Physics Reports vol.319

M.S. Volkov, D.V. Gal+tsov / Physics Reports 319 (1999) 1}83

GRAVITATING NON-ABELIAN SOLITONS AND BLACK HOLES WITH YANG}MILLS FIELDS

Mikhail S. VOLKOV , Dmitri V. GAL:TSOV
 Institute for Theoretical Physics, University of Zu( rich-Irchel, Winterthurerstrasse 190, CH-8057 Zu( rich, Switzerland
Department of Theoretical Physics, Moscow State University, 119899 Moscow, Russia Yukawa Institute for Theoretical Physics, University of Kyoto, Kyoto 606, Japan

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

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Physics Reports 319 (1999) 1}83

Gravitating non-Abelian solitons and black holes with Yang}Mills "elds Mikhail S. Volkov *, Dmitri V. Galt'sov
 Institute for Theoretical Physics, University of Zu( rich-Irchel, Winterthurerstrasse 190, CH-8057 Zu( rich, Switzerland
Department of Theoretical Physics, Moscow State University, 119899 Moscow, Russia Yukawa Institute for Theoretical Physics, University of Kyoto, Kyoto 606, Japan Received December 1998; editor: A. Schwimmer Contents 1. Introduction 2. General formalism 2.1. Einstein}Yang}Mills theory 2.2. Space-time symmetries of gauge "elds 2.3. Dimensional reduction and scalar "elds 2.4. Static EYM "elds: embedded Abelian solutions 3. Particle-like solutions 3.1. Bartnik}McKinnon solutions 3.2. Sphaleron interpretation 3.3. Solutions with K-term 3.4. Stringy generalizations 3.5. Higher rank groups 3.6. Axially symmetric solutions 4. Non-Abelian black holes 4.1. EYM black holes: the exterior region 4.2. EYM black holes: the interior structure 4.3. Non-Abelian dilaton black holes 4.4. Higher gauge groups and winding numbers

4 7 7 8 11 14 15 16 21 25 28 31 33 34 35 39 44 45

5. Stability analysis of EYM solutions 5.1. Even-parity perturbations 5.2. Odd-parity modes 6. Slowly rotating solutions 6.1. Perturbation equations 6.2. Solutions 7. Self-gravitating lumps 7.1. Event horizons inside classical lumps 7.2. Gravitating monopoles 7.3. Gravitating YMH sphalerons 7.4. Gravitating Skyrmions 8. Concluding remarks 8.1. EYM cosmologies 8.2. Cosmological sphaleron 8.3. EYM instantons Acknowledgements References

46 47 48 50 51 53 55 56 58 65 68 70 71 72 72 73 73

* Corresponding author. Present address: Institute of Theoretical Physics, University of Jena, FroK belstieg 1, D-07743, Jena, Germany. E-mail addresses: [email protected] (M.S. Volkov), [email protected]. (D.V. Galt'sov) 0370-1573/99/$ - see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 0 - 1

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Abstract We present a review of gravitating particle-like and black hole solutions with non-Abelian gauge "elds. The emphasis is given to the description of the structure of the solutions and to the connection with the results of #at space soliton physics. We describe the Bartnik}McKinnon solitons and the non-Abelian black holes arising in the Einstein}Yang}Mills theory, and consider their various generalizations. These include axially symmetric and slowly rotating con"gurations, solutions with higher gauge groups, K-term, dilaton, and higher curvature corrections. The stability issue is discussed as well. We also describe the gravitating generalizations for #at space monopoles, sphalerons, and Skyrmions.  1999 Elsevier Science B.V. All rights reserved. PACS: 04.70.Bw; 11.27.#d Keywords: Gravity; Gauge "elds; Solitons; Black holes

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M.S. Volkov, D.V. Galt'sov / Physics Reports 319 (1999) 1}83

1. Introduction Non-Abelian solitons play an important role in gauge theories of elementary particle physics [94,3,267], where in most cases, the e!ects of gravity can safely be neglected, perhaps except for the very heavy magnetic monopoles in some Grand Uni"ed Theories. Besides, it has recently become clear that solitons are equally important also in string theory, where gravity is essential. A large number of solutions for such gravitating lumps have been obtained [118], however, almost all of them are Abelian. At the same time, the gauge group in string theory is quite complicated, which suggests that non-Abelian solutions can be important as well. From the point of view of General Relativity (GR), the theory of gravitating non-Abelian gauge "elds can be regarded as the most natural generalization of the Einstein}Maxwell (EM) theory. It is therefore reasonable to study gravitating gauge "elds and, in particular, to check whether the standard electrovacuum results have natural generalizations. In view of this, the aim of the present review is to discuss non-Abelian gravity-coupled solitons and black holes. The concepts of solitons and lumps refer in this text to any particle-like solutions of a non-linear "eld theory. Such solutions are asymptotically #at, topologically trivial and globally stationary, although not necessarily stable. The "rst example of gravitating non-Abelian solitons was discovered by Bartnik and McKinnon (BK) in the four-dimensional Einstein}Yang}Mills (EYM) theory for the gauge group SU(2) [19]. This example can be regarded as canonical in the sense that solutions in other models of gravitating non-Abelian gauge "elds studied so far inevitably share a number of characteristic common features with the BK particles. Soon after the BK discovery it was realized that, apart from solitons, the EYM model contains also non-Abelian black holes [324,215,33]. As these manifestly violate the non-hair conjecture, they have attracted much attention and stimulated a broad search for other black hole solutions in models of four-dimensional Einstein gravity with non-linear "eld sources. The results obtained have led to certain revisions of some of the basic concepts of black hole physics based on the uniqueness and no-hair theorems. Speci"cally, it has been found that the violation of the no-hair conjecture is typical for gravitating non-Abelian gauge theories, especially for those models which admit solitons in the #at spacetime limit. Let us recall (see [164] for a more detailed account) that the existing classi"cation of black holes in GR is based on Hawking's strong rigidity theorem, stating that a stationary black hole is either static (with non-rotating horizon) or axisymmetric [159]. This theorem uses only fairly general assumptions like the weak energy condition (although some of them are being critically revised [91,90]). For vacuum and electrovacuum black holes with non-vanishing surface gravity at the horizon, Israel's theorems [176,177] ensure that staticity implies spherical symmetry. It follows then that the static Schwarzschild and Reissner}NordstroK m (RN) solutions, respectively, are unique. For zero surface gravity, static electrovacuum black holes belong to the Majumdar}Papapetrou family [166]. In the stationary case, a chain of uniqueness theorems asserts that the regularity of the event horizon and the asymptotic behaviour speci"ed by mass, angular momentum and Coulomb charges determine the solutions completely. As a result, stationary electrovacuum black holes are necessarily axisymmetric and should belong to the Kerr}Newman family [77,243,75]. Apart from the vacuum and electrovacuum cases, uniqueness has been established for supergravity models containing in four-dimensions sets of scalars parameterizing coset spaces and the corresponding multiplets of the U(1) "elds [62,60].

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Another well-known statement, the no-hair conjecture, claims that the only allowed characteristics of stationary black holes are those associated with the Gauss law, such as mass, angular momentum and electric (magnetic) charges [277,21]. Thus, for example, it follows that black holes cannot support external scalar "elds, since there is no Gauss law for scalars. On the other hand, black holes with a spin 3/2-"eld, say, are allowed, since they carry a conserved fermion charge given by the Gauss #ux integral [6]. The conjecture has been proven for non-interacting boson [25,24,23] and fermion "elds [157,303,304] of various spins, as well as for some special non-linear matter models [4,22,242,162,163,298]. Although the uniqueness and no-hair theorems had actually been proven only for special types of matter, the appealing simplicity of these assertions inspired a widespread belief in their possible general validity. However, after the discovery of the EYM black holes it became clear that this is not the case. First, the EYM black holes possess a short-ranged external non-Abelian gauge "eld and are not uniquely speci"ed by their mass, angular momentum and conserved charges. The no-hair conjecture is therefore violated. Second, static EYM black holes with non-degenerate horizon turn out to be not necessarily spherically symmetric [200,202]. This shows that Israel's theorem does not generalize to the non-Abelian case. A similar phenomenon has also been observed in the EYM-Higgs model [273]. Next, the perturbative considerations suggest that non-static EYM black holes with rotating horizon do not necessarily have non-zero angular momentum [327,66], which shows that the Abelian staticity conjecture [78] does not straightforwardly apply either [170,300]. In addition, the Frobenius integrability conditions for two commuting Killing vectors are not automatically ful"lled for self-gravitating Yang}Mills "elds [170,165], and so the Ricci circularity condition is not guaranteed. The standard Lewis}Papapetrou parameterization of a stationary and axisymmetric metric used for the uniqueness arguments can therefore be too narrow. However, explicit examples of the circularity violation are not known yet. Finally, the three-dimensional reduction of the Yang}Mills action in the presence of a Killing symmetry does not lead to the standard sigma-model structure essential for the uniqueness proof [161,126]. All this shows that a number of very important features of electrovacuum black hole physics cease to exists in the EYM theory, and this seems to happen generically for models with non-Abelian gauge "elds. At the same time, what might seem surprising from the traditional point of view, "nds a natural explanation in the context of #at space soliton physics [94,3,267]. For example, the existence of hairy black holes can often be directly inferred from the existence of solitons in #at space time. For small values of Newton's constant the implicit function theorem ensures that #at space solitons admit weakly gravitating, globally regular generalizations } weak gravity can be treated perturbatively. It turns out that for a large class of matter models the same argument can be used to show the existence of solutions in a small neighbourhood of the point r "0 in parameter space, where r is the event horizon radius [188]. This implies that a weakly   gravitating lump can be further generalized to replace the regular origin by a small black hole. The result is a hairy black hole whose radius is considerably smaller than the size of the lump surrounding it. One can argue in the spirit of the no-hair conjecture that a small black hole cannot swallow up a soliton which is larger than the black hole itself. However, a larger black hole must be able to do this, and indeed, the radius of a hairy black hole inside a soliton usually cannot exceed some maximal value.

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In view of the arguments above, the most surprising fact is the existence of the BK particles, which have no #at space counterparts, and of their black hole analogues, whose radius can be arbitrary. The pure EYM theory is characterized by the fact that all "elds are massless, and the solutions of the theory can therefore be thought of as bound states of two non-linear massless "elds. As a result, the EYM solitons and black holes exhibit a number of special features which distinguish them from solutions of other gravitating non-Abelian gauge models. For example, the BK particles provide the only known example of solitons, whether gravitating or not, which admit regular, slowly rotating generalizations [69]. Solitons of other models, such as the t'Hooft}Polyakov monopoles, say, do not rotate perturbatively [66,171], which means that they either cannot rotate at all or their angular momentum assumes discrete values. The EYM black holes are uniquely distinguished by their peculiar oscillatory behaviour in the interior region [115,59], which is reminiscent of some cosmological models [27]. On the other hand, some features of the EYM solutions are shared by those of other non-linear models. The BK solitons, for example, exhibit a remarkable similarity [134] with the well-known sphaleron solution of the Weinberg-Salam model [208], which allows one to call the BK particles EYM sphalerons. (Let us remind to the reader that the term `sphalerona refers to the static saddle point solution in a gauge "eld theory with vacuum periodicity [179]. Sphalerons are characterized by half-integer values of the Chern}Simons number of the gauge "eld and can be reduced to any of the nearest topological vacua via a sequence of smooth deformations preserving the boundary conditions [237,208]. Such objects are likely to be responsible for the fermion number nonconservation at high energies and/or temperatures [217].) Other properties of EYM solitons and black holes, such as their nodal structure and discrete mass spectrum, are generic and shared by practically all known solutions in models with gravitating non-Abelian gauge "elds. Such solutions can be thought of as eigenstates of non-linear eigenvalue problems, which accounts for the discreteness of some of their parameters. The list of the non-linear models with gravitating non-Abelian gauge "elds, investigated during recent years, contains, apart form the pure EYM theory, also various generalizations. These include the dilaton, higher curvature corrections, Higgs "elds, and a cosmological constant. The Einstein} Skyrme model has also been studied in detail. All these theories admit gravitating solitons and hairy black holes. The pure EYM theory has been studied most of all, some of its features being typical for all other models. For this reason, the central part of our review will be devoted to a description of the basic EYM solutions as well as their direct generalizations. However, other important solutions, such as the gravitating monopoles and Skyrmions, will also be duly described. We will concentrate only on the most important results, but our list of references is quite complete. Several review articles on the related subjects are available [140,181,254,37,232,165,21,161], but overlaps with the present text are small. The plan of the paper is as follows. In Section 2 we give the basic de"nitions, brie#y discuss symmetries of the non-Abelian gauge "elds, and derive the reduced two-dimensional Lagrangians for the spherically symmetric models. Section 3 is devoted to the BK solutions, their sphaleron interpretation and various generalizations. Basic properties of the EYM black holes are discussed in Section 4. Sections 5 and 6 contain a discussion of stability of EYM solitons and black holes and an analysis of their rotational excitations. Gravitating solutions in models admitting solitons in #at space are discussed in Section 7. These include the t'Hooft}Polyakov monopoles, the Yang}Mills}Higgs sphalerons, and the Skyrmions. Some other important results not mentioned in

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the main text are brie#y discussed in Section 8. Most of the solutions described below are known only numerically, although their existence has been established rigorously in some cases. For this reason we present a number of "gures and tables describing the numerical results. All of them have been produced with the `shooting to a "tting pointa numerical procedure, described in [266].

2. General formalism In this section the "eld-theoretical models, whose solutions will be considered in the next chapters, are introduced. These are the EYM theory and its generalizations, including the dilaton, the Higgs "elds, as well as the Einstein}Skyrme model. We brie#y discuss symmetry conditions for gauge "elds and derive the e!ective two-dimensional Lagrangians for all models in the spherically symmetric case. 2.1. Einstein}Yang}Mills theory The basic model which will be discussed below is the four-dimensional EYM system for a compact, semi-simple gauge group G. The Lie algebra of G is characterized by the commutation relations [T , T ]"if T (a, b, c"1,2, dim(G)). The basic elements of the model are (M, g , A), ? @ ?@A A IJ where M is the space-time manifold with metric g , and the Lie-algebra-valued one-form is IJ A,A dxI,T A? dxI. We choose the standard action I ? I



S " #7+



1 1 ! R! tr F FIJ (!g dx , IJ 16pG 4Kg

(2.1)

where g is the gauge coupling constant, F "R A !R A !i[A , A ],T F? , and K'0 is IJ I J J I I J ? IJ a normalization factor: tr (T T )"Kd . For G"SU(2) we choose T "q , K", f "e with ? @ ?@ ?  ?  ?@A ?@A q being the Pauli matrices. We adopt the metric signature (#! ! !), the space-time covariant ? derivative will be denoted by , and the Riemann and Ricci tensors are R? "R C? !2 and @IJ I @J R "R? , respectively. The gravitational constant G is the only dimensionful quantity in the IJ I?J action (the units "c"1 are understood). Apart from the general space-time di!eomorphisms, the EYM action is invariant with respect to the gauge transformations of the gauge "eld A: A PU(A #iR )U\, I I I

F PUF U\, IJ IJ

g Pg , IJ IJ

(2.2)

where U(x)3G. In addition, the Yang}Mills (YM) part of the action displays the conformal symmetry g PX(x)g , IJ IJ

A PA , I I

F PF , IJ IJ

(2.3)

which is not, however, a symmetry of the Einstein}Hilbert action. The variation of the action (2.1) gives the Einstein equations R ! R g "8pG ¹ IJ IJ IJ 

(2.4)

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with the gauge-invariant YM stress-energy tensor 1 tr(!F F N# g F F?@) , ¹ " IN J  IJ ?@ IJ Kg

(2.5)

and the YM equations D FIJ"0 , (2.6) I where the gauge covariant derivative is D , !i[A , ]. Owing to the conformal invariance the I I I stress tensor is traceless, ¹I"0. As a consequence of the gauge invariance, the dual-"eld tensor I *FIJ,()(!g eIJMNFMN (e"1) satis"es the Bianchi identities D *FIJ,0 . (2.7) I Asymptotically #at solutions in the theory have well-de"ned ADM mass and angular momentum [11,1]. Similarly, one can de"ne conserved Lie-algebra-valued electric and magnetic charges [339,2]. Note that the straightforward de"nition [339] 1 Q" 4p



 

*F,



1 P" 4p

F,

(2.8)

 

1 1 where the integration is over a two sphere at spatial in"nity, is not gauge invariant in the non-Abelian case [281,92], and hence requires the gauge "xing. This reminds of a similar situation for the ADM mass, which was originally de"ned only for distinguished coordinate systems [11]. On the other hand, in the presence of a time-like Killing symmetry the ADM mass can be covariantly expressed by the Komar formula [209]. For the YM charges there exists a similar gauge invariant construction [227] using the Lie-algebra-valued `Killing scalara K, where D K"0. Now, it is important that these invariant de"nitions can be generalized to the case where I the con"guration has no symmetries at all. Indeed, the symmetries always exist in the asymptotic region, and these can be used in order to put the surface integrals into the covariant form. As a result one can de"ne the ADM mass [1] and conserved gauge charges [2] for an arbitrary isolated system in the completely covariant and gauge-invariant fashion. For example, for the gauge "eld given by A"AM #a, where AM "i;M d;M \, and aP0 as rPR, one can de"ne F "DM a !DM a . Here DM is the covariant derivative with respect to AM . The conserved and gauge IJ I J J I invariant (up to global gauge rotations) charges Q and P are then given by (2.8) with F replaced by ? ? tr(F ;M T ;M \) [2]. ? 2.2. Space-time symmetries of gauge xelds The issue of symmetry for non-Abelian gauge "elds has been extensively studied in [274,29,125,154}156,236,150,160,99,20,211,210,17,213,70,72,18]. The symmetry conditions for gauge "elds in the in"nitesimal language were "rst formulated in [29,125]. An EYM "eld con"guration will respect space-time isometries generated by a set of Killing vectors m if the metric K is invariant, L K g "0 , K IJ

(2.9)

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under the action of the isometry group, while the corresponding change in the gauge "eld, A PA !eK L K A , can be compensated by a suitable gauge transformation, A PA # I I I I K I eK D = , such that [29,125] I K (2.10) L K A "D = . I K K I Here L stands for the Lie derivative. In order to actually "nd A for a given set of Killing vectors I m the procedure is "rst to solve the integrability conditions for (2.10), which gives the = 's, and K K then to solve (2.10) for A [125,20]. I Throughout this article we will assume the 2#2 block-diagonal form of the metric ds"g dx? dx@#h dx+ dx, , (2.11) ?@ +, where x?,+x, x,, x+,+x, x,, and g depend only on x?. In the case of spherical symmetry ?@ the Killing vectors m generate the SO(3) algebra, and the most general solution to (2.9) can be K parameterized as (2.12) ds"g dx? dx@!R (d0#sin 0 du) , ?@ where R depends on x?,+t, r,. The corresponding most general solution to Eq. (2.10) for the gauge group SU(2) is sometimes called Witten's ansatz [103,335,125,17]: A"aT #i(1!Re w) [T , dT ]#Im w dT . (2.13) P P P P Here the real one-form a"a dx?,a dt#a dr and the complex scalar w depend only on x?. The ?  P position-dependent gauge-group generators T "n?T , P ?

T0"R0T , P

1 T " R T , P sin 0 P P

(2.14)

where n?"(sin 0 cos u, sin 0 sin u, cos 0), obey the standard commutation relations, such that [T ,T0]"iT . The ansatz (2.13) is invariant under the U(1) gauge transformations generated by P P U"exp(ib(t, r)T ), under which P a Pa #R b, wPe @w , (2.15) ? ? ? i.e., a and w transform as a two-dimensional Abelian vector and a complex scalar, respectively. The ? four independent real amplitudes in (2.13) contain therefore one pure gauge degree of freedom. One can choose "w" and X "a !R (arg w) as the three gauge-invariant combinations. Under parity ? ? ? transformation, 0Pn!0, uPn#u, one has T P!T , T0PT0, T P!T . As a result, the P P P P e!ect of parity on (2.13) is P : a P!a , wPwH , (2.16) ? ? and hence among the four real amplitudes one is parity-even, and three are parity-odd. After the gauge transformation with U"exp(iT 0) exp(iT u) [9], an equivalent form of the   ansatz (2.13) in terms of the constant group generators T can be obtained: ? (2.17) A"aT #Im (wT ) d0!Re (wT ) sin 0 du#T cos 0 du , >   > where T "T #iT . The residual gauge freedom (2.15) is now generated by U"exp(ib(t, r) T ). >    Expression (2.17) is sometimes easier to use than (2.13), but it contains the Dirac string singularity.

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Another frequently used gauge is w"wH, in which case the gauge "eld admits the following useful representation: A"a T #i P

1!w U dU\ where U"exp(ipT ) . P 2

(2.18)

The corresponding "eld strength reads (2.19) F"!T da!i dw[T , dT ]!w a dT #T (w!1) d0  sin 0 du . P P P P P Note that the components F and F do not vanish. ?P ?F Unlike the situation in the Abelian case, in order to achieve the temporal gauge condition a "0  one has to make a gauge transformation which renders the whole con"guration time-dependent. Therefore, for static "elds one cannot set a "0, unless X "a !(arg w) "0. The second    gauge-invariant combination, X "a !(arg w), vanishes in the static case by virtue of the YM P P equations. As a result, one can always reduce a to zero by a gauge transformation, and hence the P most general static, spherically symmetric SU(2) YM "elds can be parameterized by two real functions: a and w.  In the static, purely magnetic case the SU(2) ansatz (2.17) with a"Im w"0 can be generalized to the gauge group SU(N) [333,213,18]. Such a generalization includes instead of one function w, N!1 independent real amplitudes w (r), j"1,2, N!1. The gauge "eld potential A is given by H a Hermitean N;N matrix, whose non-vanishing matrix element are A " (N!2j#1)cos 0 du, j"1,2, N , HH  A "(A )H"w H, j"1,2, N!1 . HH> H>H  H Here H"(j(N!j) (i d0#sin 0 du) and the asterisk denotes complex conjugation. In the static, axially symmetric case, the metric is given by ds"eB dt!eD du!eI(do#dz) ,

(2.20)

(2.21)

where d, f, and k depend on x?,+o, z,. The purely magnetic SU(2) gauge "eld reads [269] A"aT #+Re wT #(Im w!l) T , du , (2.22) P M  where the one-form a"a do#a dz and the complex scalar w depend on o and z. The group M X generators are given by T "cos lu T #sin lu T , T "l\R T , (2.23) M   P P M with integer l. The "eld (2.22) also has the residual U(1) invariance expressed in the form (2.15) and generated by U"exp(ib(o, z) T ). P

 In systems with a Higgs "eld the YM equations do not always imply that a vanishes in the static case. The most P general spherically symmetric YM "elds include then a , a , and w"wH.  P  This metric ful"lls the circularity condition and gives rise to non-trivial EYM solutions. It is unclear, however, whether other, non-circular solutions can exist as well.

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Note that the ansatz in Eq. (2.17) can be generalized to describe spherically symmetric YM "elds with arbitrary winding numbers: A"aT #Im (wT ) d0!l Re (wT ) sin 0 du#l T cos 0 du . (2.24)  > >  Here an integer l has the meaning of the Chern number of the U(1) bundle [17]. However, for lO1 the YM equations impose the following condition for non-vacuum "elds: w"0. This implies that all con"gurations with lO1 are embedded Abelian. Since we are interested in non-Abelian solutions, for which w does not vanish identically, we shall always assume that l"1 in (2.24). 2.3. Dimensional reduction and scalar xelds We shall mainly be considering below spherically symmetric systems. It is convenient to derive the "eld equations in the spherically symmetric case by varying the e!ective two-dimensional action (the Hamiltonian approach to the problem was discussed in [100,28]). Let us insert into the four-dimensional (4D) EYM action (2.1) the ansatz (2.13) for the gauge "eld and the metric parameterized as 1 ds"pN(dt#a dr)! dr!R (d0#sin 0u) , N

(2.25)

with p, N, a, and R depending only on x?,+t, r,. Integrating over the angles and dropping the total derivative, the result is the two-dimensional (2D) e!ective action [68]



S" +m!am #¸ ,p dx ,

(2.26)

where m is de"ned by the relation 2im !1 . (RR),R R R?R" ? R

(2.27)

From now on the coordinates x? and all other 2D quantities are assumed to be dimensionless, after the rescaling x?Px?/L, where L is a length scale. The dimensionless gravitational coupling constant i in (2.27) is proportional to G/L. For the scale-invariant YM theory L is arbitrary. It is convenient to choose L"(4pG/g, in which case i"1. The matter "eld Lagrangian ¸ in (2.26)

then reads R 1 ¸ "! f f ?@#"Dw"! ("w"!1) , 7+ ?@ 4 2R

(2.28)

where f "R a !R a and D "R !ia are the 2D Abelian "eld strength and the gauge ?@ ? @ @ ? ? ? ? covariant derivative, respectively. One can see that the residual U(1) gauge symmetry (2.15) arises naturally at the level of the reduced action.

 The change lP!l in (2.24) can be achieved by a gauge transformation.

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Note that, since the theory under consideration is purely classical, the EYM length scale L and the corresponding mass scale, M"iL/G, do not contain Planck's constant. Restoring for a moment in the formulas the speed of light, c, one has



4pG , cg

L"



M"

4p . gG

(2.29)

It is worth noting that, up to the replacement (4p/gPe, this corresponds to the classical system of units based on G, c, and the charge e that had been introduced in [182] before Planck's units were discovered. Dividing and multiplying by ( c, one obtains L"(a L and M"(a M , where . . L and M are Planck's length and Planck's mass, respectively, and a"4p/ cg. Since one can . . expect the value of (a not to be very far from unity, this shows that the units (2.29) are actually closely related to Planck's units. (a) Let us consider generalizations of the EYM theory with additional scalar "elds. The "rst one is the EYM-dilaton (EYMD) model with the 4D matter action





1 1 (

)! eA( tr F FIJ (!g dx , IJ 8pG 4Kg

S " 7+"

(2.30)

where c is a parameter. In the spherically symmetric case, with " (t, r), the 4D EYMD action reduces to the 2D form (2.26) with the matter Lagrangian ¸ given by

R ¸ " (R )#eA(¸ , (2.31) 7+" 7+ 2 where ¸ is speci"ed by (2.28). The length scale is now "xed, L"(4pG/g, and i"1. 7+ (b) Next, consider the EYM-Higgs (EYMH) model with the Higgs "eld U,U?T in the adjoint ? representation of SU(2). The 4D matter action is the sum of the YM term and




S " &



1 j tr D U DIU! (U?U?!v) (!g dx , I 2 4

where D U"R U!i[A , U]. The spherically symmetric Higgs "eld is I I I U"v T , P

" (t, r) being a real scalar. This gives the 2D matter Lagrangian in (2.26):

(2.32)

(2.33)

¸ "¸ #R (R )!"w" !e R ( !1) . (2.34)

7+   Here the length scale is L"1/M , e"M /(2M , where M "gv and M "(2jv are the 5 & 5 5 & vector boson mass and the Higgs boson mass, respectively. The gravitational coupling constant is i"4pGv. (c) For the EYMH model with a complex Higgs "eld U in the fundamental representation of SU(2) the 4D Higgs "eld action is




S " &



j (D U)R DIU! (URU!v) (!g dx , I 4

(2.35)

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where D U"R U!iA U. The spherically symmetric Higgs "eld is given by I I I U"v exp(im T )"a2 , (2.36) P where " (t, r) and m"m(t, r) are real scalars, while "a2 is a constant unit vector of the two-dimensional complex vector space, 1a " a2"1. The corresponding 2D matter "eld Lagrangian is ¸ "¸ #R"Dh"! "w!e K"!e R ( !1) , (2.37)

7+   where h" exp(im/2) and D "R !ia /2, the other parameters being the same as in the triplet ? ? ? case. (d) Finally, we shall consider the SU(2) Skyrme model. The 4D matter action is





f 1 S " tr ! A AI# (F FIJ) (!g dx , 1 I 4 32e IJ

(2.38)

where A ";RR ; and F "[A , A ]. The spherically symmetric chiral "eld is speci"ed by I I IJ I J U"exp(2is(t, r)T ) , (2.39) P which leads to the 2D Lagrangian











R sins 1 ¸ " #sin s (Rs)! R# sin s .

2 R 2

(2.40)

The length scale is L"1/ef, the gravitational coupling constant being i"4pGf . The Einstein-matter coupled "eld equations are obtained by varying (2.26) with respect to N, p, a, and the matter variables in ¸ . After varying one can set the shift function a to zero, but one

cannot do this before varying, since for time-dependent "elds one of the Einstein equations would be lost [68]. In the static case, on the other hand, one can get rid of a from the very beginning. If the Schwarzschild gauge condition R"r can be imposed, which is often the case, the Einstein equations for static "elds look especially simple. Note that the 4D Einstein equations read in the dimensionless notation G "2i ¹ . (2.41) IJ IJ For all matter models described above the matter Lagrangian in the static, purely magnetic case reduces to ¸ "!(NK#;) , (2.42)

where K and ; depend only on the matter variables, and N"1!2im/r. As a result, varying the reduced action (2.26) with respect to m and p gives the independent Einstein equations m"NK#;,r¹, (ln p)"2iK,i (r/N) (¹!¹P) , (2.43)   P from which the components of the (dimensionless) stress tensor can be read o!. The matter "eld equation are obtained by varying p¸ . The dimensionless ADM mass is given by m(R). The

dimensionful mass is (iL/G) m(R).

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We shall consider below time-dependent problems only for the EYM "elds, in which case the full system of equations following from (2.26) and (2.28) reads R (rpf ?@)"2p Im (wD@w)H , ? D (pD?w)"(p/r) ("w"!1)w , ? (ln p)"(2/r)((1/Np)"D w"#"D w") ,  P m "2N Re D w (D w)H ,  P m"!rf f ?@#(1/Np)"D w"#N"D w"#(1/2r) ("w"!1)  ?@  P with the notation of (2.28) and the asterisk denoting complex conjugation.

(2.44) (2.45) (2.46) (2.47) (2.48)

2.4. Static EYM xelds: embedded Abelian solutions For static EYM "elds the results of this section can be summarized as follows. The most general spherically symmetric SU(2) gauge "eld can be parameterized as A"a T dt#w (T d0!T sin 0 du)#T cos 0 du ,      and the space-time metric is ds"pNdt!(1/N) dr!r (d0#sin0 du) .

(2.49)

(2.50)

The amplitudes a , w"wH, m, N,1!2m/r, and p depend on r. The EYM equations (2.44)}(2.48)  assume the form ((r/p) a )"(2/pN) w a ,   (pNw)"(p/r) w(w!1)!(a/pN) w ,  m"(r/2p) a #Nw#(1/2r)(w!1) ,  p/p"(2/r) w .

(2.51) (2.52) (2.53) (2.54)

In the next two sections we shall study solutions to these equations. The simplest one is a "0, w"$1, m"M, p"1 , (2.55)  which describes the Schwarzschild metric and a pure gauge YM "eld. The simplest solution with a non-trivial gauge "eld describes colored black holes [13,87,330,261,184] a "a (R)#Q/r, w"0, N"1!(2M/r)#(Q#1)/r, p"1 . (2.56)   The metric is RN with the electric charge Q and unit magnetic charge. This solution is embedded Abelian in the sense that the SU(2) gauge "eld potential (2.49) is a product of the potential of the U(1) dyon, A"a dt#cos 0 du, and a constant matrix. The appearance of such a solution is not  surprising: since the SU(2) gauge group contains U(1) as a subgroup, the set of solutions in the EYM theory includes, in particular, the embeddings of all electrovacuum solutions. These are often found by integrating the EYM equations. In this way, for example, the colored black holes and their generalizations with rotation and cosmological term [10,189}191], as well as solutions with cylindrical and plane symmetries [245}249] have been obtained. Note that the embedded Abelian

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solutions are not completely equivalent to their U(1) counterparts. For example, the Dirac string for (2.49) and (2.56) can be globally removed by passing to the regular gauge (2.13) } the SU(2) bundle is trivial. In addition, unlike their U(1) counterparts, the colored black holes are unstable [221].

3. Particle-like solutions The rapid progress in the theory of gravitating solitons and hairy black holes started after Bartnik and McKinnon had discovered particle-like solutions for the EYM equations (2.51)}(2.54) [19]. This came as a big surprise, since it is well known that when taken apart, neither pure gravity nor YM theory admit particle-like solutions. Indeed, the existence of stationary gravitational solitons is ruled out by Lichnerowicz's theorem [224]. One can go farther than this and consider regular con"gurations with "nite mass and arbitrary time dependence, as long as they do not radiate their energy to in"nity. In particular, system which are exactly periodic in time do not radiate, and such solutions are also ruled out [139]. It is worth noting, however, that the vacuum theory of gravity admits, `quzi-solitonsa, geons, which exist for a long time as compared to the characteristic period of the system before radiating away their energy [64]. Another famous statement `there are no classical glueballsa asserts that the pure YM theory in #at space does not admit "nite energy non-singular solutions which do not radiate energy out to in"nity [94}96,105,260]. This can be readily seen in the static situation, since the spatial components of the stress-energy tensor ¹ then satisfy [105] IJ



¹ dx"0 . (3.1) GI 0 This is the consequence of the identity ¹ "R (¹Hx ) implied by the conservation law R ¹H"0. The GI H G I H G physical meaning of (3.1) is that the total stresses in an extended object must balance. It follows then that the sum of the principal pressures, p , where p "!¹G (no summation), cannot have G G G G a "xed sign. However, for the YM "eld the scale invariance implies that p "¹ '0, which G G  shows that the system is purely repulsive and the force balance is impossible. The argument can be extended to exclude solutions with arbitrary time-dependence, as long as they do not radiate [260,95]. It is interesting that the geon-type solutions in the pure YM theory are also excluded } any "eld con"guration initially con"ned in some region falls apart in the time it takes light to cross the region [96]. This can be viewed as another indication of the purely repulsive nature of the gauge "eld. Although for purely attractive or repulsive systems the balance cannot exist, the situation changes if the system includes interactions of both types. The example is provided by the YM-Higgs models (2.32), (2.35). For the scalar Higgs "eld one has p (0, which corresponds to G G pure attraction, such that the balance is again impossible. However, in a combined system with both gauge and Higgs "elds repulsion and attraction can compensate each other, which leads to the existence of soliton solutions. These will be considered in Section 7 below. In a similar way, the  Note though, that no rigorous existence proof for geons has been given so far.

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coupling of the repulsive YM "eld to gravity can lead to a force balance, as is illustrated by the BK example. It should be stressed, however, that the presence of both attractive and repulsive interactions is necessary but not su$cient for the existence of equilibrium con"gurations. This becomes especially clear when one takes into account a number of the no-go results for the EYM system [331,106,234,235,238,121]. These, for example, rule out all non-trivial EYM solutions in three spacetime dimensions [106], and large classes of solutions in four dimensions. In particular, all charged 4D EYM solitons are excluded [121]. In view of this one can wonder about the su$cient condition which ensures the existence of the BK solutions. Unfortunately, this condition is unknown. Even though the existence of the BK solutions can be deduced from a complicated analysis of the di!erential equations [293,287,55], it still does not "nd a clear explanation in physical terms. For example, the topological arguments, which proved to be very useful in the YMH case, do not apply in the EYM theory. It is interesting that, even though the topology is di!erent, there exists a very useful analogy between the EYM theory and the #at space doublet YMH model. It has already been mentioned that the BK particles are in some ways similar to the sphaleron solutions of the Weinberg}Salam model [237]. The resemblance is two-fold. First, both sphalerons and BK solitons can be thought of as equilibrium states of a pair of physical "elds one of which is repulsive (YM "eld) while another one is attractive (gravity or the Higgs "eld). Secondly, both types of solutions relate to the top of the potential barrier between distinct topological vacua of the gauge "eld. In this sense one can call the BK particles EYM sphalerons [134]. In view of their importance, we shall describe below in this section the BK solutions and the corresponding sphaleron construction in some detail. We shall consider also the generalizations of the solutions due to the dilaton, K-terms, as well as the solutions for higher gauge groups and in the case of axial symmetry. Although very interesting in their own right, these exhibit essentially the same features as the BK solutions. 3.1. Bartnik}McKinnon solutions These solutions are known numerically [19,215,55], their existence was established in [293,287,55]. The gauge "eld potential and the space-time metric are chosen in the form (2.49), (2.50), and the "eld equations are given by (2.51)}(2.54). The electric "eld amplitude a should be set  to zero, since otherwise only the embedded Abelian solutions are possible [133,121,40]. As a result, the metric function p can be eliminated from (2.52) and (2.53) [268] and there remain only two independent equations




Nw#



2m (w!1) w w(w!1) ! " , r r r r

(w!1) . m"Nw# 2r

(3.2) (3.3)

Given a solution to these, p is obtained from




p"exp !2



 w dr . r P

(3.4)

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Eqs. (3.2) and (3.3) have singular points at the origin, r"0, at in"nity, r"R, and at r"r , where  N(r )"0. Assuming that N vanishes nowhere (no horizons), consider the local power-series  solutions to Eqs. (3.2) and (3.3) in the vicinity of the singular points. Requiring that curvature is bounded one obtains at the origin w"1!br#(  b!b)r#O(r) , (3.5)   N,1!(2m/r)"1!4br#br#O(r) .  At in,nity, using the results in [133,121,40], the only nontrivial possibility are con"gurations that asymptotically approach the magnetically neutral solution (2.55):






a 3a!6aM w"$ 1! # #O(r\) , r 4r

(3.6)

a m"M! #O(r\) . r In these expressions b, a and M are free parameters, M being the ADM mass. Applying the standard methods one can show that the Taylor expansions in (3.5) and (3.6) have a nonzero convergence radius [215,55]. Inserting these expressions into (3.4), one obtains p(r)"p(0)#O(r),

p"1#O(r\)

(3.7)

at the origin and at in"nity, respectively. The next step is to integrate Eqs. (3.2) and (3.3) with the boundary conditions (3.5) and (3.6). The strategy is to extend the asymptotics numerically to the intermediate region, where the matching conditions are imposed at some point r [266]. There are three such conditions for w, w and m. This agrees with the number of the free parameters in (3.5) and (3.6). It turns out that the matching can be ful"lled, provided that the values of the parameters b, a and M are restricted to a one-parameter family +b , a , M ,, where n is a positive integer. This gives an in"nite family of globally regular L L L solutions in the interval 04r(R labeled by n"1, 2,2 . The index n has the meaning of the node number for the amplitude w: for the nth solution w(r) has n nodes in the interval r3[0,R), such that w(R)"(!1)L. For all solutions w is bounded within the strip "w(r)"41 (see Fig. 1). This can be understood as follows: Suppose that w leaves the strip, such that w(r)'1 for some r'0 (notice that the equations are invariant under wP!w). Then, taking the boundary conditions into account, w must develop a maximum outside the strip: w(r )"0,

w(r )'1, w(r )(0 for some r '0. However, Eq. (3.2) implies that in the region w'1 an

extremum can only be a minimum, which shows that the condition "w"(1 cannot be violated. The behaviour of the metric functions m and p is similar for all n's: they increase monotonically with growing r from m(0)"0 and p(0),p at the origin to m(R)"M and p(R)"1 at in"nity, L L respectively (see Fig. 2). Here M is the ADM mass of the solutions and p is the time deceleration L L factor at the origin. Qualitatively, the solutions show three distinct regions with the two transition zones; see Figs. 1}3. This distinction is due to the characteristic behavior of the e!ective charge function Q(r) de"ned by the relation [19] (see Fig. 1) 2M Q(r) . N(r)"1! # r r

(3.8)

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Fig. 1. w(r) and the e!ective charge function Q(r) for the lowest BK solutions.

Fig. 2. Metric functions m(r) and p(r) for the lowest BK solutions.

(a) In the interior region, r41, one has Q(r)+0, the functions w, m and p are approximately constant, and the metric becomes #at as rP0. The stress tensor ¹I is isotropic and corresponds to J the equation of state o"3p with oJr; note that this has the SO(4)-symmetry. The transition zone about r"1 is marked by the high energy density. The metric functions grow rapidly in this zone approaching their asymptotic values, while Q(r) reaches the unit value.

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Fig. 3. On the left: the radial energy density r¹ for the lowest BK solutions. On the right: the equations of state ¹P/¹  P  and ¹00/¹. 

(b) In the near-"eld region, 14r4R , one has Q(r)+1, the gauge "eld is well-approximated by  that for the Dirac magnetic monopole, w"0, such that o"!p "p0 with oJr\. The metric is P close to the extreme RN metric. In the `charge-shieldinga transition zone, r&R , the e!ective  charge decays to zero. (c) Finally, in the far-"eld region, r'R , the solutions are approximately Schwarzschild. The  gauge "eld strength decreases as 1/r, the components of the stress tensor are o"!p "3p0/2 P with oJr\. The polynomial decay of the functions re#ects the massless nature of the "elds in the problem. The above distinction becomes more and more pronounced with growing n. The ADM mass M increases with n from the `ground statea value M "0.828 to M "1, rapidly converging to L   the upper limit (see Table 1). The dimensionful mass is (4p/Gg M , and the BK particles are L therefore extremely heavy. Notice that the discrete energy spectrum emerging in the problem resembles that of some quantum system. The analogy becomes even more striking if one thinks of the node number n as relating to some wave function. The values of +b , a , M , p , for the lowest L L L L n are given in Table 1, while those for large n can be approximated by b "0.706!2.186e , M "1!1.081e , L L L L a "0.259e , p "0.707e (3.9) L L L L with e "exp(!pn/(3) [55]. L As nPR, more and more nodes of w accumulate in the "rst transition zone near r"1, where N develops a more and more deep minimum, closely approaching the zero value. In the near-"eld

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Table 1 Parameters of the BK solutions n

b L

M

a L

p L

1 2 3 4 5

0.4537 0.6517 0.6970 0.7048 0.7061

0.8286 0.9713 0.9953 0.9992 0.9998

0.8933 8.8638 58.929 366.20 2246.8

0.1264 0.0208 0.0033 0.0005 0.00009

L

zone at the same time, the amplitude and frequency of the oscillations decrease with growing n, while the size of the zone stretches exponentially, the whole con"gurations approaching the extreme RN solution: w"0, p"1, N"(1!(1/r)).

(3.10)

However, for "nite n the size of the near-"eld zone is "nite, and since w tends to $1 in the far-"eld region, the solutions are always neutral. The limiting solution with n"Rcan also be investigated [55,61,294,288]. This turns out to be non-asymptotically #at, which can be qualitatively understood as follows. The Schwarzschild coordinate system breaks down in this limit, and one should use the general parameterization (2.25) with R"R(r) (and a"0). The EYM "eld equations (see Eqs. (3.24)}(3.26) with K"0) then can be reformulated as a dynamical system. This system admits a separatrix that starts from the saddle point with R"0, w"N"1, which corresponds to the origin, and after in"nitely many revolutions ends up at the focal point with R"1, w"N"0, which corresponds to the degenerate horizon. The function R thus changes only in the interval [0, 1]. In contrast, the phase-space trajectories of solutions with n(Rmiss the focal point and propagate further to the region R'1, where they approach (3.10) for large n. As a result, the con"guration that the BK solutions tend to for nPR is the union of the oscillating solution in the interval R3[0, 1] and the extreme RN solution (3.10) for R3[1,R). Such a con"guration cannot be regarded as a single solution globally de"ned for all R50. For the oscillating solutions the t"const. hypersurfaces are non-compact and in the limit RP1 correspond to the cylinders R;S. The four-metric in this limit is the direct product of a unit two-sphere and a unit pseudosphere [55]. Other solutions to the EYM equations (3.2) and (3.3) with the regular boundary conditions (3.5) at the origin are also non-asymptotically #at and belong to the bag of gold type. This case is in fact generic [55]. The qualitative picture is as follows. Using again the general parameterization (2.25) with the gauge condition p"1 the Einstein equation for R(r) is R"!2w/R. The regular boundary conditions at the origin imply that R(0)"0, R(0)"1. Since the second derivative R is always negative, the "rst derivative R is always decreasing. As a result, R(r) tends to develop a maximum at some "nite r where R vanishes. After this the integral curve for R(r) bends down "nally reaching zero at some "nite r, where the curvature diverges. This behaviour is generic. Only for the special boundary conditions speci"ed by b"b in (3.5) the integral curves of R(r) bend down L slowly enough to be able to escape to in"nity. For b"b , R(r) develops an extremum at r"R,  R(R)"1. For all other values of b in (3.5), R(r) develops a maximum at a "nite value of r leading to a bag of gold solution.

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3.2. Sphaleron interpretation The BK solutions admit an interesting interpretation as EYM sphalerons [134,299,143]. This is based on some common features which they share with the sphaleron solution of the Weinberg}Salam model [237,208]. In fact, sphalerons exist also in other gauge models [123,43,255] with vacuum periodicity [179]. In such theories the potential energy is a periodic function of the winding number of the gauge "eld. The minima of the energy, which are called topological vacua, are separated by a potential barrier of a "nite height, and `sittinga on the top of the barrier there is a classical "eld con"guration called sphaleron. Since its energy determines the barrier height, the sphaleron is likely to be important for the barrier transition processes when the system interpolates between distinct vacuum sectors, at least in the electroweak theory [217,275]. The winding number of the gauge "eld changes during such processes, which leads to the fermion number nonconservation due to the axial anomaly. In what follows, we shall introduce the topological vacua in the EYM theory, and show that the odd-n BK "eld con"gurations can be reduced to either of the neighbouring vacua via a continuous sequence of static deformations preserving the boundary conditions. The solutions therefore relate to the top of the potential barrier between the vacua, which accounts for their sphaleron interpretation. 3.2.1. Topological vacua The sphaleron construction for the BK solutions starts by de"ning the topological vacua in the EYM theory as "elds +g , A, with zero ADM mass: g "g , A"iUdU\. Here g is MinIJ IJ IJ IJ kowski metric on R and U"U(xG). Imposing the asymptotic condition at spatial in"nity [179] lim U(xG)"1 , (3.11) P any U(xG) can be viewed as a mappings SPSU(2), and the set of all U's falls into a countable sequence of disjoint homotopy classes characterized by an integer winding number



1 tr k[U]" 24p

UdU\UdU\UdU\ .

(3.12)



0 As a result, the vacuum "elds split into equivalence classes with respect to the winding number of the gauge "eld k[U]. These are called topological vacua. Note that the gravitational "eld is assumed to be topologically trivial. A representative of the kth vacuum class, +g , iU dU\, with k[U ]"k, can be parameterized IJ I I I by U "exp+ib(r)T , where b(0)"0, b(R)"!2pk . (3.13) I P The gauge "eld iU dU\ is given by Witten's ansatz (2.13) with a"0, w"exp(ib(r)). I I Note that the asymptotic condition (3.11) imposes the following fall-o! requirement for the components A of the vacuum gauge "eld with respect to an orthonormal frame: ? A "o(r\) for rPR . (3.14) ? In what follows, we shall demand that non-vacuum A's should obey the same fall-o! condition.

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The vacua with di!erent k cannot be continuously deformed into one another within the class of vacuum "elds subject to (3.14). However, it is possible to join them through a continuous sequence of non-vacuum "elds, +g [j], A[j],, where the gauge "eld A[j] obeys (3.14) for all values of IJ the parameter j, while g [j] is an asymptotically #at metric on R. The crucial point is that there IJ are such interpolating sequences that pass through the BK "eld con"gurations. 3.2.2. The interpolating sequence Consider a family of static "elds [134,139] 1 dr!r(d0#sin0 du) , ds"p(r)N (r) dt! H H N (r) H 1#w 1!w U dU\#i U dU\ , A[j]"i > > \ \ 2 2

(3.15) (3.16)

where N (r)"1!2m (r)/r, and j3[0, p]. The functions m (r), p (r), and U are given by H H H H ! sinj P (w!1) m (r)" w#sinj p (r) dr , H H p (r) 2r  H  (3.17) p (r)"exp !2 sinj dr w/r , H P U "exp+ij(w$1)T , , ! P where w coincides with the gauge amplitude for the nth BK solution. The boundary conditions for w ensure that, for any n,













A [j]"O(r) as rP0, A [j]"O(r\) as rPR . (3.18) ? ? The metric (3.15) becomes #at for j"0, p. The gauge "eld (3.16) vanishes for j"0, whereas for j"p it can be represented as A[p]"iUdU\, with U"exp+ip(w!1)T , . (3.19) P Now, one has w(0)"1, w(R)"(!1)L. Comparing with (3.13) one can see that for odd values of n the pure gauge "eld (3.19) has unit winding number, whereas the winding number is zero if n is even. The "eld sequence (3.15), (3.16) therefore interpolates between distinct topological vacua for n odd, and between di!erent representatives inside the same vacuum sector for n even. 3.2.3. The Chern}Simons number It is instructive to rederive the above result in a di!erent way [208,134,253,319]. Consider an adiabatic time evolution along the family (3.15), (3.16) by letting the parameter j depend on time in such a way that j(!R)"0, j(R)"p. Introduce the Chern}Simons current,






2i 1 eIJ?@ tr A A ! A A , KI" J ? @ 3 ? @ 8p (!g

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whose divergence KN"(1/16p)tr *F FIJ. For any j(t) in (3.16) the temporal component of the N IJ gauge "eld is exactly zero implying that all spatial components of KI vanish. This gives





 

1 R dx (!g tr *F FIJ . (3.20) " IJ 16p \ \ The quantity on the left is N [t]!N [!R], where N is the Chern}Simons number of the !1 !1 !1 gauge "eld. Since for a pure gauge "eld N coincides with the winding number, one has !1 N [!R]"0. Substituting (3.16) into (3.20) and evaluating the integral on the right one "nally !1 arrives at dx (!gK

R





3 R  N [j(t)]" dt jQ sinj dr w(w!1) !1 2p \  1!(!1)L (j!sin j cos j) . " 2p

(3.21)

This shows that for odd values of n the Chern}Simons number changes from zero to one as j increases from 0 to p. If n is even then one has N "0 for any j, which corresponds to a sequence !1 of "elds within one vacuum sector. Now, for j"p/2 the "elds (3.15)}(3.17) exactly correspond to those of the BK solutions. The gauge "eld (3.15) then can be obtained from the BK "eld via a gauge rotation with U"exp+ip(w!1)T /2,. The metric function m (r) for j"p/2 satis"es the Einstein equation (3.3) P H with the appropriate boundary condition and hence coincides with the BK mass amplitude m(r), while p (r) coincides with p in (3.4). As a result, the con"guration (3.15)}(3.17) for j"p/2 is the H gauge rotation of the BK "eld. Summarizing, there are sequences of static "elds that interpolate between distinct vacuum sectors passing through the BK con"gurations with odd n. Using (3.21), one can determine the Chern}Simons number for these solutions: N ". This shows that the odd-n BK solutions reside !1  in between the two vacua and can be reduced to each of them through a continuous sequence of static deformations preserving the boundary conditions. This accounts for their interpretation as sphalerons. Note also that these solutions admit a zero energy fermion bound state [143,319,74]. Moreover, the spectrum of the Dirac operator for fermions in the background "elds (3.15)}(3.17) for odd values of n exhibits the standard level crossing structure [319]. As the parameter j varies from zero to p, the lowest positive energy level crosses zero at j"p/2 and "nally replaces the highest energy level of the Dirac see. The fermion number therefore changes. For even values of n the BK solutions are topologically trivial, since (3.21) gives in this case N "0. !1 3.2.4. The energy and the negative modes The above considerations show that the odd-n BK solutions relate to saddle points on the potential barrier separating the neighbouring topological vacua. For static ow-shell "elds in GR the potential energy is the ADM mass } provided that the con"guration is asymptotically #at and ful"lls the initial value constraints. Thus, for example, for a spherically symmetric, globally regular static "eld con"guration the energy is



 M"lim m(r)" r¹ dr .   P

(3.22)

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Fig. 4. On the left: the ADM mass along the "eld sequence (3.15)}(3.17) through the n"1 BK solution. On the right: the potential energy (3.23) in the vicinity of this solution.

Here the second equality on the right is due to the initial value constraint G"2¹ (in the   dimensionless units chosen), since one has G"2m/r. It is worth noting that for any value of j in  (3.15)}(3.17) the "elds do satisfy the initial value constraints, from which the only non-trivial one is G"2¹. In addition, the GP"2¹P Einstein equation is also ful"lled. As a result, the energy along   P P the "eld sequence (3.15)}(3.17) is obtained by simply taking the limit rPR in m (r) in (3.17). The H result is given by Eq. (3.23) below (with b"1). The energy vanishes for j"0, p and reaches the maximum at the sphaleron position j"p/2 (see Fig. 4). The last remark suggests that the BK solutions are unstable. The corresponding negative mode is given by the derivative of (3.15)}(3.17) with respect to j at j"p/2. The two facts are essential: (a) the energy is maximal at j"p/2; (b) the boundary conditions at in"nity (3.18) hold for any j, which ensures that the negative mode is normalizable. The stability analysis presented in Section 5 below gives the following result: the nth BK solution has 2n negative eigenmodes in the spherically symmetric perturbation sector, of which n modes are parity-even and n are parity-odd [323]. The described above negative mode belongs to the odd-parity sector. 3.2.5. The physical picture The fact that the BK solutions have more than one negative mode re#ects the important di!erence between the EYM and the YMH sphalerons. Sphalerons are usually associated with non-contractible loops in the con"guration space [237]. Such loops can be obtained from the interpolating sequences considered above by identifying the end points. The maximal value of energy along a loop, E , is minimized over all loops to obtain E ,inf+E ,. If (a) E exists and is

 

  positive; (b) there is a loop whose E is equal to E , then there is a saddle point solution called

 

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sphaleron [237]. By construction, this has only one negative mode. In reality, however, it is very di$cult to show that the conditions (a) and (b) above hold, and this has never been done. At the same time, these conditions are essential, since the mere existence of non-contractible loops does not imply the existence of any non-trivial solutions [237]. For example, in #at spacetime pure YM theory there is vacuum periodicity but there are no static "nite energy solutions at all. For these reasons a somewhat weaker de"nition is adopted in this text. According to this, sphalerons are only required to relate to the top of the potential barrier between the vacua. As a result, they are still topologically non-trivial, but can have more than one negative mode, and their existence is not guaranteed by the minimax argument. The following physical picture seems plausible: The vacuum-to-vacuum transitions in the EYM theory are suppressed at low energies, whereas the suppression is removed at the energy of the order of mass of the n"1 BK solution. In order to justify this picture, one should take into account the fact that, since the n"1 BK solution has more than one negative mode, its energy does not determine the minimal height of the potential barrier between the vacua. In fact, the latter is zero. In order to see this, it is instructive to consider the interpolating sequence (3.15)}(3.17) through the n"1 BK solution, and to generalize it by replacing w(r) by w(br), where b is a parameter. Since the boundary conditions do not change, the new sequence still interpolates between the distinct vacua, but, unless b"1, no longer passes through the BK con"guration. For any b the "elds ful"ll the (00) and (rr) Einstein equations, which allows one to de"ne the energy as m (R), where m (r) is given by H H (3.17) with w(r) replaced by w(br). This gives


P








 dr (w!1) w#sinj exp !2b sinj w dr , (3.23) r 2r  P where w"w(r) corresponds to the n"1 BK amplitude (see Fig. 4). For a "xed value of b the energy is maximal at j"p/2 and vanishes as bP0,R, so that the minimal barrier height is zero [318,316]. In view of the last remark, one can wonder as to why the barrier transitions in the EYM theory should be suppressed at low energies [318,316]. Note, however, that in the low-energy limit the EYM theory reduces to pure YM theory, for which transitions between distinct vacua are known to be strongly suppressed, despite the fact that inf+E ,"0. On the other hand, the existence of

 the energy scale provided by the sphaleron mass suggests that the transitions are unsuppressed above a certain energy threshold. However, the evaluation of the path integral is necessary in order to make any de"nite statements. M(b, j)"b sinj

3.3. Solutions with K-term A natural generalization of the BK solutions is provided by including the K-term into the EYM equations [328,67,307]. It is convenient from the very beginning to use the general parameterization of the metric (2.25) with p"1 and a"0. The static EYM-K equations then read 2w , R"! R

(3.24)

 At "nite temperature the total transition amplitude in the YM theory can be large due to the high number of thermal quanta participating in the transitions, although for each individual quantum the amplitude is small.

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(w!1) (NR)R"2 Nw! #1!KR , 2R

(3.25)

w(w!1) . (Nw)" R

(3.26)

There is also a "rst integral for these equations






(w!1) (NR)R"2 Nw! #1!KR . 2R

(3.27)

The equations admit, for example, the de Sitter solution: w"$1, R"r, N"1!Kr/3. For K"0 the BK con"gurations ful"ll these equations. The nth BK soliton has the typical size R (n),  for R'R the metric being almost vacuum. It follows that if the K-term is non-zero but very small,  K;1/R, then the contribution KR to the energy density is negligible. For R(R one therefore   expects that the solutions do not considerably deviate from the BK con"gurations. In the region R'R , however, the e!ect of K becomes signi"cant, which suggests that the metric approaches the  de Sitter metric. Hence } for su$ciently small values of the cosmological constant } the solutions are expected to resemble the regular BK solitons surrounded by a cosmological horizon at R&1/(K and approach the de Sitter geometry in the asymptotic region. The numerical analysis con"rms these expectations [328]. Starting from the regular boundary conditions at the origin w"1!br#O(r),

R"r#O(r) ,

(3.28)

N"1#(4b!K/3)r#O(r) , the numerical procedure breaks down at some r '0 where N vanishes, which corresponds to the  cosmological horizon. The local power-series solution at the horizon contains four parameters: r ,  R , R , and w :    w"w #w x#O(x), R"R #R x#O(x) , (3.29)     N"N x#O(x) ,  with x"r!r . Here w and N are expressed in terms of R , R , and w [328]. The strategy is to       extend numerically the local solutions (3.28) and (3.29) and impose the matching conditions at some point r , where 0(r (r . For any K;1, this leads to a family of solutions in the interval  0(r(r , which are parameterized by the node number n"1, 2,2 . For the nth solution  w oscillates n times for 0(r(r (see Fig. 5).  Since the boundary conditions at the cosmological horizon are already "xed, the next step is merely to integrate from the horizon outwards to extend the solutions to the asymptotic region. For rPR one "nds a w"w # #O(R\),  R

R"R r#O(1) , 






1 2K 2M K #O(R\) , N"1! ! R# (w !1)! a  R 3 3 R

(3.30) (3.31)

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27

Fig. 5. On the left: amplitudes w, N, m, and p for the asymptotically de Sitter solution with n"3 and K"0.0003. On the right, the conformal diagram for this solution (the black regions should not be confused with space-time singularities). The de"nition of the null coordinates ; and < is given in [328].

where the parameters w , a, R , and M are determined numerically. It turns out that w O$1,    which means that the magnetic charge does not vanish and in the string gauge (2.17) is given by 1 P" 4p



1

F"(w !1)T .  

(3.32)



The geometry in the asymptotic region is RN-de Sitter. It is interesting that the charge determined by the asymptotic behaviour of N does not coincide with the one in (3.32). Some numbers are: for the n"1 solution with K"0.01 one has b"0.452, R "16.431, w "!0.944, w "!1.001,    M"0.821 [328]. The space time is topologically non-trivial, and the conformal diagram is qualitatively identical to that for the de Sitter solution (see Fig. 6). Similar to the de Sitter case, the diagram contains two boundaries R"0, which now correspond to a pair of BK solutions. One can think of the spacetime manifold as the de Sitter hyperboloid slightly deformed by masses of two BK particles placed at the opposite sides of the spatial section. Let us consider now what happens for large values of K. When K grows, the coe$cient R in  (3.30) decreases until it vanishes for some critical value K (n). One has K (1)"0.3305. For large     r the con"gurations then approach the Nariai solution: w"$1, R"1/(K, N"!Kr. For K'K , R(r) is no longer monotone: it reaches a maximum at some "nite r and then starts   C decreasing until it vanishes at some r , where N diverges. Such solutions belong to the bag of gold   type, the &bag' containing an event horizon and a spacetime singularity. If K continues to increase, the position of the maximum of R moves towards the origin, while the horizon shifts towards the

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Fig. 6. On the left: change of the topology for the EYM-K solutions. The n"1 solution for K"K "0.3304 is  asymptotically de Sitter. The one for K"K "0.3305 is of the Nariai type, while the one for K"K "0.3306 is of the   bag of gold type. On the right: the regular compact solution for n"3.

singularity. Finally, for K"K (n), the horizon merges with the singularity: r Pr . What     remains is a completely regular manifold whose spatial sections are topologically S and have the re#ectional symmetry with respect to r . One has K (1)", in which case the solution is known C   analytically [109] N"1,

R"(2 sin(r/(2),

w"cos(r/(2) ,

(3.33)

which corresponds to the standard metric on R;S. For solutions with n'1 one has 'K (n)'K (R)", and the spatial sections are squashed three-spheres.     No regular solutions exist for K'K (n). In this case N is everywhere positive and diverges at  r which is the position of the second zero of R. All EYM-K solutions described above are   unstable [67]. 3.4. Stringy generalizations It is natural to wonder whether the BK solutions play any role in string theory. The most popular stringy generalization of the EYM theory is the EYM-dilaton theory (2.30). One can also consider a more general model with the (4D) Lagrangian (3.34) L"!R#(

)! eA( (F FIJ!bR )!;( ) .   ?IJ ? %  Here R "R RIJHO!4R RIJ#R is the Gauss}Bonnet term, b is a parameter, and ;( ) is % IJHO IJ the dilaton potential. For c"1, ;( )"0, b"1 this corresponds to the toroidal compacti"cation

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of the low-energy heterotic string e!ective action. The axion can be set to zero provided that the gauge "eld is purely magnetic. If c"1, b"0, and ;( )"(!1/8) exp(!2 ), then the theory (3.34) can be obtained via compacti"cation of ten-dimensional supergravity on the group manifold [83]. In the static, spherically symmetric case the 2D matter "eld Lagrangian following from (3.34) is

 ¸ "¸ #2cb eA( (Np)(1!NR)!R;( ) ,

7+" p where ¸ is given by Eq. (2.31). If b"0, the "eld equations in the gauge R"r are 7+" w(w!1) , (eA(pNw)"eA(p r





 

(3.35)

(3.36)

1 (rpN )"2cpeA( Nw# (w!1) !rp;( ) , 2r

(3.37)

1 r m" N #eA( Nw# (w!1) #r;( ) , 2r 2

(3.38)

2 (ln p)"r # eA(w . r

(3.39)

We shall consider these equations for a number of special cases. 3.4.1. EYMD theory Let us choose ;( )"0 and b"0. Eqs. (3.36)}(3.39) exhibit then the global symmetry

P # , rPr exp(c ) , (3.40)   which allows one to set (0)"0. The local asymptotic solutions to (3.36)}(3.39) are given by w"1!br#O(r),

m"O(r),




a w"$ 1! #O(r\), r D

" # #O(r\) ,  r

"2cb r#O(r) ,

m"M#O(r\) ,

(3.41) (3.42)

at the origin and at in"nity, respectively. Here D is the dilaton charge. The symmetry (3.40) implies the existence of the conserved quantity Npr+c ln Np!2 ,"C ,

(3.43)

where C is an integration constant. In the regular case one has C"0, which leads to a "rst integral of the "eld equations






2 g "exp ( ! ) ,   c

(3.44)

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where g ,Np. This implies the relation between the dilaton charge and the ADM mass [114]:  D"cM . (3.45) It follows also that for the string theory value, c"1, the string metric ds"e(ds is synchronous. Q Numerical integration of Eqs. (3.36)}(3.39) with the boundary conditions speci"ed by (3.41), (3.42) shows that analogues of the BK solutions exist for any value of c [112,219,35]. These EYMD solitons are also labeled by the node number n of w, and the behaviour of the metric and the gauge "eld is qualitatively the same as in the EYM case. The dilaton is a monotone function. In the limit where cP0 the dilaton decouples and one recovers the BK solutions. In the opposite limit, cPR, the gravitational degrees of freedom decouple after the rescaling cPR,

P /c,

rPcr .

(3.46)

Eqs. (3.36)}(3.39) then reduce to those of the #at spacetime YM-dilaton model: w(w!1) , (e(w)"e( r




(3.47)



(w!1) (r )"2e( w# . 2r

(3.48)

Remarkably, these also admit regular solutions with the same nodal structure as in the gravitating case [218,36]. The dilaton can therefore play a similar role like the Higgs "eld. For all values of c the EYMD solutions have 2n negative modes in the spherically symmetric perturbation sector. Consider the modi"cation of the EYMD solutions due to the Gauss}Bonnet term [113]. It turns out that when b in (3.34) is non-zero and small, the qualitative structure of the solutions does not change. However, if b exceeds some critical value b (n), the solutions cease to exist. One has  b (n)4b (1)"0.37. As a result, for the string theory value, b"1, there are no regular particle  like solutions. For bO0 the scale symmetry (3.40) still exists leading to the conserved quantity analogous to that in (3.43). For regular solutions this implies the equality M"cD, but the relation (3.44) no longer holds. 3.4.2. Gauged supergravity For c"1, ;( )"!()exp(!2 ), and b"0 Eqs. (3.36)}(3.39) are integrable [82,83]. The  action (3.34) corresponds then to the truncation of the N"4 SU(2);SU(2) gauged supergravity. As a result, one can use the supersymmetry tools to derive the following system of "rst order Bogomol'nyi equations: Np"e(\( ,

(3.49)

1#w (w!1) r N" #e( # e\( , 2 2r 8

(3.50)






(w!1) r , r " e\( 1!4e( r 8N




(3.51)



w!1 r , rw"!2w e\( 1#2e( r 8N

(3.52)

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which are compatible with the "eld equations (3.36)}(3.39). The Bogomol'nyi equations are completely integrable and admit a family of globally regular solutions which can be represented in the form ds"a

sinh o +dt!do!R(o)(d0#sin0 du), , R(o)

o R(o)"2o coth o! !1, sinh o

o w"$ , sinh o

(3.53) e("a

sinh o . 2 R(o)

(3.54)

Here o3[0,R), and the parameter a re#ects the presence of the scale symmetry (3.40). These solutions are stable, preserve  of the supersymmetries, and have unit magnetic charge. When  expressed in the Schwarzschild coordinates, the asymptotics of the solutions are r N"1# #O(r), 9a

2r Np"2e("a# #O(r) , 9

r w"1! #O(r) , 6a NJln r,

r , Np"2e(J 4 ln r

(3.55) 4 ln r wJ , r

(3.56)

at the origin and at in"nity, respectively. The geometry is #at at the origin, but it is not asymptotically #at. The space-time manifold is geodesically complete and globally hyperbolic. 3.5. Higher rank groups The simplest generalization of the SU(2) EYM(D) theory to higher gauge groups is the SU(2);U(1) theory [136,111]. However, unless the U(1) "eld vanishes, this does not admit regular particle-like solutions, although black holes are possible. The situation improves for larger groups. The SU(N) case has been studied most of all. The existence of di!erent embeddings of SU(2) into SU(N) leads to several inequivalent expressions for spherically symmetric gauge "elds. Most of these correspond to models with gauge groups being subgroups of SU(N), and there is only one genuinely SU(N) ansatz [333,213]. In the static, purely magnetic case this is given by (2.20). The EYM "eld equations read [214] rNw#2(m!rP)w "(q !q )w , H> H H H  H m"NG#P ,

(3.58)

(ln p)"2G/r ,

(3.59)

(3.57)

where j"1,2, N!1 and q "j(N!j)w!( j!1)(N!j#1)w #2j!N!1 with H H H\ w "w "0. One has  , ,\ 1 , q . (3.60) G" j(N!j)w , P" H H 4r H H

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Solutions to these equations for N"3, 4 have been studied numerically [214,204}207,194,295]; see [240] for the corresponding existence proof. Since the results do not depend considerably on whether the dilaton is included or not, we shall discuss only the pure EYM case. Consider the SU(3) theory, when there are two independent YM amplitudes w and w . The local power-series   solutions read w "1!b r#b r#O(r) ,    w "1!b r!b r#O(r),    a w "$ 1!  #O(r\) ,  r





 

a w "$ 1!  #O(r\) ,  r

m"O(r) ,

m"M#O(r\)

(3.61)

(3.62)

at the origin and at in"nity, respectively. Notice the appearance of the cubic terms in the expansions of the w 's at the origin. The numerical matching with "ve shooting parameters, b , a H H H and M, gives global solutions. For these the behaviour of m and p is similar to that of the BK case, while the gauge "eld amplitudes obey the condition "w "41 and are characterized by two H independent node numbers n and n . If w "w ,w, the equations reduce to those of the SU(2)     case } up to the rescaling rP2r, mP2m. As a result, there are solutions with the node structure (n, n), corresponding to the embedded BK solutions. There are, however, solutions with the same node structure but with w Ow , and they are slightly heavier. Finally, there are solutions with   arbitrary (n , n ), where n On . The novel feature is that one of the numbers can vanish, while     the corresponding amplitude is not constant. The lowest value of the mass M(n , n ) is   M(1, 0)"2;0.653, then comes M(2, 0)"2;0.811, followed by the doubled mass of the embedded n"1 BK solution, M(1, 1)"2;0.828. The next is the mass of the genuine SU(3) solution, M(1, 1)"2;0.847, and so on. In the limit nPR the (0, n) and (n, n) sequences display the oscillating behavior in the inner region, outside of which the geometry tends to the extreme RN solution with magnetic charges P"(3 and P"2, respectively. For comparison, in the SU(2) case the limiting solution is characterized by P"1. If the dilaton is included, then the limiting solution in the exterior region corresponds to the extreme dilatonic black holes with magnetic charge P. In the SU(4) case the nodal structure is determined by (n , n , n ) [194]. The diagonal sequence    (n, n, n) contains three di!erent types of solutions. The "rst one, with w "w "w , corresponds to    the nth BK solution rescaled by the factor (10. For solutions of the second type two of the three w 's H are equal, while for those of the third type all the w 's are di!erent. For nPR these solutions H approach in the exterior region the extreme black hole con"guration with the charge P"(10. Other sequences tend in a similar way to the extreme solutions with other charges, whose values can be classi"ed using the algebraic structure of SU(4). A similar classi"cation can also be carried out in the general SU(N) case [207]. One should have in mind, however, that for "nite n the regular solutions are always neutral. For nPRthey tend to a union of two solutions, one of which is nonasymptotically #at and exists in the interval 04r4P, while another one is the extreme RN solution in the region r'P. For the regular SU(N) solution with dilaton the equality D"cM still holds. All known regular EYM solutions for higher gauge groups are unstable [71,73].

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3.6. Axially symmetric solutions The EYM(D) "eld equations in the axially symmetric case are obtained by using the ansatz (2.22) for the gauge "eld and that in (2.21) for the metric. These are rather complicated partial di!erential equations for the seven real amplitudes in (2.21) and (2.22). Due to the residual gauge invariance of (2.22), the number of independent amplitudes is six. The equations admit interesting solutions generalizing the BK particles to higher winding numbers [199,195,201]. Similar generalizations exist also in #at space for monopoles [269] and sphalerons [196,197]. The basic idea can be illustrated as follows. The axial ansatz (2.22) for the gauge "eld covers, in particular, the spherically symmetric case. Speci"cally, choosing in (2.22) l"1 and i w(o, z)" +1#w (r)#e\ 0(1!w (r)), , 2 w (r)!1 0 a (o, z)#ia (o, z)" e\ M X r

(3.63)

with z#io"re 0 and w (r)"w H(r) and omitting the tilde sign, the expression reduces to the spherically symmetric ansatz (2.13) with a "a "0. In a similar way, the axially symmetric line  P element (2.21) can be used in the spherically symmetric case. As a result, given a static, spherically symmetric solution one obtains a solution for the axial EYM equations with l"1. The next step is to use this solution as the starting point in the numerical iteration scheme with l"1#dl. The iterative "eld con"gurations must satisfy certain regularity conditions at the symmetry axis and at in"nity. The iterations converge, and repeating the procedure one obtains in this way solutions for arbitrary l. The physical values of l are integer. This "nally gives the generalized BK solutions characterized by a pair of integers (n, l), where n is the node number of the amplitude w in (2.22). For lO1 the solutions are not spherically symmetric. The contours of equal energy density ¹ are 2-torii and squashed 2-spheres. The mass of the solutions, M(n, l), increases with l. For  example, for n"1 one has M(1, 1)"0.828 (the BK solution), M(1, 2)"1.385, M(1, 3)"1.870; while for n"2 one "nds M(2, 1)"0.971, M(2, 2)"1.796, M(2, 3)"2.527 and so on [201]. If the winding number l is "xed while the node number n tends to in"nity, then the mass approaches the value M(R, l)"l. In this limit the solutions exhibit a complicated oscillating behaviour in the interior region, outside of which region the con"gurations approach the extreme RN solution with the magnetic charge l. One can think of the solutions with higher winding numbers as describing nonlinear superpositions of l BK particles aligned along the symmetry axis. This is supported by the fact that their Chern}Simons number, which is computed in the same way as for the BK solitons, is equal to l/2. The solutions can be generalized by including a dilaton "eld for an arbitrary value of the coupling constant c. It turns out that the metric-dilaton relation (3.44) holds in the axially symmetric case too. This implies the relation (3.45) between mass and the dilaton charge. For cPR gravity switches o!, but the solutions survive in this limit [198]. Although the corresponding stability analysis has not been carried out yet, it is very likely that all these EYMD solitons are unstable. It remains unclear whether the procedure described above gives all solutions in the static, purely magnetic, axially symmetric case. It is possible that con"gurations violating the circularity

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condition can exist as well. In particular, it is unclear whether the BK solutions exhaust all possibilities for l"1.

4. Non-Abelian black holes The intuitive physical idea behind the original no-hair conjecture [277] apparently was that only exact physical symmetries like gauge symmetries can survive in a catastrophic event like a gravitational collapse. Associated with gauge "elds there are conserved charges, which can be measured using the Gauss #ux theorem. On the other hand, all physical quantities which are not coupled to the corresponding gauge "elds, like baryon number, are not strictly conserved. As a result, they either disappear during the collapse or become unmeasurable. The proof of this conjecture was given for a number of special cases. In such a proof it is not the process of collapse that is usually considered but the result of it } a stationary black hole spacetime with certain matter "elds. It turns out that black holes cannot support linear hair for scalar [86,25,303], spinor [157,304] and massive vector [24,23] "elds; see [149,22,21,242,164,239] for a more recent discussion. A similar proof was given also for some non-linear matter models [4,22,242,162,163,298]. An interesting con"rmation of the no-hair conjecture was found in the fermionic sector of the N"2 supergravity, where there are black holes with an external spin- gravitino "eld [6]. The  latter, being subject to the Gauss law, gives rise to the fermionic supercharge of the black hole, and this does not contradict the conditions of the conjecture. Note that in the bosonic sector supergravity black holes generically support external scalar "elds associated with dilaton and moduli "elds [138], and these are not subject to a Gauss law. A similar example is provided by black holes with conformal scalar "eld [42,26]. However, in all these cases the scalars are completely parameterized by black hole mass and gauge charges and hence cannot be considered as independent } sometimes they are called secondary hair. In addition, the Abelian supergravity black holes are subject to the uniqueness theorems [62,60]. All this solidi"ed the belief in the universal validity of the no-hair conjecture. The "rst example of the manifest violation of the conjecture was found in the EYM theory. In the early study of the EYM system it was observed that for any solution (g , A ) of the IJ I Einstein}Maxwell theory there is a solution (g , A ) of the EYM theory with the same metric and IJ I A "TA (x), where T belongs to the Lie algebra of the gauge group [340]. Therefore, either I I directly solving the EYM equations or starting from the Kerr}Newman con"gurations, a number of embedded U(1) black hole solutions for the EYM "eld equations had been found [13,87,330,261,184,189}191]. These were called colored black holes [261]. In the spirit of the no-hair and uniqueness conjectures, it was assumed for some time that these solutions exhaust all stationary EYM black holes. A partial con"rmation of this assumption was given by the

 Hairy black holes in the Einstein}Skyrme model had been described in [228] before the EYM solutions were found, which paper, however, has remained almost unknown.  Unfortunately, despite the fact that they are neutral, the same name stuck later to the non-Abelian EYM black holes.

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`non-Abelian baldness theorema proven for the SU(2) gauge group [133,121,40]: all static EYM black hole solutions with "nite colour charges are the embedded Abelian ones. This assertion, however, left open a possibility to have essentially non-Abelian EYM black holes in the neutral sector, and such solutions were found in [324,215,33]. For these black holes the only parameter subject to the Gauss law is their mass, and for a given value of the mass there can be several di!erent solutions. The no-hair conjecture is therefore violated. The existence of one example triggered a broad search for other hairy black holes in various non-Abelian models. This has led to their discovery inside magnetic monopoles [258,222,54,56], Skyrmions [117,38], as well as in a number of other systems. For the reader's convenience we list here the families of hairy black holes known up to date: 1. Black holes with pure Yang}Mills xelds. Apart from static and spherically symmetric SU(2) EYM black holes [324,326,215,33], this family includes also static and axially-symmetric solutions [200,202], stationary and non-static con"gurations [327,69], solutions for higher gauge groups [136,204,207], and the generalizations including a dilaton and the higher curvature terms [112,219,306,111,257,309,302,206,205,194,295,185,187,7]. 2. Black holes with Yang}Mills}Higgs xelds. These are magnetically charged solutions with the triplet Higgs "eld [258,222,54,56,5], their analogues for a more general matter model as well as the generalizations to the non-spherically symmetric case [273,271,332], and the neutral solutions with the doublet Higgs "eld [149,231,308,301]. 3. Skyrme black holes [228,117,172,38,203]. In view of this variety of hairy black holes one can wonder as to how to reconcile their existence with the original arguments behind the no-hair conjecture. One possibility, which can probably be applied for unstable solutions, is to argue that hairy black holes cannot appear via gravitational collapse. Another option, which is suggested by the examples of stable black holes inside solitons, is to assume that the topological charge cannot disappear during the collapse. This is supported by the fact that in all known examples the size of black holes inside solitons is bounded from above, otherwise no hairy solutions exist. It seems then that a black hole cannot swallow up a topological object with the typical size exceeding that of the black hole. 4.1. EYM black holes: the exterior region The EYM black hole solutions satisfy the same equations as the regular BK solutions [324,326,317,215,33]:




Nw#



2m (w!1) w w(w!1) ! " , r r r r

(w!1) m"Nw# . 2r

(4.1) (4.2)

The boundary conditions at in"nity are still given by (3.6)




a w"$ 1! #O(r\), r

m"M#O(r\) ,

(4.3)

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which are now supplemented with the requirement that there is a regular event horizon at r"r '0:  N(r )"0, N(r)'0 for r'r . (4.4)   The regularity assumption implies that all curvature invariants at r"r are "nite. The local  power-series solution to (4.1) and (4.2) in the vicinity of the horizon reads w (w!1) (r!r )#O((r!r )) , (4.5) w"w #      N  (w!1) 1 (r!r )#O((r!r )) , (4.6) N" 1!    r r   It follows from the second condition in (4.4) that r'(w!1).   For a given r '0, Eqs. (4.1)}(4.6) de"ne a non-linear boundary value problem in the interval  r 4r(R. The following simple argument is in favour of the existence of a non-trivial solution at  least for r <1. Passing everywhere to the radial coordinate x,(r!r )/r , some terms in the    equations acquire the factor 1/r and can be omitted for large r , if only they are bounded for x50.   As a result, the Einstein equation (4.2) decouples and admits the solution m"r /2 corresponding  to the Schwarzschild metric. The remaining YM equation (4.1) reads









xw  w(w!1) " , x#1 (x#1)

(4.7)

where the di!erentiation is with respect to x. The following solution to this equation is known [48]: 2x!1!(3 w"$ . 2x#5#3(3

(4.8)

This describes a regular YM hair on the Schwarzschild background. This shows that the non-trivial solutions to the full system of equations are likely to exist, at least for r <1.  In order to obtain the solutions to the full problem (4.1), (4.2) the procedure is to numerically extend the asymptotics (4.3), (4.5) to the intermediate region. The matching conditions for w, w and m determine then the three parameters M, a and w in (4.3) and (4.5). As a result, for any given  r '0, one "nds a sequence of the global solutions in the interval r 4r(R. These are   parameterized by the node number n of w. The non-Abelian black holes can therefore be labeled by a pair (r , n), where r '0, n"1, 2,2 . The existence of these solutions has been established in   [292,55]. For any (r , n) the behaviour of the amplitudes w, m and p in the exterior region is qualitatively  similar to that for the regular BK solutions (see Fig. 7). The amplitude w starts from some value 0(w (r )(1 at the horizon and after n oscillations around zero tends asymptotically to (!1)L. L  Considerations similar to those used in the regular case show that one has "w(r)"41 everywhere outside the horizon. The metric functions m and p increase monotonically with growing r from m(r )"r /2 and p(r )"p (r ) to m(R)"M (r ) and p(R)"1, respectively. In the asymptotic    L  L  region, rPR, the geometry is Schwarzschild with the mass M,M (r ) depending on r and n. L  

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Fig. 7. On the left: the amplitudes w, m and p for the non-Abelian black hole solutions with r "1. On the right: the  e!ective charge Q and the radial energy density r¹. 

Since the YM "eld strength decays asymptotically as 1/r, the Gauss #ux integral vanishes and the node parameter n cannot be associated with any kind of YM charge of the black hole. One can qualitatively distinguish between two regions in the parameter space of the solutions: (a) r '1, and (b) 0(r 41. This is due to the comparison of the `barea mass of the black hole and   its total ADM mass. The bare mass is m(r )"r /2. Since m(R) can be thought of as the total energy,   including the gravitational binding energy, con"ned in the region r(R, the bare mass is the total energy trapped inside the event horizon. Using (4.2), the ADM mass M,m(R) can be represented as the sum of the bare mass and the `dressinga mass



 r r¹ dr , (4.9) M" #  2 P the second term on the right being the contribution of the matter distributed outside the horizon. (a) r '1: In this case the matter term in Eq. (4.9) is small, M (r )+r /2, and the YM "eld almost  L   does not in#uence the geometry outside the horizon. The YM "eld energy increases slightly with growing n, the ADM mass being (see Fig. 8) 1 r . (4.10) M (r )4M (r )(M " #   L   2 2r  Here M coincides with the mass of the RN black hole with unit charge. The following approxima tion is very good for n52 and r51: M (r )+M (see Fig. 8). This is due to the fact that for large L   values of n the amplitude w oscillates in a small vicinity of zero. The charge function Q de"ned by

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Fig. 8. The ADM mass M (r ) and the temperature screening coe$cient b (r ) versus the event horizon radius. For n"0 L  L  solutions are Schwarzschild.

Eq. (3.8) is then close to unity, while metric is approximately RN. For nPRthe metric converges pointwise to the RN metric. For r <1 the matter term in Eq. (4.9) is of the order of 1/r and the back reaction of the YM "eld   on the spacetime geometry becomes negligible. The solutions in this limit reduce to the bound states of the YM "eld on the "xed Schwarzschild geometry. These are described by Eq. (4.7), whose solution for n"1 is given by Eq. (4.8), while those for n'1 are known only numerically. (b) 0(r 41: As r decreases, the e!ect of the YM "eld on the geometry becomes more and   more pronounced. For r &1 the `barea and `dressinga masses are of the same order of magnitude,  such that M (r )&r . For solutions with r (1 the role of the YM "eld is dominant. The solutions L    then look like small black holes dressed in the YM `coata. As r approaches zero the external "eld  con"guration becomes more and more close to that for the regular BK soliton with the same value of n. The ADM mass for r (1 varies within the following range:  M "0.8284M (r )(M (r )"1 . (4.11)  L    Here M is the mass of the ground state BK solution (see Fig. 8). For any r (1, M (r ) converges   L  to the unit value for nPR. In this limit an in"nite number of nodes of w accumulates near r"1. As a result, similar to the situation in the BK case, the solutions tend for nPR to a union of an oscillating solution in the interval r (r(1 and the extreme RN solution for r'1.  In the limit r P0 the event horizon shrinks to zero and the black hole solutions converge  pointwise for r'0 to the regular BK con"gurations. The "rst law of black hole physics for the EYM black holes was considered in [170,169], the thermodynamics was discussed in [252,306]. The Hawking temperature of the EYM black holes

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can be computed by analytically continuing the metric to the imaginary time and requiring the absence of the conical singularity. The result is ¹"i /2p, where the surface gravity i "p N /2     can vanish only for the extreme RN solution [289,290]. Using (4.6) one obtains 1 , (4.12) ¹"b (r ) L  4pr  where the second factor on the right is the temperature of the Schwarzschild black hole with radius r , while b (r )"p(r )(1!(w!1)/r) (see Fig. 8). The presence of b (r ) in this formula leads to  L     L  the existence of a (narrow) region in the parameter space, r (n)(r (r (n), for which the    temperature increases with growing M and hence the speci"c heat is positive [306]. All EYM black holes are unstable and have 2n negative modes in the spherically symmetric perturbation sector (see Section 5 below). Note that for black holes the analogy with sphalerons does not exist. The reason is very deep } black holes are associated with thermal states and not with excitations over pure vacuum states. For black holes one cannot de"ne classical YM vacua even formally, since the topology is di!erent and pure gauge "elds are no longer characterized by integer winding numbers. 4.2. EYM black holes: the interior structure From the conceptual point of view the problem of "nding the solutions in the interior region r(r is simple. One should merely integrate the "eld equations (4.1) and (4.2) inwards starting  from the event horizon, since the boundary conditions (4.5) at r are already speci"ed by the  exterior solutions. The result, however, turns out to be quite bizarre [115,59]. Therefore, before passing to the numerical analysis, some preliminary analytical considerations can be useful. 4.2.1. Special solutions It turns out that the interior solutions generically do not have an inner horizon. Indeed, supposing that such a horizon exists at some r (r , the solution in the vicinity of r is given by \  \ (4.5) with r and w replaced by r and some w , respectively. One can extend this solution   \ \ outwards to match the one that propagates inwards from r . Now, in order to ful"ll the three  matching conditions for w, w and m one needs at least three free parameters. At the same time, only r and w can be used for this, since the solution coming from r is completely speci"ed by the \ \  boundary conditions at r . One can use r as the third matching parameter. As a result, it might   happen that for some special values of the event horizon an inner horizon exists too. However, for an arbitrary r the matching is impossible. This shows that the non-Abeian black holes generically  do not possess inner horizons [128,130,58,57] and therefore have a spacelike singularity, as suggested by the strong cosmic censorship hypothesis. Next, one can study the power series solutions to (4.1) and (4.2) in the vicinity of the singularity at r"0. One "nds three distinct types of local solutions, but none of them are generic. First, there are local solutions with a Schwarzschild type singularity:






8b!3 r #O(r) , 30m  m"m (1!4br#8br)#2br#O(r) ,  w"$ 1!br#b

(4.13) (4.14)

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with arbitrary m and b. Since the inner horizon is generically absent, a solution of this type should  match the one propagating inwards from r . However, the number of free parameters is not enough  for this. Next, there are solutions with a RN-type singularity: w  r#cr#O(r) , w"w #  2(1!w) 

(4.15)

(w!1) m"!  #m #O(r) .  2r

(4.16)

Here the number of free parameters is three: m , c, and w O$1, which agrees with the number of   the matching condition. However, one has NP#R for rP0, such that N is positive near the singularity. This requires the presence of an inner horizon, which generically does not exist, hence this type of singularity is also generically impossible. Finally, there is a branch of local solutions containing only one free parameter, w [115,59]:  w  r#O(r) , w"w $r!  2(w!1) 

(4.17)

(w!1) m"  $2w (w!1)#O(r) .   2r

(4.18)

For these the singularity is space-like, and NJ!(w!1)/r as rP0, which formally corres ponds to a RN metric with `imaginary chargea. The geometry is conformal to R;S. It is worth noting that solutions of this type were found [259] in the study of the phenomenon of mass in#ation [262]. Summarizing, none of local power-series solutions contain enough free parameters to ful"ll the matching conditions. Note also that the possibility of having a singularity at a "nite value of r can be ruled out [291]. To recapitulate, solutions in the interior region do not generically possess inner horizons and do not exhibit a power-law behaviour in the vicinity of the singularity. One can wonder then how these solutions look like. The answer is provided by a numerical analysis. First of all, it turns out that the special solutions described above can indeed be obtained by "ne tuning of the parameters } apart from those in (4.17), (4.18). For n"1 and r (1)"0.6138 the singularity is Schwarzschild  type, such that there is no inner horizon. For all n'2 there are special solutions which have RN-type singularity with an inner horizon at some r (n). For example for n"2 one has \ r (2)"1.2737 and r (2)"0.0217; while for n"3 one "nds r (3)"1.0318 and r (3)"0.0894. It  \  \ seems that there are also other special solutions with the Schwarzschild-type singularity, but these are very di$cult to obtain numerically [59]. 4.2.2. Generic case Coming to the generic behaviour of the solutions in the interior region, this turns out to be quite unusual. Speci"cally, choosing an arbitrary external solution, with n"r "1, say, and integrating 

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Fig. 9. On the left: the mass function m for the interior solutions with r "1 and n"1, 2, 3. The amplitude w is shown for  the solution with n"3. On the right: the second oscillation cycle for the n"3 solution.

inwards from the event horizon, the metric functions m and p and the derivative w exhibit in the interior region violent oscillations, whose amplitude and frequency grow without bounds as the system approaches the singularity. At the same time, since everything happens at the very short scale, the amplitude w is almost constant in the internal region. During a typical oscillation cycle the qualitative behaviour of the solution is as follows (see Fig. 9). The kth cycle (k"1, 2,2) starts at the kth local minimum of the mass function m at some r where m(r ) is very close to zero. Then, as r decreases from r to some R (r , m grows I I I I I exponentially fast until it reaches a very large value at R . At the same time p decreases by many I orders of magnitude. In the region r(R the mass function reaches a horizontal plateau where it is I almost constant, m+M ,m(R ); similarly for p: p+p ,p(R ). The plateau stretches as far as I I I I many orders of magnitude of r towards the singularity, during which period the geometry is approximately Schwarzschild with a very large mass M . Then, at the end of the plateau, I m catastrophically falls down reaching a very deep minimum at some r , after which the next I> oscillation cycle starts. The oscillation amplitude for each subsequent cycle is exponentially large compared to that for the preceding one. As a result, it is extremely di$cult to follow numerically more than the "rst two or three cycles. It turns out, however, that a simple analytic approximation allows one to qualitatively understand the generic behavior of the system for rP0 [115,59]. Such an approximation is based on the truncation of the "eld equations (4.1) and (4.2) due to the numerical observations that for r;r one has  w+const.O$1 ,

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(w!1) <1 , r (w!1)w <w(w!1) . r

(4.19)

Neglecting in Eqs. (4.1) and (4.2) the small terms compared to the large ones the equations reduce to the following dynamical system: x "eW!1 , y "1#eW!2eV ,

(4.20)

where eV"w, eW"!(w!1)/Nr

(4.21)

and the di!erentiation is with respect to q"!ln(r/r ), with r being a constant. We are interested   in the behaviour of the solutions for qPR. First of all, let us notice that there is only one critical point of the system, (x, y)"(0, 0), which is an attractive center for qP!R. As a result, the behaviour of the solutions for large and negative values of q is known: the trajectories starting close to the critical point spiral outwards when q increases (see Fig. 10). Now, the crucial fact is that this spiraling motion can never stop. One can show that, starting from any point on the plain, the

Fig. 10. The solution to the dynamical system (4.20).

 The sign of w is de"nite for r;r , and so one can choose w'0 utilizing the symmetry wP!w. 

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trajectory cannot escape to in"nity for "nite q and performs the full revolution around the center within "nite q. The spiraling behaviour is therefore generic and persists forever. Note that the critical point around which the spiraling occurs corresponds to w"w , N"!(w!1)/r. For   small r this agrees with the local solution (4.17), (4.18). Dynamically, however, this solution cannot be reached. Indeed, the trajectories approach the critical point only for q"!ln(r/r )P!R,  such that rP#R, contradicting the assumption that r is small and that r(r .  The solution to the dynamical system (4.20) shown in Fig. 10 illustrates nicely the various phases of a typical oscillation cycle described above. After each revolution, the picture in Fig. 10 repeats itself at an exponentially blown up scale [115]. The revolutions go on forever and the system approaches the singularity via an in"nite sequence of more and more violent oscillation cycles. At the beginning of each cycle N is very close to zero, such that the system exhibits an in"nite sequence of `almosta inner horizons, but the true Cauchy horizon never appears. Note that the oscillatory nature of the solution resembles somewhat the well-known situation in some cosmological models [27]. However, the solutions are not chaotic. This follows from the existence of the e!ective description in terms of the two-dimensional dynamical system, since chaos cannot occur in two dimensions. Inside black holes, a similar oscillatory behaviour was observed in [259] for a homogeneous mass in#ation model with radial null radiation. The values of amplitudes and periods of the oscillations can be estimated analytically [131]. The essential quantities for this are x "(r /R )<1, where r and R correspond to the beginning of I I I I I the kth cycle and the kth plateau, respectively. One can show that the x 's obey the following I recurrence relation: (4.22) x "eVI/x , I I> thus constituting an exponentially diverging sequence. In terms of x one has I r /r "x e\VI , (4.23) I> I I which can also be viewed as the ratio of the neighboring oscillation periods, since r  increase in M and decrease in p can be expressed as I I M /M "eVI/x , p /p "e\VI , (4.24) I I\ I I> I respectively. One can see that the ratios of the oscillation amplitudes and frequencies for the neighbouring cycles are exponentially large, where the argument of the exponent grows exponentially for each subsequent cycle. Note "nally that the exterior structure of the EYM solutions does not change considerably after adding some additional matter, like a dilaton or a Higgs "elds, say. However, the solutions in the interior region change completely [131,59,344,127,130,129,280] and exhibit a regular power-law behaviour near the singularity. In this sense the oscillatory character of the interior solutions described above distinguishes the EYM black holes among all other known hairy black holes. Note

 This opens a possibility for numerically integrating the equations by making a logarithmic substitution of the variables after each cycle [344].

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also that, although the EYM black hole solutions in the exterior region are unstable, their interior oscillatory structure is stable with respect to non-linear spherically symmetric perturbations [116]. 4.3. Non-Abelian dilaton black holes The EYM non-Abelian black holes can be generalized to include a dilaton [112,219,306]. The "eld equations are then given by (3.36)}(3.39). The boundary conditions at in"nity are speci"ed by (3.42), while at the horizon one has w"w #w x#O(x), " #  x#O(x), N"N x#O(x) (4.25)      with x"r!r , and w ,  , N being determined by r , w and . In the black hole case the scale        transformations (3.40) change r . However, since the solutions exist for any r and , one can set   

"0 without loss of generality. The numerical integration gives for any c(Ra family of black  hole solutions parameterized by (r , n) as in the c"0 case. The behaviour of w, m, and p is  qualitatively the same as for the EYM solutions, while the dilaton is a monotone function. The conservation law following from the scaling symmetry exists in the black hole case too, but now the integration constant in (3.43) does not vanish and is given by C"cp N r. As a result, the   relation (3.45) no longer holds, but instead one "nds D"c(M!(1/4p) i A) , (4.26)  where i "p N /2 is the surface gravity and A"4pr is the area of the event horizon.     The thermodynamics of dilaton black holes depends on value of the dilaton coupling constant c [306]. For small values of c there is a region in the parameter space for which the speci"c heat of the solutions is positive. For large c this region shrinks to zero. The stability behaviour of the solutions, on the other hand, do not depend on c: the number of negative modes is the same for all values of c. What is entirely di!erent from the EYM case is the behaviour of solutions in the interior region. First of all, for cO0, inner horizons cannot exist even after "ne tuning of the parameters [280]. Indeed, the expression in (3.43) reduces at the event horizon r to C"cp N r, which is positive. At    the inner horizon r one has C"cp N r , which is negative. This contradicts the fact that C is \ \ \\ constant, and hence inner horizon cannot exist. The numerical analysis reveals the following characteristic picture in the interior region, which is generic for all cO0: the mass function m, after some oscillations near the horizon, exhibits the power-law behaviour all the way down to the singularity [127,280]. The power-series solution in the vicinity of the singularity for cO0 is generic: (4.27) w"w #ar\H, m"kr\H, "c#ln(r\H), p"p rH ,   since it contains the maximal number of free parameters: w , a, c, k, j, and p . As c decreases, the   solutions develop more and more oscillation cycles in the interior region. However, as long as cO0, the oscillations are always replaced by the power-law behaviour for rP0. This can be illustrated as follows [280]. For c"0 the interior oscillating solution can e!ectively be described by the two-dimensional dynamical system (4.20), whose trajectories evolve in the plane (x, y). For cO0 there is an additional degree of freedom due to the dilaton, and the reduced system becomes three-dimensional. As a result, after a number of revolutions in the vicinity of the plane, which

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corresponds to several oscillation cycles, the trajectory drifts away towards a critical point of the equations that lies outside the plane. This changes the character of the solution. The EYMD black holes can be generalized in the context of the theory with the Gauss-Bonnet term [187,309]. The string theory value of the coupling constant in (3.34), b"1, is allowed for black holes. The solutions are parameterized by three numbers: ( , r , n). This is because the event   horizon for bO0 exists only if r 5r '0, which implies that one cannot use the scale symmetry to   set "0. One has r &b exp(2c ). For n"0, when the YM "eld strength vanishes, the solutions    describe black holes with dilaton hair [185,186,7]. 4.4. Higher gauge groups and winding numbers The SU(2) non-Abelian black holes admit generalizations to higher gauge groups [136,111,114,214,204}207,195,194,295]. These exist for all values of the dilaton coupling constant c. In the maximal SU(N) case, when all the gauge "eld amplitudes w are non-trivial, the solutions are H neutral and can be parameterized by (r ,n ), where n is the node number of w . The regular limit is  H H H recovered for r P0. The solution space exhibits bifurcations in this limit, since for r P0 there   exist di!erent solutions with the same node structure which merge as r grows. If some of the n 's  H tend to in"nity, the solutions approach the RN (for c"0) or charged dilaton (for cO0) black hole solutions. One of the new features is that when some of the w 's vanish identically, the U(1) charges arise. H The non-Abelian baldness theorem hence does not generalize to higher gauge groups. For example, setting in the SU(3) equations (3.57)}(3.59) w "0, the solutions for w show the usual nodal   structure. However, since w does not obey the boundary condition in (3.62), there is a magne tic charge P"(3 for all n . This is re#ecting in the behaviour of N for rPR:  N"1!2M/r#P/r#O(1/r). For higher gauge groups one can obtain in this way a variety of charged solutions, in particular, solutions with an electric charge (for a more general ansatz for the gauge "eld). For charged EYM black holes the event horizon radius is bounded from below, r 5P, since otherwise the singularity is naked. The limit where r P0 for the charged solutions is   therefore impossible. Solutions with r "P are extreme black holes with degenerate horizon. For  charged solutions with dilaton the r P0 limit is allowed, but this leads to the extreme dilaton  black holes and not to regular particle-like solutions. For such extreme dilaton black holes the metric-dilaton relation Np"e(\(A is restored. At the origin these con"gurations approach the extreme Abelian solutions with the same value of charge [138], while showing the usual nodal structure for r51. The SU(2) EYM(D) black holes admit also axially symmetric generalizations [200,202]. By far, these give the only known explicit example of static and non-spherically symmetric black holes with non-degenerate horizon. The solutions can be obtained by extending the spherically symmetric black hole con"gurations to the higher values of the winding number l, similarly to the procedure in the regular case. The solutions are asymptotically #at and have a regular event horizon, but for l'1 they are not spherically symmetric and the event horizon is not a sphere but a prolate ellipsoid. Speci"cally, expressing coordinates o, z used in the axial line element (2.21) in terms of r, 0 as z"r cos 0, o"r sin 0, the event horizon is de"ned by the relation r"r , where  g (r )"0. The horizon hence has the S topolgy, but geometrically it is not a sphere, since its  

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circumference along the equator turns out to be di!erent from that along a meridian. The surfaces of constant energy density are either ellipsoidal or torus-like. It is interesting that the energy density is not constant at the horizon and depends on 0. However, the surface gravity i , where  (4.28) i "! ggGIR g R g G  I    is constant at the horizon. If a dilaton is included, then the relation (4.26) between mass, dilaton charge and the event horizon parameters holds also for l'1. The solutions are characterized by (r , l, n). Here l and n are the winding number and the node number of the amplitude w in (2.22),  respectively. The deviation from spherical symmetry increases with growing l, but decreases for large n. As nPRthe solutions tend to the Abelian spherically symmetric black hole solutions. In the limit of shrinking horizon, r P0, the solutions reduce to the regular axially symmetric  EYM(D) solitons. More details and interesting pictures can be found in [202].

5. Stability analysis of EYM solutions The stability issue for the BK solitons and EYM black holes has been extensively studied [297,296,343,342,325,323,321,220,70,71,73], both perturbatively and at the non-linear level. A novel feature arising in this analysis is the fact that, unlike the situation in the (electro)-vacuum case, a non-trivial temporal dynamics for EYM "elds exists already in the spherically symmetric sector. The Birkho! theorem thus does not apply [19,70], and the physical reason for this can be deduced from the `spin from isospina phenomenon [178,158]. The spin and isospin of the gauge "eld can combine together giving zero value for the sum, in which case the vector YM "eld e!ectively behaves as a scalar "eld. As a result, spherical waves exist in the theory. It turns out then that all known regular and black hole solutions in the EYM theory with the gauge group SU(2) are unstable with respect to small spherically symmetric perturbations [325]. This is true also for the solutions for higher gauge groups [71,73]. The numerical analysis indicates that the non-linear instability growth results in the complete dissipation of the initial equilibrium con"guration [343]. The energy is then partially carried away to in"nity by spherical waves and partially collapses to the center. It turns out that the n"1 BK solution can play the role of the intermediate attractor in the gravitational collapse of the YM "eld [88]. Near the boundary in the initial data space between data which form black holes and data which do not there is a region characterized by a mass gap. The minimal black hole mass for data from this region is equal to the mass of the n"1 BK solution. This is reminiscent of a "rst-order phase transition. There is also another region in the parameter space characterized by the echoing phenomenon and the scaling behavior for the black hole mass, which can now be arbitrarily small [88]. This is explained by the existence of another intermediate attractor in the problem. The corresponding solution to the EYM equations is time-dependent. It describes the situation when there is a constant ingoing #ux of the YM radiation which comes from in"nity, completely re#ects from the origin, and then goes back to in"nity. The system balances just on the verge of collapse but it does not actually collapse due to "ne tuning of the parameters. The approximate description of this solution was found in [151]. In what follows we shall consider only small, spherically symmetric perturbations for the BK solitons and non-Abelian black holes. The complete set of perturbations splits into two parity

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groups. The even-parity sector is obtained via perturbing the non-vanishing gauge and metric amplitudes in Eqs. (2.49) and (2.50): wPw#dw(t, r), mPm#dm(t, r), pPp#dp(t, r) .

(5.1)

The odd-parity group consists of perturbations of those amplitudes in the Witten ansatz (2.13) that vanish for static, purely magnetic background solutions. These are a , a , and Im w:  P a Pda (t, r), a Pda (t, r), Im wPd(Im w) . (5.2)   P P The parity of the modes is determined by Eq. (2.16). Since the background solutions are parityinvariant, the two groups of perturbations decouple at the linearized level. 5.1. Even-parity perturbations Substituting (5.1) into the EYM equations (2.44)}(2.48) and linearizing with respect to dw, dm, and dp gives the perturbation equations. It turns out that the gravitational amplitudes dm and dp can be expressed in terms of dw, which leads to a single SchroK dinger-like equations for dw. The procedure is as follows. Linearizing Eq. (2.47) one arrives at dm "2Nwdw with N"1!2m/r, whose solution with the appropriate boundary conditions is dm"2Nwdw .

(5.3)

Similarly, linearizing Eq. (2.46) one obtains d(p/p)"4wdw/r .

(5.4)

Finally, the Yang}Mills equation (2.45) contains after the linearization dw, dm and d(p/p), where the last two terms can be expressed using (5.3) and (5.4). As a result, one arrives at [297]






p  d 3w!1 ! #pN #2 M g"ug , (5.5) p do r M where dw(t, r)"exp(iut) g(o), and the functions w, N, and p refer to the background solution under consideration. The `tortoisea radial coordinate o is de"ned by do/dr"1/pN ,

(5.6)

where o3[0,R) for solitons and o3(!R,R) for black holes. It is worth noting that Eq. (5.5) can also be obtained via computing the second variation of the mass functional M(1, p/2) in Eq. (3.23) with respect to w [318,316,68]. The "rst variation vanishes on shell, while the second one is






d dM" dw(o) ! #; dw(o) do , do

(5.7)

where the potential ; coincides with the one in Eq. (5.5). Eq. (5.5) determines a regular SchroK dinger eigenvalue problem on a line or semi-line. The potential is bounded for black holes, while for solitons it can be represented as a sum of a bounded piece and the p-wave centrifugal term 2/o (see Fig. 11). The numerical analysis reveals the existence of n bound state solutions to Eq. (5.5) with u(0 for the nth BK or EYM black hole background (with any r ) [297,296]. This implies that the solutions are unstable. The non-linear instability 

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Fig. 11. Potentials ; and ; in the SchroK dinger eigenvalue problems (5.5) and (5.14), respectively.  

growth for the BK solitons has been analyzed numerically [343,342]. It has been found that a part of the initial static con"guration collapses forming a small Schwarzschild black hole, while the rest radiates away to in"nity. It is worth noting that for the EYM black holes with n"1 the negative eigenmode gives rise to the gauge "eld tensor dF that is unbounded at the horizon [34]. However, it turns out that this mode is nevertheless physically acceptable, because the divergence can be suppressed by making wave packets with real frequency eigenmodes [41,329]. 5.2. Odd-parity modes The odd-parity perturbation sector also contains negative modes, whose existence can be established [137,47,71,325,73] without even resorting to numerical analysis. The number of such modes can also be determined analytically [323,321]. For the BK solitons the existence of the odd-parity negative modes directly relates to the sphaleron interpretation of the solutions. This is due to the fact that the Chern}Simons number of the gauge "eld changes for pulsations of the type (5.2). The even-parity perturbations, on the other hand, preserve the Chern}Simons number. In order to see this it is instructive to compute the *FF invariant for the spherically symmetric gauge "eld (2.13): pr tr F FIJ"(a ("w"!1)#p p !p p ) !(a ("w"!1)#p p !p p ) ,           2 * IJ

(5.8)

where p ,Re w and p ,Im w. This vanishes if a "a "Im w"0.    P The perturbation equations in the odd-parity sector are obtained by linearizing the Yang}Mills equations (2.44) and (2.45) with respect to da , da and d(Im w), which gives a system of three linear  P

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equations. For time-dependent perturbations one can impose the temporal gauge condition da "0. As a result, the two independent perturbation equations can be represented in the  SchroK dinger form HK W"uW .

(5.9)

Here, introducing p( ,!id/do and c ,(2Np/r, the Hamiltonian HK and the wavefunction W are H given by









wc c w !c wip( f H H M H , W" , (5.10) c w #ip( c w p( #c (w!1) m H M H  H with da (t, r)"exp(iut)(2/rN f(o) and d(Im w(t, r))"exp(iut) m(o). The radial variable o is de"ned P by Eq. (5.6). The third perturbation equation, which is the linearized Gauss constraint (Eq. (2.44) with b"0), is a di!erential consequence of (5.9). The Hamiltonian in (5.10) is formally self-adjoint and real, which ensures the existence of self-adjoint extensions. In order to show that the spectrum has a negative part, it su$ces to "nd a function W such that HK "

u(W),1W"HK "W2/1W"W2(0 .

(5.11)

Here the scalar product 1W"U2"WRU do. For solitons the function W with the required properties is speci"ed by [47] f"w /c , m"w!1 . (5.12) M H Note that this can be obtained by di!erentiating the vacuum-to-vacuum interpolating family (3.16) with respect to the parameter j at the sphaleron position, j"p/2. The direct computation gives




1W"HK "W2"!



c w # H (w!1) do(0 , M 2

(5.13)

while the norm 1W"W2 is "nite. An expression similar to (5.12) can be found also in the black hole case [325]. As a result, all BK solitons and EYM black holes are unstable. The above variational argument can be generalized to arbitrary gauge groups. The conclusion is: all static, spherically symmetric and purely magnetic regular or black hole solutions to the EYM equations for arbitrary gauge groups are unstable [71,73]. It is interesting that this conclusion can be reached without explicit knowledge of the possible equilibrium solutions. The actual number of the odd-parity negative modes for the nth BK or EYM black hole background is n. In order to see this one makes use of the fact [323,321] that for uO0 the eigenvalue problem in (5.9),(5.10) can be mapped to the equivalent one-channel SchroK dinger problem





d c ! # H (3w!1)#2(wZ) t"ut . M do 2

(5.14)

Here




t"



d f w # M#wZ , do w c w H

(5.15)

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Table 2 The eigenvalues of the negative modes for the n"1 EYM solutions r 

u (even-parity)

u (odd-parity)

0 (soliton case) 0.1 0.5 1 5 10

!0.0524 !0.0502 !0.0402 !0.0268 !0.0021 !0.0005

!0.0619 !0.0618 !0.0600 !0.0492 !0.0029 !0.0007

and Z is a solution of an auxiliary nonlinear di!erential equation Z "wZ!c . M H The direct veri"cation shows that for u"0, Eq. (5.14) has a solution t "w exp 




M

(5.16)



wZ do . (5.17) M One can show that Z can be chosen in such a way that the potential in (5.14) is everywhere bounded, both for solitons and black holes (see Fig. 11), while t is normalizable. Due to the factor  w, the number of nodes of t is n. The well-known theorem of quantum mechanics guarantees then  that the eigenvalue problem (5.14) has exactly n negative energy eigenstates. In order to actually "nd the bound states one solves numerically Eqs. (5.9), (5.10) or Eq. (5.14) [220,321]. For the n"1 EYM soliton and black holes the bound state energies for both values of parity are presented in Table 2. Summarizing, for any n and for all values of the event horizon radius r the BK solitons and non-Abelian black holes are unstable with respect to small  spherically symmetric perturbations. The number of instabilities is 2n, of which n belong to the odd-parity sector and n are parity-even.

6. Slowly rotating solutions The di!erence between the non-Abelian EYM theory and its Abelian counterpart is emphasized further still when one studies stationary generalizations of the static BK solitons and EYM black holes [327,69,66]. It turns out that rotating EYM black holes acquire an electric charge, and not just a dipole correction to the background gauge "eld which one would normally expect in the Abelian theory. As a result, spinning up the neutral static solutions makes them charged up. The family of stationary EYM black holes includes in addition quite peculiar solutions which are

 For multi-channel SchroK dinger problems bound states can be counted by using the nodal theorem recently proven in [8].

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non-static and have non-vanishing angular velocity of the horizon, but the total angular momentum measured at in"nity is zero. This shows that the electrovacuum staticity theorem [78] does not generalize in a straightforward way to the EYM theory. The regular BK solutions admit charged, stationary generalizations too. This is even more unusual, since solitons in other "eldtheoretical models, such as the t'Hooft}Polyakov monopoles, say, cannot (slowly) rotate [66,171]. Studying stationary EYM "elds presents certain di$culties. Although solutions for static, axially symmetric deformations of the BK solitons and EYM black holes have been obtained numerically [199}202], the problem becomes much more involved in the stationary, non-static case [126]. In particular, one realizes then that the EYM equations do not imply the Frobenius integrability conditions for the Killing vectors [165]. The standard Papapetrou ansatz can therefore be too narrow. This shows that the Abelian circularity theorem also does not generalize to EYM systems in a straightforward way. Although solutions to the full stationary problem are still lacking, the perturbative analysis based on the assumption of linearization stability has been carried out [327,69,66]. The basic idea is as follows. Suppose that there is a one-parameter family of stationary black hole or regular solutions of the EYM equations, approaching the static solutions for angular momentum J"0. The tangent to this family at J"0 satis"es the linearized EYM equations. Conversely, it is reasonable to expect that for a well-behaved solution of the linearized equations around the static con"gurations there exists an exact one-parameter family of stationary solutions. Accordingly, the problem reduces to studying the linear rotational excitations for the static BK solitons and EYM black holes. 6.1. Perturbation equations Consider small perturbations around the BK solitons or EYM black holes: g Pg #h , A PA #t , (6.1) IJ IJ IJ J J J where (g , A ) refer to the background con"guration, and it is convenient to express the gauge "eld IJ J in the string gauge (2.17): (6.2) A"w(T d0!T sin 0 du)#T cos 0 du .    The perturbation equations are obtained by linearizing the EYM "eld equations (2.4)}(2.6): ! Nh !2 R h?@#RNh #RNh "4 d¹ , N IJ I?J@ I NJ J NI IJ !D DNt #RNt !2[F , tN]#h?@D F #F?@ h "0 . N J J N JN ? @J ? @J Here D , #[A , ) ], and the gauge conditions J J J

hN"hN"D tN"0 N J N N are imposed. d¹ in Eqs. (6.3) is obtained by varying the energy-momentum tensor IJ ¹ " tr(F F g?@!g F F g?Mg@N) I? J@  IJ ?@ MN IJ  with respect to the metric and the gauge "eld.

(6.3)

(6.4)

(6.5)

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6.1.1. General mode decomposition In order to identify the most general rotational degrees of freedom, one determines those amplitudes in the partial wave decomposition which can give a non-vanishing contribution to the ADM #ux integral for the total angular momentum 1 JG" 32p



e (xIR hL#dLhI) dS H . (6.6) GLI H H 1 In order to carry out the partial wave decomposition for (h , t ), it is convenient to introduce the IJ J complex 1-form basis h?: dr r , h" (d0!i sin h du), h"(h)H , h"p(N dt, h" (N (2

(6.7)

the non-vanishing components of the tetrad metric g?@,(h?, h@) being g"!g"!g"1. In addition, one introduces the new Lie-algebra basis: L "T #iT , L "T !iT , L "T . The         perturbations then expand as h "h? h@ H , t "L W?h? , (6.8) IJ I J ?@ I ? ? I and the complete separation of variables in the perturbation equations (6.3) is achieved by making the following ansatz: W?"exp(iut)U?(r) > (0, u) , (6.9) H "exp(iut)H (r) > (0, u), ? ? Q HK ?@ ?@ Q HK where > (0, u) are the spin-weighted spherical harmonics [148]. Here the quantum numbers j and Q HK m are the same for all amplitudes. The spin weight s can be di!erent for di!erent harmonics and is given by s"s for H , where ?@ ?@ s "s "s "s "0, s "s "!s "!s "1, s "!s "2 ,           and by s"s? with ? s"s"s"0, s"s"s"!s"!s"!s"1, s"!s"2            for the YM perturbation amplitudes W?. As a result, for given u and j50 in (6.9), Eqs. (6.3) reduce ? to a system of radial equations for H (r) and U?(r). Due to the spherical symmetry of the ?@ ? background "elds, the quantum number m does not enter the radial equations. 6.1.2. Rotational modes Inserting (6.9) into (6.6) the integral is computed by integrating over a two-sphere at "nite radius r and then taking the limit rPR. The angular dependence of the integrand implies that the integral vanishes for any r unless j"1, and that only the h perturbation component can give P a non-vanishing contribution. Now, the transformation behavior of the angular momentum under space and time re#ections (P, ¹) implies that only those perturbation amplitudes are relevant which are even under P and odd under ¹. For j"1 these appear only in h and in two isotopic P component of t . They decouple from the remaining modes because the background solutions are 

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P and ¹ symmetric. This leads "nally to the following most general ansatz (up to global coordinate rotations) for the stationary rotational modes: h"2S(r) sin h dt du,




t" T



s(r) g(r) sin h#T cos h dt .  r  r

(6.10)

The gauge conditions (6.4) for this ansatz are ful"lled identically and the perturbation equations (6.3) reduce to the coupled system for the radial amplitudes S, s, and g:












!rNp

S  s  4(w!1) 4(w!1) # 2N# # S#4Nrw (w s!g)"0 , p r r r

!rNp

s  w(w!1) #(1#w!2wN)s!2wg!rN(wS)# 2Nw# S"0 , p r

!rNp

g  2(1!w) #2(1#w!wN)g!4w s# S"0 . p r






(6.11)

Here w, m, and p refer to the background solutions, N,1!2m/r. Given a solution of these equations, the ADM angular momentum is




S  . (6.12) JG"dG lim r X 6r P Note that, with a suitable reparameterization of the variables S, s, and g, Eqs. (6.11) can be represented in a SchroK dinger form with a manifestly self-adjoint Hamiltonian [66,69]. 6.2. Solutions For the Schwarzschild background, N"1!2M/r, p"w"1, Eqs. (6.11) admit the solution s"g"0, S"!2JM/r. This is recognized as the linear rotational excitation of the Schwarzschild metric. One can see that this mode is bounded everywhere outside the horizon. However, since it does not vanish at the horizon, it is not normalizable. Accordingly, when looking for solutions to Eqs. (6.11) for the non-Abelian backgrounds, one considers all modes that are bounded and regular, but not necessarily normalizable. The strategy then is to "nd all well-behaved local solutions in the vicinity of the origin or horizon and at in"nity, and then match them in the intermediate region. In the far-"eld region the most general solution to (6.11) that gives rise to regular perturbations in (6.10) reads 2JM aQ # #O(r\) , S"! r r

(6.13)

s"
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In the near zone, in the soliton case, the most general local solution to (6.11) that is regular at the origin is S"c r#O(r), s"c r#O(r), g"c r#O(r) . (6.14)    This contains three independent parameters: c , c , while the third one, c , resides in the higher    order terms in (6.14). Since the coe$cients in (6.11) are continuous and regular for 0(r(R, the local solutions (6.13) and (6.14) in the vicinity of r"0 and r"Radmit extensions to the semi-open intervals [0,R) and (0,R], respectively. The total solution space is six dimensional, while the local solutions specify four and three dimensional subspaces, whose intersection is (at least) one dimensional. In other words, the matching conditions for S, s, and g give six linear algebraic equations for the seven coe$cients J, Q, <, c ,2, c , such that the matching is always possible. For   example, for the n"1 BK solution the numerical matching gives Q/J"0.960,
(6.15)

and similarly for c ,2, c . All BK soliton solutions therefore admit a one parameter family of   stationary excitations parameterized by J. For black holes the space of solutions that are regular in the vicinity of the horizon is four dimensional S"c r #c x#O(x) ,   s"!c w(r )#c x#O(x), g"!c #c x#O(x) , (6.16)      where x"r!r and c , c , c , and c are four independent integration constants. As a result, all      EYM black holes admit a two-parameter family of stationary excitation. In order to clarify the meaning of the parameters in the solutions, one uses Eq. (6.12), which shows that the constant J is the ADM angular momentum. Next, after the gauge transformation with U"exp(i(p!0)T ) the gauge "eld perturbation in the asymptotic region becomes  Q (6.17) t"T <# dt#O(r\) .  r






This suggests that Q should be identi"ed with the electric charge. The constant < determines the asymptotic value of the temporal component of the gauge "eld, A (R). In the Abelian theory it  would be possible to gauge < away. In the non-Abelian theory, however, such a gauge transformation would render the whole con"guration time-dependent. The physical signi"cance of < became clear already in the course of the matching procedure. In addition, due to the coupling to the background gauge "eld, it enters the asymptotic expansion of the "eld strength, dF, whose non-vanishing components in the same gauge as the one used in (6.17) read a< Q dF " T #O(r\), dF 0" T #O(r\) ,  P r  r  a< dF "! T sin 0 cos 0#O(r\) . P r 

(6.18)

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It is obvious that Q can be expressed in terms of the #ux of the electric "eld over a two-sphere at spatial in"nity. The constant <, on the other hand, gives no contribution to the #ux, since the corresponding piece of electric "eld in (6.18) is tangent to the sphere. Summarizing, all BK soliton solutions admit slowly rotating excitations with continuous angular momentum J and electric charge Q proportional to J. Note that solitons in other "eld theoretical models, such as the t'Hooft}Polyakov monopoles or boson stars, generically do not admit slow-rotating states [327,69,66,171]. This might be due to the fact that a soliton, being a solution to a non-linear boundary value problem, can exist only for discrete values of the parameters. The angular momentum should therefore be quantized. In this sense the situation in the EYM theory is exceptional. This is presumably because both gravitational and gauge "elds are massless, which manifests in the slow (polynomial) decay of the "elds at in"nity. The slow rotational excitations of the EYM black holes can be parameterized by any two of the three parameters J, Q, or <. Accordingly, there are three distinguished branches of solutions. First, for solutions with <"0 the charge is proportional to the angular momentum [327]. Secondly, there is an uncharged branch with Q"0. Finally, there are solutions with J"0, which are non-static, as can be seen already from (6.13). In view of this the Abelian staticity theorem asserting that stationary black holes with J"0 must be static does not apply. The non-Abelian version of this theorem [300] states that a stationary EYM black hole is static and the electric "eld vanishes, provided that the following condition holds: X J!
(6.19)

where X is the angular velocity of the horizon. Notice that this agrees with the results described & above. Indeed, (6.19) implies that stationary black holes with J"0 must be static if only
7. Self-gravitating lumps In this section we shall discuss gravitating solitons and black holes in the theories admitting particle-like solutions in the #at space-time limit. Historically, the "rst investigation of such systems was motivated by a wish to understand the structure of very heavy magnetic monopoles [312]. It was found that the gravitating generalizations for #at space monopoles exist at least for small values of Newton's constant. More systematic investigations of the problem were undertaken after the discovery of the BK solutions. These have revealed a number of the typical features. First, it has been found that, for a large class of non-linear matter models, the #at space solitons can be generalized to curved space time, provided that the dimensionless gravitational coupling constant i is small. This is not very surprising, since it is intuitively clear that for small values of i perturbation theory applies. However, and this is far less obvious, it turns out that under certain conditions gravity can be treated perturbatively even for black holes, provided that the event horizon radius r is small. As a result, the gravitating lumps can be further generalized by replacing  the regular center by a black hole with a small radius r . Below we shall review the corresponding 

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argument based on the implicit function theorem [188]. This is so simple that the existence of hairy black holes might actually have been foreseen many years ago. Secondly, in the non-linear regime, when gravity is not weak, the fundamental lumps and black holes admit a discrete spectrum of gravitational excitations. The excited solutions become in"nitely heavy for iP0 and, remarkably, reduce then to the rescaled BK solutions. The excitations thus can be thought of as gravitating solitons with small BK particles or EYM black holes in the center. Finally, the fundamental and excited lumps and black holes cannot exist beyond certain maximal values of i and r . The existence of a bound for i is easy to understand. As i is proportional to the  ratio of the gravitational radius of the object to its typical size, it cannot be too large, since otherwise the system becomes unstable with respect to the gravitational collapse. The solutions, however, do not collapse as i approaches the critical value. Instead they either become gravitationally closed or coalesce with the excited solutions. The existence of the bound for r is quite  interesting. It seems as if a small black hole could not swallow up a soliton which is larger than the black hole itself. As a result, a hairy black hole appears. However, a big black hole can completely absorb all non-trivial hair. 7.1. Event horizons inside classical lumps One can use the implicit function theorem to argue that, within a large class of non-linear models, and for small values of the gravitational coupling constant i and the event horizon radius r , the #at space solitons admit both regular and black hole gravitating generalizations. This comes  about as follows [312,188]. Consider a static, spherically symmetric gravitating system with the metric (2.50) and the stress tensor ¹I"diag (o,!p ,!p ,!p ). With the notation of Section 2, J P F F the non-trivial Einstein equations G "2i¹ are IJ IJ r m"ro, p"i (o#p )p , P N

(7.1)

with N"1!2im/r. This has to be supplemented with the condition ¹I"0, which can be I J rewritten with the use of (7.1) as the Oppenheimer}Volko! equation 2 m#rp P (o#p )# (p !p ) . p "!i P P P r F rN

(7.2)

Suppose that there is an equilibrium #at space con"guration, such that Eqs. (7.1) and (7.2) are ful"lled for i"0. In most cases one can expect that this con"guration will also survive if i is non-zero and small. Indeed, for i;1 the additional terms in the equations are small and gravity can be treated perturbatively. One will then have N+p+1. At the same time, it is less clear whether the corresponding black hole generalization will also exist. For black holes N vanishes at r implying that the deviation from the #at space value N"1 is non-small. This also implies that  one cannot use perturbation theory, since the coe$cient 1/N in (7.1) and (7.2) diverges at r"r .  However, if the matter model is such that (o#p )&N at the horizon then the blowing up of 1/N P will be canceled, in which case gravity can be treated perturbatively even for black holes [188].

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It has already been mentioned that for all matter models considered in Section 2.3 the reduced 2D matter Lagrangian in the static, purely magnetic case can be represented in the form ¸ "!(NK#;) , (7.3)

where K and ; depend on r and on the matter "eld variables, but not on the gravitational variables N and p. For this Lagrangian one has o"(NK#;)/r, p "(NK!;)/r , (7.4) P such that (o#p )"2NK vanishes at the horizon. As a result, given a #at space soliton solution, P one can expect that it will have gravitating generalizations, both in the regular and black hole cases. The argument is as follows. Inserting (7.4) into (7.1) the Einstein equations for m and p become linear, and hence can be integrated in quadratures for given K and ;. For example, one will have






 dr (7.5) K r P and similarly for m. Therefore, in order to solve the gravitating problem, it remains to determine the matter variables entering K and ;. Let us denote these variables collectively by . The equations of motion for can be obtained by varying the ADM mass functional [312]. The ADM mass is M"m(R)"m(R)p(R). Using (mp)"(K#;)p one obtains p"exp !2i








  dr r dr(K#;) exp !2i K . (7.6) M[ , i, r ]"  p(r )#   r 2 P P This expression takes gravity into account but depends only on the matter variables; notice that for r "0 it was used in Eq. (3.23). Varying M with respect to gives the matter equation of motion  with all gravitational degrees of freedom expressed using formulas like (7.5). The problem therefore reduces to studying solutions of dM F[ , i, r ], "0 .  d

(7.7)

By assumption, there exists a #at space solution , such that F[ , 0, 0]"0. Suppose that the   operator dF/d is invertible for this solution. This will be the case, for example, if the solution is a local minimum of energy, dF[ , 0, 0]'0. Then the implicit function theorem ensures that for  i and r su$ciently close to zero, there exists (i, r ) satisfying (7.7) such that (0, 0)" [188].    Summarizing, if a #at space soliton solution ful"lls the conditions speci"ed above then it will survive also in the weak gravity case, and in addition will be able to contain a small black hole inside. This happens, for example, for monopoles, sphalerons, and Skyrmions considered below. However, not all matter models are of the type (7.3). For example, systems with electric "elds are not of this type and hence are not covered by the above arguments. Another example is provided by boson stars. For the BK particles there are no #at space counterparts, and the above arguments do not directly apply. However, it follows that for iO0 the existence of regular solutions to F[ , i, 0]"0 implies the existence of corresponding black hole solutions to F[ , i, r ]"0. As  a result, the existence of the BK solitons implies the existence of the EYM black holes, at least for small r . 

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7.2. Gravitating monopoles The regular monopole solutions were discovered by t'Hooft and Polyakov in the YMH model (2.32) with the triplet Higgs "eld [305,263]. For monopoles in #at space there is the absolute lower bound for energy determined by the topology of the Higgs "eld [44]: E5(4pv/g)"l" .

(7.8)

Here l is the winding number of the Higgs "eld, which coincides with the magnetic charge. Solutions saturating this bound are topologically stable. For the hedgehog ansatz in (2.33) one has l"1, and the corresponding EYMH equations following from (2.34) in the static, purely magnetic case read (w!1) r # N #w #r<( ) , m"Nw# 2 2r




p"ip



(7.9)

2w #r  , r




(Npw)"p

(7.10)



w(w!1) # w , r

(7.11)

(rNp )"p(2w #r<( ))

(7.12)

e 2im , <( )" ( !1) . N"1! 4 r

(7.13)

with

Here i"4pGv and the length scale is L"1/gv,1/M . Note that p can be eliminated from the 5 equations in the usual way. For a given solution to these equations the total energy is the ADM mass M"m(R). The dimensionful energy is (4pv/g) m(R). The boundary conditions for "nite energy solutions are w"1!br#O(r), "cr#O(r), m"O(r) ,

(7.14)

w"Bf exp(!r), "1#Cf exp(!er), m"M#O(r\) (7.15)   at the origin and at in"nity, respectively. Here b, c, B, C, and M are integration constants, and the leading behaviour of the functions f (r) and f (r) for rPR is power law [89,192].   7.2.1. Flat space solutions The #at space limit corresponds to iP0, in which case the YMH equations (7.11) and (7.12) decouple, while the Einstein equations (7.9) and (7.10) give N"p"1. The mass of the solutions, m(R), is obtained from (7.9), which gives after simple rearrangements



M"












r w!1  er # (w#w )# # ( !1)#( (1!w)) dr . 4 2 r

(7.16)

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Using (7.14) and (7.15) the integral of the last term in the integrand is equal to one, while the "rst three terms are positive de"nite. This shows that M51 ,

(7.17)

in agreement with the bound in (7.8) for l"1. In fact, (7.8) is obtained in a similar way by picking a total derivative in the 3D mass functional. Now, if e"0 and w#w "0,

r #w!1"0

(7.18)

then the "rst three terms in the integrand in (7.16) vanish and the bound M"1 is saturated. The solution of the Bogomol'nyi equations (7.18) describes the BPS monopole [265]: w"r/sinh r,

"coth r!1/r .

(7.19)

By construction, this solution is stable. It has only one non-vanishing component of the stress tensor, ¹, and hence the condition in (3.1) is ful"lled.  For e'0 the bound in (7.17) is not attained and the "rst order equations cannot be used. One has to solve then the second-order YMH equations (7.11) and (7.12). Solutions with the boundary conditions (7.14) and(7.15) can be obtained numerically [45,193]. They describe the t'Hooft}Polyakov monopoles. For any given value of e there is one solution, the existence of which was shown in [311]. The uniqueness of the e"0 solution was established in [233]. The mass M increases with e such that 14M(e)(M(R)"1.96. In the limit ePRthe Higgs "eld becomes `frozena and there remains only the YM equation (7.11), where one sets "1 [193]. The t'Hooft}Polyakov monopoles are linearly stable [12]. For any e the monopole solutions can be generalized to include an electric charge [183]. 7.2.2. Regular gravitating monopoles For iO0 the energy condition (7.17) no longer holds due to the gravitational binding. In fact, for a given i, taking the boundary conditions into account, the mass functional M[ , i, 0] in (7.6) with (w!1) e r #w # r( !1) , K"w# , ;" 2r 4 2

(7.20)

is still bounded from below by a non-zero value [312]. However, it is unclear whether the M (i),inf M[ , i, 0] is reached in the set of solutions of the EYMH equations. For example, it

 turns out that if i is large, the "elds that minimize M[ , i, 0] do not even belong to the space of di!erentiable functions [222]. At the same time, the argument of the preceding section suggests that the gravitating monopoles exist at least for small values of i. This shows that one can study these solutions numerically, however, the issue of their stability remains open. The numerical integration of Eqs. (7.9)}(7.12) with the boundary conditions (7.14)}(7.15) reveals the following picture [258,222,54,5,56,232]. First, for i being small enough, there are the selfgravitating monopole solutions. These are qualitatively similar to the #at space t'Hooft}Polyakov monopoles and reduce to them for iP0. The amplitudes w, , m, and p are monotone functions, while N develops a minimum at some r &1, N(r )'0. One can call r the monopole radius. The

solutions are characterized by i and e. The mass, M(i), decreases with growing i due to the gravitational binding (see Table 3 and Fig. 12). At the same time, the monopole radius does not

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Table 3 Masses of the fundamental monopole and its "rst excitation (e"0) n!i

0

0.02

0.1

0.5

0.75

1

1.5

1.96

0 1

1 R

0.996 6.629

0.985 3.108

0.926 1.413

0.887 1.154

0.854 }

0.783 }

0.714 }

Fig. 12. On the left: amplitudes w , , and N for the fundamental, n"0, monopole solution (solid lines) and for its "rst, L L L n"1, excitation (dashed lines) for i"0.6 and e"0. On the right: the ADM mass versus i for the n"0 and n51 regular monopole solutions with e"0; and the mass M "1/(i of the extreme RN solution. All curves for n51 0, terminate at the point B with i", while staying very close to each other and to the RN curve for i(. For small i the   mass of the nth solution is M /(k, where M is the mass of the nth BK soliton. The n"0 curve terminates at i"1.97. At L L the point B the extreme RN solution bifurcates with in"nitely many non-Abelian EYMH black hole solutions.

change considerably. As a result, the ratio of the gravitational radius of the monopole to its radius r is proportional to i. For regular solutions i must not therefore exceed some critical value,

otherwise the system becomes unstable with respect to the gravitational collapse. In agreement with the last remark, the self-gravitating monopoles exist only for a "nite range 0(i4i (e). One has i (R)"(i (e)4i (0)"1.97 [56]. As i approaches the critical







  value, N(r ) tends to zero (see Fig. 12), the functions w and reach their asymptotic values already

at r"r , and the proper distance



P l" dr/(N 

(7.21)

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diverges for rPr . As a result, the spatial geometry on the hypersurface t"const. develops an

in"nite throat separating the interior region with a smooth origin and non-trivial YMH "eld from the exterior region where w"0, "1, and the metric is extreme RN. The throat is characterized by a constant radius, r "(i . The metric function p(r), normalized by p(0)"1, diverges for M

 rPr . The limiting solution thus splits into two independent solutions, which is similar to the

behaviour of the BK solutions for nPR. The interior solution is geodesically complete. 7.2.3. Gravitational excitations For the gravitating monopoles there is an in"nite sequence of radial excitations. These exist for i being small enough, but do not admit the #at space limit, since their masses diverge for iP0. The boundary conditions (7.14)}(7.15) are still ful"lled, but the amplitude w now oscillates (see Fig. 12) and its node number (n"1, 2,2) characterizes the solutions. This similarity with the BK case is not accidental. In fact, for iP0, the excitations correspond to the rescaled BK solutions. This comes about as follows. When i"4pGv tends to zero, this can be understood either as the weak gravity limit, GP0 with "xed v, or as the limit vP0 with "xed G. Notice that in the latter case the length scale L"1/gv diverges. Consider, however, the rescaling rP(i r, mPm/(i, P /(i .

(7.22)

The whole e!ect of this on the EYMH equations is that N and <( ) in (7.13) are replaced by N"1!2m/r, <( )"e( !i) . (7.23)  As a result, the limit iP0 corresponds now to <( ) vanishing (since 4i), and the EYMH system (7.9)}(7.12) reduces to the EYM equations, which leads to the appearance of the BK solutions. Restoring the original length scale, the excited solutions for small i consist of the very small (Planck size) BK particles sitting inside the large (1/M ) size monopole. The amplitude 5 w oscillates n times in the short interval near the origin and then tends to zero for rPR. For iP0 the mass tends to M /(i, where M is the mass of the nth BK solution. All excitations have the L L same value of i "i (e). One has i (0)"3/45i (e)'i (R)" [56]. For iPi (e)











  the excited solutions show the same limiting behaviour with a throat as the fundamental solution. The picture described above remains qualitatively the same for any 04e(R. A novel feature arises for ePR [54,5,56]. The Higgs "eld is frozen in this case, and the EYMH "eld equations reduce to those of the gravitating gauged non-Abelian O(3) sigma-model. Eq. (7.12) then should be dropped, and there remain three equations (7.9)}(7.11), where one sets "1. Note that these admit the "rst integral (7.29). The boundary conditions at the origin for w and N are given by w"1#O(r), N"1!2i#O(r) ,

(7.24)

where i3[0, ]. Since N(0)O1, there is the solid angle de"cit and a conical singularity at the origin.  This is due to the fact that the energy density ¹ contains the term w /r, which blows up,  because does not vanish at the origin. For r'0 the solutions for w and N are qualitatively

 For small e the picture is slightly more complicated; details can be found in [54].

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similar to those for e(R. In particular, they are asymptotically #at (N(R)"1) and exist for arbitrary node number n. The existence of these solutions was rigorously established in [56]. In this connection it is instructive to mention also the gravitating global monopoles [15]. These arise in the coupled Einstein}Higgs model, which corresponds to the EYMH theory with dynamics of the YM "eld being suppressed. The "eld equations in the spherically symmetric case can be obtained from (7.9)}(7.12) by ignoring the YM equation (7.11) and setting w"1 in the remaining equations. The boundary conditions for and m at r"0 are still given by (7.14) (if only e(R), and the origin is regular: N(0)"1. However, since P1 for rPRwhile w"1, the term w /r in ¹ leads to the divergence of the energy in such a way that N(R)"const.(1, which  corresponds to the solid angle de"cit at in"nity. Note that the absence of "nite-energy solutions in the model agrees with the general arguments in Section 3. For ePRthe Einstein}Higgs theory reduces to the global gravitating O(3) sigma-model. In this case there remain only two non-trivial "eld equations, (7.9)}(7.10), where one sets w" "1. The solution is p"1, N"1!2i, such that there is a constant solid angle de"cit. It is worth noting that this model admits the Bogomol'nyi bound and non-trivial exact solutions in the case of cylindrical symmetry [98]. 7.2.4. EYMH Black holes For black holes, instead of (7.14), one has the boundary conditions at the regular horizon, which are given by the expression similar to that in (4.25) with w ,  , and N being functions of w and .      The possibility of a degenerate horizon was discussed in [152,153,104,32,56]. According to the argument of the preceding section, one can expect the black hole generalizations of the regular monopoles to exist at least for small values of r . This is con"rmed by the numerical analysis for the  fundamental (n"0) monopole and for its excitations (n51). For r ;1 the solutions resemble  small black holes sitting in the center of the regular lumps, the latter being almost una!ected for r
 In both cases the solutions exist if only r is less than a maximal value, r (i), where   r (i)4r (3/4)"(3/2. For r P0 the black hole solutions tend to the corresponding regular ones.    In the opposite limit, r Pr (i), black holes with i' develop the in"nite throat and become    gravitationally closed. The metric amplitude N in this case, apart from a simple zero at r"r ,  develops also a double zero at r"(i'r . For i( the picture is more complicated. In this case   for some values of i and r one "nds more than one non-Abelian black hole solution. In addition,  for r 5(i and M51/(i there exist Abelian RN solutions:  w"0, "1, p"1, N"1!(2iM/r)#i/r .

(7.25)

As a result, the no-hair conjecture is violated. When r changes, di!erent non-Abelian solutions  bifurcate with each other or with the RN solution [54]. This leads to the existence of the upper

 The numbers are given for the case where e"0.

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63

bound for r . The value r ()"(3/2 is the least upper bound for r for all solutions (for any n    and e) with i". All of them tend for r P(3/2 to the same limiting con"guration, which is the   extreme RN solution. A more detailed discussion can be found in [54,56]. Note that the existence of an upper bound for the black hole radius is quite typical for gravitating lumps [256]. One can think that a small black hole is unable to swallow up a monopole whose radius is much larger than r ,  however a bigger black hole can do this, such that all non-Abelian structures disappear inside the horizon. This phenomenon does not exist, however, in the pure EYM theory, where there is no energy scale other than Planck's mass. The non-Abelian EYM black holes can be arbitrarily large. It is worth noting also that the fundamental and excited gravitating monopoles and black holes can be generalized to include an electric charge [63]. 7.2.5. Deformed EYMH black holes The fundamental gravitating monopoles and black holes are stable, while the excited solutions are unstable with respect to small spherically symmetric perturbations. This was shown in [173] with the use of the Jackoby criterion. The RN solution (7.25), which is stable in the Abelian theory, becomes unstable in the non-Abelian case for i( [226,41,221,54,5,56]. This comes about as  follows. Perturbing the amplitudes w, m, and p of the RN solution (7.25) as in Eq. (5.1), while

P #d (t, r), and linearizing the full system of EYMH equations, the linearized YM equation decouples from the rest of the system. It is convenient to rescale r and M in (7.25) according to (7.22), which leads to the following equation for dw(t, r)"exp(iut)g(o):






d 1 ! #N i! do r



g"ug ,

(7.26)

where N"1!2M/r#1/r and do"dr/N. Notice that for iP0, which now corresponds to the limit where the Higgs "eld decouples, this equation reduces to that in (5.5). As was observed in [56], in the extreme RN case, where M"1, this equation admits in"nitely many bound states for i(,  because the corresponding potential for oP!Rbehaves like (i!1)/o. For i' there are no  bound states [56]. This corresponds to the fact that for i" the extreme RN solution bifurcates  with in"nitely many non-Abelian black hole solutions [56]. For non-extreme RN solutions the situation is more complicated, but the result is the same: the instability arises only for small enough values of M and i. The instability of the RN solution within the context of the non-Abelian EYMH theory has important implications, since it can be viewed as indication of the existence of new non-Abelian solutions [223,273,271,272,332]. This basic idea is as follows. First of all, notice that the RN solution in (7.25) can be generalized to arbitrary integer values of the magnetic charge. For this one uses the ansatz (2.24) for higher winding numbers l, while the Higgs "eld is U" T . The solution  is given by a"w"0, "1,

p"1, N"1!(2iM/r)#il/r ,

(7.27)

 The exterior region of the extreme RN solution can also coalesce with the corresponding part of the limiting solutions with the throat. This happens for i' (see Fig. 12). However, since the interior parts are di!erent, this does not  lead to the change of stability, and the RN solution for i' is stable. 

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and the gauge "eld strength is F"T l sin 0 d0du. For any given l this RN solution can be either  stable or unstable } depending on values of i and M [223,272]. Suppose that i and M are chosen such that the solution is close to the border of instability and is only `barelya unstable. One can expect then that there is a nearby stable non-Abelian solution with lower mass that bifurcates with the RN solution when i and M approach the border. It is plausible that this new solution di!ers only slightly from the RN one, and can be approximated with the use of linear perturbation theory [273,271,332]. Now, for l"1 the static perturbations around the RN solution are spherically symmetric and approximate the non-Abelian EYMH black holes described above. However, for l'1 the solutions of the perturbation equations are no longer spherically symmetric and exhibit a complicated dependence on spherical angles. As a result, assuming that the linear modes indeed approximate some new solutions, these new black holes turn out to be not even axially symmetric. Their shape can be quite complicated, exhibiting discrete, crystal-type symmetries (see [273] for interesting pictures). It is worth noting that the deformed black holes have actually been described within a more general theory, which includes the EYMH model as a special case [223,271}273,332]. In this theory the third isotopic components of the gauge "eld is regarded as an Abelian vector "elds, A , while I the "rst two component are viewed as a complex vector "eld, = . The gravitational part of the I action is standard, while the matter Lagrangian is L"!F FIJ!HH HIJ#ad FIJ!bd dIJ#m( ) =H=I#R RI !<( ) .  IJ IJ IJ I  I  IJ (7.28) Here F "R A !R A and H "D = !D = , while d "i(=H= !=H= ). The IJ I J J I IJ I J J I IJ I J J I covariant derivative is D = ,(R !iA )= , and a, b are parameters. For a", b"4, and I J I I J  m( )" the theory reduces to the triplet EYMH model in the string gauge (2.17) with U" T .  For a range of the parameters the model admits soliton and black hole solutions, and, in particular, the deformed black holes described above. Together with the results obtained in the pure EYM theory [200,202], this suggests that the existence of deformed static black holes is typical for the gravitating non-Abelian models. 7.2.6. The Bogomol'nyi bound Unfortunately, the #at space Bogomol'nyi equations (7.18) do not generalize to curved spacetime. The existence of the Bogomol'nyi bound in #at space has a very deep physical reason: the YMH theory for e"0 can be obtained via truncation of a supersymmetric model. On the other hand, the EYMH theory is not a truncation of a supergravity model, and no Bogomol'nyi bound exists in this case. As a result, a search for Bogomol'nyi equations for iO0 leads to an almost trivial result: one can "nd "rst order equations, but they admit only the extreme RN solution

 The corresponding perturbation equations are not quite the same as in the standard perturbation analysis [273].  It has already been mentioned that all spherically symmetric YM "elds with l'1 are necessarily embedded Abelian. It follows then that non-Abelian "elds for l'1 cannot be spherically symmetric.

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[97,14]. For ePR, when the Higgs "eld is frozen, the Bogomol'nyi equations are less trivial [124,53]: i"1, Nw"w, r(pN)"p(w!1), rN#w!1"$r .

(7.29)

However, all their solutions, apart from the extreme RN one, have naked singularities. In order to obtain a physically meaningful Bogomol'nyi bound in curved space [141], one should extend the EYMH theory to make it a part of a supergravity model. Such an extension was considered in [142,146]. In addition to the EYMH "elds it contains a dilaton and an abelian vector "eld, which is typical for toroidally compacti"ed string theory models. The BPS solutions in this case can be obtained analytically. Sometimes gravitating monopoles are associated with in#ation [225,315]. This is due to the fact that, since the Higgs "eld vanishes for rP0, there is a false vacuum in the monopole core. The results of the numerical analysis of the time-dependent problem suggest that monopoles indeed in#ate if i exceeds the critical value, i'i [279,278].

 Other related topics we would like to mention include solitons for the EYMH system with the ghost Higgs "eld [93], monopole solutions on the cosmological background [244], and the Rubakov-Callan e!ect for the colored black holes [132]. 7.3. Gravitating YMH sphalerons Consider the EYMH model (2.35) with the doublet Higgs "eld. The spherically symmetric ansatz (2.36) contains two independent amplitudes, and m. One can impose the on-shell condition m"p ,

(7.30)

which implies that the con"guration is parity-even. As a result, the "eld equations following from the reduced Lagrangian (2.37) in the static, purely magnetic case are given by 1 (w!1) #rN # (w#1)#r<( ) , m"Nw# 2 2r




p"2ip



w #r  , r




(Npw)"p

(7.31)

(7.32)

 

w(w!1) 1 # (w#1) , r 2

(7.33)




(7.34)

(rNp )"p

1 (w#1) #r<( ) , 2

 The existence of these equations has no explanation at present. Note that in #at space there are no Bogomol'nyi equations for ePR.

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where N and <( ) are the same as in (7.13). The boundary conditions are w"$(1!b r)#O(r), "c r#O(r), m"O(r) ,

(7.35)

w"!1#Bf exp(!r), "1#Cf exp(!er), m"M#O(r\) (7.36)   at the origin and at in"nity, respectively, where the notation is the same as in (7.15). Let us mention the following important di!erences with the triplet EYMH case. First, (7.36) implies that the solutions are neutral. Secondly, the vacuum manifold for the Higgs "eld is now three-sphere, and since n (S) is trivial, no topologically stable solutions exist. However, as was  argued in [237,208], the non-triviality of n (S) can lead to the existence of saddle point solutions in  the #at space YMH theory, which relate to the top of the potential barrier between the topological vacua. The corresponding argument is essentially the same as the one presented in Section 3.2 for the BK sphalerons. First, one "nds sequences of static "elds +A[j], U[j], interpolating between the distinct vacua. Identifying the end points, they become non-contractible loops. Second, for each loop one computes the maximal value of energy, E , which is then minimized over all loops to

 obtain E ,inf+E ,. Finally, it is assumed that (a) E is positive; (b) there is a loop whose E is 

 

 equal to E . Even though these assumptions have not been rigorously proven, they seem to be  plausible in the case under consideration. This implies the existence of a saddle-point solution called sphaleron. The sphaleron solution to (7.31)}(7.34) in the #at space limit was found numerically in [103,46]. This corresponds to the choice of the plus sign in (7.35) and the gauge amplitude w interpolating between the values 1 and !1 } very much similar to the n"1 BK solution. The Higgs "eld amplitude is a monotone function. The existence of this solution was established in [76]. For small e there is only one solution for a given value of e. For large e there are additional solutions for which the amplitude m is non-trivial, these are called deformed sphalerons [212,338]. Sphalerons are unstable. The instability is associated with the dynamics of the "eld m. Although unstable, #at space YMH sphalerons admit gravitating generalizations [149]. First, for small values of i there is a fundamental branch of regular solutions that admit the limit iP0. These are qualitatively similar to the #at spacetime sphaleron. Next, there is a branch of excited solutions, which also look similar. In particular, the amplitude w still has one node (see Fig. 13). However, the excitations do not admit the #at space limit and for iP0 reduce to the rescaled n"1 BK solution } exactly in the same manner as for monopoles. The excited solutions can be thought of as the BK sphalerons sitting inside the YMH sphalerons. Finally, there are solutions with higher node numbers. For even values of n one chooses the minus sign in (7.35). Similar to the n"1 solutions, those for n'1 also exist in pairs, each pair containing two solutions with the same nodal structure but with di!erent masses. For iP0 the nth solution with lower mass reduces to the nth rescaled BK solution, while the one with larger mass to the (n#1)th rescaled BK solution. The change of the node number then arises as follows: for iP0 the "eld con"gurations shrink and the amplitudes w with di!erent node numbers become practically identical. For example, the amplitude w for the n"1 excited solution shown in Fig. 13 coincides for iP0 with the one for the n"2 solution, for which one has w(0)"!1. For all n the solutions exist only for a "nite range of the parameter 04i4i (n, e). The

 limiting behaviour for iPi is di!erent from that for monopoles. Notice that the RN solution

 (7.25) does not ful"ll the equations in the doublet case. As a result, the solutions cannot have

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Fig. 13. On the left: amplitudes w and for the fundamental EYMH sphaleron solution (solid lines) and for its "rst excitation (dashed lines) for i"0.04 and e"0. On the right: the ADM mass M and the parameter b in (7.35) versus i for the fundamental and the "rst excitation branches. The solid and dashed curves merge at the critical value i "0.281.

 The curves for c(i), where c is the third parameter of the solutions in (7.35), merge in a similar way.

a throat connecting an interior region and the exterior RN part. What happens instead is that the two solutions with the same n coalesce for i"i , at which point all their parameters coincide

 (see Fig. 13). For example, the two solutions shown in Fig. 13 get closer and closer as i increases and "nally merge for i (1, 0)"0.281 becoming one solution. The critical values i thus



 correspond to bifurcation points. It is worth noting that, unlike the situation in the monopole case, the limiting solutions for all n are perfectly smooth and regular. In particular, the minimal value of N is quite large. No regular solutions exist for i'i .

 For any n the regular EYMH sphalerons can be generalized to include a small black hole inside. This is possible for 04i(i (n, e) and for the event horizon radius r being bounded from above

  by some value r (i), which tends to zero for iPi . For a given n and r (r (i) there are again 

   two di!erent solutions, which coalesce as r Pr (i). For r 'r (i) only the Schwarzschild solution     is possible. All known regular and black hole solutions of the doublet EYMH theory are unstable [47,334,241]. The picture described above remains qualitatively the same for any e(R. For ePRthe Higgs "eld becomes frozen, "1, and, if the phase of the Higgs "eld is still "xed according to (7.30), one obtains a simpler theory of gravitating massive non-Abelian gauge "eld. This is sometimes called the Einstein}non-Abelian}Proca model [149,231,308,301]. The conical singularity at the origin for regular solutions can be avoided in this case by choosing the minus sign in (7.35). As a result, solutions for even values of the node number n are globally regular. For odd values of n solutions contain the conical singularity, as in the monopole case. All regular solutions for r'0, as well as

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their black hole counterparts, look qualitatively similar to those for e(R. In particular, they exhibit the same bifurcation picture. This model has also been studied in the context of the Brans}Dicke theory [302]. If the constraint (7.30) is abandoned then the Higgs "eld is allowed to rotate in the internal space spanning a unit sphere S. The EYMH theory reduces then to the gravitating gauged O(4) sigma-model, whose solutions are called local textures. The equations of motion in the static, spherically symmetric and purely magnetic case can be obtained by varying the reduced Lagrangian (2.37) with "1. Note that for mO0 one cannot set to zero the radial component of the gauge "eld a . We are unaware of any curved space solutions in this case, while those without P gravity were studied in [212,338]. For global gravitating textures static solutions with "nite energy cannot exist, however, there are non-trivial time-dependent solutions [120], which can be important in the theory of #uctuations of cosmic microwave background [310]. 7.4. Gravitating Skyrmions The gravitating Skyrme model provided one of the "rst indications of the existence of hairy black holes. In [229] it was found that the equation for the chiral "eld on the Schwarzschild background admitted a regular solution. This can be regarded as approximately describing a black hole with Skyrme hair. In [228] the self-consistent gravitating problem was treated numerically, which work for some reasons remained almost unknown. In [147] self-gravitating Skyrmions with higher winding numbers were studied as candidates for soliton stars, but no stable solutions were found. The "rst systematic investigation of the problem was undertaken in [117,168,167,172]. This revealed the existence of stable regular gravitating Skyrmions and Skyrme black holes. The solution space of the models was studied also in [38]. The thermodynamics of Skyrme black holes was considered in [306]. The equations of motion of the Einstein}Skyrme model following from the 2D Lagrangian (2.40) read m"N











r sin s 1 #sin s s# r# sin s , 2 r 2

2 p"ip r# sin s s , r

(7.37) (7.38)



sin s (pN(r#2sin s)s)"p 1#Ns# sin 2s , r

(7.39)

where N"1!2im/r and i"4pGf . The length scale is given by ¸"1/ef, and the dimensionful energy is (4pf/e)M with M"m(R). The function p can be eliminated from the equations. The boundary conditions for the regular solutions read s"nl!b r#O(r), 2m"b(1#b) r#O(r) ,

(7.40)

s"ar\#O(r\), m"M#O(r\) ,

(7.41)

at the origin and at in"nity, respectively. Here b, a, and M are free parameters and l is integer.

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The #at space Skyrme model can be regarded as an e!ective theory in the low energy limit of QCD [285,286,336]. The particle-like solutions are interpreted as baryons, and the integer parameter l in (7.40) then plays the role of the baryon number. Note that l has the meaning of the topological winding number. This is because the description of the chiral "eld in terms of the SU(2) valued function U leads to the mapping SPSU(2), provided that the asymptotic condition (3.11) is imposed. The winding number is then given by Eq. (3.12), l"k[U], which formula reduces in the spherically symmetric case to l"+s(0)!s(R),/p. For l"1 the #at space solution was obtained numerically in [180]. This solution is stable. Solutions for higher winding numbers are also known, but they turn out to be unstable. There are also stable solutions for l'1, but these are not spherically symmetric. It is worth mentioning the role of the higher-order term F in the Skyrme Lagrangian in (2.38). Due to this term, the theory contains both attractive and repulsive interactions, which leads to the existence of non-trivial solutions. Consider solutions to (7.37)}(7.39) for iO0. The numerical analysis reveals that the gravitating Skyrmions exhibit the same typical features as the solutions of the EYMH models [117,172,38,306]. In particular, since Skyrmions have no charge, their behaviour is very similar to that for the EYMH sphalerons. For small values of i there is the fundamental branch of solutions with l"1 that reduce to the #at space Skyrmion as iP0 [117]. For these solutions s, m, and p are monotone functions, while N develops one minimum at some r . These solutions are stable [168].

Let us call this branch of solutions lower branch. Next, for l"1 there are also excited solutions with larger mass, which look qualitatively similar (see Fig. 14) [38,172]. In particular, s is still

Fig. 14. The l"1 fundamental Skyrmion (solid lines) and its excitation (dashed lines). On the left: the chiral "eld s(r)/n and the metric function N(r) for the two solutions with i"0.04. For iPi "0.0437 the solid and dashed curves

 merge. On the right: the mass M and the parameter b in Eq. (7.40) versus i for the lower and upper branches of solutions.

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monotone. For iP0 the excitations become in"nitely heavy and, remarkably, similar to the situation in the EYMH case, reduce to the rescaled BK solution with n"1. Speci"cally, under rP(i r, mPm/(i, cos sPw

(7.42)

the system (7.37)}(7.39) reduces in the limit iP0 to the EYM "eld equations, whose solutions are the BK sphalerons. Let us call the branch of excited solutions upper branch. The lower and upper branches for l"1 exist only for i4i . In the limit iPi the two



 branches coalesce, exactly in the same way as this happens for the EYMH sphalerons. No regular solutions exist for i'i . At the bifurcation point the stability behaviour of the solutions

 changes: all solutions of the upper branch are unstable. This change of stability agrees with the general considerations based on the catastrophe theory [306,231,308,301]. For higher winding numbers, l'1, the picture is essentially the same: there are two branches of solutions merging at some i (l). The value i (l) decreases rapidly with growing l. For iP0 the lower branch



 reduces to the #at space Skyrmion with l'1, while the upper branch again tends to the rescaled BK solution; more details can be found in [38,172]. All solutions with l'1 are unstable. It is unknown whether the lower branch solutions with l'1 admit stable and non-spherically symmetric generalizations. All solutions described above have black hole analogues. These exist for i(i (l) and if the

 event horizon radius r does not exceed a maximal value r (i). The lower and upper branches of   solutions exist in the black hole case too, and reduce to those of the regular case for r P0. In the  opposite limit, r Pr (i), the two branches coalesce. No black holes apart from the Schwarzschild   solution exist for r 'r (i). The maximal value r (i) reduces to zero as i tends to i (l).   

 To recapitulate, the critical values i (l) for the regular solutions and r (i) with i(i (l) for

 

 black holes correspond to bifurcation points where the lower and upper branches merge. These values determine the region of the parameter space (i, r ) for which non-trivial solutions exist.  A similar bifurcation picture arises in the doublet EYMH theory. It is worth noting that the solutions speci"ed by the critical values are perfectly regular. In particular, the minimal value of N is non-zero. In the monopole case, on the other hand, the limiting solutions exhibit special geometrical structures. Note "nally that, apart from the solutions described above, for which s is monotone, there are also those for which s oscillates around the value p/2. These exist for very small values of i [38,172]. The solutions of the gravitating SU(2) Skyrme model have been generalized to the gauge group SU(3), in which case their essential features remain the same [203]. The critical collapse of Skyrmions was considered in [39].

8. Concluding remarks We have given a fairly complete account of known up-to-date solitons and black holes in the basic gravity-coupled models with gauge and/or scalar "elds which respect the non-Abelian

 The charged EYMH black holes with i( show the usual bifurcation picture. 

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symmetries. These solutions show that some of the fundamental concepts of the Abelian Einstein}Maxwell theory do not extend to the non-Abelian case. Instead, a number of new typical features arise. One of such features is the appearance of the characteristic discrete structures due to the interaction of the YM "eld with gravity. These manifest in the BK solutions and their various generalizations. For those solitons which exist in the #at spacetime limit, one observes the spectrum of gravitational BK-type excitations. Another typical phenomenon, which arises in theories containing a length scale other than the Planck scale, is the existence of the upper bound for the gravitational coupling constant i and for the event horizon radius r . Non-trivial solutions can  exist only for small values of i and r . For those approaching the bound, the solutions either  become gravitationally closed or develop bifurcations, where di!erent branches of solutions merge. Such a merging is accompanied by the change of the stability properties, which can be interpreted in terms of catastrophe theory. The existence of static and non-spherically symmetric solutions, so far demonstrated at the non-linear level only in the EYM theory, is probably also typical for gravitating non-Abelian models. This is supported by the perturbative analysis for the quite general non-linear model (7.28) carried out in [273]. Such deformed solutions can be regarded as the limit of multi-soliton/black hole con"gurations when the separation tends to zero. A generic feature associated with the interior structure of non-Abelian black holes is the absence of Cauchy horizons. Let us remind in this connection the basic idea behind the mass-in#ation scenario [262]. The Cauchy horizon inside a Kerr}Newman black hole, unlike the event horizon, exists solely due to the high symmetry of the solution. In the generic situation, when deviations from the strict axial symmetry are allowed, inner horizons do not exist. Following this idea, one can argue in the same spirit that an equally e$cient way to get rid of Cauchy horizons is to add into the system more general matter. This is con"rmed by the examples of the non-Abelian black holes. For the sake of completeness, let us mention brie#y also some other important solutions for gravitating Yang}Mills "elds which remained outside the main text, because they are not of soliton or black hole type. Most of these are obtained in the pure EYM theory. 8.1. EYM cosmologies The basic idea is as follows. For the SO(4)-invariant spacetime metric ds"a(t)+dt!dr!sin r (d0#sin 0 du), ,

(8.1)

with r3[0, n], one can "nd the SO(4)-invariant YM "eld. This can be obtained either as a solution of the symmetry conditions (2.10) [160] or by requiring that the stress tensor for the spherically symmetric gauge "eld (2.13) respect the SO(4) symmetry [135]. The result is A"i

1!w(t) U dU\ where U"exp(ir T ) . P 2

(8.2)

The YM equations reduce to w #(w!1)"E ,

(8.3)

where E is an integration constant. In view of the conformal invariance, the conformal factor of the metric, a(t), does not enter the YM equations. For any solution to (8.3) the stress tensor has the

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perfect #uid structure with o"3p"3E/2a, which allows one [80,160,174,284,283,135] to solve the Einstein equations. The above solutions can be generalized to spatially open and spatially #at cosmological models [160,135], as well as to higher gauge groups [31,250,251,276]. The corresponding superspace quantization was considered, for example, in [30,79]. Since the metric in (8.1) is conformally #at and the YM equations are conformally invariant, Eq. (8.3) gives rise also to the #at space solutions for the YM "eld describing collapsing shells of the non-linear YM radiation [230,122,145]. The homogeneous and anisotropic EYM cosmological models were considered in [101,102,16], in which case the solutions turn out to be chaotic. The inhomogeneous and spherically symmetric case was considered in [282]. 8.2. Cosmological sphaleron Eq. (8.3) describes a particle in the double-well potential <(w)"(w!1). For E"1 there is a non-trivial static solution for the particle sitting on the top of the barrier separating the two wells, w"0 [174]. This solution was considered in [144,108] as a model of collapsing BK solitons. Since the bottoms of the well, w"$1, correspond to pure gauge "elds with di!erent winding numbers, which is obvious from (8.2), the static con"guration with w"0 can be naturally interpreted as sphaleron [144,107]. This sphaleron solution is distinguished by the fact that it consists of the pure gauge "eld alone, which is possible due to the interaction with the background gravitational "eld. Another characteristic feature of the solution is its high (SO(4)) symmetry. This implies that the con"guration is not localized and "lls the whole universe. Such a sphaleron solution can be used to evaluate the fermion production rate in a closed universe. This problem was considered in [320,322] in the approximation when the universe described by the metric (8.1) is static. It was assumed that the universe is "lled with (quazi)-thermal YM quanta corresponding to excitations over the trivial YM vacuum with w"1, while the sector with unit winding number is empty. As a result, there is a net di!usion of the "eld modes between the two sectors, which is accompanied by the change in the fermion number due to the axial anomaly. The corresponding di!usion rate was computed in [320,322] at the one-loop level. 8.3. EYM instantons When continued to the imaginary time, the cosmological solutions described above ful"ll the Euclidean EYM equations [313,175,314,31,341,270,110]. The metric (8.1) then becomes conformal to the standard metric on S, while the potential <(w) in the YM equation (8.3) changes sign. Solutions with "nite Euclidean action are interpreted as tunneling geometries leading to the creation of baby universes. Note that, unless the constant E in Eq. (8.3) vanishes, the YM "eld is non-self-dual. In the self-dual case one has F "*F implying that the stress tensor is zero. As IJ IJ a result, in order to obtain EYM instantons with self-dual gauge "elds one can start from a vacuum gravitational instanton and use it as a "xed background on which the YM self-duality equations are solved. Since topology of gravitational instantons can be quite arbitrary, solutions for self-dual YM "elds can di!er substantially from those in #at space [84,85,264,119,48}52,81]. It was shown in [337] that the self-duality equations for the YM "eld in axially symmetric case in #at spacetime are equivalent to the Ernst equations in GR.

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The results mentioned above, together with those discussed in the main text, almost exhaust the list of 4D solutions for gravitating YM "elds. Relatively little is known about solutions with symmetries other than spherical. At the same time, we do not discuss here solutions for DO4, since a separate review would be necessary for this (see [106,216,65] for some results in D"3). A large number of such solutions can be obtained from the 10-dimensional heterotic "ve-brane via various compacti"cations [118].

Acknowledgements M.S.V. would like to thank Othmar Brodbeck, Marcus Heusler, and Norbert Straumann for discussions and the reading of the manuscript. His work was supported by the Swiss National Science Foundation and by the Tomalla Foundation. D.V.G. thanks the Yukawa Institute for Theoretical Physics, where a part of the work was done, for hospitality and acknowledges the support of the COE and the RFBR Grant 96-02-18899.

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FINITE NUCLEI TO NUCLEAR MATTER: A LEPTODERMOUS APPROACH

L. SATPATHY , V.S. UMA MAHESWARI , R.C. NAYAK
Institute of Physics, Bhubaneswar 751 005, India
Physics Department, G.M. College (Auto.), Sambalpur 768 004, India

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

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Finite nuclei to nuclear matter: a leptodermous approach L. Satpathy *, V.S. Uma Maheswari , R.C. Nayak
Institute of Physics, Bhubaneswar 751 005, India
Physics Department, G. M. College (Auto.), Sambalpur 768 004, India Received January 1999; editor: G.E. Brown

Contents 1. Introduction 2. Nuclear matter through various approaches 2.1. Microscopic approach 2.2. Mass formula approach 3. Leptodermous expansion of energy 3.1. Introductory remarks 3.2. Leptodermous expansion of energy in energy density formalism 3.3. Is the LDM expansion unique? 3.4. Summary 4. Leptodermous expansion of nuclear incompressibility 4.1. Present status of determination of K  4.2. Various types of breathing mode vibrations 4.3. LDM type expansion of K  4.4. LDM coe$cients and dependence on modes of vibration 4.5. Convergence behaviour and pair e!ect 4.6. Dynamical e!ects 4.7. Summary

88 91 91 98 99 99 99 105 105 106 106 108 109 110 112 114 115

5. In"nite nuclear matter model of atomic nuclei 5.1. The Hugenholtz}Van Hove theorem 5.2. HVH theorem for multi-component Fermi systems 5.3. The in"nite nuclear matter model for "nite nuclei 5.4. A gedanken experiment to test INM model versus LDM type models 6. Saturation properties from nuclear masses 6.1. Extraction of saturation properties 6.2. Resolution of &r paradox'  7. Determination of incompressibility of in"nite nuclear matter 7.1. The compression model 7.2. Extraction of K  8. Summary and conclusions Acknowledgements Appendix A. Generalised Sommerfeld lemma Appendix B. Analytical expressions for LDM energy coe$cients References

* Corresponding author. Tel.: #9-674-581203; fax: #91#674#581142. E-mail address: [email protected] (L. Satpathy) 0370-1573/99/$ - see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 1 - 3

116 116 121 122 127 128 128 132 132 132 133 137 138 139 140 140

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Abstract The liquid drop model (LDM) expansions of energy and incompressibility of "nite nuclei are studied in an analytical model using Skyrme-like e!ective interactions to examine, whether such expansions provide an unambiguous way to go from "nite nuclei to nuclear matter, and thereby can yield the saturation properties of the latter, from nuclear masses. We show that the energy expansion is not unique in the sense that, its coe$cients do not necessarily correspond to the ground state of nuclear matter and hence, the mass formulas based on it are not equipped to yield saturation properties. The defect is attributed to its use of liquid drop without any reference to particles as its basis, which is classical in nature. It does not possess an essential property of an interacting many-fermion system namely, the single particle property, in particular the Fermi state. It is shown that, the defect is repaired in the inxnite nuclear matter model by the use of generalized Hugenholtz}Van Hove theorem of many-body theory. So this model uses in"nite nuclear matter with well de"ned quantum mechanical attributes for its basis. The resulting expansion has the coe$cients which are at the ground state of nuclear matter. Thus a well de"ned path from "nite nuclei to nuclear matter is found out. Then using this model, the saturation density 0.1620 fm\ and binding energy per nucleon of nuclear matter 16.108 MeV are determined from the masses of all known nuclei. The corresponding radius constant r equal to 1.138 fm thus determined, agrees quite well with that obtained from electron scattering data, leading to the resolution of the so-called &r-paradox'. Finally a well de"ned and stable value of 288$20 MeV for the incompressibility of nuclear matter K is extracted from the same set of masses and a nuclear equation of state is thus obtained.  1999 Elsevier Science B.V. All rights reserved. PACS: 21.65.#f; 21.10.Dr; 21.10.!k

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1. Introduction The program of nuclear physics envisaged in the early 1950s has been to explain why the shell model works in nuclei, starting with realistic two-nucleon interaction determined from NN scattering data and deuteron properties. To perfect the technique of the calculation of e!ective two-body interaction operative in the nuclear medium, a hypothetical system constituting a "xed ratio of neutron number to proton number and having no surface, and no Coulomb interaction between the protons was invented. The closest resemblance to this matter called in"nite nuclear matter (INM) may be found in the matter in neutron stars, super novae and the center of heavy nuclei. Its three properties namely, the binding energy per nucleon E/A, saturation density o , and  compression modulus K are supposed to be fundamental constants of nature. Their values  determine the nuclear equations of state which govern the stellar evolution and supernova explosion, etc. in the extra terrestrial domain, and the dynamics of heavy ion collisions in the laboratory. These properties cannot be measured directly, but have to be extracted from the properties of "nite nuclei through appropriate model, in a consistent manner. This is a highly nontrivial task. This has to be accomplished by following a route from "nite nuclei to nuclear matter, which has not been unambiguously laid out yet. Traditionally, the bridge connecting the "nite nuclei to nuclear matter, has been taken to be the mass formulas based on liquid drop model [1,147] or its more generalized version the droplet model [2,148]. The oldest such mass formula [1], Z #a bA#d(A, Z) E(A, Z)"a A#a A#a @   A

(1)

of Bethe and Weiszacker (BW), which gives the energy E of a nucleus (A, Z, N) of asymmetry b"(N!Z)/(N#Z), has been historically used for this connection. However, one is faced with the following discom"tures. 1. The binding energy per nucleon in nuclear matter has been assumed to be represented by the volume coe$cient a of the above formula. Whether it really corresponds to the true ground  state of nuclear matter is far from clear. In fact, in Sections 3 and 5, it will be shown that a does  not correspond to absolute ground state but an excited state of nuclear matter, with a density somewhat lower than the saturation density. In this picture, the liquid drop does not possess the full complexity of an interacting many-fermion system like nuclear matter, and it can at best simulate the average property of the system. It has no reference to the fact that a nucleus is made up of particles. The single particle property, in particular the Fermi state which forms an important quantum mechanical attribute of the system is not contained in the liquid drop. 2. The second property, i.e., the saturation density o , should in principle be determined from the  Coulomb coe$cient a of the above mass formula. On the other hand, it is assumed that the  central density of heavy nuclei corresponds to the saturation density of nuclear matter and hence, it is determined from high energy electron scattering data on nuclei. It must be stressed here that this assumption is only a hope without any concrete proof. To our knowledge, the issue of the empirical value of density of nuclear matter has still remained an open problem as it has not yet been determined in a theoretically consistent way. The main argument in favour of taking the central density of heavy nuclei as determined [3] from electron scattering to be equal

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to o is that, it is somewhat independent of the mass number A giving rise to a nuclear radius  constant r "RA\ where R is the half density radius. The central density [4] thus determined  is about, o "0.17 nucleons/fm .  This corresponds to Fermi momentum k "1.36 fm\ . $ Taking the density to be uniform, the constant radius comes out to be r "1.12 fm .  Brueckner and Gammel [5] in 1958, argued that the density of heavy nuclei would be less than nuclear matter since, the Coulomb repulsion of proton would tend to increase the radius of the nuclei. Therefore they chose a higher value o "0.24 fm\, k "1.52 fm\ and r "1.00 fm.  $  Brandow, at the suggestion of Bethe investigated this matter and showed in 1963 [6], that the surface tension force in a heavy nucleus may nearly balance the Coulomb repulsion. So the values given above as determined from electron scattering, may be accepted as representing the nuclear matter. However, in 1970 Negle from the same Bethe school emphatically concluded [7], `The saturation density cannot be directly obtained from the semiempirical mass formula. Old arguments [6] have suggested that it is roughly equal to the density in the interior of a large nucleus, but it is di$cult to believe that Coulomb, symmetry and surface e!ects compensate su$ciently to justify this assertiona. On the basis of his study with density dependent HatreeFock theory, he determined in 1970, yet another semi-empirical value for nuclear matter density as o "0.15 nucleons/fm ,  equivalently, k "1.31 fm\ and r "1.16 fm . $  Thus it can be seen that the issue of the empirical values of the saturation properties of nuclear matter is not satisfactorily solved and still open, as these have never been convincingly determined from the properties of "nite nuclei following a clean unambiguous theoretical procedure. This has led to the following ambiguity and inconsistency. The Coulomb coe$cient a (" C) in the above formula (Eq. (1)) speci"es the density  P o"3/4nr where, r is the usual nuclear radius constant. However, it is not accepted as the density   of nuclear matter, since its value obtained [8,9] in a totally free "t to masses yields r K1.22 fm,  which is much higher than the value 1.12}1.13 fm obtained from high energy electron scattering data [10] on nuclei and HF calculation [11]. All BW-like mass formula "ts, have as yet not yielded a value of r in this range. This has been termed as &r -paradox' in the literature, which   has been the subject of investigation [12,13,149] over the years, by many authors. Thus, these two properties of nuclear matter are not derived following a common path from nuclei to nuclear matter via liquid drop model. While, energy is obtained through a mass formula, the determination of density resorts to an ad hoc prescription without using any connecting link between the two entities. Further, it may be stressed that, since these two

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properties are highly interrelated, their determination from two independent sources can be considered a serious drawback in our understanding of nuclear dynamics. 3. The case of incompressibility of nuclear matter is relatively more involved compared to the other two properties. As yet, neither it has been possible to arrive theoretically at a concensous value of K , nor there is general unanimity on the method of its extraction experimentally from  the properties of "nite nuclei. Two methods for its determination from experiment have been followed in this regard. The isoscalar giant mono-pole resonance (GMR), also called the breathing mode in even}even nuclei, has been the most popular and promising source of information on K . In the "rst approach called microscopic approach, one selects a series of  e!ective interactions and performs Hartree}Fock plus random-phase approximation calculations of GMR energies for many "nite nuclei, then the interaction for which the result agrees best with the data is chosen to calculate the incompressibility of INM. Following this approach, Blaizot et al. [14] have extracted the value of K as 210$30 MeV using the GMR data of the  three doubly closed shell nuclei Ca, Zr, and Pb. Non-uniqueness of the e!ective interaction in nuclear physics, and paucity of data on GMR, have kept the issue open, although more than twenty years have passed since this estimate was done by Blaizot. In the second approach called macroscopic approach [15], using scaling approximation, one relates the "nite nuclear incompressibility K with the energies E of the mono-pole resonance as  %+0 m (2) K " 1r2E , %+0   where m is the mass of the nucleus, and 1r2 is its mean square radius. Following the liquid drop model type expansion (Eq. (1)) of energy, one writes down a similar expression for K :  K "K #K A\#K A\#K b#K ZA\ . (3)     @  Then the volume term K is identi"ed as K . Although Eq. (3) looks to be a natural path to obtain   K from K , it will be shown a posteriori in Section 4 that, unlike the case of energy, this expansion   is beset with many di$culties in regard to its convergence. Further, the number of data on GMR being only about 50 to date, no precise value of K can be extracted [16].  Thus, we may conclude from above that, presently we do not have a well founded method, following which we can arrive at the properties of INM starting from "nite nuclei. The purpose of this article is to highlight this important missing link in our understanding of nuclear dynamics, and present a new approach in which one consistently goes from "nite nuclei to nuclear matter and extracts the three properties simultaneously from one set of data namely the nuclear masses. The rationale for "nding a unique path would be to build a mass model in which the ground-state energy of a nucleus should be expressed in terms of the saturation properties of in"nite nuclear matter. Then, it will be possible to chart out a path for going from "nite nuclei to nuclear matter. It will be shown in this article that such a scheme is indeed possible by the use of Hugenholtz}Van Hove [HVH] theorem [17] of many-body theory, through which the single particle properties of the interacting many-fermion system are taken into account. This theorem has provided important guidelines in the development of Brueckner theory [18] in the past. The paper is organized as follows. In Section 2, we present a survey of the calculation of the saturation properties of in"nite nuclear matter in the many-body theories. In this section, we will

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also discuss the limitations of the liquid-drop model (LDM) mass formulas in the determination of such properties from nuclear masses. In Section 3, we analyze the theoretical foundation of the leptodormous expansion of the energy of "nite nuclei, using extended Thomas}Fermi model and energy-density formalism. In light of this, we show that the volume term of BW-like mass formulas does not represent the nuclear matter at ground-state, which has been assumed so far. Section 4 presents an analytical study of the leptodermous expansion of incompressibility of "nite nuclei and examines how reliably, K can be extracted. In Section 5, attempt is made to establish a mass  model called INM model, for which, the usual HVH theorem has been extended to describe the asymmetric nuclear matter. The generalization of this theorem with the three-body force and also for multicomponent system is discussed. In this model, we arrive at a leptodermous type of expansion, but with the di!erence that its volume and Coulomb term, etc. are ensured to be at the ground-state of in"nite nuclear matter. The foundation of INM model is examined with microscopic nuclear Hamiltonian and shown that this model is more appropriate than the conventional BW-like mass formulas to extract the saturation properties of nuclear matter from nuclear masses. In Section 6 we show that, the mass data of all known nuclei through this model yield, a radius constant r "1.138 fm in agreement with that of [19] the electron scattering data [10,11,19,20] and  E/A"!16.11 MeV corresponding to the ground-state of nuclear matter. Thus it is shown that the long standing &r -paradox' is resolved and both the properties are simultaneously determined  from the same source namely the nuclear masses. Section 7 is devoted to the problem of the determination of K from the masses of nuclei through the INM model and the value so obtained  is 288$20 MeV. The summary and conclusions are presented in Section 8.

2. Nuclear matter through various approaches 2.1. Microscopic approach Historically, Euler [21] made the "rst attempt to calculate the saturation properties of INM in second order perturbation theory taking a two-body potential of Gaussian shape. However, after the discovery of the singular nature [22,150] of the two nucleon potential (&hard core') at short distances in 1950, the inadequacy of the conventional perturbation theory for the calculation of saturation properties was recognized. So special many-body theories had to be developed. Brueckner, Levinson and Mahmoud [23] were the pioneers in initiating a new theory, which was improved and enriched further by Bethe [24] and Goldstone [25]. Jastraw [26] suggested an alternative method based on the application of the variational principle to the many-body problem. A brief outline of these two microscopic many-body approaches, and the present status, in regard to the prediction of nuclear matter properties is given below. 2.1.1. Brueckner}Bethe approach Consider a system of &A' nucleons con"ned in a large box of volume X, that obey the non-relativistic SchroK dinger equation. The Hamiltonian for such a system is given by,

 H"! # < , GH 2m G GH G

(4)

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where < is a realistic two-body potential. The hard core repulsive component of < gives rise to GH strong correlation in the motion of a pair of nucleons. Further, since the nuclear system is a low-density system, the probability of a third or more nucleons coming simultaneously close to a given correlated pair is small. In this situation, the interactions of a pair of nucleons to all order of perturbation theory, has overriding importance than the terms that need the presence of more than two Fermi sea nucleons. Therefore, it is considered proper to assume that during the collision of two nucleons, all others continue to remain undisturbed with constant momentum in nuclear matter. At "rst instance, the cluster of three and more number of nucleons are neglected and one makes the independent pair approximation [27], also called two-hole line approximation, under which the e!ective interaction G is de"ned as 1ij"<"ab21ab"G"kl2 . (5) 1ij"G"kl2"1ij"<"kl2! E GH?@ ?@$ The summation has to take into account the Pauli exclusion principle. So the single particle states a and b lie above the Fermi level F, and the pairs of states (i, j), (k, l) lie in the Fermi sea. The energy denominator depends upon the single particle energy e and is given as ? E "e #e !e !e , GH?@ ? @ G H The single-particle energies e of the states occupied in the Fermi sea are required to satisfy the G Brueckner Hartree}Fock (BHF) equation [28],









 e d " i"! " j # 1ij"G"ij2 . G GH 2m H$ The many-body ground state energy is then,

(6)

 (7) E " i"! "i # 1ij"G"ij2 .  2m GH G Before attempting the solution of Eq. (5), it is necessary to de"ne the energies e and e of the single ? @ particle states above the Fermi sea. Here we follow the guidelines led down by Rajaraman and Bethe (RB) [29]. According to these authors based on their experience on two-body G-matrix and the associated three-body corrections, these states lie high above the Fermi level, being the high momentum Fourier components, induced by the strongly repulsive short-range part of <. The kinetic energy of such states is likely to be much larger than the potential energy. So these authors recommend for the energies of these states to be just the kinetic energies;

 e "! k ? 2m ?

(8)

Earlier Bruckner et al. [30,151] had incorporated in their calculations the self consistent energy also for the particle states a and b outside the Fermi sea. This of course means, letting a particle line in the G-matrix diagram, interact with each of the Fermi-sea nucleons. Since a pair of fermi sea particles get excited above the sea, while considering the said diagram, Bruckner et al. [30,151] had to incorporate in the energy denominator of the relevant G, the e!ect in the intermediate state of the additional energy (=) of the other particle}hole pair. This has since been called o!-energy shell

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e!ect. However, in [29] RB show that, for a consistent perturbation treatment as a series in power of density (o), one should sum all diagrams having a pair of Fermi sea particles (two-body G-matrix), and then all diagrams having three Fermi sea particles and so on, because diagrams with a given number of Fermi sea particles correspond to a given power of density (o). In the present treatment we restrict ourselves to the two-body G-matrix. For consistency, following RB, the o!-energy shell diagram for a particle above the Fermi sea, which will contain three Fermi sea particles, will correspond to a higher order in density and as such to be considered along with other three Fermi-sea particle clusters. The above scheme for the ground state energy is exact under two-hole line approximation, i.e. no three and higher particle clusters are included in the de"nition of G given by Eq. (5). Although it was envisaged in the early period, subsequently it was realized that many-particle clusters, in particular the three- and four-particle clusters are important and must be taken into account. Further, the convergence of the Brueckner}Goldstone expansion, in terms of the number of hole lines has to be ensured. The pioneering role played by the Bethe school in this regard, needs hardly to be stressed. A simultaneous development which complemented this e!ort and provided a great boost in the Brueckner}Bethe (BB) perturbation calculation is a series of study initiated by Pandaripande [31,152], using constrained variational method. A number of good review articles on BB approach [4,32,153,154] are available in literature which may be consulted by the readers. 2.1.2. Variational approach The wave function for a system of &A' particles can be written as a Slater determinant, 1 det (x ) , U" G H (A!

(9)

where 's are plane wave states. G Unlike in the BB approach, such a wave function has been considered inadequate as an unperturbed wave function to be used at the starting point for the description of a strongly interacting system like nuclear matter. Following Jastrow ansatz [33], a variational trial wave function is constructed in the form W "FU , (10)  where F represents the correlation induced by the complicated two-body potential. It is generally chosen to be the product of two-body correlation function f (r ). GH  F" “ f (r ) . (11) GH GH The f (r ) are functions of a set of variational parameters. Using W as a variational trial wave GH  function, the energy, 1W "H"W 2  , (12) E"   1W "W 2   is minimized by varying the parameter set and thus an upper bound to the ground state energy E is obtained. The expectation value E involves a diagrammatic cluster expansion, analogous to  

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the Mayer expansion in the statistical mechanics of interacting system. The two-particle cluster energy can be calculated exactly; the three and higher order clusters are evaluated using Fermi hyper-netted chain}single operator chain (FHNC}SOC) approximation. The details of this method can be seen in the review articles by Day [34] and by Pandaripande and Wiringa [35]. 2.1.3. Status of saturation property calculation Since the mid-1970s, a great deal of e!ort has been made to check the goodness of Brueckner}Bethe theory through comparative studies carried out in the framework of above variational method, also called the hyper-netted chain theory. It is now believed that Brueckner}Goldstone expansion is convergent in terms of number of hole lines. The calculations of Day [36,37], and Day and Wiringa [38] for some realistic potentials have demonstrated it. In Fig. 1, the convergence of BB results in terms of the hole line expansion, is shown for the Argonne < potential [39]. The  curves marked with BB1, BB2 and BB3 denote the results obtained with two-, three- and four-hole line approximations respectively. The error bars show the estimated uncertainties in the calculation. Near the saturation, the three-hole line diagrams contribute about 5}6 MeV and the four-hole line ones about 1 MeV. Thus they are not negligible. This result supports the view that the BB theory is convergent. The results of many-body calculation obtained for various realistic modern potentials compiled by Brokmann and Machleidt [40] are presented in Table 1, and the corresponding saturation points are shown in Fig. 2. A general feature is that, out of the two properties E/A and o, if one is

Fig. 1. Energy per nucleon E/A of nuclear matter versus Fermi momentum k in two-hole line (BB1), three-hole (BB2) $ and four-hole line (BB3) approximations calculated in Brueckner}Bethe approach using Argone < potential (from  Ref. [38]).

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Table 1 Nuclear matter saturation properties E/A and k as predicted for various NN potentials with two-hole line approxima$ tion. The results shown in the square brackets are the ones obtained when three- and four-hole line contributions are added. P is the predicted %-D state of the deuteron (from Ref. [40]) " Potential

Reference

P % "

E/A (MeV)

k (fm\) $

Reference


HJ BJ RSC <  Paris HM1 Sch UNG (Bonn)C (Bonn)B (Bonn)A

[41] [42] [43] [39] [45] [46] [47] [48] [49] [49] [49]

7.0 6.6 6.5 6.1 5.8 5.8 4.9 4.4 5.6 5.0 4.4

!7.2 !8.5[!14.2] !10.3[!17.3] !10.8[!17.8] !11.2[!17.7] !11.8[!16.9] !20.2 !23.3 !12.1 !14.0 !17.1

1.27 1.36[1.48] 1.40[1.52] 1.47[1.62] 1.51[1.63] 1.48[1.56] 1.85 1.87 1.54 1.61 1.74

[36] [36] [36,44] [38] [38,45] [38,46] [46] [46] [49] [49] [49]

References to the potentials.
References for the nuclear matter calculations. Using OBEP for J53.

closer to the imperical value the other is far o!. The results form a pattern called Coester band. It is clear that, the lowest order approximation (two-hole line) is not adequate. When contributions due to three- and four-hole line are taken into account, one obtains an improved Coester band. However, the results are still far from the empirical saturation point. At this stage, it was felt that with realistic two-body interaction, there is no more scope to improve the result in BB theory and get agreement with the empirical saturation point. The possible de"ciency of BB theory was searched with great care by carrying out comparative studies in the hyper-netted chain theory. However, extensive calculations in the latter theory show that, the results are in fair agreement with the BB theory predictions. In Fig. 3, the results of Day and Wiringa [38] for the saturation properties of nuclear matter obtained with the realistic two-body Argonne < potential [39] in both the methods are compared.  The solid line corresponds to the calculation in BB theory, and the dotted lines on either side are drawn taking into account the estimated uncertainties in the calculated values. The variational calculation has been done taking two di!erent forms of the kinetic energy, namely, the Pandaripande}Bethe (PB) and the Jackson}Feenberg (JF) forms [50]. The results obtained are shown as "lled circles and crosses respectively. The uncertainties in the variational calculation are of similar magnitude as in BB calculation. Taking this into account, Day and Wiringa have concluded that, these two methods agree reasonably well with predicted saturation points of !17.8 MeV at 1.6 fm\ for the BB theory, and !16.6 MeV at 1.7 fm\ for the variational method. Further this calculation shows that, both the methods, used to their full potential, have quantitatively failed to explain the saturation with realistic two nucleon potential.

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Fig. 2. Nuclear matter saturation for various potentials (see Table 1). Open symbols denote saturation point obtained in the two-hole line approximation; symbols with a cross stand for corresponding predictions when three- and four-hole line contributions are included. Percent number refer to the D-state probability of the deuteron as predicted for the corresponding potential. The shaded rectangle denotes the empirical value of the saturation (from Ref. [40]). Fig. 3. Energy of nuclear matter as a function of Fermi momentum. The solid line denotes the "t to the Bruckner}Bethe energy; the dotted lines show their estimated uncertainties. The data and crosses stand for the variational JF and PB energies respectively (from Ref. [38]). Empirical saturation curve with a binding energy of 16 MeV at 1.33 fm\ and an incompressibility 210 MeV is shown as dashed line.

Thus extension of many-body theory including other e!ects was clearly needed. One conceivable direction of extension is to include the mesonic degrees of freedom explicitly in the many-body theory. In low energy nuclear physics, their presence is clearly manifested in the electro-magnetic properties of nuclei. The magnetic moments of neutrons and protons are substantially di!erent from those of the Dirac values. The pion-nucleon scattering cross-section is dominated by a broad resonance at 300 MeV. This is the D(1232) isobar which plays important role in the N}N scattering at low and intermediate energies. Therefore, totally freezing-out the mesonic degrees of freedom in the many-body problem may be too simplistic. Being conscious of this fact, many groups [51}54] in the past, have attempted to develop a many-body theory for the composite system of nucleons and mesons. (For a recent review on the subject, the Ref. [49] may be seen). Introduction of the additional degrees of freedom due to mesons and isobar has essentially produced two e!ects in nuclear matter; namely the medium e!ect on the two-nucleon interaction and the contribution from many-body force. Both these e!ects are substantial, however, they are of opposite sign. The medium e!ect has been found to be repulsive, while the contribution from the many-nucleon force

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is attractive. Thus, they tend to cancel without signi"cantly a!ecting the saturation properties [49]. This may be the reason why the traditional picture of nuclear system with two-body force and nucleonic degrees of freedom is largely successful. Another direction, along which the nuclear many-body theory could evolve towards perfection, is the relativistic aspect of nuclear physics. Since 1970, this branch of nuclear physics developed with considerable pace when Clark et al. [55] proposed a new method called Dirac phenomenology to describe high energy proton-nucleus scattering. In this approach, a Dirac equation with a strong attractive (scalar) and repulsive (vector) potential is solved, and the solution is used to "t quantitatively the spin observable in the above reaction, which cannot be described well by the Schroedinger equation. Simultaneously, the relativistic mean "eld theory for nuclei developed by Miller and Green [56], Serot and Walecka [57], and Brokmann [58] produced impressive success. Inspired by these developments, Shakin and co-workers [59] made a relativistic extension of many-body Brueckner theory called Dirac}Brueckner theory. This was further improved by Brockmann and Machleidt [60], and by ter Harr and Mal#iet [61]. There is no scope for comprehensive discussion on this topic here. It will su$ce for our purpose to state, the main results of this approach i.e., it could account quantitatively the remaining discrepancy between empirical saturation point and the conventional many-body theory result, obtained with the realistic two-body interaction discussed above. Thus, the Dirac}Brueckner approach could reproduce the empirical saturation properties of nuclear matter. However, this theory in its present form is not considered a complete theory. There are many criticisms [62}65]. As yet, only lowest order G-matrix calculation has been carried out in this approach. The higher order contributions have to be calculated and the convergence has to be shown. Further, for consistency, both the positive and negative energy states have to be taken into account on equal footing in the calculation. Thus, the relativistic many-body theory is still in the process of development, and we cannot regard at this stage that the problem of saturation properties of nuclear matter is satisfactorily solved taking relativistic e!ect into account. At the present time, a three-body force is being favoured as additional physical e!ect to be included in the non-relativistic theory, to compensate the de"ciency encountered so far in the saturation of nuclear matter. The idea of three-body force is quite old in nuclear physics. However, it has acquired greater importance lately because of the failure of two-body interaction to quantitatively describe the saturation and the properties of few body systems. Its contribution to the saturation property has also been considered before [4,66}69,155]. But its proper treatment using a reliable many-body method has been accomplished [70] only lately in 1980s. The measure source of this force has been considered to be due to two-pion exchange. Lagaris and Pondaripande [70,71] argued that the three body potential can be expressed in the form < " ; u (r )u (r )P (cos h ) ,  J J  J  J G J 

(13)

where ; are the strength parameters, u (r) are functions of inter-particle distances, h are the inner J J G angles of the triangle (r , r , r ) and represents a sum of three terms de"ned by the cyclic     permutation of the indices 1, 2 and 3. In a model calculation, they added the above three-body potential to the realistic < potential and carried out variational calculation to show that  saturation can be described. They varied the strength ; to get the best overall "t to the binding J

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energy, the saturation density, and incompressibility of nuclear matter. The details can be seen in Ref. [71]. This is phenomenological and clearly unsatisfactory. With only < potential,  the saturation is found with k K1.7 fm\ and E/A"!17.5 MeV, while with the addition $ of three nucleon interaction the empirical values k "1.33 fm\ and E/A"!16 MeV could  be "tted. Recently there has been considerable progress in the determination of three nucleon force at a fundamental level using proton-deutron and neutron-deutron scattering data in the framework of various "eld theoretic models [72,156] and chiral-perturbation theory [73]. With this development, it is hoped, the saturation problem will be solved and our understanding of nuclear dynamics will be "rm. 2.2. Mass formula approach In this section, we would like to examine how far the existing mass formulas are successful in determining the properties of nuclear matter through their "ts to the nuclear masses. It may be recalled that, in the mid "fties, when nuclear many-body theory of Brueckner}Bethe was in its infancy, the only existing BW mass formula (based on liquid-drop model) was used as a guide for obtaining the empirical properties of INM. The advent of shell model in late 1940s, and its overwhelming success in describing the properties of nuclei, has acquired it the status of the standard model of the nucleus. Thus, it has been recognized that shell and liquid like features are the two main features of nuclear dynamics. Mass formulas have been developed based on either of the two features as the foundation. We will be particularly concerned here with the mass formulas based on liquid-drop picture, as they o!er the promise of the determination of saturation properties of nuclear matter. The most extensive of these models is the droplet model [2,148] of Myers and Swiatecki. It originally aimed at re"ning the BW mass formula by taking into account deformation, di!useness and shell e!ects etc. It introduces two new degrees of freedom, in addition to the usual shape variables, which are associated with the deviation of the neutron and proton densities from constant bulk values, and the deviation of e!ective boundaries of the neutron and proton distribution from a common surface. It adopts the value of r equal to 1.16 fm in its latest version [74] while in the previous  versions [75] r "1.18 was taken. In fact, in the "t to known masses, r is not treated as   a variational parameter, but assigned the above value from outside. Thus, the old problem of &r -paradox' persists. Further it must be stated that, although incompressibility K occurs in the   energy expression, it is never determined from the "t of masses as a free parameter. A value of 240 MeV has been assigned to it all along. The volume term determined through droplet model is about 16 MeV which is same as was found with "ve parameter BW liquid drop model since early days. We would like to add that the "nite range liquid-drop model of Moeller and Nix [74], which is equally successful like the droplet model of Myers and Swiatecki, does not have the bulk density dilation involving the incompressibility. However, they use the value of K "300 MeV to generate  the appropriate single-particle potentials for di!erent nuclei to calculate the shell correction for the energies. They also take a preassigned value for r "1.16 fm.  Mass models based on microscopic two-body e!ective interaction by Pearson et al. [76] and Myers and Swiatecki [77] will be discussed in Section 6.2.

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3. Leptodermous expansion of energy 3.1. Introductory remarks Nucleus is a quantal many-body system. Predictably, its energy varies in a zigzag manner from one nucleus to another. At a gross level, it can be written as a sum of two parts: a smooth part and a #uctuating part. Bethe and Weizsacker, by a rare feat of imagination, compared the nucleus with a liquid drop and proposed that the smooth part can be decomposed into volume, surface, Coulomb and asymmetry terms. This decomposition, in later days called the LDM expansion, laid the foundation for theoretical nuclear physics. Even if, supposedly, one could calculate the energy at a microscopic level, this decomposition serves the most vital purpose by de"ning the concept and parameters in terms of which nuclear phenomena could be visualized. Each term in the expansion describes a speci"c property of the nucleus, e.g., the volume and surface terms describe the bulk and surface properties respectively. Thus, the LDM expansion played a central role in the evolution of nuclear physics. The thumb rule for validity of such an expansion is the so-called leptodermous approximation, i.e. a/R;1, where a and R are the surface di!useness and the size of the system. How far this approximation is valid, and whether the various terms are really independent of each other, and are not much a!ected by the #uctuating part determines the goodness of the concept. Therefore, the validity of leptodermous approximation for nuclear systems needs to be examined at a fundamental level as far as possible, using two-body e!ective interaction. 3.2. Leptodermous expansion of energy in energy density formalism In order to lay bare the underlying nuclear dynamics and "nd the various steps and approximation involved in arriving at the leptodermous expansion of energy and incompressibility, we adopt an analytical method based on energy density formalism [78], which uses Skyrme type e!ective interaction [79]. Such type of approach has been shown [80] to be an e$cient substitute for the microscopic HF method in the calculation of the average static properties of nuclei. It has also been applied successfully to the study [81,82] of the problem of giant resonances. Normally one obtains the leptodermous expansions of energy [80] and also incompressibility [83] using the concept of semi-in"nite nuclear matter [SINM]. However, as will be shown in the following, one can also obtain [84,85] the expansions in a relatively straightforward manner using the generalized Sommerfeld lemma. This is particularly suitable to study the nature of LDM expansion and its limitations in a transparent way, which is an essential motivation of the present article. 3.2.1. An anatomical model For the sake of simplicity and clarity, we show the method for a symmetric system of A fermions, without the Coulomb interaction. For such a system o "o the total energy is given by, L N



E" e[o(r)] dr ,

(14)

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with the number constraint equation,



A" o(r) dr ,

(15)

where

 e(r)" q(r)#v(r) . 2m

(16)

For the kinetic part q(r), we use the extended Thomas}Fermi functional [86}90] up to O( ), which for a symmetric system is 3 1 ( o) 1 q(r)" (1.5n)o# # o . 5 36 o 3

(17)

For the potential part v(r), we use the standard Skyrme forces without the spin}orbit contribution, t 3t #5t 9t !5t 3t qo#   ( o) . v(r)" o#  oN>#  16 16 64 8

(18)

We do not attempt to solve the Euler}Lagrange equations for self-consistent densities as our aim is to make an analytical study to "nd out the general nature of LDM coe$cients and their possible inter-relations, for which we choose the ground state density function o(r) to be a Fermi function: o  . o(r)" 1#eP\0?

(19)

Here o , R, and a are the central density, half-density radius and the di!useness parameter  respectively. Then, using the generalized Sommerfeld lemma [91,92], as de"ned in Appendix A, one can systematically express the total energy given by Eq. (14) as a sum of volume, surface, curvature and Gauss curvature terms. This series is not an in"nite expansion and the terms up to Gauss curvature order exhausts the energy, but for an exponential term exp(!R/a). This expansion becomes exact in the limit exp[!R/a];1. Considering R/a to be large and hence neglecting the exponential term is commonly referred to as leptodermous (thin-skinned) approximation. Thus, the total energy E given by Eq. (14) results in the leptodermous expansion of E as 4nR #E 4nR#E 8nR#E 4n . E"E     3

(20)

The terms E , E , E and E are the volume, surface, curvature and Gauss curvature contributions     to the total energy E. Then, the half-density radius R can be given in terms of o , by using Eq. (15)  as





na , RKr A 1!  3rA 

(21)

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where nro "1. Substituting the above expression for R in Eq. (20), and retaining terms up to the    order of A\, we obtain E "eH#eHA\#eH A\#eH A\ .     A

(22)

The expressions for the di!erent coe$cients in the above equation are given in terms of the force parameters in Appendix B. One "nds that all the coe$cients in Eq. (22) are function of the ground-state values of central density o and the di!useness parameter a. These two parameters are  normally "xed using the energy minimization criteria. As the compression modulus of the nuclear #uid has a "nite value, both these quantities are in general, functions of the size of the system, characterized by their mass number A. Therefore, Eq. (22) cannot be identi"ed as the LDM expansion of the nuclear energies. We consider the variation of the central density o with respect to mass number A of the system,  since energy and incompressibility are most sensitive to o (A). The parameter a is "xed by using the  energy minimization criteria, and the weak dependence of a on mass number A is ignored in the present treatment. To extract the A-dependence of the energy coe$cients in Eq. (22), we use the saturation condition,



d(E/A) "0 , do  M which now becomes,

(23)

e #e A\#e A\#e A\"0 . (24)     Here, the prime denotes total di!erentiation of eH's with respect to the central density variable. To G determine all the coe$cients at the saturation density o of symmetric in"nite nuclear matter, we  use the de"nition o (A)"o #do(A) , (25)   where do contains all the possible "nite-size e!ects. Making Taylor's expansion of all the coe$cients in Eq. (24) around o up to O(do), we obtain,  e A\#e A\#e A\    . (26) do"! e#eA\#e A\#e A\#0.5(e#eA\#2)do  M       As the above equation contains do on the right-hand side also, we determined the same by an iterative method. The zeroth order expression of do is equal to the above equation without the bracketed term in the denominator. To arrive at the LDM expansion of E/A and K , we should be able to expand do in powers of  A\. Considering the in#uence of higher-order terms like e, e etc. by making a binomial   expansion of the denominator in Eq. (26) and taking terms up to third order, we have,












e \L ee K1#f (n)A\#f (n)A\#2 , 1#  !   A\#2   e 2(e)  

(27)

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where,

 



ee e f (n)"!n  !   ,  e 2(e)   ee ee n(n#1) e ee  e eee ee  !   f (n)"!n !   !   !n   !   # .  2 e 2(e) 2(e) 2(e) 6(e) e 2(e)        It may be recalled here that prime denotes total di!erentiation with respect to the central density variable. Now, the LDM expansion of E/A can be obtained in a straightforward manner by performing Taylor's expansion of each of the term in Eq. (22) around o as  E "eH(o )#eH(o )A\#2#eH (o )A\       A

 







#(e (o )A\#2#e (o )A\)do     1 # (e(o )#e(o )A\#2#e (o )A\)(do)     2   1 # (e(o )#e(o )A\#2#e (o )A\)(do)#2 .     6  

(28)

Then, by using Eqs. (26) and (27) in the above equation, and grouping terms with same power of A, the complete LDM expansion of E/A up to O(A\) is E "a #a A\#a A\#a A\ ,     A

(29)

where the various LDM coe$cients in the above equation are de"ned as a "eH(o ) ,    a "eH(o ) ,   

(30)

a "eH (o )#(d e )#(d e) ,         a "eH (o )#(d e #d e )#(d e#d e)#(d e) .                The "rst and important point is that the bulk part totally decouples itself from all the surface e!ects due to the saturation condition e "0 which is not so in the case of K -expansion as we shall see in   the next section. Because of this, the two principal coe$cients in the general LDM expansion of energy, namely the surface and volume terms, remain pure, in the sense that they do not have any contributions from do. It may be noted that unlike the leptodermous expansion given by Eq. (20), the LDM expansion of energy is an in"nite series. This in"nite nature comes due to the A\ expansion of do, which is derived by making a binomial series.

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The explicit expressions for the various factors d , d , etc. are   1 (i) d "! e ,  e   1 d "! (e #e f (1)) ,    e   1 d "! (e #e f (1)#e f (1)) ,      e   1 (ii) d " e  ,  e    1 d " (2e e #e f (2)) ,  e       1 (31) (iii) d "! e  .  e    All these factors explicitly depend upon K , which in turn controls the convergence behaviour of  the LDM expansion. However, in the case of energy, the two leading coe$cients, volume and surface, are independent of these factors, and thereby K . Only the higher-order terms like  curvature and Gauss curvature are dependent on K . The dependence of convergence of LDM  expansion on K can be well understood through the quantity, do. As it was seen, the most crucial  step in arriving at the LDM expansion of any average property is the Taylor's expansion of the bulk compression variable do/o in powers of A\. It is clear from the expression for do that the  magnitude as well as the quality of the Taylor's expansion depend upon the value of K . One "nds  in general that the bulk compression variable calculated using the exact analytical expression (26) for do shows a down-turn behaviour with respect to A\, which has been discussed elaborately in Refs. [80,93]. 3.2.2. Results We now study the convergence behaviour of the LDM expansion of energy using the Skyrme forces SkMH, SkA and S3, whose parameters and properties are listed in Table 2. Firstly, we calculated the various LDM energy coe$cients using the analytical expressions given by Eq. (30) for the three Skyrme forces. Values so-obtained are given in Table 3 for the three Skyrme forces and they agree qualitatively well with those obtained [80] in ETF-SINM approach. In regard to the nuclear curvature, it may be noted that the semi-classical ETF models [2,80,94}96] give a value of about 10 MeV, as against the value of about zero, determined [74,97] from mass formula "ts. This is the so-called nuclear curvature energy anomaly. In order to show the convergence of the LDM expansion, the total energy is calculated numerically using Eqs. (14)}(19) which is then compared with the sum of all the terms given by the right hand side of Eq. (29) calculated using the coe$cients of Table 3. The results so obtained for four representative nuclei in the range 404A4200 are presented in Table 4 for Skyrme forces

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Table 2 The force parameters (t , t , t , t and p) are tabulated for the forces SkMH, SkA and S3. The lower half shows some of     nuclear matter properties (from Ref. [80])

t  t  t  t  p

(MeV fm) (MeV fm) (MeV fm) (MeV fm>N)

o (fm\)  a (MeV)  K (MeV)  r (fm)  k (fm\) $

SkMH

SkA

S3

!2645.00 410.00 !135.00 15 595.00 1/6

!1602.78 570.88 !67.70 8000.00 1/3

!1128.75 395.00 !95.00 14 000.00 1.00

0.1603 !15.776 216.7 1.142 1.334

0.1554 !15.997 263.2 1.154 1.320

0.1453 !15.857 355.4 1.180 1.291

Table 3 Values of the various coe$cients in the LDM expansion of the energy [Eq. (29)] for the Skyrme forces SkMH, SkA and S3 using the local ETF functional with  terms and a pure Fermi-function for the density distribution. All quantities are in MeV

a  a  a  a 

SkMH

SkA

S3

!15.79 19.08 10.24 !12.21

!16.01 19.95 9.67 !11.42

!15.87 18.90 6.87 !7.24

Table 4 Comparison of the exact value of the total energy per nucleon E/A, with sum of the terms in the LDM expansion of [Eq. (29)] for four representative mass numbers A with the Skyrme forces SkMH, SkA and S3. All quantities are in MeV A

40 100 150 200

SkMH

SkA

S3

Exact

Eq. (29)

Exact

Eq. (29)

Exact

Eq. (29)

!9.72 !11.35 !11.93 !12.30

!9.64 !11.33 !11.92 !12.39

!9.70 !11.39 !12.00 !12.38

!9.64 !11.38 !12.00 !12.37

!9.96 !11.56 !12.12 !12.47

!9.94 !11.55 !12.12 !12.47

SkMH, SkA and S3. It can be seen that both the results agree extremely well, i.e. up to about second decimal place. Thus, within this analytical model, we have demonstrated that the LDM expansion of energy converges quite well. When one uses more realistic forces with generalized Fermi function for the density, and as well includes Coulomb and asymmetry e!ects, the convergence is found to be equally good [80,93].

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3.3. Is the LDM expansion unique? Starting with two-body interaction, we derived above the LDM expansion of energy and demonstrated its convergence in a transparent manner. However, in the following, we show that this expansion is not unique. We follow exactly the same method as is done in Section 3.2. The steps up to the saturation condition, given by Eq. (24), remain the same. Now, in Eq. (25), instead of expanding o around INM saturation density o , we could equivalently express   o as  o (A)"o#do(A) , (32)  where o could refer to any density other than o . Then, one might do the same exercise as before  and obtain equations equivalent to Eqs. (26)}(29). Thus, we have, the LDM expansion corresponding to Eq. (32) as E "a #a A\#a A\#a A\ .     A

(33)

The "rst thing to be noted is that, irrespective of the use of Eq. (25) or (32), the forms of "nal expansions given by Eqs. (29) and (33) are the same. However, the meaning of the various coe$cients are now totally di!erent. Because, the coe$cients in Eq. (33) do not correspond to quantities at INM saturation density o , but at a di!erent density o. Therefore, when one makes  a least squares "t of Eq. (29) to the nuclear masses, there is no guarantee that the volume and Coulomb coe$cients, corresponding to saturation density o will be obtained. There is no  mechanism to constrain the coe$cients to this desired density o . This is related to the fact that  there is no concept of single particle states in the liquid-drop used in the BW like mass formulas which is purely classical in nature. This may be the reason why it has not been possible to obtain both the binding energy and saturation density of INM from a single mass formula "t so far. Further, it is interesting to "nd that once this feature is taken into account, one is able to extract consistently both the quantities from a single mass formula. A new approach with this additional feature is described in details in Section 5. 3.4. Summary 1. The LDM expansion of energy is an in"nite series. It converges quite well. 2. The volume and surface co-e$cients are fully decoupled from the higher order co-e$cients and they are also independent of each other due to the saturation condition. This feature helps in the evolution of well de"ned leading co-e$cients in the LDM expansion of energy. 3. The expansion is not unique in the sense that its co-e$cients are not necessarily at the ground state. Before we end this section, we recall the semi in"nite nuclear matter [SINM] model of Swatecki [98] which takes into account quantal e!ect and has been used to arrive at the LDM expansion [94,99]. The SINM approximation is good only for the calculation of surface co-e$cient. However, due to Friedal oscillation, as yet, it has not been possible to get respectable result [96,100] for the curvature coe$cient in this model as the integral involved in the calculation does not converge.

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How the surface structure of "nite nuclei is in#uenced by the bulk property due to quantal e!ect has been investigated [101,157].

4. Leptodermous expansion of nuclear incompressibility The compression modulus or incompressibility of in"nite nuclear matter is uniquely de"ned as



d(E/A) d(E/A) "9o , (34)  do dk  MM $ where k is the saturation Fermi momentum. On the other hand, for a "nite system such as the $ nucleus, there is no unique de"nition of compression modulus K . Usually, one de"nes [102,103]  the "nite nuclear incompressibility K in terms of a parameter g as  d(E/A) K "g , (35)  dg K "k  $

where g has the dimension of length, and characterizes the size of the system. It is sometimes identi"ed with the root mean square radius or the half-density radius. In general, representing the size of the system as gKr A, one has  d(E/A) d(E/A) K "g "9o , (36)  dg do where 4nro/3"1.  In the context of breathing mode; the two models discussed widely are the so-called scaling model [104] and the constrained Hartree}Fock (CHF) model [105}107]. In the case of scaling model, the time dependent single particle wave functions (and, hence the densities) are derived from the static, ground state wave functions using the scaling transformation; rPjr, where j is supposed to be a periodic time dependent collective variable. Then, one de"nes a &scaled' incompressibility as



d(E(j)/A) K "  dj

, (37) H where E is the total ground state energy. In case of CHF model, the nucleus is constrained by a time-dependent external "eld to have a given value of mean square radius 1r2. Here, one de"nes a &constrained' incompressibility K!&$ as  d1r2 K!&$"41r2/ , (38)  da ? where a is the Lagrange multiplier used to constrain 1r2.




4.1. Present status of determination of K



Several attempts have been made to extract K from variety of sources such as isotope shifts and  charge density di!erences [108}111], super novae explosions [112,158], neutron star masses

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[113}116,159], heavy ion collisions [117,118], etc. However, the estimates obtained lie over a wide range of values. For example, analysis of isotope shifts and charge density di!erences in the lead region suggest K &200}350 MeV. Similar order of uncertainty is also found in heavy ion studies.  Further, to arrive at a unique value of K from astrophysical studies is rather di$cult since, the  composition of the entire star structure is quite uncertain. For instance, it was found [116] that in the presence of kaon condensation, constraints on neutron star mass suggest K &200 MeV. On  the other hand, K can be even as low as about 100 MeV if kaon condensation is disallowed.  One other source of K is the nuclear breathing mode. Among the various sources just stated,  breathing mode energies are considered the most promising source of information on K . But, the  nuclear breathing mode actually determines the "nite nuclear compression modulus K (see  Eq. (2)), and not K . Therefore, to determine the value of K , one has to extrapolate from "nite   nuclei to nuclear matter. One indirect way to determine K is, as mentioned in the introduction, to  perform HF-RPA or similar calculations for breathing mode vibrations with several e!ective nuclear interactions. Then, the force which gives the best "t to experimental data determines the value of K [14]. In the second one, a more direct approach, one makes use of the concept of LDM  expansion (see Eq. (3)) to extract K with which we are mainly concerned here.  Sharma et al. [119,15] were the "rst to use this method to extract K from experimental data on  E . They determined K to be about 300 MeV using their data on various Sn and Sm isotopes,    and on those of Mg, Zr and Pb. In order to reduce the number of free parameters in their analysis, they used a physical constraint between the Coulomb coe$cient K and the volume  coe$cient K , as given by Treiner et al. [107], 





3 e 1215 !12.5 . K"  5r K  

(39)

In addition to this, they also used a presupposed value for the curvature coe$cient K of about  375 MeV. Subsequently, Pearson [120] pointed out that the constraint given by Eq. (39) is not of general validity and depends upon the force used. He then, re-examined the "ts of Sharma et al. [15] relaxing this constraint by treating K as a free parameter, and found using the seven data on  various Sn and Sm isotopes, and on Pb that, it is impossible to obtain an unique value for K , due  to the statistical instability of the "t, which leads to more than 100% error in all coe$cients in Eq. (3). Consequently, Shlomo and Youngblood [16] made an extensive least-squares "t analysis by taking into account the available set of data on breathing mode numbering about 46, spread over the mass region 284A4232, and treating both K and K as free parameters. They concluded   that this complete data set is not adequate to determine the value of K to better than a factor of  about 1.7 [200}350 MeV]. Further, they also observed in their analysis that a free least-squares "t to Eq. (3) leads to errors exceeding 100% in all the coe$cients. Thus, even though Shlomo and Youngblood overcame the drawbacks of few number of data spread over a small mass region, as used by Pearson, they still "nd equally pathological error in all the coe$cients. As observed by them, such large errors may arise partly due to the breakdown of the scaling approximation over light nuclei, and deformation e!ects, etc. One another reason may be that the presently available data are inadequate for "xing all the parameters in Eq. (3). We feel that possible slow convergence and correlations among the coe$cients will also contribute towards this error. However, it is not

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possible to pinpoint what fraction of the error arises out of the quality and inadequacy of the data, and that which arises due to the nature of convergence of the LDM expansion of K .  With this motivation, we made an extensive theoretical study [84,85] of the nature of convergence and the e!ect of correlations. Earlier studies made in this regard can be found in Refs. [15,83,107,121]. Brack and his collaborators have shown [82,122,123] that there can exist several types of density vibration, and scaling mode being a particular type. It was also shown by them using a hydrodynamical approach that, modes of density vibration and, the degree of coupling between the bulk and surface parts of these vibrations sensitively depend upon the mass number A. This e!ect is referred to as dynamical e!ect. Therefore, it is of interest and relevance to study the convergence properties of LDM expansion of K for modes other than the scaling type of mono-pole vibration,  which will be discussed here. 4.2. Various types of breathing mode vibrations The various modes of density vibration, that may be present in a "nite nucleus, can be understood in a simple manner using the pocket model of Brack and Stocker [122]. Let us consider the static ground-state density given by Eq. (19). When a nucleus undergoes mono-pole vibration, its density pro"le will be compressed or decompressed. At the compressed (or decompressed) state, it is assumed that the density function, denoted by superscript c, can still be represented by a Fermi-function. Then, one has o  , (40) o(r)" 1#exp[(r!R)/a ]  where o and a are de"ned as   o "o q ,   a "aq@"a(o /o )@ . (41)    The parameter q represents the amount of compression or decompression with respect to groundstate density, b gives the degree of coupling between the bulk and surface parts of the density  vibration, and a is the corresponding value of the surface di!useness. It must be noted that  at every stage of vibration, o and R are constrained by the number conserving equation,  A"o(r) dr. The standard scaling mode of vibration is obtained with b "!1/3. For b "0, the surface   di!useness remains unchanged at all stages of compression or decompression; this mode is referred to as pure bulk mode of vibration. When b '0, the surface di!useness changes in phase with the  change in central density, and is the so-called anti-scaling mode of vibration. As b P$R, the  mode of vibration is that of a purely surface one. In realistic hydro-dynamical calculations [82,123], which describe the experimental data on breathing mode reasonably well, it is found that b can vary from !0.23 [Pb, scaling-like  mode] to !0.84 [Zr, non-scaling mode]. Further, b tends to $R for lighter nuclei like Ca  [b "!24] and Ca [b "#9.5], i.e. pure surface like mode. These "ndings are summarized in  

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the following: A5220(approx)Ppure bulk vibration , 1204A4220Pscaling vibration , A4120Pnon-scaling vibration . Therefore, one "nds, as expected, that the degree of correlation between the bulk and surface parts, as denoted by "b (A)", increases as A decreases. Then, the obvious question is, what is the role of  b in the convergence behaviour of K -expansion? To understand this aspect, we derive in the   following subsection, the LDM expansion of K generalized to all modes of density vibration.  4.3. LDM type expansion of K  Here, we analytically derive the LDM expansion of nuclear incompressibility K starting from  "rst principles, and using the concept of leptodermous expansion of energy. For this purpose, we consider the same energy functional used in Section 3.2, given by Eqs. (14)}(18). Since our objective is to systematically study the general nature of the LDM compressibility coe$cients and their dependence on the coupling parameter b , we take the general density distribution corresponding  to a compressed/decompressed state to be Eq. (40). Thus, our whole derivation is of general nature, and the ground-state properties can be determined by imposing q"o /o "1.   Following the same procedure as detailed in Section 3.2, the total energy E at any state of compression/decompression is obtained as, E (b , q)"eH#eHA\#eH A\#eH A\ .     A 

(42)

The various coe$cients in the above equation can be readily obtained by replacing o by o and   a by a in the corresponding equations given in Appendix B. Thus, all the coe$cients are function  of the compression variable q and the coupling parameter b , in addition to the ground-state values  of central density o and the di!useness parameter a. Then the "nite nuclear compression modulus  K is calculated using the de"nition,  dE/A dE/A (43) "9 K "9o   do dq O  MM for a given value of b . Then making use of Eq. (42), we have  K "KH(o )#KH(o )A\#KH (o )A\#KH (o )A\ , (44)          where










deH KH(o )"9o G , G   do  M deH ; i"v, s, c, and o . "9 G dq O



(45)

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Now, the LDM expansion K can be obtained in a straightforward manner by performing Taylor's  expansion of each of the term in Eq. (44) around o as  K "KH(o )#KH(o )A\#2#KH (o )A\        #(KH(o )#KH(o )A\#2#KH(o )A\)do       #(KH(o )#KH(o )A\#2#KH(o )A\)(do)        (46) #(KH(o )#KH(o )A\#2#KH(o )A\)(do)#2        Here, we have included all the terms up to O(do), so that correct estimates up to Gauss curvature order i.e. O(A\), can be obtained. Then, by using Eqs. (26) and (27) in Eq. (46) and grouping terms with same power of A, the complete LDM expansion of K up to O(A\) is  K "K #K A\#K A\#K A\ , (47)      where the various LDM coe$cients in the above equation are de"ned as K "KH(o ) ,    K "KH(o )#(d K ) ,      K "KH (o )#(d K #d K )#(d K) ,           (48) K "KH (o )#(d K #d K #d K )#(d K#d K)#(d K) .                  The explicit expressions for the various factors d , d , etc. are as de"ned before in Section 3. It   may be recalled here that prime denotes total di!erentiation with respect to the central density variable for a "xed value of b . One then has for any arbitrary function f [o , a (o )],     df 1 df , " f , do  o dq O  M  df 1 df f , " . (49) do  o dq O  M  Thus, we have derived analytically, starting from "rst principles, the LDM expansion of K corresponding to the most general case, i.e. for all modes of density vibrations. In the following,  we discuss the convergence properties of this expansion and the e!ect of b in regard to the  extraction of K from GMR energies using an LDM expansion of K .  









4.4. LDM coezcients and dependence on modes of vibration Having obtained the most general expansion of K , we are now in a position to compare the  underlying structure of LDM expansions of energy and incompressibility. In case of energy expansion, we have seen that the volume and the surface co-e$cients totally decouple from other co-e$cients, and they are also independent of each other. This is due to the saturation condition

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given by e (o )"0, which ensures that there is no contribution to a from the leading order in do.    However, in the case of K , there is no such constraint on K , and hence, only the volume   coe$cient gets decoupled from all "nite size coe$cients. Consequently, K explicitly depends upon  K through the factor d . Similarly, the other important coe$cient, K , also explicitly depends    upon K as given by Eq. (39). This fact contributes to the essential di!erence between the two  expansions. A comparison between Eqs. (30) and (48) will also show that because of this di!erence, the higher order terms in K expansion get extra contributions. Further, in the case of incompressi bility, there is an additional coupling between the bulk and surface parts arising due to the dynamical factor b .  Similar to the case of energy, we now calculate the values of various compressibility coe$cients using Eq. (48) for SkMH, SkA and S3 Skyrme interactions. Results so obtained are presented in Table 5 for three values of b . The ratio "K /K " is also tabulated and it can be seen that the    scaling model result K K!K is well satis"ed for b "!1/3. It can also be seen that as    b decreases from b "0 to b "!1/2, the ratio "K /K " gets systematically lowered. Therefore, it      may be said here that as b becomes more negative, the convergence of K -expansion gets relatively   faster. From our values of higher order co-e$cients like K and K presented in the same table, we   "nd that the value of K remains more or less unchanged with respect to b . Due to this, as   b becomes more and more negative, curvature e!ect becomes as important as the surface. Hence,  even though "K /K " decreases as "b " increases, the convergence of K -expansion may not get     relatively faster as the contributions from higher order terms become important. The importance of curvature compressibility coe$cient K was "rst discussed by Treiner et al. [107], its value  as found in literature is somewhat uncertain both in magnitude and in sign. While, Treiner et al. found K to be about #345 MeV for S3 force using ETF model, Nayak et al. [83] found  within the scaling model K to be negative and about !150 MeV for the same force, which  they attributed to the use of O( ) functionals in the ETF kinetic energy density expression. From

Table 5 Values of the various coe$cients in the LDM expansion of the incompressibility [Eq. (47)] for the Skyrme forces SkMH, SkA and S3 using the local ETF functional with  terms and a pure Fermi-function for the density distribution. Three values of b have been used. All quantities are in MeV  Force

b 

K (MeV) 

K (MeV) 

SkMH (K "217 MeV) 

0 !1/3 !1/2

!406.1 !231.0 !129.1

SkA (K "263 MeV) 

0 !1/3 !1/2

S3 (K "356 MeV) 

0 !1/3 !1/2

K (MeV) 

K /K  

!109.9 !129.3 !118.0

568.1 138.1 !69.1

!1.87 !1.07 !0.6

!484.6 !295.8 !186.4

!123.7 !145.7 !138.0

595.0 167.8 !43.8

!1.84 !1.12 !0.71

!570.3 !389.6 !285.1

!114.1 !140.1 !142.7

452.5 156.2 5.26

!1.60 !1.10 !0.8

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our study [85], it is found that K is negative for all relevant values of b even with only O( )   terms in ETF functional. In a recent study [124] made within the scaling model, the value of K obtained using a "nite range, momentum and density dependent e!ective interaction is about  !150 MeV. We, further, would like to emphasize that when one makes use of an LDM expansion of K to "t  the experimental GMR data, the various coe$cients in the expansion [Eq. (47)], like K and K ,   must be A-independent, as in the case of energy expansion, i.e. nuclear mass formulas. However, the various LDM coe$cients in Eq. (47) still have a residual mass dependence due to the A-dependence of b . To make a consistent, theoretical study of the convergence properties of the expansion of K ,   the residual mass dependence must be removed. For this purpose, one needs to consider the limit APR. In this limit, the value of b is equal to zero, which pertains to the pure bulk mode.  Therefore, in the following, the convergence properties of the K -expansion, corresponding to the  pure bulk mode is studied. This has been stated to be a well converging series [123], which is not conclusive due to the so-called pair e!ect [85], as we shall see presently. 4.5. Convergence behaviour and pair ewect Here, we investigate the convergence properties of K -expansion, and then make a comparative  study of energy and compressibility expansions. For this purpose, we determine numerically, as done earlier in Section 3, the total energy using Eqs. (14)}(18) and (40), and then K can be obtained  using Eq. (43) with b "0. The values so obtained for four representative nuclei using Skyrme  forces SkMH, SkA and S3 are then compared, in Table 6, with the sum of all terms given by the right hand side of Eq. (47). It can be seen that agreement in the case of compressibility is quite good. This result then implies that the convergence of the K -expansion is almost as good as the energy  one. To understand this inference in a better way, we have presented in Table 7a and 7b, the contributions of the successive terms in the energy and compressibility expansions respectively, for the three forces and, for two representative mass numbers A"40 and 200. It can be seen that in the case of energy, the successive terms decrease by a factor of about 5, giving rise to a rapidly converging series. In contrast to this, we "nd in the case of compressibility, the Gauss curvature term K A\ is of the same order as the curvature term K A\. In fact, for values of A below   Table 6 Comparison of the exact values of the "nite nuclear compression modulus K for b "0 with the sum of the LDM   expansion of K [Eq. (47)] for four representative mass numbers A with the Skyrme forces SkMH, SkA and S3. All  quantities are in MeV A

40 100 150 200

SkMH

SkA

S3

Exact

Eq. (47)

Exact

Eq. (47)

Exact

Eq. (47)

103.4 130.2 140.4 147.1

102.7 129.7 140.1 146.8

126.9 159.5 172.0 180.0

125.9 159.1 171.7 179.8

191.3 232.3 247.4 257.1

190.3 231.9 247.1 256.9

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Table 7 Values of the di!erent terms in the LDM expansion of (a) energy per nucleon (E/A) and (b) incompressibility K for two  representative nuclei of mass number A with the Skyrme forces SkMH, SkA and S3. All quantities are in MeV (a) Force

A

a 

a A\ 

a A\ 

a A\ 

SkMH

40 200

!15.79 !15.79

5.58 3.26

0.8755 0.2994

!0.3052 !0.0611

SkA

40 200

!16.01 !16.01

5.83 3.41

0.8268 0.2828

!0.2855 !0.0571

S3

40 200

!15.87 !15.87

5.53 3.23

0.5874 0.2009

!0.1810 !0.0362

Force

A

K 

K A\ 

K A\ 

K A\ 

SkMH

40 200

216.6 216.6

!118.7 !69.4

!9.40 !3.21

14.20 2.84

SkA

40 200

263.3 263.3

!141.7 !82.9

!10.58 !3.62

14.88 2.98

S3

40 200

355.5 355.5

!166.8 !97.5

!9.76 !3.34

11.31 2.26

(b)

about 150, K A\ even overshoots K A\. Same is also found to be true in our calculation for   the higher-order terms K A\ and K A\. Further, it is interesting to note that K A\ and    K A\ are almost equal in magnitude, but opposite in sign, which is also the case with K and K .    Hence, we "nd that the terms of higher order than the surface one in the LDM expansion of K cancel pairwise. This pair e!ect gives rise to a misleading conclusion regarding the importance  of higher-order terms in the LDM expansion of K , unless otherwise investigated. Thus, the LDM  expansion of K with b "0 shows an anomalous behaviour, in the sense that K A\K    !K A\, in contrast to the rapidly converging energy expansion. The above results were found  to remain valid even when one uses more realistic ETF functionals including  terms, and with a generalized Fermi-function for o(r). Moreover, it may be mentioned here that the convergence of K -expansion is normally studied  [83] within the scaling model, going up to curvature order. In our study, including terms up to Gauss curvature order, we arrive at similar conclusions. More importantly, there appears no pair e!ect in the scaling mode. Thus already one sees that convergence behaviour of K -expansion  depends much upon the type of density vibration involved. But, it may be recalled here that, the mode of density vibration itself di!ers from one mass region to another. Therefore, in realistic situations, the K expansion pertains neither to pure bulk mode nor to scaling mode. In such  circumstances, how good is the convergence of LDM expansion of K ? This is investigated in the  following.

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4.6. Dynamical ewects We have analyzed above, the leptodermous expansion of K for the pure bulk mode of vibration,  in which the coupling b between the bulk and surface parts is zero. However, we have shown that,  in general, the mode of vibration changes from one mass region to another in the nuclear chart, and further the degree of coupling between the surface and bulk parts depends sensitively on A. Hence, it is very much desirable to examine the convergence properties of LDM expansion of K taking  into account this dynamical e!ect. It has been shown by Brack and Stocker [123] that, for a given nucleus (A, Z), the optimum value of b that will give rise to an excitation energy E close to the experimental value can be obtained  %+0 using the empirical relation, b "bM !(bM #1 .  where the mean value of coupling parameter bM is related to the mass number A as, bM "0.685A!2.15 .

(50)

(51)

Using this, we now study the general behaviour of K versus A for which the expression for K is   taken to be K (b )"K #K (b )A\#K (b )A\#K ZA\#K b , (52)         @ where K and K are, respectively, the Coulomb and symmetry incompressibility coe$cients, and  @ as usual, the asymmetry parameter b"(N!Z)/A. The values for K and K obtained with SkMH  @ are !4.70 and !349.0 MeV. In the calculation of K , we have considered only the direct  Coulomb term a "e/r . The coe$cients K and K are calculated using Eq. (48) for each mass      number A for the optimum value of b (A) determined from Eq. (50). Therefore, the above expansion  of K pertains to realistic situations. This is because, through the b -dependence of K and K , we     have taken into account the dynamical e!ect (A-dependence of b ) present in real nuclei. Further,  the optimum value of b [Eq. (50)] used for a given nucleus is derived from a "t to experimental  data on breathing mode energies. Therefore, the above LDM expansion of K calculated with the  optimum value of b , mimics most (except for e!ects like deformation) of the important e!ects  present in realistic situations. The values of K (b )/K so obtained are presented as a function of    A\ in Fig. 4. It is interesting to see that the quantity shows a &up-turn' behaviour for mass number less than 120. To understand this behaviour, we have calculated K /K without including   the dynamical e!ect. This is done by taking "xed value of b for all nuclei in the calculation. Three  sets of calculations are done with b "0,! and !, the results of which are presented in Fig. 5.    In all these cases, we "nd linear behaviour for K versus A\ which implies that higher order  e!ects such as curvature and Gauss curvature may be omitted for these values of b . This result is in  sharp contrast to the &up-turn' behaviour observed when dynamical e!ects are taken into account (see Fig. 4). The &up-turn' behaviour or change in the slope in Fig. 4 as against linear behaviour suggests that higher order co-e$cients such as K become important over medium- and low-mass regions.  This signals the break down of the leptodermous expansion of K below mass number 120.  Indication for such an increasing nature of K can also be found in the hydro-dynamical 

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Fig. 4. Values of the ratio of "nite nuclear incompressibility K to K obtained including the dynamical e!ect (A  dependence of b ) are shown versus A\. The force used is SkMH.  Fig. 5. Values of the ratio of "nite nuclear incompressibility K to K obtained for three values of the coupling parameter   b are shown versus A\. The force used is SkMH. 

calculations. In the analysis of the data of real nuclei, one should indeed take into account these dynamical e!ects. Therefore, although the leptodermous expansion of K is found to be valid for  particular values of b (such as scaling or pure bulk mode), the general expansion including the  above discussed dynamical e!ects is not as good as the particular cases. In view of this observation, and the fact that the LDM expansion is inherently not unambiguous, we may conclude that extraction of a unique value of K from GMR energies using K -expansion is beset with serious   problem. 4.7. Summary In the K -expansion, in contrast to the energy expansion, the bulk part gets strongly coupled  to surface properties explicitly through the function do as well as implicitly through the dynamical factor b . In addition, due to the various types of breathing mode vibrations present in  real nuclei, the convergence of LDM expansion of K shows &non-leptodermous' behaviour below  mass number approximately 120. This &non-leptodermous' behaviour coupled to the fact that the LDM expansion is inherently not unambiguous, may well be a reason for the large uncertainty found, in the determination of K from GMR energies, by Shlomo and Youngblood. Therefore, in  contrast to the energy expansion, the LDM expansion of K is not quite suitable for the extraction  of K .  In the following sections, we "rst discuss a model based on many-body theoretic foundation, which yields a unique LDM type expansion of energy, and then show saturation properties of INM and K can be extracted from nuclear masses using the same expansion. 

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5. In5nite nuclear matter model of atomic nuclei The liquid drop picture of Bethe and Weiszacker which forms the basis of their mass formula and "ssion theory etc., is classical in nature. It does not have the concept of single-particle property, in particular the Fermi state, an important property of all interacting many-fermion systems. This fact is manifested in the non-uniqueness of the LDM expansion, obtained using the energy density formalism, shown in the preceding section. Therefore such an entity termed here as the BW liquid, does not possess the attributes of a liquid corresponding to quantum mechanical system, which is the case with the real nuclei. Thus it is more appropriate, and also closer to reality, to develop a mass model using the in"nite nuclear matter as the basic ingredient, which may be termed as INM liquid, in contrast to classical BW liquid. The complexity of the many-body system would be imbibed by such a matter. The present section deals with a model called INM model [125}132,160] developed in recent years for the ground state masses of nuclei exclusively built on the foundation of nuclear matter. It uses the Hugenholtz}Van Hove (HVH) theorem [17,133] of many-body theory through which the essential characteristics of the system, in particular, the single particle properties are incorporated in the model. Then LDM expansion corresponding to INM liquid for "nite nuclei is obtained in a natural manner.Thus an unambiguous way to go from "nite nuclei to nuclear matter can be found. In this section, "rst we discuss about the key input of the INM model, namely, the HVH theorem and its generalizations to many-body forces and multi-component systems. This would be followed by the details of the INM model and its uses. 5.1. The Hugenholtz}Van Hove theorem The HVH theorem [17] deals with the single-particle properties of an in"nite Fermi gas with interaction, at absolute zero of temperature. In general, it refers to a relation amongst the Fermi energy, the average energy per particle, and the pressure of the system. Before we discuss this relation quantitatively , the meaning of single particle states in an interacting many-fermion system needs to be clari"ed. To be speci"c; is the concept of single-particle states physically meaningful in such a system? Only under HF approximation, when the residual interactions are neglected, then the system can be assigned well de"ned single particle states. However, in the general case, as has been shown [17] by Hugenholtz and Van Hove, a single particle state has meaning only for particles of momentum k close to the Fermi momentum k . The single particle states in such  a system are meta-stable having a life time, which approaches in"nity in the limit k tending to k .  The energy of this speci"c state is de"ned as the Fermi energy e , and it is the only true  single-particle state of the system in the sense of being stable having in"nite life time. Then the theorem states that for a system of A particles having total energy E,




R(E/A) RE E #o " Ro X RA A

,

(53)

X

where o and X are the number density and volume of the system, respectively. The derivatives are taken at constant volume X. The celebrated work [17] of Hugenholtz and Van Hove has been to

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prove the relation




RE "e .  RA X

(54)

Then the above relation (53) which is only an identity, (can be mathematically shown) now becomes the HVH theorem



E R(E/A) #o "e .  A Ro X

(55)

This relation is general and valid at any density. The second term in this relation is an important physical quantity and can be interpreted as the pressure of the system. For a saturating system at ground state, i.e. at equilibrium, [R(E/A)/Ro]X vanishes, and one obtains as a special case, E "e .  A

(56)

Thus for an interacting Fermi system in the ground state, its average energy becomes equal to the Fermi energy. This is a rare theorem in many-body physics, which is rigorously true [17] up to all orders of perturbation. It is also remarkable that the theorem does not depend upon the precise nature of the interaction. It is in general valid for any interacting self bound in"nite Fermi system, so it is applicable to liquid He, and in particular to nuclear matter. With its help, Hugenholtz and Van Hove found [17] internal inconsistency in the early nuclear matter calculations of Brueckner [18]. In order to understand the dynamical contents of the theorem, and make use of it, we attempt to prove the above theorem under di!erent physical situations. 5.1.1. HVH theorem under HF approximation We follow Bethe [24] in this regard, who obtained this relation in 1956 using HF approximation, before Hugenholtz and Van Hove gave the general proof in 1958 considering the complete many-body theory. For this, we consider a system of A fermions con"ned in a large box of volume X. The density o"A/X remains "nite even though A and X both tend to in"nity. For simplicity, we assume the particles to be spinless. Taking the e!ective interaction G between the particles to be of two-body type, the single-particle energy e and the total energy E of the system under HF approximations are





 e " i"! "i # 1ij"G"ij2 , G 2m H





 1 E" i"! "i # 1ij"G"ij2 , 2m 2 GH G

(57)

(58)

where "i2, " j2 etc. are the occupied single particle states and 1ij"G"ij2 denotes the antisymmetrized matrix elements of G. For an in"nite Fermi system, the single-particle states can be represented as

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plane waves and hence, the above expressions written in momentum space are



X I

k # e" 1kk"G"kk2 dk , (59) I (2n) 2m  1 X I I X I k 1kk"G"kk2 dk dk . (60) dk# E" 2m 2 (2n) (2n)    In both these expressions, the second term can be easily identi"ed as the single-particle HF potential <(k) given by







X I <(k)" 1kk"G"kk2 dk , (61) (2n)  where k is the Fermi momentum related to the density o"k/(6n). Further in the total energy   expression (Eq. (60)) the 1/2 factor in the potential energy is taken to avoid double counting of the two-body mutual interactions. Now in Eq. (60), we see that the total energy is a function of the Fermi momentum k , and hence its derivative with respect to k at constant X can be obtained as   follows:






4nX k 1 4nkX I RE #  [1kk "G"kk 2#1k k"G"k k2] dk "     2 (2n) Rk X (2n) 2m   1 X I I RG # kk" "kk dk dk . (62) 2 (2n) Rk    The two matrix elements occuring in the second term are identical due to antisymmetry property and hence they can be combined with the  factor dropped. G in general depends upon k or   equivalently density. However, it has been shown by Bethe [24] that it is a good approximation to assume the variation RG/Rk to be quite small to be neglected at normal and higher densities. Thus  neglecting the third term one arrives at










(63)

,

(64)

4nX k 4nX RE I # " k 1kk "G"kk 2 dk .   2m(2n) (2n)  Rk X   Using the identity







RE RE Rk  " RA X Rk X Ro  Eq. (63) can be written as




Ro RA



X

RE

k X I 1kk"G"kk2 dk"e (65) " #  RA X 2m (2n)  which can be easily identi"ed as the Fermi energy in HF theory from Eq. (59). Consequently by using this relation in the identity (53) the HVH theorem follows in a natural way. For the special case of the ground state we impose the saturation condition R(E/A)/Ro"0, which is nothing but

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the variational condition satis"ed by the HF solution. Thus the relation (56) under HF approximation is also proved. The proof given here, can be easily generalized when spin and isospin degrees of freedom are included. It can also be shown to be valid for three body and in fact, for multi-body forces. 5.1.2. HVH theorem: exact many-body approach Here we attempt to highlight the essential features of the many-body treatment given by Hugenholtz and Van Hove who have proved the theorem exactly. For a comprehensive account of the exact treatment in terms of complete many-body theory, the interested readers may refer to the original papers of the authors [17,134}136,161,162]. The book by K. Kumar [137] on this topic may also be consulted. Here we just focus on the nature of the single particle states, in particular the Fermi state in an interacting many-fermion system, which is of fundamental importance. (a) Resolvant operator method. The approach followed by them is the solution of the time independent many-body Schroedinger equation using resolvant operator method [17,137]. In this method, the stationary state of the system is related to the pole of the expectation value of the resolvant operator obtained with the corresponding unperturbed state. In this approach, the system is visualized as a collection of N fermions enclosed in a box of volume X, such that the density of the system o"N/X remains "nite, even though both of them tend to in"nity. The complete Hamiltonian of such a system is then given by H"H #< ,  where

k G aR a , H "  2m IG IG IG 1 <" v(k , k , k , k )aRaRa a  .     I I I I 4

(66)

(67) (68)

Here aRG and a G represent creation and annihilation operators for a particle of momentum k , I G I obeying anti commutation rule X d . +a G, aRH," I I (2n) IGIH

(69)

The ground state " 2 of the unperturbed (non-interacting) system is the state, where all the particle  states of the Fermi sea are occupied up to the Fermi momentum and can be written in terms of the single particle creation operators as " 2"aRaR22aRL"02 . I I I  Any other arbitrary state of the unperturbed system can be written as

(70)

(71) "k , k 22k ; m , m 22m 2"aRaR2aRNa 22a O" 2 , I K K    N   O I I where by convention k , k ,2, etc. denote particle states and m , m ,2 denote hole states. So     "m "(k and "k "'k . (72) G  G 

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Now the question is when the interaction < is switched on, how do the unperturbed states evolve towards the stationary states of the complete system. Such questions are answered by analyzing the poles of the expectation value of the resolvant operator de"ned by R(z)"(H!z)\"(H #
(73)

R(z)"(H !z)\!(H !z)\<(H !z)\    #(H !z)\<(H !z)\<(H !z)\#2 . (74)    Then for the evaluation of the expectation value of the resolvant operator for a given unperturbed state, special diagrammatic technique has been used. Finally the state under consideration exists or not is decided by the existence or non-existence of a pole of the expectation value. For example, consider an unperturbed state (say) "a2. If the expectation value of the resolvant operator R(z), i.e., 1a"R(z)"a2 has a pole, then the interacting system is said to have a stationary state corresponding to the state "a2. Such a state would have de"nite energy and in"nite life time. As for the ground state " 2, the expectation value of the resolvant operator D (z)"1 "R(z)" 2 is found to have a pole     at the energy E , which is the ground state energy of the interacting system. Thus " 2 evolves to   the stationary state "t 2, and the ground state of the interacting system is de"ned by  H"t 2"E "t 2 . (75)    (b) Single particle properties of the interacting system. Now consider an unperturbed state of the system with an additional particle having momentum k("k"'k ) represented by  (76) "k;2"aR" 2 . I  For such a state, the corresponding expectation value of the resolvant operator can be written as 1k;"R(z)"k;2"D (z) . (77) I The evaluation of such a matrix element in terms of the perturbation theory involves the expansion of the operator R(z) as shown above (Eq. (74)) in powers of the interaction, and hence can be handled through diagrammatic approach. In doing so, one has to use two types of diagrams: Connected diagrams with one external particle line at both ends, and diagrams without external line. The latter type is called ground-state diagrams. Thus following this procedure the matrix element D (z) has been shown [17] to be I D (z)"DM (z)*D (z) , (78) I I  where * indicates the convolution product of two complex functions and DM (z) is de"ned as a series I in increasing powers of the interaction potential, each of which can be represented by means of connected diagrams of the "rst type. D (z) pertains to the ground state of the system and is  expressed as the sum of the diagrams of the second type. D (z) has been found to have no pole. I Hence the unperturbed state "k;2 of the system cannot evolve to a stationary state. Rather such a state is a decaying state and has been termed as a dissipative [17] state by Hugenholtz and Van Hove. However when k tends to k , the situation becomes di!erent which can be studied by  analyzing the nature of the DM (z) function under this limiting condition. It is found that DM (z) has I I

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a singularity which is simply a cut in the complex plane along the real axis, starting from a point e up to #R, the value of which is independent of k. The real part of DM (z) varies continuously  I across this cut, while its imaginary part is found to change its sign, thus giving rise to a discontinuity. In the limit "k"Pk , the point e approaches the branch point e , the di!erence (e !e ) being  I  I  proportional to ("k"!k ). The situation very much resembles that of a Gaussian function for which  the width C of the peak behaves as (e !e ). So that for "k"Pk , the width of the peak tends to I I   zero, resulting in a d-function. In the case C is very small [C ;(e !e )], a state vector t can be I I I  I constructed which is a meta-stable state with an approximate energy (e #E ) and a life time C\. I  I The meta stable character of the single particle states "t 2 can be seen from I 1t "e\ &R"t 2"exp[!i(E #e )t!C t] . (79) I I  I I The energy e should now be interpreted as the energy of a particle-state with momentum k'k , I  which is meta-stable. In the limit "k"Pk , the single particle energy e tends to e . This limiting value  I  of e , i.e., e is called the Fermi energy. Thus in an interacting many-fermion system, there is truly I  one single particle state with in"nite life time while others are meta-stable. Such a state with an additional particle at the surface of the Fermi sea is a stationary state of the system possessing energy E #e .   Once the Fermi energy for the system has been de"ned, then using the relation between DM (z) and I D (z), the derivative of the ground-state energy E with respect to particle number A at constant   volume has been shown [17] to be equal to e and Eq. (54) is thus obtained leading to the usual  HVH theorem. It must be emphasized that, more than proving a very useful theorem in many-body physics, Hugenholtz and Van Hove discovered the true nature of single particle states of the interacting Fermi systems. It may be recalled how this theorem found immediate application in detecting the internal inconsistency in the early calculation of Brueckner and Gammel [5] for nuclear matter, who had obtained the average energy E /A"!15 MeV and Fermi energy e "!34 MeV. This discrep  ancy of about 20 MeV was later removed when they took into account a class of diagrams containing three and more particle clusters, which were otherwise neglected. 5.2. HVH theorem for multi-component Fermi systems 5.2.1. Asymmetric Fermi systems The HVH theorem as stated above and used in literature is valid for many-body systems consisting of one kind of fermions. It is also valid for symmetric nuclear matter which has same number of neutrons and protons. However, it is more interesting and useful if this could be generalized [133,138] to asymmetric Fermi systems and in particular, asymmetric nuclear matter. Consider asymmetric nuclear matter containing N neutrons, Z protons and mass number A"N#Z. The system is visualized to be enclosed in a volume X such that when both A and X tend to R, o remains a constant. Let its ground state energy be E and its number density be o. Its total ground-state energy E can be considered as a function of (N, Z) or alternatively (A, b), b"(N!Z)/(N#Z) is the asymmetry parameter. Here unlike symmetric system we shall have two Fermi energies, namely e "(RE/RN)X and e "(RE/RZ)X for neutron and proton 8 N , L

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respectively. Then the right-hand side of Eq. (53) can be expressed as








(80)

E R(E/A) 1 #o " [(1#b)e #(1!b)e ] . L N A Ro 2

(81)

RE RE " RA X RN

RN RE RZ 1 # " [(1#b)e #(1!b)e ] , L N RA RZ X RA 2 X 8 @ , @ which enables us to write Eq. (53) in a generalized form

Obviously for a symmetric system (b"0) and also for a system containing one kind of particles (b"1) the above generalized HVH theorem reduces to the usual relation (55). Further, for such a system under equilibrium with saturating forces the pressure term vanishes and hence we arrive at E 1 " [(1#b)e #(1!b)e ] . L N A 2

(82)

Both the Eqs. (81) and (82) can be termed as the generalized HVH theorem. In our subsequent discussion, we will refer Eq. (82) as the generalized HVH theorem, which will be mostly used in developing into the in"nite nuclear matter model. Explicitly this simple relation connects three fundamental observables of an asymmetric system, namely the average energy per particle and the two Fermi energies. The proof can be generalised to the multi-component systems. 5.3. The inxnite nuclear matter model for xnite nuclei 5.3.1. The INM model We have argued in the introduction of this section, that the basic ingredient of the mass model of nuclei should be in"nite nuclear matter/INM liquid, rather than classical liquid usually used in BW like mass formulas. This view is further reinforced, when one compares quantitatively the contribution of the volume term to that of all other terms put together, in the above mass formula. Moreover, the near constancy of the nuclear radius r , as determined from the electron scattering  on heavy nuclei, provides additional supports to this view. Thus, the preponderance of the INM feature over the "nite ones in real nuclei is self-evident. With emphasis on this feature as the basis, the INM model of nuclei was proposed [125] in 1987. Over the years, its success as a mass model with unique ability to predict masses of nuclei far from stability and its potential for extraction of saturation properties of nuclear matter has been recognised [125}132,160]. In this model, the nuclei are considered to have two distinct categories of properties namely the universal and the individualistic. The liquid like behavior of nuclei can be called as their universal property. The shell, deformation and di!useness, etc. can be termed as their individualistic or characteristic properties. The nuclei di!er from one another because of these individualistic properties, which may be called as their "nger prints. 5.3.2. INM equations Consider a nucleus with neutron number N, proton number Z and mass number A"(N#Z) and asymmetry parameter b. We aim at "nding the ground state energy E$(A, Z) of this nucleus. In

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this model, the ground state energy E$(A, Z) is considered equivalent to the energy E1(A, Z) of a perfect sphere made up of in"nite nuclear matter at ground state density o with the same  asymmetry b plus the residual energy g, called the local energy, which contains all the characteristic contributions like shell, deformation, etc. So E$(A, Z)"E1(A, Z)#g(A, Z)

(83)

E1(A, Z)"E(A, Z)#f (A, Z) ,

(84)

with

where








3  Z A\#a' Ab#a' A!d(A, Z) , f (A, Z)"a' A#a' Z!5    ! 16n

(85)

denotes the "nite size e!ects and E(A, Z) is the energy of the portion of in"nite nuclear matter contained in the sphere. Its value is entirely determined by the energy density of INM and will be called as in"nite part. This sphere is hereafter referred to as the INM sphere, and the superscript I stands for the INM nature of the coe$cients. Here a' , a' , a' and a' are the surface,     Coulomb, surface-symmetry and curvature coe$cients and d(A, Z) is the usual pairing term, given by



#*A\ for even}even nuclei

d(A, Z)"

0

for odd-A nuclei

(86)

!*A\ for odd}odd nuclei

Eq. (83) now becomes E$(A, Z)"E(A, Z)#f (A, Z)#g(A, Z) .

(87)

Thus the energy of a "nite nucleus is written as the sum of three distinct parts: An in"nite part E, a "nite part f and a local part g. All these three parts are considered distinct in the sense that each of them refers to a di!erent characteristic of the nucleus and as such, is more or less independent of other. Eq. (87) is our required mass formula which provides a direct link between "nite nuclei to nuclear matter. Its three functions E, f and g are to be determined. A. Determination of the xnite part f. The term E(A, Z) being the property of nuclear matter at the ground state, must satisfy the generalized [133] HVH theorem given by Eq. (82). Using Eq. (87), the INM Fermi energies e and e can be expressed in terms of their counterparts of "nite nuclei L N denoted through superscript &F' as e "e$!(Rf/RN) !(Rg/RN) , L L 8 8 e "e$!(Rf/RZ) !(Rg/RZ) , N N , ,

(88)

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where e$"(RE$/RN) and e$"(RE$/RZ) . Using Eqs. (87) and (88), the relation (82) can be recast L 8 N , as E$ g 1 # " [(1#b)e$#(1!b)e$]#S(A, Z) L N A A 2









Rg Rg 1 #(1!b) , # (1#b) RN RZ 2 8 ,

(89) (90)

where S(A, Z)"f/A!(N/A)(Rf/RN) !(Z/A)(Rf/RZ) , (91) 8 , is a function of all the "nite-size coe$cients a' , a' , a' and a' which are global in nature. As noted     earlier, the local energy g refers to a speci"c characteristic of the nucleus and it cannot mix with the global quantities. Therefore, we assume that sum of the terms involving g on the right hand side of Eq. (90) is equal to a similar term on its left hand side. Hence we arrive at the following two relations E$ 1 S(A, Z)" ! [(1#b)e$#(1!b)e$] , L N A 2

(92)

g(A, Z) 1 " [(1#b)(Rg/RN) #(1!b)(Rg/RZ) ] . 8 , 2 A

(93)

The validity of these two equations has been well demonstrated elsewhere [125,128,130,133], which has amply justi"ed our above assumption. Eq. (92) is remarkable, as it relates three properties of a nucleus, namely, the ground state energy, the neutron and proton Fermi energies. Hereafter, these two equations (92) and (93) will be referred to as "nite-size coe$cient (FSC) equation and the local energy equation, respectively. It may be noted that Eq. (92) does not contain the in"nite part E as well as the local energy part g. Thus through Eq. (92), decoupling of the "nite component f (N, Z) from the in"nite one E and local energy g has been achieved. The "nite-size coe$cients a' , a' etc. can be determined by "tting the   function S(A, Z) with the combination of data E$/A![(1#b)e$#(1!b)e$] of "nite nuclei. L N  Eq. (92) can be cast into the mass relation (1!b) (1#b) [E$(N, Z)!E$(N!1, Z)]! [E$(N, Z)!E$(N, Z!1)]"S(A, Z) E$/A! 2 2 (94) among three neighboring nuclei with neutron and proton numbers (N, Z), (N!1, Z) and (N, Z!1), whose success [125,133] has been demonstrated elsewhere. Features of FSC equation: A critical study of the underlying features of the FSC equation (92) is given below. a. Coulomb related terms. In "nite nuclei, besides the direct Coulomb term, there are other Coulomb-related terms like exchange Coulomb, proton form factor and Nolen}Schi!er charge asymmetry which are normally taken into account in the LDM mass formulas [74]. Thus, the

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direct Coulomb term is correlated with all these terms which are likely to in#uence it. In view of this, it will be of importance to examine their roles in the present model. The e!ect of the exchange Coulomb term in the model can be studied by including it in the de"nition of f function given by Eq. (85). This will contribute to f an additional term f given by  Z , (95) f (A, Z)"a  A where a "!a' 5(3/16n). This form of the exchange Coulomb is due to Slater [139] which has   been widely used [140] in the literature. The S(A, Z) function [Eq. (91)] corresponding to the above term is





a R R S (A, Z)"  Z/A!N (Z/A)!Z (Z/A) "0 .  A RN RZ

(96)

Thus this term exactly cancels in the S function. Similarly it can be seen that the proton form factor term a (Z/A) and Nolen}Schi!er charge asymetry term a (N!Z) exactly cancel.   Thus, all the three Coulomb related terms exactly cancel in Eq. (92), hence a' can now be  properly determined. b. Surface symmetry and higher order terms. The surface-symmetry term f "a bA is the next   higher order term which as such, is much smaller compared to the leading terms Coulomb and surface. This term cancels to a major extent of 66% due to the special combination of data used in the FSC equation. Although this does not fully cancel like the exchange Coulomb, proton-form factor correction terms, etc., a' being a second order term, such cancellation makes it rather  insigni"cant compared to the principal terms like a' . At the numerical level it may be considered  virtually canceled. Similarly the curvature term a A also becomes insigni"cant.  c. Rigorous decoupling of xnite part. An important aspect of the FSC equation (92) is its practical application which involves the use of Fermi energies (RE$/RN) and (RE$/RN) for "nite nuclei. Depending upon the accuracy of the formula adopted for such quantities, important improvements could be achieved. When one uses the expressions for Fermi energies as





RE$ "[E$(A, Z)!E$(A!1, Z)] , RN 8 RE$ "[E$(A, Z)!E$(A!1, Z!1)] , (97) RZ , the decoupling of the in"nite part (asymetry term) from the "nite part is not exact and a small contribution a (b!1)/(A!1) survives (of the order of a /A) in Eq. (92). This has a!ected to some @ @ extent the coe$cients a' and a' in our earlier studies [125,128]. We use better formulas [130]   RE$ 1 " [E$(A#1, Z)!E$(A!1, Z)] , RN 2 8 1 RE$ " [E$(A#1, Z#1)!E$(A!1, Z!1)] , (98) 2 RZ , resulting in the following improvements.





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1. The decoupling occurs up to an order of a /A, which can be considered perfect at the numerical @ level. 2. The pairing term e!ectively drops away in Eq. (92), thereby rendering the determination of other coe$cients with greater accuracy due to the presence of lesser number of terms and consequently lesser correlation. Thus in addition to decoupling of the "nite part from the in"nite part and local energy g, the four "nite-size correction terms namely, exchange Coulomb, proton-form factor, Nolen}Schi!er charge asymmetry and pairing have canceled. The higher order terms like surface symmetry and curvature which are already weak have been rendered insigni"cant due to partial cancelation. These have paved the way for the accurate determination of a' and a' through Eq. (92). Therefore   the determination of saturation density of nuclear matter from nuclear masses could be a reality. B. Determination of the inxnite part E. Since E refers to the energy of nuclear matter, it must satisfy the generalized HVH theorem (Eq. (82)), whose solution is of the form E"!a' A#a' bA , (99)  @ where a' and a' , can be identi"ed as the volume and symmetry coe$cients of the INM liquid. It is  @ indeed interesting to note that these two terms which in mid-thirties were taken into account [1] in the BW liquid drop model on phenomenological grounds, form the solution of a microscopic many-body equation. Using Eqs. (99) and (88) in the left and right hand sides of Eq. (82), respectively, one obtains





1 N Rf Z Rf 1 , (100) # !a' #a' b" [(1#b)e$#(1!b)e$]! L N  @ 2 A RN A RZ 2 8 , where the contribution from the local energy part (of the order of g/A) is neglected which in the limit of large A goes to zero. This equation will hereafter be referred to as the INM coe$cient equation. Hence in the second step of the INM model, the two parameters a' and a' are determined by  @ "tting Eq. (100) using the same data set used in the "rst "t [Eq. (92)] plus the values of parameters of the S function already obtained in the same "t. Thus the "nite-size parameters a' and a' in the   "rst step and the INM coe$cients a' and a' in the second step, are determined separately in two  @ di!erent "ts and therefore are free from any mutual correlation, unlike in the case of BW like mass formulas. C. Determination of local energy term g. We feel, unlike the two functions E(A, Z) and f (A, Z), g(A, Z) cannot be expressed as a speci"c function involving the variables A and Z, since it pertains to the characteristic property of individual nucleus. They can only be determined reliably from the experimental masses whose values can now be found, since the two functions E and f are known. The experimental value of the local energy g for a given nucleus can be obtained from the mass  formula Eq. (87) as g (A, Z)"E$(A, Z)!f (A, Z)!E(A, Z) . (101)  We use the local energy equation (93), as recursion relation for g to determine the g of unknown masses in terms of the gs of the known nuclei. This relation has long range extrapolation property which contributes to the success [125}132,160] of the mass formula.

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5.4. A gedanken experiment to test INM model versus LDM type models We have pointed out in Section 3, that the nuclear liquid envisaged in LDM type mass formulas does not have all the characteristics of in"nite nuclear matter. Further, we also showed theoretically that, the volume coe$cient in such mass formulas is not necessarily at ground-state, and there is no in-built mechanism to ensure this. In contrast, in the INM model, this fact has been ensured by the use of HVH theorem. Since, in this model, the energy of the nucleus is explicitly expressed in terms of the properties of INM, it is expected that this model is well equipped to extract the saturation properties from nuclear masses. Now, the question is, how to demonstrate it in practical case, when the saturation properties are not empirically known exactly. We have contrived a gedanken experiment to verify the above assertion by confronting both the models to the experiment and carrying out a comparative study. The experiment consists of the following steps. 1. For a microscopic two-body e!ective interaction, the saturation properties of nuclear matter are calculated exactly, and refered to as exact values. Further, by performing semi-in"nite nuclear matter calculations with the same force, the surface and curvature coe$cients, etc. are obtained which are also called exact values. 2. We make use of the extended Thomas}Fermi (ETF) calculation of nuclear binding energies with Skyrme-like forces, which over the years has been "rmly established. In such calculation, one obtains the smooth part of the energy corresponding to the liquid-drop nature of the nuclei which is also called the macroscopic part. Theoretical values of energy thus obtained, for same type of force, are called synthetic mass data. 3. The synthetic mass data are then used in the mass formula "t of both the INM model and the LDM type model to obtain the corresponding coe$cients and consequently the saturation properties of nuclear matter. Then the saturation properties so obtained are to be compared with the exact values, and the goodness of one or the other can be ascertained. For the above gedanken experiment, we have chosen the following LDM type mass formula:








3  Z A\ E "!a*A#a*A#a* Z!5 *"+    16n #a*bA#a* Ab#a* A!d(A, Z) , (102) @   where the superscript L refers to LDM nature of the coe$cients. In the case of ETF calculations [80], the curvature coe$cient comes out to be 10 MeV as against the LDM type mass formula-"t to real nuclei, which gives a value close to zero. Due to this, we have included the higher order terms like surface-symmetry and curvature in both INM and LDM models. Pearson et al. [141,142] have developed a method called extended Thomas}Fermi plus Strutinsky Integral approach (ETFSI) to calculate the masses of nuclei based entirely on microscopic two-body force. In their approach, the macroscopic part of the total energy for a given nucleus is calculated using the energy density formalism with generalised Skyrme force called SkSC4 [142] as the e!ective two-body interaction. The realistic ground-state energy contains shell e!ect which has to be taken into account. For this, the HF calculation is performed using the same SkSC4 force and a set of single-particle states are obtained which are subsequently used to get the

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Table 8 Values obtained for the global parameters in the INM model [Eqs. (92) and (100)] and BW mass formula [Eq. (102)] "ts using synthetic data (see text). Exact values determined directly using in"nite nuclear matter and semi-in"nite nuclear matter calculations with SkSC4 force are also given. All quantities are in MeV Parameters

Exact values

INM

BW

a  a  a  a @ a  a 

15.87 0.757 17.3 27.0 !16.0 11.1

15.925 0.7360 18.10 29.80 !31.37 5.06

14.769 0.6945 11.15 25.41 !17.77 16.43

shell correction using Strutinsky procedure. Then the total energy E in this method [141,142] is obtained as the sum of the macroscopic part and the shell correction. Synthetic mass data of 1492 nuclei provided by Pearson et al. are used in the present comparative study. The exact values of the saturation properties, the surface and curvature coe$cients etc. for the same force have also been provided by them. The INM model Eqs. (92) and (100), and the LDM model Eq. (102) are "tted using these masses to determine the corresponding coe$cients. All these values so obtained are presented in Table 8 along with the predetermined exact values. It is indeed remarkable to "nd that the quality of agreement for the principal coe$cients a , a and   a with the exact values is far better for the INM model compared to the BW model. The agreement  for the symmetric coe$cient a in INM model is reasonably good, although it is somewhat inferior @ to LDM value. As for the higher order coe$cients like surface-symmetry and curvature, the agreement of the LDM "t is better. However, it must be noted that, because of correlation among coe$cients, the principal terms like surface and to a lesser extent other ones also are signi"cantly a!ected in the LDM "t. Further such terms in the INM model become negligible small as shown in Section 5.3 and hence, when "tted together with the principal terms, they are rendered insensitive. In any case, these terms contribute insigni"cantly in real nuclei and are normally ignored. Our interest in them is peripheral, as our primary aim is to determine the principal coe$cients particularly the volume and Coulomb coe$cients. Their close agreement with exact ones has vindicated our assertion that INM model is truly suitable for the extraction of saturation properties from nuclear masses than the LDM type model.

6. Saturation properties from nuclear masses 6.1. Extraction of saturation properties For accurate determination of saturation properties using the INM model described above we need to exercise caution in regard to the following two points. 1. The in#uence of higher order terms if any must be ascertained. 2. Numerical accuracy and the independence of results on speci"c data set must be ensured.

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We have already shown in the previous section that the Coulomb related terms like exchange Coulomb, proton-form factor and Nolen}Schi!er charge asymmetry exactly cancel in Eq. (92). E!ective cancellation of pairing term has also been shown in this equation. The two higher order terms which may indirectly a!ect the values of a , are the surface symmetry  a and curvature a coe$cients. We have already shown in the preceding section that the   surface-symmetry a term is cancelled to the extent of 66%, and it has become quite insigni"cant in  the FSC equation. Being a second order term, already its contribution is relatively much smaller than the principal terms, and coupled with it, the above feature of weakened strength, its presence has become rather symbolic at the numerical level. However, since our aim is to "nd the accurate values for the saturation properties, we are anxious to check if any semblance of the survival of the a term can a!ect the results.  With this view, the calculation is carried out retaining the a term as a free parameter in Eq. (92).  Five sets of mass data are chosen from the mass table of Wapstra and Audi [143] on the basis of experimental error associated with them. The "ve sets consists of 1252, 1294, 1371, 1404 and 1449 nuclei with maximum error 480 keV. The results are presented in Table 9. It can be seen that the values of other coe$cients remaining almost same, the value of a widely #uctuates in between  !10 MeV to !24 MeV with an accompanied error of 50}100% with the variation of input data from 1252 to 1449 in number. Thus the a term in Eq. (92) can neither in#uence other coe$cients  nor can itself be uniquely determined. To investigate further into the role of this term, we performed another set of calculation in which presupposed values for a are used while other coe$cients are  "tted to a given data set. The calculation is repeated by varying the value of a , and the one which  gives s as minimum, should be accepted. For the data set of 1371 nuclei, the value of a is found    to be around 12 MeV. However the optimum value #uctuates wildly with the variation of data set as shown in Fig. 6 resulting in no unique value. These features clearly establish that the presence of a term in Eq. (92) is rather insigni"cant and hence redundant. Therefore we have not included this  term in our model. It may be mentioned here that, theoretically the value of a calculated with  various Skyrme-like interactions vary widely [80].

Table 9 Study showing the redundancy of surface-symmetry term in the INM model. Values obtained for the global parameters [Eqs. (92) and (100)] in the INM model including the surface-symmetry coe$cient a are shown for various mass data  sets (Ref. [143]). The calculated error for each coe$cient is shown below in parenthesis. All quantities are in MeV No. of data

a' 

a' 

a' 

a' @

a' 

1252

16.05 (0.02) 16.06 (0.02) 16.09 (0.02) 16.09 (0.022) 16.06 (0.022)

19.31 (0.35) 19.29 (0.35) 19.34 (0.03) 19.36 (0.34) 19.33 (0.34)

0.7469 (0.011) 0.7498 (0.010) 0.7546 (0.009) 0.754 (0.009) 0.749 (0.009)

28.41 (0.76) 26.68 (0.76) 25.77 (0.74) 24.41 (0.73) 24.46 (0.71)

!24.60 (13.01) !15.56 (12.70) !11.15 (11.25) !14.5 (10.8) !20.6 (10.2)

1294 1371 1404 1449

130

Fig. 6. s

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versus surface symmetry coe$cient a in INM model-"ts with variation of data sets (see text for details). 

In modern BW-like mass formula [144,163], a is "xed from consideration other than ground state nuclear masses, such as "ssion barrier height. Since our overriding concern is to determine the properties of INM at ground-state, it is essential that we use ground-state masses, not any other property related to the excited states which might drive the system away from ground-state and jeopardize the accurate determination of saturation parameters. Further, in the present context, it is essential that this coe$cient is determined by treating it as a free parameters to be "xed by nuclear masses through least-squares "t, which we feel has the sanction from nuclear interactions. We would like to emphasize at this stage, that the above study showing the redundancy of a term does not mean that its contribution to the masses of real nuclei is negligible and can be  ignored. It only means in the INM model, that it has been rendered insigni"cant and irrelevant without any genuine potential to a!ect the other principal coe$cients in FSC Eq. (92) due to special combination of data used. This only helps our purpose to determine some principal coe$cients more accurately which is our prime concern. The same is also true for the curvature term. Further the curvature term in real nuclei, as noted before, is nearly zero and generally neglected. So this term is also not included. Similarly we "nd such higher order terms become insigni"cant in the INM coe$cient equation (100) except the Coulomb exchange term. Finally, we would like to remark that unlike in the LDM type mass formulas, where primary concern is to "nd all possible terms to get the mass reproduced, the perspective here, is quite di!erent being con"ned to the determination of few important coe$cients as accurately as possible. Therefore the optimum representation for the "nite-size function f (A, Z) de"ned in Eq. (85) is f (A, Z)" a' A#a' +Z!5[3/(16n)]Z,A\.  ! Now the stage is well-set for the actual determination of the saturation properties of INM and other "nite-size coe$cients. Although, the masses of about 2000 nuclei are presently known [143] experimentally, we only use those masses of 1371 nuclei with error 460 keV as our optimum data

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set as shown later, guided by the principle that data having larger error might pollute the estimation of the INM parameters. In the "rst step, we determine the "nite-size coe$cients a' and  a' by making a least-square "t to Eq. (92) using all the 1371 masses. Then these parameters so  determined are used in the second step to obtain the INM coe$cients a' and a' , by a "t to INM  @ equation (100) using the same data set. The rms deviation in the two "ts are only 371 and 372 keV, respectively. The quality of "t is undoubtedly impressive. To check the goodness and the stability of the values of the coe$cients we have repeated the above calculations by varying the number of data randomly considered throughout the mass table choosing them on the basis of experimental error ranging from 40 to 80 keV. The results are presented in Table 10, which clearly show the stability of these parameters. Especially remarkable is the stability of the two crucial parameters a' and a' ,   which determine the saturation properties. The Coulomb coe$cient a' is remarkably stable up to  second decimal place for all the "ve sets, in particular it is stable up to third decimal place for the third and fourth set. The volume coe$cient a is also stable up to third decimal place for the third  and fourth data set. Hence we have chosen the third set with 1371 nuclei as our optimum data set. We "nd the other two coe$cients a' and a' are also stable and well determined. The values of these  @ four coe$cients corresponding to the optimum data set are a' "16.108 MeV and a' "0.7593 MeV  ! a' "19.27 MeV and a' "24.06 MeV .  @ and the saturation density and the nuclear radius constant obtained from a are  o "0.1620 fm\ and r "1.138 fm.   Now we have determined all the INM coe$cients which contribute to the global part of the ground-state energy of a nucleus. These coe$cients are at the ground-state of nuclear matter. So they may be used to de"ne an expansion termed as in"nite nuclear matter model (INMM) expansion for the macroscopic part of the energy. Since the HVH theorem is valid for multi-body forces, the present result is not constrained by any assumption regarding two-body nature of the nuclear force and as such contains the e!ect of three-body or multi-body forces if at all present in the nuclear system.

Table 10 Values obtained for the global parameters in the INM model [Eqs. (92) and (100)] for various data sets (Ref. [143]) in optimum parameter space (see text). All quantities are in MeV No. of data

a' 

a' 

a' 

a' @

1252 1294 1371 1404 1449

16.112 16.097 16.108 16.108 16.077

0.7589 0.7572 0.7593 0.7594 0.7569

19.23 19.23 19.27 19.25 19.14

24.66 24.32 24.06 24.19 24.29

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6.2. Resolution of &r paradox'  In the introduction (Section 1), we have discussed about what this paradox [12,149] is. The value of r determined through the Coulomb term of LDM type mass formulas is much larger than the  value found from electron scattering data on heavy nuclei. The nuclear mass formula cannot give the saturation density of nuclear matter has been accepted as a foregone conclusion [7] and we have been living with this discomp"ture since the Hofstadter experiment [3] in the mid-1950s. However, the above value of r "1.138 fm (equivalently o "0.1628 fm\) determined through   INM model agrees well with the value 1.13 fm obtained from electron scattering data [10] on nuclei. Further it is also close to the value of 1.13 fm obtained in the HF studies [11], widely accepted in the literature. It may be noted also that our value of r is quite similar to the value  1.140$0.005 fm obtained from more recent electron scattering data [20]. Thus the value of r obtained from nuclear masses agrees with that determined from electron scattering data on  heavy nuclei. Hence the &r paradox' is satisfactorily resolved. This is essentially due to taking into  account the full many-body features of in"nite nuclear matter through the use of HVH theorem [17]. This also sets at rest the perennial doubt [5,7] if the central density of heavy nuclei could be considered same as the saturation density of nuclear matter. Thus the two important INM properties, i.e., energy and density which are interrelated are consistently determined from one kind of data namely the masses of nuclei for the "rst time. It must be mentioned here that Pearson and coworkers in their ETFSI mass formula approach have been able to "nd a set of parameters for the two-body force which give the nuclear masses and r "1.140$0.005 fm. This is no doubt commendable. However, we would like to state that, they  use [141] a priori known properties of nuclear matter like K in "xing the parameters of the force.  Further they presuppose the form of the e!ective interaction which is Skyrme-like with zero range. In contrast, we do not use any knowledge of nuclear force or saturation properties of INM and obtain these through a complete free "t to nuclear masses. In a recent mass model [77] with e!ective two-body interaction of Seylar}Blanchard type, Myers and Switacki have chosen r "1.14 and  kept it "xed in there attempt to "t the masses of nuclei by varying the parameters of the force.

7. Determination of incompressibility of in5nite nuclear matter The only remaining property of INM to be determined is its incompressibility. In Section 4, we have exhaustively discussed the various issues involved in regard to its experimental determination, as well as the theoretical approaches used for the same. Here we will attempt to determine the incompressibility in the INM model, using the masses of known nuclei numbering about 1500. Because of the quality and the abundance of such data spread over the entire nuclear chart, this determination is expected to be reliable and unique. The model proposed for the determination of K called the compression model [145] is described below.  7.1. The compression model In the preceding section we have determined energy and the density of nuclear matter at ground-state using the INM model. We have also shown convincingly that the nuclear matter

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represented by the LDM coe$cients in the LDM type mass formulas, do not correspond to ground state but to some excited state with lower density. These two species of nuclear matter must be related through the incompressibility K , and this fact can be used to determine [145] its value. We  would like to stress that although the LDM type mass formulas pertain to nuclear matter at a di!erent density, their success to describe nuclear masses has been established over the years. Hence we make use of this in the present investigation. Using the above premises, the volume term a* and the density o determined from LDM type mass formula-"ts, can be related through K , to    the corresponding ones a' and o of INM models as    K o (103) a' (o )!   !1 "a*(o ) .     18 o  The foundation of the above relation, for the determination of incompressibility coe$cients is quite sound. Using this approach we [130] had determined earlier the value of K to be 288$28 MeV.  However, we like to perform a more extensive study here in regard to the optimum parameter space as well as the variation of data so as to arrive at a still better and unique value. It must be emphasized that, the success of the present method depends upon how well the coe$cients in both the models are determined. Further we would like to stress that the value of a given coe$cient determined in the least-squares "t of masses depends upon the number of terms used, and also, inclusion/non-inclusion of speci"c term(s) in the mass formula. In particular, the LDM type mass formulas [74] like "nite-range droplet model (FRDM) and "nite-range liquid-drop model (FRLDM) have many parameters. Depending upon the taste of the authors, a particular term may occur in one formula but does not "nd a place in another, except of course, the principal terms. This is going to a!ect the values of the principal coe$cients, and thereby the value of K , hence would  give rise to di!erent values of K . As an example, the volume and surface coe$cients in the FRDM  model and FRLDM are 16.247, 22.92 MeV; and 16.00126, 21.18466 MeV, respectively [74]. Being aware of this fact and taking all possible measures, we embark upon the determination of K using  the above method. We feel, we have succeeded in arriving at a well de"ned value of K . We adopt  the following two principles in our quest for the value of K .  1. We will recognise those coe$cients in both the models as the bona"de ones, whose values can be obtained by treating them as variational parameters in the least-square "t to the masses. We feel, the values so determined have, the sanction of nuclear dynamics governed by the Hamiltonian. 2. The optimum parameter/coe$cient space must be found out in both the models. The criterion for such space is that, the least square "t should result in unique and stable values of coe$cients which should not change when the number of data is varied. The coe$cient whose value will #uctuate when the number of data is increased or the parameter space is expanded, would not qualify for inclusion. The values of such coe$cients are generally "xed by using nuclear properties other than the ground-state energy. Their contribution to the total energy of the nucleus may be important, but they have no role in the determination of the principal coe$cients which is of main concern here.






7.2. Extraction of K  Guided by the above two principles, we proceed to determine the optimum representation in the INM model and the LDM type model to obtain the value of K . 

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7.2.1. Optimum representation for INM model In Section 6, we have found the representation for the optimum parameter space in the INM model. We have shown how the FSC equation (92) which determines the "nite-size coe$cients cannot sustain surface symmetry and curvature terms. Further in this equation, exchange Coulomb, proton-form factor and charge asymmetry cancel as has been shown earlier. However, the other equation namely the INM equation (100), can have these terms in principle. We have checked that these terms except the Coulomb exchange are not sustainable in the sense that, no stable and unique values for the coe$cients can be obtained through least-squares "t. The pairing term e!ectively drops out numerically in both the FSC equation (92) and INM equation (100). Thus the optimum representation for INM model consists of the four coe$cients a' , a' , a' and a' .    @ The values of these coe$cients are already determined in the previous section and presented in Table 10 for varying data sets. 7.2.2. Optimum representation for BW like mass formula We have made a detail analysis to "nd the optimum number of parameters in the BW mass formula given by Eq. (102). We have carried out least square "ts using the optimum data set with varying number of parameters, the results of which are presented in Table 11. It can be seen that the values of all the coe$cients are well determined when the surface symmetry a and curvature  a terms are successively included. This can be seen in the values of the coe$cients and  accompanying errors shown in Table 11. From the results we "nd that the value of +!22 MeV of the surface-symmetry coe$cient a is accompanied with an error of about 1.85 MeV which is  less than 10%. So we can consider this coe$cient to be well determined. It is sustainable and can be retained. The curvature coe$cient a has the value 1.56 MeV with error of 1.51 MeV. Thus this  coe$cient is determined but with 100% error. Since the curvature is small and accompanied with larger errors, we do not include it. However, since it is sustainable as shown above, we will study the stability of our "nal result on incompressibility, by including it in the representation. Next we include the Gauss-curvature term a , the next higher order in the model. It can be seen  in Table 11 that it completely destabilizes the "t by violently disturbing the leading order coe$cients. Even the surface coe$cient a becomes negative and unphysical. It must be noted that  Table 11 Study of optimum parameter space in BW model. Values of parameters together with the errors (shown in parenthesis) obtained in BW model "ts [Eq. (102)] using mass data (Ref. [143]) are shown. All quantities are in MeV No. of parameters

a* 

a* 

a* @

a* 

D*

5

15.80 (0.03) 15.64 (0.03) 15.47 (0.15) 12.66 (0.31)

18.45 (0.10) 18.17 (0.09) 17.25 (0.90) !10.62 (2.78)

23.04 (0.08) 26.61 (0.30) 26.34 (0.40) 21.74 (0.58)

0.733 (0.002) 0.713 (0.003) 0.707 (0.006) 0.651 (0.008)

11.89 (1.01) 11.25 (0.96) 11.19 (0.97) 10.73 (0.93)

6 7 8

a* 

a* 

a* 

!22.26 (1.82) !21.92 (1.85) !7.9 (2.22)

1.56 (1.51) 102.7 (9.69)

!127.0 (12.0)

L. Satpathy et al. / Physics Reports 319 (1999) 85}144

135

this result is contrary to the common belief that the inclusion of more and more higher order terms in a LDM type expansion would result in progressively re"ned values of the leading order coe$cients. Thus the Gauss curvature term cannot qualify to be included in the optimum representation. We have checked that, other higher order terms like proton-form factor and charge asymmetry terms lead to similar unstable results. Thus for our study we take the six parameters a*, a*, a*, a* a* and D to de"ne our optimum representation.    @  Now with the above six coe$cients de"ning our parameter space in BW model, we carried out the least square "t using the same optimum data set of 1371 masses. As in the case INM model calculation, we have varied the number of data to arrive at stable values of all the coe$cients in general, but a* and a* in particular. The results are presented in Table 12. It can be seen that for the   optimum data set a* and a* have reached stability in the accuracy up to third decimal place.   Now using the coe$cients of both the models (Tables 10 and 12) determined above in Eq. (103), we have calculated the values of K for all the "ve data sets including the optimum one. The results  are presented in Table 13. For the sake of transparency and clarity, we have also presented here the appropriate mass formula coe$cients collected from Tables 10 and 12. The values for the optimum data set comes out to be 288 MeV. It is worth noting that, in spite of the variation of input data ranging from 1252 to 1449, the value of K is remarkably stable and only varies between 283 and  303 MeV. The average value thus obtained is 293 MeV which is very close to 288 MeV determined with the optimum data set. This clearly shows the stability of our result for incompressibility with respect to the variation of data.

Table 12 BW model coe$cients for the optimum representation. Values shown are obtained by "ts to the [Eq. (102)] for di!erent mass data sets [143]. All quantities are in MeV No. of data

a* 

a* 

a* 

a* @

D*

a* 

1252 1294 1371 1404 1449

15.651 15.634 15.636 15.636 15.639

0.7141 0.7128 0.7131 0.7128 0.7131

18.209 18.164 18.166 18.171 18.183

26.407 26.570 26.609 26.657 26.749

11.119 11.351 11.251 11.124 11.253

!20.746 !22.025 !22.258 !22.444 !23.097

Table 13 Values of K obtained [through Eq. (103)] using mass formula coe$cients taken from Tables 10 and 12 for INM and  BW models respectively for di!erent mass data sets. All quantities are in MeV No. of data

a' 

a' 

a* 

a* 

K 

1252 1294 1371 1404 1449

16.111 16.095 16.108 16.108 16.077

0.7589 0.7572 0.7593 0.7594 0.7569

15.651 15.634 15.636 15.636 15.639

0.7141 0.7128 0.7131 0.7128 0.7131

297 303 288 283 294

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L. Satpathy et al. / Physics Reports 319 (1999) 85}144

We have carried out an extensive study to "nd out the stability of the above value of K and  estimate the possible uncertainty associated with it. For this study, we now see how the inclusion of curvature term in the BW "t, changes our results. Besides this, we also study the e!ect of the inclusion/non-inclusion of exchange Coulomb term in both the models. In view of the arguments and justi"cations given above, only these two terms qualify to be included in the present investigation of the stability. Further, we would also like to see, how far the results remain stable when the number of input data is varied. The calculations are performed varying the parameter space mentioned above as well as the data sets and thus, we have been able to generate twenty values of K . The results of this investigation are presented in Table 14. It can be seen that the  values of K so obtained, range between 283 and 322 MeV only. We use these values to estimate  the possible error in our value of K . The error s is calculated using the expression  s" , (KG !K), where K"288 MeV and N"19 and i stands for the 19 values other   , G  than K presented in Table 14. The value of the s so calculated comes out to be 20 MeV. This    small value of s due to the variation of data and also variation of genuine parameter space,   clearly shows the stability of the value of K obtained here and thereby justi"es the soundness of  our model. Thus, it has been possible to obtain a stable and well de"ned value of 288$20 MeV for the incompressibility of nuclear matter using masses of all known nuclei numbering about 1500 in the nuclear chart with experimental error bar 480 keV. This study may be considered as an experiment in which masses are measured and the data are analyzed using standard nuclear models. Once the value of nuclear incompressibility K is known, we use the compression model  Eq. (103) to construct the nuclear equation of state (EOS) at any arbitrary density o given by






o  K (104) !1 , E(o)"E(o )#   18 o  where E(o )"!16.108 MeV and o "0.1620 fm\ correspond to the quantities at saturation   obtained in the INM model.

Table 14 Determination of possible error in the value of K . Values of K obtained with inclusion or non-inclusion of the two   sustainable terms namely, the exchange Coulomb and curvature in the optimum parameter representation (see text). INM4 stands for the INM model mass-"t in the optimum parameter space spanned by a' , a' , a' and a' while BW6 stands    @ for the BW model "t with optimum parameter space de"ned by a*, a*, a*, a' , a* and D*. Bw7 corresponds to BW6 plus    @  the curvature term. All quantities are in MeV No. of data

1252 1294 1371 1404 1449

With Exc. Coulomb

Without Exc. Coulomb

INM4&BW6

INM4&BW7

INM4&BW6

INM4&BW7

297 302 288 283 294

283 311 302 298 315

314 318 303 299 312

322 322 309 305 313

L. Satpathy et al. / Physics Reports 319 (1999) 85}144

137

Recently, Blaizot et al. [146] have reexamined the microscopic approach in an extensive study, using a set of Gogny e!ective interaction. Such interactions, being of "nite range, have been considered better than Skyrme-type interaction earlier used by them. Using more recent data on Pb and Sn isotopes, they have arrived at the value of incompressibility as K "210}220 MeV  which rea"rms their earlier result obtained in mid 1970s. Pointing out some ambiguity and inadequacy in this approach, like the determination of K , a characteristic static property  from the monopole resonances which is a dynamical one, and the coupling between single particle and collective motion, they have argued that this method is still the best available method at the present time. Recently, Mayers and Swiatecki [77] have estimated K "234 MeV in their  mass model using Seylar}Blanchard type potential. As has been known for sometime and also explicitly shown recently by Blaizot et al. [146], that many sets of e!ective interaction can "t all the static properties of nuclei and nuclear matter equally well, and yet, may give widely varying values of K .  The present study is relatively well founded, as it does not use any e!ective interaction, dynamical property like monopole resonances or the LDM expansion of K . It uses an improved  LDM expansion of energy called INMM expansion here. In general the LDM expansion of energy (also shown in the present article) is well respected in nuclear physics over the years.

8. Summary and conclusions We have reviewed the status of LDM expansions of the ground state energy and incompressibility of "nite nuclei in an analytical model starting with Skyrme-like e!ective two-body interactions. In particular, we have examined whether such expansions provide an unambiguous way to go from "nite nuclei to nuclear matter and thereby, can yield the saturation properties of the latter from nuclear masses. The LDM expansion of energy is found to converge quite well. However, the present study has clearly shown that this expansion is not unique in the sense that, its coe$cients do not necessarily correspond to the ground state of nuclear matter. Hence LDM type mass formulas based on it are de"cient and not properly equipped to give saturation properties of nuclear matter from nuclear masses. In comparison to the energy expansion, the LDM expansion of incompressibility has poorer convergence property. In pure bulk model of density vibration, this expansion shows an anomalous convergence behavior due to the pair-e!ect. Further, the various modes of density vibration vary from one mass region to another in the nuclear chart due to the coupling b between the bulk and  surface parts of the vibrations. It is shown that when the dynamical e!ect (dependence of the coupling b on mass number A) is taken into account, in the LDM expansion of K shows &upturn'   behavior below the mass number A+120, signaling the breakdown of the expansion. Therefore the extraction of incompressibility K is rather problematic from the presently available few tens of  data on monopole resonances. The de"ciency in the LDM expansion of energy has been identi"ed to its use of classical liquid drop without any reference to particles as the basis, as such does not possess the essential attribute of quantum mechanical many-fermion system, namely, the single-particle property in particular, the Fermi state. It is shown that the INM model succeeds in repairing this defect by adopting the

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generalized HVH theorem of many-body theory as its foundation. Thus the model uses true nuclear matter rather than classical liquid as its basis. The resulting expansion termed as in"nite nuclear matter model expansion (INMM) is obtained and thereby, a well de"ned path connecting "nite nuclei to nuclear matter is charted out. This bridges a long-standing gap in our understanding of nuclear dynamics. Since the INM model is built in terms of the properties of nuclear matter, it is well equipped to yield the saturation properties from nuclear masses. Using this model, the two saturation properties namely, the density o and the binding energy per nucleon a , the two highly interrelated   quantities extracted consistently from nuclear masses are 0.1620 fm\ and 16.108 MeV, respectively. The radius constant r corresponding to o determined here (r "1.138 fm) agrees quite    well with that obtained from electron scattering data. Thus the &r -paradox' is satisfactorily  resolved and the myth that nuclear mass formulas cannot give the density of nuclear matter is dispelled. This has been possible due to the decoupling of the "nite and in"nite parts of the energy and the use of special combination of data of ground-state energy and Fermi energies dictated by the HVH theorem. This has resulted in the cancelation of large number of terms, in particular, the Coulomb related terms like exchange Coulomb, proton-form-factor and the Nolen}Schi!er charge asymmetry terms, and weakening of higher order terms like surfacesymmetry and curvature through partial cancelation, rendering an accurate determination of the saturation density. The saturation properties determined here should be considered as true empirical properties since the HVH theorem, which plays the central role here, does not depend upon the form of the interaction and is valid also for multi-body forces. This has bearing on our current understanding of nuclear saturation properties requiring the presence of a three-body force in nuclear systems. The realization that BW mass formulas give the properties of nuclear matter at a di!erent density, other than the ground state, and INM model speci"cally at saturation density, has presented a unique possibility of determining K from nuclear masses, which are the best measured  and most abundant data on nuclei. Using this premise, a well de"ned and stable value of 288$20 MeV for K is determined following standard models of nuclear physics. Thus it has been  possible to obtain the nuclear equation of state.

Acknowledgements We are deeply indebted to Prof. M.K. Banerjee for valuable discussion and appreciation of the work presented here. His encouragement and support to write this article is gratefully acknowledged. We express our deep gratitude to Prof. M.K. Pal for his constant appreciation and encouragement which sustained us over the years. We are very much grateful to Prof. S.P. Pandya for his enthusiasm and encouragement during the course of the developement of this work. We would like to acknowledge with gratitude the valuable help rendered by Prof. J.M. Pearson who provided us with the synthetic data on nuclear masses. With gratitude we remember late Prof. L. Van Hove for very useful suggestion to extract incompressibility in the INM model. One of us (R.C.N) would like to thank Prof. S.N. Behera for providing hospitality at the Institute of Physics, Bhubaneswar. Finally we would like to thank Mr. C.B. Das for useful discussions and help in the preparation of this manuscript.

L. Satpathy et al. / Physics Reports 319 (1999) 85}144

139

Appendix A. Generalized Sommerfeld lemma The moments of the Fermi distribution function can be expanded in powers of a/R using the generalized Sommerfeld lemma [91,92]. In general, the energy functional contains arbitrary powers of density function o(r). If one parametrises the density o(r) to be a Fermi-function, then the energy functional contains integrals of the form,



I " JI



rI dr , (1#exp[(r!R)/a])J

(A.1)  where k is an integer with the condition k50 and l'0. Then, the generalized Sommerfeld lemma gives I to be JI RI> I k I " 1#(k#1) gI(a/R)I>#u , (A.2) JI k#1 JI k J I where






 

gI"(!1)I J



uI





1#(!1)I e\SJ !1 du , (1#e\S)J

(A.3)  and u is of the order of exp(!R/a). It may be noted that the above equation for I is not an JI JI in"nite expansion. It contains (k#1) terms other than the constant term 1 and exponential term u ; the contributions pertaining to the (k#1) terms are not necessarily decreasing when k goes JI from 0 to k. For su$ciently large values of R, u can be neglected, which is usually referred to as leptoderJI mous approximation. The resulting expansion is the so-called leptodermous expansion. We also give some useful relations between the gI's: J 1 g "g! , J> J l k gI "gI! gI\ . J> J l J

(A.4)

Some particular values of gI are presented below. J

k"0 k"1 k"2 k"3

l"1

l"2

l"3

0 n/6 0 7n/60

!1 n/6 !n/3 7n/60

!3/2 n/6#1/2 !n/2 7n/60#n/2

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Appendix B. Analytical expressions for LDM energy coe7cients Expressions for the various coe$cients appearing in Eq. (22) are given below: eH"p o#p o #p ao#p oN> ,  ?       

(B.1)




p eH"!(36n) a 0.759p o ! @ o#p o#1.359ap o ?  2a      



[p #p (b!c)]  !  o!p gN oN> ,   Q  6a





(B.2)



n p 1.517! p o# @ o 6 ?  2a 

eH "4a(6n) 













n n p ao# 0.5gN ! p oN> , # 1.973! A 6   6  







(B.3)






2n n p 2n @ o # 4.423! 1.359 p ao 2.602! 0.759 p o# ?     a 3 6 3

eH "!4na 











n [p #p (b!c)] n 1 2n  ! p o#  # o! gN ! gN p oN> , M 3   18 3  a 3 Q  

(B.4)

where p "  a, p "  b, p "R, p " R , p "R>R, p "R\R, and a"(1.5n), b"  and    ? K @ K        c". The quantities gN , gN and gN are calculated using the integral Q A M 

 

gI"(!1)I J





uI



1#(!1)Ie\SJ !1 du , (1#e\S)J

where l"p#2, and gN "g, gN "2g and gN "g. Q J A J M J

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PARTICLE INTERFEROMETRY FOR RELATIVISTIC HEAVY-ION COLLISIONS

Urs Achim WIEDEMANN , Ulrich HEINZ
 Physics Department, Columbia University, New York, NY 10027, USA
Theory Division, CERN, CH-1211 Geneva 23, Switzerland  Institut fu( r Theoretische Physik, Universita( t Regensburg, D-93040 Regensburg, Germany

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

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Physics Reports 319 (1999) 145}230

Particle interferometry for relativistic heavy-ion collisions Urs Achim Wiedemann *, Ulrich Heinz
 Physics Department, Columbia University, New York, NY 10027, USA
Theory Division, CERN, CH-1211 Geneva 23, Switzerland Institut fu( r Theoretische Physik, Universita( t Regensburg, D-93040 Regensburg, Germany Received February 1999; editor: W. Weise

Contents 1. Introduction 1.1. Historical overview 1.2. Outline 1.3. Notation and conventions 2. Particle correlations from phase-space distributions 2.1. Normalization 2.2. Classical current parametrization 2.3. Gaussian wave packets 2.4. Multiparticle symmetrization e!ects 2.5. Final state interactions 2.6. Bose}Einstein weights for event generators 3. Gaussian parametrizations of the correlator 3.1. The Cartesian parametrization 3.2. The Yano}Koonin}Podgoretskimy parametrization

148 149 150 151 152 153 155 159 164 171 177 180 181

3.3. Other Gaussian parametrizations 3.4. Estimating the phase-space density 4. Beyond the Gaussian parametrization 4.1. Imaging methods 4.2. q-moments 4.3. Three-particle correlations 5. Results of model studies 5.1. A class of model emission functions 5.2. One-particle spectra 5.3. Two-particle correlator 5.4. Analysis strategies for reconstructing the source in heavy-ion collisions 6. Summary Acknowledgements References

190 191 194 195 195 196 198 199 204 207 219 224 226 226

187

Abstract In this report we give a detailed account on Hanbury Brown/Twiss (HBT) particle interferometric methods for relativistic heavy-ion collisions. These exploit identical two-particle correlations to gain access to the space-time geometry and dynamics of the "nal freeze-out stage. The connection between the measured correlations in momentum space and the phase-space structure of the particle emitter is established, both with and without "nal state interactions. Suitable Gaussian parametrizations for the two-particle correlation

* Corresponding author. 0370-1573/99/$ - see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 3 2 - 0

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function are derived and the physical interpretation of their parameters is explained. After reviewing various model studies, we show how a combined analysis of single- and two-particle spectra allows to reconstruct the "nal state of relativistic heavy-ion collisions.  1999 Elsevier Science B.V. All rights reserved. PACS: 25.75.!q

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1. Introduction By now, a large collection of experimental data exists from the "rst relativistic collisions between truly heavy ions [135], using the 11 GeV/nucleon gold beams from the Brookhaven AGS and the 160 GeV/nucleon lead beams from the CERN SPS. The "rst relativistic heavy-ion collider RHIC at BNL will soon start taking data at (s"200 A GeV, and in the next decade, the already approved LHC program at CERN will explore relativistic heavy-ion collisions at even higher energies ((s"5.5 A TeV). The aim of this large scale experimental e!ort is to investigate the equilibration processes of hadronic matter and to test in this way the hadronic partition function at extreme energy densities and temperatures. Especially, one expects under su$ciently extreme conditions the transition to a new state of hadronic matter, the quark gluon plasma (QGP) in which the physical degrees of freedom of equilibration processes are partonic rather than hadronic [66,109,139,151]. QCD lattice simulations predict this transition to occur at a temperature of approximately 150 MeV [97]. The experimental con"rmation of a possibly created QGP is, however, di$cult, since only very few particle species, mainly leptons, can provide direct information about the initial partonic stage of the collision. The much more abundant hadrons are substantially a!ected by secondary interactions and decouple from the collision region only during the "nal `freeze-outa stage. A successful dynamical model of relativistic heavy-ion collisions should "nally explain all these di!erent observables, their dependence on the incident energy, impact parameter, and atomic number of the projectile and target nuclei. At the present stage, theoretical e!orts concentrate on discriminating between di!erent models by comparing them with characteristic observables [19,76,135,136]. The observed enhancement of strange hadron and low-mass dilepton yields and the measured J/W-suppression provide strong indications that a dense system was created in the collision whose extreme condition has signi"cantly a!ected particle production mechanisms. Furthermore, various observations signal collective (hydro)dynamical behaviour in the collision region which in turn indicates the importance of equilibration processes for the understanding of the collision dynamics. Especially, the hadronic momentum spectra show signs of both radial and azimuthally directed #ow, and two-particle correlations indicate a strong transverse expansion of the source before freeze-out. Despite the rich body of these and other observations, it remains however controversial to what extent these data are indicative for the creation of a QGP or can also be explained in purely hadronic scenarios. To make further progress on this central issue, a more detailed understanding of the space-time geometry and dynamics of the evolving reaction zone is required. The systems created in relativistic heavy-ion collisions are mesoscopic and shortlived, and the geometrical and dynamical conditions of the cauldron play an essential role for the particle production processes. For example, the maximal energy density attained in the collision, the time-dependence of its decrease, and the momenta of the produced particles relative to the collectively expanding hadronic system will a!ect the observed particle ratios. Two-particle correlations provide the only known way to obtain directly information about the space-time structure of the source from the measured particle momenta. The size and shape of the reaction zone and the emission duration become thus accessible. In combination with the analysis of single particle spectra and yields, it is furthermore possible to separate the random and collective contributions to the observed particle momenta. This permits to also reconstruct the collective dynamical state of the collision at freeze-out. These new pieces of information give powerful constraints for dynamical model calculations; they can

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also be taken as an experimental starting point for a dynamical back extrapolation into the hot and dense initial stages of the collision. The present work reviews the foundations of HBT interferometry in particle physics and discusses the technical tools for its quantitative application to relativistic heavy-ion collisions. 1.1. Historical overview HBT intensity interferometry was proposed and developed by the radio astronomer Robert Hanbury Brown in the 1950's, who was joined by Richard Twiss for the mathematical analysis of intensity correlations. Their original aim was to bypass the major constraint of Michelson amplitude interferometry at that time: in amplitude interferometry, the resolution at a given wavelength is limited by the separation over which amplitudes can be compared. Hanbury Brown started from the observation that `if the radiation received at two places is mutually coherent, then the #uctuation in the intensity of the signals received at those two places is also correlateda [74]. More explicitly, amplitude interferometry measures the square of the sum of the two amplitudes A and A falling on two detectors 1 and 2:   "A #A """A "#"A "#(AHA #A AH) . (1.1)         The last term, the `fringe visibilitya <, is the part of the signal which is sensitive to the separation between the emission points. Averaged over random variations, its square is given by the product of the intensities landing on the two detectors [22], 1<2"21"A " "A "2#1AHA2#1AA 2P21I I 2 . (1.2)    H     The last two terms of this expression vary rapidly and average to zero. According to Eq. (1.2), intensity correlations between di!erent detectors contain information about the fringe visibility and hence about the spatial extension of the source. To demonstrate the technique, Hanbury Brown and Twiss measured in 1950 the diameter of the sun, using two radio telescopes operating at 2.4 m wavelength, and determined in 1956 the angular diameters of the radio sources Cas A and Cyg A. Furthermore, they measured in a highly in#uential experiment intensity correlations between two beams separated from a mercury vapor lamp. They thus demonstrated [75,198}202] that photons in an apparently uncorrelated thermal beam tend to be detected in close-by pairs. This photon bunching or HBT-e!ect, "rst explained theoretically by Purcell [134], is one of the key experiments of quantum optics [62,197]. However, with the advent of modern techniques which allow to compare radio amplitudes of separated radio telescopes, Michelson interferometry has again completely replaced intensity interferometry in astronomy. In particle physics, the HBT-e!ect was independently discovered by Goldhaber et al. [63]. In 1960, they studied at the Bevatron the angular correlations between identical pions in pp annihilations. Their observation (the `GGLP-e!ecta), an enhancement of pion pairs at small relative momenta, was explained in terms of the "nite spatial extension of the decaying pp -system and the "nite quantum mechanical localization of the decay pions [63]. In the sequel of this work, it was gradually realized that the correlations of identical particles emitted by highly excited nuclei are sensitive not only to the geometry of the system, but also to its lifetime [95,150,204,210]. This point has become increasingly more important, and it was supplemented by the later insight that

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the pair momentum dependence of the correlations measured for relativistic heavy-ion collisions contains information about the collision dynamics [125,206]. The origins of the wide "eld of applications to relativistic heavy-ion collisions can be dated back to the works of Shuryak [150,210], Cocconi [46], Grishin, Kopylov and Podgoretskimy [65,95,96,204], and to the seminal paper of Gyulassy, Kau!mann and Wilson [68]. Important contributions in the eighties include a more detailed analysis of the role of "nal state interactions [34,35,69,94,126], the development of a parametrization [123] taking into account the longitudinal expansion of the system created in the collision [125,126] and the "rst implementation of the HBT-e!ect in prescriptions for event generator studies [185]. Also, the e!ect was seen in and analyzed for high energy collisions (see the recent review by LoK rstad [103]). In addition, there is a wealth of experimental data and theoretical work on correlations between protons and heavier fragments (pp, pd, pHe) in lower energy ((1 GeV) nuclear collisions, which are summarized in the review article of Boal et al. [32]. With the advent of relativistic heavy-ion beams at CERN and Brookhaven, many of these concepts had to be re"ned and extended to the rapidly expanding particle emitting systems created in heavy-ion collisions. The relativistic collision dynamics plays an important role in the derivation of the HBT two-particle correlator and of its modern parametrizations. It is adequately re#ected in recent model discussions of the particle phase-space density from which the two-particle correlator is calculated. Several smaller reviews [20,22,77,78,103,132] as well as a selected reprint volume [170] exist by now. The present work aims at a uni"ed presentation of the underlying concepts and calculational techniques, and of the phenomenological applications of HBT interferometry to the rapidly expanding sources created in these relativistic heavy-ion collisions. It does not provide a comprehensive review of the experimental data, for which we refer to the overview given in [82]. 1.2. Outline We start by discussing the relation between the single-particle Wigner phase-space density S(x, K) of the particle emitting source, the triple-di!erential one-particle spectrum E dN/dp and N the two-particle correlation function C(q, K) for pairs of identical bosons:



dN " dx S(x, p) , E N dp

(1.3)

"dx S(x, K) e OV" C(q, K)+1# "dx S(x, K)"

(1.4)





+1#j(K) exp ! R (K) q q . (1.5) GH G H GH The approximations are discussed in the main text; the notation used here and throughout this review is compiled at the end of this introduction. The main aim of particle interferometric methods is to extract as much information as possible about the emission function S(x, K), which characterizes the particle emitting source created in the heavy-ion collision. We discuss how the above expressions are modi"ed to include "nal state interactions and multiparticle symmetrization e!ects

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and how they apply to numerical event simulations of relativistic heavy-ion collisions. Contact between theory and experiment is made with the help of Gaussian parametrizations (1.5) of the correlator which we review in Section 3. We discuss the Cartesian Pratt}Bertsch parametrization as well as the Yano}Koonin}Podgoretskimy (YKP) parametrization where the latter is particularly adapted to the description of systems with strong longitudinal expansion. We then turn to estimates of the pion phase space density based on such Gaussian "ts. Our main focus is on the space-time interpretation of the HBT radius parameters R (K) which we establish in terms of GH space-time variances of the Wigner phase-space density S(x, K). Particle emission duration, average particle emission time, transverse and longitudinal extension of the source as well as positionmomentum correlations in the source due to dynamical #ow patterns are seen to be typical source characteristics to which identical particle correlations are sensitive. While most of our discussion is carried out for central collisions, we also review how this framework can be extended to collisions at "nite impact parameter where the HBT radius parameters depend on the azimuthal angle of the emitted particles with respect to the reaction plane. Furthermore, we discuss more advanced techniques which do not rely on a Gaussian parametrization of the correlation function but require better statistics of the experimental data. This concludes our review of existing analysis tools. Section 5 is devoted to applications of the presented framework within concrete model studies. We introduce a simple but #exible class of models for particle emission in relativistic heavy-ion collisions. These are motivated by hydrodynamical and thermodynamical considerations and allow to illustrate the main techniques discussed before. Di!erent analytical and numerical calculation schemes for the HBT radius parameters are contrasted, and we explain which geometrical and dynamical model features are re#ected by which observables. Then we discuss how resonance decay contributions to pion spectra modify these calculations, and we compare the results of this model with various other model studies, focussing on the qualitative and quantitative di!erences. All these results are "nally combined into an analysis strategy for the reconstruction of the particle emitting source from the measured one- and two-particle spectra. The method is illustrated on Pb#Pb data taken by the NA49 Collaboration at the CERN SPS. 1.3. Notation and conventions We use natural units "c"k "1. Unless explicitly stated otherwise, pairs or sets of N particles are meant to be pairs or sets of identical spinless bosons. In particular, we think of like-sign pions or kaons, the most abundant mesons in heavy-ion collisions. Most of our discussion carries over to fermionic particles by replacing the # signs in Eqs. (1.4) and (1.5) by ! signs and changing from symmetrized to anti-symmetrized N-particle states where ever they appear in derivations. In what follows, we do not mention the fermionic case explicitly. Most of our notation is introduced during the discussion. Variables in bold face denote 3-vectors. For simpler reference, we list here some of the variables used most frequently. p "(E , p ) G G G m "(m#p , ,

detected "nal state particle momenta, on-shell single particle transverse mass azimuthal angle of p ,

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E #pl y" ln N  E !pl N y p\ G p( G r\ G t[ G K"(p #p )    M "(m#K  , , U E #Kl >" ln )  E !Kl ) q"(p !p )   b"K/K C(q, K) S(x, K) o(x, p) P (p)  P (p , p )    N NK p

(roman y) single particle rapidity (italic y) coordinate in con"guration space simulated particle momenta, e.g. from Monte Carlo simulations particle momentum operator simulated particle positions simulated particle emission times average pair momentum, o!-shell transverse mass associated with K azimuthal angle of K , rapidity associated with K relative pair momentum, o!-shell velocity of particle pair (approximately) two-particle correlation function, also denoted by C(p , p )   single-particle Wigner density, emission function classical phase-space density covariant one-particle spectrum covariant two-particle spectrum normalization of the correlator number operator quantum mechanical wave packet width

2. Particle correlations from phase-space distributions There are numerous derivations of identical two-particle correlations from a given boson emitting source. An (over)simpli"ed argument starts from the observation that, after weighting the emission points of a two-particle Bose}Einstein symmetrized plane wave W (x , x , p , p )"      (e p  x> p  x#e p  x> p  x) by a normalized spatial distribution of emission points o(x), the  two-particle correlator C(q) is given by the Fourier transform of the spatial distribution:



C(q)" dx dx o(x ) o(x ) "W ""1#"o(p !p )" .       

(2.1)

Extracting the spatial information o(x) from the measured momentum spectra is then a Fourier inversion problem. The solution is unique if we assume o(x) to be real and positive. Eq. (2.1) remains, however, unsatisfactory since it does not allow for a possible time-dependence of the emitter and cannot be easily extended to sources with position}momentum correlations. Both properties are indispensable for an analysis of the boson emitting sources created in heavy-ion collisions. A sound starting point is provided by the Lorentz invariant one- and

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two-particle distributions for each particle species dN "E1a( > P (p)"E p a( p2 ,  dp

(2.2)

dN P (p , p )"E E "E E 1a( p> a( > (2.3) p a( p a( p 2 .         dp dp     These distributions involve expectation values 122 which can be speci"ed in terms of a density operator characterizing the collision process. In most applications, 122 involves an average over an ensemble of events. The two-particle correlation function of identical particles is de"ned, up to a proportionality factor N, as the ratio of the one- and two-particle spectra: P (p , p )    . C(p , p )"N (2.4)   P (p ) P (p )     In Section 2.1, we discuss its normalization as well as the experimentally used method of `normalization by mixed pairsa. Sections 2.2 and 2.3 then deal with two di!erent derivations of the basic relation (1.4) between the two-particle correlation function and the Wigner phase-space density. Final state interactions and multiparticle symmetrization e!ects are discussed subsequently in Sections 2.4 and 2.5. We conclude this chapter by discussing the implementation of this formalism into event generators. 2.1. Normalization The normalization N of the two-particle correlator (2.4) can be speci"ed by relating the particle spectra to inclusive di!erential cross sections, or by requiring a particular behaviour for the correlator (2.4) at large relative pair momentum q. Pair mixing algorithms used for the analysis of experimental data approximate these normalizations. 2.1.1. Diwerential and total one- and two-particle cross sections The one- and two-particle spectra, (2.2) and (2.3), are given in terms of the one- and two-particle inclusive di!erential cross sections as 1 dp L, P (p)"E  p dp 1 dp LL . P (p , p )"E E      p dp dp   They are normalized by

 

(2.5) (2.6)

dp P (p)"1NK 2 , E 

(2.7)

dp dp   P (p , p )"1NK (NK !1)2 ,    E E  

(2.8)

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where NK "dk aRk ak is the number operator. Two natural choices for the normalization N in Eq. (2.4) arise [68,113,176,191,192] by either taking directly the ratio of the measured spectra (2.7) and (2.8), which results in N"1 ,

(2.9)

or by "rst normalizing the numerator and denominator of Eq. (2.4) separately to unity, which gives 1NK 2 . N" 1NK (NK !1)2

(2.10)

Since in either case N is momentum-independent, it does not a!ect the space-time interpretation of the correlation function, with which we will focus primarily. For N"1, the correlator equals 1 whenever P (p , p )"P (p )P (p ). Neglecting kinematical        constraints resulting from "nite event multiplicities, one can often assume this factorization property to be valid for large relative momenta q. Since at small values of q the correlation function is larger than unity, this generally implies 1NK (NK !1)2'1NK 2, i.e., larger than Poissonian multiplicity #uctuations. This is a natural consequence of Bose}Einstein correlations. The second choice (2.10) permits to view the correlator as a factor which relates the two-particle di!erential cross section dp #/dp dp of the real world (where Bose}Einstein symmetrization LL   exists) to an idealized world in which Bose}Einstein "nal state correlations are absent, dp #/dp dp "C(q, K) dp,-/dp dp , (2.11) LL   LL   without changing the event multiplicities. Such an idealized world is a natural concept in event generator studies which simulate essentially the two-particle cross sections dp,-/dp dp . They LL   are typically tuned to reproduce the measured multiplicity distributions and thereby account heuristically for the e!ects of Bose}Einstein statistics on particle production; the quantum statistical symmetrization of the "nal state, however, is not a part of the code. With the normalization (2.10), the factor C(q, K) in Eq. (2.11) preserves the total cross sections, p #"p,-. LL LL 2.1.2. Experimental construction of the correlator From the data of relativistic heavy-ion experiments, the two-particle correlator is usually constructed as a quotient of samples of so-called actual pairs and `mixeda pairs or reference pairs. One starts by selecting events from the primary data set. Actual pairs are pairs of particles that belong to the same event. Reference pair partners are picked randomly from di!erent events within the set of events that yielded the actual pairs. The correlation function is then constructed by taking the ratio, bin by bin, of the distribution D of these actual pairs with the distribution D of the  0 reference pairs [183,184], number of actual pairs in bin (*q, *K) , D (*q, *K)"  number of actual pairs in sample

(2.12)

number of reference pairs in bin (*q, *K) D (*q, *K)" , 0 number of reference pairs in sample

(2.13)

D (*q, *K) . C(*q, *K)"  D (*q, *K) 0

(2.14)

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The number of reference pairs for each actual pair, the so-called mixing factor, is typically between 10 and 50. It has to be chosen su$ciently large to ensure a statistically independent reference pair sample while for numerical implementations it is of course favourable to keep the size of this sample as small as possible. The two-particle correlator constructed in Eq. (2.14) coincides with the theoretical de"nition (2.4) only if the reference pair distribution D coincides with an appropriately 0 normalized product of one-particle spectra. Since both actual and reference pair distributions are normalized to the corresponding total particle multiplicity, this normalization of Eq. (2.14) coincides with the normalization N"1NK 2/1NK (NK !1)2. A di!erent construction of the correlator from experimental data has been proposed by MisH kowiec and Voloshin [113] (see also [191]). Their proposal amounts to a modi"cation of the number of pairs in the sample by which Eqs. (2.12) and (2.13) are normalized and coincides with N"1. In general, the reference sample contains residual correlations, which are not of physical origin but stem typically from the restricted acceptance of experiments. For these residual correlations, corrections can be employed [183,184]. 2.2. Classical current parametrization How does one calculate the momentum correlations for identical pions produced in a heavy-ion collision? The pion production in a nuclear collision is described by the "eld equations for the pion "eld (x) [68], (䊐#m) K (x)"JK (x) .

(2.15)

This equation is obviously intractable since the nuclear current operator JK (x) couples back to the pion "eld and is not explicitly known. The classical current parametrization [68] approximates the nuclear current JK (x) by a classical commuting space-time function J(x). The underlying picture is that at freeze-out, when the pions stop interacting, the emitting source is assumed not to be a!ected by the emission of a single pion. This approximation can be justi"ed for high event multiplicities [68]. For a classical source J(x), the "nal pion state is then a coherent state "J2 which is an eigenstate of the annihilation operator a( p"J2"iJI (p)"J2 .

(2.16)

The Fourier transformed classical currents JI are on-shell. Using Eq. (2.16), the one- and twoparticle spectra (2.2) and (2.3) are then readily calculated. Usually, the classical current is taken to be a superposition of independent elementary source functions J :  , JI (p)" e (Ge NVGJI (p!p ) .  G G

(2.17)

If the phases are random, then this ansatz characterizes uncorrelated `chaotica particle emission, G and the intercept j in Eq. (1.5) equals one. In more general settings, where the phases are not G random, the intercept drops below unity. One distinguishes accordingly between a formalism for chaotic and partially chaotic sources.

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2.2.1. Chaotic sources Chaotic sources are given by a superposition (2.17) of elementary sources J , centered around  phase-space points x , p with random phases . The corresponding ensemble average specifying G G G the particle spectra (2.2) and (2.3) is [40]





Ld

 , G1J"OK "J2 , (2.18) 1OK 2" P “ dx dp o(x , p ) G G G G , 2n  , G where P is the probability distribution for the number N of sources,  P "1 and the , , , in normalized probability o(x , p ) describes the distribution of the elementary sources phase space. G G A direct consequence of the ensemble average (2.18) is the factorization of the two-particle distribution into two-point functions [40,78] 1N(N!1)2 .(P (p )P (p )#"SM (p , p )") , P (p , p )"        (   1N2 .

(2.19)

(2.20) SM (p , p ),(E E 1a( p> a( p2"(E E 1JI H(p )JI (p )2 ,     (     where 1N(N!1)2 "  P N(N!1), 1N2 "  P N and the expectation value in Eq. , . ,to,the prescription.Eq. (2.18). , Here, (2.20) is evaluated according the integer N denotes the number of sources, not the number of emitted pions. For a Poissonian source multiplicity distribution the prefactor in Eq. (2.19) equals unity. In the derivation of Eq. (2.19) a term is dropped in which both "nal state particles come from the same source. This term vanishes in the large N limit [68,78]. The factorization in Eq. (2.19) follows from the commutation relations, once independent particle emission and the absence of "nal state interactions is assumed. Due to its generality, it is sometimes referred to as `generalized Wick theorema. The emission function S(x, K) which enters the basic relation (1.4) can then be identi"ed with the Fourier transform of the covariant quantity SM (p , p ) [40,78]. The latter is given by the Wigner transform of the density matrix associated with (   the classical currents,



dy e\ )W1JH(x#y)J(x!y)2 , S (x, K)"   ( 2(2n)

(2.21)

for which the following folding relation can be derived [40]:



S (x, K)"1N2 dz dq o(z, q)S (x!z, K!q) , ( . 



dy e\ )WJH(x#y)J (x!y) . S (x, K)"      2(2n)

(2.22) (2.23)

Here 1N2 is "xed by normalizing the one-particle spectrum to the mean pion multiplicity . 1N2 "1NK 2; the distribution o and elementary source function S are normalized to unity. The .  full emission function is hence given by folding the distribution o of the elementary currents J in  phase-space with the Wigner density S (x, K) of the elementary sources. Wigner functions are  quantum mechanical analogues of classical phase-space distributions [90]. In general they are real

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but not positive de"nite, but when integrated over x or K they yield the observable particle distributions in coordinate or momentum space, respectively. Averaging the quantum mechanical Wigner function S(x, K) over phase-space volumes which are large compared to the volume (2n ) of an elementary phase-space cell, one obtains a smooth and positive function which can be interpreted as a classical phase-space density. From the particle distributions (2.19)/(2.20) one "nds the two-particle correlator [27,40,68,125,150,206,210] "dx S(x, K)e OV" , (2.24) C(q, K)"1# dx S(x, p )dyS(y, p )   if the normalization N in Eq. (2.4) is chosen to cancel the prefactor in Eq. (2.19). Adopting instead the normalization prescription (2.10) leads to a normalization of C(q, K) at qPR which is smaller than unity, since Eq. (2.10) is not the inverse of the prefactor in Eq. (2.19). 2.2.2. The smoothness and on-shell approximation The smoothness approximation assumes that the emission function has a su$ciently smooth momentum dependence such that one can replace S(x, K!q)S(y, K#q)+S(x, K)S(y, K) . (2.25)   Deviations caused by this approximation are proportional to the curvature of the single-particle distribution in logarithmic representation [42] and were shown to be negligible for typical hadronic emission functions. Using this smoothness approximation, the two-particle correlator (2.24) reduces to the expression on the r.h.s. in Eq. (1.4). Eq. (1.4) which uses the smoothness approximation, forms the basis for the interpretation of correlation measurements in terms of space-time variances of the source as will be explained in Section 3. For the calculation of the correlator from a given emission function, the smoothness approximation can be released by staring directly from (2.24). In an analysis of measured correlation functions in terms of space-time variances of the source, one can correct for it systematically [42] using information from the single particle spectra. The emission function S(x, K) depends in principle on the ow-shell momentum K, where K"(E #E ). In many applications one uses the on-shell approximation    (2.26) S(x, K, K)+S(x, E , K), E "(m#K . ) ) Again, the corrections can be calculated systematically [42] but were shown to be small for typical model emission functions for pions and heavier hadrons. The on-shell approximation (2.26) is instrumental in event generator studies, where one aims at associating the emission function S(x, K) with the simulated on-shell particle phase-space distribution at freeze-out, see Section 2.6. It is also used heavily in analytical model studies, see Section 5. 2.2.3. The mass-shell constraint Although the correlator (2.24) is obtained as a Fourier transform of the emission function S(x, K), this emission function cannot be reconstructed uniquely from the momentum correlator (2.24).

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Note that since the Wigner density S(x, K) is always real, the reconstruction of its phase is not the issue. The reason is rather the mass-shell constraint K ) q"p!p"m!m"0 , (2.27)     which implies that only three of the four relative momentum components are kinematically independent. Hence, the q-dependence of C(q, K) allows to test only three of the four independent x-directions of the emission function. This introduces an unavoidable model-dependence in the reconstruction of S(x, K), which can only be removed by additional information not encoded in the two-particle correlations between identical particles. Eq. (2.27) suggests that this ambiguity may be resolvable by combining correlation data from unlike particles with di!erent mass combinations, if they are emitted from the same source. Unlike particles do not exhibit Bose}Einstein correlations, but are correlated via "nal state interactions and therefore also contain information about the source emission function. In this review, we do not discuss unlike particle correlations, although this is presently a very active "eld of research [6,99}101,114,156,168]. It is still an open question to what extent a combined analysis of like and unlike particles allows to bypass the mass-shell constraint (2.27). The consequences of the mass-shell constraint are discussed extensively in Sections 3 and 5. 2.2.4. The relative distance distribution For several applications of two-particle interferometry it is useful to reformulate the correlator (1.4) in terms of the so-called normalized relative distance distribution









x x d(x, K)" dXs X# , K s X! , K , 2 2

(2.28)

S(x, K) , s(x, K)" dxS(x, K)

(2.29)

constructed from the normalized emission function s(x, K). Note that d(x, K)"d(!x, K) is an even function of x. This allows to rewrite the correlator (1.4) as



C(q, K)!1" dx cos(q ) x) d(x, K) ,

(2.30)

where the smoothness and on-shell approximations were used. With the mass-shell constraint in the form q"b ) q this can be further rewritten in terms of the `relative source functiona SK(r):







C(q, K)!1" dx cos(q ) x) dt d(x#bt, t; K)" dx cos(q ) x) SK(x) .

(2.31)

In the rest frame of the particle pair where b"0, the relative source function SK(x) is a simple integral over the time argument of the relative distance distribution d(x, t; K). In this particular frame the time structure of the source is completely integrated out. This illustrates in the most direct way the basic limitations of any attempt to reconstruct the space-time structure of the source from the correlation function.

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2.2.5. Partially coherent sources It is well-known from quantum optics [62,197] that, in spite of Bose}Einstein statistics, the HBT-e!ect does not exist for particles emitted with phase coherence, but only for chaotic sources. This is why in Eq. (2.17) a chaotic superposition of independent elementary source functions J (x)  was adopted. The question of possible phase coherence in pion emission from high energy collisions was raised by Fowler and Weiner in the seventies [57}59]; so far the dynamical origin of such phase coherence e!ects has however remained speculative. Their consequences for HBT interferometry can be studied by adding a coherent component to the classical current discussed above, J(p)"J

(p)#J (p) . (2.32)   An analysis similar to the one presented in Section 2.2.1 shows that as the number n (p) of  coherently emitted particles increases, the strength of the correlation is reduced [68]: j(K)"1!D(K) ,

(2.33)

n (K)  . (2.34) D(K)" n (K)#n (K)   For this reason the intercept parameter j(K) is often referred to as the coherence parameter. In practice various other e!ects (e.g. particle misidenti"cation, resonance decay contributions, "nal state Coulomb interactions) can decrease the measured intercept parameter signi"cantly. Although experimentally it is always found smaller than unity, 0(j(K)(1, this can thus not be directly attributed to a coherent "eld component. For a detailed account of the search for coherent particle emission in high energy physics we refer to the reprint collection [170]. Recent work [80] shows that the strength D(K) of the coherent component can be determined independent of resonance decay contributions and contaminations from misidenti"ed particles if two- and threepion correlations are compared, see Section 4.3. A coherent component would also a!ect the size of the HBT radius parameters and their momentum dependence [11,12,68,80,169]. The ansatz Eq. (2.32) is only one possibility to describe partially coherent emission. Alternatively, one may e.g. choose a distribution of the phases in Eq. (2.17) which is not completely random, thereby G mimicking partial coherence [148]. The equivalence of these two approaches still remains to be studied. 2.3. Gaussian wave packets It has been suggested repeatedly [51,107,110,119,120,174,176,190,192] that due to the smallness of the source in high energy and relativistic heavy-ion physics, particle interferometry should be based on "nite size wave packets rather than plane waves. This leads to an alternative derivation of the basic relations (1.3), (1.4) which replaces the classical currents from the previous section by the more intuitive notion of quantum mechanical wave packets, at the expense of giving up manifest Lorentz covariance in intermediate steps of the derivation. One starts from a de"nition of the boson emitting source by a discrete set of N phase-space points (r\ , t[ , p\ ) or by a continuous G G G distribution o(r\ , t[ , p\ ). These emission points are associated with the centers of N Gaussian G G G

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one-particle wave packets f [110,174,192] G 1  x r\  x p\ (r\ , t[ , p\ )Pf (x, t[ )" e\N  \ G e  G . G G G G G (np)

(2.35)

The wave packets f are quantum mechanically best localized states, i.e., they saturate the G Heisenberg uncertainty relation with *xl"p/(2 and *pl"1/(2p for all three spatial components l"1, 2, 3. Here, (*xl), 1 f , x( lf 2!1 f , x( lf 2, x( l being the position operator, and analogG G G G ously for *pl. We consider the free time evolution of these wave packets determined by the single particle hamiltonian HK ,  1 f (x, t)"(e\ &K R\R[ Gf )(x, t)" dk fI (k)e k  x\#IR\R[ G . (2.36) G G G (2n)



In momentum space, the free non-relativistic and relativistic time evolutions di!er only by the choices E "k/2m and E "(k#m, respectively. For the non-relativistic case, the integral I I (2.36) can be done analytically. 2.3.1. The pair approximation Here, we derive the correlator in the so-called pair approximation in which two-particle symmetrized wave functions U (x, y, t) are associated with all boson pairs (i, j) constructed from the GH set of emission points [174,176]: 1 ( f (x, t) f (y, t)#f (y, t) f (x, t)) . U (x, y, t)" H G H GH (2 G

(2.37)

The norm of this two-particle state di!ers from unity by terms proportional to the wave packet overlaps 1 f , f 2, but in the pair approximation this di!erence is neglected, 1 f , f 2+d . In Section G H G H GH 2.4 we will release this approximation and instead start from properly normalized N-particle wavefunctions. It is then seen that the pair approximation is equivalent to approximating the two-particle correlator from an N-particle symmetrized wavefunction by a sum of contributions involving only two-particle terms U . GH The two-particle Wigner phase-space density W (x, y, p , p , t) associated with U reads [90] GH   GH x y x y (2.38) W (x, y, p , p , t)" dx dy U x# , y# , t e p  xYe p  yY UH x! , y! , t . GH GH GH   2 2 2 2













Integrating this Wigner function over the positions x, y, we obtain the positive de"nite probability P (p , p , t) to measure the bosons of the state U at time t with momenta p , p . As long as "nal GH   GH   state interactions are ignored, this probability is independent of the detection time. For Gaussian wave packets it takes the explicit form (the energy factors ensure that P transforms covariantly) GH P (p , p ) 1 GH   " dx dy W (x, y, p , p , t) GH   E E (2n)   "w (p , p )w (p , p )#w (p , p )w (p , p )#w (p , p )w (p , p )  G   H    G   H   G   H   cos((r\ !r\ ) ) (p !p )!(t[ !t[ )(E !E )) , (2.39) G H   G H  



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p w (p , p )"s ((p #p )) exp ! (p !p ) , G   G    4 

161

(2.40)

s (K)"n\ pexp[!p(p\ !K)] . (2.41) G G Here, the integral over s (K) is normalized to unity, and the two-particle probability is normalized G such that its momentum integral equals one for pairs which are well-separated in phase-space. To relate this formalism to the emission function (2.22) and (2.23) of the classical current parametrization, we rewrite the two-particle probability (2.39) in terms of the Wigner densities s (x, K) of the G wave packets,











dy y y e\ K  yf x# , t[ f H x! , t[ s (x, K)"d(t!t[ ) G G G (2n) 2 G G 2 G 1 " d(t!t[ )e\x\r\ GN\NK\p\ G , G n





(2.42)





P (p , p ) 1 1 GH   " dx s (x, p ) dy s (y, p )# dx s (x, p ) dy s (y, p ) G  H  G  H  E E  2 2 N N 1 1 # dx s (x, K)e OV dy s (y, K) e\ OW# dx s (x, K) e\ OV dy s (y, K) e OW . G H G H 2 2









(2.43)

In the pair approximation, the two pion spectrum P (p , p ) for an event with N pions emitted from    phase-space points (r\ , t[ , p\ ) is a sum over the probabilities P of all N(N!1) pairs (i, j). The G G G GH  corresponding expression for a continuous distribution o(r\ , t[ , p\ ) of wave packet centers is obtained G G G by an integral over (2.43). De"ning Do "dp\ dr\ dt[ o(r\ , t [ , p\ ) , G G G G G G G



Do "1 , G

(2.44)

we "nd

 

P (p , p )" Do Do P (p , p )    G H GH  





" dx S (x, p ) dy S (y, p )# dxS (x, K) e OV     





.

(2.45)

The index on S indicates that this emission function is constructed from a superposition of  wavepackets while the emission function S in Eq. (2.22) was generated from the classical source ( currents. Similar to Eqs. (2.22) and (2.23), S (x, K) is given by a folding relation between the  classical distribution o of wave packet centers and the elementary source Wigner function s (x, K), 



S (x, K)" dp\ dr\ dt[ o(r\ , t[ , p\ ) s (x!r\ , t!t[ , K!p\ ) , G G G G G G  G G G 






E x s (x, K)" ) exp ! !pK d(t) .  n p

(2.46) (2.47)

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The normalization of Eq. (2.46) is consistent with the interpretation of the integral Eq. (1.3) as the one-particle spectrum. To determine the normalization N(p , p ) of the two-particle correlation function   P (p , p ) (2.48) C(p , p )"    ,   N(p , p )   we proceed in analogy to the experimental practice of `normalization by mixed pairsa: An uncorrelated (mixed) pair is described by an unsymmetrized product state U(x, y, t)"f (x, t) f (y, t) , (2.49) GH G H for which the two particle Wigner phase-space density and the corresponding detection probability P(p , p ) can be calculated [174] according to Eqs. (2.38)}(2.41). Taking both distinguishable GH   states U and U into account and averaging over the distribution o of wave packet centers, GH HG the normalization N(p , p ) coincides with the "rst two terms in Eq. (2.45),  





N(p , p )" dx S (x, p ) dy S (y, p ) .      

(2.50)

The two-particle correlator then coincides with the basic relation Eq. (2.24) after identifying S ,S. We note already here that starting from a discrete "nite set of N emission points  +(p\ , r\ , t[ ), , rather than averaging over a smooth distribution o, the expression for the G G G GZ ,

two-particle correlator (2.24) receives "nite multiplicity corrections [174]. These will be discussed in Section 2.6. 2.3.2. An example: the Zajc model We illustrate the consequences of the above folding relation (2.46) and (2.47) with a simple model emission function "rst proposed by Zajc [186]:








r 1 r)p p !2s # d(t) , S(r, t, p)"N exp !  2(1!s) R R P P     (2.51) N N "E , R ,R (1!s .  N(2nR P )     This emission function is normalized to a total event multiplicity N. The parameter s smoothly interpolates between completely uncorrelated (sP0) and completely position}momentum correlated (sP1) sources. In the limit sP0, this emission function can be considered as a quantum mechanically allowed Wigner function as long as R P 5 /2. In the opposite limit,   x p ! d(t) , (2.52) lim S(x, p)&d R P   Q the position}momentum correlation is perfect, and the phase-space localization described by the model is no longer consistent with the Heisenberg uncertainty relation. Inserting the model






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emission function (2.51) into the general expression (2.24) for the two-particle correlator one "nds [186] C(q, K)"1#exp(!R q) , (2.53) & 2 1 . (2.54) R "R 1! & 2  (2R P )   For su$ciently large s this leads to an unphysical rise of the correlation function with increasing q. One can argue [61,190] that the sign change in Eq. (2.54) is directly related to the violation of the uncertainty relation by the emission function (2.51). If one does not interpret (2.51) directly as the emission function S(x, p), but as a classical phase-space distribution o(r\ , p\ ) of Gaussian wave packet centers, then the correlator is readily calculated via Eq. (2.46) [61]:






C(q, K)"1#exp(!R q) , & 2N 1 , R "R 1! & 2N (2RP)






p 1 R,R# , P,P# .   2p 2

(2.55) (2.56) (2.57)

Now 2RP51 independent of the value of p, and the radius parameter is always positive. Even if the classical distribution o(r\ , p\ ) is sharply localized in phase-space, its folding with minimumuncertainty wave packets leads to a quantum mechanically allowed emission function S(x, p) and a correlator with a realistic fall-o! in q. 2.3.3. Spatial localization of wave packets Both the two-particle correlator and the one-particle spectrum calculated from (2.46) depend on the initial spatial localization p which is a free parameter. One easily sees that both limits pP0 and pPR lead to unrealistic physical situations: In the limit pP0, the wave-packets is sharply localized in coordinate space, and the momenta p\ drop out of all physical observables. The one-particle spectrum E dN/dp comes out moG N mentum-independent irrespective of the range of the wave packet momenta p\ . The momentum G correlations read [174,190]



(2.58) lim C(p , p )"1# Do Do cos[(r\ !r\ ) ) (p !p )!(t[ !t[ ) ) (E !E )] . G H G H   G H     N Due to the cosine term, the dependence of the two-particle correlator on the measured relative energy E !E and momentum p !p gives information on the initial spatial and temporal     relative distances in the source. This is the HBT e!ect. On the other hand, Eq. (2.58) shows that in this limit the correlator does not depend on the pair momentum K, since position eigenstates cannot carry momentum information. In the other limiting case pPR, the wave packets are momentum eigenstates which contain no information about the emission points r\ . In this limit, nothing can be said about the spatial G

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extension of the source, since the wave packets show an in"nite spatial delocalization. A calculation shows that also the temporal information is lost in this case: (2.59) lim C(p , p )"1#dp p .    N Clearly, physical applications of the wave packet formalism require "nite values of p. For example, one can use the Gaussian (2.51) with s"0 to generate the distribution of wave packet centers. Writing P"2m¹ to allow for an intuitive interpretation of its momentum dependence in  terms of a non-relativistic thermal distribution of temperature ¹, the one-particle spectrum shows again thermal behaviour &exp(!p/2m¹ ), but with a shifted temperature [110,174,190]  1 . (2.60) ¹ "¹#  2mp The corresponding HBT radius parameter reads p 2m¹p R "R# . & 2N  2 1#2m¹p

(2.61)

The second terms in these equations re#ect the contributions from the intrinsic momenta and spatial extension of the wave packets. This shows that repairing possible violations of the uncertainty relation in a given classical phase space distribution by smearing it with Gaussian wave packets of "nite size p, one changes both the single-particle momentum spectra and two-particle correlations. While large values of p strongly a!ect the source size and thereby the HBT-radii but have little e!ect on the slope of the single-particle spectrum, the opposite is true for small values of p. With both quantities "xed by experiment, one has therefore limited freedom in the choice of p. Di!erent attempts to give physical meaning to the parameters p can be found in the literature. For example, Goldhaber et al. [63] argued that the HBT-radius measured in pp annihilation at rest can be interpreted in terms of the pion Compton wavelength. Baym recently tried to associate p with the coherence length for phase coherence in the source [22]. On the other hand, Eqs. (2.60) and (2.61) show that (at least in Gaussian models) the physical observables have a functional dependence on only two independent combinations of the three paramters ¹, R and p. In practice,  this allows to reabsorb the wave packet width in a rede"nition of the source parameters [81,179]. 2.4. Multiparticle symmetrization ewects Multiparticle symmetrization e!ects are contributions to the spectra of Bose}Einstein symmetrized N-particle states which cannot be written in terms of simpler pairwise ones. In manyparticle systems with high phase-space density, the single- and two-particle spectra receive non-negligible contributions from multiparticle symmetrization e!ects. This complicates the interpretation of the emission function S(x, K) as reconstructed from the data. Based on strategies proposed by Zajc [184,185] and Pratt [128,130,207], there exists by now an extensive literature on these e!ects consisting of numerical [128,130,184,185,187] and analytical

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[6,29,53,176,181,188,191,212] model studies. Multiparticle symmetrization e!ects have been considered essentially in two di!erent settings. Either one starts from events which at freeze-out have a "xed particle multiplicity N [176,185,189], encoded e.g. in the model assumptions by choosing sets of N phase-space points (p\ , r\ , t[ ). Bose}Einstein correlations in the "nal state then lead to an G G G enhancement of the two-particle correlator at small relative pair momentum, but they do not a!ect the particle multiplicity. A second approach [39,51,128,188,192,207,212] does not only calculate the HBT enhancement e!ect of identical particles, but aims at accounting for the e!ects of Bose}Einstein statistics during the particle production processes as well. As a result, modi"cations of the multiplicity distribution of event samples are calculated. Here, we "rst review the formalism for "xed event multiplicities, which is tailored to calculate the "nal state HBT e!ect only. Then we discuss shortly how this formalism can be adapted to calculate changes of multiplicity distributions. 2.4.1. The Pratt formalism In his original calculation [128,132,207], Pratt starts from the (unnormalized) probability PI (p) , for detecting N particles with momenta p"(p ,2, p ). It is expressed through single particle  , production amplitudes ¹ (x) for particles with quantum numbers a and N-particle symmetrized ? plane waves ;(x ,2, x ; p ,2, p ) as follows:  ,  ,  (2.62) PI (p)" dx 2dx ¹ (x )2¹ ,(x );(x ,2, x ; p ,2, p ) ,  , ?  ? ,  ,  , , + , ?G , 1 “ exp[ip ) x H] . (2.63) ;(x ,2, x ; p ,2, p )" H Q  ,  , (N! QZS, H The sum runs over all permutations s of N particles. The main assumptions entering here are (i) the absence of "nal state interactions which allows the plane wave ansatz (2.63), and (ii) the assumption of independent particle emission which allows to factorize the N-particle production amplitude into N one-particle production amplitudes ¹ (x). It is technically convenient to change from these ? to the corresponding Wigner transformed emission function [128,132,207]











y y (2.64) S (x, p)" dy e NW ¹H x# ¹ x! . ? ? 2 2 2 ? Calculating PI (p , p ), one recovers in this formalism up to a normalization factor the usual    expression Eq. (2.19) for the two-particle spectrum. This is, however, not the relevant calculation because it gives only the two-particle spectrum from events with exactly two particles. The aim of Pratt's formalism is to compute the one- and two-particle spectra for events with multiplicity N, including all multiparticle symmetrization e!ects. They are obtained by integrating Eqs. (2.62) and (2.63) over N!1 or N!2 momenta, respectively. We use the notation P for N-particle , momentum distributions which, in contrast to Eqs. (2.5) and (2.6), are normalized to unity. Using the following building blocks [128,207]



G (p , p )" dx S (x, K) e\ OV ,    2

(2.65)

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dk dk 2 L G (p , k )G (k , k )2G (k , p ) , G (p , p )" L          L  E E  L



dp G (p, p) , C" L E L N

(2.66)

(2.67)

one obtains the desired spectra by the following algorithm [38,51,128,176,185,192,196,207]: 1 1 , P (p)" u(N!m)G (p, p) , , K N u(N) K

(2.68)

1 , 1 u(N!J) P (p , p )" ,   N(N!1)u(N) ( (\ [G (p , p )G (p , p )#G (p , p )G (p , p )] , G   (\G   G   (\G   G

(2.69)



1 CJCJ2CJL L u(N)" dp 2dp PI (p)"    , , N! “ nJL(l !) L L LJL, 1 , " u(N!m) C . K N K

(2.70)

While the sum in Eq. (2.63) runs over N! terms, this algorithm involves only sums over all partitions (n, l ) of N elements; this reduces the complexity of the problem considerably. The algorithm L, (2.68)}(2.70) is sometimes referred to as `ring algebraa [51,192], since the building blocks C K and G have a very simple diagrammatic representation in terms of closed and open rings K [128,137,176,207]. It is also referred to as Zajc}Pratt algorithm, since Zajc had analyzed essential parts of the above combinatorics in [185]. The de"nition of the C sometimes di!ers by a factor K m from the one given here, which results in appropriately modi"ed combinatorial factors in Eqs. (2.68)}(2.70). While the set of Eqs. (2.68)}(2.70) constitutes a great simpli"cation over a direct evaluation of Eq. (2.62), the high-dimensional integrations required to determine G in Eq. (2.66) still limit K its applications signi"cantly. Numerical Monte Carlo techniques have been proposed [55,128,130,207] to calculate Eq. (2.66). An alternative strategy can be applied to a small class of simple (Gaussian) models, where one can control the m-dependence of G analytically or via simple K recursion schemes. Especially for Gaussian emission functions, Eq. (2.66) allows for simple one-step recursion relations [39,128,130,188,207,212] between G and G which can be solved analytically L> L [192]. 2.4.2. Multiparticle correlations for wave packets There have been several recent attempts to combine the Pratt formalism with an explicit parametrization of the source in terms of N-particle Gaussian wave packets [51,176,192]. The strategy in these studies is to associate with each event +(r\ , t[ , p\ ), a properly symmetrized G G G GZ ,

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N-particle wave function [51,176,192]






1 , +(r\ , t[ , p\ ), PW (x, t)" “ f G(x , t) , (2.71) G G G GZ ,

, (N! QZS, G Q G where the f are the Gaussian wave packets of Eq. (2.35). Note that the normalization G NW"1/1W "W 2 of W depends on the positions +(r\ , t[ , p\ ), of the wave packet centers in , , , G G G GZ ,

phase-space. As we shall see, this prevents a straightforward application of the Pratt formalism. The normalized probability P (p; +z\ ,) for detecting N particles with momenta p"(p ,2, p ) in ,  , the speci"c wave packet con"guration +z\ ,,+(r\ , t[ , p\ ), takes the following form [176]: G G G GZ ,

NW , P (p; +z\ ,)" “ F l l(pl) , (2.72) , Q Q N! P QQYZ L J where the building blocks F (p) are given in terms of the Fourier transforms fI of the single-particle GH G wave packets as follows: F (p)"D (p, t) DH(p, t) , (2.73) GH G H D (p, t)"e\ #NR\R[ GfI (p, t[ ) . (2.74) G G G The time dependence of Eq. (2.74) drops out in F . From Eq. (2.72) the normalized one- and GH two-particle momentum distributions are obtained by integrating over the unobserved momenta [176,185]: 1 , F  (p ) “ f l l , P (p ; +z\ ,)"NW ,  QQ N! S Q Q  l QQYZ ,  1 , F  (p )F  (p ) “ f l l , P (p , p ; +z\ ,)"NW ,   QQ N! S QQ  QQ  l QQYZ , 



f " dp F (p) . GH GH

(2.75) (2.76) (2.77)

Similarly, higher order particle spectra PK contain m factors F G G in each term. , QQ The factors “ f occurring in Eqs. (2.75) and (2.76) re#ect the multiparticle symmetrization e!ects GH on the one- and two-particle spectra. They involve the overlap between pairs of wave packets f , f , G H which for the simple case of instantaneous emission, t[ "t[ , take the simple form G H (2.78) " f ""exp[!"z\ !z\ "] , H GH  G 1 (2.79) z\ " r\ #ipp\ . H H pH This overlap equals 1 for i"j and decreases like a Gaussian with increasing phase-space distance "z\ !z\ " between the wave packet centers. According to Eq. (2.79), this distance depends on the wave G H packet width p, and in the limiting cases pP0 and pPR, the overlap functions reduce to what is known as the pair approximation:

f "d . GH GH

(2.80)

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As we will see, these limits correspond to the case of in"nite phase-space volume, i.e., vanishing phase-space density of the source. Then all sums over S in Eqs. (2.75) and (2.76) are trivial and the , two-particle spectrum Eq. (2.76) reduces to a sum over all particle pairs, involving only two-particle symmetrized contributions, thus coinciding [176] with the correlator derived in Section 2.3. In order to apply the Zajc}Pratt algorithm (2.68)}(2.70), the distributions given above must be averaged over the phase-space positions (r\ , t[ , p\ ) of the wave packet centers: G G G , (2.81) PL (p ,2, p )" “ Do PL (p ,2, p ; +z\ ,) , G ,  ,  L L G where Do is de"ned in Eq. (2.44). This is the analogue of the sum over quantum numbers a in G G Eq. (2.62). At this point, one encounters the problem that the normalization NW of the N-particle wave packet does not factorize. This destroys the factorization property Eq. (2.66) of the Pratt algorithm. However, an analytical calculation becomes possible if a di!erent averaging procedure is used instead of Eq. (2.81):






, 1W "W 2 , (2.82) “ Do PDo" , , “ o(r\ , t[ , p\ ) , G G G G u(N) G G , u(N)" “ dp\ dr\ dt[ o(r\ , t[ , p\ ) 1W "W 2 . (2.83) G G G G G G , , G This modi"cation Eq. (2.82) is equivalent to working with unnormalized N-particle wave packet states, as done e.g. in Ref. [39]. ZimaH nyi and CsoK rgo [192] have tried to give this modi"cation a simple physical interpretation by noting that the factor 1W "W 2 can be interpreted as an , , enhanced emission probability for bosons (described by normalized wave packets) which are emitted close to each other in phase-space. According to Eq. (2.82), which no longer factorizes, this version of `stimulated emissiona leads to speci"c correlations among the emission points (r\ , t[ , p\ ), G G G i.e., the particles are not emitted independently. For the average in Eq. (2.82), the spectra PL can be obtained from the Zajc}Pratt algorithm , Eq. (2.68)}(2.70) using the building blocks









G (p , p )" K  





K “ Do l DH(p ) f   f  2f K\ KD K(p ) , G G  G G G G G G G  J

C " dp G (p, p) . K K

(2.84) (2.85)

One only needs to replace in Eq. (2.65) Pratt's de"nition Eq. (2.64) for the single-particle Wigner density, S (x, K), by the wave packet analogue S (x, K) given in Eq. (2.46). 2  2.4.3. Results of model studies Explicit numerical [128,130,188,207,212] and analytical [51,176,192] calculations of multiparticle symmetrization e!ects have so far only been performed for Gaussian source models. In this case the mth order Pratt terms G , see Eq. (2.66), can be calculated analytically. Writing them in the K form G (K#q, K!q)"C (gK/n) exp[!gKq!gKK] ,  K ) O ) K 

(2.86)

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the coe$cients gK and gK can then be obtained from simple recursion relations [192]. Two O ) generic features are observed in all these studies [51,128,176,185,188,207,212]: 1. For increasing m the factors gK become larger. This leads to steeper local slopes of the ) one-particle spectrum for small momenta. Multiparticle symmetrization e!ects thus enhance the particle occupation at low momentum. 2. For increasing m the factors gK decrease. This broadens the width of the two-particle correlator, O indicating that multiparticle symmetrization e!ects lead to an enhanced probability of "nding particles close together in con"guration space. While these observations are generic, their quantitative aspects are model dependent and sensitive to the particle phase-space density. Writing the single-particle spectrum Eq. (2.68) in the form [176] , u(N!m) v " C , (2.87) P (p)" v G (p, p)/C , , K K K K K u(N) K the weights v , which satisfy , v "1, can be analyzed in the limit of large phase-space volumes. K K K For a Gaussian source with width parameters R and P in coordinate and momentum space, the   phase-space density is given by o "N/(R P ). In the limit of large phase-space volume    (R P )<1 one "nds for "xed (but not necessarily small) multiplicity N [176]:   oK\ v +  . K (1#o )K 

(2.88)

Similarly, the two-particle spectrum can be written as [176] , K\ P (p , p )" u H (p , p ) , ,   K GK\G   K G G (p , p ) G (p , p ) G (p , p ) G (p , p ) H (p , p )" G   K\G   # G   K\G   . GK\G   C C C C G K\G K\G G

(2.89)

(2.90)

Again the weights u are normalized to unity, , u "1, and in the same limit as above their K K K leading behaviour is given by v oK\  " K\ . u + K (1#o )K (1#o )  

(2.91)

We "nally remark, that the correlator obtained from Eqs. (2.87) and (2.89) takes the generic form [176,191] P (p , p ) ,   "N(1#jF(p , p )) , C(p , p )"     P (p )P (p ) ,  , 

(2.92)

where j"1, N+1!(R P )\ for large phase-space volumes, and F(p , p ) approaches 1 and     0 in the limits qP0 and "q"PR, respectively.

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2.4.4. Bose}Einstein ewects and multiplicity distributions So far we only discussed multiparticle symmetrization e!ects for events with "xed multiplicity N. The question to what extent multipion correlations are also re#ected in the multiplicity distributions was asked already in the seventies [67]. Recently, it was revived in the context of Pratt's formalism [51,111,128,188,207,212] with the aim to calculate the e!ect of Bose}Einstein statistics on the particle production process. These applications typically start [128,188,192,207,212] from a multiplicity distribution p in the absence of Bose}Einstein statistics, for example a Poisson L nL distribution p"  exp[!n ] with average multiplicity n . For this case the probability p of   L L n! "nding events with multiplicity n after having accounted for Bose}Einstein correlations is then computed as [51,128,207]






 \ , (2.93) p "u(n) u(k) L I where u(k) is given in Eq. (2.70). For this particular multiplicity distribution, the multiplicity averaged one- and two-particle spectra are given by the simple expressions [38,39,191,192,196] P (p)"H(p, p) ,  P (p , p )"H(p , p ) H(p , p )#H(p , p ) H(p , p ) ,            where



(2.94) (2.95)

 H(p , p )" G (p , p ), dx S(x, K) e OV . (2.96)   K   K With the e!ective source distribution S(x, K) introduced in the second step, the correlator again takes the simple form Eq. (1.4). We expect that this source distribution coincides with the one de"ned in Eq. (2.21) in the context of the covariant classical current formalism. The reasons are that (i) both satisfy Eq. (2.24) and (ii) that the coherent states "J2 of Eq. (2.16) generate a Poissonian multiplicity distribution. According to the "rst equation in Eq. (2.96), the emission function S(x, K) contains all multiparticle symmetrization e!ects. Expressed in terms of the single-particle Wigner density S in Eq.  (2.46), it takes a complicated form. Model studies [38,39,51,128,130,188,191,196,207,212] indicate that irrespective of the particular multiplicity distribution the general features discussed below (2.86) persist: compared to the input distribution S (x, K), the multiparticle symmetrized emission  function S(x, K) is more strongly localized in both coordinate and momentum space. For the intercept parameter j one "nds results which depend on the speci"c choice for the multiplicity distribution. Cases are known where j decreases strongly with increasing phase-space density [51,128,132,188,207,212]. This discussion illustrates that for sources with high phase-space density, where multiparticle symmetrization e!ects cannot be neglected, the interpretation of the emission function S(x, K) reconstructed from the one- and two-particle momentum spectra by analyzing (1.3)}(1.5) is not straightforward. The question how Bose}Einstein e!ects on the multiplicity distribution and on the phase-space distribution can be disentangled is still open. In the remainder of this review we

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therefore concentrate on the reconstruction of S(x, K) and do not further consider its possible contamination by multiparticle e!ects. 2.5. Final state interactions Momentum correlations between identical particles can originate not only from quantum statistics but also from conservation laws and "nal state interactions. Energy}momentum conservation constrains the momentum distribution of produced particles near the kinematical boundaries. In high multiplicity heavy-ion collisions its e!ects on two-particle correlations at low relative momenta are negligible. Similarly, constraints from the conservation of quantum numbers (e.g. charge or isospin) become less important with increasing event multiplicity. Strong correlations exist between the decay products of resonances, but since resonance decays rarely lead to the production of identical particle pairs, they do not matter in practice. This leaves "nal state interactions as the most important source of dynamical correlations. For the small relative momenta q(100 MeV which are sampled in the two-particle correlator, e!ects of the strong interactions are negligible for pions. For protons, however, they dominate the two-particle correlations. On the other hand, for pions, the long-range Coulomb interactions distort signi"cantly the observed momentum correlations, dominating over the Bose}Einstein e!ect for small relative momenta. Here we discuss how Coulomb correlations are calculated for a given source function and how they can be corrected for in the data. The aim of Coulomb corrections is to modify the measured two-particle correlations in such a way that the resulting correlator contains only Bose}Einstein correlations, while the e!ects of "nal state interactions have been subtracted. For this, several simpli"ed procedures have been used in the literature, which we review in what follows. 2.5.1. Classical considerations The Coulomb interaction between particle pairs accelerates them relative to each other, thus depleting (enhancing) the two-particle correlation function at small relative momenta for like-sign (unlike-sign) pairs. In a high multiplicity environment this "nal state interaction can be reduced by screening e!ects until the particle pair has separated su$ciently from the rest of the system. Both these e!ects can be taken into account in a classical toy model which neglects the Coulomb interaction between the pairs for separations less than r and includes it for larger separations [21].  The initial and the "nally observed relative momenta q and q are then related by  (q/2) (q /2) e "  $ , 2k r 2k 

(2.97)

where k is the reduced mass. For two pions, k"m /2 and a radius r "10 fm, this results in a shift L  (q/2)"(q /2)$20 MeV. The modi"cation of the two-particle correlator is then given by the  Jacobian "dq /dq""q /q and reads [69,21]  

 

C(q)"



dq 2ke  C (q )"C (q ) 1G ,   dq   r (q/2) 

(2.98)

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where C (q ) denotes the two-particle correlator in the absence of Coulomb interactions. When   comparing Eq. (2.98) with the data, the radius r can be used to accommodate for the dependence of  the Coulomb "nal state interaction on the source size. This toy model reproduces the qualitative features of experimental data but fails to account quantitatively for the correct q-dependence of the correlator at very small relative momenta [21,22]. 2.5.2. Coulomb correction for xnite sources For a quantum mechanical discussion of "nal state Coulomb interactions, we associate to the emitted particle pairs a relative Coulomb wavefunction, given analytically by the con#uent hypergeometric function F, G

(r)"C(1#ig) e\LE e  q  r F(!ig; 1; z ) , (2.99) U q  \ (2.100) z "(qr$q ) r)"qr(1$cos h) .  !  Here, r""r", q""q", and h denotes the angle between these vectors. The Sommerfeld parameter g"a/(v /c) contains the dependence on the particle mass m and the electromagnetic coupling  strength e; we write me e k "$ , g "$ ! 4nq 4n q/2

(2.101)

where the plus (minus) sign is for pairs of unlike-sign (like-sign) particles. To illustrate the in#uence of a "nite source size in a simple case, we take recourse to the relative source function SK(r) de"ned in Eq. (2.31). This function describes the probability that a particle pair with pair momentum K is emitted from the source at initial relative distance r in the pair rest frame. For sources without x}K-correlations and neglecting the time structure of the particle emission process in the pair rest frame, the corresponding two-particle correlation for non-identical charged particle pairs reduces to [22,37]



(r)" . C>\(q, K)" dr SK(r) "U q 

(2.102)

Corrections to this expression are discussed in Section 2.5.4. For a pointlike source SK(r)"d(r), the correlator Eq. (2.102) is given by the Gamow factor G(g) which denotes the square of the (r) at vanishing pair separation r"0, Coulomb wavefunction U q  2ng G(g)""C(1#ig) e\LE"" . (2.103) eLE!1 For a Gaussian ansatz SK(r)Jexp[!r/4R], the dependence of the Coulomb correlations on the size R of the source is then determined via Eq. (2.102), see Fig. 2.1. If the particles are emitted with "nite separation r, their Coulomb interaction is weaker and the Gamow factor overestimates the "nal state interaction signi"cantly, see Fig. 2.1. The source size thus enters estimates of the Coulomb correction in a crucial way, and its selfconsistent inclusion in the correction procedure can lead to signi"cantly modi"ed source size estimates [4,5].

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Fig. 2.1. Unlike-sign correlations for 0(K (100 MeV/c and 4.0(>(5.0. (a) Dashed line: Gamow function, , corresponding to vanishing source radius. Solid line: "t of the NA49 Pb#Pb data to the function Eq. (2.104). (b) The same data compared to calculations based on Eq. (2.102) with a spherically symmetric Gaussian source, with R"0.5, 2, 4.6, and 8 fm (from top to bottom). The best "t is obtained for R"4.6 fm. Figure taken from [147].

2.5.3. Coulomb correction by unlike sign pairs Large acceptance experiments can measure like-sign and unlike-sign particle correlations simultaneously. The latter do not show Bose}Einstein enhancement but depend on "nal state interactions as well. This opens the possibility to correct for the Coulomb correlations in like-sign pairs by using the information contained in unlike-sign pairs. The q-dependence of the unlike-sign correlations C>\(q, K) is often parametrized by a q -dependent simple function [5,98]

   F(q )"1#(G(g )!1) e\O / ,  >

(2.104)

where q "q!(q) and Q is a "t parameter. q is the square of the spatial relative momentum    in the pair rest frame, where q"0. For small q , this function approaches the Gamow factor Eq.  (2.103) for a pointlike source, while it includes a phenomenological "nite-size correction for large relative momentum. More recently, one has started to avoid this intermediate step by constructing the corrected correlator C\\(q, K) for like-sign pairs directly using bin by bin the experimental  data from the measured like- and unlike-sign correlators [13,147]: C\\(q, K)"C>\(q, K) C\\(q, K) . 

 

 

(2.105)

In the absence of Bose}Einstein correlations and for pointlike sources, the lefthand side of this equation reduces to unity while the righthand side becomes a product of Gamow factors 1 . G(g ) G(g )" > \ 1#(n/3)g#O(g)

(2.106)

This expression provides an estimate for the accuracy of the correction procedure (2.105). It deviates from unity by less than "ve percent for relative momenta q'8 m/137 [147]. For pions, only the region q(10 MeV is a!ected signi"cantly, while for the more massive kaons the whole region q(25 MeV shows an error larger than 5%. Calculating the correction factors in (2.105) for extended sources, this picture does not change since the main di!erence between like-sign and unlike-sign correlations is due to the di!erent Gamow factors, and not to the r-dependent con#uent hypergeometric function in the relative wavefunctions [154]. As a consequence, one can obtain an

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improved Coulomb correction for heavier particles by dividing out these Gamow factors [154], C>\(q, K)C\\(q, K)

  C\\ (q, K)"   .   G(g ) G(g ) > \

(2.107)

This was shown to work with excellent accuracy for a wide range of source parameters [154]. 2.5.4. General formalism for xnal state interactions We now present a general formalism for the discussion of the e!ects of "nal state interactions, starting from an arbitrary two-particle symmetrized wave function W(x , x , t) which we expand in   terms of plane waves p p, 



dp dp   AW(p , p , t) p p (x , x , t) , W(x , x , t)"e\ &K R\RW(x , x , t )"            (2n) (2n)

(2.108)

p p(x , x , t)"e\ &K R\R p p(x , x , t )"e\ #>#R e px> px        " K(x, t) q (r, t)"e\ #>#R e K  x e q  r . 

(2.109)

In the last step, we have changed to center of mass coordinates x"(x #x ) and relative    coordinates r"(x !x ). The two-particle state W is evolved with the interacting hamiltonian   HK while the plane waves in which we expand follow a free time evolution, determined by HK , HK V ,   HK "HK V #HK #HK P ,    D HK V "! V ,  2M

D HK "! P ,  2k

HK P "<(r) . 

(2.110)

Here M"2m and k"m/2 are the pair and reduced mass, respectively. The two-particle state W determines the two-particle Wigner phase-space density and hence the two-particle correlator. The probability PW(p , p , t) of detecting the bosons at time t with momenta p , p is     PW(p , p , t)"AHW(p , p , t) AW(p , p , t) .      

(2.111)

Let us assume that from a time t onwards "nal state interactions have to be taken into account in  the description of the time evolution of W. The time evolution of AW(p , p , t) then reads  








r r AW(p , p , t)" dx dr HK (x, t )[e &K >&K P R\R e\ &K R\R q (r, t )]HW x# , x! , t .      2 2  (2.112) We are interested in the limit tPR of this expression. To this end, we use the M+ller operator X "lim e &K >&K P R\R e\ &K R\R , > R

(2.113)

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which determines the solution of the Lippmann}Schwinger equation for the corresponding stationary scattering problem,






D P #<(r) U q (r)"EU q (r) ,   2k

U (2.114) q (r)"X q (r, t ) .  >   Irrespective of the form of <(r), once U q  is determined, the two-particle detection probability  P (p , p , t"R) is known from Eq. (2.111). The corresponding two-particle spectrum is then given R   by summing over all pair wave functions of the event: R)AW(p , p , t"R) . (2.115) P (p , p )" AH W(p , p , t"        W   Coulomb correlations for instantaneous sources. We now illustrate the use of the two-particle spectrum Eq. (2.115), starting from the Gaussian wavefunction introduced in Section 2.3. The sum in Eq. (2.115) is then a sum over all pairs (i, j) of the set (r\ , t[ , p\ ), or an average over some G G G   W distribution o(r\ , t[ , p\ ). We restrict the calculation to instantaneous emission at time t "t "t . For G G G G H  non-identical particles (e.g. unlike-sign pions) the two-particle wave function at emission reads then  x\ x  K[ x  r\ r  q\ r W (r, x, t )"W (r, t ) W (x, t )"(np)\e\N \  >  e\N \  >  , GH       and the corresponding amplitude entering the two-particle spectrum Eq. (2.115) is

(2.116)

AW(p , p , t"R)"1 K"W 21U q "W 2      

(2.117)

H 1U q "W 2" drU q  (r)W (r) ,    

(2.118)



1UK"W 2"(np)\ e\NK\K[  e x\ K\K[  . (2.119)   For pairs of identical charged particles, the state (2.116) and the corresponding amplitude (2.117) must be symmetrized properly, adding the missing q!q terms and replacing, e.g., Uq by  1/(2(Uq $U q ). Further analytical simpli"cations of the amplitude Eq. (2.117) depend on the  \  functional form assumed for the two-particle state W. A study with Gaussian wave packets was presented in Ref. [179]. In the plane wave limit pP0 one recovers Pratt's result [126] PW(q, K)JG(g)[F(!ig; 1; iz ) FH(!ig; 1; iz ) \ \ #F(!ig; 1; iz ) FH(!ig; 1; iz )$e OP  FF(!ig; 1; iz ) FH(!ig; 1; iz ) > > \ > $e\ OP  FF(!ig; 1; iz ) FH(!ig; 1; iz )] . (2.120) > \ The "rst two lines are the Born probabilities "Uq (r)" and "U q (r)"; the exchange or interference  \  terms in the last two lines exist only for pairs of identical particles. Weighting (2.120) with the distribution SK(r), one obtains the properly symmetrized generalization of Eq. (2.102) for pairs of identical bosons. Coulomb correlations for time-dependent sources. In general, identical bosons interfering in the "nal state of a relativistic heavy-ion collision are produced at di!erent emission times t Ot . This G H

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temporal structure is neglected in the ansatz for the two-particle wavefunction Eq. (2.116) and does not appear in the corresponding result Eq. (2.120). A formalism appropriate for the calculation of two-particle spectra from arbitrary space-time dependent emission functions S(x, K) was developed in Ref. [8] (for a relativistic approach using the Bethe}Salpeter ansatz see Ref. [99]):



dp dx dy S(x, K#p)= (x!y, p) S(y, K!p) , P (p , p )" O    (2n)







dQ Q Q = (x!y, p)" e\ /V\Ws p# sH p! . O O (2n) 2 O 2

(2.121) (2.122)

Here s denotes essentially a Fourier transformed relative wavefunction times a propagator [8], O and the function = can be interpreted as the Wigner density associated with the (symmetrized) O distorted wave describing "nal state interactions. For a free time-evolution in the "nal state, the two-particle spectrum (2.121) coincides with the appropriately normalized spectrum (2.19). From the result (2.121) for the general interacting case, a simpli"ed expression can be obtained by expanding the temporal component of the phase factor in = in leading order of the small energy O transfer caused by the "nal state interaction [8]









y y P (p , p )" dx dy S x# , p S x! , p     2 2 



;[h(y)"Uq (y!* y)"#h(!y)"Uq (y!* y)"]     y y $ dx dy S x# , K S x! , K UH q (y!*y)Uq (y!*y) ,  \  2 2











(2.123)

K p p *" , * " , * "  . (2.124)   E E E ) ) ) The velocities * , * , and * are associated with the observed particle momenta p , p , and their     average K. In all three cases the argument of the FSI-distorted wave U can be understood as the distance between the two particles in the pair rest frame at the emission time of the second particle. Eq. (2.123) is obtained without invoking the smoothness approximation. Employing also the latter, both terms in Eq. (2.123) are associated with the same combination of emission functions which can then be written in terms of the (unnormalized) relative distance distribution, see also Eq. (2.28),









y y D(y, K), dx S x# , K S x! , K . 2 2

(2.125)

This function denotes the distribution of relative space-time distances y between the particles in pairs emitted with momentum K. A particularly simple expression due to Koonin is then obtained [8,36,94] in the pair rest frame, *"0"K:





P (p , p )+ dy"U q (y)" dy D(y, K) ,   ? @

(2.126)

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 (Uq $U q ) for identical particle pairs. As explained in Section 2.2.4, the last where U q "(  \    factor in Eq. (2.126) coincides up to normalization with the relative source function SK(r). With the help of the smoothness approximation the two-particle spectrum can thus be expressed by the relative source function weighted with the Born probability of the Coulomb relative wavefunction, as given before in Eq. (2.102). We "nally mention that "rst attempts have been made to include in the analysis of Coulomb "nal state e!ects the role of a central Coulomb charge [18,149] or e!ects due to high particle multiplicity [7]. It is an important open question to what extent these e!ects modify the analysis presented here. 2.6. Bose}Einstein weights for event generators Numerical event simulations of heavy-ion collisions provide one important method to simulate realistic phase-space distributions. Many such event generators exist nowadays. In principle, their output should be a set of observable momenta p with all momentum correlations (and hence the G complete space-time information) built in. However, none of the existing event generators propagates properly symmetrized N-particle amplitudes from some initial condition. As a consequence, the typical event generator output is a set of discrete phase-space points (r\ , t[ , p\ ) which one G G G associates with the freeze-out positions of the "nal state particles. This simulated event information (r\ , t[ , p\ ) lacks correlations due to Bose}Einstein symmetrization and other types of "nal state G G G e!ects. We "rst discuss di!erent schemes used to calculate a posteriori two-particle correlation functions for inputs of discrete sets of phase-space points (r\ , t[ , p\ ). We then turn to so-called shiftG G G ing prescriptions which aim at producing modi"ed "nal state momenta with correct particle correlations. 2.6.1. Calculating C(q, K) from event generator output The conceptual problem of determining particle correlations from event generators is wellknown [1,104,105]: Bose}Einstein correlations arise from squaring production amplitudes. They hence require a description of production processes in terms of amplitudes. Numerical event simulations, however, are formulated via probabilities. This implies that various quantum e!ects are treated only heuristically, if at all. Especially, event generators do not take into account the quantum mechanical symmetrization e!ects. In this sense, the event generator output is the result of an incomplete quantum dynamical evolution of the collision. The aim of Bose}Einstein weights is to remedy this artefact a posteriori by translating the phase-space information of (r\ , t[ , p\ ) into G G G realistic momentum correlations. For a set of N events of multiplicities N , this implies formally  K , NC(q, K) . (2.127) ++(r\ , t[ , p\ ), G G G GZ ,K KZ ,

The set +(r\ , t[ , p\ ), denotes the phase-space emission points of the N like-sign pions generG G G GZ ,K

K ated in the mth simulated event. The event generator simulates thus a classical phase-space distribution 1 , ,K o (p, r, t)" d(r!r\ )d(p!p\ )d(t!t[ ) .   G G G N  K G

(2.128)

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Prescriptions of the type (2.127) are not unique: a choice of interpretation is involved in calculating two-particle correlations from the event generator output. Here, we mention two di!erent interpretations of (r\ , t[ , p\ ), sometimes referred to as `classicala and `quantuma [61]. G G G `Classicala interpretation of the event generator output. In the `classicala interpretation , is interpreted as [61,178,190] the distribution of phase-space points ++(r\ , t[ , p\ ), G G G GZ ,K KZ ,

a discrete approximation of the on-shell Wigner phase-space density S(x, p), S(x, p)"o ( p, x, t) .  

(2.129)

The emission function is thus a sum over delta functions. For practical applications, it is convenient to replace the delta functions in momentum space by rectangular `bin functionsa [190] or by properly normalized Gaussians [61,178] of width eP0 (we denote both choices by the same symbol dC p\ p) G



1/e; p !e/24p 4p #e/2 H GH H dC p\ p" G 0; else,

( j"x, y, z)

1 dC exp(!(p\ !p)/e) . p\ p" G G (ne)

(2.130)

(2.131)

The one-particle spectrum and two-particle correlator then read [61,178,190]



1 , ,K dN " dx S(x, p)" dC E p\ p , G Ndp N  K G [" ,K dC e OR[ G\q  r\ G"! ,K (dC )] K G p\ GK G p\ GK . , [( ,K dC )( ,K dC )! ,K dC dC ] K G p\ Gp H p\ Hp G p\ Gp p\ Gp

C(q, K)"1#

,

(2.132)

(2.133)

The correlator (2.133) is the discretized version of the Fourier integrals in Eq. (2.24). It does not invoke the smoothness approximation, in contrast to the popular earlier algorithm developed by Pratt [129,133] (which includes "nal state interactions). The subtracted terms in the numerator and denominator remove the spurious contributions of pairs constructed from the same particles [174]. In general the result for the correlator at a "xed point (q, K) will depend on the bin width e. Finite event statistics puts a lower practical limit on e. Tests have shown that accurate results for the correlator require smaller values for e (and thus larger event statistics) for more inhomogeneous sources. In practice the convergence of the results must be tested numerically [61]. `Quantuma interpretation of the event generator output. In the `quantuma interpretation [61,174,176,178,190] the event generator output o (r\ , t[ , p\ ) is associated with the centers of   Gaussian wave packets (2.35). Neglecting multiparticle symmetrization e!ects, the corresponding Wigner function according to Eq. (2.46) reads



S(x, K)" dp\ dr\ dt[ o (r\ , t[ , p\ ) s (x!r\ , t!t[ , K!p\ ) . G G G   G G G  G G G

(2.134)

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The one- and two-particle spectra are [61,174] dN 1 , 1 , ,K E " l (p)" s (p) , (2.135) N dp N K G N  K  K G  , [" ,K s (K) e O R[ G\q  r\ G"! ,K s(K)] G G G G C(q, K)"1#e\Nq K . (2.136) , [l (p )l (p )! ,K s (p )s (p )] K K  K  G G  G  Again, the terms subtracted in the numerator and denominator are "nite multiplicity corrections which become negligible for large particle multiplicities [174]. Discussion. In both algorithms, the particle spectra are discrete functions of the input (r\ , t[ , p\ ) but G G G they are continuous in the observable momenta p , p and hence, no binning is necessary. Each of   the sums in Eqs. (2.133) and (2.135) requires only O(N ) manipulations. However, once "nal state K interactions are included, the number of numerical operations increases quadratically with N since the corresponding generalized weights [8] do no longer factorize. When also accounting K for multiparticle symmetrization e!ects, more than O(N ) numerical manipulations are typically K required [176]. The `classicala and `quantuma algorithms then di!er in two points: 1. There is no analogue for the Gaussian prefactor exp(!pq/2) of Eq. (2.136) in the `classicala algorithm. This is a genuine quantum e!ect stemming from the quantum mechanical localization properties of the wave packets. 2. For the choice p"1/e, the bin functions dC p\ p are the classical counterpart of the Gaussian G single-particle distributions s (p). Finite event statistics puts a lower practical limit on e, but in G the limit eP0 the physical momentum spectra are recovered. In contrast, in the `quantuma algorithm p denotes the "nite physical particle localization. In this case, the limit pPR (corresponding to eP0) is not physically relevant: it amounts to an emission function with in"nite spatial extension, yielding lim C(q, K)"1#dq [174].  N These algorithms have been shown to avoid certain inconsistencies arising from the use of the smoothness approximation for sources with strong position}momentum correlations [107,190]. Systematic studies indicate that violations of the smoothness approximation occur only for emission functions S(x, K) which are inconsistent with the uncertainty relation, i.e., which cannot be interpreted as Wigner densities. Pratt has shown that typical source sizes in heavy-ion collisions are su$ciently large that this problem can be neglected [133]. Extensions of these algorithms to include "nal state interactions [8] and multiparticle correlation e!ects [176] were proposed but have not yet been implemented numerically. 2.6.2. Shifting prescriptions In the previous subsection we reviewed algorithms which calculate two-particle correlation functions from a discrete set of phase-space points. The output of the algorithm is a correlator C(q, K) which denotes the probability of "nding particle pairs with the corresponding momenta; it is not a set of new discrete momenta p with the correct Bose}Einstein correlations included. The H latter is of interest e.g. for detector simulations which require on an event-by-event basis a simulated set of particle tracks to anticipate detector performance. Also, it could be used to investigate eventwise #uctuations which is not possible with an ensemble averaged correlator.

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The most direct way to achieve this goal would seem to use symmetrized amplitudes for the particle creation process. Such a scheme has been developed in the context of the Lund string model [9,10] for the hadronization of a single string. For more complicated situations there exist so far only algorithms which shift after particle creation the generated momenta p\ to their physically H observed values p H + r\ [ p\ ,  G RG G GZ ,

N p . (2.137) p\ H H While the function C(q, K) describes the two-particle correlations only for the ensemble average, the set +p , represents all measurable momenta of a simulated single event with realistic Bose}Einstein H correlations. Such a shifting prescription which employs the full phase-space information of the simulated event was developed by Zajc [184,185]. In Ref. [185] a self-consistent Monte Carlo algorithm is used for a simple Gaussian source to determine the shifts (2.137) by sampling the momentumdependent N-particle probability. Although technically feasible, this calculation of N-particle symmetrized weights involves an enormous numerical e!ort. Another class of algorithms is used in event generators for high energy particle physics. Unlike Eq. (2.137), they explicitly exploit only the momentum-space information of the simulated events. Additionally, an ad hoc weight function is employed which one may relate to the ensembleaveraged space-time structure of the source [104,105]: + p\ ,  G GZ ,

Q N Q#dQ .

(2.138)

This shifting procedure involves only particle pairs and is signi"cantly simpler to implement numerically. By decoupling the position and momentum information one looses, however, possible correlations between the particle momenta and their production points. Also, in an individual event this shifting prescription is insensitive to the actual separation of the particles in spacetime. A further problem is that the translation of dQ from Eq. (2.138) into a change of particle momenta is not unique. It changes the invariant mass of the particle pair and does not conserve simultaneously both energy and momentum. These de"ciencies are typically repaired by a subsequent rescaling of momenta; according to Ref. [105] the results show in practice little sensitivity to details of the implementation. A more sophisticated method [53}55] attempts to implement the full N-body symmetrization by a cluster algorithm. Again the weights used in this algorithm, at least in the present version, encode the space-time structure of the source only via a single ensemble-averaged radius parameter.

3. Gaussian parametrizations of the correlator In practice, the two-particle correlation function is usually parametrized by a Gaussian in the relative momentum components, see e.g. Eq. (1.5). In this chapter we discuss di!erent Gaussian parametrizations and establish the relation of the corresponding width parameters (HBT radii) with the space-time structure of the source.

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This relation is based on a Gaussian approximation to the true space-time dependence of the emission function [41,42,44,79,171] (3.1) S(x, K)"N(K)S(x (K), K) exp[!x I(K)B (K)x J(K)]#dS(x, K) . IJ  For the present discussion, we neglect the correction term dS(x, K). We discuss in Section 4 how it can be systematically included. The space-time coordinates x in Eq. (3.1) are de"ned relative to the I `e!ective source centrea x (K) for bosons emitted with momentum K [41,83,89,171] x I(K)"xI!x I(K),

x I(K)"1xI2(K) ,

(3.2)

where 122 denotes an average with the emission function S(x, K): dx f (x)S(x, K) . 1 f 2(K)" dx S(x, K)

(3.3)

The choice (B\) (K)"1x x 2(K) (3.4) IJ I J ensures that the Gaussian ansatz (3.1) has the same rms widths in space-time as the full emission function. Inserting Eq. (3.1) into the basic relation (1.4) one obtains the simple Gaussian form for the correlator C(q, K)"1#exp[!q q 1x Ix J2(K)] . (3.5) I J This involves the smoothness and on-shell approximations discussed in Section 2 which permit to write the space-time variances 1x x 2 as functions of K only. Note that the correlator depends only I J on the relative distances x I with respect to the source center. No information can be obtained about the absolute position x (K) of the source center in space-time. According to Eq. (3.5) the two-particle correlator provides access to the rms widths of the e!ective source of particles with momentum K. In general, these width parameters do not characterize the total extension of the collision region. They rather measure the size of the system through a "lter of wavelength K. In the language introduced by Sinyukov [153], this size is the `region of homogeneitya, the region from which particle pairs with momentum K are most likely emitted. Space-time variances coincide with total source extensions only in the special case that the emission function shows no position}momentum correlation and factorizes, S(x, K)"f (x)g(K). Relating (3.5) to experimental data requires "rst the elimination of one of the four relative momentum components via the mass-shell constraint (2.27). Depending on the choice of the three independent components, di!erent Gaussian parametrizations exist. In what follows, we focus on their interpretation in terms of the space-time characteristics of the source. 3.1. The Cartesian parametrization The Cartesian parametrization [25,41,123,125,206] is expressed in the out}side}longitudinal (osl) coordinate system, de"ned in Fig. 3.1. It is based on the three Cartesian spatial components

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Fig. 3.1. The osl coordinate system takes the longitudinal (long) direction along the beam axis. In the transverse plane, the `outa direction is chosen parallel to the transverse component of the pair momentum K , the remaining Cartesian , component denotes the `sidea direction.

q (out), q (side), ql (long) of the relative momentum q. The temporal component is eliminated via   the mass-shell constraint (2.27) q"b ) q,

b"K/K ,

(3.6)

b"(b , 0, bl) in the osl-system . (3.7) , This leads to a correlator of the form (1.5) with running over i, j"o, s, l. In general this GH correlator C(q, K) depends not only on K and Kl, but also on the azimuthal orientation U of the , transverse pair momentum "K ". This angle, however, does not appear explicitly in the osl-system , which is oriented for each particle pair di!erently by the angle U in the transverse plane. U has to be de"ned with respect to some pair-independent direction in the laboratory system, e.g. relative to the impact parameter b: U"L(K , b) . (3.8) , In the following we discuss both the azimuthally symmetric situation with impact parameter b"0, when all physical observables are U-independent, and the parametrization for "nite impact parameter collisions. 3.1.1. Azimuthally symmetric collisions For central collisions, b"0, the collision region is azimuthally symmetric, and the emission function and correlator are U-independent. In the osl-system, this U-invariance results in a re#ection symmetry with respect to the `sidea-direction [40]: S (x; K , U, Kl)"S (x; K , U#dU, Kl)
,
, 0S (x, y, z, t; K , Kl)"S (x,!y, z, t; K , Kl) . (3.9)  ,  , We follow common practice in dropping now the subscript osl. Whenever azimuthal symmetry is assumed, the emission function S(x, K) is speci"ed in the osl-system. Due to the yP!y re#ection symmetry of the emission function, y "1y2"0 and the three space-time variances 1x x 2(K) linear in y vanish. The symmetric tensor B (K) then has only seven I J IJ non-vanishing independent components. These combine to four non-vanishing HBT-radius

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parameters R (K) which characterize the Gaussian ansatz (1.5) for the correlator [41,89]: GH R(K)"1y 2(K) , 

(3.10)

R(K)"1(x !b tI )2(K) ,  ,

(3.11)

Rl (K)"1(z !bltI )2(K) ,

(3.12)

R (K)"1(x !b tI )(z !bltI )2(K) ,  ,

(3.13)

R (K)"0 , 

(3.14)

R (K)"0 . 

(3.15)

Obviously, the more symmetries are satis"ed by the emission function, the simpler are the expressions obtained for the HBT radius parameters. In this context we mention the case of a longitudinally boost-invariant source showing Bjorken scaling. Though not strictly satis"ed by the "nite sources created in heavy-ion collisions, this can provide a simple intuitive picture of the collision dynamics near mid-rapidity. Longitudinal boost-invariance implies a z P!z re#ection symmetry of the emission function. Thus, in addition to the space-time variances linear in y , now also those linear in z vanish, and one is left with only 5 non-vanishing independent components of B (K). In the longitudinally comoving system (LCMS), where bl"0, this leads to the further IJ simplications Rl (K)"1z 2(K) ,

(3.16)

R (K)"0 , for longitudinally boost-invariant sources in the LCMS. 

(3.17)

The general relation between the symmetries of the system, the number of its independent non-vanishing space-time variances, and the number of non-vanishing observable HBT parameters is summarized in the following table:

Symmetry

B (K) IJ

R (K) GH

None Azimuthal Azimuthal#long. boostinv. in the LCMS

10 indep. fcts. of K , , > , 7 indep. fcts. of K , > , 5 indep. fcts. of K ,

6 indep. fcts. of K , , > , 4 indep. fcts. of K , > , 3 indep. fcts. of K ,

In all cases, there are more independent space-time variances 1x x 2(K) than experimental I J observables. This arises from the mass-shell constraint (2.27) which leads to a mixing of spatial and temporal variances in the observable HBT parameters. One of the most important questions is therefore which other properties of the expanding system can be exploited to further disentangle spatial and temporal information about the emission function.

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For this, we note that independent of the particular emission function, no direction is distinguished in the transverse plane for K "0. The out- and side-components of all observables , coincide in this kinematical limit. For the space-time variances, this implies 1x 2" , "1y 2" , , )  ) 

(3.18)

1z x 2" , "1tI x 2" , "0 . )  ) 

(3.19)

As long as the limit K P0 of the emission function S(x, K) results in an azimuthally symmetric , expression (an exception is the class of opaque source models discussed in Section 5.1.2), the above relations between the space-time variances at K "0 imply that the HBT radius parameters satisfy , lim R(K)" lim R(K) ,   ), ),

(3.20)

lim R (K)"0 . (3.21)  ), In phenomenological HBT analyses one very often exploits these relations by distinguishing between an implicit K-dependence (due to the K-dependence of the space-time variances) and an explicit one (resulting from the mass-shell constraint q"q ) K/K). If the emission function features no position}momentum correlation, then all space-time variances are K-independent and Eqs. (3.18) and(3.19) hold independently of K. The di!erence between R(K) and R(K) at non-zero   K is then only due to the explicit K-dependence in Eqs. (3.10) and (3.11), i.e. the term b 1tI 2. This , implies that the explicit K-dependence dominates if the emission duration is su$ciently large [138] or if the position}momentum correlations in the source are su$ciently weak [26,127], R(K)!R(K)+b 1tI 2 .   ,

(3.22)

In this case, the di!erence between these two HBT radius parameters gives direct access to the average emission duration 1tI 2 of the source and allows to partially disentangle the spatial and temporal information contained in Eqs. (3.10)}(3.15). This gives rise to the following simple interpretation of the HBT radius parameters from the Cartesian parametrization: R measures the width of the emission region in the side direction, and  R measures the corresponding width in the out direction plus a contribution from the emission  duration which can be extracted according to Eq. (3.22) under the assumption of a weak K -dependence of the emission function. The longitudinal radius Rl "nally describes the longitudi, nal extension of the region of homogeneity in the LCMS where bl"0. No easy intuitive interpretation exists for the out-longitudinal radius parameter R . It is perhaps best understood in  terms of the linear correlation coe$cient [144] R (K)  o (K)"! ,  R (K)Rl(K) 

(3.23)

which can be positive or negative but is bounded by "o (K)"41 due to the Cauchy}Schwarz  inequality. This coe$cient was shown [144] to be of kinematical origin and useful for the

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interpretation of the longitudinal momentum distributions. We shall see in Section 3.2 that R plays a crucial role in the determination of the longitudinal velocity of the emitting source  volume element. 3.1.2. Collisions with xnite impact parameter If the azimuthal symmetry of the particle emitting source is broken, then the transverse one-particle spectrum depends on the azimuthal direction of the emitted particles. This can be quanti"ed in terms of the harmonic coe$cients v (p , y), [116,117,165] L  dN dN " dx S(x, p) E "E p dp dy d

dp   E dN  " 1#2 v (p , y) cos n( !t ) . (3.24) L  0 2n p dp dy   L The size and momentum dependence of the lowest of these harmonic coe$cients has been analyzed experimentally at both AGS and CERN SPS energies [15,16,118,124]. This allows to determine the orientation of the reaction plane for semiperipheral collisions event by event with an uncertainty of less than 303 [124,165]. Several attempts to extend this azimuthally sensitive analysis to twoparticle correlation functions exist [56,167,175,211]. The correlation measurements depend on the azimuthal direction U of the pair momentum, see Eq. (3.8), and hence allow to provide additional azimuthally sensitive information. The corresponding Gaussian radius parameters can be written formally in terms of space-time variances which are rotated via DU from the impact parameter "xed to the osl coordinate system [175]







R (K)"1[(DUx ) !(DUb) tI ][(DUx ) !(DUb) tI ]2 , G G H H GH (3.25) (DUb)"(b , 0, bl) . , We here di!er from the notation adopted in the rest of this review: the coordinates x, y and z are here given in the impact-parameter "xed system, not the osl one. As for the azimuthally symmetric case, the HBT radius parameters show implicit and explicit K-dependences. Their U-dependence thus has two di!erent origins [167,175,211]: R(K , U, >)"1x 2 sin U#1y 2 cos U!1x y 2 sin 2U ,  , R(K , U, >)"1x 2 cos U#1y 2 sin U#b 1tI 2  , , !2 b 1tI x 2 cos U!2b 1tI y 2 sin U#1x y 2 sin 2U , , , R (K , U, >)"1x y 2 cos 2U# sin 2U(1y 2!1x 2)  ,  (3.26) #b 1tI x 2 sin U!b 1tI y 2 cos U , , , Rl (K , U, >)"1(z !bltI )2 , , R (K , U, >)"1(z !bltI )(x cos U#y sin U!b tI )2 ,  , , R (K , U, >)"1(z !bltI )(y cos U!x sin U)2 .  , The explicit U-dependence denoted here is a purely geometrical consequence of rotating the x-axis from the direction of b to the direction of K . In addition, there is an implicit U-dependence of the ,

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space-time variances, 1x x 2"1x x 2(K , U, >). This U-dependence characterizes the dynamical I J I J , correlations between the size of the e!ective emission region (`region of homogeneitya) and the azimuthal direction in which particles are emitted. Both implicit and explicit U-dependences are mixed in the harmonic coe$cients

 

1 L R " R cos(mU) dU , GHK GH 2n \L

(3.27)

1 L R sin(mU) dU . R " GH GHK 2n \L

(3.28)

In models in which elliptic deformations dominate and higher than second order harmonic coe$cients can be neglected, these coe$cients satisfy the relations [175] a (K , >)+R +R +!R  ,   ,   

(3.29)

a (K , >)+R +!R +!R  .  ,   

(3.30)

The anisotropy parameter a vanishes in the absence of position}momentum correlations in the  source and thus characterizes dynamical anisotropies. On the other hand, a characterizes the  elliptical shape of the emission region. A violation of the relations (3.29)}(3.30) would rule out a large class of model scenarios considered to be consistent with the present knowledge about the space-time evolution of the collision process. Constraints of the type (3.29) and (3.30) on the harmonic coe$cients lead to a minimal azimuthally sensitive parametrization of the two-particle correlator [175]: C 0(q, K)+1#j(K)C (q, K)C (q, K, t )C (q, K, t ) ,   0  0 R

(3.31)

C (q, K)"exp[!R q !R q !Rl ql!2R q ql] ,         

(3.32)

C (q, K, t )"exp[!a (3q#q) cos(U!t )#2a q q sin(U!t )] ,  0    0    0

(3.33)

C (q, K, t )"exp[!a (q!q) cos 2(U!t )#2a q q sin 2(U!t )] .  0    0    0

(3.34)

In addition to the azimuthally symmetric part (3.32) which coincides with the Cartesian parametrization, this Gaussian ansatz involves only two additional, azimuthally sensitive "t parameters. The major di$culty in determining the harmonic coe$cients (3.27) and (3.28) from experiment is that they are expected to be statistically meaningful only for relatively large event samples while their extraction is conditional upon the eventwise reconstruction of the reaction plane. For su$ciently high event multiplicities, the probability distribution =(v , t ) of the experimentally  0 determined "rst harmonic coe$cient around the most likely, `truea values (v , tM ) is given by  0 a Gaussian of variance g [116,117,165]:






v #v!2v v cos t 1    0 . exp !  =(v , t )"  0 2g 2ng

(3.35)

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Fig. 3.2. The HBT anisotropy parameters 1a 2 as a function of the parameter mM which characterizes the event-by-event G reconstruction uncertainty in the orientation of the reaction plane. The parameters 1a 2, 1a 2 are determined by "tting   (3.31) to an event sample (3.36) of correlators whose reconstructed reaction planes #uctuate around the true impact parameter with the probability distribution (3.35). The value mM "2 corresponds to a reconstruction uncertainty of approximately 303.

An event sample with oriented reaction plane should thus be compared to a weighted average of the parametrization (3.31),



C (q, K)" v dv dt =(v , t )C 0(q, K) . RM 0   0  0 R

(3.36)

This e!ective correlator depends on the event averaged harmonic coe$cient v and the variance  g only via the ratio mM "v /g which is a direct measure of the accuracy for the reaction plane  orientation [116,117,165]. As can be seen from Fig. 3.2, for an oriented event sample with the typical 303 uncertainty in the eventwise determination of the reaction plane angle t (which P translates into mM +2 [165]), more than 80% (50%) of the anisotropy signals a (a ) survive in the   actual experimental measurement. Since the width g is given by event statistics, it is possible to reconstruct the true values a from the measured values 1a 2.   (q, K) for mM "0, i.e., for an azimuthally symmetric event sample of Furthermore calculating C RM 0 "nite impact parameter collisions, one can test to what extent the azimuthally symmetric HBT radius parameters extracted from a "t to such event samples will pick up contributions from non-zero a and a . This e!ect is, however, expected to be small [165,175].   3.2. The Yano}Koonin}Podgoretskiny parametrization The mass-shell constraint K ) q"0, explicitly given in Eq. (3.6), allows for di!erent choices of three independent relative momenta. The Yano}Koonin}Podgoretskimy (YKP) parametrization, which assumes an azimuthally symmetric collision region, uses the components q "(q#q, ,  

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q and ql and starts from the Gaussian ansatz [44,79,123,182] C(q, K)"1#j exp[!R (K)q !R(K)(ql !(q))!(R(K)#R(K)) (q ) ;(K))] . , ,   

(3.37)

Here, ;(K) is a (K-dependent) 4-velocity with only a longitudinal spatial component, ;(K)"c(K) (1, 0, 0, v(K)) ,

(3.38)

1 c(K)" . (1!v(K)

(3.39)

The combinations of relative momenta (ql !(q)), (q ) ;(K)) and q appearing in Eq. (3.37) are , scalars under longitudinal boosts, and the three YKP "t parameters R (K), R(K), and R(K) are ,   therefore longitudinally boost-invariant. In contrast to the Cartesian radius parameters, the values extracted for these YKP radius parameters do not depend on the longitudinal velocity of the measurement frame. This is advantageous in "tting experimental data. The fourth YKP parameter is the Yano}Koonin (YK) velocity v(K) which, as we will see, is closely related to the velocity of the e!ective particle emitter. The corresponding rapidity






1#v(K) 1 >7)(K)" ln 1!v(K) 2

(3.40)

transforms additively under longitudinal boosts. Since the ansatz (3.37) uses four Gaussian parameters, it is a complete parametrization for azimuthally symmetric collisions. These parameters can again be expressed in terms of the space-time variances 1x x 2 [79,180]: I J R (K)"R(K)"1y 2(K) , , 

(3.41)

R(K)"A!vC , 

(3.42)

R(K)"B!vC , 

(3.43)







A#B 2C  v(K)" 1! 1! , 2C A#B

(3.44)

where, with the notational shorthand mI ,x #iy ,






  


A"

mI  tI ! (K) , b ,

(3.45)

B"

bl  z ! mI (K) , b ,

(3.46)

C"

mI tI ! b ,

bl z ! mI b ,

(K) .

(3.47)

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In these expressions, 1y 2"1x y 2"0 since we are dealing with azimuthally symmetric sources. The kinematical limit K P0 is not free of subtleties, as one may guess by "nding b in the , , denominator of the above expressions. Indeed, for K "0 the mass-shell constraint (3.6) reads , q"blql, and the relative momenta q, ql and q on which the YKP ansatz (3.37) is based are no , longer independent. Hence, strictly speaking, the YKP parametrization exists only for K O0. In , practice this does not limit the applicability since the K P0-limit is well-de"ned for all YKP , parameters. Mathematically, the Cartesian and YKP parametrizations are equivalent and di!er only in the choice of the independent relative momentum components. The Cartesian radius parameters can therefore be expressed in terms of the YKP ones [79,180] via R"R ,  ,

(3.48)

R "R!R"b c(R#vR) ,     ,  

(3.49)

Rl "(1!bl )R#c(bl!v)(R#R) ,   

(3.50)

R "b (!blR#c(bl!v) (R#R)) .     ,

(3.51)

This set of equations provides a useful consistency check for correlation data analyzed independently with both the Cartesian and the YKP parametrizations. To invert them, one has to calculate 1 A" R , b   ,

(3.52)

2bl bl B"Rl ! R # R ,  b b   , ,

(3.53)

bl 1 C"! R # R . b  b   , ,

(3.54)

and insert them into Eqs. (3.41)}(3.44). These relations imply in particular that the YK velocity v(K) can be calculated from measured Cartesian HBT radii. In model studies [180], it was demonstrated that this velocity follows closely the velocity of the longitudinal saddle point system (LSPS) which is the longitudinally comoving Lorentz frame at the point of highest particle emissivity for a given pair momentum K. In this sense the YK velocity can be interpreted as the e!ective source velocity. Note that in the Cartesian parametrization the kinematical information associated with the YK velocity is contained in the cross-term R [123,144].  While the values extracted for R(K) and R(K) are independent of the longitudinal velocity of   the observer system, their space-time interpretation is not. For their analysis the so-called Yano}Koonin frame, which is pair momentum dependent and de"ned by v(K)"0, o!ers itself: in this frame, the terms &vC in Eqs. (3.42) and (3.43) vanish. For a class of Gaussian model emission functions including longitudinal and transverse #ow it was shown that the spatio-temporal interpretation of the "t parameters is then particularly simple [44,79]: R (K)"1y 2(K) , ,

(3.55)

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R(K)+1z 2(K) , 

(3.56)

R(K)+1tI 2(K) . 

(3.57)

In other words, the three YKP radius parameters give directly the transverse, longitudinal and temporal size of the e!ective source in the rest frame of the emitter. Especially the last equation (3.57) seems to imply that in the YKP parametrization the emission duration 1tI 2 can be accessed directly. This, however, is model-dependent: in Eqs. (3.56) and (3.57) terms were omitted on the right-hand side which can become large in certain model scenarios. For example, opaque source models with strongly surface-dominated emission give a leading geometric contribution to R  [86,160] 1 R+! (1x 2!1y 2) for opaque sources .  b ,

(3.58)

Large geometric corrections were also observed for transversely expanding sources with a boxshaped transverse density pro"le [163]. As will be discussed in Section 5.4.1, "rst checks indicate that opaque source models cannot reproduce the experimental data consistently [160,177], but they play an important role in understanding the range of validity of the approximations (3.56) and (3.57). It can happen [161,162], especially for sources with 1x !y 2(0, that the argument of the square root in Eq. (3.44) becomes negative. In this case the YKP parameters are not de"ned. For such situations a modixed YKP parametrization was suggested in [161,162] which does not have this potential problem. The corresponding modi"ed YK velocity still follows closely the #uid velocity at the point of highest emissivity, i.e. also the modi"ed YKP parametrization allows to determine the e!ective source velocity. However, the interpretation of the modi"ed parameters R , R  [162] is   less straightforward than Eqs. (3.56) and (3.57); in particular the parameter R  is in general not  dominated by the emission duration 1tI 2. So far, the YKP-parametrization has not been extended to collisions at "nite impact parameter. 3.3. Other Gaussian parametrizations A plethora of di!erent Gaussian parametrizations can be found in the literature. They belong to either of two di!erent classes. The "rst class contains parametrizations which are equivalent to the ones discussed above. A typical example for azimuthally symmetric sources is [112] C(q, K)"1#j(K) exp[!R(K) q!R(K) q!R(K) q!¹(K) (q)] . V V W W X X

(3.59)

It provides a perfectly valid azimuthally symmetric ansatz whose four "t parameters are, after insertion of the mass-shell constraint (3.6), seen to be in one-to-one correspondence with the Cartesian or YKP ones. One should keep in mind, however, that the suggestive notation ¹(K) does not warrant a physical interpretation in terms of a temporal extension; also the R(K) do not G only contain spatial information. The interpretation of these parameters has to be established again on the basis of space-time variances. Other equivalent parametrizations can be found in the literature; the relations between the various radius parameters are discussed in Refs. [162,163].

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The second class contains incomplete parametrizations: either certain terms (e.g. the outlongitudinal cross-term in the Cartesian parametrization) are neglected, or the ansatz is dimensionally reduced. The prime example is the q -parametrization (q "q!(q))   C(q, K)"1#j(K) exp[!R (K) q ] . (3.60)   Here all the di!erent spatial and temporal informations contained in the space-time variances 1x x 2 are mixed into one "t parameter R (K), and there is no possibility to unfold them again. I J  Furthermore, low-dimensional projections of a correlator which is well-described by a complete three-dimensional Gaussian parametrization in general deviate from a Gaussian shape. This is true in particular for projections on q ; in fact, it was repeatedly observed that C(q ) is better   described by an exponential or an inverse power of q . For a space-time interpretation of  correlation data such incomplete parametrizations are not suitable. It is often argued that limited statistics forces one in practice to adopt dimensional reductions in the "t parameter space. But even then it is preferable to bin the data in three independent q-components "rst and to project this three-dimensional histogram onto di!erent one-dimensional directions for "tting purposes. The parameters extracted this way can be compared to suitably averaged versions of the HBT radius parameters (3.10)}(3.15) or (3.41)}(3.44). 3.4. Estimating the phase-space density As shown by Bertsch [28,194] the correlation function can be used to extract the average phase-space density at freeze-out. In the present section we describe how this works. The phase-space density f (x, p, t) of free-streaming particles at time t is obtained by summing up the particles emitted by the source function up to this time along the corresponding trajectory:



(2n) R dt S(x!b(t!t), t; p) . (3.61) f (x, p, t)" E N \ Here b is the velocity of particles with momentum p. For large times t, f is normalized to the total event multiplicity, dx dp f (x, p, t)/(2n)"N. According to Liouville's theorem, the spatial average of any power of f is time-independent after particle production has ceased (t't ). This is in  particular true for the average phase-space density dx f (x, p, t't )  . (3.62) 1 f 2(p)" dx f (x, p, t't )  This quantity can be obtained from the measured one- and two-particle spectra. To this end one calculates, see Eq. (2.24),



P (p ) P (p )(C(p , p )!1)"P (p , p )!P (p ) P (p )" dx S(x, K) e OV             





(3.63)

and integrates over q with the mass shell constraint q ) K"0. After substituting xPx#bx (where b"K/K) one obtains



dq d(q ) K) [P (p ) P (p )(C(p , p )!1)]      

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"





dq dx dy e\ q x\y dx S(x#bx, x, K) dy S(y#by, y, K) K



(2n) + dx R(x, K) . E )

(3.64)

(3.65)

In the last step we used the on-shell approximation K+E and introduced the time-integrated ) emission function



R(x, K)"



dx S(x#bx, x, K) . \

(3.66)

It is easy to show that





(2n)L dxRL(x, K)" dx f L(x, K, t't ) .  EL )

(3.67)

Combining this with Eqs. (3.62) and (3.65) and using the smoothness approximation P (p ) P (p )+(P (K)) on the l.h.s. of Eq. (3.64), the phase-space density f can be expressed in      terms of observable quantities:



1 f 2(K)+P (K) dqd(q ) K) (C(q, K)!1) . 

(3.68)

Using the Cartesian parametrization of the correlator for zero impact parameter collisions as given in Section 3.1, the r.h.s. takes the explicit form [17,28,194] dN 1 1 f 2(K ,>)" , , d> M dM dU < (K ,>) , ,  ,

(3.69)

M cosh> R (K)(R(K) Rl (K)!(R (K)) . < (K ,>)" ,     , n

(3.70)

This expression assumes an intercept j"1 for the correlator. In reality a considerable fraction of the observed pions stems from resonance decays after freeze-out. The longlived resonances a!ect the intercept parameter j (see Section 5.3.6), and the corresponding decay pions should not be counted in the average pion phase-space density near freeze-out. This can be taken into account by substituting in Eq. (3.69) 1 f 2(K)P(j(K)1 f 2(K) ,






(3.71)

 j(K)" 1! f (K) , (3.72) P P where the sum in Eq. (3.72) runs over the resonance fractions f (K) of longlived resonances P contributing to the one-particle spectrum at K.

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Fig. 3.3. Rapidity dependence of the average pion phase-space density (3.71). The data points are for Au#Au collisions at 10.8 GeV/A measured by E877 at the AGS. The statistical errors are smaller than the symbols, the systematic normalization uncertainty is of the order of 30%. The upper (lower) lines were obtained assuming a thermal distribution (3.73) with a temperature extracted from the high (low) momentum part of the n\-spectrum. (Figure taken from [112].)

A "rst application of this approach was performed by the E877 experiment for Au#Au collisions at the AGS [17,112]. The extracted average pion phase-space density in the forward pair rapidity region 2.7(>(3.3 is shown in Fig. 3.3 as a function of the pair rapidity, averaged over K . As very forward rapidities are approached, it decreases from +0.2 to +0.1. This suggests that , the phase-space density is largest in the center of the collision at mid rapidity (here: >"1.57) and decreases towards forward and backward rapidities. More recently a large set of data, including p}p collisions at 250 GeV, S-nucleus collisions at 200 A GeV, Pb # Pb collisions at 158 A GeV and Au#Au collisions at 10.8 A GeV, was compiled in Ref. [52]. The resulting average phase-space densities are shown in Fig. 3.4. They indicate a strong dependence of the spatially averaged phase-space density on the transverse momentum K but very weak dependence on the size of the collision system and on the pion rapidity density , dN/dy. This latter aspect can be interpreted as evidence for a universal pion freeze-out phase-space density in heavy-ion collisions [52]. As a simple test of the thermalization assumption, one has compared [52,112] the measured phase-space densities in Figs. 3.3 and 3.4 to that of a thermal Bose}Einstein equilibrium distribution 1 f #(K ,>)" , exp[M cosh(>!y )/¹]!1 , Q

(3.73)

at pair rapidity > for a source rapidity y , extracting the temperature from the measured Q single-particle spectra and two-particle correlations. Dynamical #ow e!ects alter quantitative aspects of f #(K), but qualitatively the phase-space density thus calculated compares surprisingly well with the experimentally determined one, see Figs. 3.3 and 3.4.

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Fig. 3.4. Spatially averaged phase-space density in narrow rapidity windows as a function of the transverse momentum K . The heavy-ion data span more than an order of magnitude in rapidity density dN/dy, but the resulting freeze-out , densities show much less variation. The K -dependence of 1 f 2 can be well parametrized by an exponential function. The , dashed lines show Eq. (3.73) for ¹"80, 120, and 180 MeV, respectively. (Figure taken from [52].)

4. Beyond the Gaussian parametrization The space-time variances characterizing the Cartesian HBT radius parameters (3.10)}(3.15) can be written as second derivatives of the correlator (3.5) at q"0:



RC(q, K) . 1(x !b tI )(x !b tI )2"! G G H H Rq Rq q G H 

(4.1)

These curvature terms coincide with the experimentally determined half widths of C(q, K) only if the correlator is a Gaussian in q. Realistic two-particle correlation functions show, however, more or less signi"cant deviations from a Gaussian shape. The consequences are two-fold: the corresponding space-time variances do not agree exactly with the "tted radius parameters, and the Gaussian radius parameters do not contain all the information contained in C(q, K). Nevertheless, qualitatively all statements made above about mixing of spatial and temporal information in the HBT radius parameters remain valid. The reason is that the Fourier exponent q ) x in Eq. (2.24) can be written as q ) (bt!x). Hence, the q dependence of the correlator tests always the same combinations (b t!x ) of spatial and temporal aspects of the emission function S(x, K), irrespective of the shape G G of the two-particle correlator. If the correlator deviates from a Gaussian shape, one can either seek a more detailed characterization of C(q, K) supplementing the Gaussian radius parameters by a larger set of characteristic

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parameters, or one may proceed to reconstruct from the measured correlations directly information about the emission function without invoking a particular parametrization. In the following we discuss both of these strategies. 4.1. Imaging methods Instead of determining information about the emission function S(x, K), an alternative analysis strategy [37] aims at determining directly from the measured true correlator C(q, K)!1 the relative source function SK(r) introduced in Section 2.2.4:



C(q, K)!1" dr K(q, r)SK(r) .

(4.2)

For the kernel, one usually chooses K(q, r)"" q (r)"!1 where q describes the propagation of   a pair, which is created with a center of mass separation r and detected with relative momentum q. q can include two-particle "nal state interactions, cf. Eqs. (2.102) and (2.126); for free particle  propagation, K(q, r)"cos(q ) r). Eq. (4.2) can be inverted uniquely, allowing in principle for an unambiguous reconstruction of the relative source function SK(r). In practice, "nite measurement statistics on the correlation function and the strongly oscillating nature of the kernel K(q, r) render the inversion problem non-trivial. Brown and Danielewicz [37] suggest to parametrize SK(r) in terms of a "nite number of basis functions g (r), H , (4.3) SK(r)" S (K) g (r) , H H H and to determine the coe$cients S (K) from the data. In applications [36,37] some insight has been H gained on how the reconstruction can be optimized by choosing suitable sets of functions g . H 4.2. q-moments Rather than obtaining the HBT radius parameters from a Gaussian "t to the measured correlation function, a quantitative analysis of C(q, K) can be based on expectation values [g(q)\ of the true correlator C(q, K)!1 in relative momentum space [172,173]: dq q q [C(q, K)!1] G H "(R\(K)) , [q q \"  GH G H dq [C(q, K)!1]






R R R    R (K)" R R R , GH    R R Rl  

(4.4)

i, j"o, s, l ,

(4.5)

j(K)"(det R(K)/n dq [C(q, K)!1] .

(4.6)



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Similar expressions exist for the YKP parameters [172]. For a Gaussian correlator, the intercept parameter j and the HBT radius parameters R obtained via these q-moments coincide with the GH values extracted from a Gaussian "t. For non-Gaussian correlators the HBT radius parameters and the intercept can be dexned via (4.4)}(4.6). Deviations of the correlator from a Gaussian shape are then quanti"ed by higher order q-moments, which in turn can be obtained as derivatives of the generating function Z(y, K),



Z(y, K)" dq e q  y [C(q, K)!1] , (!iR)L ln Z(y, K)"y . [q 2q L\"  G G Ry 2Ry L G G It is easy to see that Z(y, K) coincides with the relative source function (2.31) Z(y, K)"SK(y) .

(4.7) (4.8)

(4.9)

So far, only the uni-directional version of these expressions has been used in theoretical and experimental investigations [98]: Restricting the correlator along one of the three Cartesian axes, CI (K, q ),C(K, q , q "0), the corresponding HBT radius parameter and intercept are de"ned via G G H$G the relations (we use the same notation as for the three-dimensional q-moments) 1 , i"o, s, l , R(K)" G 2[q\ G dq q[CI (K, q )!1] G , [q\ (K)" G G G dq [CI (K, q )!1] G G



j (K)"(R (K)/(n) dq [CI (K, q )!1] . G G G G

(4.10) (4.11) (4.12)

In the case of deviations of C(q, K) from a Gaussian form, the intercepts j from uni-directional "ts G can di!er in the di!erent directions. A quantitative measure for the leading deviation of C(q, K) from a Gaussian shape is provided by the properly normalized fourth order moments, the kurtoses [172,173] [q\(K) G !1 . (4.13) D (K)" G 3[q\(K) G The uni-directional kurtosis D (K) vanishes if the correlator at pair momentum K is of Gaussian G shape in the q -direction. The applications of q-moments to experimental data is limited by G statistics: for higher order q-moments the main contribution to the moment integral comes from larger relative momenta, while most experimental information is concentrated at small q. 4.3. Three-particle correlations The complete set of q-moments (4.8) provides a full characterization of the shape of the two-particle correlator. Alternatively, a complete characterization of this shape could be obtained

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in a Taylor expansion of C(q, K) around q"0. Extending Eq. (4.1) to arbitrary orders, the corresponding Taylor coe$cients read



iLRLC(q, K) . (4.14) 1(x !b tI ) (x !b tI )2(x L!b LtI )2" G G G G G G Rq  Rq 2Rq L q G  G G However, due to "nite momentum resolution, no detailed information about the curvature of C(q, K) at q"0 is available from the data, and Eq. (4.14) cannot be applied in practice. It illustrates, however, clearly that the two-particle correlator C(q, K) contains no information about odd space-time variances: due to the re#ection symmetry C(q, K)C(K,!q), all odd space-time variances in Eq. (4.14) vanish. The true two-particle correlator R ,C!1 depends only on the modulus o of the Fourier  GH transformed emission function but not on its phase : GH o (4.15) R (i, j)"C(p , p )!1" GH ,  G H o o GG HH



o eG(GH" dx S(x, (p #p )) e NG\NHV . GH  G H

(4.16)

Only the phase contains information about odd space-time variances [80]: GH

"q 1x2 !1(q ) x )2 #O(q ) . (4.17) GH GH GH  GH GH GH GH Here K , q denote the average and relative pair momentum for two particles with on-shell GH GH momenta p and p , respectively, and the space-time variances 1 .. 2 are calculated with respect to G H GH the emission function S(x, K ). Three-particle correlations give access to this phase as can be seen GH from [80] R (p , p , p )     r (p , p , p )" "2 cos( # # ) , (4.18)        (R (1, 2) R (2, 3) R (3, 1)    R (p , p , p )"C (p , p , p )!R (1, 2)!R (2, 3)!R (3, 1)!1 . (4.19)            Here R (r ) is the true (normalized) three-particle correlation function. To investigate the space  time information contained in the phase combination on the r.h.s. of Eq. (4.18) in more detail one can expand the emission function S(x, K ) in Eq. (4.16) around the average momentum of the GH particle triplet: KM "(p #p #p )"(K #K #K ) ,         K "KM #(q #q ), iOjOk . HI GH  GH Using q #q #q "0 one "nds [80,86]    R1x 2 R1x 2 1 I ! J  U" # # " qI qJ    2   RKM J RKM I



(4.20) (4.21)



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R1x 2 1 R1x 2 R1x 2 I # J # H  ! [qI qJ qH #qI qJ qH ]       RKM JRKM H RKM HRKM I RKM IRKM J 24 1 ! qI qJ (q #q )H 1x x x 2 #O(q) .  I J H  2   

 (4.22)

Here the averages 1 2 2 have been calculated with the emission function S(x, KM ) taken at the  momentum KM of the particle triplet. The measurable phase U depends on the odd space-time variances 1x 2, etc., and on derivatives of the point of highest emissivity 1x2 with respect to KM . These re#ect the asymmetries of the source  around its center. In the Gaussian approximation (3.1) they vanish. These considerations show that the true three-particle correlator contains additional information which is not accessible via two-particle correlations. In practice, however, it is very di$cult to extract this information. The leading contribution to U is of second order in the relative momenta q , and in many reasonable models it even vanishes GH [86]. Therefore new information typically enters r (p , p , p ) at sixth order in q. The measurement     of the phase U is thus very sensitive to an accurate removal of all leading q-dependences by a proper determination and normalization of the two-particle correlator. These general arguments are supported by model studies which found that #ow, resonance decay contributions or source asymmetries leave generically small e!ects on the phase [86]. On the other hand, it was pointed out that the intercept of the normalized true three-particle correlator r in Eq. (4.18) may provide a good test for the chaoticity of the source. Writing the  emission function for a partially coherent source as S"S #S , the intercept j of r is given in     terms of the chaotic fraction e(p) of the single-particle spectrum [30,80]: 3!2e(KM ) , j (KM ),r (KM , KM , KM )"2(e(KM )   (2!e(KM ))

(4.23)

dx S (x, p)  e(p)" . dx S(x, p)

(4.24)

In contrast to the intercept j(K) of the two-particle correlator, the intercept (4.23) of the normalized three-particle correlator is not a!ected by decay contributions from long-lived resonances which cancel in the ratio (4.18) [80]. Complete small-q expansions of R and R which generalize the Gaussian parametrization (3.5)   to the case of partially coherent sources and to three-particle correlations, improving on earlier results [30,121], can be found in [80]. In the framework of a multidimensional simultaneous analysis of two- and three-pion correlations they permit to separately determine the sizes of the homogeneity regions of the chaotic and coherent source components as well as the distance between their centers. 5. Results of model studies A completely model-independent reconstruction of the emission function S(x, K) from measured correlation data is not possible since, due to the mass-shell constraint (2.27), only certain combinations of spatial and temporal source characteristics are measurable. Only the time-integrated

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relative distance distribution in the pair rest frame SK(r) can be determined uniquely from momentum correlations. This corresponds to a whole class of emission functions S(x, K), not a unique one. In practice, the mass-shell constraint is not the only problem for the reconstruction of the emission function. The statistical uncertainties of experimental data turn even the reconstruction of the relative source function SK(r) into a complicated task [37], where additional model assumptions are employed to obtain a convergent numerical procedure. Due to these fundamental and pragmatic problems, the analysis of experimental correlation data starts from a model of the emission function S(x, K) from which one- and two-particle spectra are calculated and compared to the data. Here one can follow two alternative approaches: either, one simulates directly the kinetic evolution of the reaction zone up to the freeze-out stage, calculates the one- and two-particle momentum spectra and compares them to the data. This is the approach followed in event generator calculations. The space-time information is then extracted from within the model simulation. Alternatively, one applies the tools presented in the preceding chapters by using simple parametrizations of the emission function and adjusting the model parameters by a comparison to data. This approach makes explicit use of the relation between the measured HBT parameters and the space-time variances of the emission function, trying to obtain a direct space-time interpretation of the measurements. In this case the dynamical consistency of the extracted space-time structure of the source with the preceding kinetic evolution of the reaction zone must be established a posteriori. We will here concentrate on the second approach; reviewing a large body of model studies we illustrate the extent to which momentum correlations have a generic space-time interpretation. The insights gained in these models studies are summarized at the end of our review into an analysis strategy which allows for a simple determination of the relevant space-time aspects of the source from experimental data. This strategy is then applied to recent one- and two-particle spectra for negatively charged particles measured by the NA49 collaboration in Pb#Pb collisions at the CERN SPS. 5.1. A class of model emission functions We introduce a class of analytical models for the emission function of a relativistic nuclear collision, starting from a simple model and then discussing several dynamical and geometrical re"nements. Models of this class have been used extensively in the literature [2,42,44,45,47,48,153, 171,172,175,193]. 5.1.1. The basic model Whatever the true particle phase-space distribution of the collision at freeze-out is, we expect that its main characteristics can be quanti"ed by its widths in the spatial and temporal directions, a collective dynamical component (parametrized by a collective #ow "eld) which determines the strength of the position}momentum correlations in the source, and a second, random dynamical component in momentum space (parametrized by a temperature). A parametrization which is su$ciently #exible to incorporate these features but still allows for an intuitive physical interpretation of its model parameters assumes local thermalization prior to freeze-out at temperature ¹ and incorporates collective expansion in the longitudinal and

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transverse directions via a hydrodynamic #ow "eld u (x). The source has a "nite geometrical size in I the spatial and temporal directions, encoded in transverse and longitudinal Gaussian widths R and *g as well as in a "nite particle emission duration *q. Here q"(t!z denotes the longitudinal proper time and g"ln[(t#z)/(t!z)] the space-time rapidity. The parametrization is optimized  for sources with strong, approximately boost-invariant longitudinal expansion for which freeze-out occurs close to a hypersurface of constant longitudinal proper time q"q . It is thus more suitable  for high than for low energy collisions. The source is de"ned by an emission function for each particle species r [2,42,48,153,171,172,193]:





p ) u(x)!k 2J #1 P m cosh(y!g) exp ! S (x, p)" P P ¹ (2n)n *q ,





r g (q!q )  ;exp ! ! ! . 2R 2(*g) 2(*q)

(5.1)

For explicit calculations, it is helpful to express the particle four-momentum p using the moI mentum rapidity y and the transverse mass m "(p #m. This allows for a simple expression of , , the Boltzmann factor in Eq. (5.1), p "(m cosh y, p , 0, m sinh y) , I , , , p x p ) u(x) m " , cosh(y!g)cosh g ! , sinh g . R R ¹ ¹ r ¹

(5.2) (5.3)

For sharp freeze-out of the particles from the thermalized #uid along a hypersurface R(x)"(q cosh g, x, y, q sinh g), we would have to choose the emission function proportional to   p ) n(x), where



n (x)" dp (x) d(x!x) , I I R

(5.4)

p ) n(x)"m cosh(y!g) d(q!q ) . (5.5) ,  The four-vector n (x) points normal to the freeze-out hypersurface. The term m cosh(y!g) in the I , emission function (5.1) stems from this geometrical condition, while we have replaced the dfunction in Eq. (5.5) by a properly normalized Gaussian to allow for a "nite emission duration *q. The factor 2J #1 accounts for the spin degeneracy of the emitted particle, and a chemical P potential k allows for separate normalization of all particle yields. The ansatz implies that all P particles are assumed to freeze out with the same geometric characteristics and the same collective #ow, superimposed by random thermal motion with the same temperature. For the #ow pro"le we assume Bjorken scaling [31] in the longitudinal direction, vl"z/t; this identi"es the #ow rapidity g " log>Tl with the space-time rapidity. Assuming a linear    \Tl transverse #ow rapidity pro"le of strength g in the transverse direction, the normalized #ow "eld  u (x) reads I u (x)"(cosh g cosh g , V sinh g , W sinh g , sinh g cosh g ) ,  P   I  P r g (r)"g . (5.6)  R

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In spite of the longitudinal boost-invariance of the #ow, the source as a whole is not boostinvariant unless the longitudinal Gaussian width *gPR. The model emission function (5.1) is thus completely speci"ed by six common and one speciesdependent model parameters: ¹, g , R, *g, *q, q , k . (5.7)   P The number of model parameters can be reduced by assuming chemical equilibrium at freeze-out. This provides the following constraint between the chemical potentials: k "b k #s k . (5.8) P P P 1 Here b and s are the baryon number and strangeness of resonance r, and k , k are the two P P 1 independent chemical potentials required for baryon number and strangeness conservation in the reaction zone. 5.1.2. Model extensions Comparative model studies investigate to what extent geometrical or dynamical assumptions put into the emission function S(x, K) leave traces in the observed one- and two-particle spectra, and how this allows to distinguish between di!erent collision scenarios. To this end one compares, for example, the model (5.1) with modi"ed model assumptions about the particle production and emission processes in the collision region. Here we focus on a few possible extensions of the basic model (5.1) which all have a clear physical motivation and which have been investigated in the literature. Opaque sources. The particle production and freeze-out mechanisms in heavy ion collision are largely unknown. In particular, it is not settled whether hadronic freeze-out resembles more the surface evaporation of a hot water droplet or the simultaneous bulk freeze-out leading to the decoupling of photons and the transition from an opaque to a transparent universe in the early stages of Big Bang cosmology. The parametrization (5.1) re#ects the second type of scenario, with a bulk transition from opaqueness to transparency at proper time q $*q. Surface emission can be  included into (5.1) by multiplying this emission function with an exponential absorption factor [84,161,203] (5.9) S (x, p)"S(x, p)exp[!(8/p(l /j )] ,      VY W l "l (r, )"e\0 e\0 dx , (5.10)   V where y"r sin , x"r cos . This extra factor suppresses exponentially emission from the interior of the emission region. The particle propagates in the x (out) direction. The Gaussians in the expression (5.10) parametrize the matter density seen by the particle according to the geometrical source distribution in Eq. (5.1). Temperature gradients. The model emission function S(x, p) presented in Eq. (5.1) assumes that all volume elements freeze out at the same temperature ¹(x)"¹. Since freeze-out is controlled by a competition between the local expansion and scattering rates, and the latter have a very strong temperature dependence, this is not an unreasonable assumption [108,146,208,209]. With this assumption all position}momentum correlations in the source in Eq. (5.1) stem from the collective



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dynamics characterized by the #ow u (x) in the Boltzmann factor. To contrast this scenario with I models in which the x}K-correlations have a di!erent origin, model extensions with a particular temperature pro"le have been studied [48,160]:









1 r (q!q ) 1  " 1#a 1#d . (5.11) 2R 2q ¹(x) ¹   Hereby transverse and temporal temperature gradients are introduced via two additional "t parameters a and d. The model (5.11) implies that the production of particles with larger m is more , strongly concentrated near the symmetry axis and average freeze-out time. Emission functions for non-central collisions. The emission function (5.1) shows a y !y symmetry in the osl-coordinate system. It describes an azimuthally symmetric emission region which is adequate for zero impact parameter collisions. To investigate the new qualitative features introduced by collisions at "nite impact parameter, one can study azimuthally asymmetric extensions [87,88,175] of the model emission function (5.1). Here we take x and y to be de"ned in the laboratory system, with x in the reaction plane. An elliptic geometric deformation of the source in the transverse plane, characterized by an anisotropy parameter e , is obtained by replacing in  Eq. (5.1)









r x y Pexp ! ! , exp ! 2R 2o 2o V W o "R(1!e, o "R(1#e . V W  An elliptic deformation of the transverse #ow pattern can be introduced by u (x)"(c cosh g, u , u , c sinh g) , I , V W , x y u "g (1#e , u "g (1!e . V  R W  R

(5.12) (5.13)

(5.14) (5.15)

c "(1#u#u . (5.16) , V W These modi"cations implement in a simple way some aspects of "nite impact parameter collisions in the mid-rapidity region. They do not encode for the fact that in the fragmentation region the particle emission is peaked away from the beam axis. Hence, the total angular momentum ¸ of the system,

 

dp dx x p S(x, p) , ¸ "e [x p \"e H I G GHI H I GHI E

(5.17)

vanishes for the prescriptions (5.12)}(5.16) given above. A simple parametrization of the emission function in the fragmentation region which leads at least to a "nite expression (5.17), reads [175] uQ"g (1#e V  

x#sy . R

o "(R#sy cos u)(1!e , V  o "(R#sy cos u)(1#e . W 

(5.18)

(5.19)

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These dynamical (5.18) and geometrical (5.19) assumptions e!ectively shift the center of the particle emission in the transverse plane as a function of the longitudinal particle rapidity y with some asymmetry strength s. They break explicitly the 1803 rotation symmetry of the emission function S in the transverse plane, which is left unbroken by (5.12)}(5.16).
5.1.3. Resonance decay contributions A signi"cant fraction of the most abundant candidates for interferometric studies, charged pions, are produced by the decay of unstable resonances after freeze-out. To a lesser extent the same problem exists also for kaons. Longlived resonances can escape to quite some distance from the original freeze-out region before decaying. They then lead to HBT radius parameters which are larger than the width of the particle production region. Furthermore, due to the resonance decay phase-space, secondary points populate mainly the low momentum region and can thus introduce an additional pair momentum dependence of the two-pion correlator. To obtain realistic estimates for the geometry and dynamics of the particle emitting source, resonance decay contributions therefore must be analyzed quantitatively. In the following discussion we focus on charged pions. We include all relevant resonance decay channels r in the model emission function by writing [33,64,83,142,172,195] S (x, p)"S (x, p)# S (x, p) . (5.20) L L PL P$L The emission functions S (x, p) for the decay pions are calculated from the direct emission PL functions S (X, P) for the resonances by taking into account the correct decay kinematics for twoP and three-body decays. Capital letters denote variables associated with the parent resonance, while lowercase letters denote pion variables. In particular, M and U here denote the transverse mass , and azimuthal direction of the parent resonance, in contrast to the rest of the review where M and , U are associated with the pair momentum K . , We follow the treatment in [33,72,157,172,195]. The resonance r is emitted with momentum P at space-time point XI and decays after a proper time q at xI"XI#(PI/M)q into a pion of momentum p and (n!1) other decay products: rPn#c #c #2#c .   L

(5.21)

The decay rate at proper time q is Ce\CO where C is the total decay width of r. Assuming unpolarized resonances with isotropic decay in their rest frame, S (x, p) is given in terms of the PL direct emission function S (X, P) for the resonance r by P

 

S (x; p)"M PL

 




Q> dP d(p ) P!EHM) dq Ce\CO ds g(s) E Q\ .

P ; dX d x! X# q M



S (X, P) . P

(5.22)

Variables with a star denote their values in the resonance rest frame, all other variables are given in the "xed measurement frame. s"( L p ) is the squared invariant mass of the (n!1) unobserved G G

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decay products in Eq. (5.21); it can vary between s "( L m ) and s "(M!m). g(s) is the \ G G > decay phase-space for the (n!1) unobserved particles. For two-particle decays it reads b g(s)" d(s!m) ,  4npH

(5.23)

whereas the three-particle decay phase-space is given by Mb ([s!(m #m )][s!(m !m )]     g(s)" , (5.24) Q(M, m, m , m ) 2ns   > Q ds ((M#m)!s(s !s(s !s((m !m )!s . (5.25) Q(M, m, m , m )" > \     s Q\ Eq. (5.22) can be simpli"ed considerably: for p O0, the energy}momentum conserving d-function , constrains the angle U of the resonance momentum P "(M cosh >, P cos U, P sin U, M sinh >) I , , , , to one of two orientations if its decay product propagates in the out-direction:



d(U!U ) ! , d(p ) P!EHM)" p P sin U ! ! , , m M cosh (>!y)!EHM . cos UI " , , ! p P , , This allows to do the U-integration in Eq. (5.22), leading to






(5.26) (5.27)



 P! S (x, p)" dq Ce\COS  x! q, P! . (5.28) PL P M ! R  sums over the two azimuthal directions in Eq. (5.27), and R indicates the remaining integrations ! over the resonance momenta; for details see Ref. [172]. 5.2. One-particle spectra The one-particle momentum spectrum, determined as the space-time integral (1.3) of the emission function S(x, p), is sensitive to the momentum distribution in S(x, p) and thus allows to constrain essential parts of the collision dynamics. It contains, however, no information about the space-time structure of the source. Statistical errors on one-particle data are signi"cantly smaller than those on the two-particle spectra. In practice, exploiting the temperature, #ow and resonance mass dependence of the one-particle spectrum therefore allows to reduce the model parameter space signi"cantly even before comparing model predictions to the measured twoparticle correlations. 5.2.1. Transverse one-particle spectrum We begin by discussing the rapidity-integrated transverse momentum spectrum of the models discussed in the last subsection. Both direct `thermala pions produced in the collision region and

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those stemming from resonance decays contribute to the spectrum, see Eq. (5.28):



dN  dN dN  L" L # 2M P . P R dM dm dm , , , P$L For the model (5.1) this expression takes a compact form [145,157,172]:


 

 m dN  2J #1 P " P (2nR ) 2q *g)eIP2M e\K d  , 2 4n dM  , M P , cosh g (m) I ,sinh g (m) . K  ¹ R  ¹ R







(5.29)

(5.30)

m"r/R is the rescaled transverse radius. Obviously the geometric parameters R, *g, q of the  source enter only in the normalization of the spectrum. The product of the spatial and temporal extensions of the thermal source determines the total particle yield, but not its momentum dependence. To further constrain these parameters is not possible without using two-particle correlations. According to Eqs. (5.29) and (5.30), the m -dependence of the pion spectrum is fully , determined by the temperature ¹ (or ¹(m) if ¹ is r-dependent), the rest masses M and chemical P potentials k of the resonances, and the transverse #ow pro"le g (m)"g mL. P   To study the dependence of one- and two-particle spectra on the composition of the resonance gas, the resonance fractions f can be computed [33,172,195]: P dx S (x, p) dNP /dp PL L f (p)" " , f (p)"1 . (5.31) P P dx S (x, p) dN/dp P PL L P As we discuss in Section 5.3.6, these fractions play an important role in estimating the correlation strength j(K). In Fig. 5.1, we plot the pion transverse mass spectrum dN /dm and the resonance L , fractions f (y, p ) of the model (5.1) for two sets of source parameters. All resonance decay P , contributions are shown separately. The resonances u, g and g contribute with 3-body decays whose decay pions are seen to be particularly concentrated at small p . Comparing the cases of , vanishing and non-vanishing transverse #ow one observes the well-known #attening of the transverse mass spectrum by transverse radial #ow [102,115,145,146,152,205,208,209]. The direct pions re#ect essentially an e!ective `blueshifteda temperature [102]



1#1b 2  , (5.32) ¹ "¹  1!1b 2  where the average transverse #ow velocity 1b 2 is directly related to g . This clearly does not allow   to separate thermal from collective motion. Deviations from (5.32) are seen for the transverse mass spectra of heavier particles at low p and have been used to determine the temperature and #ow , velocity separately [136,158]. One of the main goals of two-particle interferometry is to obtain a more direct measure of the transverse expansion velocity at freeze-out. 5.2.2. Rapidity distribution The rapidity distribution is given by





dN L" m dm d d xS(x; p , , y) . , , , dy

(5.33)

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Fig. 5.1. Left: The single-pion transverse mass spectrum for the model (5.1) at ¹"150 MeV, transverse #ow g "0  (upper panel) or g "0.3 (lower panel), and vanishing chemical potentials k "0. The overall normalization is arbitrary,   the relative normalizations of the various resonance contributions are "xed by the assumption of thermal and chemical equilibrium. Right: The resonance fractions f (y, p ) according to Eq. (5.31) for the same parameters. Left column: f as P , P a function of transverse momentum at central rapidity. Right column: f as function of rapidity at p "0. P ,

For the model (5.1) its width is dominated by the longitudinal width *g of the source. This is a consequence of the assumed boost-invariant longitudinal #ow pro"le. Since the resonance decay fractions f are essentially rapidity independent (see Fig. 5.1), resonance decay contributions do not P signi"cantly a!ect the rapidity spectrum. 5.2.3. Azimuthal dependence For collisions with non-zero impact parameter the triple-di!erential one-particle spectrum (3.24) contains information about the orientation of the reaction plane. The harmonic coe$cients v which characterize this azimuthal dependence are given in terms of the Fourier transforms L [165,166]. LE ,  (cos (n ), sin (n )) d

, (5.34) (a , b )"   N L L LE ,  d

 N a "v cos (nt ), b "v sin (nt ) . (5.35) L L 0 L L 0 According to Eq. (5.34) they are normalized to the azimuthally averaged double di!erential particle distribution, and v "1. A symmetry argument similar to that employed in (3.20) and (3.21) implies 

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that in the limit p P0 the -dependent terms vanish, the emission probabilities in di!erent , azimuthal directions become equal, and lim v (p )"0 for all n51 . (5.36) L , N, Furthermore, the odd harmonic coe$cients v vanish at midrapidity for symmetric collision L systems, due to the remaining P #p symmetry in the transverse plane. A more explicit calculation depends on details of the model. For the emission function (5.12)}(5.16) one can show that v Jp for small values of p . For small transverse #ow, the leading dependence on the  , , anisotropy parameters e and e is given by   2(e !e )   . (5.37) v Jg   (1!e)(1!e)   This describes correctly the main features of a numerical study of this model [175]. Both geometric and dynamical deformations manifest themselves in the single-particle spectrum only for expanding sources with g O0. The relative minus sign in the numerator of Eq. (5.37) re#ects the di!erent  signs in the de"nitions of e and e in Eqs. (5.13) and (5.15). Once this is taken into account, Eq. (5.37)   shows that an increasing spatial deformation e or an increasing #ow anisotropy e lead to similar   e!ects on the azimuthal particle distribution. Therefore they cannot be separated without also using information from two-particle correlations. 5.3. Two-particle correlator In this section we study the question how characteristics of the emission function are re#ected in the momentum-dependence of the measured particle correlations. This was discussed already in Section 3 in the context of the model-independent relations between the space-time variances of S(x, K) and the HBT radius parameters of C(q, K). According to this discussion spatial and temporal geometric information about the source is contained in the q-dependence of the correlator, while the pair momentum dependence characterizes dynamical properties. These statements can be made more explicit in the context of speci"c model studies. For quantitatively reliable studies of `realistica (i.e. su$ciently complex) emission functions the determination of the two-particle correlator requires a numerical evaluation of the Fourier integral in Eq. (1.4). On the other hand, simple analytical approximations for the HBT radius parameters allow to summarize the main physical dependencies of the measurable quantities in an intuitive form and are quite useful for a qualitative understanding. We give a combined discussion of both the analytical approximations and the exact numerical results for the model (5.1). 5.3.1. Saddle point approximation of HBT radius parameters Characterizing the emission function S(x, K) by the Gaussian widths (3.4), (B\) (K)" IJ 1x x 2(K), is more generally applicable than a saddle point approximation around its center x (K) I J which was earlier suggested [42,153]. The tensor B (K) characterizes essential features of the IJ phase-space support of the emission function even if S(x, K) is not di!erentiable or if its curvature at the saddle point does not represent its average support su$ciently well. In all these cases the Gaussian widths B (K) still translate directly into Gaussian radius parameters, as discussed IJ

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in Section 3. Nevertheless, a saddle point approximation of S(x, K) can be technically useful for the evaluation of the integrals (3.3) when approximating the HBT radius parameters as averages over the emission function [2,42,48,153,171]. To illustrate the use and limitations of the analytical expressions thus obtained one can study a simpli"cation of the model (5.1) which is particularly amenable to analytical calculations: we neglect the resonance contributions and set *g"R and *q"0. This represents a source with exact longitudinal boost-invariance and a sharp freeze-out at time q [42,153,171]. For such  a source the cross-term R vanishes in the LCMS (see Eq. (3.17)).  Expanding the exponent of the emission function S(x, K) given in (5.1) around x "0 one obtains I in this case






g x#y K x x#y M ! ,g ! . ! , 1#  R 2 R ¹ R ¹

(5.38)

In the saddle point approximation the terms bilinear in y specify the (inverse of the) `sidea radius parameter R , while the `outwarda radius parameter receives an additional contribution from the  "nite emission duration 1tI 2"q1sinh g2 according to [42]  1 ¹  R(K )"R(K )# b q , (5.39)  ,  , ,  2 M , R R(K )" , (5.40)  , 1#(M /¹) g  , These simple expressions illustrate several of the key concepts employed in HBT interferometry: the overall size of the transverse radius parameters is determined by the transverse Gaussian widths of the collision region, and the di!erence R!R is proportional to the emission duration b 1tI 2.   , (Even a sharp freeze-out at q"q corresponds to a "nite region in t"q cosh g since the source   distribution is sampled over a "nite longitudinal range.) Most importantly, however, the radius parameters R and R are sensitive to the transverse #ow strength g of the source: the HBT radius    shrinks for "nite g since a dynamically expanding source viewed through a "lter of wavelength K is  seen only partially. This shrinking e!ect increases for larger values of M proportionally to the , ratio g/¹. The M -dependence of R is a consequence of transverse position}momentum correla ,  tions in the source which here originate from the transverse collective #ow. Saddle point integration also leads to simple expressions for the longitudinal radius parameters. Due to boost-invariance, Rl is bl-independent, and one has to calculate Rl "1z 2"q(1sinh g2  !1sinh g2). We summarize the results for di!erent approximation schemes [106,42,89]:











¹ ¹ Rl +q , to O , [106] , M M , , 1 ¹ ¹ ¹ 1 Rl +q 1# # to O , [42] , M 2 1#(M /¹)g M M , , ,  , ¹ K (M /¹)  , [exact for g "0, [89]] . Rl "q M K (M /¹)  ,  ,

 
 

(5.41) (5.42) (5.43)

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In general, the longitudinal radius is proportional to the average freeze-out time q and falls  o! strongly with increasing transverse momentum. In this case the transverse momentum dependence signals longitudinal position}momentum correlations in the source. Compared to the transverse directions, it is much stronger since the source expands predominantly in the beam direction. According to the Makhlin}Sinyukov formula (5.41), "tting an 1/M -hyperbola to Rl (extracting , the temperature, e.g. from the one-particle spectrum), the source parameter q can be determined.  Early estimates of q have been obtained by this argument. The improved calculations (5.42) and  (5.43) show that for realistic temperatures of the order of the pion mass corrections cannot be neglected. Similar remarks apply to the slopes of the `sidea and `outa radius parameters (5.39) and (5.40). Especially, all the results presented here are based on an expansion around x "0, while the I true saddle point x for "nite g is shifted in the out-direction; this can be seen from Eq. (5.38). In I  Ref. [171] an approximation scheme was developed which takes these saddle point shifts into account and allows to systematically derive improved expressions for the HBT radii (5.39)}(5.42). The resulting expressions are involved and a numerical evaluation of the radius parameters is often more convenient. The sources for quantitative uncertainties of the saddle point approximation are two-fold: The determination of the saddle point x "1x 2 is done only approximately in many model studies. I I The resulting inaccuracies turn out to be substantial and put severe limitations on the quantitative applicability of the analytical expressions quoted above. This is illustrated in Fig. 5.2. Moreover, whenever the Gaussian widths of the correlator deviate signi"cantly from its curvature at q"0, the relation between space-time variances and HBT-radii becomes quantitatively unreliable, see Section 4. In the following we therefore use the above simple analytical expressions only for qualitative guidance, basing a quantitative discussion on numerical results. Eqs. (5.39)}(5.43) have been used to extract the transverse #ow parameter g and the `freeze-out  timea q . This is not without danger. As pointed out in Ref. [42], in this particular model the  M -dependence of the HBT radius parameters re#ects the longitudinal and transverse #ow , velocity gradients. These are responsible for the reduced homogeneity lengths compared to the total source size. The parameter g in Eq. (5.40) really re#ects the transverse #ow velocity gradient  [42], and the parameter q in Eqs. (5.41)}(5.43) similarly arises from the longitudinal velocity  gradient at freeze-out. From the general considerations of Section 3.1 we know that the absolute position of the source in space and time cannot be measured. Therefore, strictly speaking, q cannot  be directly associated with the absolute freeze-out time. Such an interpretation of q makes the  additional dynamical assumption that the longitudinally boost-invariant velocity pro"le existed not only at the point of freeze-out but throughout the dynamical evolution of the reaction zone. If this were not the case and the collision region underwent, for example, longitudinal acceleration before freeze-out, the real time interval between impact and freeze-out would be longer. 5.3.2. The out-longitudinal cross-term The cross-term R of the Cartesian parametrization vanishes in the LCMS for longitudinally  boost-invariant systems or in symmetric collisions at mid-rapidity. This follows from the corresponding space-time variance (3.13) which vanishes in the LCMS if the source is symmetric under z P!z . This re#ection symmetry is broken, however, in the forward and backward rapidity regions for the systems with "nite longitudinal extension.

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Fig. 5.2. K -dependence of the HBT radius parameters R (a), R (b) and Rl (c) for the emission function (5.1) with ,   ¹"150 MeV, q "3 fm/c, R"3 fm, *g"R, *q"0. Curves for di!erent values of the transverse #ow g are shown.   Di!erences between numerically evaluated space-time variances and Gaussian widths indicate deviations of C(q, K) from a Gaussian shape. Di!erences to the dash-dotted lines re#ect the limited validity of a naive saddle point approximation around x "0. I

As "rst observed by NA35 [3], following a proposal of Chapman et al. [41,43], the Gaussian correlator is then an ellipsoid in q-space whose main axes do not coincide with the Cartesian ones: it is tilted in the out-longitudinal plane (see the l.h.s. of Fig. 5.3). This tilt is parametrized by the size and sign of R . In the forward rapidity region, R is positive in the LCMS and negative in the CMS   [13,43,147]. The sign is reversed for the backward rapidity region. Also, R has a characteristic transverse momentum dependence. At vanishing K , the x P!x  , symmetry of the source is restored and R vanishes, see (3.21) and the l.h.s. of Fig. 5.3. At large K ,  , on the other hand, the homogeneity region peaks sharply around the point of highest emissivity, and the z P!z symmetry is again approximately restored [180]. The resulting generic M dependence is seen in Fig. 5.3: "R " rises sharply at small K , reaches a maximum and then ,  , decreases again. 5.3.3. The Yano}Koonin velocity The Yano}Koonin}Podgoretskimy parametrization [79,123,180,182] describes the correlator of an azimuthally symmetric source by a longitudinal velocity v(K) and three Gaussian radius

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Fig. 5.3. Left: Contour plot of the correlation function in the q -ql-plane obtained by NA35 in S#Au collisions at 200 A  GeV. The data are for forward rapidity pairs, 3.5(>(4.5, the momentum di!erences are evaluated in a system at rapidity 3. (Figure taken from [3].) Right: M -dependence of the out-longitudinal radius parameter for the model (5.1) at , > "1.5 in the LCMS. The curves are shown for di!erent values of the transverse #ow, g "0.4 (solid) and g "1.0    (dashed). (Figure taken from [163].)

parameters R (K), R (K), and R(K). The Yano}Koonin velocity v(K) contains important informa, ,  tion about the longitudinal expansion of the source. To investigate this, a detailed study [180] was done within the framework of the class of models (5.1), of the relation among the following di!erent longitudinal reference frames: E CMS: The centre of mass frame of the "reball, speci"ed by g "0.  E LCMS (Longitudinally CoMoving System [50]): a pair-dependent frame, speci"ed by bl">"0. In this frame, only the transverse velocity component of the particle pair is non-vanishing. E LSPS (Longitudinal Saddle-Point System [48]): The longitudinally moving rest frame of the point of maximal emissivity for a given pair momentum. In general, the velocity of this frame depends on the momentum of the emitted particle pair. For symmetric sources the point of maximal emissivity (`saddle pointa) coincides with the `source centrea x (K) de"ned in Eq. (3.2). In this approximation, for a source like (5.1), the LSPS velocity is given by the longitudinal component of uI(x (K)). E YK (Yano}Koonin frame [79]): The frame for which the YKP velocity parameter vanishes, v(K)"0. Again, this frame is in general pair momentum dependent. The velocities (or rapidities) of the CMS and LCMS frames can be determined experimentally, the "rst from the peak in the single particle rapidity distribution, the second from the longitudinal momentum of the measured pion pair. The velocity of the LSPS is determined by the #ow velocity at x (K) and hence, it is not directly measurable (neither the one- nor the two-particle spectra depend on x (K)). However, the YK- and LSPS-systems coincide as long as the particle emission is symmetric around x (K) [123,180]. In model studies based on the parameterization (5.1) asymmetries are found to be small [180]. This is seen in the di!erence between the corresponding rapidities, > !> , plotted in Fig. 5.4c and d as a function of K and > respectively. The di!erence is 7) *1.1 ,

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Fig. 5.4. Calculation of the Yano}Koonin rapidity for the model (5.1) without resonances. Source parameters: R"3 fm, q "3 fm/c, *q"1 fm/c, *g"1.2 and ¹"140 MeV. (a) YK rapidity as a function of the pion pair rapidity > (both  measured in the CMS frame of the source), for various values of the transverse pair momentum K and for two values of , the transverse #ow rapidity g . (b) Same as (a), but shown as a function of K for di!erent values of >. The curves for  , negative > are obtained by re#ection along the abscissa. (c) The di!erence > !> between the rapidity of the YK 7) *1.1 frame and the longitudinal rest system of the saddle point, plotted in the same way as (a). (d) Same as (c), but shown as a function of K for di!erent values of >. ,

generally small, especially for large transverse momenta [79,180] where thermal smearing can be neglected: v(K)+v (K) . (5.44) *1.1 The observable YK velocity thus tracks the unobservable LSPS velocity. This is important since longitudinal expansion of the source leads to a characteristic dependence of the LSPS velocity v on the pair rapidity which } using Eq. (5.44) } one can con"rm by measuring the YK-velocity. *1.1 Two extreme examples illustrate this: for a static source without position}momentum correlations the rapidity of the LSPS is independent of the pair rapidity > and identical to the rapidity of the CMS: 1 1#v *1.1"const. for a static source. (5.45) > " ln *1.1 2 1!v *1.1 In contrast, for a longitudinally boost-invariant model (5.1) with *g"R, the longitudinal saddle point lies at g">. In this case, the LSPS and the LCMS coincide, > "g"> for a longitudinal boost-invariant source. *1.1

(5.46)

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For the model (5.1) the measurable YK rapidity > is plotted in Fig. 5.4a and b as a function of the 7) longitudinal pair rapidity > and transverse momentum K . The plot con"rms a linear relation , > +const. ;>, for the model (5.1) . (5.47) 7) with a proportionality constant which approaches unity for large K . Since > +> , this , 7) *1.1 linear relation between the rapidity > of the Yano}Koonin frame and the pion pair rapidity > is 7) a direct re#ection of the longitudinal expansion #ow. The linear relation shown in Fig. 5.4a has been con"rmed subsequently by experiment [13]. There are also "rst experimental hints for the K -dependence shown in Fig. 5.4a. , We emphasize that the observation of a linear relation > "> cannot be generally interpreted 7) as evidence for boost-invariant longitudinal expansion, as was suggested in Refs. [78,79,180]. In Fig. 5.4a one sees that for the model (5.1) this linear relation is the better satis"ed the larger the transverse momentum of the particle pair. This can be easily understood as a consequence of the reduced thermal smearing at large K . Whenever the random component in the momentum , distribution becomes small, the observed pair velocities track directly the #ow velocity of the emitting volume element, irrespective of the actual #ow velocity pro"le. The generic behaviour shown in Fig. 5.4a therefore indicates longitudinal expansion which is su$ciently strong to overcome the thermal smearing, but not necessarily boost-invariant. 5.3.4. Yano}Koonin}Podgoretskiny radius parameters We discussed in Section 3.2 that, as long as certain asymmetries of the source are negligible, the YKP-radius parameters have a particularly simple space-time interpretation. The transverse radius parameter always coincides with the source width in the side-direction, R (K)"1y 2, and [79,180] ,  b b ! J 1y 2+1z 2 , (5.48) R (K)" z ! J x , b b , ,  1 1 ! 1y 2+1tI 2 . (5.49) R(K)" tI ! x  b b , , The approximation in these equations results from dropping terms proportional to 1z x 2, 1x tI 2, and 1x !y 2. The "rst two of these terms vanish if the source is symmetric around its point of highest emissivity x (K). In Fig. 5.5 the e!ective emission duration (1tI 2 and longitudinal size of homogeneity (1z 2 are compared to R and R for the model (5.1). The approximations (5.48) and  , (5.49) become exact for vanishing transverse #ow while di!erences occur especially in R for large  K or signi"cant transverse #ow. These can be traced to the terms !21x tI 2/b #1x !y 2/b , , , which are neglected in Eq. (5.49). For the model (5.1) the space-time variances 1x tI 2 and 1x !y 2 indeed vanish for K P0 where the azimuthal x}y-symmetry of the source is restored. However, , when divided by powers of b , the corresponding terms in (5.49) result in small but "nite , contributions even for K P0. For opaque sources (5.9) the term !21x tI 2/b #1x !y 2/b can , , , be the dominant contribution; this can lead to large negative values of R(K ). The approximation  , (5.49) thus breaks down for such opaque sources [84,160,161,203], and the leading K -dependence , can be recast in the approximate expression (3.58). The experimental data exclude large negative values for R(K ) and thus rule out certain opaque emission functions [160,161,177].  ,





 

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Fig. 5.5. YKP radii for the same model as in Fig. 5.4. (a) R and (1tI 2, evaluated in the YK frame, as a function of  M for three values of the transverse #ow rapidity g , for pion pairs with CMS rapidity >"0. (b) Same as (a), but for ,  pions with CMS rapidity >"3. (c) and (d): Same as (a) and (b), but for R and the longitudinal length of homogeneity , (1z 2, evaluated in the YK frame. For >"0, R and (1z 2 agree exactly because b "0 in the YK frame. , J

5.3.5. Azimuthal dependence of HBT radius parameters Both the calculation of HBT-radius parameters in the saddle-point approximation and the numerical calculation have been extended [175] to the "nite impact parameter model described in Eqs. (5.12)}(5.16). For this model the "rst harmonics vanish, and the zeroth and second harmonics can be written in the saddle point approximation in terms of an average size RM and a dimensionless anisotropy parameter a :  R "RM #b 1tI 2, R "RM  , (5.50)  ,  R "!R "!R "a RM  . (5.51)     a is related to the parameter a in Eq. (3.30) by a "a RM . According to Eq. (5.51) the relations     (3.30) are exact in the saddle point approximation, with R(1#+,g(1!e)) 2   , RM " 1#2+,\CCD g#+,\C \C  g    2 2 e 1#C +,g(1!e) C 2   . a "!   2 1#+,g(1!e)  2 

(5.52) (5.53)

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Fig. 5.6. First (upper panel, R ) and second (lower panel, R ) harmonic coe$cients of the transverse HBT radius H H parameters of the model (5.18) and (5.19) and the model (5.14) and (5.15), respectively. The subscript `*a in R , R H H stands for the out (dash-dotted), side (dashed) and out-side (dotted) components. Solid thin and thick lines denote the zeroth harmonics of the out and side radius parameters, respectively. All calculations are for ¹"150 MeV, q "5 fm/c, *q"1 fm/c, *g"1.2 and R"5 fm. 

For vanishing anisotropy e "e "0 the side radius parameter in Eq. (5.52) reduces to the   corresponding expression (5.40). For non-zero e or e the average size RM depends only weakly on  D these anisotropies. In Fig. 5.6 we show the results of a numerical study of the model (5.14) and (5.15) which qualitatively con"rms the results of the saddle-point approximation. The deviations from the leading dependence R  : R  : !R "1 : !1 : !1 of Eq. (5.51) are seen to be small.    Similarly, the numerical study of the model (5.18) and (5.19) which includes directed transverse #ow in the forward rapidity region allows to con"rm the relation (3.29) between the "rst harmonic coe$cients, R  : R  : R "3 : 1 : !1.    5.3.6. Resonance decay contributions Resonance decay pions a!ect the two-particle correlator by reducing its intercept j(K) and by changing its q-dependence. Typical examples are shown in Fig. 5.7. The modi"cations of C(q, K) due to resonance decays are a consequence of the exponential decay law in (5.28), which provides the emission function S with a non-Gaussian tail in coordinate space. The latter is re#ected in PL a non-Gaussian shape of the correlator. The details depend on the lifetime of the corresponding parent resonances [172]: E Short-lived resonances, C'30 MeV: In the rest frame of the particle emitting #uid element these resonances decay very close to their production point, especially if they are heavy and have only

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Fig. 5.7. Two-pion correlations for the model (5.1) without transverse #ow (g "0), calculated according to Eq. (5.20). D Curves show the correlator without resonance contributions (thin solid lines), including pions from o decays (longdashed), other shortlived resonances *, KH, RH (short-dashed), from the u (dash-dotted), and including pions from the longlived resonances g, g, K, R, K (thick solid lines). 1

small thermal velocities. This means that the emission function S of the daughter pions has PL a very similar spatial structure as that of the parent resonance, S , although at a shifted P momentum and shifted in time by the lifetime of the resonance. Since the Fourier transform of the direct emission function is rather Gaussian and the decay pions from short-lived resonances appear close to the emission point of the parent, they maintain the Gaussian features of the correlator. E Long-lived resonances, C;1 MeV: These are mainly the g and g, with lifetimes cq+17.000 and 1000 fm, respectively, and the weak decays of K and the hyperons which on average propagate 1 several cm. Even with thermal velocities these particles travel far outside the direct emission region before decaying, generating a daughter pion emission function S with a very large PL spatial support. The Fourier transform SI (q, K) thus decays very rapidly for qO0, giving no PL contribution in the experimentally accessible region q'1 MeV. (This lower limit in q arises from the "nite two-track resolution in the experiments.) The decay pions do, however, contribute to the single particle spectrum SI (q"0, K) in the denominator and thus `dilutea the correlation. PL In this way long-lived resonances decrease the correlation strength j without a!ecting the shape of the correlator where it can be measured.

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E Moderately long-lived resonances, 1 MeV(C(30 MeV. There is only one such resonance, the u meson. It is not su$ciently long-lived to escape detection in the correlator, and thus it does not a!ect the intercept parameter j. Its lifetime is, however, long enough to cause a long exponential tail in S (x, K). This seriously distorts the shape of the correlator and destroys its Gaussian SL form. The main e!ects of resonance decay contributions on the j intercept parameter are mimicked by the `core-halo-modela [49] which assumes that the emission function can be written as a sum of two contributions. S(x, K)"S (x, K)#S (x, K) . (5.54)   Here, the `haloa source function S is regarded as the sum over the longlived resonance contribu tions which are wide enough to be unresolvable by HBT measurements. S thus a!ects only the  intercept parameter. The `corea emission function describes the contributions from the direct pions and short-lived resonance decay pions which are emitted from the same central region. The model (5.54) is thus a simpli"ed version of Eq. (5.20) and provides a simple qualitative picture for the intercept:






 (5.55) f (K) . j(K)+ 1! P P   The core-halo model neglects contributions from moderately longlived resonances, essentially the u, for which the distinction into core and halo does not apply. Due to their non-Gaussian shape these a!ect the "t parameters for the intercept j considerably and can lead to quantitative corrections of Eq. (5.55) of up to 10% [172]. An important consequence following already from Eq. (5.55) is the transverse #ow dependence of the intercept parameter, depicted in Fig. 5.8. With increasing transverse #ow, the K -dependence of the j-parameter becomes #atter [164], though it , does not vanish (see Fig. 5.8). This is important since current measurements are consistent with a K -independent intercept parameter [13,24,147]. , Aside from this (model-independent) lifetime e!ect, which generically increases the e!ective pion emission region, various model-dependent features can a!ect the degree to which resonance decay contributions change the shape of the correlator. In the model (5.1) for example, the size of the e!ective emission region in the transverse plane shrinks with increasing transverse #ow g approx imately according to Eq. (5.40). The simpli"ed core-halo model [49] neglects the proper resonance decay kinematics and thus does not describe this e!ect. As a consequence, the direct resonance emission function S  in Eq. (5.1) has a smaller e!ective emission region than that of the thermal P pions, due to the larger transverse mass M of the parent resonance (`transverse #ow e!ecta). For , "nite transverse #ow this reduces the size of the resonance emission region and counteracts the lifetime e!ect. Other models of heavy ion collisions [141,143] do not show this behaviour and lead to signi"cantly di!erent q-dependences of the correlator. 5.3.7. Kurtosis of the correlator If, as in the presence of resonance decays, the correlator deviates from a Gaussian shape, the characterization of C(q, K) via HBT radius parameters is not unambiguous. Fit results then depend

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Fig. 5.8. The intercept parameter as a function of M for di!erent transverse #ow strengths g . We use the model (5.1) ,  with temperature ¹"150 MeV. The calculation is not based on a proper "t of the two-particle correlation function, but on Eq. (5.55), including in the sum over resonances the contributions from u, g, g, and K. 1

on the relative momentum region covered by the data, on the statistical weights of the di!erent q-bins, and on details of the "ttings procedure. The q-moments discussed in Section 4.2 allow to quantify deviations of the correlator from a Gaussian shape [98,173]. They are sensitive to di!erences between model scenarios which cannot be distinguished on the basis of Gaussian radius parameters. To illustrate this point we show in Fig. 5.9 both the side radius parameter R (K ) and the corresponding one-dimensional kurtosis  , D (K ), calculated according to Eqs. (4.10) and (4.13). In the present case, the side radius parameter  , calculated from the inverted second q-moment coincides rather accurately with the radius parameters extracted from a "t to (5.56) CI (q , K)"1#j e\0G OG , i"o, s, l G in the range q 4100 MeV. For a transverse #ow between g "0 and g "0.3, R shows approximG    ately the same K -slope. Hence, once resonance decays are taken into account, scenarios with and , without transverse #ow cannot be distinguished unambiguously on the basis of Eq. (5.40). The physical origin of the K -slope of R is, however, di!erent in the two situations, and this shows up ,  in the kurtosis of the correlator. Without transverse #ow, resonance decay contributions increase R due to the lifetime e!ect. For non-zero transverse #ow, on the other hand, the K -slope arises  , from the K -dependent shrinking (5.40) of the e!ective transverse emission region which is more , prominent for resonances than for thermal pions. S  is spatially more extended in the transverse L plane than S , and thus `coversa a substantial part of the exponential tails of S [172]. As 0 0L a consequence, the total emission function (5.20) shows much smaller deviations from a Gaussian shape for the scenario with transverse #ow and results in a more Gaussian correlator. This explains why the kurtosis D (K ) plotted in Fig. 5.9 provides a clearcut distinction between the two  , scenarios.

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Fig. 5.9. The inverted uni-directional second q-variance R of (4.10) and the kurtosis D of Eq. (4.13) as a function of K at   , mid rapidity for the model (5.1) with ¹"150 MeV, R"5 fm, *g"1.2, q "5 fm/c, *q"1 fm/c and vanishing chemical  potentials. Left: g "0 (no transverse #ow). Right: g "0.3. The di!erence between the dashed and solid curves is entirely   dominated by u-decays.

5.4. Analysis strategies for reconstructing the source in heavy-ion collisions A realistic emission function of heavy-ion collisions should simultaneously reproduce the spectra and particle yields of all observable particle species. Every model implies certain constraints between all these observables. For instance, in the model (5.1) the assumption of resonance production with thermal abundances inside the same space-time geometry relates the production of di!erent particle species. While this may be su$cient to model gross properties of particle production, extensions involving additional model parameters may be needed to account for "ner details (e.g. partial strangeness saturation or rapidity dependent chemical potentials [155]). In this review, we will stay with a simple model and try to describe only the rough features of the freeze-out process. We will restrict our discussion to the single-particle spectra of negatively charged particles and to two-pion correlations, setting all chemical potentials to zero. We aim to extract from the data the phase-space properties of the pion production region, characterized by the model parameters in Eq. (5.1). 5.4.1. Determining the model parameters of analytical emission functions The model parameters of an analytical emission function should be determined by a multiparameter "t of the corresponding one- and two-particle spectra (1.3)}(1.5) to the data. For su$ciently complicated emission functions, where no accurate analytical approximations of (1.3)}(1.5) are available, this is a signi"cant numerical task. In a model like (5.1), however, certain model parameters are almost exclusively determined by particular properties of the measured particle spectra. Based on this observation we outline here a strategy for the comparison of Eq. (5.1) to data, which uses heavily the model studies presented in Sections 5.2 and 5.3. It allows to determine the model parameters ¹, g , R, *g, *q, q by a signi"cantly simpler method, according to   the following steps [177]:

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Fig. 5.10. LHS: s contour plot of a "t to the NA49 h\-spectrum [91]. Dashed lines are for constant values of g/¹. RHS:  di!erent combinations of temperature ¹ and transverse #ow g can account for the same one-particle slope. 

1. The transverse single-pion spectrum dN/dm determines the blue-shifted ewective temperature ¹ . ,  For pions with ¹+m , the slope of dN/dm is essentially given by ¹ " L ,  ¹((1#1b 2)/(1!1b 2), as argued in Section 5.2.1 and Fig. 5.1. Resonance decay contributions R R a!ect the local slope of dN/dm and thereby the "t parameter [145]. They have to be properly , taken into account. Then a "t to the corresponding one-particle spectrum determines a `valleya of parameter pairs ¹ and g (related to 1b 2) all of which can account for the same   data. This is clearly seen in the s-plot of a "t to recent NA49 h\-spectra [91], presented in Fig. 5.10. 2. Combining the single-particle spectrum dN/dm and the transverse HBT radius parameter , R (M )"R (M ) disentangles temperature T and transverse yow g . R then xxes the transverse  , , ,  , extension R. In the saddle-point approximation (5.40) the M -slope of the transverse radius R is propor, , tional to g/¹. In Fig. 5.10 we have superimposed lines of constant g/¹ onto the "t results   dN/dm . Due to the di!erent correlation between ¹ and g , the additional information provided ,  by R (M ) allows to disentangle temperature ¹ and transverse #ow e!ects [147]. Once the , , transverse #ow is "xed, the overall size of R (M ) determines the Gaussian width R of the source , , according to Eq. (5.40). Quantitative details change if the saddle-point approximation is abandoned, but the qualitative argument survives. In a numerical calculation including resonance decay contributions [177] one extracts then from the combination of R (M ) and dN/dm , , , the following values for the model parameters, see Fig. 5.12: g +0.35, ¹+130 MeV, R+7 fm.  3. The single-particle rapidity distribution dN/dy xxes the longitudinal source extension *g. The single-particle rapidity distribution (5.33) is determined by *g, the only parameter which breaks the longitudinal boost-invariance of the source (5.1). Fig. 5.11 shows that the h\ rapidity spectrum is a Gaussian with a width of 1.4 rapidity units. In the parametrization of the present model, this translates into *g"1.2.

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Fig. 5.11. The rapidity distribution of primary negative hadrons. The preliminary NA49 Pb#Pb data are taken from [91].

4. R determines q .   In principle, R and Rl depends on q , *g and *q [171]. Model calculations [161] show,   however, that for Eq. (5.1) the dependence on the emission duration *q is weak. As argued above, *g can be "xed from the single-particle rapidity distribution. The data presented in Fig. 5.12b then clearly favour a value of q +9 fm/c. In this plot *q was chosen to *q"1.5 fm/c.  From the arguments given at the end of Section 5.3.1, this value of q is likely to provide a lower  estimate for the total lifetime of the collision region. 5. R discards opaque sources.  For the model (5.1) the YKP-parameter R is mainly sensitive to the mean emission duration of  the source, since for this model the approximation (3.57) is satis"ed. The large statistical uncertainties of the NA49 data for R do not allow to constrain the model parameter space  further, see Fig. 5.12c. Certain models of opaque sources, however, which include the opacity factor (5.10), lead according to Eq. (3.58) to a negative radius parameter R, and can be excluded  [177,161] already by the present data. 5.4.2. Uncertainties in the reconstruction program We now list some sources of uncertainties in the reconstruction program described above: 1. Particle identixcation The above analysis is based on h\ (all negative hadrons) and h> (all positive hadrons) spectra and correlations. These are dominated by the corresponding charged pion contributions. The e!ect of other particle species (e.g. kaons or (anti)protons) on the one-particle spectra can be included in Eq. (5.29) and has a negligible e!ect on the shape of the "t (which is what matters) shown in Fig. 5.1. For the two-particle correlations misidenti"ed particles lead to the counting

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Fig. 5.12. Yano}Koonin-Podgoretskimy HBT-radius parameters. The "gure uses preliminary NA49 Pb#Pb data for h>h> (squares) and h\h\ (diamonds) correlations [13]. Final data have since been published in [14].

of `wrong pairsa which do not show Bose-Einstein correlations. This decreases the intercept parameter whose value was, however, not used in the reconstruction program. There are no indications that the problem of particle misidenti"cation a!ects the K -dependence of the HBT , parameters signi"cantly. Nevertheless, it would be preferable to use spectra of identi"ed particles. This would additionally allow to compare di!erent e!ective source sizes e.g. from pion}pion and kaon}kaon correlations and thereby investigate the question whether all particle species are emitted from the same source volume. Such identi"ed two-particle correlations were measured at the AGS [112] and by the NA44 Collaboration at the CERN SPS [60]. Both of these data sets are, however, restricted to narrow regions in K and > which limits their , usefulness for the above reconstruction program. If the di!erent acceptance of the two experiments is properly taken into account, the p\ p\-correlations from NA44 and the h\ h\correlations from NA49 are compatible within statistical errors (P. Seyboth and J.P. Sullivan, private communication). 2. Coulomb corrections Coulomb corrections a!ect the size of the HBT radius parameters as well as their K , dependence [13,98,147]. For example, the di!erence R!R in NA35 correlation data was   found to be non-zero only after changing from a naive Gamow correction based on Eq. (2.103) to an e!ective correction (2.104) which takes the "nite source size into account [5]. Furthermore, in the NA49 experiment a proper treatment of the Coulomb correction proved essential for a successful check of the consistency relations (3.48)}(3.54) between the Cartesian and YKP HBT parameters [13]. 3. Uncertainties awecting the K -dependence of HBT radius parameters , Particle misidenti"cation and Coulomb correction are not the only possible sources of uncertainty which may a!ect the momentum slope of HBT radius parameters. Other sources on the theoretical side are for example multiparticle symmertrization e!ects [51,176], the dependence on a central Coulomb charge [18], the mixing of anisotropy e!ects into the HBT radius parameters resulting from collisions with non-zero impact parameter in the central collision sample [165,175], or multiparticle "nal state interactions. On the experimental side, residual correlations in the mixed event sample used to normalize the two-particle correlator or the

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shape of the experimental acceptance in the K }>-plane can a!ect the presented K -depend, , ence of HBT-radius parameters. One usually argues that these uncertainties are small for the size and K -dependence of the , HBT-radius parameters. However, already small di!erences in the transverse slope of R , for , example, a!ect signi"cantly the optimal combination of "t parameters (¹, g ) in Fig. 5.10a. We  therefore consider the systematical error in the slope of R (K ) to be the most important , , uncertainty in the reconstruction program of Section 5.4.1. 5.4.3. Dynamical interpretation of model parameters Pions, as most other hadrons, rescatter during the expansion stage of a heavy-ion collision. Their phase-space distribution S (x, p) characterizes the geometrical and dynamical properties of the "nal p freeze-out stage after their last strong interaction. The emission function does not contain direct information about the hot and dense earlier stages of the collision. However, the emission function, as reconstructed from the spectra and correlation data, provides an experimentally justi"ed starting point for a dynamical extrapolation back towards the earlier stages. To illustrate this point, we discuss here the values of the geometrical and dynamical parameters, extracted for the model (5.1) from preliminary NA49 data: R+7 fm ,

(5.57)

¹+130 Mev ,

(5.58)

g +0.35 ,  q +9 fm/c ,  *g+1.3 ,

(5.59)

*q+1.5 fm/c .

(5.62)

(5.60) (5.61)

To obtain from these data a dynamical picture of the collision process, we compare "rst the two-dimensional rms width obtained from the transverse width R+7 fm, (5.63) r"(1x #y 2"(2R+10 fm ,   with the two-dimensional rms widths of a cold lead nucleus. The nuclear hard sphere radius R "r A with r "1.2 fm for lead is R.
"7.1 fm. The corresponding two-dimensional trans    verse rms width (5.64) r  .
"(1x #y 2P "(3/5R +4.5 fm .
   From this we conclude that during the collision the system has expanded by a factor +2 from the transverse size of the overlapping cold lead nuclei to the transverse extension at freeze-out. For not quite central collisions the initial nuclear overlap region is in fact expected to be somewhat smaller than given by Eq. (5.64). With an average transverse #ow velocity of about 0.35 c matter can travel over +4 fm in a time of 9 fm/c. This is barely enough to explain the observed expansion. This indicates that indeed, as argued before, the parameter q is a lower estimate of the total duration of 

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the collision. The rather low thermal freeze-out temperature of 130 MeV (other analyses indicate even lower values [92,163]) di!ers signi"cantly from the chemical freeze-out temperature (+170 MeV) needed to describe the observed particle ratios in these thermal models [23], consistent with a long expansion stage. A "rst attempt to extrapolate the "nal state characterized by (5.57)}(5.62) all the way to the beginning of the transverse expansion [81] has led to an estimate for the average energy density at this point of e+2.5 GeV/fm. 6. Summary In this work we reviewed the underlying concepts, calculational techniques and phenomenological uses of Hanbury Brown/Twiss particle interferometry for relativistic heavy-ion collisions. Compared to the astrophysical applications of HBT interferometry, its use for relativistic nuclear collisions is substantially complicated by (i) the time-dependence and short lifetime of the particle emitting source, (ii) the position}momentum gradients in the source resulting from the strong dynamical expansion of the collision region, and (iii) other dynamical origins of particle} momentum correlations which have to be subtracted properly to make a space-time interpretation of the measured correlation data possible. The Wigner phase-space density (`emission functiona) S(x, K), interpreted as the probability that a particle with momentum K is emitted from a space-time point x in the collision region, provides the appropriate starting point for the analysis of measured HBT correlations from relativistic heavy ion collisions. It accounts for the time-dependence and the position}momentum gradients of the source. Moreover, other contributions to the momentum correlations between pairs of identical particles, for example from "nal state Coulomb interactions, multiparticle symmetrization e!ects, or resonance decay contributions, can be calculated once the emission function is given. In Section 2 we discussed this in detail, after deriving the basic relation (1.4) between the phase-space emission function S(x, K) and the measured two-particle momentum correlation C(q, K). The key to a geometric and dynamical understanding of the measured two-particle correlations are the model-independent relations between the space-time variances (Gaussian widths) of the emission function and the HBT radius parameters which are extracted from Gaussian "ts (1.5) to the two-particle correlator. These were derived in Section 3. In general, the HBT radius parameters R (K) do not measure the total geometric source size, but the size of regions of homogeneity in the GH source from which most particles with momentum K are emitted. These homogeneity regions typically decrease in the presence of temperature or #ow velocity gradients which lead to characteristic position}momentum correlations in the source. Generically, the homogeneity regions are smaller for pairs with larger transverse momentum. The K-dependence of the two-particle correlator thus gives access to dynamical characteristics of the collision region. Expressing the HBT radius parameters in terms of space-time variances also shows explicitly how spatial and temporal information about the source is mixed in the measured momentum correlations. We explained this "rst for the Cartesian parametrization of C(q, K) and its extension to collisions at non-zero impact parameter. We then introduced the YKP parametrization, an alternative Gaussian parametrization of C(q, K) which is particularly well adapted for collision systems with strong longitudinal expansion: (i) Three of the four YKP "t parameters are invariant under longitudinal Lorentz boosts, i.e. their values do not depend on the observer frame. (ii) The

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225

fourth "t parameter is the Yano}Koonin velocity which measures the longitudinal velocity of the particle emitting source element. Its rapidity dependence allows to determine the strength of the longitudinal expansion in the collision region. (iii) In the particular observer frame in which the Yano}Koonin velocity vanishes, the YKP radius parameters for a large class of emission functions cleanly separate the longitudinal, transverse and temporal aspects of the source. This simpli"es the space-time interpretation of the Gaussian "t parameters considerably. An important aspect of HBT interferometry is that from a combined analysis of the singleparticle spectra and HBT correlation radii an estimate of the average phase-space density of the source at freeze-out can be obtained. Applied to heavy-ion data at the AGS and SPS, this method provided evidence for a universal freeze-out phase-space density for pions. Its transverse momentum dependence is in rough agreement with expectations based on models assuming thermalization prior to freeze-out. The connection between HBT radii and space-time variances of the emission function is based on Gaussian parametrizations of the source and the measured particle correlations. Deviations of the two-particle correlator from a Gaussian shape can contain additional space-time information which is not contained in the HBT radii. In Section 4 we described re"ned techniques which give access to such additional information. The imaging method and the method of q-moments discussed in Sections 4.1 and 4.2 characterize in di!erent ways the relative source function SK(r) which is de"ned as a time averaged distribution of relative distances in the source. Identical three particle correlations, discussed in Section 4.3, give access to odd orders of the space-time variances of S(x, K) which drop out in identical two-particle correlations. The main goal of particle interferometry for relativistic heavy-ion collisions is to extract from the measured momentum spectra as much information as possible about the emisson function S(x, K). We explained why a completely model-independent reconstruction of S(x, K) is not possible. In practice, one therefore must take recourse to a model dependent approach. This was illustrated in Section 5 for a class of analytical emission functions. Comprehensive model studies allowed to separate the generic from the more model-dependent features. Simple approximate expressions for the HBT radius parameters were given which provide a qualitative understanding of the dominant geometrical and dynamical e!ects. Their quantitative accuracy was tested by numerical means. The size and momentum dependence of the di!erent HBT radius parameters and the transverse momentum slopes of the one-particle spectra were shown to depend in general only on one or two of the model parameters of the emission function. This allows for a simple analysis strategy for the reconstruction of the emission function. It was illustrated in Section 5.4 in an application to preliminary data from the 158 A GeV lead beam experiment NA49 at the CERN SPS. The extracted source parameters are consistent with the creation of a highly dynamical system which after impact expanded in the transverse direction over a time of at least 9 fm/c with approximately one third of the velocity of light before emitting particles at a temperature of around 130 MeV. This information about the hadronic emission region provides a starting point for a dynamical back extrapolation into the hot and dense early stage of the collision, and it can be directly compared with the output of numerical event simulations of relativistic heavy-ion collisions. Compared to an analysis based on momentum space information only, the additional space-time information obtained from particle interferometry provides severe constraints for our understanding of heavyion collision dynamics.

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Remaining uncertainties in the approach were discussed. Better experimental statistics and further progress in our quantitative understanding of e!ects which in#uence the pair momentum dependence of the two-particle correlator are expected to lead in the near future to considerable improvements in the analyzing power of this method.

Acknowledgements This report grew out of the habilitation thesis submitted by one of us (UAW) to the University of Regensburg in April 1998. The present review would not have been possible without the continuous in#ux of ideas resulting from fruitful discussions with many friends and colleagues. We would like to mention explicitly J. Aichelin, D. Anchishkin, H. AppelshaK user, F. Becattini, G. Bertsch, A. Bialas, H. B+ggild, P. Braun-Munzinger, D.A. Brown, S. Chapman, J. Cramer, T. CsoK rgo , P. Danielewicz, J. Ellis, H. Feldmeier, D. Ferenc, P. Filip, P. Foka, M. GazH dzicki, K. Geiger, M. Gyulassy, H. Heiselberg, T. Humanic, B. Jacak, K. Kadija, B. KaK mpfer, H. Kalechofsky, B. Lasiuk, R. LednickyH , B. LoK rstad, M. Martin, U. Mayer, D. MisH kowiec, B. MuK ller, R. Nix, S. Padula, Y. Pang, T. Peitzmann, D. Pelte, J. Pis\ uH t, M. PluK mer, S. Pratt, P. Renk, D. RoK hrich, G. Roland, R. Scheibl, B.R. Schlei, S. SchoK nfelder, P. Scotto, C. Slotta, P. Seyboth, E. Shuryak, Yu. Sinyukov, T. SjoK strand, S. So!, J. Sollfrank, J. Stachel, R. Stock, B. TomaH s\ ik, S. Vance, A. Vischer, S. Voloshin, Y.-F. Wu, N. Xu, W.A. Zajc, K. Zalewski, Q.H. Zhang and J. ZimaH nyi. This work was supported by DFG, GSI, BMBF and DOE (contract no. DE-FG02-93ER40764).

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U.A. Wiedemann, U. Heinz / Physics Reports 319 (1999) 145}230 U.A. Wiedemann, P. Foka, H. Kalechofsky, M. Martin, C. Slotta, Q.H. Zhang, Phys. Rev. C 56 (1997) R614. U.A. Wiedemann, Phys. Rev. C 57 (1998) 266. U.A. Wiedemann, Phys. Rev. C 57 (1998) 3324. U.A. Wiedemann, B. TomaH s\ ik, U. Heinz, Nucl. Phys. A 638 (1998) 475c. U.A. Wiedemann, J. Ellis, U. Heinz, K. Geiger, in: S. Costa et al. (Eds.), CRIS '98: Measuring the Size of Things in the Universe: HBT Interferometry and Heavy Ion Physics, World Scienti"c, Singapore, 1998, hep-ph/9808043. U.A. Wiedemann, D. Ferenc, U. Heinz, Phys. Lett. B 449 (1999) 347. Y.-F. Wu, U. Heinz, B. TomaH s\ ik, U.A. Wiedemann, Eur. Phys. J. C 1 (1998) 599. J. Wosiek, Phys. Lett. B 399 (1997) 130. F. Yano, S. Koonin, Phys. Lett. B 78 (1978) 556. W.A. Zajc, Two pion correlations in heavy ion collisions, PhD thesis, University of California, 1982 (available as LBL-14864). W.A. Zajc et al., Phys. Rev. C 29 (1984) 2173. W.A. Zajc, Phys. Rev. D 35 (1987) 3396. W.A. Zajc, in: H.H. Gutbrod, J. Rafelski (Eds.), Particle Production in Highly Matter, NATO ASI Ser. B, vol. 303, Plenum, New York, 1993, p. 435. W.N. Zhang et al., Phys. Rev. C 47 (1993) 795. Q.H. Zhang, Phys. Rev. C 57 (1998) 877. Q.H. Zhang, Nucl. Phys. A 634 (1998) 190. Q.H. Zhang, U.A. Wiedemann, C. Slotta, U. Heinz, Phys. Lett. B 407 (1997) 33. Q.H. Zhang, P. Scotto, U. Heinz, Phys. Rev. C 58 (1998) 3757. J. ZimaH nyi, T. CsoK rgo , hep-ph/9705432. S.V. Akkelin, Yu.M. Sinyukov, Z. Phys. C 72 (1996) 501. G.F. Bertsch, Phys. Rev. Lett. 77 (1996) 789(E). J. Bolz, U. Ornik, M. PluK mer, B.R. Schlei, R.M. Weiner, Phys. Rev. D 47 (1993) 3860. Q.H. Zhang, W.Q. Chao, C.S. Gao, Phys. Rev. C 52 (1995) 2064. E.G.D. Cohen (Ed.), Fundamental Problems in Statistical Mechanics II, North-Holland, Amsterdam, 1968, p. 140. R. Hanbury Brown, R.Q. Twiss, Nature 178 (1956) 1046. R. Hanbury Brown, R.Q. Twiss, Proc. Roy. Soc. A 242 (1957) 300. R. Hanbury Brown, R.Q. Twiss, Proc. Roy. Soc. A 243 (1957) 291. R. Hanbury Brown, R.Q. Twiss, Proc. Roy. Soc. A 248 (1958) 199. R. Hanbury Brown, R.Q. Twiss, Proc. Roy. Soc. A 248 (1958) 222. H. Heiselberg, A. Vischer, Phys. Lett. B 421 (1998) 18. G.I. Kopylov, M.I. Podgoretskimy , Sov. J. Nucl. Phys. 19 (1974) 215. S. Nagamiya, Prog. Part. Nucl. Phys. 15 (1985) 363. S. Pratt, Phys. Rev. D 33 (1986) 72. S. Pratt, Phys. Rev. C 50 (1994) 469. E. Schnedermann, U. Heinz, Phys. Rev. C 47 (1993) 1738. E. Schnedermann, U. Heinz, Phys. Rev. C 50 (1994) 1675. E. Shuryak, Sov. J. Nucl. Phys. 18 (1974) 667. S.A. Voloshin, W.E. Cleland, Phys. Rev. C 54 (1996) 3212. Q.H. Zhang, Phys. Lett. B 406 (1997) 366.

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ELECTRONIC STRUCTURE OF CONJUGATED POLYMERS: CONSEQUENCES OF ELECTRON}LATTICE COUPLING

W.R. SALANECK *, R.H. FRIEND
, J.L. BRED DAS Department of Physics, IFM, LinkoK ping University, S-581 83 LinkoK ping, Sweden
The Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK Service de Chimie des MateH riaux Nouveaux, UniversiteH de Mons-Hainaut, Place du Parc 20, B-7000 Mons, Belgium

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

Physics Reports 319 (1999) 231}251

Electronic structure of conjugated polymers: consequences of electron}lattice coupling W.R. Salaneck *, R.H. Friend
, J.L. BreH das Department of Physics, IFM, Linko( ping University, S-581 83 Linko( ping, Sweden
The Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK Service de Chimie des Mate& riaux Nouveaux, Universite& de Mons-Hainaut, Place du Parc 20, B-7000 Mons, Belgium Received May 1999; editor: D.L. Mills Dedicated to Prof. Eli Burstein, on the occasion of his 80th birthday Contents 1. Introduction 2. The nature of conjugated polymers 3. The electronic structure of linear conjugated polymers 4. Polarons and bipolarons in non-degenerate ground state conjugated polymers

233 234 237 239

4.1. Charge transfer doping 4.2. Optical excitations 5. Model molecular systems: conjugated oligomers 6. Summary References

239 244 247 248 249

Abstract Conjugated organic polymers can be doped, via oxidation or reduction chemistry or via acid}base chemistry, to induce very high electrical conductivity. Conjugated polymers are beginning to "nd uses, in both the neutral and the doped states, in prototype molecular-based electronics applications and in electronic and opto-electronic devices. The physical basis for the many of the unusual properties of these new materials is discussed, at a su$cient level of approximation to enable an understanding of the important issues by the general condensed matter physicist. In particular, emphasis is placed on the interconnections of the electronic, geometric and chemical structures, in the ground state and especially in the excited states. The important role of electron}electron and electron}lattice interactions are pointed out, and justi"ed through a combined experimental}theoretical approach.  1999 Elsevier Science B.V. All rights reserved. PACS: 71.20.Rv; 36.20.Kd; 42.70.Jk Keywords: Conjugated polymers; Conjugated oligomers; Electron}lattice interactions; Electron}electron interactions; Optical absorption; Photoelectron spectroscopy

* Corresponding author. Fax: #46-13137568. E-mail address: [email protected] (W.R. Salaneck) 0370-1573/99/$ - see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 5 2 - 6

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1. Introduction Conventional polymers, plastics, have been used traditionally because of their attractive chemical, mechanical, and electrically insulating properties, and not for their electronic properties. Although the idea of using polymers for their electrically conducting properties dates back at least to the 1960s [1], the use of organic `n-conjugateda polymers as electronic materials [2,3] in molecular-based electronics is relatively new. While behaving as insulators or semiconductors in the pristine form, conjugated polymers can reach metallic-like electrical conductivity when `dopeda (in chemical terminology, when oxidized or reduced) [4}7]. Since 1977, `2 the dream of combining the processing and mechanical properties of polymers with the electrical and optical properties of metals 2a [8] has driven both the science and technology of conducting polymers [7,9,10]. In addition to the exploitation of the high electrical conductivities, e.g., in the manufacture of conducting transparent plastics [11] and conducting fabrics [12], the fast and high non-linear (both second-order and third-order) optical response displayed by conjugated organic compounds is also a topic of major interest [13]. More recently, conjugated polymers are "nding use in perhaps the area of highest activity to date for these materials, that of electronic applications. In particular, conjugated polymers as well as n-conjugated oligomers [14] play a central role in organic-based transistors and integrated circuits [15}18], photo-voltaic devices [19,20] and especially in organic-based light emitting devices [21]. Even solid-state lasers are under development [22}25]. A central issue in the physics of these n-conjugated polymers (and the corresponding oligomers) is the strong coupling between the electronic structure, the geometric structure, and the chemical structure, that is, the bonding pattern of the atoms in the molecular system. The latter might be called the `latticea in parallel with terminology in condensed matter physics [3,5,6,26}28]. In the 1980s, the concepts of polarons, bipolarons, and solitons were developed, in the context of both transport properties [28,29] and optical properties [3,5,9,27,29,30]. In fact, in the case of polymerbased LEDs, the development of device structures has lead to the establishment of hi-tech companies, as well as development programs within large industries, in at least a dozen countries around the world, focused mainly, but not exclusively, on a variety of display-type applications. Information on such activities may easily be obtained by searching on the Internet, in particular, the site www.cdtltd.co.uk and the links therein. During the past few years, the perception of the nature of the physical basis of the unique electronic properties of these conjugated polymers, both as isolated `moleculesa and as molecular solids, has developed to a somewhat more sophisticated level of understanding. The essential ideas about the nature of the unusual charge bearing species, and of the excited states of conjugated systems, have been discussed intensely over the past twenty years. Recently, however, there has been a re"nement of these ideas, which enables a better understanding of certain features of the electronic structure of conjugated polymers. From a computational point of view, the determination of the electronic structure of conjugated systems might, at "rst glance, be considered relatively easy. One of the simplest quantum-chemical methods, the one-electron HuK ckel technique, is mainly designed for conjugated molecules. On the other hand, a description of the essential features of properties such as luminescence, electron}hole separation, or nonlinear optical response, requires a proper description of electronic excited states as well as inter-chain interactions, for which many-body e!ects are important. In this context, the understanding of the nature of n-conjugated systems is complicated by the presence of

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electron}electron interactions, and the strong connection between, and mutual in#uence of, the electronic and geometric structures. In this review, the role of the electron}lattice coupling in determining the nature of charge bearing species, as well as the nature of optically excited states in conjugated polymers and oligomers, is discussed. By `electron}lattice couplinga is meant the strong in#uence of the presence of an extra electron, a hole, or an excitation on the (local) geometry of the molecules, that is, on the nuclear coordinates (i.e., the `latticea in solid-state or condensed matter physics terminology). A central point is that, although there is a strong coupling of electrons or holes to the underlying lattice, the charge bearing species and the neutral species in the excited state, are surprisingly mobile, increasing from modest mobility values, at low electric "elds, to values approaching 0.1 cm/V-s at high "elds [31]. This overview begins by reiterating the generalized phenomenological descriptions of charge bearing species in conjugated polymers, focusing on charge transfer doping in one of the most common conjugated polymers studied today. Then, the nature of optical excitations is discussed, making particular use of recent work on oligomers of conjugated polymers, where more detailed quantum chemical calculations may be employed to gain insight into ideas of localization resulting from strong `electron}lattice couplinga. The descriptions focus on `ideal polymer chainsa, with one conjugated repeat-unit de"ning the periodicity of the chain; but some comments on more realistic materials are included near the end of the article. An overview of the state-of-the-art as it existed just a few years ago may be found in Refs. [28,29,32]. The experimental results discussed are focused on photoelectron spectroscopy, optical absorption spectroscopy and photoluminescene measurements. No discussion of the experimental details is provided, however, as these techniques are su$ciently standard, and speci"cs may be found in the references provided. A complete historical perspective is not attempted. In the remainder of the review, only references to the most central issues are given.

2. The nature of conjugated polymers The geometric structure of several common polymers discussed below are sketched in Fig. 1, where, as is conventional, only the monomeric repeat units, or `unit cellsa, are indicated. Since carbon has the electronic structure, 1s2s2p, carbon atoms form four nearest-neighbor bonds. In p-bonded polymers, the C-atoms are sp hybridized, as in polyethylene (PE), polymer I in Fig. 1, and each C-atom has four p-bonds. In such non-conjugated polymers, the electronic structure of the chain of atoms (or chemical groups) which comprises the backbone of the macromolecule consists of only p-bands (possibly with n-electronic levels localized on side groups as, for example, in polystyrene). The large electron energy band gaps in p-bonded polymers, E (p), renders these  polymer materials electrically insulating, and generally non-absorbing to visible light. In polyethylene, for example, which consists of a monomeric repeat unit (a `unit cella in solid-state physics terminology) de"ned by }(CH }CH )}, the optical band gap is on the order of 8 eV.   In conjugated polymers, however, there exists a continuous network, often a simple chain, of adjacent unsaturated carbon atoms, i.e., carbon atoms in the sp (or sp) hybridized state. Each of these sp C-atoms has three p-bonds, and a remaining p atomic orbital which exhibits n-overlap X with the p -orbitals of the nearest neighbor, sp hybridized C-atoms. This chain of atoms with X

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Fig. 1. The structural formulae of several common polymers chains are illustrated. Each carbon atom can have four bonds. At each bonding vertex there are four bonds. Where less than four bonds are shown explicity, it is understood that hydrogen atoms are present to account for the un-shown bonds. (I) polyethylene, or PE; (II) trans-polyacetylene, or PA; (III) poly(para-phenylenevinylene), or PPV; (IV) poly(para-phenylene), or PPP; and (V) poly(thiophene), or PT. Note that the R groups may be hydrogen atoms, alkyl-chains, or even oxygen containing groups such as methoxy-, ethoxy- or decyloxy-groups.

n-overlap of the atomic p -orbitals leads to the formation of n-states delocalized along the polymer X chain. In a system with one-dimensional periodicity, these n-states form the frontier electronic bands, with a n-band gap, E (n)(E (p), accounting for optical absorption at lower photon   energies. The essential properties of the delocalized n-electron system, which di!erentiate a typical conjugated polymer from a conventional polymer with p-bands, are as follows: (i) the electronic band gap, E , is relatively small (&1}4 eV), leading to low-energy electronic excitations and  semiconductor behavior; (ii) the polymer chains can be rather easily oxidized or reduced, usually through charge transfer with molecular dopant species; (iii) carrier mobilities are large enough that high electrical conductivities are realized in the doped (chemically oxidized or reduced) state; and (iv) charge carrying species are not free electrons or holes, but quasi-particles, which may move relatively freely through the material, or at least along uninterrupted polymer chains. Although both positively and negatively charged species can exist, we discuss mostly negatively charged species, formed by the addition of electrons, because these species are directly accessible through photoelectron spectroscopy (while hole states are not easily studied by photoelectron spectroscopy). Finally, since few polymers are crystalline, macroscopic electrical conductivity in "nite samples requires hopping between chains. Conjugated polymers are essentially quasi-one dimensional, in the sense that there occurs covalent bonding within the chains while interactions between chains are of van der Waals type, and `softa. The intrinsic low-dimensional geometrical nature of polymer chains, and the general property of conjugated organic molecules that the geometric structure is dependent upon the ionic state of the molecule, leads to the existence of the unusual charge carrying species. The charge bearing species are not free electrons or holes, but may be any one of several di!erent types of essentially well-de"ned quasi-particles, each consisting of a coupled charge-lattice deformation entity. The charge bearing species are `self-localizeda, in the sense that the presence of electronic charge leads to local changes in the atomic geometry (the lattice), which, in turn, leads to localized

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Fig. 2. Energy level scheme of the self-localized states in conjugated polymers: (A) Band edges of a neutral polymer, where both physics and chemistry terminologies are used to label the band edges; (B) polaron state formed upon the addition of an extra electron; and (C) spinless bipolaron state formed upon the addition of a second electron, which also corresponds to the combination of two polarons.

changes in the electronic structure. These species are generated, for example, through optical absorption in the neutral system, or through charge transfer doping. Associated with these species are localized electronic states with energy levels within the otherwise forbidden electron energy gap, E , the so-called `gap statesa.  Note, in Fig. 1, that for polymer II, trans-polyacetylene, an interchange of the carbon}carbon single (!) and double (") bonds reproduces the identical ground-state geometry. Thus, transpolyacetylene is termed a `degenerate ground-statea system. This geometric symmetry has consequences for the nature of the self-localized charge bearing species in trans-polyacetylene, which are termed solitons, after the mathematics and wave equations which describe their behavior [3,28,33]. In the remainder of the conjugated polymers shown in the "gure, III, IV, and V, a simple interchange of the carbon}carbon single and double bonds does not reproduce the same groundstate geometric con"guration, but produces higher-energy geometric con"gurations [27,29,34]. Thus, these polymer systems are termed `non-degenerate ground-statea systems. This symmetry also has consequences for the type of charge bearing species (and the type of optical excitations) which occur in these polymer chains. Excess electrons added to any conjugated polymer chain lead to new electronic states within the otherwise forbidden electron energy gap, `gap statesa, as sketched for a non-degenerate groundstate system in Fig. 2. In principle, the very "rst electrons (extremely dilute concentrations) added to any conjugated polymer chain form singly charged polarons which have an esr signal [3,27,28,34}36]. In chemical terminology, a polaron is a radical-ion in association with a local geometry relaxation. Polarons are also self-localized states, as diagrammed in Fig. 2. At higher concentrations, the mobile polarons may pair up and form entities called either solitons or bipolarons. In the case of trans-polyacetylene, it has been established that pairing up of polarons leads to spinless, singly charged solitons that represent the lowest energy eigenstates of the coupled electron (hole)}lattice systems, and are responsible for the unusual electrical, magnetic and linear as well as non-linear optical properties. On the other hand, in non-degenerate ground-state conjugated polymers, the polarons can pair up to form spinless, doubly charged bipolarons. A bipolaron is thus a di-ion around which occurs a strong, localized lattice relaxation.

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3. Electronic structure of linear conjugated polymers First, the electronic (band) structure of two linear polymers is described in terms of energy band theory in solid-state physics. A wide variety of literature exists discussing energy bands in polymeric materials from di!erent points of view [26,37]. The essential feature involves extending the molecular orbital eigenvalue problem to systems with one-dimensional periodic boundary conditions, i.e., over regularly repeating monomeric units. The translational symmetry of the polymer chain implies that the solutions of the SchroK dinger equation must be of the Bloch form, W(k, r)"u (r)exp(ikx), where u (r) has the period of the lattice (the unit repeating along only I I one-dimension, here, x), and k is the crystal momentum along the direction x. Periodicity in reciprocal space implies that W(k, r)"W(k#K, r), where K is the reciprocal lattice vector. The "rst Brillouin zone is de"ned by the region between !n/a4k4n/a, where a is the magnitude of the real space lattice vector. The "rst Bragg re#ections and the "rst forbidden electron energy gap occur at the "rst Brillouin zone. The electronic energy band structure, o(E), of an arbitrary one-dimensional system will contain many overlapping bands, usually of di!erent symmetry, and spread out over di!erent binding energies. The N electrons of the system will occupy the lowest (deepest binding energy) bands. For comparison with ultra-violet photoelectron spectra of polymers which lack any crystal symmetry, i.e., disordered polymers, the density-of-valence states, or DOVS, are computed from the band structure in the usual way, o(E)J(RE/Rk)\. Trans-polyacetylene, often denoted PA or (CH) , is the stable stereoisomer of polyacetylene at V room temperature, and is illustrated in Fig. 1. Trans-polyacetylene is the geometrically simplest of the conjugated polymers. Since the carbon atoms are sp-hybridized, the network (chain) of overlapping p atomic orbitals, with the periodic boundary conditions imposed by the unit cell, X results in a dispersed n-band. For a given inter-nuclear distance, the n-overlap of two parallel p -orbitals, S , is normally less than that of p-overlap of two 2p -atomic orbitals, S , and of two X L V NV 2s-atomic orbitals, S , that is, S (S (S . The overall width of the n-band is then less than that NQ L NV NQ of the p-bands. As a result, the binding energy of the occupied n-band is lower than that of the occupied p-bands, while the destabilization of the unoccupied n-band is lower than that of the unoccupied p-bands, i.e., the n-bands form the frontier electronic stucture [37]. Because the ground-state geometical structure of trans-polyacetylene is dimerized [3,26,28,35], there are alternating single and double bonds, and the translational unit cell consists of two CH-units. The n-overlap of two parallel p -orbitals, S , is larger for the slightly shorter `doublea bonds (&1.36 As ) X L and is smaller for the slightly `longera single bonds (1.44 As ), but is "nite for both bonding cases. A lucid discussion may be found in a book by Ho!man [26]. The energy band structure of trans-polyacetylene, calculated using the Valence E!ective Hamiltonian quantum chemical method [29,37], is compared with the measured UPS spectrum [38] in Fig. 3. The band structure in the "gure is rotated by 903, relative to the typical orientation of presentation, to enable a more direct comparison with the experimental spectrum. To facilitate the comparison between the UPS data and the calculated band structure, the density-of-valence-states (the DOVS), or o(E), is included. The highest occupied energy band is the n-band, which can be seen clearly as the band edge in the UPS spectra. This comparison of UPS spectra using VEH results indicates the level of information which can be obtained from this type of electronic structure study [39]. The position in energy of the n-band edge, and changes in the electronic structure in the region of the edge, are of central importance in studies of conjugated polymer

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Fig. 3. The electronic band structure of trans-polyacetylene, PA, based on the VEH model (bottom), is compared with the DOVs computed from the VEH band structure, and with the experimental DOVs as obtained from UPS (top) [38].

surfaces and interfaces. Clearly, the UPS spectra, sensitive to only the top most molecules of the "lm, are appropriate in studies of the deposition of metal atoms on polymer surfaces, as models of the early stages of formation of the polymer-on-metal interface and of the metal electrode [39]. Poly(para-phenylenevinylene), or PPV, is a conjugated polymer which has been intensively used in the development of light-emitting devices [21]. The chemically substituted derivatives of PPV are soluble in common organic solvents, and are of importance because they are processable using standard techniques, such as spin coating [21]. Unsubstituted PPV, on the other hand, is not soluble in common solvents, nor is it processable into thin "lms by any other simple means [40]. The chemical structure of PPV (for R"hydrogen) is shown in Fig. 1. In Fig. 4 are illustrated the most recent experimental He I and He II UPS valence band spectra of PPV [41] compared with the DOVS caluclated from the VEH band structure [38]. The density of valence states, shown in direct comparison with the UPS spectra, is obtained from the band strucutre, which is shown at the bottom of the "gure. The frontier electronic structure is at the low binding energy region, to the right of the "gure. The peaks at higher binding energies, peaks E, F and D, originate from electrons in di!erent p-bands, while peak C is composed of contributions from the four highest p-bands, the lowest n-band, and a small portion from the relatively #at part of

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Fig. 4. The electronic structure of poly(p-phenylenevinylene), or PPV, based on the VEH model (bottom) is compared with the DOVs computed from the VEH band structure, and with the experimental DOVs as obtained from UPS (top) [41].

the second n-band. The two peaks at lowest binding energies, peaks B and A, are derived from the three topmost n-bands. Peak B is dominated by the next highest n-band, which is extremely #at, since it corresponds to electronic levels fully localized on the bonds between ortho-carbon atoms within the phenylene rings. In general, a #at (dispersionless) band results in a high-intensity peak in the DOVS, since it corresponds to a large number of discrete states per unit energy. The larger disperion of the top n-band results in lower intensity in the UPS data.

4. Polarons and bipolarons in non-degenerate ground-state conjugated polymers 4.1. Charged species Electrons can be added to polymer chains in several ways. One common situation is the case of charge transfer doping from, for example, metal atoms with electrons in low binding energy states, as for the alkali metals. Depending upon the nature of the polymer chain, di!erent species are formed. Total energy estimates indicate that two extra electrons may go either into two independent singly charged polarons or into one doubly charged, spinless bipolaron [30,42,43]. The balance between the two situations is very subtle. Therefore, speci"c extrinsic factors can determine which will be the lower-energy con"guration: two independent (spatially separated) singly charged polarons, or one spinless, doubly charged bipolaron. When poly (p-phenylenevinylene) is doped by

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Fig. 5. The UPS spectra of the low binding energy portion of the UPS spectrum of PPV during charge transfer from rubidium atoms are shown [45].

charge transfer from alkali metal atoms, the presence of the counter-ions is enough to lead to bipolaron formation. Two spectroscopic studies are described below. The doping of PPV by the physical vapor deposition of sodium atoms in UHV was reported by Fahlman and co-workers [44]. The sodium atoms donate electrons to the PPV system, which combine in pairs to form bipolarons, leading to new electronic states in the otherwise forbidden energy gap at high doping levels. Of course, the very "rst electron added to the non-degenerate ground-state polymer chain must go into a polaron state (when there are no other electrons in the vicinity available for pairing). At higher doping levels, the addition of one more electron results in the combination of polarons into bipolarons, if in the given system, the bipolaron state is energetically favorable over two independent polarons. For sodium-doping of PPV, only high doping levels were studied. Bipolaron states, as diagrammed in Figs. 2 and 6, are observed in the UPS spectra. A more systematic study of doping of PPV via alkali atoms was carried out using rubidium. The polymer was doped in small stages such that the polaron to bipolaron transition could be followed using UPS [45]. The low-energy electronic structure seen in the UPS changes in a special way, as shown in Fig. 5. Starting at the bottom, i.e., the spectrum for the undoped polymer, the strong feature, A, corresponds to electrons emitted from the dispersionless #at band, which is localized on the phenylene groups [38,46]. The n-band edge is seen clearly as feature B. The curve next to the bottom corresponds to a doping level so low that only polarons are formed. The average distance between electrons, donated to the polymer chain from widely dispersed (in space) rubidium atoms, is large enough that combination (pairing) to bipolaron states does not readily occur. The Fermi energy, E , indicated by the bold vertical bars, has moved from about 4.3 eV (relative to the vacuum $ level) in the undoped case (lowest curve) to about 3.3 eV. Note that there is a "nite density of states

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Fig. 6. An idealized diagram of the interaction of two polarons to form a bipolaron.

Fig. 7. The relationship of the energy levels in bipolaron states in PPV to the self-localized geometrical structure. Note the quinoid structure of the central rings in the doubly charged chain (sketched in the middle).

`ata the Fermi level (which appears to spill over to the low binding energy side of E . This $ apparently unphysical e!ect has to do with the nature of the spectroscopy [39,47,48]). In the case of negatively charged, electron}polaron states, the upper band is only half "lled; because the Fermi level should lie in the middle of the half-"lled band, a "nite density of states is expected `ata E , as $ observed. As the level of doping with rubidium atoms is increased, the polarons combine to form bipolarons, as diagrammed in Fig. 6. The Fermi level moves to about 2.4 eV, and two well-de"ned spectral features evolve in the energy region of the original forbidden energy gap (the binding energy region from approximately 5.4}2.3 eV in the lowest curve in Fig. 5), as estimated from optical data [43]. Since the original HOMO moves into the energy gap to become the lower (higher binding energy) polaron state, only the remaining portion of the n-band edge should be observed. A hint of the remainder is seen as a weak indication of a peak to the left (to higher binding energies) of peak C in the "gure. Note also that the apparent resolution of the spectra is determined by the nature of the photo-ionization process in molecular solids [47], and not by energy resolution of the equipment [39].

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The electronic structure of bipolarons was modeled in the early 1980s [27,34,36]. More recent work is outlined here [43,49]. An e$cient computational approach is to have the geometry of the ionic state of the polymer chain calculated by the Austin Model One, or AM1, method [50], which provides good estimates of ground-state molecular geometries, ionic-state geometries, as well as other properties. AM1 modeling of molecular ionic states includes e!ects of electron}electron interactions. The geometric structures in the vicinity of the added charges are as indicated schematically in Fig. 7. The electronic band structure and the corresponding DOVS, are then calculated using the Valence E!ective Hamiltonian pseudopotential method, with the AM1 geometrical parameters as input. The electronic structure of the in-gap states associated with singly charged polymer chains, polarons, and doubly charged chains, bipolarons, are reproduced in excellent agreement with the experimental UPS spectra [45]. The in-gap states observed in the UPS spectra are derived from the quinoid-like geometric structure appearing near the charges. Similar results are obtained for the doping of PPV from sodium atoms. In analysis of the UPS results, the VEH-level calculations, with AM1-level geometry optimization of the ionic state, appear to be su$cient to account for the experimentally observed electronic structure. Since, in order to obtain agreement with experiment, it is necessary to (1) include electron}electron interactions in the geometry optimization; and (2) to use the ionic, relaxed geometry in order to obtain the positions of the new (bipolaron) energy levels in the energy gap, it is clear that both electron}electron and electron}lattice interactions are involved in the bipolaron formation process. The precise energy splitting of the bipolaron states in the otherwise forbidden energy gap is in#uenced by details of electronic structure of the polymer. In the case of non-conjugated substituents attached to the phenylene-groups to render the polymer soluble in common organic solvents, torsion angle e!ects as well as e!ects of bonding to the side groups, typically alkyl chains and alkoxy groups, are observed [46]. In addition, CN-groups attached to the vinylene moieties a!ect the electronic structure such that the bipolaron entities are more localized and the energy splitting of the in-gap states adjusts accordingly [51]. Further insight into the nature of self-localized bipolaron states can be obtained by considering, in a simpli"ed phenomenological model, the changes in the pattern of alternating single- and double-bonds which occur with the addition of two charges. In Fig. 7 is illustrated the idealized bonding pattern in the neighborhood of two added electrons on a segment of a chain of PPV. In a somewhat more realistic picture, the changes in bond alternation pattern would occur slowly as a function of distance along the bipolaron, with a half-width of several phenylene-groups. In moving from left to right in over a bipolaron (in Fig. 7), a variation would be seen from completely aromatic-like on the left to a more-or-less quinoid-like structure in the center of the bipolaron, and then back to aromatic-like on the right. In the simple description of Fig. 7, the electrons are shown as point charges, at the two idealized extremes of the geometrically altered portion of the chain. Coulomb repulsion tends to separate the two like-charges. However, in moving the isolated charges farther away from one another (thus extending the bipolaron domain), the bonding pattern must change from aromatic-like to quinoid-like, in order to keep four bonds at each carbon atom all along the chain. The quinoid structure is a higher energy con"guration, costing elastic energy to generate from the aromatic structure. Thus, separation of the two excess electrons costs mechanical (strain) energy and saves in Coulomb repulsion energy, until a balance is achieved, determining the size of the bipolaron, a FWHM of about 3}4 rings [27,28].

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Fig. 8. A conventional diagram of an electron polaron and a negative bipolaron, with possible optical transitions shown. Note that (in oligomers) transitions P3, P4 and BP2 are symmetry forbidden, since the wavefunctions involved are of the same symmetry [42].

The introduction of charges to the polymer chain through chemical doping brings with it the potential due to the counterions, which maintains overall charge neutrality after doping. The role of the counterions in the energetics and localization of polarons or bipolarons on the polymer chain is not well established, though it is certainly expected to play an important role. This role is of particular importance in the stabilization of doubly charged excited states (bipolarons), since it can screen out the repulsive Coulomb interaction between the two electronic charges on the polymer chain. There has therefore been interest in creating charges on the polymer chain without chemical doping. Two methods have been studied in some detail: photoexcitation and charge-injection in semiconductor device structures. Photoexcitation can produce separated charges, via a number of mechanisms, including direct charge photogeneration, exciton ionization at electron traps, exciton}exciton collisions, among others [52,53]. This creates charge-separated states, which in time will recombine. The presence of these charged excitations is studied through optical absorption spectroscopy, both at short time scales [53,54], and also at much longer time scales of milliseconds [55]. The view now generally held is that the optical absorption of the charge carriers at short times is characteristic of the `intrinsica charges on the chain, but that the long-time response is due to charges which have become immobilized or trapped at defects in the polymer, such as dopant ions. In general, strong `polaronica absorption bands are seen. For example, in PPV, an induced absorption is observed at short times due to the photogeneration of charges. This induced absorption has a maximum near 1.6 eV, similar in energy to the dopant-induced absorption seen in this polymer [55]. Charge injection in device structures provides a more recent method for introducing charge onto the polymer chains. The most convenient device structure is the "eld-e!ect transistor or diode, which has been used recently for studies of poly(3-hexylthiophene) and its model oligomers [56,57].

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Di!erential optical absorption carried out as a function of bias voltage provides a sensitive probe for the change in optical response with charge density, and a wide range of sub-gap absorptions have been measured. The role of disorder in the possible stabilization of the injected charges is probably important. For example, the studies reported in [57] on the hexamer of thiophene, a-sexithienyl, reveal a wide range of optical transitions, which were assigned to at least three distinct charged states. The optical data show clearly transitions involving polarons and bipolaron levels, as indicated schematically in Fig. 8. In addition, since inter-molecular interactions can be strong for this material, interchain excitations can be found. These are termed n-dimers and are formed when two polarons on adjacent chains couple together, the energy levels showing splitting. More recent work, on well-ordered poly(3-hexyl thiophene) [56], shows the presence of only one type of charge; this is considered to be a singly charged polaron, and the absence of other charged species is taken to indicate that disorder is much less signi"cant for this material than for most other conjugated polymer samples. Note that thin "lms oligomers of polythiophene do form highly ordered structures, when prepared under the proper conditions [18]. Charges can also be injected into light-emitting diode structures, and observations of the transient optical absorption of excited-states have been reported [31]. For the case of PPV [31], the same absorption band near 1.6 eV, that is seen for photoexcited charges at short times, is found. 4.2. Optical excitations Optical absorption corresponds to di!erences in energy states, and thus is an indirect measure of the electronic structure. It has been known for many years [58}60] that strong electron}phonon (or exciton}phonon) coupling can cause the exciton to become self-trapped. Thus, both electron}electron correlation e!ects and strong electron}lattice interactions can also be important here. In trans-polyacetylene, electron correlation e!ects lower the energy of the lowest (two-photon optically allowed) singlet excited state with A symmetry, so that it falls below the one-photon  optically allowed 1Bu excited state [61,62]. An important consequence is that polyacetylene does not belong to the class of luminescent conjugated polymers [63]. The photoluminescence spectrum of poly(p-phenylenevinylene) is red shifted relative to the optical absorption spectrum [40,64,65], as can be seen in Fig. 9. It is presently generally accepted that this di!erence in photon energies is related to the optical generation of geometrically relaxed self-localized excited states [40,43] (although alternative proposals, have been put forth [66,67]), and that the photoluminescence emission comes from the radiative decay of weakly bound polaron}excitons with a binding energy of a few tenths of an electron volt [43,55,68}70]. Here, it should be mentioned that the term `exciton binding energya has often been used in the literature on conjugated polymers as arising from the (electron}electron) Coulomb interactions, while possible contributions form electron}lattice coupling were not considered [68,69]. A typical manifestation of electron}lattice coupling in conjugated polymers, however, is the appearance of vibronic progressions in the optical absorption spectra. For rigid, fully delocalized bands, vibronic e!ects would be on the order of 1/N, where N is the number of atoms in the structure (the larger the number of electrons, the smaller the expected in#uence of an electron excitation on the geometrical structure) [43]. Because of the self-localization of the excitations, vibronic features actually are observed even for macromolecules. Also, since the n-electrons are highly delocalized and polarizable, signi"cant electron correlation e!ects occur, as the n-electrons redistribute in the presence of

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Fig. 9. A "gure of optical absorption spectrum and photo-luminescence in PPV. This data may be found in a variety of publications, only a few of which are referenced here [40,64,65].

Fig. 10. An idealized representation of the interaction of a hole polaron, P>(h), and an electron}polaron, P\(e), to form a polaron}exciton, P}E.

additional charges. In the formation of an electron}hole pair from an optical absorption event, there are two sources of binding, one related to electron}electron interaction, and one related to electron}lattice coupling. Hence, a bound electron}hole pair is referred to as a polaron}exciton. The combination of an electron (electron}polaron) and a hole (hole}polaron) to form a polaron}exciton is shown schematically in Fig. 10. The vibrational structure and the implications for electron}lattice coupling have been analysed in detail [43]. Relaxation energies on the order of 0.2}0.3 eV have been deduced from spectra of PPV, as in Fig. 9, and of oligomers of PPV [71]. Similar values of the relaxation energy are obtained directly from quantum chemical semiempirical geometry optimizations of the 1Bu excited state of long oligomers of poly(pphenylenevinylene)s [72]. Note that the discussions above are relative to excitations in the absence of counter ions which are present when these materials are doped. Optical excitations in doped polymers or oligomers are not discussed here. The coupling to vibrations, which in#uences the optical absorption, also in#uences the photoluminescence emission process. The red shift in the photoluminescence spectrum of PPV (relative to the optical absorption spectrum) corresponds to a Stokes' shift. By a Stokes' shift is meant the di!erence in energy between the 0}0 vibrational peak in the optical absorption relative to the

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0}0 peak in the photoluminescence spectrum. When a vibrational progression is observed in the optical absorption, the 0}0 transition is to the relaxed excited state [43]. There are several possible relaxation processes (such as ring rotations, in addition to the polaron e!ects sketched in Figs. 2 and 6) as well as disorder e!ects, which can be involved in the Stokes' shift; these complicate data analysis, but will not be discussed here [49]. In studies of the Stokes' shift in the 2-methoxy-5-[(2ethyl-hexyl)oxy]-derivative of PPV (MEH-PPV, a special case of III in Fig. 1), the shift is measured to be about 0.2}0.3 eV in typical samples [73]. On the other hand, when MEH-PPV chains are more aligned and oriented within an ultrahigh molecular-weight polyethylene matrix, the Stokes' shift decreases to less than 0.1 eV, while at the same time the 0}0 vibrational peak in the optical absorption spectrum is red shifted from 2.25 to 2.13 eV [73,74]. Since a bound electron}hole pair is stabilized with respect to two fully separated polarons [70], the polaron}exciton binding energy may be de"ned as the di!erence between the creation energy of two fully separated, geometrically relaxed charge carriers of opposite sign, i.e., one positive polaron and one negative polaron, and the energy of a neutral polaron}exciton, including electron}electron and electron}lattice interactions [43]. Numerous studies have been carried out, which, in comparison with optical absorption spectra, con"rm the values of the binding energy. In this context, one can address the estimates of the forbidden energy gap in conjugated polymers obtained from optical absorption spectroscopy and ultraviolet photoelectron spectroscopy. For example, in PPV, the 0}0 transition in the optical absorption spectrum is at 2.45 eV. Since the valence band edge in UPS is about 1.5 eV below the Fermi energy [46], then if the Fermi energy lies in the middle of the n}nH gap, E , the gap may be estimated to be two times 1.5 eV, or about 3.0 eV. UPS data do not  include either lattice relaxation e!ects or interactions between the departed electron and the remaining hole [48]. Thus, the di!erence between the E estimated from UPS data and the 0}0  transition in the optical absorption should be about twice the polaron binding energy, yielding for the latter a value of the order of 0.3 eV, within the error bar range of the numbers cited above (although there are additional possible uncertainties in this simple analysis which will not be discussed here). The relative contributions to polaron}exciton binding energy from electron correlation e!ects and electron}lattice coupling can be addressed from the vibronic analysis of the experimental n}nH optical absorption data for PPV (both the polymer and model oligomers) [71] and from direct calculations of the excited-state geometry [72]. Since in the case of polarons, there is no simple experimental measurement of the lattice relaxation energy, the results of theoretical modeling of the geometrical relaxation in the ionic excited state must be used, using the same approach as for the neutral excited-state calculations [72]. Single polaron relaxation energies of about 0.15 eV are obtained for long, coplanar oligomers of PPV, where the lattice modi"cations also lead to semi-quinoid-like structures [43]. Comparing the relaxation energies for two polarons with the relaxation energy for the neutral polaron}exciton, indicates that in PPV at least, there is very little lattice contribution to the polaron}exciton binding energy, even, perhaps, a slightly negative (repulsive) contribution. The absence of a positive lattice contribution to the polaron}exciton binding energy would not be expected at the one-electron level, where a polaron}exciton becomes equivalent to a (doubly charged) bipolaron. When electron}electron interactions are taken into account, however, the geometry of a polaron}exciton (which can be thought of as a neutral bipolaron) is di!erent from that of a doubly charged bipolaron [43,72]. Note that both the singlet and triplet polaron}excitons

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are stabilized with respect to the free electron}hole pair. Since the singlet polaron}exciton binding energy turns out to be about a few tenths of an electron volt [43,70,75], an important conclusion is that this small polaron exciton binding energy can be thought of as arising from a (near) cancellation of the electron}electron and electron}lattice contributions, which occurs indirectly via the e!ect of correlation on the excited-state geometry.

5. Model molecular systems: conjugated oligomers One approach to studying conjugated polymers is to focus on the properties of more well de"ned oligomers, thus avoiding problems such as defects in ideal polymer chains or disorder. Model oligomer systems consist of molecules which are all the same, of de"nite structure, and which can be handled in ways not applicable to polymers [14]. In addition, it is often possible to carry out quantum chemical calculations on "nite-size molecules using more sophisticated computational approaches, whereas calculations of long (periodic or disordered) structures are more di$cult [49]. The redox states of oligomers of polythiophene, V in Fig. 1, ranging from 6 to 9 to 12 rings per molecule, have been studied in a combined experimental}theoretical work [76]. Each of the molecules carried one dodecyl side chain per three thiophene units to ensure solubility in common organic solvents. It was found that for molecules with 6 or 9 rings, single charges are stored in the form of polarons, while for two charges, bipolarons are favored. On the other hand, for oligomer molecules with 12 rings, two charges are stored in two independent polaron states rather than a single bipolaron. The rationalization of this behavior is that on short molecules, the polaron wavefunctions are forced to overlap because of spatial con"nement. On the longer molecule, the polarons are free to move apart su$ciently that the wavefunctions do not overlap. The implication of this work is again to point out the subtle balance between polarons and bipolarons on su$ciently long polymer chains. In studies of oligomers of poly(p-phenylene), IV in Fig. 1, ranging from two rings per molecule to 12 rings per molecule, results similar to the results on the oligomer molecules of polythiophene are obtained [49]. For short molecules, the stable con"guration for the storage of two charges is the bipolaron. For longer molecules, in particular with 12 rings, the stable con"guration for the storage of two charges is two independent polarons. It should be mentioned here that early studies of chain}chain interactions were carried out on oligomers representing segments of trans-polyacetylene [77], using the SSH Hamiltonian [3,28]. The studies focused on localization e!ects, and did not include electron}lattice or electron}electron interactions. The results of the studies did show, however, the importance of molecule}molecule (that is, chain}chain) interactions, which may indeed change the nature of the optically excited species which occur in conjugated polymers. There are implications from a combination of the results on polymers compared with the results on oligomer molecules. The discussions above are based on ideas applicable to ideal polymer chains. Often the experimental situation is more complicated. First, polymer chains in real samples are not the uniform, completely periodic, perfect linear chains, of the type represented in Fig. 1, which are often assumed in the interpretation of optical data. In reality, complications such as molecular weight distribution, conjugation lengths, polymerization defects such as branching and

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cross-linking, and even order, up to a certain degree, must be considered in extracting physical electronic structural information from real samples [7,78,79]. Thus the stability of bipolarons in certain non-degenerate ground-state conjugated polymers may indeed be because the real samples are composed of chains where the ideal segments are less than that necessary for formation of independent polarons. The e!ects of spatial con"nement in determining the nature of the charge storage species occur in other, similar systems. Charges stored on linear polyenes ("nite linear molecules which are oligomers of polyacetylene) should lead to the formation of solitons [28]. Chemically stable polyenes have phenyl-groups as end groups. In studies of charge transfer doping of a diphenylpolyene with seven C"C double bonds in the polyene portion of the molecule, a,u-diphenyltetradecaheptane, it was observed that bipolaron (like) states were formed [80]. The polarons are con"ned to the "nite molecule, and forced to interact by virtue of the con"nement, which results in the formation of bipolaron-like entities on the polyene portion of the molecule. Two bipolaron-like states (rather than one soliton state) are seen clearly in the UPS spectra. On the other hand, in some cases, where longer oligomer molecules are studied in solution, it may not always be clear that the molecules are extended and are really `rigid rodsa; the molecules can be bent or folded, thereby exhibiting actual persistence lengths which may be shorter than what is expected for straight molecules [7]. Such e!ects also may be important in oligomeric molecular solids, where the extent of crystalline regions may determine to a great extent the properties observed. One has to make sure that e!ects studied are not functions of structural imperfections. This latter point has apparently plagued this "eld for some time, especially in the previous decade, warranting such a precautionary note here.

6. Summary Conjugated polymers are soft, essentially one-dimensional molecular systems. Consequently, the addition of an extra electron (or hole) leads to self-localization e!ects (relaxation e!ects) which result in new electronic states with energy levels within the otherwise forbidden electron energy gap. In addition, optical absorption leads to the generation of corresponding self-localized species. The addition of the "rst charge to a polymer chain leads to the formation of a polaron. The addition of a second charge generally leads to spinless charge bearing species, either solitons or bipolarons, depending upon the symmetry of the ground state geometry of the polymer chains. Both electron}lattice and electron}electron interactions are important in determining the nature of the self-localized charged or optically excited species. The stability of bipolarons over two independent polarons in non-degenerate ground-state systems depends upon extrinsic e!ects, such as disorder or the presence of counterions when charges are added by charge transfer doping. In studies of non-degenerate ground state conjugated oligomers of varying length, in the absence of extrinsic in#uences, bipolarons may be unstable to the formation of two independent polarons. As materials improve it will likely become clear that use of oligomers will increase, not only as test structures for studies of the formation of species such as polarons and bipolarons, but also in parallel with polymers directly for electronic applications. In the future, most likely both polymers and oligomeric molecules will each play their own important roles in organic-based electronics applications.

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Acknowledgements The authors are indebted to J. Cornil, Mons, for helpful discussions, and F. Cacialli, N. Greenham and R. Gymer for help with the "gures. The Cambridge-Mons-LinkoK ping collaboration is supported by the European Commission within a Training and Mobility of Researchers (TMR) Network (SELOA, project number 1354) and within a Brite/EuRam project (OSCA, project number 4438). The work in Mons is partly supported by the Belgian Federal Government `InterUniversity Attraction Pole on Supramolecular Chemistry and Catalysis (PAI 4/11)a, and FNRS-FRFC. Research on condensed molecular solids and polymers in LinkoK ping is supported in general by grants from the Swedish Natural Science Research Council (NFR), the Swedish Research Council for Engineering Sciences (TFR), and the Carl Tryggers Foundation.

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253

CONTENTS VOLUME 319 M. Volkov, D.V. Gal'tsov. Gravitating non-Abelian solitons and black holes with Yang}Mills "elds L. Satpathy, V.S. Uma Maheswari, R.C. Nayak. Finite nuclei to nuclear matter: a leptodermous approach U.A. Wiedemann, U. Heinz. Particle interferometry for relativistic heavy ion collisions W.R. Salaneck, R.H. Friend, J.L. BreH das. Electronic structure: consequences of electron}lattice coupling

Nos. 1}2, p. 1

No. 3, p. 85

Nos. 4}5, p. 145

No. 6, p. 231

CONTENTS VOLUMES 311}318 I.V. Ostrovskii, O.A. Korotchcenko, T. Goto, H.G. Grimmeiss. Sonoluminescence and acoustically driven optical phenomena in solids and solid}gas interfaces R. Singh, B.M. Deb. Developments in excited-state density functional theory D. Prialnik, O. Regev (editors). Processes in astrophysical #uids. Conference held at Technion } Israel Institute of Technology, Haifa, January 1998, on the occasion of the 60th birthday of Giora Shaviv S.J. Sanders, A. Szanto de Toledo, C. Beck. Binary decay of light nuclear systems B. Wolle. Tokamak plasma diagnostics based on measured neutron signals F. Gel'mukhanov, H. Agren. Resonant X-ray Raman scattering J. Fineberg. M. Marder. Instability in dynamic fracture Y. Hatano. Interactions of vacuum ultraviolet photons with molecules. Formation and dissociation dynamics of molecular superexcited states J.J. Ladik. Polymers as solids: a quantum mechanical treatment D. Sornette. Earthquakes: from chemical alteration to mechanical rupture S. Schael. B physics at the Z-resonance D.H. Lyth, A. Riotto. Particle physics models of in#ation and the cosmological density perturbation R. Lai, A.J. Sievers. Nonlinear nanoscale localization of magnetic excitations in atomic lattices A.J. Majda, P.R. Kramer. Simpli"ed models for turbulent di!usion: theory, numerical modelling, and physical phenomena T. Piran. Gamma-ray bursts and the "reball model E.H. Lieb, J. Yngvason. Erratum. The physics and mathematics of the second law of thermodynamics (Physics Reports 310 (1999) 1}96) G. Zwicknagel, C. Toep!er, P.-G. Reinhard. Erratum. Stopping of heavy ions at strong coupling (Physics Reports 309 (1999) 117}208) F. Cooper, G.B. West (editors). Looking forward: frontiers in theoretical science. Symposium to honor the memory of Richard Slansky. Los Alamos NM, 20}21 May 1998 D. Bailin, A. Love. Orbifold compacti"cation of string theory W. Nakel, C.T. Whelan. Relativistic (e, 2e) processes D. Youm. Black holes and solitons in strong theory J. Main. Use of harmonic inversion techniques in semiclassical quantization and analysis of quantum spectra N. KonjevicH . Plasma broadening and shifting of non-hydrogenic spectral lines: present status and applications M. Beneke. Renormalons A. Leike. The phenomenology of extra neutral gauge bosons R. Balian, H. Flocard, M. VeH neH roni. Variational extensions of BCS theory F.T. Arecchi, S. Boccaletti, P. Ramazza. Pattern formation and competition in nonlinear optics F.A. Escobedo, J.J. de Pablo. Molecular simulation of polymeric networks and gels: phase behavior and swelling N.A. Obers, B. Pioline. U-duality and M-theory F.C. Michel, H. Li. Electrodynamics of neutron stars

311, No. 1, p. 1 311, No. 2, p. 47 311, Nos. 3}5, p. 95 311, No. 6, p. 487 312, Nos. 1}2, p. 1 312, Nos. 3}6, p. 87 313, Nos. 1}2, p. 1 313, No. 3, p. 109 313, No. 4, p. 171 313, No. 5, p. 237 313, No. 6, p. 293 314, Nos. 1}2, p. 1 314, No. 3, p. 147 314, Nos. 4}5, p. 237 314, No. 6, p. 575 314, No. 6, p. 669 314, No. 6, p. 671 315, Nos. 1}3, p. 1 315, Nos. 4}5, p. 285 315, No. 6, p. 409 316, Nos. 1}3, p. 1 316, Nos. 4}5, p. 233 316, No. 6, p. 339 317, Nos. 1}2, p. 1 317, Nos. 3}4, p. 143 317, Nos. 5}6, p. 251 318, Nos. 1}2, p. 1 318, No. 3, p. 85 318, Nos. 4}5, p. 113 318, No. 6, p. 227