occur in both molecular and nuclear physics. The prominent nuclear examples, Li and He, are often called neutron halos [13}15], which in other publications [44] are used as synonymous with Borromean system. A more suitable distinction and a precise de"nition is given in [43,45], where halos denote systems with a substantial cluster decomposition and a signi"cant part of the wave function outside the classically allowed region. For short-range interactions the Exmov ewect [46] may appear, i.e. a three-body system has in"nitely many bound states if at least two of the two-body subsystems have an in"nite s-wave scattering length. The E"mov e!ect is mathematically the same as the Thomas ewect [47], i.e. a three-body system with zero-range two-body interactions has in"nitely many bound states. Extremely large scattering lengths are necessary to obtain just a few E"mov states and only the "rst is predicted in the atomic helium trimer [17,30]. They may also appear by chance in neutron dripline [37]. 1.2. Theoretical few-body methods The various few-body methods are in general each optimized for computation of di!erent quantities. Quantum Monte Carlo variational computations are most e$cient in calculations of ground-state energies. The Green function Monte Carlo method gives high-precision and the quantum di!usion Monte Carlo version e$ciently improves the ground-state energy of an approximately known wave function [48,49]. The method is less suited for spatially dilute systems [48], for wave functions with nodes and when derivatives are involved. It is easily generalized to a higher number of particles, e.g. 112 He atoms [48], thermodynamic properties of helium liquids [50], the "rst excited state of the e!ective four-body molecular system SF }He [51] and nuclear systems with mass numbers up to 7 [49,52]. For nuclei with very complicated two- and threenucleon potentials the e!ects arising from uncertainties in the interactions and from de"ciencies of the methods are not easily separated. The sudden approximation applies for high-energy scattering of two- and three-body projectiles on a one-body target provided the reaction time is much faster than the intrinsic projectile motion [53], e.g. fragmentation of Li on C [54] and e\ on He [55]. The Faddeev equations [56] were originally intended for momentum space, where the three Faddeev components are expanded on the partial angular momenta related to the Jacobi coordinates. Most applications have been within nuclear physics, bound states as well as scattering [6], but also applied on atomic helium trimers [57]. The Faddeev equations have also been generalized to N-body systems [58] and used in both momentum [59] and coordinate space [60]. The numerical problems are substantial already for the four-body problem, e.g. p#HPp#H, n#He and d#d [61]. There are also ongoing attempts to make realistic calculations for N'4 [62].
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The Faddeev procedure is o! hand most suited for short-range interactions [36,63]. However, treating the large distance part of the Coulomb interaction as the kinetic energy avoids the large couplings between partial angular momenta related to di!erent Jacobi coordinates allowing computations of e.g. bound state energies of dt\ [32] and scattering processes like e>#HPPs#p [64]. The partial wave expansion may be avoided by directly solving the coupled three-dimensional equations for the lengths of the two Jacobi coordinates and the angle between their directions [65], e.g. p#p\, d#d\, t#t\, d#t\ and t#d [65] and N#d [66]. Hard cores present di!erent problems [67]. The hyperspherical harmonics expansion uses the eigenfunctions of the six-dimensional angular momentum operator as an angular basis for the three-body wave function. The expansion coe$cients, depending on hyperradius, are then determined by a set of coupled di!erential equations. This approach is useful for simple two-body potentials [68] and especially for Borromean systems of relatively small spatial extensions. When a two-body subsystem has a bound state an increasing number of hyperspherical harmonics are needed for increasing hyperradius. Including the two-body bound state wave function explicitly in the basis introduces di$culties related to the two non-orthogonal parts of the new basis [69]. This expansion is also not very useful for complicated potentials with repulsive cores and for highprecision calculations. Kohn's variational principle has been used for the recombination process e>#HPPs#p for energies below the threshold for three-body breakup [70] and in three-body nuclear scattering processes [71]. Extension to energies above the three-body threshold, also including Coulomb interactions, employs Faddeev type of components each expanded on hyperspherical harmonics modi"ed with factors describing two-body correlations [72,73]. The e!ort is shifted from a large number of relatively simple basis states to fewer complicated basis states designed to describe parts of the correlations, e.g. the atomic helium trimer ground state [74] and nuclear scattering n#d and p#d [71,74]. The hyperspherical adiabatic expansion was originally introduced to describe the autoionizing states in the helium atom [75]. Although these states are sharp resonances in the continuum they appear as bound states or as shape resonances in the individual adiabatic potentials. The couplings between these states are responsible for the decays. The method is speci"cally powerful for low-energy scattering and bound states. It provides physical insight, e.g. H\ [76], but is less accurate than variational methods, which on the other hand cannot predict resonances such as the autoionizing states in helium. In atomic and chemical physics this method has been successfully applied to many reactions, e.g. scattering e#He> [77], e!#Ps [78], H#H [79], and reactions e\#HPPs#p [80,81], p#\H>, H #FPHF#H [26] and D #FPDF#D [27]. 1.3. The philosophy and structure of the report This report employs the hyperspherical adiabatic expansion allowing thorough, yet transparent, general analyses of a number of physical systems within essentially all sub"elds of physics. The analytic formulations of the method are directly suited for numerical implementations. The distinct advantages are accurate treatment of both large and small distances, in contrast to the generally accepted belief, and identical procedures for bound and continuum states. The method, "rst developed for s-states [21], proved its power by supplying numerical properties of the E"mov
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states [20]. Detailed knowledge of the basic three-body method is used in applications to study few-body correlations within a number of genuine many-body systems [82}85]. The report is intended to describe the method in details and as complete as possible, on a level allowing all physics researcher to follow and judge the arguments. We have generalized the formulation to d dimensions and focused on the mathematical structure, especially at large distances, and extracted as many model independent results as possible. These are often concerned with weakly bound and spatially extended systems, where the details of the interactions are less important. The numerical examples are molecular and nuclear halos. Interesting e!ects are halo properties and E"mov occurrence conditions in d (not necessarily integer) dimensions. Comparison between properties in two and three dimensions are particularly interesting [39,86]. We believe the report will be useful due to the detailed basic descriptions, the general conclusions, the physical insight and the topical systems used as illustrations. We shall "rst in Section 2 sketch a derivation of the angular and radial equations of motion. In Section 3 we explain analytical and numerical details of our method including improvements to deal with a strong repulsive core. In Section 4 we pay special attention to the (sometimes crucial [13,43,39,87,88]) large-distance structure of the Faddeev equations and derive analytically as far as possible the asymptotic behaviour of the corresponding adiabatic potentials and the couplings between them. The exotic E"mov e!ect is discussed in Section 5 for d (perhaps non-integer) dimensions and for arbitrary total angular momentum [11,39,46,86,89]. The practical examples begin with a brief sketch of the properties of two-dimensional systems in Section 6. Then we discuss in Section 7 the bound state energies and structures of the atomic helium trimers [17,30,67]. They provide severe tests of both the numerical method and the possible model independence as well as information about properties and occurrence of E"mov states [30,89]. In Section 8 we study the nuclear halo structures exempli"ed by Li (Li and two neutrons) [54], He (alpha particle and two neutrons) [90] and the hypertriton (proton, neutron and -particle) [91]. Other examples could have been isospin and beta decay [92], the solar neutrino problem [93], the dense helium plasma changing the properties of the three- system with strong consequences for the triple -rate [94] or few-body correlations producing new structures on, or even outside, the neutron dripline [95,96]. Application to the continuum structure and scattering involving three particles in both initial and "nal states are discussed in Section 9. The examples are the continuum structure of nuclear halo systems [97}101]. Other examples could have been the high-energy fragmentation processes of He and Li [40,54,90,102}104]. We shall also discuss the atomic helium trimers. First subject to an external electric "eld as a way of controlling the E"mov e!ect [105]. Second by computing recombination rates of three helium atoms into a dimer and a third helium atom by employing hidden crossing theory [106] combined with the hyperspherical adiabatic expansion [107]. Finally, we included three appendices containing independent results used in the derivations while Section 10 contains a brief summary and the conclusion.
2. Hyperspherical description of three-body systems Three points in a d-dimensional space always de"ne a plane. A three-body problem is therefore basically a planar problem independent of the number of dimensions d of the space, provided
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of course that d52. For d(2 the internal con"guration of the system is restricted. For d52 the number of available dimensions only in#uences the centrifugal barrier terms. This has profound e!ects on both structure and dynamics of the three-body system. These observations are valid for all types of interactions. The total number of degrees of freedom is 3d including the centre of mass. As we shall see the three-body problem can then be formulated in general for any dimension d52. We shall use Jacobi and hyperspherical coordinates and exploit the hyperspherical adiabatic expansion. We shall assume that d is an integer in all the derivations, even though this restriction is unnecessary. In fact, it is possible to carry out analytic continuations of some of the "nal results and thereby provide the generalization to non-integer dimensions. Similar techniques are used other places, e.g. dimensional 1/d expansion in chemical physics [108,109], analytical continuations in the dimension parameter and dimensional regularization in mathematical and particle physics [110,111]. We shall restrict ourselves to consider systems with short-range two-body interactions as de"ned by Eq. (C.5). It is tedious, perhaps technically di$cult, but straightforward to add three-body short-range forces as well as the spin-dependent spin}orbit, spin}spin and tensor forces. The present formulation may also be useful for long-range interactions. 2.1. Hyperspherical coordinates Let us consider 3 particles in d-dimensions, i.e. 3d degrees of freedom of which d and 2d are related to centre of mass and relative motion, respectively. The masses, coordinates and momenta of the particles are m , r and p , i"1, 2, 3. The total mass and momentum are M " G G G R m #m #m and P "p #p #p . The Hamiltonian is, after subtraction of the centre of mass R energy, given by P p (1) H" G ! R # < (r !r )#< (r , r , r ) , G H I @ 2M 2m R G G G where i, j, k is as an even permutation of 1, 2, 3 such that i is associated with the particle pair ( j, k). The two-body interaction between the pair ( j, k) is then denoted < , which here is assumed to G be central and short range as de"ned in Eq. (C.5). The three-body interaction < is added for @ later use. 2.1.1. Dexnition of the coordinates Let us for each i"1, 2, 3 de"ne the ith set of Jacobi coordinates (x , y ) as G G mm H I , x " (r !r ), " G HI H I HI m(m #m ) H I
m r #m r m (m #m ) I I , G H I y " r! H H " , G GHI G GHI m #m (2) m(m #m #m ) H I G H I where m is a normalization mass. Each of the sets, i, j, k"1, 2, 3, 2, 3, 1, 3, 1, 2, combined with the centre of mass coordinate, describes the system. The space-"xed hyperspherical coordinates
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(, , , ) are [112] G VG WG x " sin , y " cos , (3) G G G G where is the hyperradius and is the hyperangle con"ned by 04 4/2. The angular parts of G G x and y describing their directions are denoted and , each therefore representing (d!1) G G VG WG angles. The total number of coordinates is thus 2d of which only one carries the dimension length. The total set of the 2d!1 angular coordinates ( , , ) is denoted by or simply . G VG WG G The volume element corresponding to the relative motion is given by dBx dBy "B\d sinB\ cosB\ d d d ,B\dd . G G G G G VG WG The kinetic energy operator in the centre of mass system is now given by
R 2d!1 R K
! ! # , ¹" R R 2m
(4)
(5)
R lK lK R !2(d!1) cot(2 ) # VG # WG , (6) K "! G R sin cos R G G G G and lK and lK are the angular momentum operators corresponding to x and y respectively, see VG WG G G Appendix B for de"nitions when dO3. Here K is the square of the grand angular momentum operator in 2d dimensions, see Eq. (B.3). 2.1.2. The kinematic rotation The connection between di!erent sets of Jacobi coordinates are [113,114] x "!x cos #y sin , y "!x sin !y cos , (7) H G GH G GH H G GH G GH where the rotation angle is con"ned by !/24 4/2 and given by GH m (m #m #m ) I (8) "arctan i, j, k GH mm G H and i, j, k is the sign of the permutation i, j, k. This transformation is usually called the kinematic rotation [115]. The six possible rotations corresponding to iPj, iOj, and the identity operation form a group. Successive rotations of 1P2, 2P3 and 3P1 then return all vectors back to their initial positions and therefore # # ". The hyperradius is independent of the choice of Jacobi coordinates whereas the di!erent hyperangles are related by G sin "sin cos #cos sin !2 cos sin cos sin cos , (9) H G GH G GH G G GH GH G where is the angle between x and y . For a "xed value of , can assume any value in the G G G G G interval between 0 and 2. Then we obtain the constraints
! 4 4 ! ! ! . G GH GH G H 2 2
(10)
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We may use a body-xxed set of coordinates, e.g. (, , ), where the angles describing the G G orientation of the plane of the particles, the three Euler angles for d"3, are removed. A number of coupled equations increasing with angular momentum and particle spins appears with coordinate and frame singularities related to the Coriolis force for non-vanishing angular momentum [115]. We shall use the laboratory system and the hyperspherical coordinates. Restrictions on the total angular momenta and its partial wave decomposition arise instead from the choice of basis functions. The total number of basis functions needed presumably turn out to be roughly the same in both these procedures. 2.2. Hyperspherical adiabatic expansion We shall use the adiabatic hyperspherical expansion [75], where we "rst solve the angular part of the SchroK dinger equation and then expand the full wave function on the complete set of these angular basis functions. For "xed the set of eigenvalues and eigenfunctions are then obtained as solutions to
2m K ! ()# < (\ sin ) (, )" () (, ) , (11) G HI G L L L
G where we assumed that < only depends on . We also introduced () as an arbitrary function of @ to be chosen later for numerical e$ciency. Then we expand the total wave function on this complete set of solutions " \B\ f () (, ) , (12) L L L where we included the radial phase-space factor \B\ . The spectrum arising from Eq. (11) is discrete due to the "nite intervals con"ning the angular variables or, alternatively, their periodic nature. The corresponding set of solutions is complete for each value of as shown directly for d"3 in [116]. Inserting Eq. (12) into the SchroK dinger equation with the Hamiltonian in Eq. (1) we obtain by use of Eqs. (5) and (11) the coupled set of hyperradial equations
R 1 (2d!3) (2d!1) 2m(E!< () ) @ ! #
()# ()# !Q ! f () L LL L R 4
R #Q f () , " 2P LLY R LLY LY LY$L where E is the three-body energy and the functions P and Q are de"ned by
R (, ) P (), (, ) L LLY R LY
,
(13)
(14)
R (, ) Q (), (, ) L LLY R LY
,
(15)
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where these angular matrix elements for an operator O are de"ned by
OI , d H()OI () ,
(16)
with d from Eq. (4). All () are normalized and P "0. The expansion in Eq. (12) is e$cient L LL when the e!ective diagonal potentials in Eq. (13),
()# ()#(2d!3) (2d!1)/4 L !Q #< () , < , LL @ L 2m
(17)
dominate over the coupling terms P and Q , where nOn . LLY LLY The equations in Eq. (13) decouple completely in the adiabatic limit where R R " "0 . L R L R
(18)
Therefore < can provide substantial insight [75] in close analogy to the corresponding L potentials obtained in the Born}Oppenheimer approach [25]. Di!erentiating Eq. (11) with respect to give P and Q as LLY LLY P "0 , (19) LL R< L R LY for nOn , (20) P "! LLY
!
L LY
Q " P P , (21) LL LK KL K$L
!
L KP P Q "2 LLY
! LK KLY LY K$L K$LY L R< 1 R( ! ) L LY #2P for nOn , (22) ! L LY LLY R
!
R L LY 2m 1 (23) <(, ), < (\ sin )! () . G
G HI 3 G These equations can immediately be used numerically and analytically to obtain P as described LLY in Section 4.4. It is more di$cult to compute Q as the expression in Eq. (22) includes a sum over LLY all the angular eigenstates. When high numerical precision is needed, the adiabatic hyperspherical method may be modi"ed into the diabatic-by-sector or hyperspherical close coupling method [8,27,78,80,81], where a set of a priori chosen values of , , , 2, , is selected and the wave function for a given is , expanded on the angular eigenfunctions corresponding to the closest point k"k() in such a set.
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2.3. Faddeev equations We have so far assumed a procedure where we selected one of the three sets of hyperspherical or Jacobi coordinates de"ned in Eq. (2). One choice is the democratic coordinates providing a completely symmetric treatment of each Jacobi set [79,117]. We shall use the Faddeev equations and wave functions with three components related to the three sets of Jacobi coordinates, i.e. (, )" (, )# (, )# (, ) ,
(24)
where each component, i"1, 2, 3, satis"es the angular Faddeev equations
1 2m (K ! ) # < (\ sin )! () [ # # ]"0 , G G HI G 3
(25)
where we omitted the index n introduced in Eq. (11) to distinguish di!erent angular solutions. By adding the three equations in Eq. (25) we obtain the original angular SchroK dinger equation (11). A three-body state where the particles j and k are in a two-body bound state and particle i is far away is described by " "0 and essentially as the two-body bound state wave function, see H I G Section 4.3. In general, describes the correlations between particles j and k including the correct, G sometimes crucial, behaviour at large hyperradii. In Fig. 1 is indicated how strong correlations between two particles suggest the use of coordinates including one describing the corresponding relative two-body motion. The necessary corresponding angular momenta could be dramatically smaller. Beside the correct asymptotics, another advantage of the Faddeev equations over the SchroK dinger equation is that couplings to higher partial angular momenta in each Faddeev component is of second order in the potentials. To understand this we partial-wave expand each angular Faddeev component on angular momentum eigenfunctions coupled to a total orbital angular momentum ¸ and projection M. Omitting as an argument we then get for d"3 , (26) *+()" JV JW ( ) [> V ( ) > W ( ) ] G G J VG J WG *+ G V W J J where JV JW ( ) are expansion coe$cients and l and l are the partial angular momenta related to G G V W the directions and of the coordinates x and y . This familiar coupling of angular momenta VG WG G G in three dimensions is generalized to dO3 in Appendix B. Finite intrinsic spins are discussed in Section 3.4.
Fig. 1. When the two particles 1 and 2 are strongly correlated it is an advantage to use Jacobi set number 3 (right) rather than Jacobi set number 1 (left).
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With Eq. (26) we rewrite Eq. (25) as
R R l (l #d!2) l (l #d!2) ! !2(d!1) cot(2 ) #V V #W W ! JV JW ( ) G R G G R sin cos G G G G
2m 1 < (\ sin )!
G HI G
3
JV JW ( )# RK JV JW J V J W [J V J W ] ( ) "0 , (27) G G GH G H H$G J V J W where RK JV JW J V J W is a projection operator transforming or `rotatinga from the basis of Jacobi set GH j to that i [113,114], i.e. #
RK JV JW J V J W [J V J W ] ( )" d d [> V ( ) > W ( ) ]H GH H G VG WG J VG J WG *+ ;J V J W ( ) [> V ( ) > W ( ) ] . H H J VH J WH *+
(28)
In this integral ( , , ) depend on ( , , ) through Eqs. (7) and (9). The operator H VH WH G VG WG `rotatinga from one set i to the same set i is the identity operator, i.e. RK JV JW J V J W [J V J W ] ( )" GG G G J V J W ( ) V V W W . G G J J J J In Eq. (27) JV JW is coupled to JV JW ( jOi) by a term of "rst order in the potential < and G H G JV JW in turn is coupled to J V J W by a term of "rst order in < . Therefore the coupling between H G H
JV JW and JV JW is of second order in the potentials when (l , l )O(l , l ). This concludes the G G V W V W argument showing that the coupling of di!erent angular momentum states in each of the Faddeev components at most is of second order in the two-body short-range potentials. In this context we can now understand why the method is less suited for long-range Coulomb potentials where polarization e!ects are very important. Consider for example the system H\. Let us assume a structure where one electron is bound to the proton and the other is far away. The hydrogen atom is signi"cantly polarized and therefore excited into higher angular momentum states. In the Faddeev picture each partial angular momentum in the electron}electron system contributes substantially to the interaction causing the polarization. Thus the partial angular momentum convergence is very slow. Returning now to short-range potentials. The coupling between di!erent angular momentum states in the original SchroK dinger equation is of "rst order in the potentials. Therefore to describe correlations arising from < and < a much larger number of partial angular momenta in Jacobi set H I i is often needed when the SchroK dinger equation is used instead of the Faddeev equations, see Fig. 1. This is especially pronounced when more than one of the two-body subsystems has a bound state. The two simultaneously crucial two-body correlations are then much harder to describe with the SchroK dinger equation. The asymptotic behaviour is decisive and is simply described with a few partial waves in the Faddeev equations. The extreme E"mov states [46,118] are in this way easily computed with only s-waves in all three Jacobi sets [20,37]. A disadvantage of the Faddeev equations is that the equations are integro-di!erential G equations, see Eq. (27). Another disadvantage is that Eq. (27) is not variational in the sense that the lowest angular eigenvalue might be smaller than the exact eigenvalue when a limited number of partial angular momenta is included. For instance, if we do not include any basis states of the Faddeev component , the potential < is not present in the Faddeev equations and the e!ects of
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< cannot be contained in their solution. Consequently, the computed lowest eigenvalue must be too small when < is repulsive. The advantage of variational equations can be regained by computing the energy as the expectation value of the Hamiltonian in the state obtained from the Faddeev equations. The variation should then consist of inclusion of di!erent sets of basis functions. The numerical expense may easily be too high. It is more advantageous to parametrize the wave function in complete analogy to the Faddeev decomposition and use the SchroK dinger equation directly. All physical properties require computations of matrix elements. We therefore need to calculate the inner product between two wave functions in terms of the Faddeev components. Using the relation in Eq. (24) between the SchroK dinger wave functions, and , and their Faddeev components we obtain
I " I G H G H " d sinB\ cosB\ JV JW H( )RK JV JW J V J W [I J V J W ] ( ) , G GH H G G G G G G H JV JW J V J W
(29)
where the rotation operator RK is de"ned in Eq. (28). GH Using the full space spanned by the three Faddeev components the resulting SchroK dinger wave function may be vanishing, i.e. " # # "0, even when the Faddeev components individually are non-zero [21,63,119]. These solutions with zero norm are unphysical and called G spurious solutions. The corresponding Faddeev components may be obtained as eigenfunctions to the kinetic energy operator, since " # # "0 implies that the potential term vanishes as seen from Eq. (25). Spurious states do not appear in momentum space Faddeev calculations [6] either because the total energy is negative or because the boundary conditions exclude vanishing wave functions. In conclusion, the Faddeev equations provide a number of advantages although they also introduce complications. When all two-body interactions are of short range the angular Faddeev equations are simpler and more transparent to solve than the SchroK dinger equation. We shall use the Faddeev equations and in the following illuminate the previous statements by numerical examples.
3. The hyperspherical adiabatic potentials The key quantities in our adiabatic expansion are the angular eigenvalues () de"ned by L Eq. (11) or equivalently by Eq. (27), and the coupling constants P () and Q () de"ned LLY LLY by Eqs. (14) and (15). These quantities are needed as functions of the hyperradius from zero to an upper limit depending on the nature of the problem. This information is for
short-range interactions most e$ciently obtained by directly solving the Faddeev equations in Eq. (27). We shall concentrate on potentials where r< (r)P0 for rP0. Then all < (r) disappear G G from Eq. (27) in the limit "0. Moderately diverging potentials may also be treated, see [42] and Section 8.
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3.1. The free angular solutions Let us start with "0 and assume that < (\ sin )"0 for all i. Then the total angular G HI G wave functions as well as each of the Faddeev components are eigenfunctions of the kinetic energy operator for a constant , see Eqs. (5), (11) and (25). These free solutions may be found from the partial-wave expansion in Eq. (27), which by de"ning x"cos 2 reduces to the di!erential G equation for the Jacobi functions in Eq. (A.1). The complete set of normalized free solutions to Eq. (25), regular at both "0 and "/2, are then for given quantum numbers l and l by use G G V W of Eq. (26) given as , * + ) JV JW ( )"JV JW ( ) [> V ( ) > W ( ) ] G G G G J VG J WG *+
(30)
JV JW ( )"NJV JW sinJV cosJW P B\ >JV B\ >JW (cos 2 ) , G G L G G L G
(31)
"K(K#2d!2),
(32)
K"2n#l #l , V W
where n is a non-negative integer, P B\ >JV B\ >JW are Jacobi polynomials and the quantum L numbers n (or K), l and l are non-negative integers limited by l #l 4K. The normalization V W V W constants NJV JW are derived from Eq. (A.4): L
NJV JW " L
(2n#d!1#l #l )(n#1)(n#d!1#l #l ) V W V W . 2(n#d/2#l )(n#d/2#l ) V W
(33)
The free solutions in Eq. (30) form a convenient basis. They can be ordered into degenerate subspaces corresponding to each value of K"0, 1, 2, 2 with the parity (!1)JV >JW "(!1)), see Eq. (32) and Appendix B. The spectrum in Eq. (32) is the same as for the angular momentum operator for one particle in a space of dimension 2d, see Appendix B. This is because the number of degrees of freedom determines the strength of the centrifugal barrier term as seen by transforming the kinetic energy operator from Cartesian to hyperspherical coordinates. The result is then obvious when the degrees of freedom are reduced by d e!ectively removing the centre of mass motion. The familiar example is the spectrum K(K#4) for three particles in three dimensions. 3.1.1. Degeneracy of the free solutions The eigenvalues are determined by the non-negative integer values of K obtained from n, l and V l as in Eq. (32). The eigenstates related to each K-value are separated into states of given total W angular momentum ¸. The remaining degeneracy related to each of the Faddeev components is denoted D(d, ¸, K), where we do not include the trivial degeneracy of 2¸#1 due to angular momentum projection. From Eq. (32) we have in general that l #l 4K. For d"3 each ¸-value, V W limited by l !l 4¸4l #l , corresponds to one state, where the di!erent angular momentum V W V W projections still are not counted. For d"2 the restrictions are instead ¸"l !l or ¸"l #l , V W V W see Appendix B. For d'3 the triangular inequalities and consequently the restrictions on K and ¸ remain the same as for d"3 [120].
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Fig. 2. The possible combinations of (l , l ) for "xed values of the quantum numbers K and ¸ are shown as every second V W of the points with integer coordinates within the rectangle de"ned by l !l 4¸4l #l 4K. Only every second V W V W point is allowed since l #l must have the same parity as K. For d"2 we have the restriction ¸"l !l or V W V W ¸"l #l implying that only points on three sides of the rectangle are possible. The number of states for d53 is thus V W (¸#1) (K!¸#2)/2 when K#¸ is even and ¸(K!¸#1)/2 when K#¸ is odd. For d"2 we get instead the number of states 2(K!¸)#(¸#1)"K#1.
In Fig. 2 we sketch and compute the allowed sets of (l , l )-values for "xed K and ¸ arriving at the V W number of states D(d"2, ¸, K)"K#1 ,
D(d, ¸, K)"
(34)
(¸#1) (K!¸#2)/2 for K#¸ even ,
¸(K!¸#1)/2
for K#¸ odd .
(35)
3.1.2. The kinematic rotation of the free solutions The free solutions in Eq. (30) can be expressed in any of the three sets of Jacobi coordinates. The operator RK JV JW J V J W describing this kinematic rotation from system j to system i is given in Eq. (28). GH The transformation relating the corresponding free solutions can be de"ned as an overlap matrix RI : GH RI *+)JVG JWG *+)JVH JWH , *+)JVG JWG ( ) *+)JVH JWH ( ) , (36) GH G G H H which is diagonal in K, ¸ and M due to (kinetic) energy and angular momentum conservation. The matrix elements are also independent of M. The remaining part, RI JVG JWG JVH JWH , is a square matrix of GH dimension D(d, ¸, K). The free solutions constitute a complete orthonormal basis for each of the Faddeev components as well as for the total wave function, i.e. the sum of the three Faddeev components. Each choice of Jacobi coordinate system selects the set of basis functions in Eq. (30). The transformation between these di!erent sets must therefore be unitary and two subsequent transformations must be identical to one connecting the same initial and "nal basis sets, i.e. (37) RI JVG JWG JVH JWH RI JVH JWH JYVG JYWG " VG VG WG WG . GH HG J J J J JVH JWH The matrix elements RI JVG JWG JVH JWH for d"3 are called the Raynal}Revai coe$cients [113,121]. For GH d"2 and ¸"0 they can be found in [39]. RI RI "RI , GH HI GI
RI \"RI R , GH GH
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The total wave function may be obtained from the three Faddeev components, expressed in any set of Jacobi coordinate, by use of the Hermitian matrix RI of dimension 3D(d, ¸, K) de"ned in terms of the matrices RI : GH RI RI RI . (38) RI " RI RI RI RI RI RI Using the group properties in Eq. (37) we then get RI "3RI from which it follows that the eigenvalues of RI must be 0 or 3. Let us now for given (K, ¸, M) expand each Faddeev component of a wave function " # # on the free solutions in Eq. (30) with expansion G coe$cients CJV JW . The inner product in Eq. (29) between and I is then G I " CJV JW HRI JV JW J V J W CI J V J W , CRI CI , (39) G GY GY G G GY JV JW J V J W where RI is de"ned in Eq. (38) and C and CI are the states described by the set of coe$cients CJV JW and CI JV JW , respectively corresponding to and I . G G Let us assume that C is a non-trivial eigenstate of RI . Then its norm is zero according to Eq. (39) if the eigenvalue is 0 and non-zero if the eigenvalue is 3. Therefore the space spanned by the eigenstates corresponding to the eigenvalue 3 must be identical to the space of normalizable physical states. This space consisting of the free solutions to the SchroK dinger equation has dimension D(d, ¸, K). Since the full space has the dimension 3D(d, ¸, K) the remaining space spanned by eigenstates corresponding to the eigenvalue 0 must have the dimension 2D(d, ¸, K). This is then the space of non-normalizable states, identically vanishing functions named spurious solutions in Section 2.3. The simplest example corresponds to K"¸"0, where the dimension is D(d, 0, 0)"1 and the free solutions are constants independent of all angular coordinates. Expressing these constants in other coordinate systems give the same constants and the transformation matrix in Eq. (38) must then be
1 1 1
RI " 1 1 1 .
(40)
1 1 1
Thus, the unnormalized eigenfunctions of RI corresponding to the eigenvalue 3 is ( , , )" (1, 1, 1) and a complete set corresponding to the eigenvalue 0 is for instance ( , , )"(1,!1, 0) and (, ,!1). Therefore, the physical space has dimension 1 and the space of spurious solutions has dimension 2. 3.2. Small distance solutions We shall now formulate a method to solve the Faddeev equations in Eq. (27) for general short-range potentials for relatively small values of , i.e. from zero and roughly up to the order of the ranges of the potentials.
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3.2.1. Expanding on the free solutions We shall expand each angular Faddeev component *+( ) on the complete set of free G G solutions in Eq. (30), i.e. *+( )" CL JV JW * + ) JV JW ( ) , G G G G G L JV JW
(41)
where ¸ and M are the conserved angular momentum, CL JV JW are the expansion coe$cients. G The dependence and subsequent summation over n is needed, since non-vanishing interactions imply that K"2n#l #l no longer, in contrast to ¸ and M, is a conserved quantum V W number. Let C denote the column vector consisting of the expansion coe$cients in Eq. (41) of all three Faddeev components sequentially ordered from i"1 to 3. The angular Faddeev equations in Eq. (27) can then be written as [¹I #
(42)
where we assumed an appropriate truncation of the expansion in Eq. (41), used Eqs. (32) and (38), de"ned ¹I as the diagonal matrix with the matrix elements K(K#2d!2) given by the eigenvalues of the kinetic energy operator and "nally de"ned
;
d cosB\ sinB\ P B\ >JV B\ >JW (cos 2 ) G G G G L
2m 1 ; < (\ sin )! () P B\ >J V B\ >J W (cos 2 ) . G HI G LY G
3
(43)
In
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3.2.2. Handling spurious solutions and symmetries We showed in Section 3.1.2 that the eigenvalues of RI are 0 and 3, where the eigenfunctions corresponding to 0 are the spurious solutions. However, if the basis for a given K-value is incomplete, i.e. not all partial angular momenta with l #l 4K are included, the eigenvalues V W calculated using this truncated basis are no longer strictly 0 and 3, but somewhere in the interval [0, 3]. To see this we denote the transformation RI in the reduced basis by RI . The column vector C is P assumed to be an eigenfunction with the eigenvalue x, i.e. RI C"xC and the norm C "1. P Now we may write C"C #C , where C and C are column vectors in the eigenspace of the full RI corresponding to the eigenvalues 3 and 0, respectively. Then x" CRI C " CRI C "3 C 3[0, 3]. The eigenfunctions in the incomplete space correP sponding to 0 are still unphysical, since the norm computed by Eq. (39) remains zero. Choosing an incomplete basis is highly desirable, because selection of basis states according to importance provide the #exibility of using high values of K and only the lowest angular momentum states, i.e. an incomplete basis. Let us then introduce a new basis obtained as the eigenfunctions of RI diagonalized in the truncated basis. Rewriting the matrix equation in Eq. (42) in the new basis maintains the structure, since RI commutes with the diagonal kinetic energy matrix and these operators can therefore be simultaneously diagonalized. Now RI is diagonal with zero for all matrix elements corresponding to the eigenvalue zero. The new matrix
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are given by
1 0 0
V
W
XG L JV JW GY LY J J " V V W W (!1)JV 0 0 1 , (45) L LY J J J J 0 1 0 GGY where we used that the e!ect of X are x C !x and the Jacobi coordinate systems 2 and 3 are interchanged, but with the opposite ordering of the two particles implying that the signs of the vectors x and x also are changed, see Eq. (2). Here we assumed that the truncation in the basis is symmetric for the components for identical particles 2 and 3, i.e. if one term in the expansion of Faddeev component number 2 is included the same term in the expansion of Faddeev component number 3 must also be included, and vice versa. Similarly, if all three particles are identical the projection operator is (46) P"(1#X X #X X $X $X $X ) , where the exchange operators X are de"ned in analogy to X and (#) again applies for bosons GH and (!) for fermions. The corresponding matrix elements are obtained by use of Eq. (45). Clearly, the same basis truncation for all three Faddeev components are also required for this highly symmetric case. It is here worth emphasizing that X in Eq. (45) is diagonal in the quantum number n and furthermore also independent of n. Therefore, X can easily be generalized when other basis function than Jacoby polynomials are used for the motion. This illustrates that symmetry comes G in quite naturally in the Faddeev equations. In contrast, symmetrization when using the SchroK dinger equation is more di$cult unless symmetric coordinates are chosen from the beginning, e.g. the Smith}Whitten coordinates in [122]. In any case the procedure described above simultaneously reduces the size of the matrix, removes the unphysical spurious solutions and restores the correct symmetries. 3.2.3. Perturbative treatment for small We can solve the Faddeev equations in Eq. (42) for small in perturbation theory. Let us assume that the column vector C is a solution to Eq. (42) for
"K(K#2d!2). Then only components with this K are present in the perturbative solution. Then Eq. (42) reduces to
G 3
(47)
(48)
Let us for simplicity assume that < (0)"< . Then the diagonal part of
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Thus, C is an eigenvector of RI with the eigenvalue x and consequently "x
x 2m < ! () !1 . 3
(50)
If x"3 the dependence disappears and the expression is precisely the "rst order perturbation result derived directly from the SchroK dinger equation in Eq. (11), provided () is equivalently treated, i.e. 2m (51)
"! ()#K(K#2d!2)# < (0) , G
G where we lifted the restriction of identical strengths of the potentials. The value x"3 can only be obtained if the basis with the K-space is complete otherwise x(3 and the eigenvalue will behave as if the potentials were weaker. This derivation can rather easily be generalized for potentials diverging as r\ when rP0. Then the dependence remains as sin\ , the strength becomes proportional to and vanishing in the G G limit of "0. The result is that the terms < (0) in Eq. (51) must be replaced by the expectation G value with the unperturbed wave function of the small limit of < (\ sin ). G HI G Potentials diverging as r\ when rP0 do obviously not vanish in the limit of "0. However, they approach a constant times sin\ . The solution in space may then be obtained exactly in G G this limit. If the wave function is concentrated at small the potential is simply adding a term G proportional to \, i.e. of the form as a centrifugal barrier or attraction. This may then be treated G more precisely resulting in a non-vanishing constant to be added to the eigenvalue spectrum. If only one of the two-body potentials has this long-range behaviour on top of otherwise short-range interactions the constant simply rede"nes the partial angular momentum quantum number l [42]. VG 3.2.4. The minimal basis size The "nal perturbation expression in Eq. (51) only holds if the eigenvalues of RI are 0 or 3. This is only ful"lled if the basis for each K-value is complete, i.e. all possible partial angular momenta must be included in the basis. Let us aim at computing all angular eigenvalues up to a certain limit, i.e.
# ("0)4 ,K (K #2d!2), where K is de"ned by this equation for "0.
Then the bases are complete for all K4K for "0, provided the summation in Eq. (41) for
each Faddeev component includes all 04l #l 4K and all 04n4n "(K !l !l ). V W V W
With increasing , couplings between di!erent K-values become important and "rst-order perturbation breaks down. This implies that values of n, l and l higher than for "0 are needed V W in the expansion in Eq. (41). On the other hand, compared to the potential energy term in Eq. (27) the repulsive angular momentum barriers become increasingly important as l and l increases. V W This suppresses the contributions from high partial waves and only the smaller partial angular momenta are needed for a given K-value. In fact, only the lowest partial angular momenta are needed for large , see Section 4. For "nite the basis then has to be complete for all K4K , where K (K #
d!2)&
roughly de"nes the largest angular eigenvalue
we want to compute accurately.
For a repulsive core the angular eigenvalues must become very large for small -values. However, K is not necessarily large, since we can exploit the freedom in choosing () close to the
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anticipated -dependent value of the most important (lowest) eigenvalue . In principle, the results are independent of (), even when we exploit the allowed -dependence. This rede"nition of zero point energy can reduce considerably the number of K-values in which a complete basis is needed. The numerical procedure for highly repulsive cores is then to choose () such that the lowest eigenvalue () is as close to zero as possible. Then only eigenvalues ()4
corresponding to
values of K4K , where K (K #d!2)& ! (), require a complete basis. This
method is used in accurate computations of the atomic helium trimers discussed in Section 7. It could also be useful for hard-core nucleon}nucleon potentials. Increasing we see from Eq. (27) that the e!ective potential, <(\ sin ), becomes both HI G increasingly deeper and more narrow as a function of . The wave function then must vary rapidly G from zero for "0 to a "nite value at the edge of the short-range potential after which a smooth G dependence is obtained in the (much larger) interaction free region of -space. These potentials G may also support bound states in an amount roughly independent of , since the potential depth increases as while the range decreases as \. Thus, to describe the strong variation and the possible nodes of the components JV JW ( ) in Eq. (41), we need a basis with an increasing number G G of polynomials (large n), but not necessarily with large partial angular momenta. The overall conclusions are then that we must choose a complete basis for small K-values, for intermediate K we only need to include some of the possible partial angular momenta, and for high values of K we can decrease the number of partial angular momentum components and concentrate on the lowest partial angular momenta. This prescription of increasing the number of polynomials for each (l , l ) as increases is limited in practice to values of the hyperradius V W comparable to the ranges of the interactions. 3.3. Intermediate distance solutions The three particles all simultaneously interact when the hyperradius is su$ciently small. Increasing we arrive at intermediate distances corresponding to con"gurations where at least one particle must be outside the interaction ranges of both the other two, see Fig. 3. Overlapping
Fig. 3. The shaded areas of non-zero potentials in the ( , )-plane, where is the angle between x and y . For a large G G G G G value of the hyperradius (left) the potentials < and < are only non-zero close to the point "0, i.e. " and "0 H I H G GH G or ", see Eq. (7). For a smaller hyperradius the potentials start to overlap (right). G
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potentials require a de"nition, at least for anything else than schematic square well potentials, of a distance r outside which the potential < is su$ciently small to be neglected in this context. G G We shall tentatively use Eq. (C.5) and assume that < (r)K0 for r'r . The corresponding G G value outside which the potential < (\ sin ) is negligible as function of for "xed is then G G HI G G given by "arcsin( r /)+ r / , G HI G HI G
(52)
where the last approximation only is valid for large . The assumption that all three potentials do not overlap simultaneously for a given may be expressed by the statement that if one point is inside the range of potential i, then it must be outside the ranges of both the potentials j and k, or, in an equivalent mathematical formulation, if 4 G G then ' and ' , where (i, j, k) is a permutation of (1, 2, 3). From Eq. (10) this implies that H H I I # ( for all i and j, iOj. By use of Eq. (52) we then obtain a su$cient condition for G H GH non-overlapping potentials, i.e. '( r # r )/sin HI G GI H GH
for all pairs (i, j) where iOj ,
(53)
which in the following shall serve as the de"nition of intermediate distances. 3.3.1. The Faddeev components outside the potentials We can now solve the angular Faddeev equations in Eq. (27) for ' , i.e. in the regions where G G the potentials are zero. We shall from now on assume that "0, since we have no use for the associated #exibility at larger distances. The solutions, regular at "/2, are then according to G Appendix A given by JV JW ( )"AJV JW sinJV cosJW (#1) G G G G G ;P B\ >JW B\ >JV (!cos 2 ) , J G
(54)
where AJV JW is an arbitrary constant and G ,!(d!1#l #l )#((d!1)#
V W
(55)
or, equivalently, expressing in terms of
"(2#l #l ) (2#l #l #2d!2) . V W V W
(56)
If is a non-negative integer Eq. (54) reduces to the free solutions of Eq. (31). There is an ambiguity in the de"nition of when assumes too negative values. We specify fully by choosing the branch of the square root in Eq. (55) such that is either real or complex with positive imaginary value. In the following we shall maintain this de"nition of as a function of , l and l . V W The derivation assumed that the potential < (\ sin ) is exactly zero for ' . The G HI G G G correction to the eigenvalue arising from this assumption is to "rst order in the deviation from zero
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of < (\ sin ) for ' found as G HI G G G 2m d sinB\ cosB\ < (\ sin ) 4MJV JW G G G G G HI G ?G M IHI 2m dr rB\ < (r) , 4MJV JW B \B G HI
G PG where we used the substitution r"\ sin and the de"nition HI G MJV JW ,AJV JW (#1) G G
(57)
(58) ;max sinJV cosJW PB\ >JW B\ >JV (!cos 2 ) G G J G ?G Y?G of the maximum probability for the solution in Eq. (54) for 5 . Since MJV JW is a "nite number G G G for a given we can for short-range potentials make arbitrarily small by choosing r su$ciently G large, see Eq. (C.5). However, the accuracy is limited by through the upper bound of r de"ned G in Eq. (53). 3.3.2. The Faddeev components inside the potentials We now turn to the angular Faddeev equations in Eq. (27) for ( , i.e. in the regions where G G the potential < di!ers from zero while < and < for intermediate distances therefore must be G H I vanishingly small. In other words, we have both ' and ' . Then the two Faddeev H H I I components, and , are both eigenfunctions of the kinetic energy operator, see Eq. (25), i.e. H I (K ! ) ( )"0, (K ! ) ( )"0 , (59) H G I G where the dependence of indicates that we used the freedom to choose i as the most convenient G Jacobi system. Then for 4 we see that "! ! is one particular solution to Eq. (25), G G G H I since each of the potential and kinetic energy terms independently vanish due to the trivial factor of zero and by use of Eq. (59). Thus is an eigenfunction of the kinetic energy operator. G We want to express this particular solution explicitly as function of . Then we must G G transform the other two Faddeev components to Jacobi set i. We "rst note that and must be H I regular for "0, because this point expressed in the other Jacobi coordinates must be within the G regions ' and ' , where both functions are regular solutions. Projection of these H H I I Faddeev components on the partial angular momentum states of Jacobi set i maintain these properties, i.e. the projected functions are still regular (at "0) and eigenfunctions of the kinetic G energy operator. Thus the kinematic rotation operator in Eq. (28) acting on in Eq. (54) must be proportional to H another eigenfunction, regular at "0, of the kinetic energy operator. Using the properties in G Eqs. (A.1) and (A.2) we then "nd RK JV JW J V J W [sinJ V cosJ W ( #1)P B\ >J W B\ >J V (!cos 2 ) ] ( ) GH H H JY H G
"RJV JW JV JW ( )sinJV cosJW (#1)P B\ >JV B\ >JW (cos 2 ) , GH G G J G
(60)
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where , and are related by Eqs. (55) and (56) through energy conservation (same ). The coe$cients RJV JW J V J W ( ) may depend on , but not on . Using the de"nition in Eq. (28) of the GH G operator RK JV JW J V J W we "nd GH RJV JW J V J W ( ) GH sinJ V cosJ W ( #1)PB\ >J W B\ >J V (!cos 2 ) H H JY H " d d VG WG sinJV cosJW (#1)PB\ >JV B\ >JW (cos 2 ) G G J G ;[> V ( ) > W ( ) ]H [> V ( ) > W ( ) ] , (61) J VG J WG *+ J VH J WH *+ where ( , , ) are functions of ( , , ) through Eq. (7). This result formally depends on the H VH WH G VG WG value of , but if the point of divergence "0 is excluded from the integral in Eq. (61) G H the coe$cients are independent of . This condition of excluding "0 is ful"lled for ( , G H G GH see Eq. (10). The coe$cients in Eq. (61) are related to an analytic continuation to non-integer of the matrix elements RI de"ned in Eq. (39). For integer "n and "n we get the precise relation from Eqs. (30) and (A.7) to be
(n#1)NJ V J W LY RI L JV JW LY J V J W "
(!1)LY G H L>JV >JW LY>JV >JW (n #1)NJV JW L ;RJV JW J V J W ((2n#l #l ) (2n#l #l #2d!2)) (62) GH V W V W for iOj and n, n 50. The Kronecker re#ects the conservation of K. We are now ready to express the above particular solution to Eq. (27), "! ! , G H I explicitly in terms of , i.e. G
JV JW ( )"! RJV JW J V J W ( )AJ V J W G G GH H H$G J V J W ;sinJV cosJW (#1)PB\ >JV B\ >JW (cos 2 ) . (63) G G J G The complete solution to Eq. (27) for 4 is this particular solution added to the complete G G solution to the corresponding homogeneous equation, i.e.
R R l (l #d!2) l (l #d!2) ! !2(d!1) cot(2 ) #V V #W W G R R sin cos G G G G 2m # < (\ sin )! JV JW ( )"0 . G HI G G G
A change of variable from to r,\ changes Eq. (64) into G HI G R cot(2r /) R l (l #d!2) HI ! !2(d!1) HI # V V Rr (/ )sin (r /) Rr HI HI r l (l #d!2) 2m
W W # # < \ sin HI ! HI uJV JW (r)"0 , G HI (/ ) cos (r /) G HI HI HI JV JW ( )"BJV JW uJV JW (\ ) , G G G G GH G
(64)
(65) (66)
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where the arbitrary constant BJV JW expresses the relative weight of the homogeneous solution in the G ith Faddeev component. Eq. (65) reduces to the two-body radial equation in Eq. (C.3) when
JV JW ( )"BJV JW uJV JW (\ )! RJV JW J V J W ( )AJ V J W G G G G GH G GH H H$G J V J W ;sinJV cosJW (#1)PB\ >JV B\ >JW (cos 2 ) . (67) G G J G However, the total wave function, " # # , inside the region of potential i, i.e. for G H I 4 , is given by the even simpler expression G G , ( , , )" BJV JW uJV JW (\ ) [> V ( ) > W ( ) ] G G GH G J VG J WG *+ G VG WG JV JW arising because # precisely cancels the particular solution in Eq. (63). H I
(68)
3.3.3. Matching and solving the angular eigenvalue problem The solutions in Eqs. (54) and (67) must be continuous at " , i.e. G G uJV JW (\ ) GH G "(#1)PB\ >JW B\ >JV (!cos 2 )AJV JW BJV JW G G sinJV cosJW J G G G G # (#1)P B\ >JV B\ >JW (cos 2 ) RJV JW J V J W ( )AJ V J W . J G GH H H$G J V J W By using Eq. (A.7) we can rewrite this condition in a more transparent form
(69)
uJV JW (\ ) G GH G BJV JW "!Q AJV JW sin() G (#1)sinJV cosJW J G G G # P AJV JW cos()# RJV JW J V J W ( )AJ V J W , (70) J G GH H H$G J V J W where P ,P B\ >JV B\ >JW (cos 2 ) and Q ,Q B\ >JV B\ >JW (cos 2 ). J J G J J G Continuity of the derivative gives a similar equation obtained by deriving with respect to on G both sides of Eq. (70). Eliminating BJV JW from Eq. (70) gives a matrix equation for the coe$cients G AJV JW valid for all i"1, 2, 3, i.e. G RQ R ln uJV JW G sin() !l cot !l tan Q ! J AJV JW V G W G J G R Rr G HI R ln uJV JW RP G !l cot !l tan P ! J " V G W G J R Rr HI G
; cos()AJV JW # RJV JW J V J W ( )AJ V J W , G GH H H$G J V J W where r"\ in the logarithmic derivative of uJV JW (r). G HI G
(71)
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This equation is linear in the coe$cients AJV JW and the corresponding determinant must be zero G to allow non-trivial solutions. This in turn de"nes a non-linear eigenvalue equation in . The information about the potentials is contained in the logarithmic derivative of uJV JW at the energy G ( /2m) \, which then in general has to be found numerically by solving Eq. (65). This has two important implications. First, solving Eq. (71) is reduced to "nding solutions to a number of two-body equations. In practice, both angular eigenvalues and eigenfunctions can be computed by this procedure, see Section 7. Secondly, when is a few times larger than r , the angular G eigenvalues are essentially model independent in the sense that they are identical for di!erent potentials provided these logarithmic derivatives are identical. This can be rephrased by saying that potentials resulting in the same two-body phase shifts produce the same angular eigenvalues at large . 3.4. Finite spins of the particles For simplicity of notation we have so far assumed that the particles either were spin zero bosons or that the spin degrees of freedom were totally uncoupled and therefore could be ignored. However, this is not possible for a number of interesting systems, perhaps especially in nuclear physics, where the spin}orbit, spin}spin and tensor forces are very important two-body interactions. The previous chapters and sections demonstrated that the orbital part of the wave functions is decisive for the large-distance asymptotical behaviour. The essential ingredients are therefore already established. On the other hand, a two-body bound state located far away from a third particle must at least asymptotically conserve the total (not orbital) angular momentum of that two-body state. Thus the spins cannot simply be factorized away by including another coupling of the total spin to the orbital part in Eq. (26). Instead it is natural to employ the ls coupling scheme for the individual pairs of particles. Incorporating the corresponding couplings at smaller distances is then rather straightforward. Details can be found in [23]. Several practical examples are discussed in the published literature [40,91,100,123].
4. Large-distance asymptotic behaviour The previous sections discussed solutions for intermediate distances roughly understood as hyperradii larger than the ranges of the two-body interactions r or as de"ned by Eq. (58). In this G section we shall discuss very large distances including the asymptotic limit of in"nite hyperradii. The behaviour of the adiabatic potentials and the coupling terms in this limit is necessary in order to understand the method of the adiabatic expansion. In numerical calculations of bound state properties this knowledge is revealing but not essential, since the wave functions are exponentially decreasing with distance. However, for scattering problems the boundary conditions at "R de"ne the process and the large-distance information is absolutely essential, see Section 9. 4.1. Expansion of eigenvalue equation The overall assumption is that the hyperradius is much larger than the range of the potentials, i.e.
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asymptotically approaching zero, or at least not diverging, in the limit of "R. Furthermore, we shall in the expansions in this section assume that the hyperradius also is much larger than all two-body scattering lengths aJV corresponding to the angular momentum l of system i as G V de"ned in Eq. (C.6), i.e. <aJV . In Section 5 we shall investigate the intriguing limit, where G r ;;aJV . G G The approximation in Eq. (52) is now valid and ( implies that r,\ ;. Then G HI G Eq. (65) reduces to the two-body radial equation in Eq. (C.3) and the solution uJV JW (r) becomes the G two-body radial wave function at the energy ( /2m) \. If this energy is small uJV JW is simply G HI the two-body wave function at zero energy. The logarithmic derivative needed in Eq. (71) is therefore for small obtained by using the asymptotic form in Eq. (C.6) as G R ln uJV JW G !l cot !l tan V G W G Rr HI r B\>JV \ r d!2#2l V G !1 (72) #O G , " r aJV G G HI where O( f () ) is a function at most of the same order as f () when PR. Inserting Eqs. (72), (A.10) and (A.11) into Eq. (71) we obtain after a little algebra for odd dimensions d that
d sin() !cot # 2
d!2 d d #l #l # #l (#1) V V W 2 2 2 d # #l (#d!1#l #l ) V V W 2
B\>JV B\>JV #O AJV JW G aJV r HI G G r " 1#O G cos()AJV JW # RJV JW J V J W ( )AJ V J W . G GH H H$G J V J W For even d and d#2l 54 we obtain instead V ;
d!2 d d #l #l # #l (#1) V V W 2 2 2 sin() (#B #l )(#d!1#l #l ) V V W O log for d#2l "4 V r B\>JV G ; # A JV JW G aJV B\>JV HI G O for d#2l '4 V r G r cos()AJV JW # RJV JW J V J W ( )AJ V J W , " 1#O G G GH H H$G J V J W
(73)
(74)
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401
and for the last case of even d and d#2l "2 (d"2, l "0) we obtain V V
sin() a ! 2 # (#l #1)# (#1)#2 log HI G W
AJV JW G
(75) " cos()AJV JW # RJV JW J V J W ( )AJ V J W , G GH H
V W H$G J J which may be obtained either from Eq. (73) by approaching the limit dP2 from d'2 or by using Eq. (A.11) directly in Eq. (71). The key quantities in these asymptotic equations, which immediately lead to the asymptotic eigenvalue equations, are the coe$cients RJV JW J V J W ( ) de"ned in Eq. (61). They are conGH tinuous functions of because both ( #1);P B\ >J W B\ >J V (!cos 2 ) and JY H (#1)P B\ >JV B\ >JW (cos 2 ) are continuous functions of and therefore of . FurtherJ G more, the denominator in the integrand of Eq. (61) is not zero provided is su$ciently small, see G Eq. (A.10). Thus, we proceed by exercising our freedom to choose an value close to zero. G Since is con"ned by Eq. (55) the right-hand side of Eq. (73) remains "nite unless P!(d!1#l #l )#iR, whereas the left-hand side behaves as sin() (/aJV )B\>JV AJV JW V W G G when PR. This means that non-trivial solutions to Eq. (73) only can exist if approaches an integer n, or if P!(d!1#l #l )#iR. The "rst of these cases corresponds to continuum V W con"gurations with all three particles far away from each other, see Section 4.2. The second case corresponds to two of the particles in a two-body bound state and the third particle far away from these two, see Section 4.3. This analysis of Eq. (73) also applies directly to Eqs. (74) and (75), except when d"2 where powers of the form xB\ must be replaced by log x, see [39]. 4.2. Three-body continuum states We shall extract the properties of those solutions to Eq. (73), where approaches an integer when PR, i.e. eigenvalues or and coe$cients, AJV JW and BJV JW , in the corresponding wave G G functions in Eqs. (54) and (67). Let us "rst assume that one of the coe$cients AJV JW is non-zero in the limit "R. Then the G corresponding must approach an integer n, see Eq. (73). For another component of the same solution, but with partial angular momenta l and l , the related must be V W "#(l #l !l !l ) as seen from Eq. (55), since the energy ( ) is unchanged. If AJ V J W is V W V W G non-zero in the limit "R, also approaches an integer and l #l !l !l must be an even V W V W number. Only components with the same parity contribute to a given solution. For a coe$cient AJV JW , non-vanishing at "R, the related value of must in this limit behave G as "n#O(\B\JV ), again seen from Eq. (73). Assuming "n#b\B\J , where b is a constant and l and n are integers, we get "K(K#2d!2)#4(K#d!1)b\B\J with K"2n#l #l . For another contributing component AJ V J W we "nd the same form V W G "n #b\B\J with n "n#(l #l !l !l ). V W V W For a component with l #l 'K we have that Pn(0. The right-hand side of Eq. (73) V W remains "nite, but the left-hand side behaves as sin()(#1)B\>JV AJV JW "O(B\>JV )AJV JW , G G
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since the "rst-order pole in the -function cancels the zero point in sin(). Therefore, this coe$cient must vanish accordingly as AJV JW "O(\B\JV ) implying that AJV JW (R)"0. G G For a component with l #l 4K we have Pn50 and Eq. (73) becomes V W d d!2 d #l #l n# #l (n#1) V V W 2 2 2 bJV \J AJV JW ( aJV )B\>JV G (n#B #l )(n#d!1#l #l ) V V W HI G
(76) "AJV JW #(!1)L RJV JW J V J W ( )AJ V J W . G GH H
V W H$G J J From this equation we see that AJV JW "O(J \JV ) and AJV JW (R)"0 for l 'l . For l (l the G G V V left-hand side of Eq. (76) vanishes for PR and therefore the right-hand side must also vanish. Therefore we obtain (77) DJV JW ,cos()AJV JW # RJV JW J V J W ( )AJ V J W "O(JV \J ) . G G GH H
H$G JV JW We conclude that only components with l 4l and l #l 4K can be non-zero in the limit V V W "R. These results are summarized in Table 1. We can now "nd the constant b by solving the equation in Eq. (76) at "R with inclusion of only the contributing components of l 4l and l #l 4K. As and are dimensionless and V V W b has the dimension of length raised to the power d!2#2l we must "nd that b"aB\>J , where a is an average over the only available lengths in the problem, i.e. the scattering lengths aJ , G i"1, 2, 3. The other scattering lengths cannot enter in the eigenvalue behaviour, because Eq. (76) dictates that those with l (l disappear for large and those with l 'l require that the V V coe$cient AJV JW vanishes. G For K"¸"l "0 for three identical bosons in three dimensions a turns out to be 12/ times the two-body scattering length [20,21]. We then get
"n#
\B\J , a
(78)
\B\J . (79) a These equations exhibit the asymptotic behaviour of the angular eigenvalues. The features of a given value are the speci"c -dependence characterized by an angular momentum quantum number l and the approach towards a constant recognized from the free spectrum.
"K(K#2d!2)#2(K#d!1)
Table 1 The asymptotic large-distance behaviour the Faddeev components in the limit PR, i.e. the coe$cients AJV JW , DJV JW G G and BJV JW de"ned in Eqs. (54), (77) and (66). The eigenvalues behave as "K(K#2d!2)#2(K#d!1)b\B\J . G The integers K and l characterize how the eigenvalues approach the asymptotic value at "R
l #l 4K and l 4l V W V l #l 4K and l 'l V W V l #l 'K V W
AJV JW G
DJV JW G
BJV JW G
O() O(J \JV ) O(\B\JV )
O(JV \J ) O() O()
O(JV \J ) O(\JV ) O(\JV )
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With this knowledge we can now use Eqs. (67), (70), (A.10), (A.11) and the results in Table 1 to calculate the wave function inside the potential regions, i.e. the coe$cients BJV JW related to the G region 4 , see Table 1. G G 4.2.1. Normalization of the asymptotic wave function The normalization of the asymptotic wave function, de"ned by Eqs. (54), (67) and (68), can now be computed when approaches an integer in the limit PR. All contributions from regions, 4 , inside the potentials vanish, because the corresponding coe$cients BJV JW vanish or G G approach a constant as seen in Table 1 and the sizes of the intervals also approach zero, i.e. P0. G The norm is therefore entirely due to the contributions from the regions, ' , outside the G G potentials. From Eqs. (54), (30), (39), (A.7), (62), (77) and Table 1 we obtain for a solution characterized by l that (n#1)RI L JV JW LY J V J W (n #1) G GY + (!1)L>LY AJV JW HAJ V J W G H NJV JW NJ V J W
L LY G GY JV JW JV JW (n#1) " AJV JW H(!1)L G NJV JW L G JV JW
; (!1)LYAJV JW # RJV JW J V J W ( )AJ V J W G H GH H$G J V J W (n#1) (!1)LAJ JW HDJ JW , + (80) G G NJ JW W L G J where we used that AJV JW DJV JW vanishes in this limit for l Ol . The sum over l in Eq. (80) is G G V W restricted by l #l 4K, since AJV JW vanishes in the large -limit for l #l 'K. From Eq. (76) we G V W W see that if b"0 then also DJ JW "0. In this case also components with l "l do not contribute to G V the norm, which consequently vanish in this large-distance limit. Thus, solutions with b"0 have zero norm and must be spurious. We can now see that there must exist at least one non-vanishing component, AJ JW , with G l #l 4K. Otherwise no non-zero terms would be left in the sum in Eq. (80). The triangular W inequalities l !l 4¸4l #l imply that l 5l !¸, where ¸ is the angular momentum W W W obtained by coupling of l and l . Therefore we also have K5l #l 5l #l !¸ and W W l 4(K#¸). If we now consider a component with l #l 'K then the triangular inequality V W gives us that l 5l !¸'K!l !¸ and therefore l '(K!¸)5(l #l !¸!¸). Thus, V W V V for ¸"0 all components with l #l 'K also have l 'l . V W V For ¸"0 states we can also conclude that inside the potential regions the Faddeev components with l "l fall o! slower with than all other components. This is consistent with the fact that the V scattering lengths aJ , i"1, 2, 3, determine the low-energy properties of any given system at large G distances. This conclusion does not hold in general for ¸'0, where a component with minimum l (l and l #l 'K could fall o! even slower inside the potential regions. For such an l -value V V W V we have l 4¸#l and l "(K!¸)#1(l , which corresponds to a slower fall o! than for W V V that of l "l , see Table 1. V
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4.2.2. The asymptotic degeneracy The asymptotic states in the limit of "R are identical to the free states obtained without interactions. The angular spectrum of the free solutions are given by (R)"K(K#2d!2) and the degeneracy for each of these values is 3D(d, ¸, K), but only D(d, ¸, K) of these are physically acceptable while 2D(d, ¸, K) are spurious states, see Section 3.1.1. We shall now try to "nd how many physical solutions are there for a given K and ¸ for each l between 0 and the maximum value (K#¸). Let ¹(k) denote the number of Faddeev components with l 4k and l #l 4K, where each l -value in each Jacobi set has to be counted. Let S(k) denote V V W V the number of spurious solutions, which can be obtained from these ¹(k) Faddeev components by combining into identically vanishing total wave functions. The total number of components are ¹((K#¸) )"3D(d, ¸, K) and the total number of spurious solutions are S((K#¸) )"2D(d, ¸, K). A solution corresponding to l "k is a linear combination of the ¹(k) components with l 4k, see V Section 4.2. The total number of solutions with l "0, 1, 2, k is thus ¹(k), but S(k) of these are spurious. Therefore P(k)"¹(k)!S(k) is the number of physical solutions with l 4k and thus P(k)!P(k!1)"¹(k)!S(k)!P(k!1) is the number of physical solutions with l "k. We may understand this by stepwise increasing l : the number of physical states with l "0 is the number of components with l "0 minus the number of ways these can be combined into V spurious states with zero norm. The number of physical states with l "1 is given by the number of components with l 41 minus the number of spurious states obtained by combining components V with l 41 minus the number of physical states with l "0. We may continue in this way until we V have obtained the total number of D(d, K, ¸) physical states and 2D(d, K, ¸) spurious states, all found from l 4(K#¸). 4.2.3. Numerical accuracy and the potential cutows r G The accuracy of the numerical procedure is closely related to the choice of r , i.e. the point G outside which the potential < is assumed to be zero. Then < (\ sin )"0 for ' , see G G HI G G G Section 3.3. Using the asymptotic wave function of a state characterized by l and the approxima tion in Eq. (80) give sinB\>J cosB\>JW 4 AJ JW (n#1) G G G G ? G JW 2m ;P B\ >J B\ >JW (cos 2 ) < (\ sin ) DJ JW . (81) L G G HI G G
Using the asymptotic behaviour in Table 1 we obtain, after the change of integration variable to r"\ sin 4\, the simple accuracy estimate HI G HI 2m O(\B\J ) dr rB\>J < (r) for l 'l ,
G PG (82) 4 2m O(\B\J ) dr rB\>J < (r) for l 4l ,
G PG where we introduced the maximum short-range angular momentum quantum number l 50 for which the integrals in Eq. (82) are "nite, see Eq. (C.5). These estimates can be as small as desired by choosing r su$ciently large. Except for the solutions of K"0, where l "0 and "O(\B), we G can even conclude that both absolute and relative error / vanish when PR.
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The freedom in the choice of r can be exploited to obtain a faster convergence of the G eigenvalues, i.e. faster vanishing of with increasing . For r increasing with as r "O(\C), G G where is a small positive number, we use
dr rB\>J
PG
2m 2m < (r) 4rJ \J dr rB\>J < (r) G G
G PG
(83)
and Eq. (82) to obtain the accuracy "O(\B\J >C ) for l (l , where "2(l !l ). For potentials with l "R, e.g. Gaussian, Yukawa and square well potentials, we can therefore choose r J\C causing to vanish faster than any power of . Choosing r as a function of G G does not alter the leading-order terms in Table 1 and the -dependent correction term in Eq. (79). The reason for this is that as long as P0 and Eq. (73) consequently is valid the length scale G decisive for the convergence of each solution is aJ . G The -dependent term in Eq. (79), 2(K#d!1) (/a )\B\J , vanishes faster than in Eq. (82) when l 'l . Dividing the solutions into categories according to the dominating l at large distance is therefore only meaningful for l 4l . Other solutions with l 'l and the same limiting value "K(K#2d!2) couple and all contain a piece of the slowest converging component with the same resulting large-distance behaviour "K(K#2d!2)#O(\B\J ). 4.3. Two-cluster continuum states Let us investigate solutions to Eq. (73) corresponding to large-distance con"gurations with a two-body bound state and the third particle far away. Then we have "O(!), "!(d!1#l )#it and from Eq. (56) therefore "!4t!(d!1), where tP#R. We W assume "rst that l "0 and d(4. V 4.3.1. Weakly bound two-body states The -function for zPR and arg z( may be approximated by [124]
(z)K(2 exp
1 z! log z!z [1#O(z\) ] , 2
(84)
which inserted into Eq. (84) along with the expression for leads to
d i !exp t#i (d!1#l ) cot W 2 2 2
d 2 # exp t#i (1#l ) W (d!2) 2 2
\B A JW G at HI G
1 " exp t#i (d!1#l ) A JW # RJV JW J V J W (!4t!(d!1))AJ V J W . (85) W G H GH 2 2 H$G J V J W
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The argument in RJV JW J V J W ( ) diverges towards !R. By choosing in Eq. (61) su$ciently G GH small and using Eqs. (A.9), (A.10), (10) and (84) we get (86) RJV JW J V J W (!4t!(d!1))"O(exp( (!2 #)t) ) , GH GH where is an arbitrarily small positive number. The terms in Eq. (85) containing RJV JW J V J W (!4t!(d!1)) therefore fall o! faster than the other terms. Thus for non-zero GH coe$cients A JW we "nd to leading order that G d d!2 B\ sin 2 2 t+C . (87) , C, a d!2 HI G 2
Since t must be a positive real number also the scattering lengths must be positive, i.e. a'0. If G a(0 a bound two-body state is not present in the subsystem, see Appendix C. Using Eq. (C.9) we G now obtain
2m " E , (88) a
HI G where E is the (small) two-body energy for angular momentum 0. In this derivation we have assumed that \r ;1 to allow the use of uJV JW appearing in G G Eq. (70) as the wave function of zero energy of subsystem i. The solution in Eq. (88) is only consistent with this assumption when a
+!4C
4.3.2. Bound two-body states The result in Eq. (88) is general and applies to all two-body bound states for any dimension and any partial angular momentum. For each two-body bound state with energy E (0 one solution must approach "(2mE / ) as PR. Let JV (r) be the normalized radial wave function of a two-body bound state between particles G j and k with angular momentum l . Then the function V J V J W ( )" V V W W BJV JW JV (\ sin ) (1#O() ) (89) H G H G J J J J G G HI G G is a solution to Eq. (27) for large for any l allowed by a given total ¸. (O( f (x) )/f (x) is a function W remaining "nite as xP0.) To see this we "rst conclude that an exponential fall for large r of JV (r) G implies that RK J V J W JV JW [JV JW ] ( )"O(exp(!(!2mE / ) ) for 4 . Thus for large HG G H H H a trivial solution to Eq. (27) is J V J W "0 with jOi, l Ol or l Ol . Then we de"ne H V V W W r"\ sin and insert Eq. (89) into Eq. (27) arriving at HI G R d!1 R l (l #d!2) 2m
HI < (r)! HI ! ! #V V # G Rr r Rr r
HI r R R # r #(2d!1)r #l (l #d!2) #O(r\) BJV JW JV (r) 1#O "0 , W W G G Rr Rr (90)
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which is valid to leading order in if "(2m/ )E because JV is a solution to Eq. (C.3). Thus G Eq. (89) is a solution to Eq. (27) with the desired properties. That is for each two-body bound state of particles j and k with angular momentum l and for all V possible values of l allowed by the restrictions imposed by the total angular momentum ¸, Eq. (27) W has solutions, where only one component, JV JW , is non-zero asymptotically for PR. FurtherG more, the contributions from the other components fall o! exponentially because the corresponding mixing only occurs through the kinematic rotation. The leading order eigenvalue related to this solution is "(2m/ )E and the next order term may be derived from Eq. (90) by perturbation theory, i.e.
R R 2m
" E # dr rB\JV H(r) r #(2d!1)r #l (l #d!2) JV (r) . G W W G Rr Rr
Normalization of the total angular wave function, ,1, gives us that
1"
(91)
d sinB\ cosB\ BJV JW JV (\ sin ) G HI G G G G G
KBJV JW B \B G HI
B dr rB\JV (r) "BJV JW HI , G G B
(92) because JV is normalized. Thus the coe$cients are given by BJV JW "\B B . For this solution G G HI we can now calculate the diagonal coupling term Eq. (15), i.e.
R R d(d!2) 1 dr rB\JV H(r) r #dr # JV (r) . Q " G G LL Rr Rr 4 The diagonal e!ective potential in Eq. (17) is then given by
2m (d!1) (d!3)#4l (l #d!2) W W # < , < "E # @ L 4
(93)
(94)
which is the same potential as in the two-body radial equation when the phase space factor is included in the wave function, see Eq. (C.4). Thus, the three-body system for large distance has solutions behaving as a two-cluster system with one of the clusters as a bound state of particles j and k and the other `clustera being simply particle i. To leading order these solutions are independent of the properties of the short-range potentials relying only on the two-body wave function as exponentially falling o! with distance [75]. 4.4. Asymptotic behaviour of the non-diagonal coupling terms Let us use Eq. (20) to calculate the asymptotic form of P , where n and n are asymptotically LLY either two-cluster states or three-body continuum states. Only the wave function in regions of non-zero potentials enters into Eq. (20). Eqs. (68) and (89) give for PR inside the region of non-vanishing < that the total wave function (not the Faddeev components) to "rst order in is G G , ( , , )" BJV JW JV (\ ) [> V ( ) > W ( ) ] G L G L GH G J VG J WG *+ L G VG WG JV JW
(95)
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Table 2 The leading-order behaviour of the coupling terms P between angular eigenstates n and n for PR for total angular LLY momentum ¸"0. The two-cluster states correspond to two-body bound states in system i and i with partial angular momenta (l , l ) and (l , l ), respectively. The eigenvalues of the three-body continuum states, characterized by hyperV W V W spherical quantum numbers K and K and dominated by the partial angular momentum l and l , approach
(R)"K(K#2d!2) and (R)"K (K #2d!2) for PR, see Section 4.2. These results are also valid for ¸'0 L LY if both l 4(K!¸) and l 4(K !¸) n
n
Condition
P LLY
Cluster
Cluster
O(\) O(exp(!) )
Cluster
Continuum
Continuum
Continuum
(i, l , l )"(i , l , l ) V W V W (i, l , l )O(i , l , l ) V W V W l 5l V l (l V KOK K"K
O(\B \JV \) O(\B \J >JV \) O(\B\ J J ) O(\\ J \J )
where the two-body wave functions JV correspond to the energy ( /2m) \. The quantities G L L
, BJV JW and JV now carry the index n to discriminate between di!erent angular solutions. Using G L L G L Eq. (95) we then rewrite Eq. (20) as
BJV JW HBJV JW ?G G LY d sinB\ cosB\ P "! G L G G G LLY
!
L LY G JV JW 2m R< (\ sin ) G HI G JV (\ ) (1#O() )HJV (\ ) (1#O() ) ; G L HI G G G LY HI G G
R BJV JW HBJV JW B G LY HI "! G L
!
B\ V W L LY G J J 2m PG R< dr rB\JV (r)H 2< (r)#r G JV (r)#O(\) . ; (96) G L G GLY
Rr The leading-order behaviour of P may now be obtained from Eqs. (79), (91), (92) and Table 1. We LLY have furthermore for ¸"0 that the components with l "l exhibit the slowest fall o! of BJV JW as G V a function of , see Section 4.2. These results are collected in Table 2, which also is valid for continuum states with l 4(K!¸). For ¸'0 and l '(K!¸) other components than l "l V may fall o! even slower and therefore become dominant for su$ciently large , see Section 4.2. Then the results in Table 2 do not apply. Less detailed results for the ¸"0 and d"3 are obtained in [116]. The coupling terms Q in Eq. (22) corresponding to matrix elements of the second derivative are LLY also needed. Unfortunately, they cannot be calculated in the same way because Eq. (22) contains a sum over all the angular eigenstates. It is however tempting to use Q "RP /R# P P L LL L LY LLY LLY obtained by di!erentiating Eq. (14) as an analogue of the operator expression Q"RP/R#P found from [116]. Then we "nd the asymptotic behaviour to be Q "O(\)P . LLY LLY
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It should however be noted that the in"nite sum hidden in P, (P) " P P , in general LY LLY LYL LL cannot predict the Q J\ behaviour of the diagonal term for a two-cluster state as obtained in LL Eq. (93). As seen from Table 2, all couplings to a two-cluster state fall o! faster than \ except the ("rst derivative) couplings to other two-cluster states corresponding to another two-body bound state in the same subsystem with the same angular momentum. Therefore, if the two-body subsystem does not have another bound state the diagonal Q must from this argument fall o! LL faster than \. Since this is wrong, we can conclude that the in"nite sum over angular eigenvalues in the expression for Q must be taken before the limit PR. Each of the coupling terms do converge for large , but the convergence in and the summation over the in"nitely many angular eigenvalues cannot always be interchanged. This problem is worse if one of the two-body systems has in"nitely many bound states as for particles of both positive and negative charge. Then there must be in"nitely many adiabatic potentials each corresponding to bound two-body subsystems behaving as ! for large . We cannot interchange the in"nite sum in the adiabatic expansion of Eq. (12) with the limit PR. In practice, the hyperspherical adiabatic expansion cannot immediately be used to describe threebody scattering above the three-body breakup threshold for oppositely charged particles. On the other hand, the present treatment may be applicable to systems with the same charges on all particles.
5. The E5mov and Thomas e4ects in d dimensions In 1970 E"mov discovered that a three-body system in three dimensions with short-range interactions and at least two of the two-body subsystems with an s-state of zero energy, must have in"nitely many bound states [46]. A zero-energy two-body s-state is equivalent to an in"nitely large s-wave scattering length a. However, less extreme conditions of "nite scattering length and G a resulting "nite number of bound states are already interesting. These bound states with rather characteristic properties arise in systems where the absolute values of a are much larger than the G ranges of the potential r . G The `E"mov limita of in"nitely many bound states is reached for a/r PR. It is only G G occasionally recognized [125,126] that this is exactly the same limit giving rise to the Thomas e!ect discovered already in 1935 [47]. This e!ect, occurring in three-body systems with zero-range two-body potentials, produces an in"nite number of strongly bound states. Also here a/r "R simply because r ,0 for such potentials. G G G 5.1. Three identical bosons in 3 dimensions The analyses in Section 4 are valid in the limit of <aJV but the key equations, Eqs. (73)}(75), G are correct already when JV sin() in these G G equations is therefore not necessarily large for
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Under these conditions all solutions, dominated by non-zero angular momentum l '0, have already approximately reached the free eigenvalues when
1 R ( )" d d VG WG (4 GH
(#1) sin[ (#1) (!2 ) ] (#1) sin(2(#1) ) \ G H , ; (#1) sin(2 ) (#1) sin(!2 ) G H
(97)
where we can use "0 according to Eq. (10). Then " and the integrand in Eq. (97) is G H GH independent of and immediately resulting in VG VG sin[ (#1) (!2 ) ] GH . R ( )" GH (#1) sin(!2 ) GH
(98)
The three particles have the same mass, then "$/3, see Eq. (8) and we get GH 2 sin( (#1)/3) . R ( )" GH (#1)(3
(99)
We can further simplify Eq. (73) by using the boson symmetry, A "A "A , and the equal scattering lengths a"a"a. Then we obtain 2(#1) cos()#( sin((#1)/3) " . a sin() HI G
(100)
The lowest solutions, "4(#2), to this equation are shown in Fig. 4 as a function of /a for G negative as well as for positive scattering length. The free spectrum, "K(K#4), is approached for <a with the exception of "12 corresponding to K"2 for which no completely G symmetric state exists. For positive scattering length, a'0, the lowest eigenvalue diverges as G
"!(/a)(m/m ) as given by Eq. (88) corresponding to a two-body bound state. At "0 the G G spectrum is shifted downwards compared to the free spectrum where the lowest solution then has "!1#0.503i and "!5.01. The partial angular momenta are zero for the interesting component and the -dependence of the angular wave function is found from Eqs. (A.12) and (54): (#3/2) sin(( !/2)) G . "! A G sin(2 ) (3/2) G G
(101)
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Fig. 4. The lowest angular eigenvalues for 3 identical bosons of mass m in 3 dimensions interacting with zero-range potentials with scattering length a,a. The unit mass in Eq. (2) is chosen as m"m such that ,1. The plot HI G contains both the solutions for positive a (full curve) where the two-body systems have bound states, and negative a (dashed curve). For P0 the spectra for negative and positive scattering length are the same. The lowest eigenvalue is
("0)"!5.01. For PR all the eigenvalues approach the values for no interactions, either from above (positive a) or from below (negative a).
The diagonal e!ective potential in Eq. (17) is < "( /2m) (15/4# () )\, where Q is neglect LL ed as unimportant for the distances of interest in the present context, i.e. r ;;a where G G
"!5.01. These neglected terms arise due to the variation of the angular wave functions, which in our case only vary slowly with , because and nearly are constants. In fact, all the couplings de"ned in Eqs. (14) and (15) are negligibly small, since the angular eigenvalues and therefore also the Faddeev components remain nearly constant over this large region of space. These e!ective potentials are shown in Fig. 5, where the dominating feature is the strong variation for the interesting region of small . From we then obtain the simple small distance behaviour of the e!ective potentials, i.e.
(!1.26)
(!!1/4) < ()" , , 2m 2m
(102)
where "1.006 corresponding to "!5.01. The unnormalized radial wave functions for small energy are solutions for the potentials in Eq. (102), i.e. f ()"( sin( ln(/r )#) , (103) G where is a phase depending on the boundary condition at &r . These solutions are not G coupled to components related to the higher potentials, since the coupling constants are negligibly small. We can therefore assume that f "0 for n52. By counting the number of nodes between L r and a in f we get a fairly accurate estimate of the number N of bound states, i.e. G G a . (104) NK ln G r G
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Fig. 5. The e!ective potential, ; "< 2m/ , corresponding to the lowest angular eigenvalue for 3 identical bosons of mass m in 3 dimensions interacting with zero-range potentials. The diagonal coupling term Q is not included. The LL s-wave scattering length a is used to scale the axes into dimensionless quantities. The plot exhibits the solutions for both positive a, where the two-body systems have bound states, and negative a. For ;a the potential approaches !1.26\.
Thus for a"$R or r "0 we have an in"nite number of bound states. These two limits are G G precisely where the E"mov or Thomas e!ects occur. Improved radial wave functions corresponding to these bound states with "nite energies can be found by solving Eq. (13) with only one angular eigenfunction included, neglect of the coupling terms and use of the small distance expression in Eq. (102). We then get f ()"(K (), K
,
2m(!E) ,
(105)
where K is the modi"ed Bessel function of the second kind, which is exponentially decreasing for L large arguments. When the energy approaches zero the wave function in Eq. (105) reduces to Eq. (103). As the contributions from both small and large are small we use Eq. (105) for every in ]0, R[ to obtain
2 , ; " (1#) 3 2m(!E)
(106)
which is similar to Eq. (C.10) for two-body systems deviating only by the constant factor depending on instead of d. The binding energies may be determined as soon as an appropriate boundary condition is provided at a small -value. Such a boundary condition is typically that f ( )"0 for "0 or that the logarithmic derivative of f must equal a speci"c value for &r slightly outside the G range of the short-range potential. With K ( )"0 the bound state energies are found from K "z / , where z are the zero points of the Bessel functions, i.e. K (z )"0, n"0, 1, 22 . L L L K L Using ;1 the wave function in Eq. (103) is an accurate approximation and we obtain that
E. Nielsen et al. / Physics Reports 347 (2001) 373}459
z Kz exp(!n/) and therefore L 2 E "exp ! n E , L
413
(107)
where E is the energy of the "rst of these E"mov state. Size and energy are related by Eq. (106) and the size therefore increase exponentially, i.e.
2 "exp n . L
(108)
As is almost constant over the region in in which the E"mov states exist, they all have almost identical angular wave functions as given in Eq. (101). In other words, these three-body states exhibit the same angular correlations and di!er essentially only by size and number of nodes in the radial wave function. From a general non-relativistic SchroK dinger equation of N particles,
, ! #<(r , 2, r )!E (r , 2, r )"0 , , , 2m G G G we see that the simultaneous transformations
(109)
r C r , E C E , < C < , G G r ,tr , E ,t\E, < (r , 2, r ),t\<(t\r , 2, t\r ) (110) G G , leave the SchroK dinger equation unchanged for any positive number t. Thus, the SchroK dinger equation is invariant under scalings, where all lengths are multiplied by a number t and all energies are multiplied by t\. The E"mov states follow these general scaling rules for length and energy in quantum mechanics, but now in addition also within the E"mov series, i.e. E C E "Et\,
C " t ,
(111)
where t only can be an integer power of exp(/), i.e. t"exp(n /). In this case the scaling applies within the spectrum itself without any scaling of the two-body potentials, i.e. by applying the scaling in Eq. (111) on energy and size of a single E"mov state, the same quantities of another E"mov state in the same E"mov series is obtained. The reason for this additional symmetry is that the kinetic energy and the e!ective potential in Eq. (102) both scale as t\ under the transformation in Eq. (110). Only the cuto! at small , which de"nes the energy of the lowest E"mov state, is not invariant under this scaling. 5.2. Occurrence conditions for the Thomas and Exmov ewects Let us now try to determine the restrictions on dimension, angular momenta and masses, which allow the E"mov and Thomas e!ects. The requirement is simply that the lowest radial potential is of the form given in Eq. (102) for r ;;aJV . This is ful"lled if the lowest angular eigenG G value is less than !(2d!3) (2d!1)/4!1/4"!(d!1) such that "!(d!1)!, where is a positive number, see Eqs. (17) and (102). The number of E"mov states for a system of three bosons in three dimensions increases proportional to for a given ratio between the
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scattering length and the range of the potential. This dependence is expected to hold in general although may di!er from the value 1.006 found in Eq. (102). Also the angular part of the wave function of the E"mov states may di!er from Eq. (101) in the general case. 5.2.1. Dimensional requirements Let us return to Eq. (73) and again study the limit JV sin() is then large G V G unless is an integer. Thus, the free spectrum is obtained even when ;aJ unless the ampliG tude AJV JW vanishes. The only di!erence compared to the result in Eq. (78) is that we now have G "n#O(\B\JV ) instead of "n#O(\B\JV ). Analogously, for d#2l "4 we "nd from V Eq. (74) that now "n#O([ln() ]\). Therefore for a solution to Eq. (73), where a component with l "l is non-zero for ( ) > ( ) ] "(!1)*[> ( ) > ( ) ] and " . The integrals in Eq. (7) are now VH * WH * WG * WG * H GH easily computed and we "nd (#1)P B\ >* B\ (!cos 2 ) J GH (!cos )* R * *( )" GH GH (#1)P B\ B\ >*(1) J d d # #¸ 2 2 d F !, #¸#d!1; #¸; cos (!cos )* , " GH GH d d 2 #¸ # 2 2
(112)
where we used that > is normalized and Eq. (A.2). We rewrite Eq. (73) as * d d d (#1) # #¸ sin() sin # B\ 2 2 2 ! # A * G a d!2 d d sin # (#d!1#¸) HI G 2 2 2
d d # #¸ 2 2 d " F !, #¸#d!1; #¸; cos (!cos )*A * , GH GH H d d 2 H$G #¸ # 2 2 (113)
where only components with a
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Fig. 6. The lowest angular eigenvalue ("0) (full curve) for 3 identical bosons with total angular momentum ¸"0 obtained from Eq. (113) is shown as a function of the dimension d and compared with !(2d!1) (2d!3)/4!1/4" !(d!1) (dashed curve). The E"mov e!ect occurs if (0)(!(d!1), i.e. for 2.3(d(3.8.
Again the dimension d"2 needs special treatment. We can either take the limit of dP2 in Eq. (113) or equivalently insert Eq. (112) into Eq. (75). In both cases we obtain the same result for d"2, i.e.
sin() cos()# (#1#¸)# (#1)#2#2 ln A * G a HI G (#1#¸) F(!, #¸#1; 1#¸; cos ) (!cos )*A *"0 , (114) # GH GH H ¸!(#1) H$G which is a generalization of the result for ¸"0 obtained in [39]. Again the free spectrum is found for <a but also for ;a, because ln(/( a)) diverges in both cases. Therefore the G G HI G lowest eigenvalue approaches 0 both when
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Fig. 7. The lowest angular eigenvalue ("0) for a three-body system with total angular momentum ¸"0 in d"3 dimensions in the limit r ;;a as a function of the mass ratios m /m and m /m when m 4m 4m . G G
rotation angles explicitly given in Eq. (8) are functions of the two mass ratios m /m and m /m GH both then between 0 and 1. We can solve Eq. (113) for ;a for given (m /m , m /m ) assuming G that all three s-wave scattering lengths are large compared to the ranges of the potentials. The resulting lowest eigenvalue is shown in Fig. 7. Three equal masses correspond to the point (m /m , m /m )"(1, 1), which has the largest (least negative) value of ("0)"!5.01, and the e!ects are then least pronounced. As one mass increases compared to the other two, i.e. m /m decreases below 1, then ("0) decreases. When one mass decreases compared to the other two, i.e. m /m decreases below 1, ("0) also decreases. As m /m P0, ("0) diverges to minus in"nity, because both and then converge towards /2 and K(m /m (1#m /m P0. This in turn leads to divergent o!-diagonal terms in Eq. (113), cf. Eq. (98). This con"rms that the E"mov and Thomas e!ects are possible for all mass combinations in d"3 dimensions, if all three subsystems have a resonance at zero energy or if all the potentials are of zero range. In addition, Fig. 7 illustrates that the most favourable condition, the largest value of or equivalently the smallest ("0), occurs when one of the particles is light compared to both the other two. Then the largest number of E"mov states is present for a given ratio of the scattering lengths to the range of the potentials. 5.2.3. Two resonant subsystems From Eq. (104) we see that the scattering length must increase by a factor of exp(/)&20 to make room for one more E"mov state in a system of three identical bosons in 3 dimensions. That is to have, for example, 2 E"mov states the s-wave scattering length must be at least 400 times the range of the potential. In physical units this means for nuclei or atoms with potential ranges of about 5 fm or 5 As that the scattering length must be at least 2000 fm or 2000 As , respectively. The resonance energy of about /(2a) must correspondingly at least be smaller than about 1 eV or 10\ eV, where we used a light nuclear mass of 5 nucleons. Thus, the resonance energy must be
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extremely close to zero. To "nd systems where three di!erent scattering lengths simultaneously are su$ciently large is very unlikely. However, the chances increase substantially if one or more of the two-body subsystems are identical. (The advantage is lost if all three particles are identical fermions, since then the totally symmetric E"mov states are not physically allowed.) The pertinent question in this connection is then how many of the two-body interactions must simultaneously produce resonances at extremely small energies. To answer this question we "rst assume that only one scattering length is very large compared to the range of the potential. Then only the corresponding Faddeev component is non-zero for
#d 2 A *"0 . G sin d 2
sin
(115)
Then must be real and Eq. (56) gives "(2#¸) (2#¸#2d!2)5!(d!1), which excludes occurrence of both the E"mov and Thomas e!ects. Therefore, at least two of the two-body subsystems must have an s-wave resonance at zero energy. This conclusion is valid for all dimensions and angular momenta. The next step is to assume zero-energy resonances in precisely two subsystems, for example between particles 1, 2 and 1, 3, but not between 2, 3. Then A *"0 in Eq. (113), and the 2;2 determinant in the limit P0 becomes
d sin # 2 d sin 2
d d #¸ # 2 2 d d # #¸ 2 2
d (!cos )* "0 . ! F !, #¸#d!1; #¸; cos 2
(116)
This equation in general and in particular the value of the lowest eigenvalue now only depends on the masses through "arctan(m (m #m #m )/m m 3[0, /2]. This decisive dependence is shown in Fig. 8 which exhibits ("0) as a function of for d"3 and both ¸"0 and ¸"1. For ¸"0 the ("0) function is always below the critical value of !4 except in the point "/2. However, ("0)(!4 only for (0.37 for ¸"1. We conclude that for all mass combinations the E"mov e!ect is possible in three dimensions except for "/2 where ("0)"!4. This value corresponds to particle 1 being in"nitely heavy relative to at least one of the particles 2 and 3. Thus, the E"mov conditions are hard to meet for a light particle unless its interaction with one of the heavier particles produces a resonance at zero energy. Chances improve as () diverge towards !R as P0, i.e. in the limit when both m ;m and m ;m .
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Fig. 8. The lowest angular eigenvalue extracted from Eq. (116) as a function of given by Eq. (8) for a three-body system with angular momentum ¸"0 and 1 in three dimensions in the limit where r ;;a for j"2, 3 and H H <a.
The same trends are found for ¸"1, where (0) only is su$ciently negative ( (0)(!4) to produce the E"mov e!ect when (0.37. That is the E"mov e!ect only occurs for ¸"1 when particle 1 is light enough, i.e. when m ((!m !m #((m #m )#0.60m m ). The obvious next question is now if we can derive similar conditions for arbitrary ¸. To achieve this we "rst try to "nd a value of such that "!(2d!3) (2d!1)!1/4! "!(d!1)! is a solution to Eq. (116) for a value of '0. Then the related "!(d!1#¸)#i, see Eq. (55) and the derivation in Section 4.3. With this Eq. (116) can now be rewritten as
d 1!¸ #i sin (1!¸#2i) #¸ 2 2 2 $ d 1#¸ d sin #i 2 2 2
"(!cos )*F
d!1#¸ d!1#¸ d !i, #i; #¸; cos , 2 2 2
(117)
where both sides of the equation are real when is real, see Eq. (A.3). For ¸51 and any value of 50, Eq. (117) has a solution with one of the signs in Eq. (117). This follows, since the left-hand side is a constant in and the right-hand side is a continuous function of , which takes the value 0 for "/2 and diverges towards $R for P0. To compensate on the right-hand side when P0 we must have PR and consequently P!R. It is thus proven that an arbitrary low value of can be obtained for any ¸ by choosing the value of su$ciently small. The symmetry of the wave function must follow the sign in Eq. (117), because the corresponding coe$cients in Eq. (113) must obey the relation A *"$A *. The interchange of particles 2 and 3 must therefore give the phase (!1)*, i.e. even ¸ for bosons and even or odd ¸ for fermions depending on the symmetry of the spin wave function.
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5.3. Conclusion The E"mov and Thomas e!ects only exist in spaces with a dimension between 2 and 4 or more precisely between 2.3 and 3.8. Thus, the only integer dimension in which the e!ects exists is d"3. At least two of the two-body subsystems must have in"nite s-wave scattering lengths to obtain the E"mov e!ect, i.e. one of the particles must have in"nite scattering length when interacting with both of the others. Let us call this particle the resonant particle. Large scattering length for higher angular momenta cannot produce the e!ect. The Thomas e!ect only occurs when at least two of the potentials are of zero range. For given dimension, total angular momentum and mass ratios, all E"mov states only di!er in size and number of nodes in the radial wave function. Within each E"mov series the mean square radius increases by a constant factor from one state to the next whereas the three-body binding energy decreases by the same factor. This factor is constant within each E"mov series, but otherwise it depends on dimension, angular momentum and mass ratios. All mass combinations allow the e!ects for d"3 and total angular momentum ¸"0, but most E"mov states appear for a "xed ratio of scattering length and interaction range when the resonant particle is light compared to both the two others. In the opposite case, where the resonant particle is much heavier than one or both of the two others, the attractive e!ective potential is only barely above the critical value and therefore very few E"mov states are present. For ¸'0 the E"mov e!ect only exists when the mass of the resonant particle is su$ciently small compared to the other two masses. Turning this argument around the maximum total angular momentum, where an in"nite E"mov series exists for a given set of mass ratios, increases with decreasing mass of the resonant particle. For molecular systems with two heavy particles and one light resonant particle an E"mov series exists for each total angular momentum up to a maximum value depending on the mass ratios.
6. Three-body systems in two dimensions In this section we shall brie#y review the most pertinent universal properties for two dimensional systems and illustrate with atomic helium trimers. Further details and discussions can be found in [39]. For d"2 we conclude from Eq. (114), valid for
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Three identical bosons in the weak binding limit give two bound states, ground (g) and excited states (ex), with energies E and root mean square radii R, i.e. E"16.52E , R"0.111a,
E"1.267E ,
E "!4 exp(!2) , 2a
R"0.927a ,
(118)
where a is the s-wave scattering length, the reduced mass and as Euler's constant, see Eq. (C.9). Hence, the properties of the bound states only depend on the scattering length contrary to the E"mov states in 3 dimensions, where the range of the two-body potentials also enters, see Section 5.1. This model independence in two dimensions implies that the same scattering length produces approximately identical states. The generalization to systems with three di!erent particles requires more parameters. However, a similar model independence still arises in the limit of weak binding, but the number of bound three-body states could now be one as well as two. The three-body binding energies still only depend on the inverse square of a scattering length, but the exact combination could be a complicated function of the ratios of masses and scattering lengths. All short-range potentials, except a central attraction and an outer repulsive barrier, cannot produce bound three-body states unless one of the two-body subsystems has a bound state. Thus, Borromean systems are unlikely in two dimensions and their properties di!er very much from those in 3 dimensions. As examples we use the atomic helium trimers with the realistic LM2M2 potential [130]. The energies of the three-body bound states are given in Table 3. Both He and He have two bound states. The ratios between the three-body binding energies and the two-body binding energies are smaller than predicted in Eq. (118). The scattering lengths of He}He and He}He are then too small to produce the model-independent results. The asymmetric systems, HeHe and HeHe , each only have one bound state with ¸L"0>. When one of the atoms in He is replaced by He the kinetic energy is increased whereas the potential energy is unchanged. Therefore, the very weakly bound excited state of He must move to an energy above the two-body threshold, which also remains unchanged. When another He is Table 3 Bound state energies for the four helium trimers in 2 dimensions in absolute values in units of mK and relative to the binding energy of the most strongly bound two-body subsystem, which therefore provides the threshold. The second and third columns give the smallest s-wave scattering length (in atomic units) and the two-body energy (in mK) of that two-body subsystem. The He atom is treated as a spin 0 boson and He is in a fully symmetric state. The symmetric trimers each have two bound states whereas the asymmetric systems only have one System
a
E (mK)
E (mK)
E /E
He
42.312
!38.384
HeHe He He He
42.312 131.96 2343.4
!38.384 !3.8190 !0.013134
!172.1 !38.61 !68.06 !12.57 !0.1884 !0.01616
4.48 1.0059 1.773 3.29 14.32 1.230
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replaced by He the most strongly bound two-body subsystem is He}He, which is far less bound than He . Therefore, there is still room for a bound state in HeHe although it lies above the two-body breakup threshold in HeHe . The He system is symmetric with two bound states. When one of the He atoms is replaced by He the two-body threshold decreases. The lowest angular eigenvalue is then much lower, but receives asymptotically only contributions from those two Faddeev components corresponding to the two identical He}He subsystems, see Section 4. In contrast a symmetric system receives contributions from all three components. Therefore the attractive pocket in the corresponding e!ective radial potential has to be more shallow for asymmetric than for symmetric systems. In the present case this di!erence is su$cient to remove the three-body excited state.
7. Helium trimers: accurate numerical calculations The examples in this section are the weakly bound atomic helium trimers in three dimensions [17,29,30,67,131]. Precise calculations of the energies and wave functions should "rst of all reveal detailed information about the structure of these small molecules and in particular of the suggested E"mov state. Second, such investigations should provide insight into the general structure of weakly bound three-body systems and of the nature of E"mov states. Third, it is a severe test of our method for two reasons, i.e. the very long-range correlations present in E"mov states are naturally described in our method, but very hard to obtain numerically with conventional methods, and also the strong repulsion at short distance in the two-body interactions increases the numerical di$culties. These computations provide a good starting point for investigations of the possibility of adjusting the two-body interactions by an external "eld designed to sweep the region close to in"nite scattering length. This would allow controlled observation of the E"mov e!ect. In this report we shall build on [30,105] and restrict the detailed discussions to illustrate the method and to extract the universal properties. 7.1. The two-body phenomenological potentials The atomic helium}helium interaction is extremely weak and of the van der Waals type. It is therefore at large distance dominated by induced dipole}dipole interactions behaving as r\, where r is the inter-atomic distance, i.e. the potential is of short range and the maximum short-range angular momentum is l "1 according to Eq. (C.5). The dimer, He , is bound [132,133] and the size measured to be r !r "120$20 a.u. (atomic units of length, 1 a.u."0.524177 As ), which is consistent with the theoretical predictions for the wave function and the binding energy of around 1 mK"0.0861735 eV"3.16679;10\ a.u. Such a small number requires high precision and the interaction is very di$cult to calculate with su$cient accuracy from "rst principles. The atomic interaction models used in this context are therefore often "ts to various measured thermal properties of the helium dimer. We shall use the LM2M2 potential [130], which has an extremely large short-range repulsion, <(0)"2.1;10 K"6.6 a.u., compared to the attractive minimum, <(5.6 a.u.)"!10.97 K" 3.47;10\ a.u. The narrow attractive pocket is followed by a quickly decreasing tail. We shall also
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Table 4 The s-wave scattering length a and e!ective range r of the LM2M2 interaction [130] in 3 dimensions for three atomic dimers He}He
He}He
He}He
a
r
a
r
a
r
189.054
13.843
!33.261
18.564
!13.520
25.717
use simpler model potentials, i.e. Gaussian <(r)"S exp(!r/b), exponential <(r)" S exp(!r/b) and square well <(r)"S (r(b), to study the model dependence and especially the e!ect of the repulsive core. These simple potentials have shallow attractions extending all the way to the center in contrast to the repulsive core and the narrow attractive pocket for the LM2M2 potential. The potential strength and range parameters are adjusted to give the same s-wave scattering length, a, and e!ective range, r , as for the LM2M2 potential shown in Table 4. These parameters [30] depend on the reduced mass of the two-body system and therefore on the isotope content of the dimer even though the initial two-body interaction remains unchanged. The large positive scattering length for He re#ects the presence of a barely bound state whereas the other two negative scattering lengths show that these two-body systems are unbound. The e!ective ranges are roughly 3}5 times larger than the diameter of the atoms. How far the simple potentials can be exploited is an interesting question, i.e. which accuracies can be obtained with the corresponding relatively fast estimates. In dimers and trimers with more than one He we should account for the Pauli principle. With only two of these fermions we can treat the systems as consisting of spin zero bosons, since the ignored spin degree of freedom, having exceedingly little in#uence on the interaction, is taking care of the antisymmetry. For the He system this is no longer possible, but now the proper symmetry results in a system far from being bound. To learn more about general structures we study instead the arti"cial system of three identical spin zero bosons with the mass as He. The three-body interaction in Eq. (1) is very small [134,135] and does not contribute at the present level of accuracy. We shall neglect it in the investigations of the atomic helium trimers. 7.2. The adiabatic potentials We can now solve the angular part of the three-body problem for di!erent isotope combinations and di!erent potentials. For (100 a.u. we solve Eq. (42) whereas for '100 a.u. we solve Eq. (71), where we only include components with l "0. For the Gaussian and exponential V potentials we switch to Eq. (71) at "300 a.u. The sizes of the bases in the expansion in Eq. (41) give full convergence for the 4 most signi"cant digits of the ground state binding energy. We need to include partial angular momenta up to l "14 for LM2M2 and 4 for Gaussians. For l "0 we use V V 150 and 80 basis states for the two interactions. For LM2M2 and l "14 we use 20 polynomials V and 30 for the Gaussian for l "4. For ¸"0 the total number of basis states is 840 and 1500 for V He and HeHe with LM2M2 and 405 and 590 with Gaussians.
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Fig. 9. The four lowest normalized s-wave angular Faddeev components as functions of for He in three dimensions G at "100 a.u. for the LM2M2 potential and the total angular momentum ¸"0. All three Faddeev components are identical.
For illustration corresponding to the four lowest angular eigenvalues is plotted in Fig. 9. G L The wave function for the lowest of these -values has a peak at small and falls rapidly o! re#ecting the bound two-body state. The higher lying eigenfunctions exhibit an increasing number of oscillations in . The angular eigenvalues are all extremely large for small hyperradii re#ecting the strongly repulsive core, see the examples in Figs. 10 and 11. One eigenvalue diverges parabolically for He and HeHe when PR due to the two-body bound state, see Eq. (91). The down wards divergence begins shortly before "1000 a.u. in agreement with the value of a " 12/a+760 a.u in Eq. (78). The remaining eigenvalues approach K(K#4) in agreement with Eq. (79). Only odd (even) values of K appear as the free spectrum at large distance due to the odd (even) parity. Furthermore, there is no totally symmetric states with K"1, 2 for He . For He one eigenvalue corresponding to l "0 approaches each K(K#4) value from above, because all 3 Faddeev components are identical. Thus, only one component couples to itself. For HeHe we now "nd a much denser spectrum due to fewer symmetry restrictions. In other words, additional levels are present. These `extraa eigenvalues approach their asymptotic values from below due to the negative scattering length of the He}He subsystem. For HeHe the features of the previous spectra remain except that now the diverging level has disappeared, since no more bound subsystem is present. All two-body scattering lengths are negative and therefore all the eigenvalues approach their asymptotic values from below, compare with Eq. (76). The eigenvalues corresponding to the LM2M2 potential have many avoided crossings and increase dramatically for small due to the repulsive core in that potential. The Gaussian potential is "nite at the center and the eigenvalues obtain instead their free value according to Eq. (51) for "0. This should in principle also occur for the LM2M2 potential, but the value of this potential at "0 is extremely large and in practise an almost diverging potential appears.
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Fig. 10. The angular eigenvalues as function of hyperradius for the He -trimer calculated with the LM2M2 potential (solid curves) and the Gaussian model potentials (dashed curves) for angular momentum and parity ¸L"1\.
Fig. 11. The angular eigenvalues as function of hyperradius for the HeHe -trimer calculated with the LM2M2 potential (solid curves) and the Gaussian model potentials (dashed curves) for angular momentum and parity ¸L"1\.
The angular eigenvalue spectra for the Gaussian potential coincide with those of the LM2M2 potential for approximately '100 a.u. strongly indicating that only the low-energy scattering properties are important when is larger than a few times the range of the potentials. At distances outside the e!ective range the two spectra of these wildly di!erent potentials are remarkably similar as predicted. On the other hand, for (100 a.u. major di!erences occur. Clearly, "nite and smooth two-body potentials allow much faster and more accurate numerical computations. Whether this is realistic and su$cient for a given problem has to be evaluated for each case.
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We can analyze the degeneracy of ¸L"0> for the asymptotic spectrum for the asymmetric systems, HeHe and HeHe , by noting that for each K, which must be even due to parity, we have K/2#1 possible values of l "l varying from 0 to K/2, since 04l #l 4K. If we now V W V W choose Jacobi system i such that particle j and k are identical we can see that l has to be even. V Therefore the degeneracy is the nearest integer below K/4#1. Due to symmetry we have only two diwerent s-wave Faddeev components, ( ) and ()" (). Therefore at most two of G G H I the K/4#1 asymptotic states are of the type l "0. Similarly, we have one Faddeev component with l "l "1 (one is forbidden due to symmetry), two with l "l "2 etc. We can therefore "ll V W V W up each K-level asymptotically: First we have two states of the type l "0, then one of the type l "1, then two with l "2, etc., until the number of possible K/4#1 states is reached. For He the degeneracy is somewhat more complicated, because of the higher symmetry requirements. Formal group theory must be used to obtain the degeneracy for each K-level. However, the degeneracy for each l within each K-level is simple, because only even values of l "l are allowed for all the Faddeev components and furthermore all components must be V W identical. Therefore we have one state with l "0, none with l "1, one with l "2, none with l "3, etc., until the entire K-level is full. The recipe for "nding the degeneracy and l -splitting at "R for ¸L"1\ is the same as for ¸"0. Let us "rst study HeHe . To write a basis for the wave function it is simplest to choose the Jacobi system with the x-coordinate connecting the two He. The allowed components in this system are now (l , l )"(0, 1), (2, 1), (2, 3), 2 up to l #l "K for each value of odd K-values. V W V W That is each K-level has (K#1)/2 physical solutions. These solutions can be built up of two independent components with (l , l )"(0, 1), one component with (l , l )"(1, 0) which asympV W V W totically couples to one with (l , l )"(1, 2), two with (l , l )"(2, 1) which couples to the two with V W V W (l , l )"(2, 3), etc. Hence, there will be two states with l "0, two with l "1, four with l "2, two V W with l "3, etc., until (K#1)/2 states are reached. For He this derivation is again more complicated, but we can conclude that the solutions are a subset of the solutions for HeHe , simply due to the increased symmetry requirements. Speci"cally, we see that components with odd l and thus solutions with odd l are forbidden. V 7.3. Bound states: energy and structure The calculated angular eigenvalues and coupling constants are now used in Eq. (13) to obtain the bound state energies and wave functions. In three dimensions He has two bound state with angular momentum and parity ¸"0>, HeHe has one bound state with ¸"0> and HeHe is unbound. If treated with proper account of the Pauli principle also He is unbound. None of these trimers are bound for ¸"1\ and it is therefore also expected that none are bound for ¸"1> and for higher total angular momenta. The computed energies of the bound states are given in Table 5. In the numerical procedure to solve the radial equation we need more adiabatic potentials for HeHe than for He due to the denser angular spectrum. We also need many more adiabatic potentials to obtain the same precision for the LM2M2 potential than for the Gaussian and the exponential potentials. This is directly related to the repulsive core which introduces extra avoided crossings into the angular eigenvalues. The radial wave function must be very small under the small distance repulsive barrier. Unfortunately, the contribution to the energy is still not negligibly small,
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Table 5 Bound state energies in mK for the helium trimers in 3 dimensions using various two-body potentials. The number in the brackets, , is the minimum number of adiabatic potentials included in the calculation to obtain this result. The He has two bound states whereas HeHe has only one. No other trimer with stable isotopes has bound states in 3 dimensions. For the LM2M2 potential we give the results for only one adiabatic potential to compare with the result in [17] He
Potential
HeHe
Ref. [17]
!105.9 !125.2 !106.1
1 8 1
!2.121 !2.269 !2.118
1 8 1
!9.682 !13.66 !10.22
1 10 1
Gaussian
!150.7
3
!2.485
3
!18.82
5
Exponential
!174.2
3
!2.731
3
!24.61
5
LM2M2
since the small probability has to be multiplied by the large values of the adiabatic potentials. High-accuracy therefore requires careful treatment also of this region. To improve the accuracy we have here exploited the arbitrary -dependent constant () appearing in Eq. (13) and discussed in Section 3.2.4. In Table 5 we compare our results, where only the lowest adiabatic potential is included, with those of [17] obtained with the same approximations. For He we "nd very good agreement for both ground a excited state, while the result for HeHe deviates by 6% of the three-body energy of about !10 mK. A much bigger basis for the asymmetric systems con"rmed that our results are converged and our computations are more accurate than this discrepancy. In [17,29] it was concluded that the excited state of He behaves as an E"mov state. If the strength of the LM2M2 potential is decreased by 3%, i.e. multiplied by 0.97, the potential is no longer strong enough to bind this state. On the other hand, if the potential is increased by 20% stronger, i.e. multiplied by 1.2, the state moves into the continuum. Of course, this state becomes more bound as the potential is strengthened, but the binding energy of the two-body bound state increases even faster. Consequently, the continuum threshold catches up with the energy of the excited state, which then becomes unbound with respect to disintegration into a dimer and a free helium atom. Therefore, by studying this excited state, the ground state and the bound state of HeHe we expect to obtain valuable information about E"mov states. The geometric structure of the ground states of He and HeHe both resemble equilateral triangles where HeHe is slightly prolonged in one direction. The distribution of the He atom in HeHe is qualitatively similar to that of one of the He atoms in He . The lighter mass of He, the resulting larger reduced masses or the larger kinetic energies, explains why HeHe is less bound than He and simultaneously twice as big. The He atom has a tendency to be further away than He causing an extra asymmetry and an additional prolongation of HeHe . The excited state of He , the E"mov state, is about 10 times as spatially extended as the ground state and the size is of the same order as the two-body scattering length of 189 a.u. This is in agreement with the interpretation of the state as the least bound E"mov state in a potential similar
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to that of Fig. 5. According to Eq. (108) this size should approximately be 23 times the size of the ground state. Similarly from Eq. (107) we conclude that the binding energy should be 23"520 times less than that of the ground state. In fact, by replacing 23 by 10 we obtain very comparable numbers for both size and energy. This is a rather good agreement, since the approximations used in deriving Eqs. (107) and (108) are not really valid in the present case, where the state both is the "rst and the last of the E"mov series. 7.4. Helium trimers in external xelds We have seen in the previous subsection that even in the He trimer, where the two-body scattering length is 10 times the e!ective range, there is only room for one E"mov state. It is therefore very unlikely that nature should provide systems with more than a few E"mov states. External "elds may provide manipulation possibilities as for Bose}Einstein condensates [84,136] of Rb and Na, where an external magnetic "eld via the hyper"ne structure of the atoms moves down a resonant state in one of the higher lying two-body channels thereby producing a Feshbach resonance at zero energy in the two-body system [137,138]. For E"mov states one major problem is that these two-body subsystems have a lot of bound states below the resonance at zero energy. Then the E"mov states are must appear as resonances above these two-body thresholds and slightly below the three-body threshold. 7.4.1. Sweeping the threshold region with a perturbing potential A static external electric "eld E polarizes an atom and induces a dipole E, where is the polarizability of the atom. To lowest order the two-body potential then receives an additional contribution [105] 16 > ( ) , <(r)"! E (119) 5 r
where r is the vector separating the two atoms, is the angle between E and r, and and are the polarizabilities of the two atoms. The two atoms within our three-body system are in a relative s-state. Therefore the "rst-order contribution from the potential < vanishes, since it is proportional to > > > "0. The contribution in second-order perturbation arising from <, which then is fourth order in both E and polarizability, always lower the ground-state energy. Therefore, the potential in Eq. (119) only contributes to second order and decreases always the ground-state energy. This means that higher order correction terms to the potential itself might produce lower order changes (in polarizability and "eld strength) to the ground-state energy. The terms arise as the "eld induces a dipole in one atom, which in turn induces a dipole in the other atom and these two dipoles interact [105]: > ( )#(1/(5)> ( ) E , <(r)"! ( # )(4 r
(120)
which is second order in E, but third order in the polarizabilities. The contribution, proportional to the matrix element > <> , is now negative. Thus, to move a resonance of two helium
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atoms towards the threshold of zero energy, we must use the unbound systems He or HeHe. Here HeHe is closer to the threshold and the most promising combination of helium trimers is therefore HeHe , where a resonance at zero energy in the two identical He}He subsystems would produce an in"nite E"mov series of bound states. The unperturbed LM2M2 potential is now modi"ed by adding < and < with a cuto! at distances smaller than about 1 a.u. The s-wave scattering length a of He}He is "rst calculated as a function of the "eld strength. We integrate the coupled two-body equations of the di!erent angular momentum channels from a small distance r +1 a.u. to a large distance r outside the e!ective range. Then we match the s-wave function to the form (r)J1!a/r and the higher angular momentum wave functions to (r)Jr\\J. The value of r as well as the number of J angular momenta included is varied in the computation to reach stability for a. For HeHe the scattering length decreases from a"!33.261 a.u., the value without "eld, to !R as the "eld strength increases to E"0.053 a.u., where the system becomes bound. Increasing E above 0.053 a.u. the scattering length decreases from #R towards zero. For He the divergence is at E"0.067 a.u. The three-body binding energies are remarkably similar for the LM2M2 and Gaussian potentials, see Table 5. We shall therefore use the Gaussian potential in the calculations of the HeHe trimer in the external electric "eld, i.e. <(r)"S ( E ) exp(!r/b) with constant range and the depth as function of the strength of the "eld adjusted to reproduce the computed scattering length. The three-body system becomes bound at E"0.045 a.u. while the two-body subsystems still remain unbound below E"0.05319 a.u. For this strength interval we then have a Borromean system. Increasing the "eld strength above 0.045 a.u. we obtain an excited state in the three-body spectrum at E"0.053 a.u. This strength is very close to the two-body threshold of E"0.05319 and it is rather di$cult to determine accurately the energies of the in"nitely many excited states, which successively must appear below this threshold. We "nd the "rst excited three-body state at E"0.05315 a.u., at E"0.05316 a.u. there are both a two-body bound state and two excited states of the trimer. At E"0.05319 a.u. the second excited three-body state is no longer found in the computations. Its binding energy is less than !3;10\ mK, beyond which our calculations are unreliable. 7.4.2. Appearance of Exmov states in external electric xelds The characteristic features of E"mov states around the two-body threshold are obtained numerically, but disappointingly only a few of these states are found. The reason is that only two of the subsystems are close to having a resonance at zero energy and furthermore these identical subsystems correspond to two light and one heavy particle. Let us in the following label the He atom by 1. The decisive parameter is in Section 5.2.3 found to be " arctan(4.003(4.003#2 ) 3.016)/3.016"1.127. According to Eq. (116) the lowest angular eigenvalue is "!4.118 when r ;;a and G G therefore the critical parameter determining the distance between the E"mov states is "0.343 as discussed in Section 5. Hence, a new E"mov state would occur when the scattering length is increased by a factor exp(/)"9442. This clearly demonstrates that an extreme "ne tuning is required to allow a larger number of E"mov states. Since a "eld strength of 0.053 a.u. corresponds to the very high value of 2.7 V/As , we conclude that it is rather unrealistic to hope for more than one E"mov state in this system.
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The knowledge and experience collected so far may, however, be used to determine the optimum requirements for observing E"mov states by "ne tuning with an external electric "eld. First, the atoms or molecules should interact via short-range interactions, which undisturbed almost are able to bind the dimers. Second, the atoms should have a large polarizability, but not so large that the dimers become bound via the generated attractive van der Waals forces. Third, according to Fig. 8 it would be an advantage, if one of the atoms is light compared to the other two, while these two heavier atoms should be unable to form a two-body bound state. Under these special circumstances the trimer of one light particle and two heavy particles would be a good candidate for observing the E"mov states. An alternative is to relax the restrictions and search for E"mov states as resonances instead of bound states [139]. Then there are many more possibilities to "nd tunable systems both with a static electric "eld through the polarization of the atoms and with magnetic "elds through a Feshbach resonance.
8. Nuclear three-body halos The examples in this section are nuclear halos in three dimensions, i.e. the hypertriton H (n#p#), a three-body system weakly bound below the n#p bound subsystem [91], and the Borromean halos He (n#n#He) and Li (n#n#Li) without bound subsystems [5,40,54,123]. The hypertriton provides information about the strange sector of the strong interaction. Neutron halos are found at the neutron dripline [13}15,43,45,87]. They have been rather successfully described in terms of few-body models [5,33}36]. Antisymmetrization between core (He, Li) particles and the neutrons is treated by adding a short-range twobody potential behaving as r\ at short distances [42]. This strictly diverging repulsion, the small binding and the large spatial extension all require careful treatment and our method is designed for that. 8.1. Hypertriton: the simplest strange halo The hypertriton ( H) is the lightest nucleus with "nite strangeness [91]. This three-body system may roughly be described as a -particle bound to a deuteron, where the binding energy is B "0.13$0.05 MeV and the total angular momentum and parity are J">. If the spatial extension of a two-body system is large compared to the range of the interaction the root mean square radius is given by r + /(4B "10.2 fm, where is the reduced mass and the parameters are for the -deuteron system. Thus, r is roughly 5 times larger than the deuteron root mean square radius of about 2 fm. The hypertriton is the most pronounced nuclear halo found so far. 8.1.1. Two-body potentials We need two-body interactions reproducing the low-energy properties of the potentials, i.e. basically scattering length and e!ective range. For the nucleon}nucleon interaction the main component corresponds to the quantum numbers of the deuteron, i.e. the triplet s- and d-states
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with the total angular momentum and parity J"1>, but also nucleon}nucleon relative p-states are possible in H. For the -nucleon interaction we also include s, p and d-states. For each isospin we parametrize the spin dependence of the interactions with central, spin}spin, tensor and spin}orbit terms each with Gaussians as the radial shapes. We use GC1 for ¹"0 (N}N) and G5r for ¹"(N}) both from [91] and ¹"1 (N}N) from [90]. This set of parameters reproduce the nucleon}nucleon scattering lengths and e!ective ranges for the singlet and triplet s-states. In addition, the computed deuteron properties, binding energy B "2.224575 MeV, root mean square radius R "1.967 fm, d-state admixture P &5.66%, B B B asymptotic ratio of the d to s-wave component "0.0233, and the electric quadrupole moment B Q "0.273 fm, are all in agreement with the experimental data. The -nucleon potential is B adjusted to the Nijmegen SC interaction [140], but with the strength of the central potential reduced by 10% to reproduce the all important H binding energy [91]. 8.1.2. The hypertriton structure Only s, p and d-waves contribute. The partial wave Faddeev components are "rst chosen consistent with parity, angular momentum coupling and isospin conservation or symmetries of the system. We include up to 100 basis states for each component. The angular eigenvalues are then for small hyperradii computed by solving Eq. (42) with the basis modi"ed to include spins as indicated in Section 3.4. For large hyperradii we use the diagonal part of the large-distance asymptotic expansion [23,91]. The angular spectrum equals the free spectrum K(K#4) at both "0 and "R, since the potentials are "nite and of short range. In the Jacobi system, where the x-coordinate connects the nucleons, we "nd one level for K"0 corresponding to (l , l , ¸, s , S)"(0, 0, 0, 1, 1/2). For K"2 V W V we get "ve levels related to (l , l , ¸, s , S)"(0, 0, 0, 1, 1/2), (1, 1, 0, 0, 1/2), (1, 1, 1, 0, 1/2), V W V (2, 0, 2, 1, 3/2), (0, 2, 2, 1, 3/2). All eigenvalues decrease for small consistent with Eq. (51) re#ecting the overall attraction. The lowest eigenvalue diverges parabolically for PR as
"!2B m / , due to the bound state in the neutron}proton two-body subsystem. The free B , spectrum is recovered for PR, since one eigenvalue originating from each of the higher values at "0 decreases and replaces the missing level at in"nity. The calculated angular eigenvalues and the related coupling constants are now used in Eq. (31) to obtain the bound state energies and radial wave functions. The hypertriton binding energy is determined with a relative accuracy of about 10\ already by including the lowest three adiabatic potentials. We obtain then B "0.108 MeV, root mean square radii r "5.88 fm for the three-body system and r "11.73 fm for the }deuteron two-body system. The main B contribution to the wave function is by far given by the lowest component, which is peaked at around 3 fm with a rather slow exponential decay at larger distances. Signi"cant probability is still found at 30 fm strongly indicating the need for an accurate treatment at these relatively large distances. The total wave function is a sum of the three Faddeev components, where each is expressed in its Jacobi coordinates. Transforming all components into the Jacobi coordinates of the set, where the -particle is a spectator (x-coordinate between the two nucleons), we "nd that 95% of the probability is from the component with l "0. The remaining 5% almost completely arises from V the component with l "2 representing the d-wave contribution to the deuteron. The deuteron V structure is almost maintained in the presence of the -particle.
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Fig. 12. The hypertriton probability arising from the dominating component. The coordinates are the nucleon}nucleon distance r and the }deuteron distance r . The volume element is included in the probability. V W
Transforming the total wave function into one of the other sets of Jacobi coordinates, where one of the nucleons is the spectator, we "nd that components (l , l , ¸, s , S)"(0, 0, 0, 0, 1/2), V W V (0, 0, 0, 1, 1/2), (1, 1, 0, 0, 1/2), (1, 1, 0, 1, 1/2) contribute by 55.0%, 18.5%, 18.7% and 6.0% of the probability in the wave function, respectively. All of these correspond to s or p-waves, and the -nucleon relative angular momentum 2 therefore gives negligible contributions. The dominating component (95%) in the hypertriton wave function corresponds to a relative neutron}proton s-state, and a relative }deuteron s-state. The contribution of this component to the hypertriton probability distribution is shown in Fig. 12 as a function of the neutron}proton distance r , and the }deuteron distance r . This distribution is elongated along the r -direction, V W W re#ecting the large value of the -deuteron root mean square radius compared to the deuteron root mean square radius. In other words, the -particle is essentially always found far outside the deuteron. 8.1.3. Information about the strange strong interaction The previous conclusions all arise from the chosen set of two-body interactions. However, they are not completely determined by the imposed constraints and small uncertainties are present especially in the theoretical }nucleon interaction. We have therefore varied the shapes of the radial formfactors still maintaining the same scattering lengths and e!ective ranges. The hypertriton properties are only marginally a!ected. The binding energy at most changes by 50 keV and usually much less. This is a substantial fraction of the hypertriton binding energy of 130 keV, but very insigni"cant in comparison with the strengths of the two-body interactions typically 10}100 MeV. The singlet s-wave in the -nucleon potential is the dominating component. Thus, the correis the most signi"cant individual parameter. To maintain the sponding scattering length a , by measured binding energy within an acceptable uncertainty interval we can at most vary a , about 10% around the value !1.85 fm. It is then tempting to conclude that the hypertriton binding energy, via these accurate calculations, determines this parameter with this precision.
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This is also a rather safe conclusion, but we want here to emphasize that an additional uncertainty should be included from a possibly more realistic weaker attraction producing the usual small underbinding, which subsequently is compensated by the three-body attractive potential. In any case precise experimental values and accurate computations of the hypertriton give access to information about the strong interaction, both the o!-shell behaviour and the strange sector. Again it is worth emphasizing that the precision must be very high, because otherwise only the simplest low-energy properties like scattering lengths can be extracted. 8.2. Borromean two-neutron halo nuclei: He and Li Large interaction cross-sections of some light nuclei close to the neutron dripline [141] are indications of a spatially extended neutron halo around an ordinary nuclear core [142]. Large Coulomb and nuclear two-neutron dissociation cross-sections [142}145] and narrow momentum distributions of the fragments after breakup [54,146] are other signs of this peculiar structure. The neutrons are mainly in classically forbidden regions at distances much larger than the range of the neutron-core interaction. Examples of Borromean two-neutron halo nuclei are He (#n#n) and Li (Li#n#n) [5]. Their three-body binding energies or equivalently their two-neutron separation energies, 973.4$1.0 keV [147] and 295$35 keV [148], are much larger than for beta-stable nuclei. Unlike the hypertriton, these systems have no two-body bound subsystems. On the other hand, the presence of neutrons within the core forbids some of the additional three-body neutron states due to the Pauli principle. This has to be carefully considered. 8.2.1. Two- and three-body potentials The neutron}neutron potential is the isospin 1 part of the nucleon}nucleon interaction given in [90]. The neutron}He interaction reproduces the s, p and d phase shifts from zero and up to 20 MeV [42,98]. The experimental data on neutron}Li is much more limited and uncertain than that of neutron}He. There is strong evidence for a low-lying p-resonance at 500$60 keV with a width of 400$60 keV [149}151]. There is also accumulating evidence for an even lower lying somewhat uncertain virtual s-state approximately at 0.15$0.15 MeV [150,151]. Both these states are necessary to explain some of the fragmentation data [40,54,144,145,152,153]. The Li-core has a spin of , which inevitably produces two levels for each orbital angular momentum arising from the neutron spin of . The measurements refer to these actual (spin-split) levels. We then choose a realistic model with spin splitting reproducing the available data [102,123]. To bene"t from the fairly accurate approximation assuming zero core spin we then simply reduced the spin splitting parameter to zero [42,98]. This implies a p-resonance at 0.75 MeV and an s-state energy of 0.58 MeV, which then necessarily must be above the measured values. Accurate computions with these sets of two-particle interactions lead to three-body binding energies of He and Li too small by 0.5 MeV and 150 keV compared to the experimental values [7]. Unfortunately, the correct three-body binding energy is crucial, since essentially all other properties are strongly correlated with this energy. Adjusting the two-body interactions simultaneously alter the two-body continuum properties like resonances. Three-body continuum properties therefore cannot be expected to be described correctly [101]. To "ne tuning the energy we use a three-body interaction as in Eqs. (1) and (13). This is not a genuine three-body force, but a phenomenological way of accounting for those polarizations of
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the particles, which are beyond that described by the two-body interactions. Thus, this interaction must be of short range, since it only contributes when all three particles interact simultaneously. So far our model assumed complete decoupling of relative and intrinsic particle degrees of freedom. This assumption is inconsistent with the full antisymmetrization required when the di!erent particles contain identical fermions. Using the same deep neutron-core s-wave potential as appropriate for the neutrons in the core, the halo neutrons would naturally occupy these s-states, which already are occupied by the neutrons within the core. To overcome this problem one should project out the undesired overlap of the three-body wave function with the Pauli-forbidden two-body neutron-core state [42,90]. To achieve this we investigated three methods, which only di!er insigni"cantly for weakly bound systems. The "rst consists of excluding the lowest adiabatic potential, which essentially entirely otherwise would support the forbidden state. The second consists of modifying the two-body interaction to exclude the lowest bound forbidden two-body states, while still maintaining the scattering length and e!ective range of the potential. The third consists of adding a short-range two-body potential with a speci"c r\ divergence at small distance still maintaining the same phase shifts for all energies, but without the undesired Pauli-forbidden two-body bound states in the resulting `phase-equivalenta two-body potential [42]. The third method basically removes the lowest adiabatic potential both at large and at small distances. In the numerical computations we shall use the third method for s-waves while the p -states for the Li-core are removed from the active space by a strong repulsion incorporated as an unphysically large spin}orbit interaction. The di!erent methods produce almost indistinguishable results, but the computing time could di!er substantially. The interaction parameters are from [42,98] resulting in binding energy and root mean square radius of 1.0 MeV and 2.50 fm for He, and 0.30 MeV and 3.35 fm for Li. The energies agree clearly with the measured values and the radii are consistent with 2.57$0.10 fm [154] and 3.1$0.3 fm [155] obtained from the data for He and Li, respectively. 8.2.2. The two-neutron halo structure We include s, p and d-waves for He and only s and p-waves for Li, where we furthermore assume zero spin for the Li-core. The basis size of up to 50 states in the expansion in Eq. (41) is substantially smaller than for the hypertriton, since the spatial extension is much smaller and no Pauli allowed two-body bound states are present. The angular eigenvalues are "rst computed by solving Eq. (42) with the basis modi"ed to include spins as in [23,98]. The "ve lowest eigenvalues of the calculated spectrum are shown in Fig. 13 for the three-body halo nuclei He and Li. The use of the phase equivalent potential for the s-wave neutron-core interaction automatically excludes the -function originating from "0 and diverging towards !R for large . This excluded level corresponds to the Pauli forbidden bound states occupied by core neutrons. Therefore, the lowest originates instead from 12. This -function is mainly of p-wave character in the neutron-core relative wave function. The eigenvalues approach the values of the free spectrum for large distances. The lowest of these converges towards zero at !R. The lowest decisive eigenvalue for He is around the minimum dominated by the K"2 contribution. For Li one of the levels originating from 32 at "0 decreases very fast, passes through one sharp and one smooth avoided crossing, and becomes the lowest level at values of larger than 2 fm [42]. This eigenvalue is of s-wave character in the neutron-core relative wave
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Fig. 13. The angular eigenvalues as function of hyperradius for He (solid lines) and Li (dashed lines) calculated from the interactions speci"ed in [42,98,90].
function. Therefore, the Li wave function is dominated by contributions originating from the K"4 level with one node, i.e. mainly of s-wave character. The variation in the small distance behaviour is consistent with Eq. (51) when we consider the moderate p-wave attraction and the strong s-wave attraction overcompensated at short distances by the r\ repulsion. The degeneracy 2 for the bunch originating from 12 arises from s or p-waves coupled to 0 or 1. The calculated angular eigenvalues and the related coupling constants are now used in Eq. (13) to obtain the bound state energies and radial wave functions of these two Borromean systems. The binding energies are obtained with a relative accuracy of about 10\ by including only the lowest two adiabatic potentials in Fig. 13. We "nd the energies and root mean square radii given above in agreement with the measured values. The radial wave functions for the lowest two adiabatic potentials are shown in Fig. 14. These radial wave functions related to the lowest adiabatic potential give for both nuclei more than 90% of the total probability. The wave function is spatially more extended for Li than for He, re#ecting the smaller binding energy and the s-wave versus the more con"ned p-wave character. To investigate the spatial distribution we again transform two of the Faddeev components into the Jacobi coordinates of the third. Using the Jacobi coordinates, where the core is the spectator, the main contributions for both He and Li correspond to relative s-states both between the two neutrons and the core and the dineutron centre of mass. This component contributes 90% and 83% of the He and Li total probability, respectively. In the other Jacobi coordinate, where one of the neutrons is the spectator, He receives 90% of the total probability from components with relative p-waves in the -neutron motion. For Li the Li-neutron relative s-state gives 60% of the probability and the relative p-state provides the remaining 40%. The probability distribution of the dominating component in the total three-body wave function for He is shown in Fig. 15 as a function of the neutron}neutron distance r and the core-nn V
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Fig. 14. The radial wave functions as functions of hyperradius for He (solid lines) and Li (dashed lines) corresponding to the lowest two adiabatic potentials in Fig. 13. The relative normalization is correct.
Fig. 15. The He probability distribution arising from the dominating component of s-waves in the Jacobi coordinate system where the neutron}neutron distance is r and the }nn distance r . The volume element is included in the V W probability.
distance r . We observe two maxima, where the position of the highest corresponds to a triangular W three-body geometry with the two neutrons separated by about 2 fm and the -particle slightly more than 2 fm away from the two-neutron centre of mass. The second maximum is close to an aligned geometry with the two neutrons around 4 fm apart and the -particle at the centre of mass of the two neutrons.
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Fig. 16. The Li probability distribution arising from the dominating component of s-waves in the Jacobi coordinate system where the neutron}neutron distance is r and the }nn distance r . The volume element is included in the V W probability.
In Fig. 16 we show the completely analogous probability distribution for Li. Only one dominating maximum appears, but the distribution is in general more irregular. The position of this maximum again corresponds to a triangular three-body geometry with a separation between the neutrons of around 2 fm and the same distance between the Li-core and the centre of mass of the neutrons. The maximum is clearly wider for Li than for He and consequently the three particles in Li have a larger probability of being far apart than in the He case. This re#ects again, as in Fig. 14, the di!erent binding energies and the di!erence in partial wave character for these systems.
9. Continuum structure and scattering The hyperspherical adiabatic expansion is in the previous sections applied in investigations of bound state properties. The method is also well suited for studies of low-energy continuum spectra and scattering processes. When at most one particle is charged the method is directly applicable and in particular exploited for Borromean systems [98}100] and -scattering on a deuteron [91]. When more than one particle is positively charged the Coulomb interaction is inevitable and the detailed picture of Section 4.2 is not applicable. Still the general numerical method can be used as seen in a three-alpha model calculation of wave function and resonance position and width of the astrophysically interesting second 0> state of C [22]. Other processes like neutral atom three-body recombination, B#B 3B, in#uences the stabil ity of Bose}Einstein condensates. A full three-body calculation must include all bound two-body subsystems, e.g. 1464 bound states in Rb when each atom is in a triplet state [156]. To gain physical insight we shall use the method to investigate the recombination rate for the reactions
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He#He 3He in three dimensions. Other applications are tempting both in molecular, atomic and nuclear physics. However, before speci"c applications we must formulate the general procedure. 9.1. Scattering in the hyperspherical adiabatic approach The radial wave functions must be labeled by two channel indices n, the usual index on the components, and n , an additional index labeling di!erent total wave functions in Eq. (12). We have G to solve the hyperradial equation (13) with the boundary conditions at large distance given by [99,158], i.e.
f G ()" L L
m ( H\( )!S G H> ( ) ) for E5E , L L IL L L 4 L LG IL L
m K ( ) 4 IL L
(121)
for E(E , L
where E is the energy of the three-body system, E ,lim ( /2m) \ is the asymptotic energy L M L of channel n, ,(E!E )2m/ , H!(x) is the Hankel function, K (x) the modi"ed Bessel L L I I function of the second kind and S G are the elements of the S-matrix. The order of the Bessel L L L and Hankel functions is "K#2 if channel n is a continuum channel with the angular L eigenvalue asymptotically approaching "K(K#4) and "#l if channel n is a two-cluster W L L channel where the third particle has orbital angular momentum l relative to the remaining W two-body system. The energy E is always larger than E when the two-body system is unable to L form bound states. The coupling constants P between two-cluster states in Eq. (13) only fall o! as LLY \, see Table 2. Therefore numerical integration is necessary up to a very large hyperradius before reaching the asymptotic form in Eq. (121). Scattering experiments involving only the three particles in a Borromean system are not possible. However, positions and widths of resonances in the continuum spectrum often carry information about physical reaction processes. The importance is ampli"ed for Borromean systems, where only one or perhaps a few bound states are present. Resonances can be computed from the energy dependence of the phase shifts of the radial wave functions. The S-matrix is "rst calculated and diagonalized for each energy resulting in eigenfunctions and eigenvalues. The eigenvalues are then converted into eigenphases by assigning the form exp(2i ). The identifying feature of a resonL L ance is a rapid variation of the eigenphase shifts with energy while passing the value of /2 [157]. This procedure is rather di$cult and not very accurate due to the inherent ambiguities of such phase analyses in multichannel systems. A more precise and robust method is to solve the radial equations in Eq. (13) directly by using a complex energy E"E !i/2 with the boundary conditions in Eq. (121). These solutions correspond to poles of the S-matrix at these complex energies [158]. For Borromean systems no two-cluster channels are present, i.e. we have always E5E . The resonances are then those L solutions, where only the H> function of Eq. (121) is present in the asymptotic region, where H> turns into an outgoing (hyper)spherical wave, i.e.
2 3 exp #i#i K# (H> ()P )> 2 2
.
(122)
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The complex energy E and the corresponding wave number " !i produce time and large-distance spatial wave functions proportional to exp(iEt/ )"exp[ (iE !/2)t/ ], exp(i)"exp[ (i #)] , (123) which routinely is interpreted as a spontaneously decaying, exp(!t/2 ), pre-formed state with the constituents escaping towards in"nity and their probability distribution therefore increasing asymptotically with the distance . The complex energy method can be further supplemented with the phase shift analysis. Narrow resonances with small imaginary values of the energy must necessarily result in rapid variations of the phase shifts. On the other hand, large imaginary values of the energy produce smooth variations of the phase shifts. The three-body poles obtained in this way reveal properties of the three-body system and may therefore contribute to measurable quantities. The connection between the three-body continuum structure and the underlying continuum structure of the two-body subsystems is not properly established. 9.2. Continuum structure of nuclear halos The bound state properties were discussed in connection with the hypertriton H and the two Borromean nuclei He and Li. The hypertriton has the deuteron as one bound two-body subsystem, He and Li have spin zero or 3/2\, respectively of both the core and the three-body ground states. Scattering of a -particle on a deuteron provides further information about the }nucleon interaction. For energies below the deuteron breakup threshold the radial functions f () decrease L exponentially except of that corresponding to the lowest adiabatic eigenvalue. This function is at large distances turning into a phase-shifted sine function in the variable . When the }deuteron relative kinetic energy approaches zero we obtain the scattering length a +18 fm and the B e!ective range r +4 fm, where we used the interaction in Section 8.1 for this spin 1/2 scattering B channel. When is the }deuteron reduced mass we have the approximate connection to the binding energy of B +0.13 MeV as
1 B " . (124) 2a 1!r /a B B B The Borromean two-neutron halo nuclei He and Li are easily excited into the continuum by distant Coulomb collisions. The dipole mode is then the leading-order excitation and the 1\ continuum structure becomes decisive for the size of the corresponding Coulomb dissociation cross sections [14,98]. None of these nuclei have documented 1\ resonances, but He has a narrow 2> resonance at 0.82$0.025 MeV with the width 0.113$0.020 MeV [159]. We "rst compute the angular eigenvalue spectra for the JL"2> and JL"1\ channels for He with the interactions speci"ed in [42,98,90]. There is a pronounced attractive pocket in the lowest adiabatic potential for JL"2> and a smaller pocket for JL"1\ [98]. The spin dependence of the two-body interactions is responsible for that di!erence. The lowest poles of the S-matrix is computed as E"(0.82!i0.093/2) MeV for JL"2> (consistent with measurements) and E"(0.95!i0.38/2)MeV for JL"1\.
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Fig. 17. Real and imaginary parts (left) and absolute square (right) of the radial wave function of He for JL"2> for the energy E"(0.82!i0.093/2)MeV, where the S-matrix has a pole. The dashed lines are full numerical calculations and the solid lines are the asymptotic Hankel functions H>(). Courtesy of A. Cobis [160]. L
Fig. 18. The same as Fig. 17, but for JL"1\ and the S-matrix pole energy of E"(0.95!i0.38/2)MeV. Courtesy of A. Cobis [160].
In Fig. 17 we show the dominating component of the radial wave function for the relatively narrow JL"2> resonance. The wave function resembles that of a bound state with a large concentration of probability at small distances in the pocket region of the potential. The wave function begins to fall o! outside the pocket and only then the exponential growth, characteristic for decaying states, takes place. The absence of nodes inside the potential pocket indicates that this is the ground state for JL"2>. A small increase of the potential pocket converts this resonance into a bound state with the corresponding pole shifting "rst towards zero and then taking a real negative value. We show in Fig. 18 the corresponding wave function for the lowest pole of JL"1\, where the attractive region disappears in the total e!ective radial potential leaving an extended #at part at
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Fig. 19. The eigenphases corresponding to the lowest angular eigenvalues for He obtained after diagonalization of the S-matrix for the JL"2> (left) and JL"1\ (right) channels. Courtesy of A. Cobis [160].
distances above "5 fm [98]. Any resonance width should therefore be large. The concentration of probability in the pocket region is now much smaller re#ecting the much larger width. Still a small bump is found at small hyperradii. This rather modest enhancement is responsible for a substantial increase of the J L"1\ strength function [98]. We also calculate the S-matrix and extract the related eigenphases shown in Fig. 19. For JL"2> the rapid rise of the phase and the crossing of /2 is a convincing con"rmation of the resonance at about 0.9 MeV found in the complex energy method. Another pole in this channel appears around 1.8 MeV producing many avoided crossings, but now the phases are rather smooth at the crossings consistent with the large imaginary value of the corresponding complex energy. The JL"1\ channel shows the same general picture, although now the energy variations are less strong, but still it is possible to connect the lowest resonance unambiguously to a complex energy solution. The Li halo nucleus has ground-state spin and parity 3/2\ and electric dipole excitations can therefore produce states of JL"1/2>, 3/2>, 5/2>. If the Li core with JL"3/2\ is truly inert in the process and all interactions are spin independent the resulting spectra must be highly degenerate. However, the two-body interactions depend on the spin couplings of the core and the neutron resulting in ground-state spins of Li of either 1 or 2. Both states are expected to be present as resonances in the low-energy two-body spectrum. This means that the continuum spectrum of Li could be rather complicated. With a modest spin splitting of the JL"1\, 2\ of Li the channels with JL"3/2> would be a typical 1\ excitation and contain the largest number of adiabatic low lying states. The corresponding con"guration must contain one neutron in an s-state and the other neutron in a p-state around the core. The 1\ excitation is basically lifting one of these neutrons from s to p or from p to s states. These excitations are strongly dependent on the spin splitting of the neutron-core interaction and therefore the "nite spin of the core is essential. We shall use JL"3/2> as illustration. The corresponding angular eigenvalue spectrum is "rst computed revealing an attractive pocket suggesting low-lying structures in this 1\ channel. The lowest pole of the S-matrix gives the energy E"(0.68!i0.33/2) MeV. The dominating component
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Fig. 20. Real and imaginary parts (left) and absolute square (right) of the radial wave function of Li for JL"3/2> for the energy E"(0.68!i0.33/2) MeV, where the S-matrix has a pole. The dashed lines are full numerical calculations and the solid lines are the asymptotic Hankel functions H>(). Courtesy of A. Cobis [160]. L
Fig. 21. The same as Fig. 19 for Li for JL"3/2>. Courtesy of A. Cobis [160].
of the radial wave function for this resonance is shown in Fig. 20. The concentration of probability in the pocket region is small re#ecting the relatively large imaginary part of the energy. As for JL"1\ in He, the exponential growth sets in leaving only a small trace of small-distance structure. Still this small bump, combined with similar contributions from the continuum states of JL"1/2>, 5/2>, produce substantial e!ects in the low-energy dipole strength function [98]. The eigenphases obtained after computation and diagonalization of the S-matrix show relatively rapid variations and crossings of /2 at about 0.7 MeV and 0.9 MeV, see Fig. 21. The two other crossings of /2 at higher energies show much less energy variation. In all cases are complex energy
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solutions unambiguously related. The near degeneracy is directly related to two similar attractive adiabatic potentials. The wave functions do not show pronounced probability peaks at small distances for these energies. In conclusion the complex energy method allows relatively simple investigations of resonances in three-body systems. Resonances exhibit all the characteristic features found for two-body resonances, i.e. an increased probability in the pocket region of the potential and an asymptotic exponentially diverging tail. The eigenphases increase around the resonance energy and crosses /2. 9.3. Atomic recombination reactions The continuum properties may strongly in#uence scattering reactions. We shall here concentrate on the recombination or the dissociation process involving three helium atoms, i.e. He#He 3He. If we aim at the process from left to right we can numerically integrate Eq. (13), match the solution to the form in Eq. (121) and extract the S-matrix. We use the Gaussian two-body potentials in Section 7 and the two lowest adiabatic channels. The transition probability for breakup of the two-body bound state producing the lowest three-body continuum state, where all three atoms at "R are non-interacting, is then given by P(E)" S " S . The calculated transition probability in Fig. 22 behaves as P(E)JE at low energy and grows smoothly until approximately 9E , where a striking dip of about two orders of magnitude appears. This process is the prototype of one of the mechanisms of decay of Bose}Einstein condensates and it is therefore of practical importance to understand whether this behaviour is speci"c to the helium atoms or perhaps a general phenomenon. This problem is discussed in [107]. 9.3.1. The WKB approximation: hidden crossing theory The hyperspherical version of the hidden crossing theory is used in [106] for positronium formation in positron}hydrogen collisions and in [161] to give both an approximate expression for
Fig. 22. The transition probability S for the reaction He #HeP3He using di!erent methods: The solid line is for coupled channels, the dotted line is the upper limit from hidden crossing theory for the zero-range model, the short and long-dashed lines are the mixed Gaussian and short-range model with and without the Langer correction, respectively. The energy unit is chosen to be E "!1.3 mK.
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Fig. 23. The real part of <() in Eq. (125) as a multi-valued function of in the complex plane for three identical bosons in three dimensions interacting with zero-range potentials with positive scattering length a. The energy unit is again G E "!1.3 mK. We show only the two lowest branches, which asymptotically (P#R) represent the two-cluster state and the lowest three-body breakup channel. Also shown is one of the two paths, which contributes to the breakup reaction in the hidden crossing approach by going around the branch point at "(2.5918#2.9740i) a. HI G
Wannier's threshold law and a rigorous derivation for Coulomb interacting particles. We shall here only consider three dimensions and three identical bosons with total angular momentum 0 and zero-range two-body interactions. We assume that the three scattering lengths a are identical and G positive such that there is one bound state in each two-body subsystem. Therefore, the angular eigenvalues are given by Eq. (100), where is understood as an analytic single valued complex function of with "rst order poles at all integers, except for "$1 where the function is "nite. Eq. (100) implicitly de"nes the multi-valued function () in the entire complex plane. The diagonal potentials in Eq. (17) ( "4(#2), "0, Q +0) LL 4(#2)#15/4 4(#1)!1/4 2m <()" "
(125)
is therefore also a multi-valued function of . On the positive real axis each of these values or branches corresponds to the adiabatic potentials in Eq. (13), which for asymptotically large correspond to initial or "nal states for the reaction. The two lowest branches of this function are shown in Fig. 23. The function <() has a branch point " ,(2.5918#i2.9740) a found by Taylor HI G expansion of Eq. (100), ()K( )#(R/R) (! )#(R/R) (! ), with " " 0.46719#i0.86782 as the solution to R/R"0. Then ()+ $(2(R/R)\(! ), where ,( ) and the second derivative is for " . Going around the branch point changes the branch of (), the sign in front of the square root, and through Eq. (125) <() changes accordingly. The breakup reaction corresponds to a `particlea in an initial state at large moving towards smaller values on the lowest branch of <(). The "nal state is this particle moving from small towards larger -values on the second branch of <(). The transition probability is in the
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hidden-crossing theory obtained as an integral along a continuous path connecting the corresponding points by going around the branch point. Let and denote the classical turning R R points, <( )"E, on lower and upper branch. Then we have [161] RG 2m (E!<() ) , S " exp(iQ (E)#i ), Q (E)" d A A A (126)
A A Q (E)"iS(E)#(E)/2 , (127) A where S and are real, "$/2 is the topological phase [162]. The sum is over all possible A continuous paths c connecting "R on "rst and second branches. There are basically two such paths: one of them connects "R on the "rst branch with "Re , continues away from the real axis and around the branch point and back to the real axis at "Re on the second branch, then up to the classical turning point " and "nally to R "R on the second branch, see Fig. 23. The second path connects "R on the "rst branch with the classical turning point at " , continues back to "Re on the "rst branch, then up R and around the branch point to "Re on the second branch and from there to "R. The square root in the integrand of Eq. (126) changes sign when the path passes the classical turning points. When (Re , i.e. E5<(Re ) on the second branch, only the part of the path, from R "Re on the "rst branch and around the branch point to "Re on the second branch, contributes to S in Eq. (127). All other contributions are real, because then is real and <()(E. The phase di!erence between the two possible paths contributing to Eq. (126) is de"ned in Eq. (127), since the parts of the integrals outside Re contribute equally to both paths and therefore cancel in the phase di!erence. Then c is the path that goes from " , out to "Re , R around the branch point to "Re on the second branch and then up to " . By adding R coherently the two paths contributing in Eq. (126) we get the transition probability
S "4 exp(!2S(E) )sin((E)/2)44 exp(!2S(E) ) , (128) where we used the relative topological phase of between the two paths [162]. For small energies, 0(E(<(Re ), the classical turning point on the second branch is outside the position of the branch point, i.e. 'Re . Then <()'E for -values between Re and R . This part of the integrals in Eq. (126) are then imaginary corresponding to tunneling and R contributes therefore to S(E) and not to (E) in Eq. (128). As EP0 the main contribution to S(E) arises from tunneling through this barrier, i.e.
MR
2m R (<()!E)Kv log , (129)
Re 0 M where we used 2m<()/ Kv \ with v "15/4 on the second branch for large distance. Since "v /(2Em), we get S JET from Eq. (128). R This low-energy dependence is only nearly correct as seen by integrating over all "nal states, constraint by energy conservation, in the breakup channel, i.e. S(E)&
d
S (E) J dp dp V W
p#p E m V W !E J "E G [a] , 2m E
G
(130)
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where p and p are the conjugated momenta of the Jacobi coordinates, see [157] p. 500. Therefore, V W we must have v "4 instead of as used above. The discrepancy is removed by adding < ()" /(8m), the Langer correction to the potential <() [163]. This extra term arises * automatically in the rigorous derivation in [161]. It is a general correction providing substantial improvements in WKB calculations [164,165]. The energy E in Eq. (130) must for zero-range potentials be the only available energy, i.e. the two-body binding energy E " /(m [a]), see Eq. (C.9). To "nd the proportionality constant in G G Eq. (130) we could turn to Eq. (128). However, according to Eq. (102) the lowest branch of <() diverges as <()"!1.26\ for P0 and there is therefore no classical turning point on the "rst branch. Consequently, "R in a zero-range model and sin((E)/2) is undetermined and in this way we can only "nd the upper limit given in Eq. (128), see the dotted curve in Fig. 22 exceeding all the other results. The phase di!erence can only be computed by use of a "nite range potential. We use the Gaussian potentials from Section 7 to provide adiabatic potentials when is on the real axis, while we maintain the potentials of the zero-range model with the same scattering length when is away from the real axis. We thereby get a classical turning point on the lower branch and is calculated as function of energy with and without the Langer correction, see Fig. 24. Both curves passes through 3, i.e. at E"2.8E and E"7.9E , where the calculated transition probabilities in Eq. (128) vanish due to this destructive interference between the two paths. This gives the origin of the dip in Fig. 22. The computed hidden crossing transition probabilities in Eq. (128) are shown in Fig. 22. The position of the minimum only coincides with the coupled channel result without the Langer correction. This is unfortunate, since then neither the correct amplitude nor the correct threshold law is obtained. On the other hand, the results are very good with the Langer correction for energies below 0.2E , but the dip then occurs at too small an energy. The fairly good agreement is somewhat surprising as the model is rather crude.
Fig. 24. Half of the phase di!erence in Eq. (128) as function of energy in units of E . The two-body interaction is the Gaussian found in Section 7 with the same scattering length and e!ective range as LM2M2 [130]. The dashed line is with and the solid line is without the Langer correction.
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9.3.2. The three-body recombination rate The three-body recombination rate at zero temperature is given by [157] S (E)
R "2(2)3 n [a] lim (E/E ) m G G # exp(!2S(E) ) (0)
"8(2)3 n [a] lim sin . (131) (E/E ) 2 m G G # where n is the atomic density [105] and Eq. (128) was used for S (E) . If a is large compared G to the range of the potential we can use the zero-range model, i.e. Eq. (129), v "4 and Re "2.5918 a. This gives [107] HI G
(0)
R "68.4 an sin "7.38 an , (132) m G 2 m G G G where we inserted the computed (0)/2, which is fairly close to 3, see Fig. 24. Therefore a 3% decrease in (0)/2 would shift the result to exactly 3 and change the recombination rate by 100% to zero. This sensitivity is perhaps better appreciated by the comparison in Table 5 of ground state energies for Gaussian and LM2M2 potentials amounting to a 20% relative di!erence. We have from Eq. (102) that the potential behaves as <()"!1.26\ /2m when is between , which is comparable to the range of the two-body potential, and , which is about the R R scattering length a. We may divide the integral in Eq. (127) into three contributions, i.e. from G short range (( ), middle range ( (( ) and long range (Re ' ), where the latter R R R R also includes the path around the branch point. Let us now imagine a system, where the two-body potential can be tuned around the point of in"nite scattering length. The short-distance contribution must be nearly constant for relatively small variations of the potentials. At zero energy this also holds for the large-distance contribution as everything in this region scales with the scattering length. All lengths and energies must be proportional to respectively a and [a]\ leaving the integral in Eq. (126) invariant. With the G G Langer correction we therefore get from Eq. (127)
MR
d (1.01\# "2.01 ln a#constant. G MR Thus in a tuning experiment [166] the recombination rate should behave as (E"0)" #2
(133)
R "68.4 [a]n sin(1.01 ln(a)#) , (134) G m G G where is an unknown phase, nearly independent of small variations of the potential when a is G positive and large. This phase depends on the other hand on the adiabatic potentials at short distance and thereby on the radial shape of the two-body potentials. This is analogous to the E"mov states, located in the interval between and , but still depending on the boundary R R condition provided by the shape of the potentials for ( , see Section 5. R We now consider negative scattering lengths, where the zero-range model is insu$cient for estimates of recombination rates due to the lack of bound states. For large negative a, the G
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large-distance behaviour of the adiabatic potentials, corresponding to three-body continuum states, equals that of the zero-range potential, but for "nite range potentials there may be lower lying adiabatic potentials corresponding to two-body bound states. Branch points connecting two-body and continuum branches must therefore be within a few times the range of the two-body potentials. Thus, the hidden crossing result for recombination rates must be highly sensitive to the shape of the two-body potentials. However, as noticed in [167], the lowest adiabatic potential corresponding to a three-body continuum state has a barrier at large distance, see the dashed curve in Fig. 5. At very low energies the system has to tunnel through this barrier and we can use the WKB approximation to estimate this penetration probability. We "rst add the Langer correction to the potential in Eq. (125) e!ectively removing !. The inner turning point of the barrier, where <( )"E, is for R R very low energies close to the point where <( )"0 and therefore "!1, i.e. R "("!1)"!0.903 a as found from Eq. (100). At large distance K0 and the outer R HI G turning point is then about K2 /(mE. The tunneling probability in the limit of zero energy R is then
P"exp !2
MR
d
4(#1) 2m m ! E K69.5 G [a]E ,
G
(135)
M which has both the E behaviour and the factor [a], see Eq. (134). G Thus in an experiment, where the potentials are tuned around the point where the scattering length is in"nitely large, we must again "nd the behaviour expressed in Eq. (134), but with a di!erent proportionality factor. In fact, the proportionality constants for negative and positive scattering lengths are entirely independent as the recombination processes are completely di!erent. R
10. Summary and conclusions We have investigated quantum mechanical three-body problems with short-range interactions in d dimensions. We have formulated and discussed a general method, derived a number of useful analytical properties, extracted occurrence conditions for the exotic but interesting E"mov structures, applied the method in realistic numerical computations of bound states and scattering properties. The distinguished features of the method are the e$cient treatment of large distances allowing analytical deduction of universal conclusions about spatially extended and weakly bound systems, the formulation of how to deal with bound and scattering states on the same footing and the possibility of high accuracy when both small and large distances turn out to be important. The method combines hyperspherical adiabatic expansion with the Faddeev equations. This expansion divides the degrees of freedom into hyperradial and hyperangular parts. For "xed hyperradius we solve the angular part of the Faddeev equations and obtain a complete basis set of eigenfunctions. The corresponding discrete eigenvalues are functions of hyperradius and related to one of the adiabatic potentials. We eliminate a priori states of unwanted symmetries and spurious
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solutions with zero norm by diagonalizing the kinematic rotation operator and subsequently removing the eigenstates corresponding to the eigenvalue 0. An optimal incomplete basis is used for small hyperradii whereas simpli"ed equations are used at intermediate distances when at least one particle is not interacting with both the other two. Analytical behaviour of the adiabatic potentials are extracted at very large distances. The mathematical formulations are tested on physical systems in realistic numerical calculations. First the atomic helium trimers where the interaction has a strongly repulsive core. The ground state of He is a state of a size comparable to the e!ective range of the interactions. The ground state of HeHe is a weakly bound halo system. The excited state of He is the "rst (and last) E"mov state. It could be classi"ed as a halo state on top of a E"mov state. These trimers are investigated in two dimensions where all asymmetric trimers have one bound state and the symmetric bosons have two bound states. We use the method for the breakup reaction, He#He P3He, since this type of process is relevant for the stability of Bose}Einstein condensates. The second set of examples are the nuclear three-body halos of the hypertriton and the Borromean nuclei He and Li. The hypertriton structure is basically a two-cluster system with the -particle and a deuteron far apart and weakly bound. We derive limits for the -nucleon singlet s-wave scattering length. The bound and continuum state structure of the Borromean nuclei are surveyed. The Pauli principle are accounted for by phase equivalent potentials. The peaks of the density distributions correspond roughly to a distance of 2 fm between the particles. Low-lying 1\ continuum structure increase the Coulomb dissociation cross sections substantially. We discuss the Thomas and E"mov e!ects existing when the s-wave scattering length divided by the e!ective range is an in"nitely large number. We derive occurrence conditions for arbitrary angular momenta and dimensions. For three identical bosons the e!ects are only possible for dimensions between 2.3 and 3.8. We use an external electric "eld to tune to the conditions. For HeHe the "eld strength needed to reach the E"mov limit is 2.7 V/As . A number of the conclusions in this report are universal in the sense that they are independent of the details of the short-range two-body potentials. Beyond a few times the range of the potentials the angular spectrum only depends on the logarithmic derivatives of the radial two-body wave functions for energies given by the angular eigenvalues and the hyperradius. In other words two-body potentials with the same low-energy scattering properties produce identical intermediate and large-distance behaviour. For d(4 the lowest part of the angular spectrum only depends on the s-wave scattering lengths and the two-body binding energies provided the hyperradius is a few times the range of the potentials. For d54 and hyperradii much larger than the absolute values of the scattering lengths these determine exclusively the adiabatic potentials. At smaller hyperradii, still larger than the ranges of the potentials, the eigenvalues depend on the shapes of the potentials. For d"2, 3, but not for d54, there may in addition exist a region in hyperradius where the low energy part of the angular spectrum is model independent but still not converged to the free spectrum. This region extends from a few times the range of the potentials up to the absolute values of the scattering lengths. Thus in such cases the large-distance behaviour of the angular spectrum is independent of the exact shapes of the potentials and only depends on the low energy scattering properties. However, even then the wave function is not necessarily positioned at the
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corresponding large hyperradii. For d"3 the energies and wave functions of the E"mov states depend upon the behaviour at short distance, because the long-range model independent part of the adiabatic potentials does not provide a repulsive barrier that would keep the wave function in this long-distance region. Instead the lowest adiabatic potential is attractive and proportional to !1/ in the region between the range of the two-body potentials and the absolute values of the scattering lengths. In contrast for d"2 a repulsion proportional to #1/ is present. The two bound states, always existing for three identical particles and overall attractive two-body potentials, are independent of the details of the potential in the limit of weak binding. Their energies and root mean square radii are then proportional to the two-body bound state energy and the two-body scattering length, respectively. The large proportionality constant related to the three-body ground-state energy reminds us about the non-existing, but still not very distant, Thomas e!ect. The radius of the excited state is close to the scattering length reminding about the also non-existing E"mov e!ect. For three di!erent particles and d"2 this type of model independence remains but the number of bound states may be reduced to one. The three-body recombination processes in three dimensions also depend on the shapes of the two-body potentials, since the lack of repulsion in the lowest adiabatic potential allows the scattering wave function in the inner region, where the lowest adiabatic potentials are model dependent. For positive scattering length, still in the low-energy regime, the recombination rate is proportional to the fourth power of the scattering length. The in#uence of the details of the potentials on this rate is an oscillating factor depending on the shapes of the potentials. For negative scattering length the process of recombination can only take place at small distance, where the adiabatic potentials and the couplings are strongly dependent on the shapes of the two-body potentials. On the other hand, for d"2 and very large scattering length, there is a centrifugal barrier e!ectively screening o! the short-distance region. Therefore, the zero-temperature recombination rate should be independent of the shape of the two-body potentials even in the limit of large scattering length. Our method may be extrapolated to more complicated systems. First, it could be extended explicitly to 4, 5 or 6 particles. This is again most appealing when the large distances are decisive. Second, the large-distance treatment may be exploited for many-body systems described by appropriate generalized hyperspherical coordinates. This seems to be most interesting for processes like chemical reactions where large distances are essential. Third, it is possible to use the method in descriptions of few-body correlations within genuine many-body systems. Vaguely speaking, this could be two correlated particles interacting with the average of all remaining, possibly many, particles. Fourth, with hyperspherical coordinates for many particles the centrifugal barrier term maintains the usual form, but with a di!erent strength, i.e. l(l#1) is replaced by ( f!1) ( f!3)/4, where f"d(N!1) is the number of intrinsic degrees of freedom for N particles in d dimensions. Fifth, apart from three dimensions, two dimensions attract in particular increasing interest especially in condensed matter physics. Three-body correlations in two dimensions di!er substantially from those of three dimensions. It is quite conceivable that large-distance correlations, the hall mark of our method, could mediate otherwise very unlikely reactions and thereby exhibit catalytic behaviour. In conclusion, we "nd many speci"c applications of our method and in addition a number of possible extensions.
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Appendix A. Jacobi functions The Jacobi functions, P? @(x) and Q? @(x), as well as the Jacobi polynomials repeatedly appear in J J this report. We shall here collect some of their basic properties derived from [124]. They are solutions to the di!erential equation (1!x)y#[b!a!(a#b#2)x]y #(#a#b#1)y"0 ,
(A.1)
where a and b are real constants, is a complex number, and primes denote derivatives with respect to the variable x3]!1, 1]. The Jacobi functions P and Q are regular and irregular at x"1, respectively. We can express P as (#a#1) P? @(x)" [1#O(1!x) ] J (#1)(a#1)
(#a#1) 1 " F !, #a#b#1; a#1; (1!x) , (#1)(a#1) 2
(A.2)
de"ning normalization of P as O(t)/t is a function remaining "nite for tP0. Here F is the hypergeometric function expressed as a power series in x, i.e. a(a#1)b(b#1) ab x#2 , F(a, b; c; x)"1# x# 2!c(c#1) 1!c
(A.3)
where the convergence radius is 1 unless either a or b is a non-positive integer, in which cases the series terminates and the resulting P? @(x) is simply a polynomial of order n with the orthogonality L relation
2?>@>(n#a#1)(n#b#1) . dx(1!x)?(1#x)@P? @P? @" LY L LLY (2n#a#b#1)n!(n#a#b#1)
(A.4)
\ We want to rewrite P in a more suitable form around x"!1. First
(!b) 1 P? @(x)" F !, #a#b#1; b#1; (1#x) J (#1)(!!b) 2
1#x \@ (b)(#a#1) # 2 (#1)(!)(#a#b#1)
1 ;F #a#1,!!b; 1!b; (1#x) . 2
(A.5)
Using (x)(1!x)"/sin(x) the irregular Jacobi function Q is given by (b)(#a#1) Q@ ?(!x)"!cot(b)P@ ?(!x)# J J (#a#b#1)
1 ;F #a#1,!!b; 1!b; (1#x) 2
1#x \@ 2
(A.6)
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and we then obtain the simple relation P? @(x)"cos()P@ ?(!x)!sin()Q@ ?(!x) , J J J
(A.7)
which is valid even when b is a non-negative integer, where (x) has poles and Eqs. (A.5) and (A.6) invalid. This implies that (#b#1) Q@ ?(!x)" J (#1)(b#1)
(!) (#a#b#1) L L (1#n)# (b#1#n) (b#1) n! L L ! (1#!n)! (#a#b#n#1)!ln
1#x 2
1#x L 2 for b"0 ,
0
!
(b)(#a#1) 1#x \@@\ (!!b) (#a#1) 1#x L L L for b51 , (#a#b#1) 2 (1!b) n! 2 L L
(A.8)
where (x) ,x(x#1)2(x#n!1) and is the di-gamma function, i.e. the logarithmic derivaL tive of the gamma function. The function Q@ ?(!x) diverges for xP!1, as (1#x)\@ for b'0 and as ln(1#x) for b"0. J Thus P? @(x) is only regular for x"!1 when is an integer. If is a negative integer then J P? @ vanishes due to the pole in (#1), see Eqs. (A.2) and (A.5). Then the term J (#1)sin()Q@ ?(!x) does not vanish when P!n, see Eq. (A.7) and (#1)P? @(x) still J diverges at "!1. Thus only solutions to Eq. (A.1) with "!n are regular at x"$1. For large absolute values of and 0(arg ( we approximate F and "nd 1 e\ ?2?>@(1!e\ A)\?\ (1#e\ A)\@\ P? @(cos )" J ( (!)\ (e\ J>?>@>A#e ?> e JA) (1#O( \) ) .
(A.9)
We need a"(d!2)/2#l , b"(d!2)/2#l , x"cos(2), where d is the spatial dimension, l and V W V l are the partial angular momenta and is the hyperangle, see Section 2. From Eqs. (A.2), (A.6) and W (A.8) we then get that (#d/2#l ) V (1#O() ) PB\ >JV B\ >JW (cos 2)" J (#1)(d/2#l ) V QB\ >JV B\ >JW (cos 2) J
(A.10)
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1 ! (2 # (1#)# (1##l )#2 ln ) W #O( ln )
for d#2l "2 , V
d!2 d #l # #l V W 2 2 \B\JV (1#O() ) (#d!1#l #l ) V W #O(ln ) for d even and d#2l 54 , V " d!2 d #l # #l V W 2 2 \B\JV (1#O() ) (#d!1#l #l ) V W d # #l V 2 d !cot (1#O() ) else , d 2 (#1) #l V 2 (A.11)
where "! (1) is Euler's constant. A case of special interest is d"3, l "l "0 and therefore V W a"b". The corresponding P is given by J (#) sin(2(#1)) . (A.12) P (cos 2)" J (#1)() (#1)sin(2) Appendix B. Spherical coordinates in d dimensions We shall here give some pertinent details and derive key formulae for an integer dimension d52, see [108] for the details. We divide the Cartesian coordinates (x , 2, x ) into the "rst d!1 B coordinates (x , 2, x ) and the last coordinate x . We de"ne angles and radii r recursively B\ B I I /x )3[0, ]. Thus by r "(x#r "(x #2#x and "arctan(r B B B\ B B B B\ x "r cos , B B B x "r sin "r sin cos , B\ B\ B\ B B B\ x "r cos "r sin 2 sin cos , B B x "r sin 2 sin sin , B B x "r sin 2 sin cos . (B.1) B B The parity transformation is given by r C r , C ! for 34i4d and C # . The B B G G Laplace operator is RB d!1 R K " # ! B , B Rr r Rr r B B B B
(B.2)
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453
where the square of the angular momentum operator for d52 is given by R 1 R !(d!2)cot # K , K "! B R B sin B\ R B B B
K "0 .
(B.3)
The angular eigenfunctions to K "!R/R can be chosen as (1/(2)exp(!il s ), where l is a non-negative integer and s"$1 correspond to the two degenerate solutions related to angular momentum projections of opposite sign. Then the eigenvalues of K are l and the parity of the eigenfunctions are (!1)J . For d'2 we now assume the recursive form of the eigenfunction to be ( , , )"f ( )>B\ ( , , ), >B B JB\ 2J Q B\ 2 JB 2J Q B 2
(B.4)
where >B\ ( , , ) is an eigenfunction of K with the eigenvalue l (l #d!3). JB\ 2J Q B\ 2 B\ B\ B\ We then get the eigenvalue equation and the solutions as
R R l (l #d!3) ! !(d!2) cot # B\ B\ f ( )" f ( ) , B B B R R sin B B B
(B.5)
f ( )"sinJB\ P B\ >JB\ B\ >JB\ (cos ) , B B L B
(B.6)
"(n#l
(B.7)
B\
) (n#l #d!2)"l (l #d!2) , B\ B B
where n is a non-negative integer and l ,n#l . The parity of this solution is B B\ (!1)L(!1)JB\ "(!1)JB . Thus the eigenvalues of K are l (l #d!2), where l are non-negative B B B B integers, and the parity of the eigenfunctions are (!1)JB . The complete set of (unnormalized) angular eigenstates in d dimensions are > (),> B B\ 2 ( , 2, ) J J J J Q B B G\ B\ >JG\ (cos ) , (B.8) "exp(!isl ) sinJG\ PGB\ >J G J \JG\ G G where l 5l 5l 50 and s"$1. The two quantum numbers of three dimensions lm B B\ 2 (l"l , m"sl ), are in d dimensions generalized to d!1 quantum numbers lm , where l"l and B m "(l ,l , l , s). B\ B\ 2 The two angular momenta lK and lK associated with the Jacobi vectors x and y couple to a total V W angular momentum ¸K "lK #lK . For d"3 the simultaneous eigenfunctions of lK , lK and ¸K are V W V W given by C*+ > ( )> ( ) , (B.9) [> V ( ) > W ( ) ] , JV KV JW KW JV KV V JW KW W J W *+ J V KV >KW + are the Glebsch}Gordan coe$cients. where C*+ JV KV JW KW For d"2 the eigenfunctions of the two angular momentum operators lK and lK are V W (1/(2)exp(!is l ) and (1/(2)exp(!is l ), where l and l are non-negative integers and VV V WW W V W
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s and s both are $1. The total angular wave function is an eigenfunction of lK , lK and V W V W ¸K "iR/R #iR/R (eigenvalue s l #s l ), i.e. V W VV WW 1 " exp(!is l !is l ) 2 VV V WW W
1 1 1 " exp !i(s l #s l ) ( # )!i (s l !s l ) ( ! ) . VV WW 2 V W WW V W 2 2 VV
(B.10)
We can now assign the quantum numbers ¸S where ¸"s l #s l and S"sign(s l #s l ). VV WW VV WW For each l 50, l 50 and ¸51 there is a degeneracy of 2, S"$1, corresponding to the same V W size of ¸. Thus, S plays the same role for d"2 as the projection quantum number M for d"3. For each ¸"0 and l , l 51 the degeneracy is still 2, namely arising from s "!s "1 and V W V W !s "s "1. These two states belong to the same eigenspace of the total angular momentum V W ¸K and even rotationally invariant two-body potentials may therefore couple them. For ¸"l "l "0 there is no degeneracy. V W We can generalize Eq. (B.9) to d'3, see [120] for details. The corresponding summations over the two sets of the d!2 projection quantum numbers, m and m , each related to a partial angular V W momentum, then produce the state of total angular momentum quantum number ¸ with d!2 projections MM . Appendix C. Basic properties of two-body systems in d dimensions The SchroK dinger equation for two particles of masses m and m described by the relative coordinates r and interacting via a central potential <( r ) is
R ! #<(r)!E (r)"0 , Rx 2 G G where "m m /(m #m ) and x is the ith component of the relative coordinate. G Expanding the wave function in angular momentum eigenfunctions gives
(C.1)
(r)" (r)> () , (C.2) J J J where "(l , l , , l , s) and are de"ned in Eq. (B.8). By use of Eqs. (B.2) and (B.7) we then B B\ 2 obtain the equation
(C.3)
(C.4)
R (d!1) R l (l #d!2) 2 ! ! #B B # (<(r)!E) (r)"0 , J Rr r Rr r
where we from now on shall use l"l rather than to label the wave functions. B Introducing R (r),r B\ (r) leads to the equation J J (d!3) (d!1)#l(l#d!2) 2 R ! ! (<(r)!E) R (r)"0 , J
Rr r where the centrifugal barrier for d"2 and l"0 is negative.
E. Nielsen et al. / Physics Reports 347 (2001) 373}459
455
Let us de"ne the following moments of the potential:
I, J
dr <(r) rB\>J .
(C.5)
P The maximum short-range angular momentum l is now de"ned as the maximum angular mo mentum l for which I exists and can be made arbitrarily small by increasing r to a su$ciently J large value. The potential is of short range if l 50. For l4l the e!ect on a plane wave from such potentials is small in the region r'r . For potentials falling of exponentially we must have l "!R. The de"nition of short range in [116] is equivalent to l "R. Assuming that <(r)"0 for r'r we "nd for E'0 that is a linear combination of the Bessel J functions r\B J (kr) and r\B > (kr), where k"(2E/ . For E"0 we get B\ >J B\ >J rJ and r\J\B (1 and ln r for d"2, l"0). The scattering length aJ is de"ned by (r) for large r for E"0 by J aJ B\>J r , (r)Jln for d"2 , (C.6) (r)JrJ 1! J r a
where a is positive for d"2. Thus, the scattering length is the node in the zero-energy wave function outside the potential. If aJ3]r , R[ the zero-energy wave function has a node and the system must have at least one bound state with angular momentum l. If the system for d"2 and l"0 does not have any bound state, a must be in the interval [0, r ]. A more detailed discussion of scattering length and e!ective range for d"2 can be found in [168]. The perturbative e!ect of the non-zero potential for r'r is dr rB\ (r) <(r) , (C.7) J P which diverges for l'l , see Eq. (C.6). However, if we assume <(r)"0 for r'r , we rede"ne a short-range potential for all l. Then the scattering length aJ looses any direct physical meaning as it diverges for r PR when l'l . A bound state of energy E(0 must for r'r have the wave function (r)"r\B K (r) , (C.8) J B\ >J
where "(!2E/ and K is the modi"ed Bessel function of the second kind. When r ;1 J the logarithmic derivatives of Eqs. (C.8) and (C.6) must for d#2l(4 approximately be equal at r , i.e. d d!2 l# sin l# 2 2 B\>J 4 , (C.9) E"! d!2 2aJ l# 2
which is valid for zero-range potentials with r ,0. For d#2l54 the bound state energy depends on both aJ and r .
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Let us de"ne a zero-range potential < (r)"lim f (t)<(tr) as the limit of an arbitrary potential R <(r), where the function f (t) is adjusted to maintain a constant scattering length independent of the parameter t. This limiting procedure in general only allows one constraint: the scattering length fully de"ning < (r). The mean square radius for l"0 and d(4 is given by
dr rB>(r\B K (r) ) (4!d)d B\ " (C.10) r " 2(!E) 6 dr rB\(r\B K (r) ) B\ in the zero-range approximation. This expression is therefore more generally valid in the limit of very weakly bound systems, where 2(!E) \r ;1. It is interesting that r as function of d has its minimum for d"2.
References [1] M.C. Gutzwiller, Rev. Mod. Phys. 70 (1998) 589. [2] M.J. Holman, N.W. Murray, Astron. J. 112 (1996) 1278. [3] H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One- and Two-electron Atoms, Springer Verlag, Berlin, New York, 1957. [4] J.M. Richard, Phys. Rep. 212 (1992) 1. ** [5] M.V. Zhukov, B.V. Danilin, D.V. Fedorov, J.M. Bang, I.J. Thomson, J.S. Vaagen, Phys. Rep. 231 (1993) 151. ** [6] W. GloK ckle, H. Witala, D. HuK ber, H. Kamada, J. Golak, Phys. Rep. 274 (1996) 107. [7] J. Carlson, R. Schiavilla, Rev. Mod. Phys. 70 (1998) 743. [8] C.D. Lin, Phys. Rep. 257 (1995) 1. ** [9] G. Tanner, K. Richter, J.M. Rost, Rev. Mod. Phys. 72 (2000) 497. [10] J. Adamowski, B. Szafran, S. Bednarek, B. SteH be, Few-Body Systems 10 (Suppl.) (1998) 189. [11] V.M. E"mov, Com. Nucl. Part. Phys. 19 (1990) 271. * [12] A.S. Jensen, A. Cobis, D.V. Fedorov, E. Garrido, E. Nielsen, Few-Body Systems 10 (Suppl.) (1999) 19. [13] K. Riisager, Rev. Mod. Phys. 66 (1994) 1105. [14] P.G. Hansen, A.S. Jensen, B. Jonson, Ann. Rev. Nucl. Part. Sci. 45 (1995) 591. [15] I. Tanihata, J. Phys. G 22 (1996) 157. [16] B. Jonson, K. Riisager, Phil. Trans. R. Soc. Lond. A 356 (1998) 2063. [17] B.D. Esry, C.D. Lin, C.H. Greene, Phys. Rev. A 54 (1996) 394. * [18] F. Robicheaux, Phys. Rev. A 60 (1999) 1706. * [19] Y. Li, C.D. Lin, Phys. Rev. A 60 (1999) 2009. [20] D.V. Fedorov, A.S. Jensen, Phys. Rev. Lett. 71 (1993) 4103. *** [21] A.S. Jensen, E. Garrido, D.V. Fedorov, Few-Body Systems 22 (1997) 193. ** [22] D.V. Fedorov, A.S. Jensen, Phys. Lett. B 389 (1996) 631. ** [23] E. Nielsen, D.V. Fedorov, A.S. Jensen, E. Garrido, The three-body problem with short-range interactions, preprint, IFA, University of Aarhus, 2000. [24] R. Martin, J.M. Richard, T.T. Wu, Phys. Rev. A 52 (1995) 2557. [25] J.M. Rost, J.S. Briggs, J. Phys. B 24 (1992) 4293. * [26] J.M. Launay, M.Le. Dourneuf, J. Phys. B 15 (1982) L455. [27] P. Honvault, J.M. Launay, Chem. Phys. Lett. 287 (1998) 270. [28] J. Yuan, C.D. Lin, J. Phys. B 31 (1998) L637. [29] T. Cornelius, W. GloK ckle, J. Chem. Phys. 85 (1986) 3906. [30] E. Nielsen, D.V. Fedorov, A.S. Jensen, J. Phys. B 31 (1998) 4085. *** [31] T. Yamazaki, Few-Body Systems 10 (Suppl.) (1999) 151. [32] Chi-Yu Hu, A.A. Kvitsinsky, Phys. Rev. A 46 (1992) 7301. [33] L. Johannsen, A.S. Jensen, P.G. Hansen, Phys. Lett. B 244 (1990) 357. *
E. Nielsen et al. / Physics Reports 347 (2001) 373}459 [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84]
457
J.M. Bang, I.J. Thompson, Phys. Lett. B 279 (1992) 201. ** D.V. Fedorov, A.S. Jensen, K. Riisager, Phys. Rev. C 49 (1994) 201. * D.V. Fedorov, A.S. Jensen, K. Riisager, Phys. Rev. C 50 (1994) 2372. * D.V. Fedorov, A.S. Jensen, K. Riisager, Phys. Rev. Lett. 73 (1994) 2817. ** S.K. Adhikari, A. Del"no, T. Frederico, L. Tomio, Phys. Rev. A 47 (1993) 1093. E. Nielsen, A.S. Jensen, D.V. Fedorov, Few-Body Systems 27 (1999) 15. *** E. Garrido, D.V. Fedorov, A.S. Jensen, Phys. Rev. C 53 (1996) 3159. L. Tomio, T. Frederico, A. Del"no, A.E.A. Amorim, Few-Body Systems 10 (Suppl.) (1999) 203. E. Garrido, D.V. Fedorov, A.S. Jensen, Nucl. Phys. A 650 (1999) 247. ** K. Riisager, D.V. Fedorov, A.S. Jensen, Europhys. Lett. 49 (2000) 547. * J. Goy, J.-M. Richard, S. Fleck, Phys. Rev. A 52 (1995) 3511. * A.S. Jensen, K. Riisager, Phys. Lett. B 480 (2000) 39. V.M. E"mov, Phys. Lett. B 33 (1970) 563. ** L.H. Thomas, Phys. Rev. 47 (1935) 903. ** R.N. Barnett, K.B. Whaley, Phys. Rev. A 47 (1993) 4082. B.S. Pudliner, V.R. Pandharipande, J. Carlson, S.C. Pieper, R.B. Wiringa, Phys. Rev. C 56 (1997) 1720. J. Boronat, J. Casullera, Phys. Rev. B 59 (1999) 8844. D. Blume, M. MladenovicH , M. Lewerenz, K.B. Whaley, J. Chem. Phys. 110 (1999) 5789. V.R. Pandharipande, Nucl. Phys. A 654 (1999) 157c. C.J. Joachain, Quantum Collision Theory, North-Holland Physics Publishing Company, Amsterdam, 1983. E. Garrido, D.V. Fedorov, A.S. Jensen, Phys. Rev. C 59 (1999) 1272. *** E. Garrido, E. Moya de Guerra, Nucl. Phys. A 650 (1999) 387. L.D. Faddeev, Sov. Phys. JETP. 12 (1961) 1014. ** M.I. Haftel, T.K. Lim, J. Chem. Phys. 77 (1982) 4515. O.A. Yakubovsky, Yad. Fiz. 5 (1967) 1312 [Sov. J. Nucl. Phys. 5 (1967) 937.] Ch. Elster, W. GloK ckle, Phys. Rev. C 55 (1997) 1058. S.P. Merkuriev, S.L. Yakovlev, Teor. Mat. Fiz. 56 (1984) 60 [Theor. Math. Phys. 56 (1984) 673]. ** F. Ciesielsky, J. Carbonell, Phys. Rev. C 58 (1998) 58. N. Barnea, M. Viviani, Phys. Rev. C 61 (2000) 034003. E. Nielsen, D.V. Fedorov, A.S. Jensen, Phys. Rev. C 59 (1999) 554. A.A. Kvitsinsky, J. Carbonell, C. Gignoux, Phys. Rev. A 51 (1995) 2997. A.A. Kvitsinsky, C-Y. Hu, J.S. Cohen, Phys. Rev. A 53 (1996) 255. C.R. Chen, G.L. Payne, J.L. Friar, B.F. Gibson, Phys. Rev. 44 (1991) 50. E.A. Kolganova, A.K. Motovilov, Yad. Fiz. 62 (1999) 1253 [Phys. At. Nucl. 62 (1999) 1179.] B.V. Danilin, I.J. Thompson, J.S. Vaagen, M.V. Zhukov, Nucl. Phys. A 632 (1998) 383. L.V. Grigorenko, B.V. Danilin, V.D. Efros, N.B. Shul'gina, M.V. Zhukov, Phys. Rev. C 60 (1999) 044312. J.W. Humberston, P. van Reeth, M.S.T. Watts, W.E. Meyerhof, J. Phys. B 30 (1997) 2477. A. Kievsky, S. Rosati, M. Viviani, Phys. Rev. Lett. 82 (1999) 3759. A. Kievsky, M. Viviani, S. Rosati, Nucl. Phys. A 551 (1993) 241. *** M. Viviani, A. Kievsky, S. Rosati, Few-Body Systems 18 (1995) 25. ** A. Kievsky, Few-Body Systems 10 (Suppl.) (1999) 27. J. Macek, J. Phys. B 1 (1968) 831. ** B.L. Christensen-Dalsgaard, Phys. Rev. A 29 (1984) 470. Jz. Tang, S. Watanabe, M. Matsuzawa, Phys. Rev. A 46 (1992) 2437. A. Igarashi, I. Shimamura, Phys. Rev. A 58 (1998) 1166. L. Wolniewicz, J. Hinze, A. Alijah, J. Chem. Phys. 99 (1993) 2695. A. Igarashi, I. Toshima, Phys. Rev. A 50 (1994) 232. Y. Zhou, C.D. Lin, J. Phys. B 28 (1995) 4907. P.O. Fedichev, M.W. Reynolds, G.V. Shlyapnikov, Phys. Rev. Lett. 77 (1996) 2921. M. Lewerenz, J. Chem. Phys. 106 (1997) 4596. J.L. Bohn, B.D. Esry, C.H. Greene, Phys. Rev. A 58 (1998) 584.
458 [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130]
E. Nielsen et al. / Physics Reports 347 (2001) 373}459 K.C. Engvild, Fusion Technol. 34 (1998) 253. E. Nielsen, D.V. Fedorov, A.S. Jensen, Phys. Rev. A 56 (1997) 3287. ** D.V. Fedorov, A.S. Jensen, K. Riisager, Phys. Lett. B 312 (1993) 1. K. Riisager, A.S. Jensen, P. M+ller, Nucl. Phys. A 548 (1992) 393. * E. Nielsen, A.S. Jensen, D.V. Fedorov, Few-Body Systems 10 (Suppl.) (1999) 277. E. Garrido, D.V. Fedorov, A.S. Jensen, Nucl. Phys. A 617 (1997) 153. A. Cobis, D.V. Fedorov, A.S. Jensen, J. Phys. G 23 (1997) 401. *** P.G. Hansen, A.S. Jensen, K. Riisager, Nucl. Phys. A 560 (1993) 85. K. Riisager, A.S. Jensen, Phys. Lett. B 301 (1993) 6. A.S. Jensen, D.V. Fedorov, K. Langanke, H.-M. MuK ller, in: M. de Saint Simon, O. Sorlin (Eds.), Int. Conf. ENAM 95, Edition Frontieres, 1995, p. 677. * A.S. Jensen, K. Riisager, Phys. Lett. B 264 (1991) 238. A.S. Jensen, K. Riisager, Nucl. Phys. A 537 (1992) 45. E. Garrido, A. Cobis, D.V. Fedorov, A.S. Jensen, Nucl. Phys. A 630 (1998) 409c. A. Cobis, D.V. Fedorov, A.S. Jensen, Phys. Rev. C 58 (1998) 1403. *** A. Cobis, D.V. Fedorov, A.S. Jensen, Phys. Rev. Lett. 79 (1997) 2411. * A. Cobis, D.V. Fedorov, A.S. Jensen, Phys. Lett. B 424 (1998) 1. * D.V. Fedorov, A. Cobis, A.S. Jensen, Phys. Rev. C 59 (1999) 554. E. Garrido, D.V. Fedorov, A.S. Jensen, Europhys. Lett. 36 (1996) 497. E. Garrido, D.V. Fedorov, A.S. Jensen, Europhys. Lett. 43 (1998) 386. E. Garrido, D.V. Fedorov, A.S. Jensen, Phys. Rev. C 58 (1998) R2654. E. Nielsen, D.V. Fedorov, A.S. Jensen, Phys. Rev. Lett. 82 (1999) 2844. *** S.J. Ward, J.H. Macek, S.Yu. Ovchinnikov, Phys. Rev. A 59 (1999) 4418. E. Nielsen, J. Macek, Phys. Rev. Lett. 83 (1999) 1566. *** D.R. Herschbach, J. Avery, O. Goscinsky, Dimensional Scaling in Chemical Physics, Kluwer Academic Publishers, Dordrecht, 1993. M. Dunn, D.K. Watson, Phys. Rev. A 59 (1999) 1109. * A. Gonzalez, Few-Body Systems 13 (1992) 105. G. Leibbrandt, Rev. Mod. Phys. 47 (1975) 849. L.M. Delves, Nucl. Phys. 9 (1959) 391; 20 (1960) 275. J. Raynal, J. Revai, Nuovo Cimento 68 (1970) 612. * S.I Vinitskii, S.P. Merkuriev, I.V. Puzynin, V.M. Suslov, Yad. Fiz. 51 (1990) 641 [Sov. J. Nucl. Phys. 51 (1990) 406]. R.G. Littlejohn, K.A. Mitchell, Phys. Rev. A 58 (1998) 3705. A.A. Kvitsinsky, V.V. Kostrykin, J. Math. Phys. 32 (1991) 2802. ** B.R. Johnson, J. Chem. Phys. 73 (1980) 5051. V.M. E"mov, Nucl. Phys. A 210 (1973) 157. P. NavraH til, B.R. Barrett, W. GloK ckle, Phys. Rev. C 59 (1999) 611. M. Dunn, D.K. Watson, in: D.R. Herschbach, J. Avery, O. Goscinsky (Eds.), Dimensional Scaling in Chemical Physics, Kluwer Academic Publishers, Dordrecht, 1993, p. 375. Ya.A. Smorodinskii, V.D. Efros, Yad. Fiz. 17 (1973) 210 [Sov. J. Nucl. Phys. 17 (1973) 107]. ** R.C. Whitten, F.T. Smith, J. Math. Phys. 9 (1968) 1103. D.V. Fedorov, E. Garrido, A.S. Jensen, Phys. Rev. C 51 (1995) 3052. *** A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions (Bateman Manuscript Project), McGraw-Hill, New York, 1953. ** S.K. Adhikari, T. Frederico, I.D. Goldman, Phys. Rev. Lett. 74 (1995) 487. A.E.A. Amorim, L. Tomio, T. Frederico, Phys. Rev. C 56 (1997) R2378. * S.A. Vulgal'ter, G.M. Zhislin, Teor. Mat. Fiz. 55 (1983) 269 [Theor. Math. Phys. 55 (1983) 493.] T.K. Lim, P.A. Maurone, Phys. Rev. B 22 (1980) 1467. S.K. Adhikari, A. Del"no, T. Frederico, I.D. Goldman, L. Tomio, Phys. Rev. A 37 (1988) 3666. R.A. Aziz, M.J. Slaman, J. Chem. Phys. 94 (1991) 8047. *
E. Nielsen et al. / Physics Reports 347 (2001) 373}459
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[131] T. GonzaH lez-Lezana, J. Rubayo-Soneira, S. Miret-ArteH s, F.A. Gianturco, G. Delgado-Barrio, P. Villarreal, Phys. Rev. Lett. 82 (1999) 1648. [132] W. SchoK llkopf, J.P. Toennies, J. Chem. Phys. 104 (1996) 1155. * [133] F. Lou, C.F. Giese, W.R. Gentry, J. Chem. Phys. 104 (1996) 1151. * [134] C.A. Parish, C.E. Dykstra, J. Chem. Phys. 101 (1994) 7618. [135] I. Roeggen, J. AlmloK f, J. Chem. Phys. 102 (1995) 7095. [136] A. Gri$n, D.W. Snoke, S. Stringari (Eds.), Bose}Einstein Condensation, Cambridge University Press, Cambridge, 1995. [137] Ph. Courteille, R.S. Freeland, D.J. Heinzen, F.A. van Abeelen, B.J. Verhaar, Phys. Rev. Lett. 81 (1998) 69. [138] J. Stenger, S. Inouye, M.R. Andrews, H.J. Miesner, D.M. Stamper-Kurn, W. Ketterle, Phys. Rev. Lett. 82 (1999) 2422. [139] E. Nielsen, B.D. Esry, submitted for publication. [140] P.M.M. Maessen, T.A. Rijken, J.J. de Swart, Phys. Rev. C 40 (1989) 2226. [141] I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida, N. Yoshikawa, K. Sugimoto, O. Yamakawa, T. Kobayashi, Phys. Rev. Lett. 55 (1985) 2676. [142] P.G. Hansen, B. Jonson, Europhys. Lett. 4 (1987) 409. ** [143] C.A. Bertulani, G. Baur, Nucl. Phys. A 480 (1988) 615. [144] E. Garrido, D.V. Fedorov, A.S. Jensen, Phys. Lett. B 480 (2000) 33. [145] E. Garrido, D.V. Fedorov, A.S. Jensen, Europhys. Lett. 50 (2000) 735. [146] T. Kobayashi, O. Yamakawa, K. Omata, K. Sugimoto, T. Shimoda, N. Takahashi, I. Tanihata, Phys. Rev. Lett. 60 (1988) 2599. [147] G. Audi, A.H. Wapstra, Nucl. Phys. A 595 (1995) 409. [148] B.M. Young et al., Phys. Rev. Lett. 71 (1993) 4124. [149] B.M. Young et al., Phys. Rev. C 49 (1994) 279. [150] S.N. Abramovich, B.Ya. Guzhovskij, L.M. Lazarev, Phys. Part. Nucl. 26 (1995) 423. [151] J.A. Caggiano, D. Bazin, W. Benenson, B. Davids, B.M. Sherill, M. Steiner, J. Yurkon, A.F. Zeller, B. Blank, Phys. Rev. C 60 (1999) 064322. [152] E. Garrido, D.V. Fedorov, A.S. Jensen, Phys. Rev. C 55 (1997) 1327. [153] G.F. Bertsch, K. Hencken, H. Esbensen, Phys. Rev. C 57 (1998) 1366. [154] L.V. Chulkov, B.V. Danilin, V.D. Efros, A.A. Korsheninnikov, M.V. Zhukov, Europhys. Lett. 8 (1989) 45. [155] I. Tanihata et al., Phys. Lett. B 287 (1992) 307. [156] B.D. Esry, private communication, 1999. [157] R.G. Newton, Scattering of Waves, Particles, Springer-Verlag, New York, 1982. [158] J.R. Taylor, Scattering Theory, Wiley, New York, 1972 (Chapter 20). [159] F. Ajzenberg-Selove, Nucl. Phys. A 490 (1988) 1. [160] A. Cobis, Ph.D. Thesis, IFA, University of Aarhus, 1997. [161] J.H. Macek, S.Yu. Ovchinnikov, Phys. Rev. A 54 (1996) 544. *** [162] R.K. Janev, J. Pop-Jordanov, E.A. Solov'ev, J. Phys. B 30 (1997) L353. [163] R.E. Langer, Phys. Rev. 51 (1937) 669. [164] L.D. Landau, E.M. Lifschitz, Quantum Mechanics, 3rd Edition, Pergamon Press, Oxford, 1977 (Chapter VII). [165] H. Friedrich, J. Trost, Phys. Rev. A 59 (1999) 1683. [166] S. Inouye, M.R. Andrews, K. Stenger, H.J. Miesner, D.M. Stamper-Kurn, W. Ketterle, Nature 392 (1998) 151. [167] B.D. Esry, C.H. Greene, Y. Zhou, C.D. Lin, J. Phys. B 29 (1996) L51. [168] B.J. Verhaar, J.P.H.W. van den Eijnde, M.A.J. Voermans, M.M.J. Scha!rath, J. Phys. A 17 (1984) 595.
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INTEGRAL-GEOMETRY MORPHOLOGICAL IMAGE ANALYSIS
K. MICHIELSENa, H. De RAEDTb a
Laboratory for Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands b Institute for Theoretical Physics and Materials Science Centre, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO
Physics Reports 347 (2001) 461}538
Integral-geometry morphological image analysis K. Michielsen , H. De Raedt * Laboratory for Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, Netherlands Institute for Theoretical Physics and Materials Science Centre, University of Groningen, Nijenborgh 4, 9747 AG Groningen, Netherlands Received August 2000; editor: M.L. Klein Contents 1. Introduction 2. Quick start 2.1. Computation of the image functionals 2.2. Computer program 2.3. Point patterns 2.4. Digitized and thresholded images 2.5. Minkowski functionals and Ising spins 3. Morphological image processing 3.1. Preliminaries 3.2. Gray-scale images 3.3. Black-and-white images 3.4. Miscellaneous operations 3.5. Mapping gray-scale to black-and-white images 4. Scanning electron microscope images of nano-ceramics 5. Integral geometry 5.1. Preliminaries 5.2. Convex sets and Minkowski functionals 5.3. Convex rings and additive image functionals 5.4. Relation to topology and di!erential geometry
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5.5. Application to images 5.6. Reducing digitization errors 5.7. Normalization of image functionals 6. Illustrative examples 6.1. Regular lattices 6.2. Triply periodic minimal surfaces [35] 6.3. Klein bottle 7. Random point sets 7.1. Two dimensions 7.2. Three dimensions 7.3. Percolation 8. Block copolymers 8.1. Micellar lattices [35] 8.2. Vesicles and worm-like micelles 8.3. Complex surfaces 9. Summary Acknowledgements Appendix A. Programming example Appendix B. Algorithm Appendix C. Minkowski functionals for elementary bodies Appendix D. Proof of (47) References
* Corresponding author. Tel.: #31-50-363-4852; fax: #31-50-363-4947. E-mail addresses: [email protected] (K. Michielsen), [email protected] (H. De Raedt). 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 0 6 - X
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Abstract This paper reviews a general method to characterize the morphology of two- and three-dimensional patterns in terms of geometrical and topological descriptors. Based on concepts of integral geometry, it involves the calculation of the Minkowski functionals of black-and-white images representing the patterns. The result of this approach is an objective, numerical characterization of a given pattern. We brie#y review the basic elements of morphological image processing, a technique to transform images to patterns that are amenable to further morphological image analysis. The image processing technique is applied to electron microscope images of nano-ceramic particles and metal-oxide precipitates. The emphasis of this review is on the practical aspects of the integral-geometry-based morphological image analysis but we discuss its mathematical foundations as well. Applications to simple lattice structures, triply periodic minimal surfaces, and the Klein bottle serve to illustrate the basic steps of the approach. More advanced applications include random point sets, percolation and complex structures found in block copolymers. 2001 Elsevier Science B.V. All rights reserved. PACS: 07.05.Pj Keywords: Morphology; Euler characteristic; Integral geometry; Minkowski functionals; Electron microscopy; Mesostructures; Polymers; Morphological image processing
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1. Introduction Geometrical patterns are encountered in many di!erent "elds of science and technology [1,2]. Very often these patterns come in the form of pictures. In general, the purpose of image analysis is to "nd out what is in these pictures [3}6]. Describing this information in words is one extreme form of characterizing the image, another extreme form is to assign one or more numbers to the image. In this paper we only consider the latter. In this paper, we will analyze two-dimensional (2D) and three-dimensional (3D) patterns by numerical representions of the corresponding images in terms of two-valued functions. Therefore, a numerical characterization of features in the image requires that the image has been digitized, i.e. that the image has been converted to numerical form [3}6]. This conversion may include additional digital image processing steps [3}6] to enhance the quality of the images. If the image contains color or gray-level information, the digitization process should include the mapping of the spatial and color/brightness information in the image onto a collection of black-and-white picture elements [3}6]. For simplicity, we will use the term pixel to refer to both 2D and 3D picture elements. Numerical functions on the set of black-and-white images are called image functionals [7]. An image functional performs a measurement of certain properties or features in the image, such as the brightness, or location of objects, their surface, perimeter, size distribution, etc. An example of an image functional is the area of black pixels on a background of white pixels. If P and P are two patterns of black pixels we obviously have (P P )"(P )#(P )!(P P ) . (1) The last term in (1) compensates for the double counting of black pixels that are common to P and P . Image functionals that share property (1) are called additive. Intuitively, it may seem obvious to require image functionals to be additive: In general, one would like to avoid counting a feature in an image more than once. However, in image analysis there is no fundamental reason to stick to additive image functionals. In fact, there is a large collection of non-additive image functionals that yield useful information on speci"c features of an image [3]. For instance, the two-point correlation function of the positions of the black pixels (i.e. the Fourier transform of the structure factor) is a non-additive functional but it certainly yields very useful information about the spatial distribution of the black pixels. Morphology is a branch of biology dealing with the form and structure of animals and plants. The same word is used for the study of the geometry and topology of patterns. Integral-geometry morphological image analysis (MIA for short from now on) employs additive image functionals to assign numbers to the shape and connectivity of patterns formed by the pixels in the image. Integral geometry [8}10] provides the rigorous mathematical framework to de"ne these image functionals. A fundamental theorem (discussed below) of integral geometry [8] states that under certain conditions, the number of di!erent additive image functionals is equal to the dimension of the pattern plus one. Thus, in the case of a 2D (3D) image there are exactly 3 (4) of these functionals, called quermassintegrals or Minkowski functionals. For a given image the "rst step in MIA is to compute these functionals themselves. The second step is to study the behavior of the three or four numbers as a function of some control parameters, such as time, density, etc. A remarkable feature of MIA is the big contrast between the simplicity of implemention and use and the level of sophistication of the mathematical theory. Indeed, as explained below, the
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calculation of the image functionals merely amounts to the proper counting of, e.g. faces, edges and vertices of pixels. The application of MIA requires little computational e!ort. Another appealing feature of MIA is that the image functionals have a geometrically and topologically intuitive and hence perceptually clear interpretation: For 2D images they correspond to the area, boundary length, and connectivity number. The four functionals for 3D images are the volume, surface area, integral mean curvature and connectivity number. This paper gives an overview of the various aspects of MIA, with an emphasis on the practical application. MIA has proven to be very useful to describe the morphology of porous media and complex #uids [11}14], the large-scale distribution of matter in the Universe [14}16], regional seismicity realizations [17], quantum motion in billiards [18], microemulsions [14,19], patterns in reaction}di!usion systems [14,20], spinodal decomposition kinetics [14,21,22], and the dewetting structure in liquid crystal and liquid metal "lms [23], and in polymer "lms [24]. In many cases additional information can be extracted from the pattern by making assumptions about size, shape and distribution of the objects. Usually, this involves making a probabilistic model of the pattern and comparing the Minkowski functionals of the model with those of the images. Applications of this stochastic-geometry approach to model natural phenomena can be found in [10]. The paper is structured as follows. Section 2 gives a brief introduction to MIA in practice (including examples of computer code), for those who want to start right away. Section 3 gives a brief introduction to morphological image processing (MIP), a digital image processing technique to enhance the quality of images while preserving the morphological content of the images as much as possible. In Section 4 we present some examples of MIP applied to scanning electron microscopy images of nano-ceramics. The mathematical framework on which MIA is based is reviewed in Section 5. In Section 6 we illustrate how MIA works in practice, using well-known point patterns and geometrical objects. Section 7 discusses the application of MIA to random point sets and percolation. In Section 8 we use MIA to analyze various structures found in block copolymers. A summary is given in Section 9.
2. Quick start In this section we give a brief overview of MIA. Thereby we focus on the practical aspects. For the sake of brevity we omit most mathematical justi"cation, references to relevant work, and discussions of examples. These can be found in the sections that follow. We only consider the analysis of black-and-white images. Extensions are discussed below. In the next two subsections we describe a simple and e$cient procedure to compute the image functionals for a given image. The following two subsections focus on the second step of MIA: The study of the dependence of these functionals on some control parameters. In the last subsection we give an alternative interpretation of the Minkowski functionals in terms of correlation functions of Ising spins. 2.1. Computation of the image functionals Consider a 2D lattice "lled with black pixels on a white background (see Fig. 1). For simplicity, we will assume that the pixels are squares and that the linear size of each square has been
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Fig. 1. Decomposition of the pixels of a two-dimensional black-and-white pattern (left) into squares, edges and vertices (right). For this example: number of squares n "8, number of edges n "24 and number of vertices n "16.
normalized to one. We want to characterize the geometry and topology of the pattern formed by the black pixels. According to integral geometry, there are three additive image functionals, called Minkowski functionals, that describe the morphological content of this 2D pattern, namely the area A, the perimeter ; and the Euler characteristic . The latter describes the connectivity (topology) of the pattern: In 2D equals the number of regions of connected black pixels minus the number of completely enclosed regions of white pixels. Two black pixels are `connecteda if and only if they are nearest neighbors or next-nearest neighbors of each other or can be connected by a chain of black pixels that are nearest and/or next-nearest neighbors. Using this de"nition we "nd that the Euler characteristic of the pattern shown in Fig. 1 is zero. Conceptually, the procedure (that easily extends to three dimensions) to compute these three numbers consists of two steps. First, we decompose each black pixel into 4 vertices, 4 edges and the interior of the pixel (see Fig. 1). Then we count the total number of squares n , edges n and vertices n and we compute the area A, perimeter ; and Euler characteristic from A"n , ;"!4n #2n , "n !n #n .
(2)
For the example shown in Fig. 1 we "nd A"8, ;"16 and "0. For a 3D cubic lattice "lled with black and white pixels (we do not distinguish between voxels and pixels) the four additive image functionals (Minkowski functionals) are the volume <, the surface area S, the mean breadth B (see Section 5), and the Euler characteristic . In 3D equals the number of regions of connected black pixels plus the number of completely enclosed regions of white pixels minus the number of tunnels, i.e. regions of white pixels piercing regions of connected black pixels. As in the 2D case, the "rst step in the calculation of these four numbers is to consider each black pixel as the union of 8 vertices, 12 edges, 6 faces and the interior of the cube. It can be shown that <"n ,
S"!6n #2n ,
2B"3n !2n #n , "!n #n !n #n ,
(3)
where n and n are the number of cubes and faces, respectively. Thus, as in the 2D case, the morphological characterization of a 3D pattern reduces to the counting of the elementary geometrical objects (vertices, edges, faces, cubes) that constitute the pattern.
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2.2. Computer program Technically, the only real `problema with the procedure described above is to avoid counting, e.g. an edge or vertex more than once. However this problem is easily solved, as illustrated by the algorithm we will brie#y discuss now. In Appendix A we list a computer program to compute A, ; and (<, S, B, ) for a 2D (3D) black-and-white pattern. We discuss the 2D case only because the 3D program only di!ers in the details. Conceptually, what these programs do is to build up the whole image using vertices, edges, etc. In practice, this is accomplished by adding active ("black in the example above) pixels to an initially empty ("white in the example above) image (held in array tmp( ) )) one by one. Just before adding the active pixel to the current image (in tmp( ) )) subroutine `minko}2D}freea determines the change in A, ; and that would result if this pixel is actually added to the current image. This change is calculated by "rst decomposing this square pixel as discussed above and then checking whether, e.g. an edge overlaps with an edge of another active pixel in the current image. Then the pixel is made active in the current image and the changes are added to the current values of A, ; and . Inspection of `minko}2D}freea shows that all it does is check to see if the pixel-to-beadded has active nearest neighbors and/or next-nearest neighbors and count the number of edges and vertices accordingly. Clearly, the number of arithmetic operations required to compute A, ; and (or <, S, B and ) scales linearly with the number of black pixels of the image. Thus the numerical procedure is e$cient. Some applications, notably those where the patterns are the result of computer simulation, make use of periodic boundary conditions. There is no need to adapt the programs given in Appendix A to deal with this situation. One can embed the original image into a larger one, formed by surrounding the original image by one extra layer of pixels, the value of which is determined by making use of the periodic boundary conditions. 2.3. Point patterns Many systems observed in nature may be modeled by point patterns. For example, a system of particles may be viewed as a collection of points de"ned by the position of the particles. These points are usually called the germs of the model [10,25]. In order to study the morphological properties of the set of points (degree of randomness, clustering, periodic ordering, etc.) it is useful to attach to the points discs (spheres) of radius r. Those discs (spheres) are called the grains of the model [10,25]. The study of the coverage of the image by the grains gives information about the distribution of the germs. Mapping the point pattern onto a square (cubic) lattice yields a black-and-white picture. Black pixels represent the germs of the model. On the pixel lattice we can construct the grains of the model in two di!erent ways. In the "rst method we consider the germs to be discs (spheres) of radius r"0. We enlarge the discs (spheres) by making black all pixels that are positioned at a distance smaller or equal to r'0 from the germs. The grains form discrete approximations to discs (spheres) in the Euclidean space. An example of this graining procedure in two dimensions is shown on the left-hand side of Fig. 2 for grains of radius r"3. The right-hand side of Fig. 2 illustrates the second graining procedure (for d"2), where we take the germs to be squares (cubes) of edge length r"1 and the grains to be enlarged squares (cubes) of edge length 2r#1, r'0. Note
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Fig. 2. Graining procedure of a point pattern in two dimensions. Left: the grains are discrete approximations to a sphere with radius three in the Euclidean space. Right: the grains are squares of edge length seven. The light grey pixels indicate the positions of the germs.
that the growing of the cubic grains leads to a faster complete coverage of the image than the growing of the circular grains. In the following, the cubic grains will be used for special cases only (see Section 7). For this type of problem MIA consists of the calculation of the three (four) numbers A, ; and (<, S, B and ) as a function of the grain size r. A schematic representation of this procedure for the case of 2D point patterns is shown in Fig. 3. 2.4. Digitized and thresholded images In general, the intensity (or gray level) in experimental images may be thought of as a continuous function of the position in the image. In order to analyze such images by computer we "rst have to digitize them [3}6]. The digitization process requires the mapping of the image on a grid and a quantization of the gray level. Usually, 2D (3D) images are partitioned into square (cubic) regions. Each square (cube) is centered at a lattice point, corresponding to a pixel. In general, the range of gray levels is divided into intervals and the gray level at any lattice point is required to take only one of these values. The output of image analysis should be a description of the given picture. Thus, we have to de"ne the various objects building up the picture, i.e. we need a method to distinguish objects from the background [3,6]. The simplest method of reducing gray-scale images to two-valued images or black-and-white pictures is to make use of a threshold. If the given picture P(x) with x31B has gray level range [a, b], and q is any number between a and b, the result of thresholding P(x, q) at q is the two-valued picture P(x, q) de"ned by [3}6]
P(x, q)"
1, P(x)5q , 0, P(x)(q .
(4)
By de"nition if P(x, q)"0, x is part of the background, and if P(x, q)"1, x is part of an object. In practice not all thresholds q yield useful P(x, q). If q is too large too many objects are classi"ed as
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Fig. 3. Schematic representation of how to use MIA to analyze the morphological properties of (patterns that can be interpreted in terms of) 2D point patterns.
background or if q is too small the opposite happens. Other thresholding operations may also be considered [3}6]. For this type of problem MIA consists of the calculation of the three (four) numbers A, ; and (<, S, B and ) as a function of the threshold q. A schematic representation of this procedure for the case of 2D gray-scale images is shown in Fig. 4.
2.5. Minkowski functionals and Ising spins It is instructive to represent black-and-white images as a set of Ising spins and to express the Minkowski functionals in these variables. This exercise is useful for two purposes. First, it shows that in certain cases, Minkowski functionals have a direct physical interpretation and second it gives insight into the kind of correlations of pixels the Minkowski functionals actually measure. We restrict ourselves to the 2D case, the extension to 3D is trivial. The standard procedure to map a black-and-white picture onto a lattice of Ising spins is to assign a spin "#1(!1) to the black (white) pixel at lattice position (i, j). Starting from the G H
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Fig. 4. Schematic representation of how to use MIA to analyze the morphological properties of 2D digitized images.
expressions given in Appendix B, a straightforward calculation gives 1 *V *W ¸ ¸ (5a) A" V W # , G H 2 2 G H 1 *V *W ;"¸ ¸ ! ( # ), (5b) V W 2 G H G> H G H> G H 1 *V * W 1 *V * W ¸ ¸ # ) "! V W ! # ( G H G H G> H G H> 4 8 16 G H G H 1 *V *W ! ( # ) G H G> H> G> H\ 16 G H 1 *V *W ( # ) # G H G> H G H> G> H> 16 G H 1 *V *W ( # ) # G H> G> H G H G> H> 16 G H 1 *V *W , (5c) ! G H G> H G> H> G H> 16 G H where (¸ , ¸ ) denote the number of lattice sites in the (x, y)-direction. Readers familiar with the V W Ising model recognize immediately that up to irrelevant constants, the area A and perimeter ; correspond to the magnetization and energy of the Ising model with nearest-neighbor
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interactions. The Euler characteristic is a weighted sum of all possible correlations of up to four neighboring spins. Furthermore, it is clear that the Minkowski functionals (A, ; and ) and the two-spin correlation function (or structure factor) 1 *V *W S(k, l)" (6) G H G>I H>J ¸ ¸ V W G H perform di!erent, hence complementary measurements on the con"guration of spins (or pixels).
3. Morphological image processing In the preceding section we took for granted that the digitized images are free of noise and other artifacts that may a!ect the geometry and topology of the structures of interest. Such perfect images are easily generated by computer and are very useful for the development of theoretical concepts and models (see e.g. Sections 6 and 7). Unfortunately, as we all know, genuine pictures or patterns obtained from computer simulations (e.g. a polymer solution, see Section 8) are all but perfect. Therefore, some form of image processing may be necessary before attempting to make measurements of the features in the image. Digital image processing is very important for many industrial, medical and scienti"c applications. There is a vast amount of literature on this subject so we can only cite a few books here [3}6]. There is also a huge number of di!erent processing steps and methods. The type of measurements that will be performed on the image is an important factor in making a selection of the most appropriate processing steps. In morphological image analysis the geometric and topological content of the image are of prime importance and this should be re#ected in the operations that are used to enhance the image quality. The morphological image processing (MIP) technique reviewed below is well adapted for this purpose. This is because MIP and MIA are based on the same mathematical concepts (see below). Most importantly it is #exible, fast and easy to use. Pioneering work in this "eld was carried out by Matheron [25] and Serra [26]. We have found the book of Giardina and Dougherty [7] a very valuable source of information and inspiration. Most of the material of Sections 3.1}3.3 can be found in [7], albeit in di!erent form. We have chosen to present the material in the same order as MIP is actually performed: From a gray scale to a black-and-white image. The emphasis is on the practical application, much less on the mathematical foundations which are given in [7,25,26]. 3.1. Preliminaries In this section we introduce the basic concepts of MIP. We start by giving a more precise de"nition of an image. For simplicity, we will discuss MIP of 2D images only. Extension to 3D is trivial, also in practice. As usual a 2D image will be represented by an ¸ ;¸ array I(i, j) of gray V W values, intensities represented by integers, ¸ (¸ ) is the number of pixels in the x (y) direction. We V W will see below that some operations may refer to pixels that are out of bounds of this array, meaning that they refer to pixels that are not de"ned. It is convenient to assign the value minus
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in"nity to such pixels [7]. Hence, we will write I(i, j)"!R whenever the pixel at (i, j) (within or outside the bounds of the array) is unde"ned. The structuring element or template is a key concept in MIP. A template is a predetermined geometrical structure, hence also an image, such as a square, a disc or star. Consistency of notation would suggest the use of the symbol ¹(i, j) to denote the image corresponding to the template ¹ but we will not do so. Instead we de"ne a template by specifying the displacement k, l relative to its 2 origin (0, 0) together with the value ¹(k, l). The size of a template is de"ned as maxk, l . A template cannot contain pixels that are unde"ned. Some examples of templates are shown in Fig. 5. Very often templates are chosen to be symmetric (with respect to the symmetry operations of a square lattice). In essence MIP is the study of how a template (or several templates) "t into an image [7,25,26]. A template represents the viewer's a priori knowledge or expectation about the morphological content of the image. Finally, we need a de"nition of an object. For reasons of consistency with the integral geometry approach discussed below an object is de"ned as a collection of pixels that satisfy the following criteria: (i) they all have the same intensity, and (ii) they are nearest neighbors or next-nearest neighbors of each other or can be connected by a chain of pixels that are nearest and/or next-nearest neighbors. It may seem strange that it is necessary to include next-nearest neighbors in counting objects but in fact it is not. This can already be seen by looking at a very simple example: A pattern that consists of two squares that touch each other at the vertex yield an Euler characteristic of one (one connected component), since n "2, n "8 and n "7 (see (1)). Clearly, there are no holes in this pattern. Hence the number of objects must be equal to the Euler characteristic (recall, for 2D patterns the Euler characteristic is equal to the number of connected components, i.e. objects, minus the number of holes, see Section 2). The only way to get a consistent procedure of counting objects and computing the Euler characteristic is to include next-nearest neighbors. 3.2. Gray-scale images We now have all the ingredients to de"ne the two basic MIP operations: Dilation and erosion of an image. Dilation D transforms an input image I(i, j) as follows: D(I, ¹)(i, j),max [I(i!k, j!l)#¹(k, l)] . 6I J72
(7)
Fig. 5. Some examples of templates used in MIP. As in most practical image processing work we adopt the convention that the intensity is digitized in the range [0, 255]. Template (1): ¹(0, 0)"64; (2): ¹(0, 0)"64, ¹(1, 0)"128, ¹(0, 1)"128, ¹(!1, 0)"192, ¹(0,!1)"255; (3): ¹(0, 0)"64, ¹(1, 0)"128, ¹(0,!1)"192; (4): ¹(0, 0)"¹(1, 0)"¹(0, 1)" ¹(!1, 0)"¹(0,!1)"255; (5): ¹(0, 0)"2"¹(1,!1)"110; (6): ¹(0, 0)"2"¹(2,!2)"255.
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Erosion E uses the minimum instead of the maximum: (8) E(I, ¹)(i, j), min [I(i#k, j#l)!¹(k, l)] . 6I J72 The maximum and minimum are to be taken over all values of displacement k, l of the template 2 ¹. In general for some (i, j), (i!k, j!l) may well go out of the bounds of array I(i, j), a situation we already anticipated for by setting I(i, j)"!R whenever (i, j) is out of bounds. A similar argument applies to erosion: If one of the pixels I(i#k, j#l)"!R, in the output image the pixel at position (i, j) will be unde"ned too. Usually unde"ned pixels are displayed in background color (black on a display, white on paper). In Fig. 6 we show some illustrative examples of D and E. We used three di!erent templates to perform dilate D and erode E on a rather schematic picture of a rabbit. The original image is shown in the top left panel of Fig. 6(D and E). The top right image of Fig. 6(D) is obtained by replacing a pixel by its most intense nearest neighbor. This has the e!ect of transforming gray pixels at the boundaries of the gray objects into white pixels. The same happens to black pixels touching white and gray objects, hence the rabbit gets in#ated a little. The bottom left panel of Fig. 6(D) shows the e!ect of changing the intensity during the process of dilation. In this case we use D to remove all the gray objects of the rabbit, increase the size of the rabbit and change the background color. The bottom right panel of Fig. 6(D) shows the result of using a square (5;5) template. Apparently, this template is so large that D replaces all gray objects by white ones, except for the legs of the rabbit, which get severely distorted. If our intention was to extract certain features from the original image of the rabbit, using the large square template obviously is not the right thing to do. Indeed, as
Fig. 6. Illustration of Dilate D and Erode E of a gray-scale image (top left panel). Top right: star-shaped template of size 1; bottom left: star-shaped template of size 2; bottom right: square-shaped template of size 2. The values of the templates is zero in all cases.
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mentioned earlier, the choice of the template is directly linked to the viewers expectation about the morphological content of the image. A similar sequence of images, obtained by employing E instead of D, is shown in Fig. 6(E). Not surprisingly, the `mina operation generally reduces the number of non-black pixels, i.e. the rabbit shrinks. However E can increase the area of gray objects too. The top-right image shows that E tends to emphasize internal structures: Eyes, inner ears, legs and other features became larger. The `stara template of size 2 (see Fig. 5 (5)) reduces the gray level of all de"ned pixels (viewed on a computer screen `blacka and `unde"neda are synonymous). Also, it reduces the number of dark-gray objects. As in the case of D, using an oversized square template (bottom right panel) yields a fairly distorted image of the rabbit. The basic morphological operations D and E can be used to construct other operations that perform more complicated "ltering operations. There are two other operations called Open (O) and Close (C) that play a central role in MIP [7]. Open and Close are de"ned as O(I, ¹),D(E(I, ¹), ¹)
(9)
and C(I, ¹),!O(!I,!¹) ,
(10)
where !I,!I(i, j) and !¹,!¹(k, l). Open and Close have all the mathematical properties that are required for MIP [7]. In particular, O and C are idempotent, i.e. O(O(I, ¹), ¹)"O(I, ¹) and C(C(I, ¹), ¹)"C(I, ¹), implying that in practice it does not help to `opena or `closea an image twice or more using the same template. In Fig. 7 we illustrate the e!ect of O and C, again using the image of the rabbit as an example. Open and Close act as "lters, the exact result of the "ltering operation depending on the template. Open O generally rounds corners from the inside of the objects (see the legs of the rabbit in Fig. 7(O)
Fig. 7. Illustration of Open O and Close C of a gray-scale image (top left panel), using the same templates as in Fig. 6.
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for example). Close C, on the other hand, smooths from the outside. Objects that do not "t the template are removed from the image (see Fig. 7(C)). It is instructive to compare, e.g. E and C (top right panel of Figs. 6(E) and 7(C)). We see that E generally increases the size of the gray features whereas C removes the small gray features but leaves other gray objects intact. This is most clearly seen by comparing the bottom right panel of Figs. 6(E) and 7(C) that show the results of E and C using the square template. Whereas C does not change the overall image very much, E makes the rabbit look like a cat. Our experience is that in practice O and C are more useful than D and E. 3.3. Black-and-white images Obviously, a black-and-white image B=,B=(i, j) may be considered as special case of the gray-scale images treated earlier. As such a discussion of black-and-white MIP may seem super#ucious. However, in practice, it is often necessary to perform MIP on the gray-scale image, convert it to black-and-white, and carry out some further MIP on the black-and-white image before the image can be used as input for MIA. Therefore, it is worthwhile to discuss MIP on black-and-white images in more detail. On a computer display a black pixel may be considered as being unde"ned [7]. Instead of assigning unde"ned pixels the value !R, in this case it is more convenient to assign to a black pixel the traditional value of zero. A white pixel takes the value one. Hence, a black-and-white image B= is represented by an array of Boolean variables B=(i, j). In analogy with gray-scale MIP the four basic operations dilate D, erode E, open O and close C are de"ned by D(B=, ¹)(i, j), B=(i!k, j!l) , 6I J72
(11a)
E(B=, ¹)(i, j), B=(i#k, j#l) , 6I J72 O(B=, ¹),D(E(B=, ¹), ¹) ,
(11b)
C(B=, ¹),E(D(B=,!¹),!¹) ,
(11d)
(11c)
respectively. Operations (11a)}(11c) are Boolean versions of (7), (8) and (9), respectively, but this is not the case for pair (10) and (11d) [7]. Operations (11a) and (11b) are digital versions of set-theoretic operators known as Minkowski addition and subtraction [7]. The latter are basic concepts in point-set geometry and integral geometry [8]. This correspondence suggests that MIP and MIA are closely related and indeed they are [7]. The collection of images shown in Figs. 8 and 9 serve to illustrate the e!ect of these four operations on the thresholded image of the rabbit (top left panel). The threshold is chosen such that the gray pixels are converted to black ones. The examples shown would suggest that MIP of gray-scale images followed by thresholding yields pictures that are almost identical to the corresponding morphological imaging processed black-and-white images. As a matter of fact comparison of the bottom left panel of Figs. 6(D) and 8(D) already shows that interchanging the order in which thresholding and MIP are performed changes the output. Indeed after MIP of the black-and-white image, some of the internal features remain visible, notably legs and eyes. In
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Fig. 8. Illustration of Dilate D and Erode E of a black-and-white image (top left panel), using the same templates as in Fig. 6.
Fig. 9. Illustration of Open O, Close C and Filter F of a black-and-white image (top left panel), using the same templates as in Fig. 6.
contrast MIP of the gray-scale image yields a completely smoothed image of the rabbit. For the input image of the rabbit used in the examples, interchanging thresholding and Erode (or Open or Close) yields the same output image. In general, this will not be the case unless the images have a very simple gray-scale structure, as the ones considered here. As a "nal example of MIP we consider a more complicated "lter F de"ned as [7] F(B=, ¹),C(O(B=, ¹), ¹) .
(12)
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Also this "lter is e!ective in removing small noisy structures (small with respect to the template ¹), and leaves larger objects intact, whereas O has the tendency to tear objects apart (see bottom right panel of Fig. 9(O)) and C has the opposite e!ect. Filter F may give a more satisfactory output image in some cases. 3.4. Miscellaneous operations Averaging an image using a template can sometimes help to remove artifacts. In our notation this operation reads 1 I(i#k, j#l) , (13) I (i, j)" C¹ 6I J72 where C¹ denotes the number of elements of template ¹. As before, in computing this average, we use the template to express our a priori knowledge or expectation about the shape and size of the objects in the image. Often it is useful to enhance the contrast of a gray-scale image. Again we can use a template to I(i#k, j#l) perform this task. For each pixel in the image I(i, j) we determine M(i, j),max 6I J72 I(i#k, j#l) and then replace each pixel in the image by invoking the rule: and m(i, j),min 6I J72 m if I(i, j)!m(M!I(i, j) , I (i, j)" (14) M if I(i, j)!m'M!I(i, j) .
Note that the value of the template does not play any role in this operation. 3.5. Mapping gray-scale to black-and-white images Excluding applications of MIA that use the threshold as a control parameter (see Section 2.4), in many situations it may be expedient to reduce the number of di!erent gray values in an image. For instance, to determine the number of objects in a gray-scale image, we will have to group pixels according to their gray value. Fluctuations in the gray values due to noise and other experimental limitations may prevent us from making the correct identi"cation if we use the full resolution of gray values (typically 256 values). Clearly, a procedure that reduces the number of gray-scale levels may be very useful. Thus, we would like to have a procedure to map the original gray-scale image onto another one with only a small number N of distinct gray levels (e.g. N"2, 4). A simple approach would be to use histogram equalization to optimize the dynamic range of the gray levels, followed by thresholding to classify pixels as either background or objects [3,4]. Clearly, it is much better to use a scheme that computes a nearly optimal distribution of the N gray levels from the original image itself. The method we will describe next performs very well in practice. It is a gray-scale version of a scheme that is used to determine nearly optimal color pallets [27]. The "rst step of the algorithm consists of making a histogram of the gray-scale image. This we can easily do at full gray-scale resolution. Let us consider the case of a reduction by a factor of two (i.e. N"128). We want to group gray levels but keep the image quality as high as possible. Which gray level should we remove "rst? A natural choice would be to select from the histogram the gray
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level k with the lowest count, say 04k4255. Then we merge the bins 2[k/2] and 2[k/2]#1 ([k/2]"k/2 if k is even, [k/2]"(k!1)/2 if k is odd). This we do by adding the count of bin 2[k/2]#1 to the count of bin 2[k/2] and then clearing bin 2[k/2]#1. This process of merging bins is repeated until we have 128 empty bins (which could be rather exceptional) and we can stop the whole procedure or until we conclude that all posibilities to merge two bins have been exhausted. In the latter case we repeat the procedure by grouping the bins 4[k/4], 4[k/4]#2 (note that in the previous step the counts in bins 4[k/4]#1 and 4[k/4]#3 have been set to zero). Thereby care has to be taken to group the same four bins only once, a technical but crucial point. Again we repeat this process, always working with groups of four bins, until the number of bins with a count larger than zero is 128 (in which case the procedure terminates) or we keep restarting the grouping of bins using increments of 8, 16,2 and so on. Clearly, this procedure terminates as soon as the number of distinct gray levels becomes equal to the desired number of gray levels. Then it is a straightforward matter to assign new gray-scale values to the pixels of the original image. Although there is some ambiguity in chosing the strategy for grouping bins, experience has shown that the procedure outlined here yields very satisfactory gray-scale images, and can be used to automatically reduce a gray-scale image to a black-and-white picture.
4. Scanning electron microscope images of nano-ceramics As an example of an application of MIP we consider the problem of identifying objects in scanning electron microscope (SEM) images of nano-ceramic materials. These materials may exhibit physical properties such as ductility, toughness and hardness of both metals and ceramics and are useful for a number of technological applications that demand good mechanical behavior and good resistance against the degrading e!ects of high temperature, corrosive environments, etc. These materials can be manufactured by di!erent techniques, for instance by covering a surface by layers of nano-sized ceramic particles. The mechanical and other properties of these materials depend on the morphology, the microstructure and the initial stress due to the use of dissimilar materials. Di!erent preparation techniques and additional (heat) treatments often yield materials that have di!erent morphologies [28}30]. The changes in the morphology during the sintering process can be monitored by means of high-resolution low-voltage scanning electron microscopy (HRSEM) [31]. In Fig. 10 we show two SEM images of SiO particles on a substrate of fused silica, before (top panel) and after (bottom panel) a heat treatment [31,32]. From Fig. 10 it is clear that the latter causes particles to aggregate. Although their size does not seem to change much, the voids get larger. A more quantitative analysis of such images requires the identi"cation of objects (i.e. particles) in the image. MIP is well suited for this purpose. At the right-hand side of Fig. 10 we depict the images obtained by MIP. The #uctuations in the intensity (i.e. gray value) within what our eyes would consider to be one particle can be rather large. This experimental artifact can be removed from the image by means of O (open) and contrast enhancement operations, both using as a template a disc with a radius of 10 pixels. The size of the template re#ects our rough guess about the size of the objects. Then we use the algorithm described in Section 3.5 to map the gray-scale image onto a black-and-white picture. The "nal step consists of removing some minor artifacts of the size of one pixel by means of a C (close) operation. For this
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Fig. 10. Electron microscope micrographs of silica at two stages of the sintering process before (left) and after (right) MIP.
purpose we use a single pixel as a template. Clearly the quality of these black-and-white pictures is su$ciently good for further analysis of the particle size, distribution etc. In Fig. 11 we present SEM images of another material, TiO on the same substrate. Depending on the heat treatment grains of TiO grow in size, leading to the mosaic-like coverages shown in Fig. 11 [31,33]. Also in this case gray-scale O and contrast-enhancement operations are used to remove noisy features from the image. Here the template is a 4-pixel-radius disc, smaller than in the previous example, but consistent with our expectation that the images of individual grains are smaller. Then the images are converted to black-and-white pictures, using the same procedure as the one described above. Also in this case the "nal pictures are of su$cient quality so that objects can easily be identi"ed and analyzed. As a "nal example we consider a rather di!erent type of system, namely small Mn O precipitates in Ag observed by high-resolution transmission electron microscopy (HRTEM). The top-left panel of Fig. 12 shows a high-resolution picture of a small part of the sample shown in the bottom-left panel. From the former we would like to extract information about the geometrical properties of the individual grains, from the latter we want to learn how the particles are distributed over the surface. It is somehow remarkable that the same MIP procedure can be used for both, apparently quite di!erent, tasks. The top-left image has very low contrast. Moreover, due to experimental conditions, the averaged intensity at the left-hand side of the image di!ers signi"cantly from the one at the right-hand side. After correcting for this artifact, repeated averaging and contrast enhancement operations with a 15-pixel-radius template followed by the standard of mapping to black-and-white yields the image shown in the top-right panel of Fig. 12. The bottom-left image is processed in the same manner, except that instead of a 15 pixel-size disc we use
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Fig. 11. Electron microscope micrographs of zirconia at two stages of the sintering process before (left) and after (right) MIP.
Fig. 12. High-resolution electron microscope image of Mn O precipitates before (left) and after (right) MIP.
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a radius of 4 pixels and we added a D (Dilate) step to remove some sharp edges from the objects. Physically, relevant information about the particle size and spatial distribution is easily extracted from these pictures. The examples discussed above illustrate that MIP is a #exible and powerful tool for enhancing image quality and object identi"cation, without destroying the morphological content of the image. Of course, depending on the type of image technique used, additional non-morphological image processing steps may be required to produce patterns that are suitable for morphological image analysis. As the emphasis of this paper is on image analysis rather than on image processing an in-depth discussion of the latter is outside the scope of the paper and we refer the reader to standard treatises on the subject [3}6]. We now review the theory that provides a rigorous framework for the quantitative characterization of the morphological properties of black-andwhite images.
5. Integral geometry In this section we present the mathematics that lies at the heart of integral-geometry-based morphological image analysis. The reader who is not interested in the mathematics can skip this section and resort to Section 2. 5.1. Preliminaries Consider the set of points of a line ¸ of length a embedded in one-dimensional (1D) Euclidian space. We take a similar line of length 2r and put the center of this line at each point of the line ¸. How does the union of all these points look like? Obviously, it is another line that is longer than ¸. The sets ¸ (black line) and ¸ (union of black and light gray lines), the result of this operation, are P shown in Fig. 13. The length l of ¸ is given by P l(¸ )"a#2r"l(¸)#2r . P
(15)
The set ¸ is called the parallel set of ¸ at a distance r. P The one-dimensional case easily extends to two and three dimensions. We consider a circular disk D of radius a, a square Q of edge length a and a equilateral triangle ¹ of side length a embedded in the 2D space. Now, we use a disc of radius r and perform the same operation as we did for the 1D case: We put the center of the disc of radius r at each point of D (or Q or ¹) and consider the union of all points. The resulting parallel sets at a distance r are shown in Fig. 13. The area A of D , Q and ¹ is given by P P P A(D )"a#2ar#r , P
(16a)
A(Q )"a#4ar#r , P
(16b)
(3 a#3ar#r . A(¹ )" P 4
(16c)
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Fig. 13. Parallel sets K (union of black and grey area) at a distance r of the sets K (black area). Top: parallel set of a line P segment ¸ of length a embedded in one dimension; bottom: parallel set of a circular disk D of radius a, a square Q of edge length a and an equilateral triangle ¹ of side length a embedded in the two-dimensional space.
Formulae (16) suggest that there may be a general relationship between the area of the original set and its parallel set at a distance r. It is not di$cult to see that the areas of the three parallel sets can be written as A(K )"A(K)#;(K)r#r , (17) P where ;(K) denotes the boundary length (or perimeter) of the geometrical object K. The similarity between the construction of the parallel sets and the dilation of an image by means of a template of `radiusa r is not an accident: Dilation on a black-and-white image (see Section 3.3) is a digital equivalent of building the parallel set in Euclidian space [7]. This again shows that MIA and MIP have common roots. As a last example we consider a cube C of edge length a embedded in 3D space. A simple calculation shows that the volume < of the parallel set C can be written as P 4 (18) <(C )"a#6ar#3ar# r . P 3 Again, under certain restrictions on the shape of the 3D object K, (18) suggests the generalization 4 <(K )"<(K)#S(K)r#2B(K)r# r , P 3
(19)
where S(K) is the surface area and B(K) is the mean breadth. What is the point of all this? The examples presented above suggest that for a su$ciently simple geometrical object, the change in the volume (area) can be computed from the original volume, area, and mean breadth (area and perimeter), as long as we in#ate or de#ate the object without changing its topology. This is the key to the morphological characterization of sets of points in Euclidian space. Obviously, sets of pixels can be analyzed using these concepts too but in order to
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be useful in practice, there should be no constraints on the shape of the objects. The purpose of the next two subsections is to discuss the generalization of the above concept to objects of arbitrary shape. 5.2. Convex sets and Minkowski functionals The simple geometrical bodies used in the previous subsection are special in the sense that the set of points making up the body is convex. In fact, the general relations alluded to above only hold for convex bodies. Convex bodies play an important role in integral geometry. Therefore, we will "rst review some of their basic properties. A collection of points K in the d-dimensional Euclidean space 1B is called a convex set if for every pair of points in K, the entire line segment joining them also lies in K. A convex set with nonempty interior is called a convex body. A single point x31B is also a convex set and convex body. We will only consider convex sets that are bounded and closed, i.e. that are compact. The class of all compact convex sets is denoted by *. The parallel set K of a compact convex set K3* at a distance r is the union of all closed balls of P radius r, the centers of which are points of K [9]. The operation of taking a parallel set preserves the properties of convexity and compactness, i.e. K 3* [10]. Clearly the notion of a parallel set, P introduced in the previous subsection, agrees with this de"nition. The general expression for the volume vB of the parallel body K at a distance r of a convex body P K, is given by the Steiner formula [8]
B d =B(K)rJ , vB(K )" J P
J where the =B(K) are called quermassintegrals or Minkowski functionals and are given by J =(K)"l(K), =(K)"2, d"1 , =(K)"A(K), =(K)";(K), =(K)", d"2 , 2 =(K)"<(K), =(K)"S(K), =(K)" B(K) , 3 4 =(K)" , 3
d"3 .
(20)
(21a) (21b)
(21c)
Clearly (21) generalizes the results for the simple examples given above and con"rms that the Minkowski functionals have an intuitively clear meaning. It can be shown [8] that the Minkowski functionals are E Motion invariant: A functional is motion invariant if (gK)"(K) for K3* and g3G. Here G denotes the group of all translations and rotations in 1B. If we think of as an image functional, this condition assures that the result of the measurement does not depend on the choice of the coordinate system. E C-additive: A functional is C-additive if (K K )"(K )#(K )!(K K ) for K , K 3* and K K 3*. The notion of C-additive (means additive on the set *) is not just
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a technical one because the union of two convex sets is not necessarily convex, although the intersection is. E Continuity: is continuous if lim (K )"(K) whenever K is a sequence of compact sets J J J such that lim K "K in the Hausdor! metric [7]. Intuitively, this continuity property of J J means that whenever the compact convex sets K approach the compact convex set K, also J (K ) approaches (K). This is a rather technical condition that is sati"ed when we limit J ourselves to sets of pixels. A fundamental result in integral geometry is the completeness of the family of Minkowski functionals. A theorem by Hadwiger [8] states that every motion invariant, C-additive and continuous functional over * can be written as B (K)" a =B(K) (22) J J J with suitable coe$cients a 31. In other words, the d#1 Minkowski functionals form a complete J system of morphological measures on the set of convex bodies [8]. What is the relevance of the above to image functionals? In the Introduction we mentioned that MIA uses additive image functionals. Of course, we prefer to use motion invariant, additive image functionals. However, there is no reason why an image should be a convex set, so if we could replace `C-additivea by `additivea then Hadwiger's theorem would tell us that there are no more, no less than d#1 di!erent additive image functionals. This would be a nice result because it implies that we would have to switch to non-additive or coordinate-system-dependent image functionals to "nd additional non-morphological structure in the image. However, the extension of Hadwiger's theorem to additive instead of C-additive image functionals requires further consideration. 5.3. Convex rings and additive image functionals The results of the previous subsection can be generalized to a much more general class of objects by considering the convex ring [8] R, the class of all subsets A of 1B which can be expressed as "nite unions of compact convex sets J A" K ; K 3* . (23) G G G If A and A both belong to R then so do A A and A A . As before, an additive functional has the property (A A )"(A )#(A )!(A A ). Motion invariance of on R is de"ned as for on *. Obviously, an image is an instance of the convex ring R, the pixels being the convex sets and elements of *. Fundamental to the extension from * to R is the Euler characteristic or connectivity number de"ned as [8]
(K)"
1, KO , 0, K"
(24)
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for all K3*. The Euler characteristic is an additive, motion invariant functional on R [8]. For an element A of the convex ring R, the use of the property of additivity of yields
J (25) (A)" K " (K )! (K K )#2#(!1)J>(K 2K ) . G G G H J G G GH The value of (A) is independent of the representation of A as a "nite union of compact convex sets [8]. Note that all sets appearing on the r.h.s. of (25) are convex so that we can use (24) to compute the numerical (integer) value of (A). The Euler characteristic can be used to de"ne the Minkowski functionals for all elements of the convex ring A3R [8]. Recalling that a single point x31B is a convex set, we can write the characteristic function of the set A as I (x)"(Ax). Then the volume of A is given by =B(A)"G I (gx) dg. Here dg denotes the motion-invariant kinematical density [8,9] and the integration is over all elements of G [8,9]. The expression of the volume suggests the following de"nition [8] of the Minkowski functionals on R:
=B(A)" J
G
(AgE ) dg "0,2, d!1 , J
=B(A)" (A) "B /(1#d/2) , (26) B B B where E is a -dimensional plane in 1B. The normalization is chosen such that for a d-dimensional J ball B (r) with radius r, =B(B (r))" rB\J where denotes the volume of the unit ball ( "1, B J B B B "2, ", "4/3) [14]. The Minkowski functionals inherit from the property of additivity
J =B(A)"=B K " =B(K )! =B(K K )#2 G J J J G J G H G G GH #(!1)J>=B(K 2K ) (27) J J and motion invariance. Hadwiger [8] has shown that representation (22) is also valid for elements of the convex ring R. The d#1 Minkowski functionals form a complete system of additive functionals on the set of objects that are unions of a "nite number of convex bodies [8]. In the translation of these abstract mathematical results to MIA it is essential to keep in mind the conditions under which the mentioned theorems hold. Fortunately, in practice, this is easy to do. The crucial step is to decompose the image into a union of convex sets so that we can use the theoretical results that hold on the convex ring. We address this issue in Section 5.5. For completeness and also because we make use of it in Section 7, we state one more important result in integral geometry, the so-called kinematic formulae. These are very useful tools in stereology and stochastic geometry [8,9,25]. They play a key role in deriving averages of Minkowski functionals (see Appendix D for an example). Hadwiger's principal kinematic formulae read [8]
J MB(AgB) dg" MB (B)MB(A) , J J\I I G I MB(A)" B\J =B(A), "0,2, d . J J J B
(28a) (28b)
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Using (21), (28b) and the fact that (K)"1 for K3* we de"ne the normalized Minkowski functionals as M(K)"l(K), M(K)"(K), d"1 , 1 M(K)"A(K), M(K)" ;(K) , 2 1 M(K)" (K), d"2 , M(K)"<(K), M(K)"S(K) , 1 3 M(K)" B(K), M(K)" (K), d"3 4
(29a)
(29b)
(29c)
for all K3*. 5.4. Relation to topology and diwerential geometry The Euler characteristic is identical to the one de"ned in algebraic topology [8]. For d"2, (A) equals the number of connected components minus the number of holes. In three dimensions (A) is given by the number of connected components minus the number of tunnels plus the number of cavities. Some examples are shown in Fig. 14. The Euler characteristic describes A in a purely topological way, i.e. without reference to any kind of metric. Very often one is interested in the topology of the surface RA of A [2,15]. The Euler characteristic of RA is directly related to that of A, namely (RA)"(A)[1!(!1)L], where n is the dimension of the body A (n4d) [19]. The principal curvatures of a surface are useful quantities for the numerical characterization of the surface of a 3D body. They are de"ned as follows. Consider a point on the surface and the vector through this point, normal to the surface. A plane containing this normal vector intersects the surface. This intersection is a planar curve with a curvature called the normal curvature. Rotation of the plane about the normal produces various planar curves with di!erent values of normal curvature. The extreme values of the normal curvatures are called the principal curvatures and of a surface. These two curvatures can be combined to give two useful measures of the curvature of a surface, namely the Gaussian and mean curvature de"ned as G" and H"( # )/2, respectively. The integral mean curvature H and integral Gaussian curvature G are given by 1 H(A)" 2
1 1 # df R R .
(30)
and
G(A)"
1 df , R R .
(31)
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Fig. 14. Two- and three-dimensional "gures with various connectivity numbers or Euler characteristics .
respectively. Here R "1/ and R "1/ are the principal radii of curvature of A and df is the area element on A. For H and G to be well de"ned the boundary RA should be regular. The mean breadth is proportional to the integral mean curvature: H(A)"2B(A) .
(32)
The Euler characteristic of RA is closely related to the integral Gaussian curvature G and the genus g (number of handles): G(A)"2(RA), (RA)"2(1!g) .
(33)
Note that integral geometry imposes no regularity conditions on the boundary RA of the objects: H(A) and (A) are always well de"ned. 5.5. Application to images Each pixel in a 2D (3D) black-and-white image is a convex set. Therefore, such images may be considered as an element of the convex ring R and we can invoke integral geometry to build additive image functionals to measure features in the image. However, as mentioned before, some care has to be taken because the Minkowski functionals take known values on convex sets only. The key to the practical application of integral geometry to images is the additivity of (see (25)): We can compute the Minkowski functionals of an image A by decomposing A into convex sets K . G However, if we would take for K all black pixels (assuming the background consists of white G pixels), then we would have to compute all the intersections that appear in (25). Although this can be done, it is much more expedient to take a slightly di!erent route.
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First, we write each pixel K as the union of the disjoint collection of its interior body, interior faces (in 3D only), open edges and vertices [19]. We will denote the interior of a set A by Ax "ARA. The values of the Minkowski functionals of the open interior of an n-dimensional body A3R embedded in 1B (n4d) are given by [19] =B(Ax )"(!1)B>L>J=B(A), "0,2, d . (34) J J By making use of the additivity of the Minkowski functionals (see (27)) and the fact that there is no overlap between open bodies on a lattice, the values of the Minkowski functionals on the whole pattern P"P(x, q) may be obtained from =B(P)" =B(Nx )n (P), "0,2, d , (35) J J K K K where n (P) denotes the number of the open bodies Nx of type m present in P. On a square and K K cubic lattice there are d#1 open bodies Nx : Nx corresponds to a vertex, Nx to an open line K segment, Nx to an open square on both the 2D square and the 3D cubic lattice, and Nx to an open cube on the 3D cubic lattice. The values of the Minkowski functionals for the building blocks Nx of K a 2D square and a 3D cubic lattice are given in Tables 1 and 2, respectively. Their derivation is given in Appendix C. The procedure to calculate n (P) is described in Appendix B. This completes K the construction of the method to compute the d#1 additive image functionals for a d-dimensional lattice "lled with black and white pixels. In essence the method boils down to the simple procedure of counting vertices, edges, etc., as described in Section 2. We illustrate the procedure to compute the Minkowski functionals by considering the 2D checkerboard pattern with an even number ¸ of cells, of edge length one, in each direction. We consider free and periodic boundary conditions (see Fig. 15). The left picture in Fig. 15 shows the 4;4 checkerboard lattice with free boundary conditions, i.e. the pattern is completely surrounded by white pixels. The right picture shows the same pattern but with periodic boundary conditions. For the ¸ ;¸ checkerboard P with free boundary conditions we "nd n (P )" $ $ (¸ #1)!2, n (P )"2¸ , n (P )"¸ /2 and hence A(P )"¸ /2, ;(P )"2¸ and $ $ $ $ (P )"¸ /2!(¸ !1). Note that this value of corresponds to the value we "nd if we calculate $ as the number of connected components minus the number of holes, since the number of connected components (black structure) equals one and the number of holes equals (¸ /2!1)(¸ !2). For the ¸ ;¸ checkerboard P with periodic boundary conditions we "nd . Table 1 Minkowski functionals =( "0,2, d"2) for the open bodies N[ , the basic building blocks of a two-dimensional J K square lattice. Q[ : open square of edge length a; ¸[ : open edge of length a; P[ : vertex. A denotes the covered area, ; the perimeter and the Euler characteristic m
N[ K
="A
=";/2
="
0 1 2
P[ ¸[ Q[
0 0 a
0 a !2a
!
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Table 2 Minkowski functionals =( "0,2, d"3) for the open bodies N[ , the basic building blocks of a three-dimensional J K cubic lattice. C[ : open cube of edge length a; Q[ : open square of edge length a; ¸[ : open edge of length a; P[ : vertex. < denotes the covered volume, S the surface area, B the mean breadth and the Euler characteristic m
N[ K
="<
="S/3
="2B/3
="4/3
0 1 2 3
P[ ¸[ Q[ C[
0 0 0 a
0 0 2a/3 !2a
0 a/3 !2a/3 a
4/3 !4/3 4/3 !4/3
Fig. 15. 4;4 checkerboard pattern. The black line denotes the boundary. Left: free boundaries; right: periodic boundaries.
n (P )"¸ , n (P )"2¸ , n (P )"¸ /2 which yields A(P )"¸ /2, ;(P )"2¸ . . . . . (P )"!¸ /2. Note that (P )/¸ "lim (P )/¸ "!1/2. . . * $
and
5.6. Reducing digitization errors The Minkowski functionals computed on the lattice of pixels, will be called digital Minkowski functionals. They yield approximate values of the Minkowski functionals of these objects in Euclidean space. By digitizing the 2D (3D) image we have introduced square (cubic) distortions in the objects, causing a directional bias. For example, digitizing a 2D (3D) image transforms a smooth contour (surface) to a more stepwise contour (surface). The more complicated the image the better the digital approximations are likely to become since the parts of the stepwise boundary or surface will exhibit each orientation more often. The most problematic structures may be isotropic ones. We will not treat the problems of digitization to large extent but will only give a simple method to obtain a better approximation to the Euclidean perimeter and area (covered area and volume) in two (three) dimensions. There are several methods to correct for the systematic error, caused by digitization of the image [3,26]. A correction to the digital Minkowski functionals that leads to a better approximation of the area and perimeter in 2D (volume and covered area in 3D) can be made by explicitly taking into account the number of `stepsa n (P) in the pattern P"P(x, q). In 2D (3D) patterns, `stepsa are 1
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Fig. 16. Steps as de"ned on a 2D (3D) square (cubic) lattice.
de"ned by two pixels that share only one vertex (edge). Examples of steps are shown in Fig. 16. The corrected digital Minkowski functionals corresponding to the perimeter and area (covered area and volume) in 2D (3D), may be written as = B(P)"=B(P)!CBn (P), "0, 1 , J J J 1
(36)
where =B(P) is given by (35) and CB denotes a correction factor. On a square and cubic lattice J J C"C"1, C"(2!(2)/2 and C"(2!(2)/3. The procedure to calculate n (P) is 1 described in Appendix C. As an illustration we use this correction procedure to calculate the area and perimeter for the triangle shown in Fig. 17a. The Euclidean perimeter and area are 20.35 cm and 16.47 cm, respectively. As usual, we "rst digitize the image of the triangle by mapping the triangle on square grids. Some results for various grid spacings are shown in Figs. 17b}d. The digitization transforms the straight and smooth boundaries of the triangle to more stepwise boundaries. Using the procedure outlined in Section 2 we calculate the digital perimeter and area of the objects in Figs. 17b}d. Then by making use of (36), we compute the corrected digital perimeter and area. The results are summarized in Table 3. As seen from Table 3 the values of the digital perimeter and area obtained using (35) are always larger than their Euclidean counterparts. Reducing the grid spacing does not lead to a fast convergence of the values of the digital perimeter and area to the values of the Euclidean ones. If we compute the digital perimeter and area using (36) the values are still larger than the Euclidean ones but the improvement is substantial.
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Fig. 17. Digitization process for a triangle. Triangle in Euclidean space (a); triangle mapped on a square grid with grid spacing 2 (b); 1 (c) and 1/2 (d). Table 3 Digital and corrected digital perimeter and area of the objects shown in Figs. 17b}d. ¸ denotes the grid spacing and ;, A (; , A ) denotes the digital (corrected digital) perimeter and area, respectively. The Euclidean perimeter and area are 20.35 cm and 16.47 cm, respectively Fig. 17
¸
; (cm)
A (cm)
; (cm)
A (cm)
(b) (c) (d)
2 1 1/2
36.00 30.00 30.00
48.00 31.00 23.50
33.07 27.36 26.92
28.00 22.00 18.25
5.7. Normalization of image functionals In the following sections, we calculate Minkowski functionals for 2D (3D) for a variety of square (cubic) lattice systems. For practical purposes it is convenient to introduce the following quantities: AI "A/¸, ;I ";/¸N , "/N, d"2
(37a)
(37b)
and
where ¸ denotes the linear size of the square (cube) and N denotes the number of germs.
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In the case of 2D (3D) periodic structures the structures may be divided into equivalent regions bounded by a square (cubic) unit cell of space. Let us assume that the 2D (3D) periodic structure of total area ¸ (volume ¸) is composed of several unit cells of typical length scale ¸ . Then the Euler characteristic of the whole system is given by
¸ B ,M , (38) ¸ where M,(¸/¸ )B denotes the number of unit cells. The other morphological quantities of the whole system may be written as "
A"AM ¸ M, ;";M ¸ M, d"2 , (39a) <"<M ¸ M, S"SM ¸ M, B"2BM ¸ M, d"3 . (39b) The quantities AM , ;M , <M , SM , BM and characterize the structure within one elementary unit of the periodic structure.
6. Illustrative examples We "rst apply MIA to simple cubic, face-centered cubic and body-centered cubic lattice structures with and without imperfections. We use the method described in Section 2.3 to analyze these point patterns. In Sections 6.2 and 6.3 we employ the method described in Section 2.4 to compute the value of the Minkowski functionals of some complex 3D surfaces, namely some triply periodic minimal surfaces and the Klein bottle. 6.1. Regular lattices The face-centered cubic (FCC) and body-centered cubic (BCC) lattices are of great importance, since an enormous variety of solids and several complex #uids [34] crystallize in these forms. The simple cubic (SC) form, however, is relatively rare. The SC lattice may be generated from the following set of primitive vectors: a "¸ (1, 0, 0), a "¸ (0, 1, 0), a "¸ (0, 0, 1) , (40) where ¸ denotes the lattice constant. A symmetric set of primitive vectors for the FCC cubic lattice is ¸ ¸ ¸ a " (0, 1, 1), a " (1, 0, 1), a " (1, 1, 0) 2 2 2
(41)
and for the BCC cubic lattice is ¸ ¸ ¸ a " (1, 1,!1), a " (1,!1, 1), a " (!1, 1, 1) . 2 2 2
(42)
To compute the Minkowski functionals for the SC, FCC, and BCC lattices we place them on a cubic lattice with lattice constant one, making use of (40)}(42), and we put one black pixel at each
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Fig. 18. Graining procedure for the SC lattice with periodic boundary conditions and ¸ "4. The thick solid line indicates the dimensions of the unit cell.
point of the SC, FCC or BCC lattice, respectively. By making use of the procedure described in Section 2.3 we transform the resulting point pattern into a pattern of `sphericala grains of radius r and study the behavior of the Minkowski functionals as a function of r. An example of the graining procedure is shown in Fig. 18 for the SC lattice with periodic boundary conditions and ¸ "4. The thick solid line indicates the dimensions of the conventional unit cell, simply called the unit cell from now on. Fig. 19 shows the Minkowski functionals
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Fig. 19. Minkowski functionals as a function of r for the perfect SC (dotted curve), FCC (solid curve) and BCC (dashed curve) lattice with M"1 and ¸ "32 with periodic boundary conditions.
Crystal structures formed in materials are not perfect. Therefore, it is of interest to study the in#uence of defects on the curves shown in Fig. 19 for the BCC lattice. Imperfections in the crystal structure may be formed by the absence or by small displacements of some of the basic lattice points. Also the presence of impurities, creating extra lattice points, causes an imperfect crystal structure. In Fig. 20 we show the Minkowski functionals as a function of r for perfect and imperfect BCC lattice structures. The solid curve depicts the data for a perfect BCC lattice containing M"8 unit cells of linear dimension ¸ "16. The dashed curve shows the data for the same BCC lattice to which $30% of defects have been added at randomly chosen positions. The dotted curve depicts the results of displacing $30% randomly chosen basic lattice points over a random distance 0 or 1. Apart from some minor changes the three curves behave in the same way. Only if we move all the lattice points over a random distance 0 or 1 (dash}dotted lines), the curves for BI and di!er qualitatively from the ones of the perfect BCC lattice. Therefore, we may conclude that the presence of small amounts of defects in the crystal structure does not alter the characteristic behavior of the Minkowski functionals as a function of r. 6.2. Triply periodic minimal surfaces [35] A minimal surface in 1 is de"ned as a surface for which the mean curvature (see Section 5.4) is zero at each of its points. As a consequence, at every point of a minimal surface the two principal curvatures are equal, but opposite in sign. Hence the Gaussian curvature is always non-positive.
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Fig. 20. Minkowski functionals as a function of r for BCC lattice structures with M"8 and ¸ "16 with periodic boundary conditions. Solid line: perfect BCC lattice; dashed line: BCC lattice to which $30% of impurities have been added at randomly chosen positions; dotted line: BCC lattice of which $30% of randomly chosen basic lattice points have been moved over a randomly chosen distance 0 or 1; dash}dotted line: BCC lattice of which all the basic lattice points have been moved over a randomly chosen distance 0 or 1.
For every closed circuit on the surface, the area is a minimum. We will consider the triply periodic minimal surfaces (TPMS), minimal surfaces that are periodic in three independent directions. During the last years these TPMS and similar interfaces have been elaborately discussed in literature since structures related to TPMS may form spontaneously in physico-chemical and in biological systems [2,36]. Examples may be found in various crystal structures [2,37,38], lipidcontaining systems [39}42], microemulsions [43], block copolymers [44}57], skeletal elements in sea urchins [58,59] and cell membranes [60]. A TPMS is either free of self-intersections or may intersect itself in a more or less complicated way. Each TPMS without self-intersections is two-sided and subdivides 1 into two in"nite, connected but disjunct regions. These two regions, or labyrinths, are not simply connected and they interpenetrate each other in a complicated way. The two labyrinths may di!er in shape or they may be congruent, i.e. there exist symmetry operations mapping one labyrinth onto the other. In the latter case the surface is called a balance surface [61]. The symmetry of a balance surface is described by a group}subgroup pair H/I of spacegroups, where H contains all isometries of 1 which map the surface onto itself. An isometry of H maps each side of the surface and each labyrinth either onto itself or onto the other side and the other labyrinth [61]. I contains only those isometries which map each side of the surface and each labyrinth onto itself. If the two sides of
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a balance surface are `coloreda so that they are symmetrically distinct, black}white space groups instead of the group-subgroup pairs with index 2 may be used to describe its symmetry [61]. In this case the surface is called oriented. The periodic surfaces can be divided into equivalent regions bounded by a unit cell of space. There are two common choices of unit cells, the lattice fundamental region and the crystallographic cell [62]. The lattice fundamental region contains the smallest region of the surface that reproduces the complete surface upon translation of this unit cell alone. The crystallographic cell is the smallest cube generating space by the lattice and can contain many lattice fundamental regions. We give our data for the crystallographic cell, simply called the unit cell from now on, and consider the bicontinuous structure of total volume ¸ to be composed of several unit cells of typical length scale ¸ . Then the Minkowski functionals for one elementary unit may be calculated from (38) and (39). Here we only consider the P (primitive) [63] the D (diamond) [63] and the G (gyroid) [64] surfaces, which are TPMS free of self-intersections. The P, D and G surfaces, with group-subgroup pairs of space groups with index 2 Im3 m/Pm3 m, Pn3 m/Fd3 m and Ia3 d/I4 32, respectively, divide space into two equal labyrinths related by a translation (for P and D) or an inversion (for G), thereby generating a bicontinuous geometry. The Bravais lattices for the P, D, and G surfaces are BCC, SC and BCC, respectively. For the oriented P, D and G surfaces the Bravais lattices are SC, FCC and BCC, respectively. The oriented P, D and G surfaces may be approximated by the periodic nodal surfaces [65}67] cos x#cos y#cos z"0 ,
(43a)
sin x sin y sin z#sin x cos y cos z#cos x sin y cos z#cos x cos y sin z"0 ,
(43b)
sin x cos y#sin y cos z#sin z cos x"0 .
(43c)
In Fig. 21a}c we show the nodal P (43a), D (43b) and G (43c) surfaces, in their unit cell. Tables 4}6, summarize the results for the Minkowski functionals of the thresholded nodal oriented P, D and G surfaces for several numbers of unit cells. As seen from Tables 4}6, the values calculated using the integral-geometry approach are in good agreement with the numbers found in literature [62,68,69]. Note that in contrast to many works in literature, we compute (P) and not (RP)"2(P) (see also
Fig. 21. Unit cube for the nodal primitive P surface (a), the nodal double diamond D surface (b) and the nodal gyroid G surface (c). The surfaces are generated from Eq. (43).
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Table 4 Minkowski functionals of the P surface (at threshold q"0.5) obtained from (43a) for M"(¸/¸ ) unit cells, where ¸ denotes the edge length of the total cube and ¸ the edge length of a unit cube. The values found in the literature [62,68,69] are given in parenthesis ¸
¸
M
<M
SM
16 32 32 64 64 128
64 64 128 64 128 128
64 8 64 1 8 1
0.502 0.501 0.501 0.501 0.500 0.500 (0.500)
3.800 3.715 3.715 3.671 3.671 3.675 (2.345)
!2 !2 !2 !2 !2 !2 (!2)
Table 5 Minkowski functionals of the D surface (at threshold q"0.5) obtained from (43b) for M"(¸/¸ ) unit cells, where ¸ denotes the edge length of the total cube and ¸ the edge length of a unit cube. The values found in the literature [62,68,69] are given in parenthesis ¸
¸
M
<M
SM
16 32 32 64 64 128
64 64 128 64 128 128
64 8 64 1 8 1
0.500 0.500 0.500 0.500 0.500 0.500 (0.500)
6.000 6.000 6.000 6.000 6.000 6.000 (3.838)
!8 !8 !8 !8 !8 !8 (!8)
Table 6 Minkowski functionals properties of the G surface (at threshold q"0.5) obtained from (43c) for M"(¸/¸ ) unit cells, where ¸ denotes the edge length of the total cube and ¸ the edge length of a unit cube. The values found in the literature [62,68,69] are given in parenthesis ¸
¸
M
<M
SM
16 32 32 64 64 128
64 64 128 64 128 128
64 8 64 1 8 1
0.500 0.500 0.500 0.500 0.500 0.500 (0.500)
4.970 4.900 4.900 4.857 4.857 4.847 (3.092)
!4 !4 !4 !4 !4 !4 (!4)
Section 5.4). Only the numbers for the area are about a factor of 1.6 larger than the numbers quoted in literature. This systematic error is due to the digitization of the picture. This operation transforms the smooth surface to a more stepwise surface which enlarges the covered area.
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Table 7 Corrected digital volume and covered area of the P, D and G surfaces (at threshold q"0.5) obtained from (43a) to (43c) for one unit cell with edge length ¸ "128. The Euclidean values found in the literature [62,68,69] are given in parenthesis TPMS
<M
SM
P D G
0.481 (0.500) 0.470 (0.500) 0.474 (0.500)
2.204 (2.345) 3.782 (3.839) 2.916 (3.092)
Table 8 Digital (HI) and corrected digital (HI ) homogeneity index of the P, D, and G surfaces (at threshold q"0.5) obtained from (43a) to (43c) for one unit cell with edge length ¸ "128. The Euclidean values found in the literature [70] are given in the last column TPMS
HI
HI
Ref. [70]
P D G
1.4053 1.4658 1.5051
0.6527 0.7336 0.7023
0.7163 0.7498 0.7667
In Table 7 we show the results of using the method described in Section 5.6 to reduce this error. The results are for one unit cell and ¸ "128. As seen from Table 7 the numbers for the volume and the covered area are now about a factor 0.94}0.98 smaller than the numbers quoted in literature [62,68,69]. Hence for the examples shown in this section the method described in Section 5.6 underestimates the Euclidean volume and covered area. From Tables 4}7 the values for the non-oriented surfaces may be obtained using the following relationships [62]: (i) one unit cell of the oriented P, D and G surfaces contains one, eight and one unit cell(s) of the non-oriented P, D and G surfaces, respectively; (ii) the area numbers of the oriented surfaces must be multiplied by a factor of 2 in the case of the P and G surface and a factor of 1 in the case of the D surface to obtain the area numbers for the non-oriented surfaces. A useful dimensionless measure of the surface of bicontinuous structures is the homogeneity index HI de"ned as [69]
HI"
!SM . 4
(44)
A discussion of other dimensionless quantities can be found in [62]. For `homogeneousa minimal surfaces for which the Gaussian curvature is constant everywhere on the surface, HI"0.75 [70]. The TPMS have homogeneity indices close to 0.75: HI"0.7163 for the P-surface, HI"0.7498 for the D-surface and HI"0.7667 for the G-surface [70]. The values for HI for the TPMS mapped on a cubic grid, as calculated using the method described in Section 5.5 are given in the second column of Table 8. The results are for one unit cell of edge length ¸ "128. The third column of Table 8 shows the results if we use the correction method described in Section 5.6 to calculate HI . These values are much closer to the Euclidean ones found in the literature [70] (see last column of Table 8).
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In summary: Integral-geometry MIA is a convenient tool to characterize the morphological properties of complex surfaces such as the TPMS. In particular, to study the topology of the TPMS, MIA does not require the use of labyrinth graphs or surface tilings [71]. 6.3. Klein bottle The Klein bottle is a well-known non-orientable (see Section 6.2) surface in algebraic topology. It is obtained if two holes cut in the sphere are closed up with MoK bius bands (closed circular strip with a twist) or if two MoK bius bands are pasted together along their boundaries [72]. The Klein bottle is a single-sided bottle without boundary. A real Klein bottle cannot exist in 3D since the surface has to pass through itself without a hole, but it can be immersed in 3D. The parametric equation is given by (45a) x" (3 sin 2t#4)sin s(sin t# cos t(4 cos t!3) , (45b) y"!cos t sin t# (3 sin 2t#4) sin t cos s , z"cost# (3 sin 2t#4)(4 cos t!3) cos t cos s (45c) with t"0,2, and s"0,2, 2. Some pictures of a Klein bottle are given in Fig. 22. The surface of the Klein bottle looks rough and steplike due to the digitization process. The Klein bottle has
Fig. 22. Various orientations of the digitized Klein bottle.
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Euler characteristic zero [72]. If we study the topology of the Klein bottle by means of MIA we also "nd "0. This demonstrates once more that MIA is a fast, reliable and convenient tool to study the topology of weird surfaces. In the following two sections we present examples that show that MIA does not loose these attractive properties when we analyze much more complex patterns.
7. Random point sets We consider a collection of N points p , with positions generated from a uniformly uncorrelated G random distribution, in a convex domain L1B. The mean density of points equals "N/, where denotes the volume of . We attach to every point p (germ) a grain A 3R. A con"guraG G tion of the grains A gives rise to a set A 3R G , , A " g A , (46) , G G G where g 3G, under the assumption that the translations are restricted to . This random G distribution of grains includes the Boolean model [25], a basic model in stereology and stochastic geometry [9,10]. We are interested in the mean values of the normalized Minkowski functionals of A . In the , bulk limit N, PR, "N/ "xed, the averages M /N , "0,2, d, with the notation J , M ,MB, are known exactly [11,73,74] and are given by J J M /N "(1!e\MK )/ , ,
(47a)
M /N "m e\MK , ,
(47b)
M /N "(m !m )e\MK , ,
(47c)
M /N "(m !3m m #m )e\MK , ,
(47d)
where M denote the average of the Minkowski functionals of the ensemble with density and J , m denote the mean values of the Minkowski functionals of a single grain. In Appendix D we give J a derivation of (47) which di!ers from the one given in [11] in that no use is made of the so-called Minkowski polynomials [74]. On a regular d-dimensional lattice it is more natural to work with hypercubes instead of digital approximations of the corresponding Euclidean shapes. This suggests the construction of a genuine discrete integral geometry, without making reference to Euclidean space. We therefore consider a collection of N pixels p in a hypercubic domain L1B of volume "¸B . The positions of the G V pixels are generated from a uniformly uncorrelated random distribution. The mean density of pixels equals "N/. We attach to every germ p a hypercubic grain C . In the bulk limit G G N, PRwith "xed, the averages of the morphological quantities of the ensemble of con"gurations of the hypercubic grains C are still given by (47) but the explicit expressions for M and G J m , "0,2, d are di!erent from those in Euclidean space [75]. J
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7.1. Two dimensions In 2D Euclidean space and in the bulk limit (47) yields A()/N "AI ()/"(1!e\M?)/ , , ;()/N ";I ()/ "ue\M? , , 1 ()/N "()" 1! u e\M? , , 4
(48a) (48b) (48c)
where we made use of (29b) and (37a) and where a and u denote the mean values of the area and perimeter of a single grain. The Euler characteristic of a single grain equals one. In the case that the grains are circular discs of radius r, we substitute in (48) a"r and u"2r. This leads to AI "1!e\L, ;I "2 re\L, "(1!n)e\L
(49)
with n"r. As we are working on a square lattice anyway it might be more bene"cial to use square grains (instead of discrete approximations to circular discs) to study the morphological properties of random point sets and to make no reference to Euclidean space at all. Then we can still use (47) but we need expressions for the normalized Minkowski functionals on a lattice, analogous to the Euclidean ones given in (29b). In analogy with Sections 5.1 and 5.2 on a square lattice the area of a square C , parallel to a square C of edge length a at a distance is given by C A(C )"(a#2)"a#4a#4 C "A (C)#; (C)#4 (C) , (50) A A A where A (C)"a, ; (C)"4a and (C)"1. Using Steiner's formula (20) and substituting for A A A the volume of the unit cube in d dimensions (i.e. "1) we "nd that B B M "A , M "S /2, M "4 . (51) A A A Finally using (47) we obtain AI "1!e\L, ;I "4a e\L, "(1!n)e\L (52) A A A with n"a. We adopt the procedure outlined in Section 2.3 to compute the morphological properties of a uniform random distribution of points in a square of edge length ¸ with periodic boundaries. First, we transform the point pattern into a pattern of disc-like grains of radius r and then investigate the behavior of the Minkowski functionals as a function of r. The solid lines in Fig. 23 show AI , ;I and as a function of r for a single realization of N"10 240 pixels on a square lattice of linear size ¸"1024, subject to periodic boundary conditions. For small r the disc-like grains are isolated. This gives rise to a small covered area, perimeter and to a positive Euler characteristic. For large r the discs cover almost completely the whole square leading to a large covered area, a small perimeter and a negative Euler characteristic which approaches zero in the case of the completely covered square. For intermediate r, the coverage has a net-like structure with a negative Euler characteristic and a large perimeter. The dotted lines in
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Fig. 23. Minkowski functionals as a function of r for random point sets with `disc-likea grains, using periodic boundary conditions. Solid lines: 10 240 points in a square of edge length 1024; dotted lines: "t to data, using the expressions given by (53).
Fig. 23 are the results obtained by "tting AI "1!e\L, ;I "ue\L, "(1!f u/4)e\L , (53) $ $ $ to data with n"f r and u"2f r. The functional behavior of AI , ;I and is chosen to be $ $ $ the same as for grains that are circular discs in Euclidean space. The "tting parameters f , f and f have been introduced to take into account that in practice we are working on a square lattice and are approximating circular discs by discrete structures. We "nd f "0.22, f "0.66 and f "0.80 for the dotted line by "tting the solid line. If we use square grains instead of disc-like grains we obtain the results shown in Fig. 24 (solid line). The dotted lines are the results obtained from (52). As seen from Fig. 24 the agreement between the data and the theoretical result (52) is excellent. Note the absence in (52) of any adjustable parameter. 7.2. Three dimensions In 3D and in the bulk limit we "nd [14] <()/N "
(54a) (54b)
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Fig. 24. Minkowski functionals as a function of r for a random point set with square grains, using periodic boundary conditions. Solid lines: 10 240 points in a square of edge length 1024; dotted lines: results obtained from discrete integral geometry (see (52)), without "tting.
s B()/N "BI ()/2 "b 1! e\MT , , 64 b
1 s e\MT , ()/N "()" 1! sb# , 2 384
(54c) (54d)
where use has been made of (29c) and (37b) for d"3 and where
(55a)
SI "4 re\L ,
(55b)
3 n e\L , BI "4 r 1! 32
3 n e\L " 1!3n# 32 with n"4r/3.
(55c) (55d)
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We now consider the case that the grains are cubes on a regular 3D lattice. In analogy to the 2D case we "rst compute the volume of the cube C parallel to a cube C of edge length a at a distance C <(C )"(a#2) , C "< (C)#S (C)#8B (C)#8 (C) , A A A A
(56a) (56b)
where < (C)"a, S (C)"6a, B (C)"3a/2 and (C)"1. Then by using the Steiner formula (20) A A A A and substituting "1 we "nd that B M "< , M "S /3, M "8B /3, M "8 . A A A A
(57)
Finally, using (47) we obtain
(58a)
SI "6a e\L , A
(58b)
BI "3a (1!n)e\L , A
(58c)
"(1!3n#n)e\L A
(58d)
with n"v"a. Note the absence in (58) of any adjustable parameter. We will now study the Minkowski functionals for sets of points which are randomly positioned in a cube of edge length ¸ subject to periodic boundaries. Again we follow the procedure described in Section 2.3: We transform the point pattern into a pattern of `sphericala grains of radius r and study the behavior of the Minkowski functionals as a function of r. Fig. 25 shows the Minkowski functionals
(59a)
SI "s e\L , $
(59b)
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Fig. 25. Minkowski functionals as a function of r for random point sets with `sphericala grains, using periodic boundary conditions. Dotted line: 1024 points in a cubic box of edge length 128; solid line: 128 points in a cubic box of edge length 64; dashed line: 16 points in a cubic box of edge length 32.
s BI "2 b! e\L , $ 64
sb s " 1! f # f e\L $ 2 384
(59c)
(59d)
to the data with n"4f r/3, s"4f r and b"2f r. The functional behavior of
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Fig. 26. Minkowski functionals as a function of r for random point sets with `sphericala grains, using periodic boundary conditions. Solid lines: 1024 points in a cubic box of edge length 128; dash}dotted lines: "t to 1024-point data, using the expressions given by (59); dashed lines: 512 points in a cubic box of edge length 128; dotted lines: "t to 512-point data, using the expressions given by (59).
7.3. Percolation Consider a square lattice of linear dimension ¸ of which a certain fraction of squares (sites) is colored black, whereas the others are colored white. We assume that the sites are colored randomly, that is each site is colored black or white independent of the color of its neighbors. We call p (04p41) the probability of a site being colored black. Hence, the average number of black sites is given by p¸ and the average number of white ones by (1!p)¸. Black pixels can be grouped into clusters: Two black pixels belong to the same cluster if they are nearest neighbors or can be connected by a chain of black pixels that are nearest neighbors (note that a `clustera is not the same as an `objecta, see Section 3.1). Percolation theory deals with the number and geometric properties of these clusters. The origin of the mathematical theory of percolation goes back to a publication by Broadbent and Hammersley [76]. Since percolation processes and their applications have been discussed by so many authors we can only make reference to a few books which give a general introduction to percolation theory, deal extensively with its mathematical aspects, give applications and/or give an extensive bibliography [77}81]. From percolation theory it is known that for in"nite square lattices there exists a critical value p , the percolation threshold, such that all clusters are "nite when p(p , but there exists an in"nite cluster when p'p . The latter
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Fig. 27. Minkowski functionals as a function of r for a random point set with cubic grains, using periodic boundary conditions. Solid lines: 1024 points in a cubic box of edge length 128; dotted lines: results obtained from discrete integral geometry (see (58)), without "tting.
cluster is called a percolation cluster since it percolates through the system like water percolates through a porous stone. There are two distinct types of percolation problems. An example of a site percolation problem (as the one described above) is shown in Fig. 28a. The clusters are encircled. Another type of percolation process is bond percolation, in which the edges rather than the sites are colored black or white at random. A cluster is then a group of sites connected by black bonds. An example of bond percolation is shown in Fig. 28b. It is well known that every bond model may be reformulated as a site model on a di!erent lattice, but not vice versa. One of the interesting problems in percolation theory is the determination of the value of the percolation threshold p . This value depends on the percolation problem studied and on the lattice type (including its dimensionality). The percolation threshold p , de"ned as the concentration at which an in"nite cluster appears in the in"nite lattice, for site percolation on the square lattice is not known exactly. Several authors have given (rigorous) upper and lower bounds [82}88] 0.556(p (0.679492 . (60) Computer evaluations of the percolation threshold give the estimate [89,90] p "0.5927460. It is A also known that p "1/2 for site percolation on the in"nite triangular lattice. For bond percolation on the in"nite square lattice p "1/2, as suggested by Sykes and Essam [91], and proven rigorously by Kesten [92]. For the bond problem on the triangular lattice and honeycomb lattice
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Fig. 28. Percolation problems on a square lattice: (a) site percolation problem (nearest neighbor), (b) bond percolation problem. The clusters are encircled.
Sykes and Essam proposed that p "2 sin(/18), and p "1!2 sin(/18), respectively [91]. Rigorous proofs of these conjectured values are given by Wierman [93]. In the remainder of this section we will consider only the site percolation problem on a square or cubic lattice. Clearly, this is yet another example of a random point (or pixel) set. We consider a square (cubic) lattice of which a certain fraction of squares (cubes), with positions generated from a uniformly uncorrelated random distribution, is colored black. The probability of a site being colored black is p (04p41). In Fig. 29 (30) we show the Euler characteristic as a function of p (diamonds) for a 2D (3D) square (cubic) lattice with periodic boundaries and linear dimension ¸"512 (¸"64). In 2D (see Fig. 29), (p) is positive and increases with p for small p. At some value of p, (p) starts decreasing and becomes negative. (p)"0 at p+0.39. For large p, (p) starts increasing again and becomes zero for the completely covered square (periodic boundaries). In 3D (see Fig. 30), the behavior of as a function of p is similar but (p) has one more change of sign: "0 at p +0.16 and p +0.61. Intuitively, one would think one could use the Euler characteristic to de"ne percolation [11,14,94}96]. Namely in 2D the Euler characteristic is de"ned as the number of connected components (objects) minus the number of holes. In the case of site percolation, one then expects to be positive and equal to p for small p while for large p one expects to be negative. The question then arises when one can relate the change of sign of to the percolation threshold. Calculation of the Euler characteristic of a pattern of active (here considered to be black) pixels requires the consideration of both the nearest and next-nearest neighbors of the active pixels and the nearest neighbors of inactive (white) pixels only (see also Section 3.1). In 2D we may write (61) 夹(p)"N夹(p)!N (p) , 5 where N (N夹) denotes the mean number of white (black) objects on the simple quadratic lattice 5 with nearest (and next-nearest) neighbors. The distinction between both lattices (¸at and ¸at夹) is
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Fig. 29. Percolation on a 2D square lattice with periodic boundaries (¸"512). Diamonds: Euler characteristic as a function of p; solid line: matching polynomial. Fig. 30. Percolation on a 3D cubic lattice with periodic boundaries (¸"64). Diamonds: Euler characteristic as a function of p. The solid line is a guide to the eyes.
denoted by the star. ¸at and ¸at夹 form a matching pair of lattices [91]. Note that all Euler characteristics computed in this work are, in fact, calculated on ¸at夹 and should for reasons of consistency have been denoted by 夹. However, for the sake of clarity we have omitted the 夹 in all sections except this one. According to Sykes and Essam [91], at density p the mean number of black (white) clusters on the simple quadratic lattice with nearest neighbors N (p) [N (p)] di!ers 5 from the mean number of white (black) clusters on the simple quadratic lattice with nearest and 夹 夹 夹 next-nearest neighbors N (p) [N (p)] by (p) [! (p)]. Or in other words 5 (62a) N (p)"N夹 (p)#(p) , 5 (62b) N (p)"N夹(p)!夹(p) , 5 where (p)"p!2p#p ,
(63a)
夹(p)"p!4p#4p!p ,
(63b)
are called the matching polynomials [91]. Property (62) is called cross-matching. From (61) and (62b) it follows that 夹(p)"夹(p) ,
(64)
i.e. in 2D the matching polynomial coincides with the Euler characteristic [94]. As can be seen from Fig. 29, where we plot the Euler characteristic (diamonds) and the matching polynomial 夹(p) (solid line) as a function of p, our numerical results are in perfect agreement with (64). Note that the value of p for which 夹(p)"0 (i.e. p+0.39) does not correspond to the percolation threshold p夹+0.41 for site percolation on the square lattice with nearest and next-nearest neighbors. The property of cross-matching (62) leads to p #p夹"1 [91]. To determine p and p夹 a second relation is required.
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For some cases Sykes and Essam calculated the exact values of percolation thresholds using the matching polynomials (see above) [91]. If ¸at"¸at夹 the lattice is called self-matching [91]. For self-matching lattices [91] N (p)"(p)#N (p) , 5 where (p)"p!3p#2p .
(65)
(66)
The triangular lattice is an example of a self-matching lattice. For a self-matching lattice p "1/2 and hence, in this particular case, (p )"0. 8. Block copolymers Block copolymers consist of a successive number of sequences of the same type of monomer, called blocks. The polymer block types, the composition and overall molecular size can be varied and precisely controlled [97]. AB block copolymers, for example, consist of a sequence of type A monomers covalently bonded to a chain of type B monomers. In equilibrium AB block copolymers assemble into a variety of phases, creating domains of component A and component B. The type of domain structures that are formed is determined by three experimentally controllable factors: The overall degree of polymerization, the relative volume fraction of the A and B components, and the A}B segment}segment (Flory}Huggins) interaction parameter [97]. The "rst two factors contribute to the entropic chain-conformational energy and the latter one to the enthalpic interaction energy. These two energies are important in the formation of interfaces in block copolymers. If the product of the overall degree of polymerization and is large, narrow interfaces separate nearly pure A and B domains [97]. In this regime, the inter-material contact area is minimized under the constraint of "xed volume fraction [47]. These conditions lead to interfacial surfaces of constant mean curvature (CMC) [47]. A minimal surface, examples of which are described in Section 6.2, is a special kind of a CMC surface. The mean curvature of a minimal surface is identically zero everywhere on the surface. Such a surface minimizes the area without any volume constraint [47]. For diblock copolymers the following phases can be identi"ed: A spherical micellar phase, a hexagonally packed cylinder phase, a lamellar phase, and a gyroidal phase [57]. Spheres and cylinders are surfaces with non-zero CMC, lamellae and gyroids are minimal surfaces. The "nal domain morphology determines the properties of the polymeric material and hence its end-use capabilities. Usually, the morphology of polymer systems is studied by analyzing the structure factor obtained from scattering experiments, such as small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS), and/or direct observation using transmission electron microscopy (TEM), scanning electron microscopy (SEM) or laser scanning confocal electron microscopy (LSCM). Structure factors, which describe the scattering length distribution of the phase-separated structure of polymer blends, give information about the average domain size and in the case of an ordered lattice about the lattice type and the space group. To get an idea of the real 3D morphology from these data the modi"ed Berk theory [98] has been used to analyze for example
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the scattering data in bicontinuous phase-separated polymer blends and to generate the 3D morphology corresponding to the scattering data [53]. TEM and SEM micrographs are two dimensional and give information about the size, shape and connectivity of the domains. A characterization of the morphology of the underlying 3D structure might be obtained by making assumptions about the structure and comparison of the TEM or SEM images with computer generated 2D crystallographic projections of these assumed structures. Experimentally, 3D images of the morphology of a real polymer blend can be obtained by LSCM. Similar data is also obtained from computer simulations of polymer systems. A method to estimate the mean and Gaussian curvature from 3D digital images by means of di!erential geometry is presented in [70]. Tools of integral and di!erential geometry have also been used to describe quantitatively the morphology of homopolymer blends at di!erent spinodal decomposition stages [99]. In the following sections, we apply the method based on integral geometry and described in Section 5, to analyze 3D morphologies observed in computer simulations of various block copolymer systems. 8.1. Micellar lattices [35] In `soft materialsa, which exhibit both temporal and spatial structural #uctuations over many length and time scales, the underlying lattice is formed under certain thermodynamic conditions and is far less rigid than the lattice of atomic crystals. The application of conventional crystallographic techniques to identify the mesostructures in these `softa materials may be rather di$cult. As an example we calculate the Minkowski functionals and the structure factor for computer-simulation data of a 50% aqueous solution of a triblock copolymer surfactant (ethylene oxide) (propylene oxide) (ethylene oxide) [or (EO) (PO) (EO) ]. The data are generated with a three-dimensional dynamic mean-"eld density functional method [100], a numerical method for the simulation of coarse-grained morphology dynamics in polymer liquids. The experimental [101] and simulated [102] phase diagram in the 50}70% surfactant concentration interval agree well and consist of four di!erent phases: a micellar, an hexagonal, a bicontinuous and a lamellar phase. The orientation of the copolymers is always such that the hydrophilic ethylene oxide shields the more hydrophobic propylene oxide from contact with water. Fig. 31 shows the morphology of propylene oxide in a 50% aqueous solution of (EO) (PO) (EO) . The simula tion box is a cube of edge length 16 (32) with periodic boundaries for the "gure on the left (right). The polymer solution is micellar. Visual inspection of the pattern suggests that for a small system the micelles are organized in a BCC-like lattice. For the larger systems it is more di$cult to identify the structure visually. In order to study the Minkowski functionals as a function of r for these polymer systems we "rst threshold the cubic image. Then we determine the centers of the micelles in the black-and-white picture and use the same graining procedure for these centers (germs) as the one described above (see Fig. 2, left). In Fig. 32 we depict the Minkowski functionals as a function of r for the propylene oxide in the cubic simulation box of edge length 16 (solid line). For comparison we also show the Minkowski functionals for the perfect BCC lattice with ¸ "8 and M"2 (dashed line), for the same BCC lattice but all lattice points displaced over a randomly chosen distance 0 or
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Fig. 31. Morphology of propylene oxide in an aqueous solution (50% polymer surfactant) of (EO) (PO) (EO) [(ethylene oxide) (propylene oxide) (ethylene oxide) ] Left: cubic simulation box of edge length 16; right: cubic simulation box of edge length 32. Periodic boundary conditions were used.
Fig. 32. Minkowski functionals as a function of r. Solid line: simulation data of a 50% aqueous solution of (ethylene oxide) (propylene oxide) (ethylene oxide) in a cubic box of edge length 16; dashed line: perfect BCC lattice with M"2 and ¸ "8; dotted line: BCC lattice with M"2, ¸ "8 and all lattice points displaced over a randomly chosen distance 0 or 1; dash}dotted line: random set of 16 points in a cubic box of edge length 16. Periodic boundary conditions were used.
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1 (dotted line) and for a random set of 16 points in a cubic box of edge length 16 (dash}dotted line). From Fig. 32 we may conclude that the micelles are organized in a BCC lattice structure with ¸ "8 and M"2 and of which the lattice points are somewhat displaced. The Euler characteristic per unit cell for the micellar phase equals two, which is also characteristic for a BCC lattice structure. From the Euler characteristic as a function of r we may see that the radius of the micelles has to be smaller than three lattice units. Otherwise the micelles glue together and di!ers from one. Fig. 33 demonstrates that it is much harder to draw a similar conclusion from the structure factor S(r) of the same system. The structure factor of the polymer solution exhibits additional, pronounced peaks, peaks that are absent in the case of a perfect BCC lattice, and also does not resemble the structure factor of a BCC lattice with random distortions. Fig. 34 shows the Minkowski functionals as a function of r for the propylene oxide in the cubic simulation box of edge length 32 (solid line). The number of micelles, as derived from equals 100. For comparison we also depict the Minkowski functionals for a random set of 100 points in a cubic box of edge length 32 (dashed line), for the perfect BCC lattice with ¸ "8 and M"4 (dash}dotted line) and for the same BCC lattice but all lattice points displaced over a randomly chosen distance 0 or 1 (dotted line). From this "gure we may conclude that in the bigger simulation box the micelles are no longer organized on a BCC lattice structure, neither that their distribution is random. Again, the latter conclusion is di$cult to draw from the structure factor, as can be seen in Fig. 35, where we give S(r) for the polymer solution (solid line) and for a random set of 100 points in a cubic box of edge length 32 (dotted line). The data of S(r) suggests that the micelles are randomly distributed in the cube. In summary, MIA of structures formed in soft materials such as polymer solutions provides information about the mesostructures that is hard to obtain by other methods.
Fig. 33. Structure factor. Solid line: simulation data of a 50% aqueous solution of (ethylene oxide) (propy lene oxide) (ethylene oxide) in a cubic box of edge length 16; dashed line: perfect BCC lattice with ¸ "8 and M"2; dotted line: BCC lattice with ¸ "8 and M"2 and all lattice points displaced over a randomly chosen distance 0 or 1. Periodic boundary conditions were used.
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Fig. 34. Minkowski functionals as a function of r. Solid line: simulation data of a 50% aqueous solution of (ethylene oxide) (propylene oxide) (ethylene oxide) in a cubic box of edge length 32; dashed line: random set of 100 points in a cubic box of edge length 32; dash}dotted line: perfect BCC lattice with ¸ "8 and M"4; dotted line: BCC lattice with ¸ "8 and M"4 and all lattice points displaced over a randomly chosen distance 0 or 1. Periodic boundary conditions were used.
8.2. Vesicles and worm-like micelles Block copolymers are materials that are capable of forming mesoscale structures whose morphology can be tailored by controlled synthesis. Identi"cation and quanti"cation of the morphology of these mesoscale structures may be rather di$cult. In this section we consider an example for which conventional crystallographic techniques do not work and for which MIA proves to be very valuable. We perform a MIA on computer-simulation data of the time evolution of a spherical droplet of the diblock copolymer AB in water (W). The data are generated with a three dimensional dynamic mean-"eld density functional method [100], a numerical method for the simulation of coarse-grained morphology dynamics in polymer liquids. The simulation box is a cube of edge length 32 with periodic boundaries. The initial spherical droplet has a radius of 10 lattice units. Fig. 36 shows the morphologies of the B-block after 2000 time steps of di!usion as a function of the interaction exchange parameters " and . Depending on the values of and 5 5 5 the spherical droplet remains a solid sphere, becomes a vesicle or a worm-like micelle. For 5 "1.0 and independent of the value of the polymer droplet dissolves in water what results 5 5 in a rather homogeneous polymer}water mixture.
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Fig. 35. Structure factor. Solid line: simulation data of a 50% aqueous solution of (ethylene oxide) (propy lene oxide) (ethylene oxide) in a cubic box of edge length 32; dotted line: random set of 100 points in a cubic box of edge length 32. Periodic boundary conditions were used.
Fig. 36. Morphologies of the B-block at t"2000 as a function of and . If the polymer forms a vesicle only half of 5 5 it is shown.
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Fig. 37. Morphologies of the B-block as a function of time for "1.5. Top: "!3.5; bottom: "!4.0. If the 5 5 5 polymer forms a vesicle only half of it is shown.
The formation of the worm-like micelles as the dynamic result of di!use phenomena is illustrated in Fig. 37 in terms of some snapshots of the B-block morphology. The time series at the top (bottom) of the "gure is for "1.5 and "!3.5 ( "!4.0). For "!3.5 5 5 5 5 ( "!4.0) and 0(t(600 (0(t(400) the polymer forms a vesicle. A schematic picture of 5 the polymer layers constituting the vesicle is shown in Fig. 38a. For "!3.5 and 5 600(t(2000 the polymer organizes in structures that look like hollow spheres with several holes and the number of holes seems to increase with time. In the end the polymer forms a hollow sphere-like structure with many holes. For "!4.0 and 400(t(2000 the polymer shows 5 a behavior that is similar to that described for "!3.5 but now the number of holes decreases 5 as a function of time. In the end the polymer forms a worm-like structure with one or more loops. In both cases the B-block is always shielded from contact with water by the A-block. A quantitative description of the various polymer morphologies may be obtained from MIA. We calculate the Minkowski functionals as a function of time for the B-block for " "1.5 and 5 several choices of . The results are shown in Fig. 39. For t"0 and all the B-block is 5 5 concentrated in a solid sphere-like structure leading to an Euler characteristic of one. For "!1.5 (dotted line) the B-block remains organized in a solid sphere-like structure for all 5 times. All Minkowski functionals are approximately constant. For "!3.0 (dash}dotted line) 5 and t'200 the polymer forms a vesicle, leading to a decrease of the covered volume and the integral mean curvature. The surface area increases and the Euler characteristic equals two. The same happens for "!3.5 (dashed line) [ "!4.0 (solid line)] and 0(t(600 5 5 [0(t(400]. For "!3.5 and intermediate t (600(t(1500) the polymer organizes in 5 structures that look like hollow spheres with several holes. The number of holes increases with
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Fig. 38. Schematic picture of the AB polymer vesicle (a) and double-layered micelle (b).
Fig. 39. Minkowski functionals as a function of t for "1.5. Solid line: "!4.0; dashed line: "!3.5; 5 5 5 dash}dotted line: "!3.0; dotted line: "!1.5. 5 5
time. The Euler characteristic is negative and decreases while the integral mean curvature increases. The volume further decreases and the surface area "rst decreases (for t(800) and remains approximately constant afterwards. For "!4.0 and 400(t(1500 the Minkowski func5 tionals show behavior very similar to that described for "!3.5 except for the fact that the 5
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number of holes (Euler characteristic) decreases (increases) with time. For "!3.5 and 5 "!4.0 and large t (t'1500) the polymer structures di!er from each other. For 5 "!3.5 the polymer forms a hollow sphere-like structure with many holes. The integral mean 5 curvature is large and positive and the Euler characteristic is strongly negative. For "!4.0 5 the B-block forms a worm-like structure with one or more loops. This gives rise to a large integral mean curvature and an Euler characteristic that is zero or slightly negative. Given the behavior of the Euler characteristic as a function of time (see Fig. 39) we expect that for t'2000 the polymer will organize in a worm-like structure without loops ("1). To gain more insight into the formation mechanism of the polymer vesicle, we study the polymer morphology for "!4 and "1.5 in more detail for 0(t(200. In Fig. 40 we depict the 5 5 Minkowski functionals as a function of t for the B-block. For 0(t(64 the B-block is concentrated in a solid sphere-like structure leading to an Euler characteristic of one. All other Minkowski functionals are approximately constant. For 64(t(100 cavities and tunnels are formed in the B-block morphology. The Euler characteristic #uctuates between positive and negative values. The surface area strongly increases. The integral mean curvature "rst decreases and becomes negative but increases and becomes positive again afterwards. Around t"100 the polymer system seems to stabilize and the Euler characteristic becomes equal to three. The polymer is now organized in a double-layered micelle. A schematic picture of the latter is given in Fig. 38b. When time evolves the excess of B-block material in the core moves to the outer layer of B-block material and "nally around t"180 a vesicle is formed (see Fig. 38a for a schematic picture).
Fig. 40. Minkowski functionals as a function of t (0(t(200) for "1.5, "!4. 5 5
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In summary, MIA gives the morphological properties of complex structures formed in polymer solutions and provides detailed information about the structure of the material. 8.3. Complex surfaces As already mentioned above, depending on the experimental conditions, complex domain structures may be formed in block copolymers. Since the morphology of the "nal domain structure determines the properties of the polymeric material, the study of the statics and dynamics of these complex structures is a key issue in polymer science. Several computer simulation techniques have been developed to simulate the formation of the domain structures. Once the domain structures are obtained the problem of characterization of the structures arises. Usually, the domain structures are characterized in terms of the average domain size which may be calculated from the "rst moment of the structure factor or the "rst zero in the radial distribution function [103,104]. As computer simulations are limited to rather small system sizes a structure factor analysis su!ers from severe "nite-size e!ects. Disregarding this problem, the average domain size alone cannot give a complete morphological characterization of the domain structure. For this purpose MIA is very useful [14,21,22,96,99,105]. In this section, we will use MIA to characterize the complex domain structures observed in computer simulation data of an AB binary polymer blend containing A}B-type block copolymer. Adding a small amount of block copolymer to a polymer blend leads to a variety of stable complex domain structures on the mesoscopic scale, such as irregular bicontinuous domains, hexagonally aligned cylinders, and cubic crystalline structures of spherical domains [106]. The domain structure formation in the polymer system is studied by the self-consistent "eld (SCF) dynamic density functional (DDF) method, a combination of the SCF path integral formalism of the polymer conformations and the mesoscopic transport equations [107}109], and the Ginzburg Landau (GL) method, a more phenomenological approach. For a quantitative comparison between the SCFDDF and the GL method see [106]. Fig. 41 shows the 2D domain structures, obtained by the GL and SCF-DDF method, of an A B polymer blend containing 20% volume fraction of A !B block copolymer [106] at L \L time t"100 and 3000. In Fig. 41 a block copolymer with block ratio f"0.2 (left) and 0.5 (right) is added, the block ratio being de"ned as the ratio of the length of block A to the total chain length of the block copolymer. The domain structures are obtained after a sudden temperature quench from a high-temperature equilibrium state [106]. Initially, (at t"0) the polymer systems were homogeneous mixtures. The A}B segment}segment interaction parameter (so-called Flory}Huggins interaction parameter ) is set to 0.5. The simulation box is a square of edge length 128 subject to periodic boundaries. The pictures in Fig. 41 suggest that for f"0.5 the resulting domain structure (at t"3000) is bicontinuous while for f"0.2 it is globular. In order to study the dynamics of the domain structure formation in more detail we compute the Minkowski functionals as a function of time for all cases. The results are shown in Fig. 42 ( f"0.2, GL method: solid line; f"0.2, SCF-DDF method: dashed line; f"0.5, GL method: dotted line and f"0.5, SCF-DDF method: dash}dotted line). Both methods give roughly the same results. The area A/¸ is more or less constant as a function of time. The perimeter ;/¸ decreases as a function of time. For f"0.2, the Euler characteristic is positive and slightly decreases as a function of time. This means that for f"0.2 the domain structure
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Fig. 41. Two-dimensional domain structures of an A B polymer blend containing 20% volume fraction of A !B block copolymer. Left: A !B , f"0.2; right: A !B , f"0.5. L \L
Fig. 42. Minkowski functionals as a function of t for simulation data of an A B polymer blend containing 20% volume fraction of A !B block copolymer in a square of edge length 128. Solid line: f"0.2, GL method; dashed L \L line: f"0.2, SCF-DFF method; dotted line: f"0.5, GL method; dash}dotted line: f"0.5, SCF-DFF method.
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Fig. 43. Three-dimensional domain structures of an A B polymer blend containing 20% volume fraction of A !B block copolymer. (a) A !B , f"0.2, SCF-DDF method; (b) A !B , f"0.2, GL method; L \L (c) A !B , f"0.5, SCF-DDF method; (d) A !B , f"0.5, GL method.
consists of separate objects and that the number of separate objects is slightly decreasing as a function of time. For f"0.5, the Euler characteristic is always negative pointing to a bicontinuous domain structure. The resulting domain structure depends on the architecture of the block copolymer. For comparison, the results for the average domain size as a function of time can be found in [106]. In Fig. 43 we show the 3D domain structures, obtained by the SCF-DDF method ((a) and (c)) and the GL method ((b) and (d)), of an A B polymer blend containing 20% volume fraction of A !B block copolymer [106] at time t"1000. Also for this example "0.5. The L \L simulation box is a cube of edge length 32 with periodic boundaries. In Figs. 43(a), (b) ((c), (d)) a block copolymer with block ratio f"0.2 ( f"0.5) is added. In both cases, interconnected bicontinuous structures are observed. Initially (at t"0) the polymer systems were homogeneous mixtures. In order to study the evolution of morphology of the domain structures in more detail we compute their Minkowski functionals as a function of time. The results are depicted in Fig. 44 ( f"0.2, GL method: solid line; f"0.2, SCF-DDF method: dashed line; f"0.5, GL method: dotted line and f"0.5, SCF-DDF method: dash}dotted line). As in the case for the 2D system also for the 3D system both methods give roughly the same results. As a function of time the volume
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Fig. 44. Minkowski functionals as a function of t for simulation data of an A B polymer blend containing 20% volume fraction of A !B block copolymer in a cubic box of edge length 32. Solid line: f"0.2, GL method; dashed L \L line: f"0.2, SCF-DFF method; dotted line: f"0.5, GL method; dash}dotted line: f"0.5, SCF-DFF method.
the 3D system the bicontinuous phase can be seen in a much larger region (0.16(p (0.84). Therefore, a slight asymmetry between the two phases is not enough to induce the morphological transition for a system with a nearly symmetric composition. The Minkowski functionals for cases (b) and (d) are [105] (b): N"8; <M "0.47; SM "3.65; BM "0.27; "!1.88 , (d): N"8; <M "0.50; SM "4.55; BM "0.023; "!3.75
(67)
and are close to the values of oriented P and G surfaces (at threshold q"0.5, one unit cell and ¸ "32) P: <M "0.50; SM "3.71; BM "0; "!2 , G: <M "0.50; SM "4.90; BM "0; "!4 .
(68)
For cases (a) and (c) we "nd (a): N"1; <M "0.47; SM "6.37; BM "1.00; "!10.00 , (c): N"8; <M "0.50; SM "3.50; BM "0.13; "!1.63
(69)
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which resemble the values of oriented D and P surfaces (at threshold q"0.5, one unit cell and ¸ "32) D: <M "0.50; SM "6.00; BM "0; "!8 , P: <M "0.50; SM "3.71; BM "0; "!2 .
(70)
These examples show that MIA can be used to characterize complex, computer generated surfaces with modest e!orts.
9. Summary Integral-geometry morphological image analysis characterizes patterns in terms of numerical quantities, called Minkowski functionals. These morphological descriptors have an intuitively clear geometrical and topological interpretation. Integral-geometry morphological image analysis yields information on structure in patterns. In most cases this information is complementary to the one obtained from two-point correlation functions. As the examples presented in this review and many other applications mentioned in the introduction show, MIA is a very versatile and powerful method to study structure in (complex) patterns. For some problems, e.g. the characterization of complex surfaces, the same morphological descriptors can also be computed within the framework of di!erential geometry. In contrast to the di!erential geometry approach, the application of integral geometry does not require the surface to be regular, nor is there any need to introduce labyrinth graphs or surface tilings to compute derivatives. A remarkable feature of MIA is the big contrast between the level of sophistication of the underlying mathematics and the ease with which MIA can be implemented and used. MIA is applied directly to the digitized representation of the patterns, it can be implemented with a few lines of computer code, is computationally inexpensive and is easy to use in practice. Therefore, we believe it should be part of one's toolbox for analyzing geometrical objects and patterns.
Acknowledgements We are grateful to Prof. H. Wagner for very helpful advice and to Prof. J.Th.M. De Hosson for providing us with the SEM images and for helpful discussions. We would like to thank Dr. G.J.A. Sevink, Prof. J.G.E.M. Fraaije and Dr. T. Kawakatsu (work partially supported by the national project, which has been entrusted to the Japan Chemical Innovation Institute by the New Energy and Industrial Technology Development Organization (NEDO) under MITI's Program for the Scienti"c Technology Development for Industries that Creates New Industries) for providing us the simulation data of the Pluronic-water mixtures, of vesicles and worm-like micelles, and of the complex surfaces in A !B block copolymers respectively. Part of this research has been L \L "nancially supported by the Council for Chemical Sciences of the Netherlands (CW/NWO) and Unilever Research Laboratories.
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Appendix A. Programming example ! ! ! ! ! ! !
Minkowski}functionals}2D computes the Minkowski functionals (area,perimeter,euler) for a 2D image, represented by the 1D array LATTICE(.). A pixel at (jx, jy) is active if LATTICE(jx;jy*Lx)"1, otherwise LATTICE(jx;jy*Lx)"0. Here 0(jx(Lx and 0(jy(Ly. The array TMP(.) is used as work space.
&
subroutine minkowski}functionals}2D(Lx,Ly,lattice,tmp, area,perimeter,euler) ! 2D implicit integer (a}z) integer lattice(0:Lx*Ly!1),tmp(0:(Lx;2)*(Ly;2)!1) sur"0 cur"0 eul"0 tmp(0:(Lx;2)*(Ly;2)!1)"0 ! work space do jy"0,Ly!1 do jx"0,Lx!1 i"jx;Lx*jy if( lattice(i) ' 0 ) then ! active pixel call minko}2D}free(Lx;2,Ly;2, jx;1, jy;1,tmp,s,c,e) tmp(jx;1;(Lx;2)*(jy;1))"1 ! add pixel to image sur"sur;s cur"cur;c eul"eul;e endif enddo enddo area"sur perimeter"cur euler"eul end
! ! ! ! !
Minkowski}functionals}3D computes the Minkowski functionals (volume,surface,integral mean curvature,euler) for a 3D image, represented by the 1D array LATTICE(.). A pixel at (jx, jy, jz) is active if LATTICE(jx;Lx*(jy;Ly*jz))"1, otherwise
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! LATTICE(jx;Lx*(jy;Ly*jz))"0. ! Here 0(jx(Lx, 0(jy(Ly, and 0(jz(Lz. The array TMP(.) ! is used as work space. ! subroutine minkowski}functionals}3D(Lx,Ly,Lz,tmp, & lattice,volume,surface,curvature,euler) implicit integer (a}z) integer lattice(0:Lx*Ly*Lz!1),tmp(0:(Lx;2)*(Ly;2)*(Lz;2)!1) vol"0 sur"0 cur"0 eul"0 tmp(0:(Lx;2)*(Ly;2)*(Lz;2)!1)"0 ! work space do jz"0,Lz!1 do jy"0,Ly!1 do jx"0,Lx!1 i"jx;Lx*(jy;Ly*jz) if( lattice(i)'0 ) then ! active pixel call minko}3D}free(Lx;2, Ly;2, Lz;2, jx;1, jy;1, jz;1,tmp,v,s,c,e) tmp(jx;1;(Lx;2)*(jy;1;(Ly;2)*(jz;1)))"1 ! add pixel to image vol"vol;v sur"sur;s cur"cur;c eul"eul;e endif enddo enddo enddo volume"vol surface"sur curvature"cur euler"eul end
&
subroutine minko}2D}free(Lx,Ly, jx, jy, lattice,surface,perimeter,euler2D) implicit integer (a}z) integer lattice(0:LX*LY!1) parameter(
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& & & & & &
surface}face"1 perimeter}face"!4 perimeter}edge"2 euler2D}face"1 euler2D}edge"!1 euler2D}vertex"1)
, , , , ,
!(a*a, open face) !(!4*a, open face) !(2*a, open line) !(open face) !(open line) !(vertices)
nedges"0 nvert"0 do i0"!1,1,2 jxi"jx;i0 jyi"jy;i0 kc1"1!lattice(jxi;Lx*jy) nedges"nedges;kc1;1!lattice(jx;Lx*jyi) do j0"!1,1,2 k4"Lx*(jy;j0) nvert"nvert;kc1*(1-lattice(jxi;k4))*(1-lattice(jx;k4)) enddo ! j0 enddo ! i0 surface"surface}face perimeter"perimeter}face;perimeter}edge*nedges euler2D"euler2D}face;euler2D}edge*nedges;euler2D}vertex*nvert end subroutine minko}3D}free(Lx,Ly,Lz, jx, jy, jz, lattice, & volume,surface,curvature,euler3D) implicit integer (a}z) integer lattice(0:LX*LY*Lz!1) parameter( &
volume}body"1 ,
!(a*a*a, where a is lattice displacement)
& &
surface}body"!6 , !(!6*a*a, open body) surface}face"2 , !(2*a*a, open face)
& & & & &
curv}body"3 , !(3*a, open body) curv}face"!2 , !(!2*a, open face) curv}edge"1 , !(a, open line) euler3D}body"!1 , !(open body) euler3D}face"1 , !(open face)
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& &
euler3D}edge"!1 , !(open line) euler3D}vertex"1) !(vertices) nfaces"0 nedges"0 nvert"0
do i0"!1,1,2 jxi"jx;i0 jyi"jy;i0 jzi"jz;i0 kc1"1!lattice(jxi;Lx*(jy;Ly*jz)) kc2"1!lattice(jx;Lx*(jyi;Ly*jz)) kc3"1!lattice(jx;Lx*(jy;Ly*jzi)) nfaces"nfaces;kc1;kc2;kc3 do j0"!1,1,2 jyj"jy;j0 jzj"jz;j0 k4"Lx*(jyj;Ly*jz) k7"Lx*(jy;Ly*jzj) kc7"1!lattice(jx;k7) kc1kc4kc5"kc1*(1!lattice(jxi;k4))*(1!lattice(jx;k4)) nedges"nedges;kc1kc4kc5 & ;kc2*(1!lattice(jx;Lx*(jyi;Ly*jzj)))*kc7 & ;kc1*(1!lattice(jxi;k7))*kc7 if(kc1kc4kc5.ne.0) then do k0"!1,1,2 jzk"jz;k0 k9"Lx*(jy;Ly*jzk) k10"Lx*(jyj;Ly*jzk) nvert"nvert;(1!lattice(jxi;k9))*(1!lattice(jxi;k10)) & *(1!lattice(jx;k10))*(1!lattice(jx;k9)) enddo ! k0 endif ! kc1kc4kc5 enddo ! j0 enddo ! i0 volume"volume}body surface"surface}body;surface}face*nfaces curvature"curv}body;curv}face*nfaces;curv}edge*nedges euler3D"euler3D}body;euler3D}face*nfaces & ;euler3D}edge*nedges;euler3D}vertex*nvert end
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Appendix B. Algorithm We describe a procedure to determine how the number of open bodies of each type changes when one adds (removes) one active pixel (see Section 2.2) to (from) a given pattern. Using this procedure it is easy to compute the Minkowski functionals for a given pattern, namely by adding the active pixels one by one to an initially empty picture (see Appendix A). In 2D, the number n (P) of open squares building up the objects in the ¸ ;¸ picture V W P(x, q)"P(i, j, q); (i"1,2, ¸ , j"1,2, ¸ ) increases (decreases) with one if one adds (removes) V W one active pixel at the position x"(i, j) to (from) the image. Therefore, if we add an active pixel, n (P)"1 ,
(B.1)
where we introduce the symbol to indicate that we compute the di!erence. Similarly, the change in the number of open edges, n (P) is given by n (P)" [Q(i#, j, q)#Q(i, j#, q)] , ?!
(B.2)
where Q(x, q)"1!P(x, q). For the change in the number of vertices, n (P), we "nd n (P)" Q(i#, j, q)Q(i#, j#, q)Q(i, j#, q) . ? @!
(B.3)
The change in the number of steps, n (P), may be computed from 1 n (P)" P(i#, j#, q)(Q(i#, j, q)#Q(i, j#, q)) . 1 ? @!
(B.4)
In 3D, the number n (P) of open cubes building up the objects in the ¸ ;¸ ;¸ image V W X P(x, q)"P(i, j, k, q); (i"1,2, ¸ , j"1,2, ¸ , k"1,2, ¸ ) increases (decreases) with one if one V W X adds (removes) one active pixel to (from) the image at the position x"(i, j, k), i.e. n (P)"1. The change in n (P), the number of open faces, may be computed from n (P)" [Q(i#, j, k, q)#Q(i, j#, k, q)#Q(i, j, k#, q)] . ?!
(B.5)
The change in n (P), the number of open edges, reads n (P)" [Q(i#, j, k, q)Q(i#, j#, k, q)Q(i, j#, k, q) ? @! # Q(i, j#, k, q)Q(i, j#, k#, q)Q(i, j, k#, q) # Q(i#, j, k, q)Q(i#, j, k#, q)Q(i, j, k#, q)] .
(B.6)
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For the change in n (P), the number of vertices, we "nd n (P)" Q(i#, j, k, q)Q(i#, j#, k, q)Q(i, j#, k, q) ? @ A! ;Q(i#, j, k#, q)Q(i#, j#, k#, q) ;Q(i, j#, k#, q)Q(i, j, k#, q) .
(B.7)
The change in n (P), the number of steps, is given by 1 n (P)" [P(i#, j#, k, q)(Q(i#, j, k, q)#Q(i, j#, k, q)) 1 ? @! # P(i#, j, k#, q)(Q(i#, j, k, q)#Q(i, j, k#, q)) # P(i, j#, k#, q)(Q(i, j#, k, q)#Q(i, j, k#, q))] .
(B.8)
Appendix C. Minkowski functionals for elementary bodies We derive the values of the Minkowski functionals for the open bodies building up a blackand-white image. A 2D image consists of square pixels. Each pixel is a closed square Q31 of edge length a and is the union of its interior Qx , its four open edges ¸x , and its four vertices P (points are open as well as closed convex bodies). The parallel volume v(Q ) is given by P v(Q )"a#4ar#r , (C.1) P which by making use of (21b) and (34) implies =(Q)"=(Qx )"a, =(Q)"!=(Qx )"2a , =(Q)"=(Qx )" ,
(C.2)
and A(Q)"A(Qx )"a, ;(Q)"!;(Qx )"4a, (Q)"(Qx )"1 .
(C.3)
Next, we consider the closed edge ¸31 of length a. The parallel volume v(¸ ) is obtained from P v(¸ )"2ar#r , (C.4) P which leads to =(¸)"!=(¸x )"0, =(¸)"=(¸x )"a , =(¸)"!=(¸x )"
(C.5)
and A(¸)"!A(¸x )"0, ;(¸)";(¸x )"2a,
(¸)"!(¸x )"1 .
(C.6)
Finally, we consider the vertices P31. Their parallel volume reads v(P )"r . P
(C.7)
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Hence, =(P)"=(Px )"0, =(P)"!=(Px )"0 , =(P)"=(Px )"
(C.8)
and A(P)"A(Px )"0, ;(P)"!;(Px )"0,
(P)"(Px )"1 .
(C.9)
A 3D image contains closed cubes C, each being the union of its interior Cx , its six open faces Qx , its twelve open edges ¸x and its eight vertices P. For the closed cube C31 of edge length a the parallel volume reads 4 v(C )"a#6ar#3ar# r , P 3
(C.10)
from which it follows that =(C)"=(Cx )"a, =(C)"!=(Cx )"2a , 4 =(C)"=(Cx )"a, =(C)"!=(Cx )" , 3
(C.11)
and <(C)"<(Cx )"a, S(C)"!S(Cx )"6a , B(C)"B(Cx )"a, (C)"!(Cx )"1 .
(C.12)
The parallel volume for the closed square Q31 of edge length a is given by 4 v(Q )"2ar#2ar# r , P 3
(C.13)
which implies =(Q)"!=(Qx )"0, =(Q)"=(Qx )"a , 4 2a , =(Q)"=(Qx )" =(Q)"!=(Qx )" 3 3
(C.14)
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and <(Q)"!<(Qx )"0,
S(Q)"S(Qx )"2a ,
B(Q)"!B(Qx )"a, (Q)"(Qx )"1 .
(C.15)
For the closed line ¸31 of length a the parallel volume can be obtained from 4 v(¸ )"ar# r . P 3
(C.16)
This yields for the Minkowski functionals =(¸)"=(¸x )"0, =(¸)"!=(¸x )"0 , 4 a =(¸)"=(¸x )" , =(¸)"!=(¸x )" 3 3
(C.17)
and hence <(¸)"<(¸x )"0, S(¸)"!S(¸x )"0 , a B(¸)"B(¸x )" , (¸)"!(¸x )"1 . 2
(C.18)
Finally, for the vertices P31 we have for the parallel volume 4 v(P )" r , P 3
(C.19)
which leads to =(P)"!=(Px )"0, =(P)"=(Px )"0 , 4 =(P)"!=(Px )"0, =(P)"=(Px )" , 3
(C.20)
and <(P)"!<(Px )"0, S(P)"S(Px )"0 , B(P)"!B(Px )"0, (P)"(Px )"1 .
(C.21)
Appendix D. Proof of (47) The aim is to compute the mean value of MB(A ), i.e. the con"gurational average with respect to J , the product density element [11] 1 , dg , d(g ,2, g )" G , , G where dg " denotes the volume of . G
(D.1)
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We "rst consider the con"gurational average of a single grain. We rewrite (46) as A "A g A (D.2) , ,\ , , and make use of the properties of additivity and motion invariance of the Minkowski functionals, and of the kinematic formula (28). The con"gurational average for the single grain A reads , dg dg MB(A ) , "MB(A )#MB(A )! MB(A g A ) , J , J ,\ J , J ,\ , ,
1 I "MB(A )#MB(A )! MB (A )MB(A ). (D.3) J ,\ J , I\J , I ,\
J At this point it is expedient to switch to a matrix notation. For d"3 and with the notation M ,MB, (D.3) reads J J dg , "Q M M #R , (D.4) , , ,\ ,
where the matrices M , Q and R are given by , , , M (A ) M (A ) , , M (A ) M (A ) M " , , R " , , , , M (A ) M (A ) , , M (A ) M (A ) , , 0 0 0 , 0 0 , Q " , , 2 0 , , , 3 3 , , , , with
(D.5a)
(D.5b)
M (A ) M (A ) "1! , , "! , , , , M (A ) M (A ) "! , , "! , . , ,
(D.6)
Repeating the steps that lead to (D.4) the con"gurational average over two grains A and , A reads ,\ dg dg dg , ,\ " Q M ,\ #R M , , ,\ ,
"Q Q M #M R #R , ,\ ,\ , ,\ ,
(D.7)
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and the average over all possible con"gurations can be written as
2 M,
dg 2dg , "Q 2Q R #Q 2Q R #2#Q R #R . , , , ,\ , ,
(D.8)
Exact result (D.8) is of little practical value unless we make additional assumptions about the properties of the individual grains. If all grains are identical we have Q"Q and R"R for G G i"1,2, N. Likewise if the distribution of size and shape of the grains is the same for all grains, averaging (D.8) over this distribution also yields Q"Q and R"R for all i. Evidently, the latter G G case contains the former. Hence averaging (D.8) over the size and shape of the grains yields
M , 2 M , ,
dg 2dg , "(1#Q#2#Q,\)R . ,
(D.9)
By mathematical induction it can be shown that
a ,\ b 1#Q#2#Q,\" ,\ c ,\ d ,\ where a
0
0
0
a ,\ 2b ,\ 3c ,\
0
0
,\ 3b ,\
0
a
a ,\
,
(D.10)
Ra 1!, , b " ,\ , " ,\ ,\ 1!, R
c " ,\
Ra Ra ,\ # ,\ , R R
d
Ra Ra Ra ,\ #3 ,\ # ,\ R R R
,\
"
(D.11)
and m m "1! , "! , m m "! , "!
(D.12)
and where m ,M (A ) denote the averages over size and shape of the Minkowski functional for J J G a single grain A . G With expressions (D.10)}(D.12) we obtain from (D.9) M "m a , , ,\ m m Ra , M "! ,\ #m a ,\ , R
(D.13a) (D.13b)
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m m Ra 1 Ra ,\ ! (2m #m m ) ,\ #m a M " , , ,\ R R
(D.13c)
m Ra m m Ra ,\ # (m #3m m ) ,\ M "! R , R 1 Ra ! (6m m #m m ) ,\ #m a , R ,\
(D.13d)
where M denote the mean values of the Minkowski functionals of the ensemble with density . J , The expressions in (D.13) may be further simpli"ed by putting z,(1!)a "1!, . ,\ We "nd
(D.14)
Rz Ra "(1!) ,\ !a "!N,\ , ,\ R
(D.15a)
Rz Ra Ra ,\ !2 ,\ "!N(N!1),\ , "(1!) R R
(D.15b)
Rz Ra Ra ,\ !3 ,\ "!N(N!1)(N!2),\ . "(1!) R R
(D.15c)
Substituting (D.14), (D.15) and m /"1!, in (D.13) "nally gives M "(1!,) , , M "m N,\ , , m M "! N(N!1),\#m N,\ , , m m m M " N(N!1)(N!2),\!3 N(N!1),\#m N,\ . ,
(D.16a) (D.16b) (D.16c) (D.16d)
In the bulk limit N, PR with "N/ "xed, we have
m , lim ," lim 1! "e\MK , ,
(D.17)
and M /N "(1!e\MK )/ , , M /N "m e\MK , , M /N "(m !m )e\MK , , M /N "(m !3m m #m )e\MK . , Note that the expressions in (D.18) are valid for any dimension d.
(D.18a) (D.18b) (D.18c) (D.18d)
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References [1] P. Ball, The Self-Made Tapestry: Pattern Formation in Nature, Oxford University Press, Oxford, 1998. * [2] S. Hyde, S. Andersson, K. Larsson, Z. Blum, T. Landh, S. Lidin, B.W. Ninham, The Language of Shape; The Role of Curvature in Condensed Matter: Physics, Chemistry and Biology, Elsevier, Amsterdam, 1997. * [3] J.C. Russ, The Image Processing Handbook, CRC Press, Florida, 1995. * [4] K.R. Castleman, Digital Image Processing, Prentice-Hall, Englewood Cli!s, 1996. * [5] R.C. Gonzalez, R.E. Woods, Digital Image Processing, Addison-Wesley, Reading, MA, 1993. * [6] A. Rosenfeld, A.C. Kak, Digital Picture Processing, Academic Press, New York, 1982. * [7] C.R. Giardina, E.R. Dougherty, Morphological Methods in Image and Signal Processing, Prentice-Hall, Englewood Cli!s, 1988. *** [8] H. Hadwiger, Vorlesungen uK ber Inhalt, Ober#aK che und Isoperimetrie, Springer, Berlin, 1957. *** [9] L.A. SantaloH , Integral Geometry and Geometric Probability, Addison-Wesley, Reading, MA, 1976. ** [10] D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and its Applications, Akademie Verlag, Berlin, 1989. ** [11] K.R. Mecke, H. Wagner, Euler characteristic and related measures for random geometric sets, J. Stat. Phys. 64 (1991) 843}850. * [12] K.R. Mecke, A morphological model for complex #uids, J. Phys.: Condens. Matter 8 (1996) 9663}9667. [13] K.R. Mecke, Morphological thermodynamics of composite media, Fluid Phase Equilibria 150}151 (1998) 591}598. [14] K.R. Mecke, Integral geometry in statistical physics, Int. J. Mod. Phys. B 12 (1998) 861}899. *** [15] A.L. Mellot, The topology of large-scale structure in the universe, Phys. Rep. 193 (1990) 1}39. [16] K.R. Mecke, T. Buchert, H. Wagner, Robust morphological measures for large-scale structure in the Universe, Astron. Astrophys. 288 (1994) 697}704. [17] N.G. Makarenko, L.M. Karimova, A.G. Terekhov, A.V. Kardashev, Minkowksi functionals and comparison of discrete samples in seisomology, Phys. Solid Earth 36 (2000) 305}309. [18] J.S. Kole, K. Michielsen, H. De Raedt, Morphological image analysis of quantum motion in billiards, Phys. Rev. E 63 (2001) 016201. [19] C.N. Likos, K.R. Mecke, H. Wagner, Statistical morphology of random interfaces in microemulsions, J. Chem. Phys. 102 (1995) 9350}9361. [20] K.R. Mecke, Morphological characterization of patterns in reaction}di!usion systems, Phys. Rev. E 53 (1996) 4794}4800. [21] K.R. Mecke, V. Sofonea, Morphology of spinodal decomposition, Phys. Rev. E 56 (1997) R3761}R3764. [22] V. Sofonea, K.R. Mecke, Morphological characterization of spinodal decomposition kinetics, Eur. Phys. J. B 8 (1999) 99}112. [23] S. Herminghaus, K. Jacobs, K. Mecke, J. Bischof, A. Fery, M. Ibn-Elhaj, S. Schlagowski, Spinodal dewetting in liquid crystal and liquid metal "lms, Science 282 (1998) 916}919. [24] K. Jacobs, S. Herminghaus, K.R. Mecke, Thin liquid polymer "lms rupture via defects, Langmuir 14 (1998) 965}969. [25] G. Matheron, Random Sets and Integral Geometry, Wiley, New York, 1975. * [26] J. Serra, Image Analysis and Mathemathical Morphology, Academic Press, London, 1982. * [27] M. Gervautz, W. Purgathofer, A simple method for color quantization: Octree quantization, in: A.S. Glassner (Ed.), Graphics Gems, Academic Press, Boston, 1990. [28] D.J. Taylor, B.D. Fabes, Laser processing of sol}gel coatings, J. Non-Cryst. Solids 147}148 (1992) 457}462. [29] D.J. Shaw, T.A. King, Densi"cation of sol}gel silica glass by laser irradiation, in: J.D. Mackenzie, D.R. Ulrich (Eds.), Sol}Gel Optics, Proceedings of SPIE, Vol. 1328, 1990, pp. 474}481. [30] J.Th.M. De Hosson, D.H.J. Teeuw, Nano-ceramic coatings produced by laser treatment, Surf. Eng. 15 (1999) 235}241. [31] J.Th.M. de Hosson, M. de Haas, D.H.J. Teeuw, High resolution scanning electron microscopy observations of nano-ceramics, in: D.G. Rickerby et al. (Eds.), Impact of Electron Scanning Probe Microscopy on Materials Research, Kluwer Academic Publishers, 1999, pp. 109}134.
536
K. Michielsen, H. De Raedt / Physics Reports 347 (2001) 461}538
[32] J.Th.M. De Hosson, Laser synthesis and properties of ceramic coatings, in: N.B. Dahotre, T.S. Sudarshan (Eds.), Intermetallic and Ceramic Coatings, Marcel Dekker, New York, 1999, pp. 307}441. [33] D.H.J. Teeuw, M. de Haas, J.Th.M. De Hosson, Residual stress "elds in sol}gel derived thin TiO layers, J. Mater. Res. 14 (1999) 1896}1904. [34] A.P. Gast, W.B. Russel, Simple ordering in complex #uids, Colloidal particles suspended in solution provide intriguing models for studying phase transitions, Phys. Today 51 (1998) 24}30. [35] K. Michielsen, H. De Raedt, J.G.E.M. Fraaije, Morphological characterization of spatial patterns, Prog. Theor. Phys. 138 (Suppl.) (2000) 543}548. [36] J. Klinowski, A.L. Mackay, H. Terrones, Curved surfaces in chemical structure, Philos. Trans. Roy. Soc. Lond. A 354 (1996) 1975}1987. [37] H.G. von Schnering, R. Nesper, How nature adapts chemical structures to curved surfaces, Angew. Chem. 26 (1987) 1059}1080. [38] A.L. Mackay, Equipotential surfaces in periodic charge distributions, Angew. Chem. Int. Edn. Engl. 27 (1988) 849}850. [39] V. Luzzati, A. Tardieu, T. Gulik-Krzywicki, E. Rivas, F. Reiss-Husson, Structure of the cubic phases of lipid}water systems, Nature 220 (1968) 485}488. [40] W. Longley, T.J. McIntosh, A bicontinuous tetrahedral structure in a liquid-crystalline lipid, Nature 304 (1983) 612}614. [41] K. Larsson, Two cubic phases in monoolein-water system, Nature 304 (1983) 664}664. [42] P. Mariani, V. Luzzati, H. Delacroix, Cubic phases of lipid-containing systems, J. Mol. Biol. 204 (1999) 165}189. [43] A. Ciach, Phase diagram and structure of the bicontinuous phase in a three-dimensional lattice model for oil}water}surfactant models, J. Chem. Phys. 96 (1992) 1399}1408. [44] E.L. Thomas, D.B. Alward, D.J. Kinning, D.C. Martin, D.L. Handlin Jr, L.J. Fetters, Ordered bicontinuous double-diamond structure of star block copolymers: a new equilibrium microdomain morphology, Macromolecules 19 (1986) 2197}2202. [45] H. Hasegawa, H. Tanaka, K. Yamasaki, T. Hashimoto, Bicontinuous microdomain morphology of block copolymers. 1. Tetrapod-network structure of polystyrene-polyisopropene diblock polymers, Macromolecules 20 (1987) 1651}1662. [46] D.M. Anderson, E.L. Thomas, Microdomain morphology of star copolymers in the strong-segregation limit, Macromolecules 21 (1988) 3221}3230. [47] E.L. Thomas, D.M. Anderson, C.S. Henkee, D. Ho!man, Periodic area-minimizing surfaces in block copolymers, Nature 334 (1988) 598}601. [48] Y. Mogi, K. Mori, Y. Matsushita, I. Noda, Tricontinuous morphology of triblock copolymers of the ABC type, Macromolecules 25 (1992) 5412}5415. [49] Y. Matsushita, M. Tamura, I. Noda, Tricontinuous double-diamond structure formed by a styrene-isopropene-2vinylpyridine triblock copolymer, Macromolecules 27 (1994) 3680}3682. [50] M.F. Schulz, F.S. Bates, K. Almdal, K. Mortensen, Epitaxial relationship for hexagonal-to-cubic phase transition in a block copolymer mixture, Phys. Rev. Lett. 73 (1994) 86}89. [51] D.A. Hajduk, P.E. Harper, S.M. Gruner, C.C. Honeker, G. Kim, E.L. Thomas, L.J. Fetters, The gyroid: a new equilibrium morphology in weakly segregated diblock copolymers, Macromolecules 27 (1994) 4063}4075. [52] D.A. Hajduk, P.E. Harper, S.M. Gruner, C.C. Honeker, E.L. Thomas, L.J. Fetters, A reevaluation of bicontinuous cubic phases in starblock copolymers, Macromolecules 28 (1995) 2570}2573. [53] H. Jinnai, T. Hashimoto, D. Lee, S.-H. Chen, Morphological characterization of bicontinuous phase-separated polymer blends and one-phase microemulsions, Macromolecules 30 (1997) 130}136. [54] S. Sakurai, H. Irie, H. Umeda, S. Nomura, H.H. Lee, J.K. Kim, Gyroid structures and morphological control in binary blends on polystyrene-block-polyisopropene diblock copolymers, Macromolecules 31 (1998) 336}343. [55] M.W. Hamersky, M.A. Hillmeyer, M. Tirrell, F.S. Bates, T.P. Lodge, E.D. von Meerwall, Block copolymer self-di!usion in the gyroid and cylinder morphologies, Macromolecules 31 (1998) 5363}5370. [56] M.E. Vigild, K. Almdal, K. Mortensen, I.W. Hamley, J.P.A. Fairclough, A.J. Ryan, Transformations to and from the gyroid phase in a diblock copolymer, Macromolecules 31 (1998) 5702}5716.
K. Michielsen, H. De Raedt / Physics Reports 347 (2001) 461}538
537
[57] F.S. Bates, G.H. Fredrickson, Block copolymers-designer soft materials. Advances in synthetic chemistry and statistical theory provide unparalleled control over molecular scale morphology in this class of macromolecules, Phys. Today 52 (1999) 32}38. [58] G. Donnay, D.L. Pawson, X-ray di!raction studies of echinoderm plates, Science 166 (1969) 1147}1150. [59] H.-U. Nissen, Crystal orientation and plate structure in echinoid skeletal units, Science 166 (1969) 1150}1152. [60] K. Larsson, Cubic lipid}water phases: structures and biomembrane aspects, J. Phys. Chem. 93 (1989) 7304}7314. [61] W. Fischer, E. Koch, On 3-periodic minimal surfaces, Z. Kristallogr. 179 (1987) 31}52. [62] K. Grosse-Brauckmann, On gyroid interfaces, J. Colloid Interface Sci. 187 (1997) 418}428. * [63] H.A. Schwarz, Gesammelte Mathematische Abhandlungen, Springer, Berlin, 1890. [64] A.H. Schoen, In"nite periodic minimal surfaces without self-intersections, NASA Technical NOTE No. D-5541, 1970. [65] A.L. Mackay, Crystallographic surfaces, Proc. Roy. Soc. Lond. A 442 (1993) 47}59. [66] C.A. Lambert, L.H. Radzilowski, E.L. Thomas, Triply periodic level surfaces as models for cubic tricontinuous block copolymer morphologies, Philos. Trans. Roy. Soc. Lond. A 354 (1996) 2009}2023. [67] A. Ciach, R. Ho"yst, Periodic surfaces and cubic phases in mixtures of oil, water, and surfactant, J. Chem. Phys. 110 (1999) 3207}3214. [68] W.T. GoH zH dzH , R. Ho"yst, Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions, Phys. Rev. E 54 (1996) 5012}5027. [69] S.T. Hyde, Swelling and structure. Analysis of the topology and geometry of lamellar and sponge lyotropic mesophases, Langmuir 13 (1997) 842}851. [70] Y. Nishikawa, H. Jinnai, T. Koga, T. Hashimoto, S.T. Hyde, Measurements of interfacial curvatures of bicontinuous structure from three-dimensional digital images. 1. A parallel surface method, Langmuir 14 (1998) 1242}1249. [71] W. Fischer, E. Koch, Genera of minimal balance surfaces, Acta Crystallogr. A 45 (1989) 726}732. * [72] P.S. Aleksandrov, Combinatorial topology, Vol. 1, Graylock, Rochester NY, 1956. [73] P. Davy, Projected thick sections through multi-dimensional particle aggregates, J. Appl. Probab. 13 (1976) 714}722. [74] H.G. Kellerer, Minkowski functionals of Poisson processes, Z. Wahr. Verw. Gebiete 67 (1984) 63}84. [75] K. Michielsen, H. De Raedt, Morphological Image Analysis, Comput. Phys. Commun. 132 (2000) 94}103. [76] S.R. Broadbent, J.M. Hammersley, Percolation processes, Proc. Cambridge Phil. Soc. 53 (1957) 629}641. [77] D. Stau!er, A. Aharony, Introduction to Percolation Theory, Taylor&Francis, London, 1991. * [78] H. Kesten, Percolation Theory for Mathematicians, BirkhaK user, Boston, MA, 1982. [79] G. Deutscher, R. Zallen, J. Adler, Percolation Structures and Processes, A. Hilger, Bristol, 1983. [80] M. Sahimi, Applications of Percolation Theory, Taylor&Francis, London, 1994. [81] G. Grimmett, Percolation, Springer, Berlin, 1999. [82] J.M. Hammersley, Percolation processes. Lower bounds for the critical probability, Ann. Math. Stat. 28 (1957) 790}795. [83] T.E. Harris, A lower bound for the critical probability in a certain percolation process, Proc. Cambridge Philos. Soc. 56 (1960) 13}20. [84] B. ToH th, A lower bound for the critical probability of the square lattice site percolation, Z. Wahr. Verw. Gebiete 69 (1985) 19}22. [85] S.A. Zuev, Bounds for the percolation threshold for a square lattice, Theor. Probab. Appl. 32 (1987) 551}553. [86] M.V. Menshikov, K.D. Pelikh, Percolation with several defect types. An estimate of critical probability for a square lattice, Math Notes Acad. Sci. USSR 46 (1989) 778}785. [87] J.C. Wierman, Substitution method critical probability bounds for the square lattice site percolation model, Combin. Probab. Comput. 4 (1995) 181}188. [88] J. van den Berg, A. Ermakov, A new lower bound for the critical probability of site percolation on the square lattice, Random Structures and Algorithms 8 (1996) 199}212. [89] R.M. Zi!, B. Sapoval, The e$cient determination of the percolation threshold by a frontier-generating walk in a gradient, J. Phys. A 19 (1986) L1169}L1172. [90] R.M. Zi!, Spanning probability in 2D percolation, Phys. Rev. Lett. 69 (1992) 2670}2673.
538
K. Michielsen, H. De Raedt / Physics Reports 347 (2001) 461}538
[91] M.F. Sykes, J.M. Essam, Exact critical percolation probabilities for site and bond percolation problems in two dimensions, J. Math. Phys. 5 (1964) 1117}1127. [92] H. Kesten, The critical probability of bond percolation on the square lattice equals 1/2, Commun. Math. Phys. 74 (1980) 41}59. [93] J.C. Wierman, Bond percolation on honeycomb and triangular lattices, Adv. Appl. Probab. 13 (1981) 293}313. [94] B.L. Okun, Euler characteristic in percolation theory, J. Stat. Phys. 59 (1990) 523}527. [95] H. Tomita, C. Murakami, Percolation pattern in continuous media and its topology, in: R. Takaki (Ed.), Research of Pattern Formation, KTK Scienti"c Publishers, Tokyo, 1994, pp. 197}203. [96] A. Aksimentiev, R. Ho"yst, K. Moorthi, Kinetics of the droplet formation at the early and intermediate stages of the spinodal decomposition in homopolymer blends, Macromol. Theory Simul. 9 (2000) 661}674. [97] F.S. Bates, G.H. Fredrickson, Block copolymer thermodynamics: theory and experiment, Annu. Rev. Phys. Chem. 41 (1990) 525}557. [98] S.-H. Shen, D. Lee, S.-L. Chang, Visualization of 3D microstructure of bicontinuous microemulsions by combined SANS experiments and simulations, J. Mol. Struct. 296 (1993) 259}264. [99] A. Aksimentiev, K. Moorthi, R. Ho"yst, Scaling properties of the morphological measures at the early and intermediate stages of the spinodal decomposition in homopolymer blends, J. Chem. Phys. 112 (2000) 6049}6062. [100] J.G.E.M. Fraaije, B.A.C. van Vlimmeren, N.M. Maurits, M. Postma, O.A. Evers, C. Ho!mann, P. Altevogt, G. Goldbeck-Wood, The dynamic mean-"eld density functional method and its application to the mesoscopic dynamics of quenched block copolymer melts, J. Chem. Phys. 106 (1997) 4260}4269. [101] P. Alexandridis, U. Olsson, B. Lindmann, Self-assembly of amphiphilic block copolymers: the (EO) (PO) (EO) -water-p-xylene system, Macromolecules 28 (1995) 7700}7710. [102] B.A.C. van Vlimmeren, N.M. Maurits, A.V. Zvelindovsky, G.J.A. Sevink, J.G.E.M. Fraaije, Simulation of 3D mesoscale structure formation in concentrated aqueous solution of the triblock polymer surfactants (ethylene oxide) (propylene cxide) (ethylene oxide) and (ethylene oxide) (propylene oxide) (ethylene oxide) , Application of dynamic mean-"eld density functional theory, Macromolecules 32 (1999) 646}656. [103] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Clarendon Press, Oxford, 1971. [104] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford University Press, Oxford, 1993. [105] K. Michielsen, H. De Raedt, T. Kawakatsu, Morphological image analysis, in: D.P. Landau et al. (Eds.), Computer Simulation Studies in Condensed-Matter Physics XIII, Springer Proceedings in Physics, Springer, Berlin, in press. [106] T. Kawakatsu, R. Hasegawa, M. Doi, Dynamic density functional approach to phase separation dynamics of polymer systems, Int. J. Mod. Phys. C 10 (2000) 1531}1540. [107] J.G.E.M. Fraaije, Dynamic density functional theory for microphase separation kinetics of block copolymer melts, J. Chem. Phys. 99 (1993) 9202}9212. [108] R. Hasegawa, M. Doi, Adsorption dynamics. Extension of self-consistent "eld theory to dynamical problems, Macromolecules 30 (1997) 3086}3089. [109] T. Kawakatsu, E!ects of changes in the chain conformation on the kinetics of order-disorder transitions in block-copolymer melts, Phys. Rev. E 56 (1997) 3240}3250.
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CONTENTS VOLUME 347 D. Atwood, S. Bar-Shalom, G. Eilam, A. Soni. CP violation in top physics
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V.K.B. Kota. Embedded random matrix ensembles for complexity and chaos in "nite interacting particle systems
223
C. Song. Dense nuclear matter: Landau Fermi-liquid theory and chiral Lagrangian with scaling
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E. Nielsen, D.V. Fedorov, A.S. Jensen, E. Garrido. The three-body problem with short-range interactions
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FORTHCOMING ISSUES D.R. Grasso, H.R. Rubinstein. Magnetic "elds in the early Universe V.M. Loktev, R.M. Quick, S. Sharapov. Phase #uctuations and pseudogap phenomena T. Nakayama, K. Yakubo. The forced oscillator method: eigenvalue analysis and computing linear response functions S.O. Demokritov, B. Hillebrands, A.N. Slavin. Brillouin light scattering studies of con"ned spin waves: linear and nonlinear con"nement I.M. Dremin, J.W. Gary. Hadron multiplicities C.O. Wilke, C. Ronnewinkel, T. Martinetz. Dynamic "tness landscapes in molecular evolution V. Narayanamurti, M. Kozhevnikov. BEEM imaging and spectroscopy of buried structures in semiconductors J. Richert, P. Wagner. Microscopic model approaches to fragmentation of nuclei and phase transitions in nuclear matter H.J. Drescher, M. Hladik, S. Ostapchenko, T. Pierog, K. Werner. Parton-based Gribov}Regge theory D.G. Yakovlev, A.D. Kaminker, O.Y. Gnedin, P. Haensel. Neutrino emission from neutron stars V.A. Zagrebnov, J.-B. Bru. The Bogoliubov model of weakly imperfect Bose-gas G.E. Volovik. Super#uid analogies of cosmological phenomena H. Heiselberg. Event-by-event physics in relativistic heavy-ion collisions C.N. Likos. E!ective interactions in soft condensed matter physics C. Ronning, E.P. Carlson, R.F. Davis. Ion implantation into gallium nitride B. Gumhalter. Single and multiphonon atom}surface scattering in the quantum regime H. Stark. Physics of colloidal dispersions in nematic liquid crystals B. Ananthanarayan, G. Colangelo, J. Gasse, H. Leutwyler. Roy analysis of pi-pi scattering R. Alkofer, L. von Smekal. The infrared behaviour of QCD Green's functions J.-P. Minier, E. Peirano. The PDF approach to turbulent polydispersed two-phase #ows
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