Physics at the Japan Hadron Facility: Proceedings of the Workshop Adelaide, Australia, 14-21 March 2002

respectively. As an upper bound on the NSI, we assume for the coupling strengths that GSLSL < O.QlGLL. The neutrino energy distribution, assuming that the electron mass can be neglected, is d2N„

2 z

'TLLTTLL,

Tll—UpUtf{3-2z),

dzd£lv

(22)

where T^ is the muon decay rate, z = LJ/E^ and E^ is the muon beam energy. The tau production cross section is

•Mb =

(23)

f"J«£&

The differential cross section expressed in terms of the Bjorken scaling variables x' and y1 has the general form d2aT dx'dy'

_ \rLL\lTjLLTjLL^ - H* I "ri urj

+ GLLGS^

a dx,dyl

LL

\(~iSLSLi2TTSLSLTrSLSL I" 1 ^ I "ri "rj

{UtfU%s<- + UTfU?^)

d

^^r

^

a SLSL fa,dyl

.

The dependence on i and j has been made explicit, the standard term

(24) dxf

j;h ,

the interference term d J;ffi , and the scalar term ^dy'L m a y 'De calculated using a parton model. Note that the interference term vanishes in the limit where the parton and tau masses are zero. The sensitivity of a future neutrino factory to NSI is investigated by defining a x 2 function which determines the required detector mass and number of

100 useful muon decays in order to claim new physics. The x 2 function for a detector mass and number of muons (N^ • Mdet = 10 2 lkt) is defined as |7VSM — NNSII2 XNM21 = 2 ^ jfsW ' k

(25)

*

where N^M is the expected number of tau producing charged current events in energy bin k in the absence of NSI for this detector mass and number of muons, while A ^ * 7 is the number expected with the NSI present. The required number of detector mass-number of muon units, NMrec, is

NMrec > 4 ^ ,

(26)

X/VM21

where xlo¥ ls ^ e c m s c l u a r e c l value at 90% confidence level, and one detector mass-muon number unit corresponds to 10 21 kt. The proposed muon storage ring at CERN for example would provide ~ 10 21 useful muon decays per year 13 . The number of detector mass-muon number units required to detect NSI at 90%CL versus the non-standard mixing angle is shown in Fig. 1. The values of the parameters used in producing this plot are explained in the caption. The main result is the variation of NMree over the full range of . This simplified analysis indicates that for some values of a detector mass of ~ 30kt would be in a good position to either detect NSI or increase the upper bound on the non-standard coupling strength. Points of particular interest are <j> = 7r/4, 57r/4 and (j> = 3^/4, 77r/4. The first case corresponds to the situation V!*LSL = v^h or XJ^SL = U^L, up to a phase. For this case NMrec is at a maximum. In the second case, V^LS^ = u^L or U^LSL — U^L, up to a phase, these points corresponding to direct flavour violation. In this case NMrec is at a minimum. This result is to be expected, since at this energy and over a medium base line the neutrino beam will comprise mostly of nonoscillated i^'s. Any flavour violating non-standard coupling will in essence be picked out over the standard term. This effect was noted in previous work by Datta et al.14 5. Conclusion In this study we have shown that in order to arrive at a fully normalised experimental event rate resulting from neutrino oscillations with non-standard interactions one needs to utilise a field theoretic description of the process. This formalism has the added advantage of providing an explicit treatment of oscillations of non-chiral neutrinos. The numerical study of a generic neutrino factory has shown the importance of allowing for an arbitrary mixing when looking for NSI. This is exemplified

101 10000 F

1000 r-

Figure 1. is the non-standard mixing angle and NMrec is the required detector massmuon number units defined in the text. The beam energy is taken to be 50 GeV, the mass difference is Sm2 — 2.5 x 1 0 - 3 eV 2 , the base line is \L\ — 732 Km and the standard mixing angle is taken to be maximal, sin(20) = 1.

by the results showing t h a t the neutrino flux and detector mass required to detect new physics is greatest when the non-standard interaction eigenstate is also an eigenstate of the corresponding weak interaction. T h e converse is true for NSI t h a t are not eigenstates of the weak interaction. Any results obtained from precision experiments directly measuring NSI or from oscillation experiments must include the possibility t h a t a weaker force may be present and may not be diagonal in the weak basis. More d r a m a t i c results may be expected from the non-standard interaction of the electron neutrino with a fermion background (a non-standard M S W effect), however we leave this to future work.

References 1. Y. Fukuda et al., Super-Kamiokande Collaboration, Phys. Rev. Lett. 8 1 , 1562 (1998). 2. M. C. Gonzalez-Garcia, P. C. de Holanda, C. Pena-Garay and J. W. Valle, Nucl. Phys. B573, 3 (2000). 3. M. Apollonio et al., CHOOZ Collaboration, Phys. Lett. B466, 415 (1999). 4. Y. Itow et al., hep-ex/0106019. 5. M. Fritschi, E. Holzschuh, W. Kundig, J. W. Petersen, R. E. Pixley and H. Stussi, Phys. Lett. B173, 485 (1986); R.G. Robertson, T.J. Bowles, G.J. Stephenson, D.L. Wark, J. F. Wilkerson and D.A. Knapp, Phys. Rev. Lett. 67, 957 (1991);

102 A.I. Belesev et al, Phys. Lett. B350, 263 (1995); V.M. Lobashev et al., Phys. Lett. B460, 227 (1999); C. Weinheimer et al., Phys. Lett. B300, 210 (1993). 6. A. Rouge, Eur. Phys. J. C18, 491 (2001). W. Fetscher and H. J. Gerber, Eur. Phys. J. C 1 5 , 316 (2000). 7. J.D. Jackson, S.B. Treiman and H.W. Wyld, Jr., Phys. Rev. 106, 517 (1957). 8. C. Weinheimer et al., Phys. Lett. B460, 219 (1999); Talk by V. Aseev et. al., at the International Workshop on Neutrino M asses in the sub-eV Range, Bad Liebenzell, Germany, January 18th-21st, 2001. See also homepage: http://wwwikl.fzk.de/tritium/. 9. B. H. McKellar, M. Garbutt, G. J. Stephenson and T. Goldman, hep-ph/0106122; G. J. Stephenson, T. Goldman and B. H. McKellar, Phys. Rev. D62, 093013 (2000); G. J. Stephenson and T. Goldman, Phys. Lett. B440, 89 (1998); Y. Farzan, O. L. Peres and A. Y. Smirnov, Nucl. Phys. B612, 59 (2001); F. Vissani, Nucl. Phys. Proc. Suppl. 100, 273 (2001). 10. L. M. Johnson and D. W. McKay, Phys. Rev. D 6 1 , 113007 (2000); L. M. Johnson and D. W. McKay, Phys. Lett. B433, 355 (1998); T. Ota, J. Sato and N. a. Yamashita, hep-ph/0112329. 11. M. Beuthe, hep-ph/0109119; C. Y. Cardall, Phys. Rev. D 6 1 , 073006 (2000); W. Grimus, S. Mohanty and P. Stockinger, hep-ph/9909341; W. Grimus and P. Stockinger, Phys. Rev. D54, 3414 (1996); C. Giunti, C. W. Kim, J. A. Lee and U. W. Lee, Phys. Rev. D48, 4310 (1993). 12. C. Y. Cardall and D. J. Chung, Phys. Rev. D60, 073012 (1999). 13. B. . Autin, A. . Blondel and J. R. Ellis, "Prospective study of muon storage rings at CERN," CERN-99-02. 14. A. Datta, R. Gandhi, B. Mukhopadhyaya and P. Mehta, hep-ph/0105137.

N E U T R I N O OSCILLATION SEARCHES AT ACCELERATORS A N D REACTORS

S. N. T O V E Y Research Centre for High Energy Physics, University of Melbourne, Parkville, VIC. 3010, Australia E-mail: S. [email protected]

This talk will review recent searches for neutrino oscillations using neutrinos produced at accelerators and reactors.

1. Introduction The theory of neutrino oscillations will be briefly reviewed. Results will be presented on the following experiments : (1) (2) (3) (4)

NOMAD and CHORUS at CERN, KARMEN at Rutherford and LSND at Los Alamos, K2K in Japan, CHOOZ as a typical reactor experiment.

2. The Theory of Neutrino Oscillations These results are well known and only a few equations will be quoted without derivation. Oscillations can only occur if the neutrino flavour or weak eigenstates differ from the neutrino mass eigenstates. The two representations (ue,i/^,i>T) and (v\,U2,vz) a r e related via a unitary mixing matrix. For simplicity consider a two-neutrino scenario in which we ignore (say) the electron neutrino. Then the mass and flavour eigenstates are related via a simpler (unitary) matrix :

103

104 cosO sinO —sinO cosO

Z) •

(I

»

If one produces a beam of one flavour (say v^) then after a distance L the probability that it has become a vT is : P{yT,L)

= sin2 (26) x sin2 (nL/Losc),

(2)

where the characteristic oscillation length is given by : Losc = (47r^)/Am 2 ,

(3)

and Am2 = m\—m\.

(4)

This equation is in natural units (h = c = 1 ) . If we measure L in km, E in GeV and Am 2 in eV 2 , then : Losc » ( 2 . 5 ^ ) / A r a 2 .

(5)

By conservation of particles, the probability that the v^ remains a v^ is : P(I/„,L)=1.0-P(I/T,I).

(6)

3. The N O M A D and CHORUS Experiments at CERN These experiments were situated in the West Area Neutrino Facility (WANF) at CERN. Neutrinos were produced by allowing a beam of 450 GeV/c protons to strike a Berylium target. The resulting spectrum of neutrinos at NOMAD is shown in Fig. 1. The spectrum at CHORUS is slightly harder as the crosssectional area of that experiment is smaller than NOMAD and neutrinos near the beam axis have slightly higher energies. The methods used by NOMAD and CHORUS are completely complementary, and they are described below. Both seek to observe the appearance of vT via the reaction : vT + N->T~ 3.1. The NOMAD

+ X.

(7)

Experiment

A full description of the NOMAD detector can be found in Altegoer et al.1. NOMAD does not have the spatial resolution to see the track of a T~ which typically travels less that 1 mm. Instead it relies on Kinematic Cuts, Topological Constraints and Likelihood Methods. And it uses the novel (at

105

Neutrino Flux

80

100

120

Neutrino Energy (GeV)

Figure 1.

The energy spectra of the various neutrino flavour states at NOMAD.

least to HEP) techniques of Blind Analyses and Frequentist Statistics. A full description and final results can be found in Astier et al.2. In summary NOMAD sees no evidence for oscillations. The number and shape of the observed events are consistent with background. The number of candidates is 52 events, to be compared to a total predicted background of 51.1 ± 5.4 events. The signal and background are compatible in each decay channel. In the most sensitive region (i.e. that with the smallest background) NOMAD observes 1 event, as compared to a calculated background of 1.62 (+1.89-0.38).

3.2. The CHORUS

Experiment

CHORUS had a target of photographic emulsion with a total mass of 770 kg. The spatial resolution is very good and would allow the tracks produced by

106

a T~ to be seen. Potential r vertices are located via a Fiber target tracker, with an angle resolution of about 2 mrad and a position resolution of 150 mm. A calorimeter and spectrometer are situated downstream of the target and tracker. More details can be found in Eskut et al.3 CHORUS has completed what they term their "Phase I" analysis, and no candidates were observed with negligible backgrounds, see Eskut et al.4 A complete rescanning of the emulsion (Phase II) is underway. This will allow a signicant improvement of their current published limits. 3.3. The CERN

Results

The final NOMAD limit and the Phase I CHORUS limits are summarised in the plot below in Fig. 2.

T

1—i

i ill"

^Tio> u

1 0 '2

_ NOMA©

10

1

T

V —» V

H x 90% C.L. CDIIS .1

10

10

j

i

i i i i i il

10

-J

• • i t •i

10

-1

sin2 28 Figure 2.

The regions of parameter space excluded by the two CERN experiments.

As can be seen from the exclusion plot, at high values of Am 2 the NOMAD

107

experiment produced a limit about an order of magnitude better than previous results. CHORUS (Phase I) is not quite as good but in Phase II it is expected to better the NOMAD limit. It is also clear that the CERN experiments are only sensitive for values of Am 2 > 1 eV 2 . For smaller mass differences significant oscillations would not have had time to develop.

4. Experiments with Stopped Pion Beams Two experiments have searched for oscillations using neutrinos produced by stopping a beam of n mesons in a massive target. The results are controversial. The LSND experiment 5 at Los Alamos in the U.S.A. claims to have seen a signal. The KARMEN experiment 6 at the Rutherford Appleton Laboratory in the U.K. does not see any sign of oscillations A careful comparison (see Fig. 3) shows that the two groups are not quite in conflict. There is a very small region of parameter space where a signal from LSND cannot be excluded by KARMEN.

Figure 3.

Results from the KARMEN and LSND experiments.

However the jury is still out. A new experiment MiniBoone, which will use the 8 GeV booster at Fermilab in the U.S.A., should resolve this issue soon. It has a custom-designed beam stopper, as does KARMEN, whereas the beam stopper at Los Alamos was not optimised for neutrino oscillation searches.

108

5. Long Baseline Experiments The low values of Am 2 suggested by positive results for oscillations using neutrinos produced in the Sun and in the atmosphere have led to a number of proposals to send accelerator-produced neutrinos to distant detectors. Two of these, from Fermilab to the Soudan mine and from CERN to Gran Sasso in Italy, have yet to collect data. Coincidently both have baselines of about 750 km. One experiment in Japan is already producing results, and will be discussed here. 5.1.

K2K

The K2K experiment 7 sends neutrinos produced in a 12 GeV proton accelerator at the KEK laboratory to the giant SuperKamiokande (SuperK) detector in western Japan. The baseline is about 250 km. Unfortunately K2K has ceased operations due to an accident at SuperK in which many photomultipliers were destroyed. The prior results were tantalising. The mean energy of the neutrinos (mostly u^) produced at KEK means that, should they oscillate into i/T, then those vT would be below threshold for producing charged tauons via a charged current reaction. K2K is thus essentially a disappearance experiment in which the "signal" would be too few neutrinos arriving at SuperK. The preferred values of the oscillation parameters to explain the SuperK atmospheric neutrino data are Am 2 = 0.3 x 1 0 - 3 eV2 with maximal mixing. If those values are assumed then the K2K experiment expects to observe a deficit of events at a neutrino energy of about 600 MeV. Preliminary data, see Fig. 4, show just such a deficit. As noted K2K has been halted by the accident at SuperK. Reconstruction of that detector is proceeding at pace, and K2K hopes to resume operations by the end of 2002. 6. Reactor Experiments Nuclear reactors produce copious numbers of neutrinos, in particular the anti-iv In the absence of oscillations the flux should fall off like an inverse square law. A departure from such behaviour would indicate oscillations into another type of neutrino which, at these low energies, would not interact. The two most powerful reactors being used are at CHOOZ in France and at Palo Verde in the U.S.A. Figure 5 shows the CHOOZ detector. Neither experiment sees a departure from the fluxes predicted in a no-oscillation scenario.

109

m

15

Note: Am2=3xMF aV2 corresp onds to 600 MeV Ev

1

10

4 Figure 4. The spectrum of i/ M Charged Current neutrino interactions observed in the SuperKamiokande detector using a ^ b e a m from KEK. See the text for a discussion.

Results from CHOOZ 8 are shown below in Fig. 6. The Palo Verde experiment has very similar limits. 7. Conclusions Positive results (discussed by other speakers at this workshop)from experiments studying neutrinos produced in the Sun and in the Earth's atmosphere do disappear before arriving at a detector. By far the most likely scenario is that they have oscillated into another neutrino species which cannot be detected. These experiments indicate that the differences in the neutrino masses (or more precisely the quantities Am 2 ) are small. Short baseline experiments at high energy accelerators have all yielded negative results, which is now thought to be because they cannot explore the region of small Am 2 . Planned long baseline experiments at CERN and Fermilab should (just) be able to explore that region.

110

CZZZZZTZ. .'*>'

1

.. j 2

~ n x r z r r r i . . ._..; „u „;.:;:•••. :i 3

4

5

6fn

Figure 5. The CHOOZ detector in France

The one positive hint of an oscillation signal has come from a low-energy accelerator produced beam in a medium baseline experiment from KEK to SuperK. The resumption of that experiment is eagerly awaited. Finally it should be noted that n o experiment has seen the appearance of a new species of neutrino in a beam that was originally of a different species. When, or if, observed that will be the decisive evidence that oscillations do occur.

Acknowledgements I would like to thank Dr. Jaap Panman, the spokesman of the CHORUS experiment for providing me with the powerpoint version of a recent CHORUS presentation.

111

— analysis C YA 90% CL Kamiokande (multi-GeV) H 90% CL Kamiokande (sub+multi-GeV) I . . . .

0

i

0.1

i . . , . i .. • .

0.2

Figure 6.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

sin2(28)

Oscillation limits from CHOOZ

References 1. 2. 3. 4. 5. 6. 7.

J. Altegoer et al., NIM A 4 0 4 , 96 (1998). P. Astier et al., Nucl. Phys. B 6 1 1 , 3 (2001). E . E s k u t et a.l, NIM A 4 0 1 , 7 (1997). E . E s k u t et al., Phys. Lett. B 4 9 7 , 8 (2001). C . A t h a n a s s o p o l u s et al., Phys. Rev. C 5 4 , 2658 (1996). B . A r m b r u s t e r et al., Phys. Rev. C 5 7 , 3414 (1998). Y. O y a m a , K E K P r e p r i n t 2001-7, copy of a talk a t t h e "Cairo Int. Conference", J a n u a r y 2001. 8. M. Apollonio et al., Phys. Lett. B 4 6 6 , 415 (1999).

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4. Hadron Structure and Properties

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LATTICE QCD A N D H A D R O N S T R U C T U R E

A. W. T H O M A S Special Research Centre for the Subatomic Structure of Matter and Department of Physics and Mathematical Physics, Adelaide, SA 5005, Australia E-mail: [email protected]

One of the great challenges of lattice QCD is to produce unambiguous predictions for the properties of physical hadrons. We review recent progress with respect to a major barrier to achieving this goal, namely the fact that computation time currently limits us to large quark mass. Using insights from the study of the lattice data itself and the general constraints of chiral symmetry we demonstrate that it is possible to extrapolate reliably from the mass region where calculations can be performed to the chiral limit.

1. Introduction At the present time we have a wonderful conjunction of opportunities. Modern accelerator facilities such as Jlab, Mainz, DESY and CERN are providing data of unprecedented precision over a tremendous kinematic range at the same time as numerical simulations of lattice QCD are delivering results of impressive accuracy. It is therefore timely to ask how to use these advances to develop a new and deeper understanding of hadron structure and dynamics 1 . Let us begin with lattice QCD. Involving as it does a finite grid of spacetime points, lattice QCD requires numerous extrapolations before one can compare with any measured hadron property. The continuum limit, a —• 0 (with a the lattice spacing), is typically under good control 2 . With improved quark and gluon actions the 0(a) errors can be eliminated so that the finite-a errors are quite small, even at a modest lattice spacing 3 - say 0.1 fm. In contrast, the infinite volume limit is much more difficult to implement as the volume, and hence the calculation time, scales like TV4. Furthermore, this limit is inextricably linked to the third critical extrapolation, namely the continuation to small quark masses (the "chiral extrapolation"). The reason is, of course, that chiral symmetry is spontaneously broken in QCD, with the pion being a massless Goldstone boson in the chiral limit. As the lattice volume must contain the pion cloud of whatever hadron is under study one expects that the box size, L, should be at least 4m" 1 . At the physical pion mass this is a box

115

116

5.6 fm on a side, or a 56 4 lattice, with a — 0.1 fm. This is roughly twice as big as the lattices currently in use. Since the time for calculations with dynamical fermions (i.e. including quark-antiquark creation and annihilation in the vacuum) scale4 as m~36, current calculations have been limited to light quark masses 6-10 times larger than the physical ones. With the next generation of supercomputers planned to be around 10 Teraflops, it should be possible to get as low as 4 times the physical quark mass, but to actually reach the physical mass on an acceptable volume will need at least 500 Teraflops. This is 10-20 years away. Since a major motivation for lattice QCD must be to unambiguously compare the calculations of hadron properties with experiment, this is somewhat disappointing. The only remedy for the next decade at least is to find a way to extrapolate masses, form-factors, and so on, calculated at a range of masses considerably larger than the physical ones, to the chiral limit. In an effort to avoid theoretical bias this has usually been done through low-order polynomial fits as a function of quark mass. Unfortunately, this is incorrect and can yield quite misleading results because of the Goldstone nature of the pion. Once chiral symmetry is spontaneously broken, as we have known for decades that it must be in QCD and as it has been confirmed in lattice calculations, all hadron properties receive contributions involving Goldstone boson loops. These loops inevitably lead to results that depend on either logarithms or odd powers of the pion mass. The Gell-Mann-Oakes-Renner relation, however, implies that mn is proportional to the square root of mq, so logarithms and odd powers of m , are non-analytic 5 in the quark mass, with a branch point at mq = 0. One simply cannot make a power series expansion about a branch point. On totally general grounds, one is therefore compelled to incorporate the non-analyticity into any extrapolation procedure. The classical approach to this problem is chiral perturbation theory, an effective field theory built upon the symmetries of QCD 6 . There is considerable evidence that the scale naturally associated with chiral symmetry breaking in QCD, A X SB, is of order 4irfn, or about 1 GeV. Chiral perturbation theory then leads to an expansion in powers of m ff /A x sB and P / A X S B , with p a typical momentum scale for the process under consideration. At 0(p4), the corresponding effective Lagrangian has only a small number of unknown coefficients which can be determined from experiment. On the other hand, at 0(p6) there are more than 100 unknown parameters, far too many to determine phenomenologically. Another complication, not often discussed, is that there is yet another mass scale entering the study of nucleon (and other baryon) structure 7,8 . This scale is the inverse of the size of the nucleon, A ~ R"1. Since A is naturally more like

117 a few hundred MeV, rather t h a n a 1 GeV, the n a t u r a l expansion parameter, mn/A, is of order unity for mn ~ 2 - 3mP h y s - the lowest mass scale at which lattice d a t a exists. This is much larger than mK/AxsB ~ 0.3 — 0.4, which might have given one some hope for convergence. As it is, the large values of m,r / A at which lattice d a t a exist make any chance of a reliable expansion in chiral perturbation theory (xPT) fairly minimal 9 . Even though one has reason to doubt the practical utility of xPT, the lattice d a t a itself does give us some valuable hints as to how the dilemma might be resolved. T h e key is to realize t h a t , even though the masses may be large, one is actually studying the properties of Q C D , not a model. In particular, one can use the behaviour of hadron properties as a function of mass to obtain valuable new insights into hadron structure. T h e first thing t h a t stands out, once one views the d a t a as a whole, is just how smoothly every hadron property behaves in the region of large quark mass. In fact, baryon masses behave like a + bmq, magnetic m o m e n t s like (c + t / m q ) - 1 , charge radii squared like (e + / m q ) _ 1 and so on. T h u s , if one defined a light "constituent quark mass" as M = Mo + cmq (with c ~ 1), one would find baryon masses proportional to M (times the number of u and d quarks), magnetic moments proportional to M _ 1 and so on - just as in the constituent quark picture. There is simply no evidence at all for the rapid, nonlinearity associated with the branch cuts created by Goldstone boson loops. Indeed, there is not even any evidence for a statistically significant difference between properties calculated in quenched versus full Q C D ! How can this be? T h e n a t u r a l answer is readily found in the additional scale, A ~ Br1, mentioned earlier. In Q C D (and quenched Q C D ) , Goldstone bosons are emitted and absorbed by large, composite objects built of quarks and gluons. Whenever a composite object emits or absorbs a probe with finite m o m e n t u m one m u s t have a form-factor which will suppress such processes for m o m e n t a greater t h a n A ~ Br1. Indeed, for m* > A we expect Goldstone boson loops to be suppressed as powers of A / m r , not mn/A (or mn/Axss). Of course, this does not mean t h a t one cannot in principle carry through the program of xPT. However, it does mean t h a t there may be considerable correlations between higher order coefficients and t h a t it may be much more efficient to adopt an approach which exploits the physical insight we j u s t explained. Over the past three years or so, our group in Adelaide has worked with a number of colleagues around the world to do just this. T h a t is, we have developed an efficient technique, using very few free parameters, to extrapolate every h a d r o n property which can be calculated on the lattice from the large mass region to the physical quark mass - while preserving the most i m p o r t a n t non-analytic behaviour of each of those observables. This task is not trivial,

118

in that various observables need different phenomenological treatments. On the other hand, there is a unifying theme. That is, pion loops are rapidly suppressed for pion masses larger than A (mff > 0.4 — 0.5 GeV). In this region the constituent quark model seems to represent the lattice data extremely well. However, for mw below 400-500 MeV the Goldstone loops lead to rapid, nonanalytic variation with mq and it is crucial to preserve the correct leading non-analytic (LNA) and sometimes the next-to-leading non-analytic (NLNA) behaviour of xPT. In order to guide the construction of an effective, phenomenological extrapolation formula for each hadron property, we have found it extremely valuable to study the behaviour in a particular chiral quark model - the cloudy bag model (CBM) 8 ' 10,11 . Built in the early 80's it combined a simple model for quark confinement (the MIT bag) with a perturbative treatment of the pion cloud necessary to ensure chiral symmetry. The consistency of the perturbative treatment was, not surprisingly in view of our earlier discussion, a consequence of the suppression of high momenta by the finite size of the pion source (in this case the bag). Certainly the MIT bag, with its sharp, static surface, has its quantitative defects. Yet the model can be solved in closed form and all hadron properties studied carefully over the full range of masses needed in lattice QCD. Provided one works to the appropriate order the model preserves the exact LNA and NLNA behaviour of QCD in the low mass region while naturally suppressing the Goldstone boson loops for m^ > A ~ R~1. Finally, it actually yields quite a good description of lattice data in the large mass region for all observables in terms of just a couple of parameters. With this lengthy explanation of the physics which underlies the superficially different extrapolation procedures, we now summarise the results for some phenomenologically significant baryon properties. 2. Electromagnetic Properties of Hadrons While there is only limited (and indeed quite old) lattice data for hadron charge radii, recent experimental progress in the determination of hyperon charge radii has led us to examine the extrapolation procedure for extracting charge radii from the lattice simulations. Figure 1 shows the extrapolation of the lattice data for the charge radius of the proton, including the In mff (LNA) term in a generalised Pade approximant 12 ' 13 : {r}

*~

l + c2ml

•

{i}

Here ci and c2 are parameters determined by fitting the lattice data in the large mass region (m 2 > 0.4 GeV 2 ), while \i, the scale at which the effects of

119 pion loops are suppressed, is not yet determined by the data but is simply set to 0.5 GeV. The coefficient \ is model independent and determined by chiral perturbation theory. Clearly the agreement with experiment is much better if, as shown, the logarithm required by chiral symmetry is correctly included rather than simply making a linear extrapolation in the quark mass (or ml). Full details of the results for all the octet baryons may be found in Ref. [12]. The situation for baryon magnetic moments is also very interesting. The LNA contribution in this case arises from the diagram where the photon couples to the pion loop. As this involves two pion propagators the expansion of the proton and neutron moments is: MP(")

= A (J ( n ) T am, + O K )

(2)

Here //Q is the value in the chiral limit and the linear term in mn is proportional to m | , a branch point at m g = 0. The coefficient of the LNA term is a = 4.4//jvGeV _1 . At the physical pion mass this LNA contribution is 0.6pjv, which is almost a third of the neutron magnetic moment. 1.2 1.0 0.8

*A

v C o

"o u

0.6

-

0.4

•--+-

—•

- - 1 - * - . -

0.2 0.0 }r

A J_

-0.2 0.0

0.2

0.4 m

2

0.6 (GeV 2 )

0.8

1.0

Figure 1. Fits to lattice results for the squared electric charge radius of the proton - from Ref. [12]. Fits to the contributions from individual quark flavours are also shown (the u-quark results are indicated by open triangles and the d-quark results by open squares). Physical values predicted by the fits are indicated at the physical pion mass. The experimental value is denoted by an asterisk.

Just as for the charge radii, the chiral behaviour of fj.p^ is vital for a correct extrapolation of lattice data. As shown in Fig. 2, one can obtain a very

120

satisfactory fit to some rather old data, which happens to be the best available, using the simple Pade approximant 14 : pin)

^

=

li-jfom^+^m?

(3)

Existing lattice data can only determine two parameters and Eq. (3) has just two free parameters while guaranteeing the correct LNA behaviour as m^ —¥ 0 and the correct behaviour of HQET at large ml. The extrapolated values of fj,p and n" at the physical pion mass, 2.85 ± 0.22/JJV and —1.90 ± 0.15pjv are currently the best estimates from non-perturbative QCD 14 . (Similar results, including NLNA terms in chiral perturbation theory, have been reported recently by Hemmert and Weise15.) For the application of similar ideas to other members of the nucleon octet we refer to Ref. [16], while for the strangeness magnetic moment of the nucleon we refer to Ref. [17]. The last example is another case where tremendous improvements in the experimental capabilities, specifically the accurate measurement of parity violation in ep scattering 18 , is giving us vital information on hadron structure. Hs-

3 +->

^ome

c

-s o -1-5

CI)

aM

3.5 3.0 2 5 2.0 1.5 1.0 0.5 ().()

cO

-0 5 -1 0 C o -1.5 CD U -2 0

S

3 55

-2.5

Figure 2. Extrapolation of lattice QCD magnetic moments for the proton (upper) and neutron (lower) to the chiral limit. The curves illustrate a two parameter fit to the simulation data, using a Pade approximant, in which the one-loop corrected chiral coefficient of mn is taken from xPT. T h e experimentally measured moments are indicated by asterisks. The figure is taken from Ref. [14].

121 In concluding this section, we note t h a t the observation t h a t chiral corrections are totally suppressed for m^ above about 0.5 GeV and t h a t the lattice d a t a looks very like a constituent quark picture there suggests a novel approach to modelling hadron structure. It seems t h a t one might avoid m a n y of the complications of the chiral quark models, as well as m a n y of the obvious failures of constituent quark models by building a new constituent quark model with u and d masses in the region of the strange quark - where SU(3) s y m m e t r y should be exact. Comparison with d a t a could then be m a d e after the same sort of chiral extrapolation procedure t h a t has been applied to the lattice d a t a . Initial results obtained by Cloet et al. for the octet baryon magnetic m o m e n t s using this approach are very promising indeed 1 9 . We note also the extension to A-baryons, including the NLNA behaviour, reported for the first time in these proceedings 2 0 .

3. M o m e n t s of Structure Functions T h e m o m e n t s of the parton distributions measured in lepton-nucleon deep inelastic scattering are related, through the operator product expansion, to the forward nucleon m a t r i x elements of certain local twist-2 operators which can be accessed in lattice simulations 1 . T h e more recent d a t a , used in the present analysis, are taken from the Q C D S F 2 1 and M I T 2 2 groups and shown in Fig. 3 for the n = 1, 2 and 3 moments of the u — d difference at N L O in the MS scheme. To compare the lattice results with the experimentally measured m o m e n t s , one must extrapolate in quark mass from about 50 MeV to the physical value. Naively this is done by assuming t h a t the moments depend linearly on the quark mass. However, as shown in Fig. 3 (long dashed lines), a linear extrapolation of the world lattice d a t a for the u — d m o m e n t s typically overestimates the experimental values by 50%. This suggests t h a t important physics is still being omitted from the lattice calculations and their extrapolations. Here, as for all other hadron properties, a linear extrapolation in m ~ m 2 must fail as it omits crucial nonanalytic structure associated with chiral symmetry breaking. T h e leading nonanalytic (LNA) t e r m for the u and d distributions in the physical nucleon arises from the single pion loop dressing of the bare nucleon and has been shown 2 3 , 2 4 to behave as m 2 log mn. Experience with t h e chiral behaviour of masses and magnetic m o m e n t s shows t h a t the LNA t e r m s alone are not sufficient to describe lattice d a t a for m^ > 200 MeV. T h u s , in order to fit the lattice d a t a at larger m^, while preserving the correct chiral behaviour of m o m e n t s as mn —>• 0, a low order, analytic expansion in m 2 is also included in the extrapolation and the m o m e n t s of u — d are fitted

122 1

0.4

1

1

QCDSF Gockeler e t al.. 1996 QCDSF Gockeler et al., 1997 QCDSF Best et al., 1997 MIT-DD60Q (Quenched) MIT-SESAM (Full)

Experiment

Meson Cloud Model

0.3

0.1 •+-

h

-l

h

-i

h

0.12-

0.04 H

1-

•H

h

0.06-

-jbn-iL0.02-

o.o

0.2

0.4

0.6

0.8

[GeV2]

1.0

Figure 3. Moments of the u — d quark distribution from various lattice simulations. The straight (long-dashed) lines are linear fits to this data, while the curves have the correct LNA behaviour in the chiral limit — see the text for details. The small squares are the results of the meson cloud model and the dashed curve through them best fits using Eq. (4). The stars represent the phenomenological values taken from NLO fits in the MS scheme. The figure is taken from Ref. [25].

with the form 25 : (xn)u-d

= an + K ml + an c L N A ml In

ml

ml + (i<

(4)

123 where the coefficient 24 , cLNA = - ( 3 ^ + l)/(47r/ f f ) 2 . T h e parameters a „ , bn are determined by fitting the lattice data. T h e mass /j determines the scale above which pion loops no longer yield rapid variation and corresponds to the upper limit of the m o m e n t u m integration if one applies a sharp cut-off in the pion loop integral. Consistent with our earlier discussion of this scale it is taken to be 550 MeV. Multi-meson loops and other contributions cannot give rise to LNA behaviour and thus, near the chiral limit, Eq. (4) is the most general form for m o m e n t s of the P D F s at (^(m 2 ) which is consistent with chiral symmetry. We stress t h a t \i is not yet determined by the lattice d a t a and it is indeed possible to consistently fit both the lattice d a t a and the experimental values with \i ranging from 400 MeV to 700 MeV. This dependence on \i is illustrated in Fig. 3 by the difference between the inner and outer envelopes on the fits. D a t a at smaller quark masses, ideally m 2 ~ 0.05-0.10 G e V 2 , are therefore crucial to constrain this parameter in order to perform an accurate extrapolation based solely on lattice d a t a . 4.

Conclusion

T h e next few years will see tremendous progress in our understanding of hadron structure. In combination with the very successful techniques for chiral extrapolation, which we have illustrated by j u s t a few examples, lattice Q C D will finally yield accurate d a t a on the consequences of non-perturbative Q C D . Furthermore, the physical insights obtained will guide the development of new quark models which provide a much more realistic representation of Q C D . Acknowledgments This work was supported by the Australian Research Council and the University of Adelaide. References 1. A. W. Thomas and W. Weise, "The Structure of the Nucleon," ISBN 3-527-402977 Wiley-VCH, Berlin 2001. 2. D. G. Richards et al. [LHPC Collaboration], Nucl. Phys. Proc. Suppl. 109, 89 (2002) [arXiv:hep-lat/0112031]. 3. J. M. Zanotti et al., Nucl. Phys. Proc. Suppl. 109, 101 (2002) [arXiv:heplat/0201004]. 4. T. Lippert, S. Gusken and K. Schilling, Nucl. Phys. Proc. Suppl. 83, 182 (2000). 5. L. F. Li and H. Pagels, Phys. Rev. Lett. 26, 1204 (1971). 6. J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984). 7. W. Detmold et al., Pramana 57, 251 (2001) [arXiv:nucl-th/0104043]. 8. A. W. Thomas, Adv. Nucl. Phys. 13, 1 (1984).

124 9. T. Hatsuda, Phys. Rev. Lett. 65, 543 (1990). 10. G. A. Miller, A. W. Thomas and S. Theberge, Phys. Lett. B91, 192 (1980). 11. S. Theberge, G. A. Miller and A. W. Thomas, Can. J. Phys. 60, 59 (1982). 12. E. J. Hackett-Jones et at., Phys. Lett. B494, 89 (2000) [hep-lat/0008018]. 13. G. V. Dunne et al., Phys. Lett. B531, 77 (2002) [hep-th/0110155]. 14. D. B. Leinweber et al., Phys. Rev. D60, 034014 (1999). 15. T. R. Hemmert and W. Weise, arXiv:hep-lat/0204005. 16. E. J. Hackett-Jones et al., Phys. Lett. B489, 143 (2000). 17. D. B. Leinweber and A. W. Thomas, Phys. Rev. D62, 074505 (2000). 18. K. S. Kumar and P. A. Souder, Prog. Part. Nucl. Phys. 45, S333 (2000). 19. I. C. Cloet, D. B. Leinweber and A. W. Thomas, arXiv:hep-ph/0203023. 20. I. C. Cloet, D. B. Leinweber and A. W. Thomas, "Baryon Resonance Phenomenology" , in these proceedings. 21. M. Gockeler et al, Nucl. Phys. Proc. Suppl. 53, 81 (1997). 22. D. Dolgov et al., Nucl. Phys. Proc. Suppl. 94, 303 (2001). 23. A. W. Thomas et al., Phys. Rev. Lett. 85, 2892 (2000). 24. D. Arndt and M. J. Savage, nucl-th/0105045; J. Chen and X. Ji, hep-ph/0105197. 25. W. Detmold et al., Phys. Rev. Lett. 87, 172001 (2001). 26. P. A. Guichon, Phys. Lett. B163, 221 (1985).

BARYON RESONANCE PHENOMENOLOGY

I.C. CLOET, D.B. LEINWEBER AND A.W. THOMAS Special Research Centre for the Subatomic Structure of Matter and Department of Physics and Mathematical Physics, University of Adelaide, SA 5005, Australia E-mail: dleinweb6physics.adelaide.edu.au; icloetQphysics.adelaide.edu.au; [email protected]

The Japan Hadron Facility will provide an unprecedented opportunity for the study of baryon resonance properties. This paper will focus on the chiral nonanalytic behaviour of magnetic moments exclusive to baryons with open decay channels. To illustrate the novel features associated with an open decay channel, we consider the "Access" quark model, where an analytic continuation of chiral perturbation theory is employed to connect results obtained using the constituent quark model in the limit of SU(3)-flavour symmetry to empirical determinations.

1. Introduction The Japan Hadron Facility will present new opportunities for the investigation of baryon resonance properties. In particular, access to the hyperons of the baryon decuplet will be unprecedented. This contribution serves to highlight the novel and important aspects of QCD that can be explored through an experimental program focusing on decuplet-baryon resonance phenomenology. To highlight the new opportunities, it is sufficient to address the magnetic moments of the charged A baryons of the decuplet. The magnetic moments of these baryons have already caught the attention of experimentalists and hold the promise of being accurately measured in the foreseeable future. Experimental estimates exist for the A + + magnetic moment, based on the reaction 7T+ p ->• 7T+ 7' p. The Particle Data Group 1 provides the range 3.7-7.5 /z;v for the A + + magnetic moment with the most recent experimental result 2 of 4.52 ± 0.50 ± 0.45 p.^. In principle, the A + magnetic moment can be obtained from the reaction y p —^ 7r° 7' p, as demonstrated at the Mainz microtron 3 . An experimental value for the A + magnetic moment appears imminent. Recent extrapolations of octet baryon magnetic moments 4,5,6 have utilized an analytic continuation of the leading nonanalytic (LNA) structure of Chiral Perturbation Theory (xPT), as the extrapolation function. The unique feature of this extrapolation function is that it contains the correct chiral behaviour

125

126

as mq —y 0 while also possessing the Dirac moment mass dependence in the heavy quark mass regime. The extrapolation function utilized here has these same features, however we move beyond the previous approach by incorporating not only the LNA but also the next to leading nonanalytic (NLNA) structure of x P T in the extrapolation function. Incorporating the NLNA terms contributes little to the octet baryon magnetic moments, however it proves vital for decuplet baryons. The NLNA terms contain information regarding the branch point at m^ = M A — MN, associated with the A —> Nn decay channel and play a significant role in decuplet-baryon magnetic moments. 2. Leading and Next-to-Leading Nonanalytic Behaviour We begin with the chiral expansion for decuplet baryon magnetic moments 7 . The LNA and NLNA behaviour is given by
+ \ra{G* - QK) ,

(i)

where qt and 1^ are charge and isospin respectively, and Gj (j = T , K) is given by Sj

=

3

^ \^H2T%milN)

+Cin-6Ntmj,N)}

.

(2)

% describes the meson coupling to decuplet baryons and C describes octetdecuplet transitions. We take 7i = —2.2 and C = —1.2. Here we omit the Roper 7 as this transition requires significant excitation energy and is strongly suppressed by the finite size of the meson source. The octet-decuplet mass splitting <5jv is assigned its average value SN = Mio - M8 = 1377 - 1151 = +226 MeV. We take fn - 93 MeV and fK = 112 MeV. The function F(5,m,fi) form a

has the

m > | 6 |, H5,m,„)

= - £ l n (%)

+V ^ ^ f j

-tan"1 -

7

4 = ^ \ >

m<\&\, HS,rn,,) = J - l n ( 24 ) 7T \fi J

1 +

- V s ^ ^ l n ( ' - V ^ 2Z ^ )2 . n ys + y/S - m J

(3)

"This definition for F(8, m, n) corrects a sign error in Ref. [7]. It differs by an overall minus sign and suppresses additive constants which are irrelevant in our analysis. As S —y 0, f ( J , m i , / i ) ->• rn.Tr.

127

Hence the LNA behaviour of decuplet magnetic moments is given by XTT mn + XK mK + X'„ ^(-^N,mn,n7r)

+ X'K

F{-6N,mK,nK),

(4)

where Xvr

=

MNH2

MNC2

XK

* CM 2

108 2

MNn

*(fi<)2

MNC

qi_

V108

48

2

XK

72

=

T

(/x) 2 V48

32

(5)

The values for the above chiral coefficients, Eq. (5), describing the strength of various meson dressings of the A baryons, are summarized in Table 1 for the four A baryons of the decuplet. Table 1. The baryon chiral coefficients for the four A baryons of the decuplet. Coefficients are calculated with H — - 2 . 2 and C = - 1 . 2 , where H is the decuplet-decuplet coupling constant and C is the decuplet-octet coupling constant. We have suppressed the kaon loop contribution by using SK = 1-2 /jr and jn — 93 MeV.

X*r XK X'* XK

A++

A+

A0

A~

-2.33 -1.61 -1.56 -1.08

-0.777 -1.070 -0.518 -0.719

+0.777 -0.535 +0.518 -0.361

+2.33 0 + 1.56 0

3. Analytic Continuation of x P T It is now recognized that in any extrapolation from the heavy quark regime (where constituent quark properties are manifest) to the physical world, it is imperative to incorporate the quark-mass dependence of observables predicted by x P T in the chiral limit. However, as results are often obtained using methods ideally suited for heavy quark masses, it is imperative for the extrapolation function to correctly reflect the behaviour of the physical observable in the heavy quark mass regime as well. An extrapolation function for the A-baryon magnetic moments satisfying these criteria is ^~

l-T(mn)/^lQ

+ |3m^,

W

where JIQ and /3 are parameters optimized to fit results obtained near the strange quark mass and r(m 7r ) is taken from the chiral expansion for decuplet magnetic moments in Sec. 2

128

+x'n {^r{SN,mir,/ir)

IXm*) = X-K m* +XK (mK - m^) ^v^ A

+XK

-

,

v

N

v

B

-

Tv) ,

C

{.H-&N,mK,pK)-TK\,

(7)

where m^- , Tw and ^ x are constants defined to ensure that each term A, B, C and D, vanishes in the chiral limit. Utilizing the following relations provided byXPT mK2 = m{°)

+-mv2,

(8)

mf = yjim^y-lim^y

,

(9)

the four terms of Eq. (7) vanish in the chiral limit provided

' v" / *

v2

M 0 ) ) 8 -^v

(10) Figure 1 presents a plot of each of the four terms of Eq. (7) (without the chiral coefficient pre-factors) as a function of m j . Figure 2 presents a plot of the four terms summed with the appropriate weightings of Table 1 for each of the four charge states of the A. The extrapolation function of Eq. (6) is designed to reproduce the leading and next-to-leading nonanalytic structure expressed in Eq. (7) for expansions about mn = 0. Equation (6) may be regarded as an analytic continuation of Eq. (7), preserving the constraints imposed by chiral symmetry and introducing the heavy quark mass regime behaviour to the extrapolation function. The LNA behaviour of Eq. (7) is complemented by terms analytic in the quark mass with fit parameters po and 0 adjusted to fit additional constraints on the observable under investigation. Hence the extrapolation function guarantees the correct nonanalytic behaviour in the chiral limit. Further, as mn becomes large, Eq. (6) is proportional to l/m£. As mw2 oc mq over the applicable mass range, the magnetic moment extrapolation function decreases as \/mq for increasing quark mass, precisely as the Dirac moment requires. This extrapolation function therefore provides a functional form bridging the heavy quark mass regime and the chiral limit.

129 1.6

i

1

1

_

1.4 3 in

a 0

1.2

-

C ^ ——=

1.0

^ ^^\^^^~~k

(IH

-

-t->

o

0.6

3

0.4

"ffl

0.2

IH

£1

u

!)___

7/

r^^-^

ET^

^-r^^^

-

0.0

-

-0.2 i

-0.4 0.0

0.2

i

0.4 m

2

0.6 (GeV 2 )

i

0.8

1.0

Figure 1. Plots of the four chiral expansion functions (without the chiral coefficient prefactors) of Eq. (7), labelled A, B, C, D in Eq. (7).

0.4 m / Figure 2.

0.6 (GeV2)

Plots of the sum of all four chiral expansion terms of Eq. (7), for each A baryon.

4. R e s u l t s The method employed to obtain our theoretical predictions is analogous to that presented in our previous analysis of octet baryon magnetic moments 4 . We take the established input parameters, the strange-constituent and strange-current

130

quark masses (Ms and c m^hys respectively 6 ), obtained by optimizing agreement between the AccessQMc and octet baryon magnetic moments. There, Ms = 565 MeV and c ra.Phys = 144 MeV provides optimal agreement. The constituent quark model (CQM) provides the following formulas that relate the constituent quark masses to the delta magnetic moments: HA++ = 3/iu ,

fiA+ = IVu + Hd ,

f^^o = flu + 2/J.d ,

^A-=3^d,

(11)

with 2MW

^ =3 1 ^ ^ '

^

=

IMN

"3M7m'

\MN

^-SMT^'

,10.

(12)

These formulas are used to obtain two magnetic moment data points near the SU(3)-flavour limit where u and d quarks take values near the s-quark mass. To fit Eq. (6), which is a function of mn, to the magnetic moments given by the CQM in Eq. (11) with constituent-quark masses Mu = Md = Mi (i = 1,2), we relate the pion mass to the constituent quark mass via the current quark mass 4 . Chiral symmetry provides m - ^ - = ' hys m? (m£ h y s )2 '

(13) l

;

where mj?hys is the quark mass associated with the physical pion mass, m f y s . From lattice studies, we know that this relation holds well over a remarkably large regime of pion masses, up to m r ~ 1 GeV. The link between constituent and current quark masses is provided by M = Mx + cmq,

(14)

where Mx is the constituent quark mass in the chiral limit and c is of order 1. Using Eq. (13) this leads to phys

M

c m\

= Mx + ^t^2ml-

(15)

(m? r The link between the constituent quark masses M; and mn is thus provided by

The parameter c is expected to be the order of 1. T h e name indicates the mathematical origins of the model: Analytic Continuation of the Chiral Expansion for the SU(6) Simple Quark Model.

c

131 Table 2. Theoretical predictions for the charged A baryon magnetic moments. The fit parameters po and /3 are given for each scenario. The only currently known experimental value for the A baryon magnetic moments is the A++ moment, where recent measurements provide 2 /i A ++ = 4.52 ± 0.50 ± 0.45 /ijy. Baryon

A++ A+ A"

Mo

0

AccessQM (AIJV)

+5.67 +2.69 -3.22

0.16 0.20 0.06

+5.39 +2.58 -2.99

where Ms — c m p h y s = Mx encapsulates information on the constituent quark mass in the chiral limit, and c rafys provides information on the strange current quark mass. We use the ratio rnPhys -phys

= 24.4 ± 1 . 5 .

(17)

provided by x ? T 8 to express the light current quark mass, mP hys , in terms of the strange current quark mass, m f y s , in Eq. (16). The analytic continuation of xPT, Eq. (6), is fit to the constituent quark model (CQM) as a function of mn2. Results are presented in Figs. 3 and 4. The magnetic moments given by the CQM either side of the SU(3)-flavour limit are indicated by a dot (•) and the theoretical prediction is indicated at the physical pion mass by a star (*). These results along with the parameters Ho and /3, are summarized in Table 2. The interesting feature of these plots is the cusp at mw2 = &M2 which indicates the opening of the octet decay channel, A -> Nir. The physics behind the cusp is intuitively revealed by the relation between the derivative of the magnetic moment with respect to ml and the derivative with respect to the momentum transfer q2, provided by the pion propagator l/(q2 + ml) in the heavy baryon limit. Derivatives with respect to q2 are proportional to the magnetic charge radius in the limit q2 -> 0, (r2M) = -^GM(q2)\q,=0.

(18)

If we consider for example A++ ->• pn+ with | j , rrij) = |3/2, 3/2), the lowestlying state conserving parity and angular momentum will have a relative Pwave orbital angular momentum with \l,mj) — |1,1). Thus the positivelycharged pion makes a positive contribution to the magnetic moment. As the opening of the p n+ decay channel is approached from the heavy quark-mass regime, the range of the pion cloud increases in accord with the Heisenberg uncertainty principle, AE At ~ h. Just above threshold the pion cloud extends

132 6

M

i

—i

1

1

1

1

0.3

0.4

0.5

r

-

0.0

0.1

0.2

mj1 (GeV2)

0.6

0.7

Figure 3. The extrapolation function fit for A++ and A+ magnetic moments. The magnetic moments given by the CQM either side of the SU(3)-flavour limit are indicated by dots (•) and the theoretical prediction for each baryon is indicated at the physical pion mass by a star (*). The only available experimental d a t a is for the A + + and is indicated by an asterisk (*). The proton extrapolation 4 (dashed line) is included to illustrate the effect of the open decay channel, A —¥ N TT, in the A"*" extrapolation. The presence of this decay channel gives rise to a A+ moment smaller than the proton moment.

towards infinity as AE -» 0 and the magnetic moment charge radius diverges. Similarly, (d/drn^)GM —*• —oo. Below threshold, GM becomes complex and the magnetic moment of the A is identified with the real part. The imaginary part describes the physics associated with photon-pion coupling in which the pion is subsequently observed as a decay product. It is the NLNA terms of the chiral expansion for decuplet baryons that contain the information regarding the decuplet to octet transitions. These transitions are energetically favourable making them of paramount importance in determining the physical properties of A baryons. The NLNA terms serve to enhance the magnitude of the magnetic moment above the opening of the decay channel. However, as the decay channel opens and an imaginary part develops, the magnitude of the real part of the magnetic moment is suppressed. The strength of the LNA terms, which enhance the magnetic moment magnitude as the chiral limit is approach, overwhelms the NLNA contributions such that the magnitude of the moments continues to rise towards the chiral limit. The inclusion of the NLNA structure into octet baryon magnetic moment

133

-1.0 -1.5

^r -2.o c

6 o-2.5 -3.0 -3.5 0.0

0.1

0.2

0.3 0.4 0.5 m 2 (GeV2)

0.6

0.7

Figure 4. The extrapolation function fit for the A - magnetic moment. The magnetic moments given by the CQM either side of the SU(3)-flavour limit are indicated by dots (•) and the theoretical prediction is indicated at the physical pion mass by a star (*). There is currently no experimental value for the A - magnetic moment.

extrapolations is less important for two reasons. The curvature associated with the NLNA terms is negligible for the N and E baryons and small for the A and S baryons. More importantly one can infer the effects of the higher order terms of xPT, usually dropped in truncating the chiral expansion, through the consideration of phenomenological models. If one incorporates form factors at the meson-baryon vertices, reflecting the finite size of the meson source, one finds that transitions from ground state octet baryons to excited state baryons are suppressed relative to that of xPT to finite order, where point-like couplings are taken. In xPT it is argued that the suppression of excited state transitions comes about through higher order terms in the chiral expansion. As such, the inclusion of NLNA terms alone will result in an overestimate of the transition contributions, unless one works very near the chiral limit where higher order terms are indeed small. For this reason octet to decuplet or higher excited state transitions have been omitted in previous studies 4 ' 5 ' 6 . In the simplest CQM with mu = md, the A + and proton moments are degenerate. However, spin-dependent interactions between constituent quarks will enhance the A + relative to the proton at large quark masses, and this is supported by lattice QCD simulation results 9 . As a result, early lattice QCD predictions based on linear extrapolations 9 report the A + moment to be

134 greater t h a n the proton moment. However with the extrapolations presented here which preserve the LNA behaviour of x P T , the opposite conclusion is reached. We predict the A + and proton magnetic m o m e n t s of 2.58 //JV a n d 2.77 /ijy respectively. The proton magnetic moment extrapolation 4 is included in Fig. 3 as an illustration of the importance of incorporating the correct nonanalytic behaviour predicted by %PT in any extrapolation to the physical world. An experimentally measured value for the A + magnetic m o m e n t would offer i m p o r t a n t insights into the role of spin-dependent forces and chiral nonanalytic behaviour in the quark structure of baryon resonances.

5.

Conclusion

An extrapolation function for the decuplet baryon magnetic m o m e n t s has been presented. This function preserves the leading and next-to-leading nonanalytic behaviour of chiral perturbation theory while incorporating the Dirac-moment dependence for moderately heavy quarks. Interesting nonanalytic behaviour of the magnetic m o m e n t s associated with the opening of the n N decay channel has been highlighted. It will be interesting to apply these techniques to existing and forthcoming lattice Q C D results, and research in this direction is currently in progress. An experimental value exists only for the A + + magnetic m o m e n t where P A + + = 4.52 ± 0.50 ± 0.45 (IN- This value is in good agreement with the prediction of 5.39 HN given by our AccessQM as described above. Arrival of experimental values for the A + and A - magnetic m o m e n t s are eagerly anticipated and should be forthcoming in the next few years. More importantly, these techniques may be applied to the decuplet hyperon resonances where the role of the kaon cloud becomes i m p o r t a n t . We look forward to new J H F results in this area in the future.

Acknowledgement This work was supported by the Australian Research Council.

References 1. Particle Data Group, Eur. Phys. J., C15, (2000). 2. A. Bosshard et al., Phys. Rev. D44, 1962 (1991). 3. M. Kotulla [TAPS and A2 Collaborations], Prepared for Hirschegg '01: Structure of Hadrons: 29th International Workshop on Gross Properties of Nuclei and Nuclear Excitations, Hirschegg, Austria, 14-20 Jan 2001. 4. I. C. Cloet, D. B. Leinweber and A. W. Thomas, Phys. Rev. C65, 062201 (2002). 5. D. B. Leinweber, D. H. Lu and A. W. Thomas, Phys. Rev. D60, 034014 (1999).

135 6. E. J. Hackett-Jones, D. B. Leinweber and A. W. Thomas, Phys. Lett. B489, 143 (2000). 7. M. K. Banerjee and J. Milana, Phys. Lett. D54, 5804 (1996). 8. H. Leutwyler, Phys. Lett. B378, 313 (1996). 9. D.B. Leinweber, R.M. Woloshyn, T. Draper, Phys. Rev. D46, 3067 (1992).

LATTICE QCD, G A U G E FIXING A N D THE T R A N S I T I O N TO THE P E R T U R B A T I V E REGIME

A. G. W I L L I A M S A N D M. S T A N F O R D Special Research Centre for the Subatomic Structure of Matter, University of Adelaide, SA 5005, Australia E-mail: Anthony. [email protected]; [email protected]

The standard definition of perturbative QCD uses the Faddeev-Popov gauge-fixing procedure, which leads to ghosts and the local BRST invariance of the gauge-fixed perturbative QCD action. In the nonperturbative regime, there appears to be a choice of using nonlocal Gribov-copy free gauges (e.g. Laplacian gauge) or of attempting to maintain local BRST invariance at the expense of admitting Gribov copies and somehow summing or averaging over them. It should be recognized that the standard implementation of lattice QCD corresponds to the former choice even when only physical (i.e. colour singlet) observables are being calculated. These issues are introduced and briefly explained.

1. Introduction Perturbative quantum chromodynamics (QCD) is formulated using the Faddeev-Popov gauge-fixing procedure, which introduces ghost fields and leads to the local BRST invariance of the gauge-fixed perturbative QCD action. These perturbative gauge fixing schemes include, e.g. the standard choices of covariant, Coulomb and axial gauge fixing. These are entirely adequate for the purpose of studying perturbative QCD, however, they fail in the nonperturbative regime due to the presence of Gribov copies, i.e. gauge-equivalent gaugefield configurations survive in the gauge functional integration after gauge fixing. One could define nonperturbative QCD by imposing a non-local Gribovcopy free gauge fixing (such as Laplacian gauge) or, alternatively, one could attempt to maintain local BRST invariance at the cost of admitting Gribov copies. We will see that by definition the standard ensemble-averaging technique of lattice QCD corresponds to the former definition. We begin by reviewing the standard arguments for constructing QCD perturbation theory, which use the Faddeev-Popov gauge fixing procedure to construct the perturbative QCD gauge-fixed Lagrangian density. The naive La-

136

137

grangian density of QCD is £QCD

= —F^F^

+ -£qj(iP-

msMh

(1)

where the index / corresponds to the quark flavours. The naive Lagrangian is neither gauge-fixed nor renormalized, however it is invariant under local SU(3)C gauge transformations g(x). For arbitrary small uia{x) we have

g(x) = expi-igs

(j)

ua(x)\

£ SU(Z),

(2)

where the \a/2 = ta are the generators of the gauge group SU(3) and the index a runs over the eight generator labels a = 1, 2,..., 8. Consider some gauge-invariant Green's function (for the time being we shall concern ourselves only with gluons)

wnmm = sv;™£*.

...

where 0[A] is some gauge-independent quantity depending on the gauge field, A^(x). We see that the gauge-independence of 0[A] and S[A] gives rise to an infinite quantity in both the numerator and denominator, which must be eliminated by gauge-fixing. The Minkowski-space Green's functions are defined as the Wick-rotated versions of the Euclidean ones. The gauge orbit for some configuration A^ is defined to be the set of all gauge-equivalent configurations. Each point A9 on the gauge orbit is obtained by acting upon A^ with the gauge transformation g. By definition the action, S[A], is gauge invariant and so all configurations on the gauge orbit have the same action, e.g. see the illustration in Fig. 1.

.gauge orbit Figure 1. Illustration of the gauge orbit containing Ap and indicating the effect of acting on A^ with the gauge transformation p. The action S[A] is constant around the orbit.

138

The integral over the gauge fields can be written as the integral over a full set of gauge-inequivalent (i.e. gauge-fixed) configurations, fVAs{, and an integral over the gauge group fVg. In other words, fVAst is an integral over the set of all possible gauge orbits and fDg is an integral around the gauge orbits. Thus we can write

I VA= fvA6t

fvg.

(4)

To make integrals such as those in the numerator and denominator of Eq. (3) finite and also to study gauge-dependent quantities in a meaningful way, we need to eliminate this integral around the gauge orbit, fT>g. 2. Gribov Copies and the Faddeev-Popov Determinant Any gauge-fixing procedure defines a surface in gauge-field configuration space. Figure 2 is a depiction of these surfaces represented as dashed lines intersecting the gauge orbits within this configuration space. Of course, in general, the gauge orbits are hypersurfaces and so are the gauge-fixing surfaces. Any gaugefixing surface must, by definition, only intersect the gauge orbits at distinct isolated points in configuration space. For this reason, it is sufficient to use lines for the simple illustration of the concepts here. An ideal (or complete) gaugefixing condition, F[A] — 0, defines a surface that intersects each gauge orbit once and only once and by convention contains the trivial configuration A,, = 0. A non-ideal gauge-fixing condition, F'[A] = 0, defines a surface or surfaces which intersect the gauge orbit more than once. These multiple intersections of the non-ideal gauge fixing surface(s) with the gauge orbit are referred to as Gribov copies 1 ' 2,3,4 ' 5 ' 6 . Lorentz gauge (dflA^(x) = 0) for example, has many Gribov copies per gauge orbit. By definition an ideal gauge fixing is free from Gribov copies. We refer to the ideal gauge-fixing surface ^[^4] = 0 as the Fundamental Modular Region (FMR) for that gauge choice. Typically the gauge fixing condition depends on a space-time coordinate, (e.g. Lorentz gauge, axial gauge, etc.), and so we write the gauge fixing condition more generally as F([Aj;a:) = 0. Let us denote one arbitrary gauge configuration per gauge orbit as A° and let this correspond to the "origin" of gauge configurations on that gauge orbit, i.e. to g — 0 on that orbit. Then each gauge orbit can be labelled by A° and the set of all such A°„ is equivalent to one particular, complete specification of the gauge. Under a gauge transformation, g, we move from the origin of the gauge orbit to the configuration, A3^, where by definition A° - ^ A9^ — gA°^g^ - (i/gs)(d^g)g^. Let us denote for each gauge orbit the gauge transformation, g = #[J4°], as the transformation which takes us

139

F'[A]=0 Figure 2.

Ideal, F[A], and non-ideal, F'[A], gauge-fixing.

from the origin of that orbit, A° , to the configuration, Aff-, which lies on the ideal gauge-fixed surface specified by F([A];x) = 0. In other words, we have F{[A];x)\Ai = 0 for A« = A**- G FMR. The inverse Faddeev-Popov determinant is defined as the integral over the gauge group of the gauge-fixing condition, i.e.

-,l[A^f} =

jvg8[F[A}]jv98{g-g) det

fSF([A];x" V Sg(y)

-1

(5)

Let us define the matrix Mf [A] as SFa([A];x) (6) Sgb(y) Then the Faddeev-Popov determinant for an arbitrary configuration A^ can be defined as Ap[A] = |det Mf[^4]|. (The reason for the name is now clear). Note that we have consistency, since A ^ 1 ^ 6 ^ ] = A ^ 1 ^ ] = fVg S(g — g)AFl\A\. We have 1 = fVg AF[A] £[F[y4]] by definition and hence MF([A];x,y)ab

I VAef-

= f VA^- f Vg AF[A] S[F[A]] = f VA AF[A] S[F[A]].

(7)

Since for an ideal gauge-fixing there is one and only one g per gauge orbit, such that -F([A|; x)\§ = 0, then |detMf[^4]| is non-zero on the FMR. It follows that if there is at least one smooth path between any two configurations in the FMR and since the determinant cannot be zero on the FMR, then it cannot change sign on the FMR. The Gribov horizon is defined by those configurations with det M F [ A ] = 0 which lie closest to the FMR. By definition the determinant can change sign on or outside this horizon. Clearly, the FMR is contained within the Gribov horizon and for an ideal gauge fixing, since the sign of the determinant cannot change, we can replace IdetM^I with det M^, [i.e. the overall sign of the functional integral is normalized away in Eq. (3)].

140

These results are generalizations of results from ordinary calculus, where d e t

(|^)

_ = jdx1---dxnS^(f(x)),

(8)

and if there is one and only one x which is a solution of f(x) — 0 then the matrix Mjj = dfi/dxj is invertible (i.e. non-singular) on the hypersurface f(x) = 0 and hence det M ^ 0. 3. Generalized Faddeev-Popov Technique Let us now assume that we have a family of ideal gauge fixings F([i4];i) = f([A];x) — c(x) for any Lorentz scalar c(x) and for /([A]; x) being some Lorentz scalar function, (e.g. d^A^x) or n^A^x) or similar or any nonlocal generalizations of these). Therefore, using the fact that we remain in the FMR and can drop the modulus of the determinant, we have fvA«f

= JVA det MF[A] S[f[A] - c].

(9)

Since c(x) is an arbitrary function, we can define a new "gauge" as the Gaussian weighted average over c(x), i.e., fvA**-

oc fvcexpl-^-

fd4xc(x)2\

oc / x > y l d e t M F [ ^ ] e x p | - ^ f ex jvAVXVx

expl.-ijd4xd4y

fVA detMF[A] S[f[A] - c] d4xf{[A];x)2\ x x(x)MF([A];x,y)X{y)\

xexp{~Jd4xf([A];x)2},

(10)

where we have introduced the anti-commuting ghost fields \ and \. Note that this kind of ideal gauge fixing does not choose just one gauge configuration on the gauge orbit, but rather is some Gaussian weighted average over gauge fields on the gauge orbit. We then obtain

where \Fa,u,F%

- 1 (f([A];x))2

+ J d4xdAy x(x)MF([A]; x, y)X(y).

+ Y,9f(ip-

"»/)«/ (12)

141

4. S t a n d a r d G a u g e Fixing We can now recover standard gauge fixing schemes as special cases of this generalized form. First consider standard covariant gauge, which we obtain by taking f([A];x) = dliA>1(x) and by neglecting the fact that this leads to Gribov copies. We need to evaluate Mf[A] in the vicinity of the gauge-fixing surface for this choice: b MJIWx MF([A],x,y) vY -

8 F a

^

x

Sgb{y)

)

-- *[^A«>(x)-c{x)\ Sgb{y)

m ) -_0 „ ^SA°»{*) • (13) ( t f ) ]

Under an infinitesimal gauge transformation g we have A$(x)Sm-^9

(A%(x)

= Al(x) + 9sfab^b(x)Al(x)

- d^a{x)

+

0(J){\A)

and hence in the neighbourhood of the gauge fixing surface (i.e. for small fluctuations along the gauge orbit around A& f •), we have

MF{[A];x,yyb=d*JAail{x)

(15)

6u»(y)} u-0

= d; ( [ - c W +g,fabeAcHx)]

x S^(x

- y)) .

We then recover the standard covariant gauge-fixed form of the QCD action

+(dl>Xa)(d"6ab-gfabcA2)Xb.

(16)

However, this gauge fixing has not removed the Gribov copies and so the formal manipulations which lead to this action are not valid. This Lorentz covariant set of naive gauges corresponds to a Gaussian weighted average over generalized Lorentz gauges, where the gauge parameter £ is the width of the Gaussian distribution over the configurations on the gauge orbit. Setting £ — 0 we see that the width vanishes and we obtain Landau gauge (equivalent to Lorentz gauge, dl'All(x) = 0). Choosing £ = 1 is referred to as "Feynman gauge" and so on. We can similarly recover the standard QCD action for axial gauge, where n^A^(x) = 0. Proceeding as for the generalized covariant gauge, we first identify /([.A]; a;) = ntiA>i(x) and obtain the gauge-fixed action

SdQ,q,A] =
\Fa^F;u

- ^ (n.A^x))2

+Y,q,(iP

-

mj)qf (17)

142 Taking the "Landau-like" zero-width limit £ —> 0 we select nltA,t(x) = 0 exactly and recover the usual axial-gauge fixed Q C D action. Axial gauge does not involve ghost fields, since in this case MF{[A^);x,yYb=n,8-£^

_ = „ „ {[-d^Sab]S^(x

- yj)

,

(18)

which is independent of A^ since n^Aff(x) = 0. In other words, the gauge field does not appear in Mp[A] on the gauge-fixed surface. Unfortunately axial gauge suffers from singularities which lead to significant difficulties when trying to define perturbation theory beyond one loop. A related feature is t h a t axial gauge is not a complete gauge fixing prescription. While there are complete versions of axial gauge on the lattice, these always involve a nonlocal element, or reintroduce Gribov copies at the boundary so as not to destroy the Polyakov loop. 5. D i s c u s s i o n a n d C o n c l u s i o n s There is no known Gribov-copy-free gauge fixing which is a local function of A^x). In other words, such a gauge fixing cannot be expressed as a function of A^(x) and a finite number of its derivatives, i.e. F([yt];;c) ^ F{d,j,,A^{x)) for all x. Hence, the gauge-fixed action, 5 j [• • •], in Eq. (12) becomes non-local and gives rise to a nonlocal q u a n t u m field theory. Since this action serves as the basis for the proof of the renormalizability of Q C D , the proof of asymptotic freedom, local BRS symmetry, and the Schwinger-Dyson equations (to n a m e but a few) the nonlocality of the action leaves us without a reliable basis from which to prove these features of Q C D in the nonperturbative context. It is well-established t h a t Q C D is asymptotically free, i.e. it has weakcoupling at large m o m e n t a . In the weak coupling limit the functional integral is dominated by small action configurations. As a consequence, momentum-space Green's functions at large m o m e n t a will receive their dominant contributions in the p a t h integral from configurations near the trivial gauge orbit, i.e. the orbit containing A^ = 0, since this orbit minimizes the action. If we use s t a n d a r d gauge fixing, which neglects the fact t h a t Gribov copies are present, then at large m o m e n t a J VA will be dominated by configurations lying on the gauge-fixed surfaces in the neighbourhood of each of the Gribov copies on the trivial orbit. Since for small field fluctuations the Gribov copies cannot be aware of each other, we merely overcount the contribution by a factor equal to the number of copies on the trivial orbit. This overcounting is normalized away by the ratio in Eq. (3) and becomes irrelevant. T h u s it is possible t o understand why Gribov copies can be neglected at large m o m e n t a and why it is sufficient to use standard gauge fixing schemes as the basis for calculations

143

in perturbative QCD. Since renormalizability is an ultraviolet issue, there is no question about the renormalizability of QCD. Lattice QCD has provided numerical confirmation of asymptotic freedom, so let us now turn our attention to the matter of Gribov copies in lattice QCD. Since the observable 0[A] and the action are both gauge-invariant it does not matter whether we sample from the FMR of an ideal gauge-fixing or elsewhere on the gauge orbit. The trick is simply to sample at most once from each orbit. Since there is an infinite number of gauge orbits (even on the lattice), no finite ensemble will ever sample the same orbit twice. This makes Gribov copies and gauge-fixing irrelevant in the calculation of colour-singlet quantities on the lattice. The calculation of gauge-dependent Green's functions on the lattice does require that the gauge be fixed. The standard choice is naive lattice Landau gauge, which selects essentially at random between the Landau gauge Gribov copies for the gauge orbits represented in the ensemble. This means that, while the gauge fixing is well-defined in that there are no Gribov copies, the Landau gauge-fixed configurations are not from a single connected FMR. For this reason lattice studies of gluon and quark propagators are now being extended to Laplacian gauge for comparison. Laplacian gauge is interesting because it is Gribov-copy-free (except on a set of configurations of measure zero) and it reduces to Landau gauge at large momenta. Lattice calculations of the Laplacian gauge and Landau gauge quark and gluon propagators converge at large momenta and hence are consistent with this expectation. In conclusion, it should be noted that throughout this discussion there has been the implicit assumption that nonperturbative QCD should be defined in such a way that each gauge orbit is represented only once in the functional integral, i.e. that it should be defined to have no Gribov copies. This is the definition of nonperturbative QCD implicitly assumed in lattice QCD studies. We have seen that this assumption destroys locality and the BRS invariance of the theory. An equally valid point of view is that locality and BRS symmetry are central to the definition of QCD and must not be sacrificed in the nonperturbative regime, (see, e.g. Ref. [3,4,5,6]). This viewpoint implies that Gribov copies are necessarily present, that gauge orbits are multiply represented, and that the definition of nonperturbative QCD must be considered with some care. Since these nonperturbative definitions of QCD appear to be different, establishing which is the one appropriate for the description of the physical world is of considerable importance.

144

References 1. L. Giusti, M. L. Paciello, C. Parrinello, S. Petrarca and B. Taglienti, Int. J. Mod. Phys. A16, 3487 (2001) [arXiv:hep-lat/0104012] and references therein. 2. P. van Baal, arXiv:hep-th/9711070. 3. H. Neuberger, Phys. Lett. B183, 337 (1987). 4. M. Testa, Phys. Lett. B429, 349 (1998) [arXiv:hep-lat/9803025]. 5. M. Testa, arXiv:hep-lat/9912029. 6. R. Alkofer and L. von Smekal, Phys. Rep. 353, 281 (2001) [arXiv:hep-ph/0007355] and references therein.

Q U A R K MODEL A N D CHIRAL S Y M M E T R Y A S P E C T S OF EXCITED B A R Y O N S

A. H O S A K A Research Center for Nuclear Physics, Osaka University, Ibaraki 567-0047 Japan E-mail: [email protected]

We investigate properties of excited baryons from two different points of view. In one of them, the argument is based on a quark model with an emphasis on flavour independent nature of the baryon spectra. This easily indicates positions of missing resonances. In the other, we discuss how baryon resonances are classified by the chiral symmetry group. It is shown that there are two baryon representations. We present nir production experiments to distinguish the two baryon representations.

1. Introduction It is important that we are able to control meson-baryon dynamics quantitatively as much as possible. This will provide not only a clue to understand fundamentals of hadron physics but also a basis for the description of more complicated system such as hadronic and/or quark matter in extreme conditions. The latter is one of the important subjects of the JHF project 1 . Quark models provide now standard methods to describe hadrons, as we know empirically that they work well over a wide range of phenomena 2 . Flavour symmetry is implemented by regarding quarks as fundamental representations of SU(6) spin-flavour symmetry. By choosing constituent quark masses appropriately, baryon magnetic moments are well reproduced. Introducing another scale parameter for quark distributions (oscillator parameter for a quark confining force), numbers of excited states are predicted which can be compared with data, although fine tuning is necessary to get reasonable results. We also know that chiral symmetry dictates much of hadron dynamics 3 . Chiral symmetry with its spontaneous breaking is indeed one of important aspects of low energy QCD. Interactions of pions and kaons, which are the Nambu-Goldstone bosons, are described well by the current algebra and chiral perturbation theory 4 . To the present date, it is not obvious how chiral symmetry and the quark

145

146

model aspects can merge into a single framework. Nevertheless, here we attempt to demonstrate examples based on the above two aspects for excited baryons. In Sec. 2, we study masses and transition amplitudes of excited baryons in a quark model. We point out that baryon spectra resemble very much rotational bands of a deformed harmonic oscillator model. As an illustration, we compute not only masses but also pion transition amplitudes between the same rotational band. In Sec. 3, we propose another point of view where positive and negative parity baryons are considered as members of a multiplet of chiral symmetry. Two distinguished assignments of chiral symmetry of baryons are discussed from a group theoretical point of view. Experiments which can observe the two assignments are proposed. The final section is devoted to a brief summary of this report. 2. Quark Model Description and Deformed Excited States 2.1.

Masses

Let us first look at experimental data as shown in Fig. 1. We take 49 states out of 50 states of three and four stars, and several states with one and two stars 5 . In showing experimental data we follow the prescriptions: (1) Masses are measured from the ground states in each spin-flavour multiplet, in order to subtract the spin-flavour dependence which is well established by the GellMann-Okubo mass formula. (2) Masses of 2 8 M S , 4&MS states for positive parity, and of 48JWS states for negative parity are reduced by 200 MeV. Then the resulting mass spectra show a very simple systematics, which remarkably resembles rotational bands. It is also important that the regularity is common to all channels independent of flavour. Hence, we consider a quark model with a deformed harmonic oscillator potential 6 : HDOQ -

7^ + 2 m^lx2i + "&? + w ^. 2 )

22 »=l

L

(1)

J

Here we ignore interactions due to gluons and mesons. The only dynamics here is the shape change which is described by the deformed oscillator potential, u)x ^ uiy 7^ <JJZ . The Hamiltonian looks very simple, but it can indeed reproduce the structure of the mass spectra in Fig. 1. After removing the centre of mass motion, we find an intrinsic energy Eint(Nx,Ny,Nz) = (Nx + l)cux + [Ny + l)w„ + {Nz + l)w z , where Nx,Ny, Nz are the sum of principal quantum numbers for the internal degrees of freedom for the p and A coordinates. Imposing the volume conservation condition LOxwyu)z = w 3 , the minimum energy occurs when the system is deformed. When

147

Nx = Ny = 0,NZ = N, one obtains a prolate deformation. Our discussions in what follows are based on this prolate deformation, since it is energetically most favourable. Physical states are obtained by rotating the deformed states, whose energies are computed by the standard cranking method 6
— f^int

2000 1500

'-H,„(2420) F^(2390) Fl9<2000) , J

P„(1720)

—"

P„('M°)

F

»'2"°»

->,,,w-p^r;—;•"=""—-500

P„(14«)

"

P0,(1M0>

P01(1810)

P,a(2M0) F„(1950) F„(20701 .P M (1«0) . .

F1S(1S15) P„(1880)

P„(1W0)

""~~ F17<2030)

L=0

13

P^deoo)

P,3(1385)

o -

L-2

P},(1910)

P33I1232)

'10,

10 0

N

JJ1500 Dlg(2200) • 1^(2350) " •

G^piOO)

q,(210O)

£1000 S„(153a

D „(1700)

DtelieSOl „

(1B3()1

S„(1750)

I) 0^(1520)

B 500

(1800) -D13(1670) •

S„(1650)

D1S(1775)

8

MS

%

Ss,(1620)

s„,
10„

N Figure 1.

SU(3) baryon masses.

148 The only parameter in the DOQ model, w, is determined by the average of the excitation energies of the first l / 2 + excited states in the 2 8 multiplet. We find u — 644 MeV. The resulting mass spectra are compared with the observed spectrum in Fig. 1. The agreement between the theoretical and observed states is remarkable. In Fig. 1 it is easily recognized that there are many missing states in the region of L — 4 and L = 3. It would be interesting if those states are observed in experiments.

2.2. Pion

Transition

If the deformation of the system is large, it has significant influence on transition amplitudes. A familiar example is the E2 transition between the same rotational band of deformed nuclear states, which is called intra-band transitions (see Fig. 2). Here a characteristic feature is that the decay rates are written as products of common factors of intrinsic moments and geometric factors of rotation (Clebsch-Gordan coefficients). Consequently, ratios between decay rates are given solely by the geometric factors. This is called the Alaga rule in nuclear physics 7 . In the case of baryon decays, the strong interaction is dominant and, therefore, we investigate decays accompanied by a pion emission. We are interested in the intra-band transitions where the orbital angular momentum changes by AL = 2. Due to the pseudoscalar nature, the angular momentum and parity of the pion are l^ = 1 + or 3 + . The selection rule of the transitions for positive parity excited states is shown in Fig. 2. Selection rules (even parity)

Figure 2. Inter- and intra-band transitions. Selection rules for even parity transitions are also shown.

Assuming the semiclassical nature of large deformation, we compute the amplitudes in the collective coordinate cranking method. Let us briefly outline how the decay amplitudes are computed 8 . Write a deformed excited state as V'intC*)- Here, for simplicity, x denotes the internal coordinates of three quarks. In the semiclassical picture, the deformed intrinsic state V'int(^) rotates

149

adiabatically: rj;TOt(A,x) — R(A)ipint(x), where R(A) is the rotation matrix of angle A. By taking a matrix element of the nqq interaction Hamiltonian, hwqq(x) = —#/(2m)V • ara (g is the nqq coupling constant, m the constituent quark mass and a the isospin index of the pion.), in the rotating state i>TOt, we obtain the collective Hamiltonian for the decay : Hco\\{A) = J d3xi>lot{A,x)Kqq{x)j;rot{A,x).

(3)

Now a collective rotational wave function of orbital angular momentum L is given by the standard D function of rank L, which is combined with a spin wave function Xm to form the state of total spin (J, M): ^JM(A) = [D%l0(A),xli]JM • I The decay amplitude of N*(JM) -»• •KN*(J'M ) is then given by = jd[A]

&J,M,(A)HCO]](A)*JM(A)

.

(4)

Computation of this matrix element is tedious but straightforward. The final expression is rather lengthy, but as anticipated, it is expressed as products of geometric factors and intrinsic moments of the type : Q'-(A) = (mt\jlw(kx)Y,w0(x)\int).

(5)

In Table 1 the decay widths for even parity intra-band transitions (L = 4 —> V — 2) are summarized. There is one large number, T(7/2 + —> 5/2 + ), but others are small. In particular, those of L — 2 —> V = 0 are negligible due to small phase space. The large decay width of T(7/2 + -» 5/2 + ) may explain the missing nucleon resonance around the mass 2200 MeV in the G71 pion channel. Experimentally, observation of intra-band transitions requires measurements of at least two pions, since the daughter resonance of the intra-band transition still decays into the ground state. Table 1. MeV.

Decay widths of L = 4 nucleon resonances in

9/2 -4 5/2 18.9

9/2 -> 3/2 35.3

7/2 -4 5/2 115

7/2 ->• 3/2 0.5

Apart from the absolute values of the various decay widths, it is of interest to see, in the long wave length limit, the ratios of transitions which are dominated by the geometric factors. Consider, for instance, 1% = 1 + transitions of L + 2 (Ji = L + 3/2) -J- L(Jf = L+ 1/2) and of L (J; = L - 1/2) -> L - 2 {Jf = L - 3/2). When L = 2 these transitions are L = 4 ( J f = 7/2+) ->• L = 2 (Jf = 5/2+) and L = 2 (Jf = 3/2+) -> L = 0 (jf = 1/2+) (see the

150 GeV

Baryons

Mesons

,_ AT<1535)

1.5 [.°l(1260)

/,(1285) ''

_ 1(1300) °

AT, (1400) I*

1(1295)

j-

£(1190)

K,<1270)

/1(1120)

,a 0 (980)

1 0

p(770)

OX 780)

° /- 0 (975)

1


, K (890)

5"

, _

j.

,."('232) 2

. W(939) «—^

0.5 „- *(139)

Figure 3. Mass splittings of positive and negative parity hadrons in various channels. The uncertain mass of sigma (
right part of Fig. 2. Apart from the k7 (pion momentum) dependence due to the phase space, the ratio of the decay widths becomes ri+2(J,=L+3/2)-)-L(J / =L+l/2) _ (L + 1)(2L - 1) ^L(Ji=L-l/2)-yL-2(Jf=L-3/2)

(L ~ 1)(2L + 3)

-> 9/7

(for L = 2).

(6)

This is analogous to the Alaga rule in nuclear physics7. 3. Chiral Symmetry of Baryons 3.1. Chiral Partner

Baryons

Let us now turn to another topic : the role of chiral symmetry. Naively, if chiral symmetry is a good symmetry, one would expect that hadrons are classified by linear representations of chiral symmetry, where positive and negative parity states form a representation. In reality, however, due to spontaneous breaking, such a symmetry pattern may not necessarily be realized. Still, one can imagine that observed hadrons would belong to linear representations when chiral symmetry is restored. This is suggested by a common pattern of the mass splitting between the positive and negative parity states as shown in Fig. 3. In particular, all the mass splittings are of the same order of about half GeV, which could be generated by the spontaneous breaking. In this way, one could imagine, for instance, that the ground state nucleoli belongs to the representation, ( | , 0) © (0, ^), of the isospin chiral group SU(2)/{X SU(2)i. The pions are assigned to ( | , i ) which is supplemented by the still controversial particle, sigma. Another example is a representation for p and ai, which can be identified with (1,0) ® (0,1). In this report, we would like to discuss particle classifications for baryons 9,10,11,12 . So far, such an investigation has not been done very often, although it is a very fundamental question in hadron physics.

151

3.2. Naive

and Mirror

Assignments

of

Baryons

Chiral symmetry is a flavour symmetry for the right and left handed fermions. In QCD, the current quarks possess this symmetry. The right and left handed components (of the Lorentz group) are defined in terms of a four component nucleon field TV by N = Nr + Nr,

Nr = ^ ^ N ,

Nl=]—^N.

(7)

For SU(2).RX S U ( 2 ) L , chiral (symmetry) transformations are defined as flavour transformations for the right and left handed components: Nr -> giiNr , Wj —>• 9LNI , where gR and gL are elements of SU(2)i{ and SU(2)^, respectively. Now consider two fermions and their transformation rules. Naively, one would expect the same transformations for both N\ and N2 > Nir - • gRNlr,

Nu -> gLNu ;

N2r -»• gnN2r,

N2l -> gLN2i.

(8)

13

As first noticed by Lee , it is also possible to assign the opposite transformation for the second nucleon, Nir

- > gRNlr

, Nu

- > 9LNu

\

A ^ ^ O L # 2 r , N2, - > gRN2l

•

(9)

The point is that it is sufficient to assign different internal symmetries for the right and left handed components. It does not matter whether which of SU(2)ft or SU(2)x is assigned. Since the transformations of the right and left handed components are interchanged, the second assignment of fiq. (9) is called mirror, while the first assignment of Eq. (8) is called naive. From Eqs. (8) and (9), it is obvious that the sign of the axial charge of N2 is opposite to that of Ni in the mirror assignment. The sign of g& was discussed by Weinberg for the case of single fermion 15 . It becomes significant when there are two fermions (nucleons here). Another feature of the mirror assignment is that it is possible to introduce the chiral invariant mass term in the form of interaction between Ni and N2: mo(NiN2 + N2Ni), where mo is a parameter which can not be determined within the present theoretical arguments. Such a mass term was used by DeTar and Kunihiro 14 in order to explain a possibility of finite mass of baryons when chiral symmetry is restored. 3.3. iv and rj Productions

at Threshold

Region

In order to distinguish the two chiral assignments, we propose an experimental method to study the two chiral assignments 10 . As discussed in the preceding sections, one of the differences between the naive and mirror assignments is the relative sign of the axial coupling constants of the positive and negative

152 (1)

(2)

/

VJV_/ N—

1

i

4

(3)

\ — w

SnNN

/ / a

% — 3

-it

&AI*W

/ / >•

1

gxNN

Figure 4. Dominant three diagrams for 7r and 77 productions. The incident wavy line is either a pion of photon. In fact, there are in total 12 resonance dominant diagrams.

parity nucleons. In the following discussions, we identify N+ ~ JV(939) and N_ ~ 7V(1535) = N*. Strictly, the identification of the negative parity nucleon with the first excited state Af(1535) is no more than an assumption. From the experimental point of view, however, ./V(1535) has a distinguished feature that it has a strong coupling with an rj meson, which can be used as a filter for the resonance. In practice, we observe the pion couplings which are related to the axial couplings through the Goldberger-Treiman relation gnN±N±fw — SuM± . Let us consider n and 77 productions induced by a pion or photon. Suppose that the two diagrams of (1) and (2) as shown in Fig. 4 are dominant in these processes. Modulo energy denominator, the only difference of these processes is due to the coupling constants g^NN and gnN'N'- Therefore, depending on their relative sign, cross sections are either enhanced or suppressed. In the pion induced process, due to the p-wave coupling nature, another diagram (3) also contributes substantially. In actual computation, we take the interaction Lagrangians : L„NN = g-wNNNi-frT • TTN , LvNN.

= gnNN' {Nr)N* + N*T)N),

L*NN> = g*NN> (NT • TTN* + N*T • nN),

LnN'N' - gnN'N'{N*if5T

(10)

• nN*) .

We use these interactions both for the naive and mirror cases with empirical coupling constants for gVNN ~ 13, g^NN* ~ 0.7 and gVNN* ~ 2. The coupling constants g-xNN' ~ 0.7 and gVNN' ~ 2 are determined from the partial decay widths 5 , rjv(i535)->irjv ^ rw(i535)-n?N ~ 70 MeV. The unknown parameter is the gwN*N' coupling. One can estimate it by using the theoretical value of the axial charge g*A and the Goldberger-Treiman relation for ./V*. When g*A = ± 1 for the naive and mirror assignments, we find gnN'N' = ^ " ^ w / / ^ ~ ±17. Here, just for simplicity, we use the same absolute value as gnNN • The coupling values used in our computations are summarized in Table 2. Several remarks are in order here: • We assume resonance (N*) pole dominance. This is considered to be good, particularly for the t] production at the threshold region, since •q is dominantly produced by N*.

153 Table 2. Parameters used in our calculation. mN 938 (MeV)

m^i' 1535 (MeV)

TJV*

140 (MeV)

9TTNN

13

I Tip -> JtriN ]

9nNN*

9r)NN'

0.7

9nN'N'

2.0

13 (naive) - 1 3 (mirror)

>JtT)N 1

Total Cross Section ~ 60 - 40 <j

| 20 H

0 540

560 580 Pen, ( M e V / <=)

600

560 580 600 620 640 660 680 Photon energy [MeV]

Pion momentum distribution [mbf

141 = 575 MeV/c 12- IP 1 am. 1

Angular distribution

E = 590 MeV

Mirror ^s.

8-

Naiv

6Naive

4-0.5

0.0 „„_D

cm.

0.5

Mirr 10

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 cog

Figure 5. Various cross sections for w and r\ productions.

• There are in total twelve resonance dominant diagrams. Due t o t h e energy denominator, t h e three diagrams in Fig. 4 are d o m i n a n t . • Background contributions, in which two meson (seagull) or three meson vertices appear, are suppressed due t o G-parity conservation. We show various cross sections for the pion and photon induced processes in Fig. 5. We observe t h a t : (1) Total cross sections are of the order of micro barn t h a t is well accessible by experiments. (2) In t h e photon induced process, t h e diagrams (1) and (2) interfere destructively. In the pion induced case, due t o the m o m e n t u m dependence of the initial vertex, t h e third term (3) becomes d o m i n a n t . (3) In the pion induced reaction, the angular distribution of the final s t a t e pion differs crucially depending on t h e sign of the wNN a n d nN*N* couplings.

154 4.

Summary

In this report, we have discussed two i m p o r t a n t aspects of baryon dynamics: one is a quark model aspect and the other is t h a t of chiral symmetry. In baryon mass spectra, it is emphasized once again t h a t a simple non-relativistic quark model of spatial defamation works well for almost all SU(3) baryon masses independent of flavour. This suggests t h a t this i m p o r t a n t dynamics has gluonic nature. At the same time, chiral symmetry and relevant dynamics, especially due to the chiral mesons, have also significant roles. Evidently the coupling of the chiral mesons renormalizes the simple quark model calculations. A more fundamental question is how multiplets of chiral symmetry are related to those of the quark model states. In particular, from the chiral s y m m e t r y argument, we have shown t h a t two different types of baryons are possible which are distinguished by the sign of the axial charge. It is interesting to investigate a microscopic origin of such states, perhaps from meson effects, relativistic quark effects and etc.

Acknowledgments T h e author thanks his collaborators, Miho Koma, Hiroshi Toki, Daisuke Jido and Makoto Oka on the subjects presented here. References 1. http://jkj.tokai.jaeri.go.jp/ 2. N. Isgur and G. Karl, Phys. Rev. D18, 4187 (1978); ibid. D19, 2653 (1979). 3. A. Hosaka and H. Toki, "Quarks, baryons and chiral symmetry", World Scientific (2001), Singapore. 4. J. Gasser and H. Leutwyler, Ann. Phys. 158, 142 (1984). 5. Particle Data Group, Review of particle physics, Eur. Phys. J. C15 (2000). 6. M. Takayama and H. Toki and A. Hosaka, Prog. Theor. Phys. 101, 1271 (1999). 7. G. Alaga and V. Paar, Phys. Lett. B61, 129 (1976). 8. M. Koma, PhD. Thesis, Osaka University (2002). 9. D. Jido M. Oka and A. Hosaka, Prog. Theor. Phys. 106, 873 (2001). 10. D. Jido M. Oka and A. Hosaka, Prog. Theor. Phys. 106, 823 (2001). 11. D. Jido, Y. Nemoto, M. Oka and A. Hosaka, Nucl. Phys. A671, 471 (2000). 12. D. Jido, T. Hatsuda and T. Kunihiro, Phys. Rev. Lett. 84, 3252 (2000). 13. B. W. Lee, Chiral Dynamics, Gordon and Breach, New York, (1972). 14. C. DeTar and T. Kunihiro, Phys. Rev. D39, 2805 (1989). 15. S. Weinberg, Phys. Rev. Lett. 65, 1177 and 1181 (1990).

Q U E N C H E D CHIRAL PHYSICS IN B A R Y O N MASSES

R. D . Y O U N G , D. B . L E I N W E B E R A N D A. W . T H O M A S Special Research Centre for the Subatomic Structure of Matter, and Department of Physics and Mathematical Physics, University of Adelaide, Adelaide SA 5005, Australia E-mail: [email protected]; [email protected]; athomas ©physics, adelaide.edu.au S. V. W R I G H T Division of Theoretical Physics, Department of Mathematical University of Liverpool, Liverpool L69 3BX, U.K. E-mail: [email protected]

Sciences,

Recent work has identified t h a t the primary differences between quenched and dynamical spectroscopy can be described by chiral loop effects. Here we highlight the features of this study.

1. Introduction Chiral symmetry has long been known to play an important role in the low energy properties of QCD. Theoretical studies of the non-perturbative features of QCD, expressed in terms of the fundamental theory, have proven to be most successful in the field of lattice gauge theory. Lattice studies are typically restricted to relatively large values of the u and d masses, where chiral effects are highly suppressed. As a consequence, direct observation of chiral properties is a challenge in lattice simulations. The study of low-lying baryon spectroscopy in both quenched and full QCD has provided a direct connection between lattice results and chiral physics 1,2 . The fact that one is restricted to quark masses much larger than the physical values means that, in addition to all the usual extrapolations (e.g. to the infinite volume and continuum limits), if one wants to compare with empirical hadron observables, one must also have a reliable method of extrapolation to the chiral limit. Any such extrapolation must incorporate the appropriate chiral corrections, arising from Goldstone boson loops, which give rise to rapid, non-linear variations as the chiral limit is approached. The importance of incorporating such behaviour has been successfully demonstrated for a number of hadronic observables, see Ref. [3] for a review and the references contained

155

156

therein. The quenched approximation is a widely used tool for studying nonperturbative QCD within numerical simulations of lattice gauge theory. With an appropriate choice of the lattice scale and at moderate to heavy quark masses, this approximation has been shown to give only small, systematic deviations from the results of full QCD with dynamical fermions. Although no formal connection has been established between full and quenched QCD, the similarity of the results has led to the belief that the effects of quenching are small and hence that quenched QCD provides a reasonable approximation to the full theory 4 . Under a more reliable choice of lattice scale, where chiral effects are negligible, clear differences are observed between quenched and dynamical results 5 . 2. Chiral Physics and the Quenched Approximation Chiral symmetry is spontaneously broken in the ground state of QCD. As a consequence, the pion is a Goldstone boson characteristic of this broken symmetry. The pion mass then behaves as m^ oc mq, the well known Gell-MannOakes-Renner (GOR) relation. In principle, this relation is only guaranteed for quark masses near zero. Explicit lattice calculations show that it holds over an enormous range, as high as m , ~ 1 GeV. These almost massless Goldstone bosons couple strongly to low-lying baryon states — particularly the nucleon (N) and delta (A). Meson-loop diagrams involving Goldstone bosons coupling to baryons give rise to non-analytic behaviour of baryon properties as a function of quark mass. At light quark masses these corrections are large and rapidly varying. At heavy quark masses these contributions are suppressed and hadron properties are smooth, slowly varying functions of quark mass. The scale of this transition is characterised by the inverse size of the pion-cloud source. Below pion masses of about 400-500 MeV these non-analytic contributions become increasingly important, while they rapidly become negligible above this point. Since lattice simulations are typically restricted to the domain where mn > 500 MeV, these rapid-varying chiral effects must be incorporated phenomenologically. Within the quenched approximation dynamical sea quarks are absent from the simulation. As a consequence the structure of meson-loop contributions is modified. In the physical theory of QCD, meson-loop diagrams can be described by two topologically differing types. A typical meson-loop diagram may be decomposed into those where the loop meson contains a sea quark, such as Fig. 1(a), and those where the loop meson is comprised of pure valence quarks, see Fig. 1(b). These diagrams involving the sea quark, type (a), are obviously absent in the quenched approximation and consequently only

157

(a)

Figure 1.

(b)

Quark flow diagrams of pion loop contributions appearing in QCD.

(a)

(b)

Figure 2. Quark flow diagrams of chiral vf loop contributions appearing in QQCD: (a) axial hairpin, (b) double hairpin.

a subset of the contributions to the physical theory are included. This type of argument, together with SU(6) symmetry, is precisely that described for the evaluation of non-analytic contributions to baryon magnetic moments by Leinweber6. This quark flow approach is analogous to the original approach to chiral perturbation theory for mesons performed by Sharpe 7 ' 8 . In addition to the usual pion loop contributions, quenched QCD (QQCD) contains loop diagrams involving the flavour singlet rf which also give rise to important non-analytic structure. Within full QCD such loops do not play a role in the chiral expansion because the rf remains massive in the chiral limit. On the other hand, in the quenched approximation the rf is also a Goldstone boson 8 and the rf propagator has the same kinematic structure as that of the pion. As a consequence there are two new chiral loop contributions unique to the quenched theory. The first of these corresponds to an axial hairpin diagram such as that indicated in Fig. 2(a). This diagram gives a contribution to the chiral expansion of baryon masses which is non-analytic at order m^. The second of these new rf loop diagrams arises from the double hairpin vertex as pictured in Fig. 2(b). This contribution is particularly interesting because it involves two Goldstone boson propagators and is therefore the source of more singular non-analytic behaviour - linear in m^. In studying the extrapolation of quenched lattice results it is essential to treat these contributions on an equal footing to the pion-loop diagrams discussed earlier.

158

3. Chiral Extrapolations In general, the coefficients of the leading (LNA) and next-to leading nonanalytic (NLNA) terms in a chiral expansion of baryon masses are very large. For instance, the LNA term for the nucleon mass is 8rrvN ' = —5.6 m% (with m , and 5mN in GeV). With mw = 0.5GeV, quite a low mass for current simulations, this yields Sm^N ' — 0.7 GeV — a huge contribution. Furthermore, in this region hadron masses in both full and quenched lattice QCD are found to be essentially linear in rn\ or equivalently quark mass, whereas 8rrvN ' is highly non-linear. The challenge is therefore to ensure the appropriate LNA and NLNA behaviour, with the correct coefficients, as mn —• 0, while making a sufficiently rapid transition to the linear behaviour of actual lattice data, where ra, becomes large. A reliable method for achieving all this was proposed by Leinweber et al.9 They fit the full (unquenched) lattice data with the form: MB =aB+pBml

+J2B(mn,A),

(1)

where Y,B is the total contribution from those pion loops which give rise to the LNA and NLNA terms in the self-energy of the baryon. The extension to the case of quenched QCD is achieved by replacing the self-energies, S s , of the physical theory by the corresponding contributions of the quenched theory 2 ,

tB. The linear term of Eq. (1), which dominates for mn S> A, models the quark mass dependence of the pion-cloud source — the baryon without its pion dressing. This term also serves to account for loop diagrams involving heavier mesons, which have much slower variation with quark mass. The diagrams for the various meson-loop contributions are evaluated using a phenomenological regulator. This regulator has the effect of suppressing the contributions as soon as the pion mass becomes large. At light quark masses the self-energies, T,B, provide the same non-analytic behaviour as xPT, independent of the choice of regulator. Therefore the functional form, Eq. (1), naturally encapsulates both the light quark limit of %PT and the heavy quark behaviour observed on the lattice. We consider the leading order diagrams containing only the lightest Goldstone degrees of freedom. These are responsible for the most rapid non-linear variation as the quark mass is pushed down toward the chiral limit. In the physical theory we consider only those diagrams containing pions, as depicted in Fig. 3. In QQCD there exist modified pion loop contributions and the additional structures arising from the rf behaving as a Goldstone boson. The diagrams contributing to the nucleon self-energy in the quenched approximation are shown in Fig. 4, the A can be described by analogous diagrams.

159

+ EJV=

L

:

'

+ F i g u r e 3 . Illustrative view of the meson-loop self-energies, S s , infullQCD. These diagrams give rise to the LNA and NLNA contributions in the chiral expansion. Single (double) lines denote propagation ofaJV (A).

+ -'N

+ —

-

*

—

—

<

•

i

—

Figure 4. Illustrative view of the meson-loop self-energies, Eg, in quenched QCD. These diagrams give rise to the LNA and NLNA contributions in the chiral expansion. A cross represents a hairpin vertex in the 77' propagator. Single (double) lines denote propagation of a AT (A).

For the evaluation of the quenched quantities we assume that the parameters of the chiral Lagrangian exhibit negligible differences between quenched and 3-flavour dynamical simulations. This is a working hypothesis with no better guidance yet available, but the successful results of this work demonstrate the self-consistency of such an assumption 1 . Only with further accurate lattice simulations at light masses will one be able to determine the extent to which our hypothesis holds. To highlight the differences in the self-energy contributions we show the net contributions to the A in full ( S A ) and quenched ( E A ) QCD in Fig. 5. For details of the breakdown of the individual contributions we refer the reader to our longer article 2 . The significant point to note is that whereas the meson cloud of the A is attractive in full QCD, it exhibits repulsive behaviour within the quenched approximation. 4. Fitting Lattice Data The lattice data considered in this analysis comes from the recent paper of Bernard et al.h These simulations were performed using an improved KogutSusskind quark action, which shows evidence of good scaling 10 . Unlike the standard Wilson fermion action, masses determined at finite lattice spacing

160

0.2 0.1

P

0.0

O,

-0.2 -0.3 0.0

0.1

0.2 mj*

Figure 5. 0.8 GeV.

0.3 0.4 (GeV 2 )

0.5

0.6

Net contributions to the A self-energy evaluated with dipole regulator at A =

are good estimates of the continuum limit results. We are particularly concerned with the chiral extrapolation of baryon masses and how their behaviour is affected by the quenched approximation. In such a study, it is essential that the method of scale determination is free from chiral contamination. One such method involves the static-quark potential. As low-lying pseudoscalar mesons made of light quarks exhibit negligible coupling to hadrons containing only heavy valence quarks, the low energy effective field theory plays no role in the determination of the scale for these systems. In fixing the scale through such a procedure one constrains all simulations, quenched, 2-flavour, 3-flavour etc., to match phenomenological static-quark forces. Effectively, the short range (0.35 ~ 0.5 fm) interactions are matched across all simulations. A commonly adopted method involving the static-quark potential is the Sommer scale 11,12 . This procedure defines the force, F(r), between heavy quarks at a particular length scale, namely TQ ~ 0.5 fm. Choosing a narrow window to study the potential avoids complications arising in dynamical simulations where screening and ultimately string breaking is encountered at large separations. The lattice data analysed in this report uses a variant of this definition, choosing to define the force5 at r\ = 0.35 fm via rlF(r\) = 1.00. The non-analytic chiral behaviour is governed by the infrared regions of the self-energy integrals. Due to the finite volume of lattice simulations much of

161 Table 1. Best fit parameters for both full and quenched d a t a sets with dipole regulator, A = 0.8 GeV. All masses are in GeV. Simulation Physical Quenched

apf 1.27(2) 1.24(2)

0ff 0.90(5) 0.85(6)

»A 1.45(3) 1.45(4)

/?& 0.74(8) 0.72(11)

this structure will not be captured. For this reason we evaluate the self-energy corrections with pion momenta restricted to those available on the particular lattice 2,13 . In this way we get a first estimate of the discretisation errors in the meson-loop corrections. In no way does this account for any artefacts associated with the pion-cloud source. Current lattice data is insufficient to reliably extract the dipole regulator parameter, A. We fit all data choosing a common value to describe all vertices, A = 0.8 GeV. This choice has been optimised 2 to highlight the main result of this analysis. We note that the value of A which we find is indeed consistent with phenomenological estimates which suggest that this should be somewhat less than 1 GeV. We fit both quenched and dynamical simulation results to the form of Eq. (1) with appropriate discretised self-energies. These fits are shown in Fig. 6. It is the open squares which should be compared with the lattice data. These points correspond to evaluation of the self-energies on the discretised momentum grid. The lines represent a restoration of the continuum limit in the self-energy evaluation. Discrepancies between the continuum and discrete version only become apparent at light quark masses, this corresponds to the Compton wavelength of the pion becoming comparable to the finite spatial extent of the lattice. The important result of this study is that the behaviour of the pion-cloud source is found to be quite similar in both quenched and dynamical simulations. Once the self-energies corresponding to the given theory are incorporated into the fit, the linear terms are found to be in excellent agreement. Our best fit parameters, for the selected dipole mass A = 0.8 GeV, are shown in Table 1. Here we observe the remarkable agreement between quenched and dynamical data sets for N and A masses over a wide range of pion mass. This leads to the interpretation that the primary effects of quenching can be described by the modified chiral structures which give rise to the LNA and NLNA behaviour of the respective theories. The success of fitting the iV and A data sets with a common regulator lends confidence to an interpretation of the mass splitting between these states. Examination of the self-energy contributions in full QCD suggests that only about 50 MeV of the observed 300 MeV N-A splitting arises from pion loops 2 .

162

0.8 ' 0.0

' 0.1

' 0.2

mwB

' ' 0.3 0.4 (GeV 3 )

' 0.5

' 0.6

Figure 6. Fit (open squares) to lattice data 6 (Quenched A, Dynamical A) with adjusted self-energy expressions accounting for finite volume and lattice spacing artifacts. The infinitevolume, continuum limit of quenched (dashed lines) and dynamical (solid lines) are shown. The lower curves and d a t a points are for the nucleon and the upper ones for the A.

The dominant contribution to the hyperfine splitting would then naturally be described by some short-range quark-gluon interactions. 5. Conclusions We have demonstrated the strength of fitting lattice data with a functional form which naturally interpolates between the domains of heavy and light quarks. The extrapolation formula gives a reliable method for the extraction of baryon masses at realistic quark masses. Although the quenched approximation gives rise to more singular behaviour in the chiral limit, these contributions are quickly suppressed with increasing pion mass. Within the quenched approximation only limited curvature is observed for the N down to low quark masses. In contrast, we find some upward curvature of the A mass in the light quark domain. The observation that the source of the meson cloud has remarkably similar behaviour within both quenched and physical simulations is of considerable importance. One can describe the primary effects of quenching by the mesonloop contributions which give rise to the most rapid, non-linear variation at light quark masses. This leads one to the possibility of applying this result to obtain more physical results from quenched simulations. The structure of

163 the meson-cloud source can be determined from quenched simulations and then the chiral structures of t h e physical theory can be incorporated phenomenologically. Natural extension of this work leads to the analysis of further hyperons to investigate the applicability over a range of particles.

References 1. R. D. Young, D. B. Leinweber, A. W. Thomas and S. V. Wright, hep-lat/0111041. 2. R. D. Young, D. B. Leinweber, A. W. Thomas and S. V. Wright, hep-lat/0205017. 3. W. Detmold, D. B. Leinweber, W. Melnitchouk, A. W. Thomas and S. V. Wright, Pramana 57, 251 (2001); nucl-th/0104043. 4. S. Aoki et al., CP-PACS Collaboration, Phys. Rev. Lett. 84, 238 (2000). 5. C. W. Bernard et al., Phys. Rev. D64, 054506 (2001). 6. D. B. Leinweber, Nucl. Phys. Proc. Suppl. 109, 45 (2002). 7. S. R. Sharpe, Phys. Rev. D 4 1 , 3233 (1990). 8. S. R. Sharpe, Phys. Rev. D46, 3146 (1992). 9. D. B. Leinweber, A. W. Thomas, K. Tsushima and S. V. Wright, Phys. Rev. D 6 1 , 074502 (2000). 10. C. W. Bernard et al., MILC Collaboration, Phys. Rev. D 6 1 , 111502 (2000). 11. R. Sommer, Nucl. Phys. B411, 839 (1994). 12. R. G. Edwards, U. M. Heller and T. R. Klassen, Nucl. Phys. B517, 377 (1998). 13. D. B. Leinweber, A. W. Thomas, K. Tsushima and S. V. Wright, Phys. Rev. D64, 094502 (2001).

QCD AT NON-ZERO CHEMICAL POTENTIAL AND TEMPERATURE FROM THE LATTICE

C.R. ALLTON* Department

of Physics,

Department

University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, U.KJ and of Mathematics, University of Queensland, Brisbane 4072, Australia E-mail: [email protected] S. E J I R I , S.J. H A N D S

Department

of Physics,

University of Wales Swansea, Singleton Park, Swansea, SA2 U.K. E-mail: [email protected] ; [email protected]

8PP,

O. K A C Z M A R E K , F . K A R S C H , E . L A E R M A N N , C H . S C H M I D T Fakultat fur Physik, Universitat Bielefeld, D-33615 Bielefeld, Germany E-mail: [email protected] ; [email protected] ; [email protected] ; [email protected] L. S C O R Z A T O Department

of Physics,

DESY

University

of Wales Swansea, Singleton Park, Swansea, SA2 U.K. and Theory Division, Notkestrasse 85, D-2260S Hamburg, Germany. E-mail: [email protected]

A study of QCD at non-zero chemical potential, 11, and temperature, T, is performed using the lattice technique. The transition temperature (between the confined and deconfined phases) is determined as a function of /i and is found to be in agreement with other work. In addition the variation of the pressure and energy density with [i is obtained for n > 0. These results are of particular relevance for heavy-ion collision experiments.

'speaker at the workshop 'permanent address

164

8PP,

165

1. Introduction The QCD phase diagram has come under increasing experimental and theoretical scrutiny over the last few years. On the experimental side, very recent studies of compact astronomical objects have suggested that their cores contain "quark matter", i.e. QCD in a new, unconfined phase where the basic units of matter are quarks, rather than nuclei or nucleons 1 . More terrestrially, heavy ion collision experiments, such as those performed at RHIC and CERN, are also believed to be probing unconfined QCD 2 . On the theoretical side, the study of QCD under these extreme densities and temperatures has proceeded along several fronts. One of the most promising areas of research is the use of lattice techniques to study either QCD itself, or model theories which mimic the strong interaction 3 . Clearly the most satisfying approach would be the former, i.e. a direct lattice study of QCD at various coordinates (T, p) in its phase space (fi is the chemical potential for the quark number). However, until recently, this has proved intractable at a practical level for very fundamental reasons. This is because the Monte Carlo integration technique, which is at the heart of the (Euclidean) lattice approach, breaks down when \i ^ 0. This work summarises one new approach which overcomes this problem and has made progress for /i ^ 0 and T =£ 0. In the next section a summary is given of the lattice technique and the problem incurred when p ^ 0. Section 3 describes the method used to overcome these difficulties, and Sec. 4 outlines the simulation details. The next two sections apply the method to variations in m and fi, and section Sec. 7 describes calculations of the pressure and energy density as functions of fi. A full account of this work is published elsewhere4.

2. Lattice Technique On the lattice, the quark fields, 4>(x), are defined on the sites, x, and the gluonic fields, U^(x), on the links x —• x + fi. Observables are then calculated via a Monte Carlo integration approach :

<"> = A T ^ — £ ' "(v^.to.

(i)

where ^ represents a sum over configurations {U, ^,r(>} which are selected with probability proportional to the Boltzmann weight P{{U,il>,xl>}) oc e-s({u,i>,i>}) w ith S a suitably defined (Euclidean) gaugeinvariant action. The fermionic part of this action is

166

SF = YJ^)iP+rn)^{x).

(2)

= M For /i = 0 it can be shown that this action produces a (real-valued) positive Boltzmann weight. Calculations at non-zero temperature, T ^ 0, can be performed by using a lattice with a finite temporal extent of Nta — 1/T, where Nt is the number of lattice sites in the time dimension. In practice, T is varied by changing the gauge coupling, go, and hence (through dimensional transmutation) the lattice spacing, a, rather than by changing Nt (which can only be changed in discrete steps!). The chemical potential is introduced into the system via an additional term in the quark matrix M, proportional to the Dirac gamma matrix, 70,

M^M + mo-

(3)

For fi ^ 0, this leads to a complex-valued Boltzmann "weight" which can therefore no longer be used as a probability distribution, and, hence, the Monte Carlo integration procedure is no longer applicable. This is known as the Sign Problem and has plagued more than a decade of lattice calculations of QCD at fj, 7^ 0. 3. Reweighting This section outlines the Ferrenberg-Swendsen reweighting approach 5 which is used to overcome the sign problem detailed in the previous section. Observables at one set of parameter values {fi,m,pi) (where /? = Q/gl, and m is the quark mass) can be calculated using an ensemble generated at another set of parameters (/3o,mo,fio) as follows,


l m

v

JDUQdetM(m,^e-s^

e (lndetM(m,/i)-lndetM(mo,/Jo)) e -S,,(/?)+S,,(/?o)\

/ e (lndetM(m,M)-lndetM(mo,/Jo)) e - S 9(f 8 )+ S <j(/ 8 <>)\

' ^ '

Here Sg is the gauge action. In principle, Eq. (4) can be used to map out the entire phase diagram of QCD. However, it has been found that its naive application fails since the relative size of fluctuations in both numerator and denominator tend to grow exponentially with the volume of the system studied. This is a signal that the

167

overlap between the ensemble at (/3o,m0,//o) is small compared with that at (P,m,n). One study which has had success in using Eq. (4) is that of Fodor and Katz 6 . They apply the reweighting approach not at an arbitrary point, (T,//), in the phase diagram, but rather they trace out the phase transition line Tc (/•*)• This method is successful presumably because the overlap between the ensembles remains high along the "coexistence" line which defines the transition. This paper utilises an alternative approach which Taylor expands Eq. (4) as a function of fj, (or m) and hence estimates the derivatives of various quantities w.r.t. fi (or m). Derivatives up to the second order are considered, thus for (ft) (in the case of a Taylor expansion in /J.) we have . {

>iM

_ ((ftp + Qifi + ft2p2) expjRin + R2H2 ~ Ag g )) (/?0i<
W

where [io = 0. In Eq. (5), Rn is the n—th derivative of the fermionic reweighting factor in Eq. (4), i.e. /detM(/i)\ _ ^ v

v

''

^a"lndetM(0)_ r

n—\

^ n=l

The Q„ are similarly the nth derivatives of the observable fi. Two observables are studied: the chiral condensate (tp1^) and the Polyakov Loop, L. Since L is a pure gluonic quantity, defined as Nt

1

(L) = (VJ2Trl[Ut(x,t)), X

(7)

t= U

all of its derivatives are zero. (Here, Vs is the spatial volume.) However, the expansion of (iprp) is more challenging since it is defined as (W) ~ ( t r M - 1 ) ,

(8)

and hence the application of Eq. (5) requires determinations of dnM~1/dfin. The susceptibilities, x, of both (iptp) and L are defined as usual by their fluctuations, e.g. XL = (volume factor) x {{L2) - (L)2).

(9)

These susceptibilities have a maximum at the transition point, j3c, and hence can be used to determine the transition point /J c (m,//).

168

4. Simulation Details The lattice calculations were performed using a "p4-improved" discretisation of the continuum action which is a sophisticated lattice action 7,8 maintaining rotational invariance of the free fermion propagator up to 0(p4) (p is the momentum here). Two dynamical flavours of quarks were used with a 16 3 x 4 lattice. Simulations were performed at quark mass, m = 0.1 and 0.2 which correspond to (unphysically heavy) pseudoscalar-vector meson mass ratios of MPS/MV « 0.70 and 0.85. Approximately 400,000 configurations were generated in total using around 6 months of a 128-node (64Gflop peak) APEmille in the University of Wales, Swansea. 5. Results for Mass Reweighting As a check of our method, we first use mass reweighting, since, unlike the \i ^ 0 case, there is no theoretical difficulty in simulating at virtually any value of m, and hence there are published data at a number of different ra values readily available for comparison. Reweighting in quark mass is simply a matter of setting \i = ^o = 0 in Eq. (4) and Taylor expanding in m rather than fi in Eqs. (5,6). (Note that dM/dm = 1 and dnM/dmn = 0 for n > 1.) We use the peak position in both the chiral condensate and Polyakov Loop susceptibilities, x ^ L, to determine the phase transition point /? c ("i). Figure 1 illustrates this by plotting XM as a function of ft (for m = 0.2). This shows the variation in the peak position of xsw, as ro is changed in steps of 0.01 around mo = 0.2.

Figure 1. The quark mass dependency of xr. m = 0.2 for the Chiral Susceptibility.

as a function of P in the neighbourhood of

169

Once these peak positions have been determined (for both XM a n d XL), 0C can be plotted as a function of m and a comparison can be made with other determinations. This is done in Fig. 2 where the f3c values from earlier work7 are also shown. The line segments around our data points at m = 0.1 and 0.2 indicate the gradient of (5C as a function of m using our Taylorexpanded reweighting technique. Shown in Fig. 2 are results from both the chiral condensate and Polyakov loop. These are both in perfect agreement (as expected). Furthermore, our method agrees with previously published data confirming the validity of the approach. i

3.75

1

1

V

J.O

-

3.7 CO." 3.65

-

/ O 8x4 (previous) A 16 x4 (previous) • 163x4 (chiral) • 163x4 (Polyakov)

3.6 3.55

"8 i

0

,

0.05

i 0.1

,

l

0.15

.

l

0.2

.

l

.

0.25

0.3

m Figure 2. work .

The transition "temperature", f)c, as a function of m in comparison with previous

6. Results for ft Reweighting We now turn to reweighting in chemical potential, fi. As in 6 , rather than applying the method to arbitrary parameter values, we trace out the transition point /3c(fi). Using Eq. (5), (ipij)) and the Polyakov Loop, L, are calculated as a function of y, together with their susceptibilities, x ^ , t ( s e e Eq. (9)). In Fig. 3, we plot XZ& against ft for various fi. Note that the peak position moves as fi changes. The determination of the transition point ftc from both the chiral condensate and the Polyakov loop (not shown here) are found to be in agreement. Because we have calculated all quantities to 0(/i 2 ) we can extract /3C and fit it to a quadratic in fi. (In fact, it can be shown4 that the first derivative

170

0.8

, , ___,_ — |lH).05 .U&£&gi

Figure 3. The n dependence of x x ^ as a function of /? in the neighbourhood of m = 0.2 for t h e chiral susceptibility.

d/Hc/dfi = 0.) We find d 2 /3 c /d^ 2 = 1.1 ± 50% for both quark masses m and 0.2. We now use

d2re

1

d 2 /? c

0.1

d£

(10) d^ / ( • da to convert d 2 /? c /d/i 2 into physical units, with the beta-function dp /do coming from string tension data 7 . We find ?c^npf « —0.14 at ma = 0.1. Figure 4 shows the phase transition curve Tc(n) obtained from this method. On this graph we have plotted the Fodor-Katz point 6 which is within our errors, confirming our method. Also shown is the n value corresponding to RHIC. It is interesting to extrapolate the curve Tc(fj) to the horizontal axis (as shown). It is known that the transition (at T fn 0) between ordinary hadronic matter and quark matter occurs at around ft ss 400 MeV. This is at a smaller value of fj. than the horizontal intercept of our data indicating (not surprisingly) the presence of higher order terms in the Taylor expansion and/or a breakdown in our method at these large values of fi. This motivates the question: for what range in fi do we expect our method to be accurate (and converge to the correct answer)? We have studied this issue by calculating the complex phase, 9, of the fermionic determinant (which enters in the reweighting factor in Eq. (4)), i.e. d/j,

2

N?TC

detM=|detM|e".

(11)

171 200

150

u £100

50

"0

200

400

600

(i, (Mev) Figure 4. The transition temperature, T c , as a function of fj.. The diamond symbol is the endpoint of the first order transition obtained by Fodor and Katz 6 .

The reweighting method will fail when the fluctuations (standard deviation) in 9 are larger than 0(7r/2). Taylor expanding Eq. (11) and noting that only odd derivatives contribute to the complex phase 6, we find that the standard deviation A0 ~ 0(TT/2) at around p/Tc ~ 0.5. Since this is around five times the fi value of RHIC, we can confirm that our method is applicable for RHIC physics. An interesting dynamical quark effect can be uncovered when studying the Polyakov Loop susceptibility, \L- For n < 0, anti-quarks are dynamically generated which screened colour charge. This leads to a reduction in the free energy of a single quark, and a corresponding reduction in the strength of the singularity at the transition. We observe this effect by noting that the peak height of XL is smaller for p, < 0 compared with // > 0 in Ref. [4].

7. Pressure and Energy Density Of great interest for heavy ion collision experiments is the study of the pressure, p, and energy density e and their p dependence. We can obtain estimates of p by employing the integral method 9 : P=^\nZ.

(12)

172

The first derivative of p w.r.t. p. is related to the quark number density, TdlnZ

=

^ v-aT'

,

x

(13)

and the second derivative to the singlet quark number susceptibility, xs • Both nq and xs can be calculated in terms of the quark matrix, M. Using the above to estimate p at the RHIC point we find that p increases by around 1% from its p. = 0 value. The energy density, e, can be obtained from e — 3p _ 4

T

1 dlnZ (I da\~

VT3 dp yadp

The derivatives of Eq. (14) can be expressed in terms of nq and xs- Combining this with the above calculation of p, we obtain a value for e alone. We find that, at the RHIC point, there is again only a 1% deviation from e(p = 0). Finally we study the variation of p and e along the transition line Tc (fi). (The above calculations were performed at fixed T.) Our aim is to determine whether p and e are constant along Tc(p). The constant p line is defined as Ap=^AT+-^-)A(^) = 0,

(15)

with a similar expression for the constant e line. Using the above and the value determined earlier for the rate of change of Tc with fi, we find that the value of both p and e along the transition line Tc(p) is consistent with zero with our current precision4. 8. Conclusions This work (which is published in full elsewhere4) has outlined a new method of determining thermal properties of QCD at non-zero chemical potential from the lattice. This approach is based on Taylor expanding the Ferrenberg-Swendsen re weighting scheme. Using this method, the susceptibilities in the chiral condensate and Polyakov loop were determined and their peak positions used to define the transition point, Tc. As a warmup exercise, the re weighting technique was used to determine the transition point as a function of the quark mass, m confirming earlier work. The method was then applied to obtaining Tc as a function of chemical potential, fi, confirming the work of Fodor and Katz 6 . Very recent work of de Forcrand and Philipsen, who studied the transition temperature for imaginary \i and then analytically continued these results to real-valued p,, also confirm our results 10 .

173

The region of applicability of our method was studied by calculating the fluctuations in the phase of the reweighting factor. This region was found to be substantial and easily covers the physically interesting values of fj, appropriate for RHIC physics. We also extracted information about the pressure, p, and energy density, e, as a function of chemical potential. We found that the variation in these quantities from their values at p. = 0 is tiny. This leads us to conclude that RHIC physics well approximated by p — 0 physics. Furthermore we find that p and e are approximately constant along the transition line Tc(p). The success of this work motivates the use of lighter, more physical quark masses, and the study of (2+1) dynamical flavours to correctly model real world physics. A cknowledgment s The authors would like to acknowledge the Particle Physics and Astronomy Research Council for the award of the grant PPA/G/S/1999/00026 and the European Union for the grant ERBFMRX-CT97-0122. CRA would like to thank the Department of Mathematics, University of Queensland, Australia for their kind hospitality while part of this work was performed. References 1. See http://wwwl.msfc.nasa.gov/NEWSROOM/news/releases/2002/02-082.html (NASA Press release); and J. Drake et al., Ast. J., June 20 2002 to appear. 2. For a recent announcement from CERN, see http://cern.web.cern.ch/CERN/Announcements/2000/NewStateMatter. 3. For a review, see S.J. Hands, Nucl. Phys. Proc. Suppl. 106, 142 (2002), hep-lat/0109034. 4. C.R. Allton, S. Ejiri, S.J. Hands, O. Kaczmarek, F. Karsch, E. Laermann, Ch. Schmidt, L. Scorzato, hep-lat/0204010 . 5. A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988); Phys. Rev. Lett. 63, 1195 (1989). 6. Z. Fodor and S.D. Katz, hep-lat/0104001; JHEP 0203, 014 (2002), hep-lat/0106002. 7. F. Karsch, E. Laermann, and A. Peikert, Phys. Lett. B478, 447 (2000); Nucl. Phys. B605, 579 (2001). 8. U.M. Heller, F. Karsch, and B. Sturm, Phys. Rev. D60, 114502 (1999). 9. J. Engels, J. Fingberg, F. Karsch, D. MiUer and M. Weber, Phys. Lett. B252, 625 (1990). 10. Ph. de Forcrand and O. Philipsen, hep-lat/0205016.

H A D R O N MASSES FROM A NOVEL FAT-LINK F E R M I O N ACTION

J. M. ZANOTTI, S. BILSON-THOMPSON, F. D. R. BONNET, D. B. LEINWEBER, A. G. WILLIAMS AND J. B. ZHANG Special Research Center for the Subatomic Structure of Matter, and Department of Physics and Mathematical Physics, Adelaide University, 5005, Australia E-mail: [email protected]; [email protected] [email protected]; [email protected] [email protected]; [email protected] W. M E L N I T C H O U K Special Research Center for the Subatomic Structure of Matter, and Department of Physics and Mathematical Physics, Adelaide University, 5005, Australia, and Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, U.S.A. E-mail: [email protected]

F. X. LEE Center for Nuclear Studies, Department of Physics, The George Washington University, Washington, D.C. 20052, U.S.A., and Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, U.S.A E-mail: [email protected]

Hadron masses are calculated in quenched lattice QCD in order to probe the scaling behaviour of a novel fat-link clover fermion action in which only the irrelevant operators of the fermion action are constructed using APE-smeared links. Light quark masses providing a m^/mp mass ratio of 0.36 are considered to assess the exceptional configuration problem of clover-fermion actions. The simulations are performed with an order a 2 mean-field improved plaquette plus rectangle gluon action on a 16 3 X 32 lattice with a lattice spacing of a = 0.125 fm. We compare actions with n = 4 and 12 smearing sweeps with a smearing fraction of 0.7. The n = 4 Fat Link Irrelevant Clover action provides scaling which is superior to meanfield improvement, and offers advantages over nonperturbative 0 ( a ) improvement including a reduced exceptional configuration problem.

1. Introduction Understanding the generation of hadron mass from first principles has proved to be challenging. The only method for deriving hadron masses directly from

174

175

QCD is a numerical calculation on the lattice. The high computational cost required to perform accurate lattice calculations at small lattice spacings, however, has led to an increased interest in quark action improvement. To avoid the famous doubling problem, Wilson fermions1 introduce additional terms which explicitly break chiral symmetry at 0(a). To extrapolate reliably to the continuum, simulations must be performed on fine lattices, which are computationally very expensive. The scaling properties of the Wilson action at finite a can be improved by introducing any number of irrelevant operators of increasing dimension which vanish in the continuum limit. The Sheikholeslami-Wohlert (clover) action 2 introduces an additional irrelevant dimension-five operator to the standard Wilson1 quark action, Ssw = Sw

j — i>(x)(r^Fllvip(x)

,

(1)

where S\y is the standard Wilson action and Csw is the clover coefficient which can be tuned to remove 0(a) artifacts. Nonperturbative (NP) 0(a) improvement 3 tunes Csw to all powers in g2 and displays excellent scaling, as shown by Edwards et al.4 who studied the scaling properties of the nucleon and vector meson masses for various lattice spacings (see also Sec. 4 below). In particular, the linear behaviour of the NP-improved clover actions, when plotted against a2, demonstrates that 0(a) errors are removed. It was also found in Ref. [4] that a linear extrapolation of the mean-field improved data fails, indicating that 0(a) errors are still present. A drawback to the clover action, however, is the associated problem of exceptional configurations, where the quark propagator encounters singular behaviour as the quark mass becomes small. In practice, this prevents the use of coarse lattices (/3 < 5.7 ~ a > 0.18 fm) 5 ' 6 . Furthermore, the plaquette version of F^, which is commonly used in Eq. (1), has large 0(a2) errors, which can lead to errors of the order of 10% in the topological charge even on very smooth configurations7. The idea of using fat links in fermion actions was first explored by the MIT group 8 and more recently has been studied by DeGrand et al.6'9, who showed that the exceptional configuration problem can be overcome by using a fat-link (FL) clover action. Moreover, the renormalization of the coefficients of action improvement terms is small. A drawback to conventional fat-link techniques, however, is that in smearing the links gluon interactions are removed at the scale of the cutoff. While this has some tremendous benefits, the short-distance quark interactions are lost. As a result decay constants, which are sensitive to the wave function at the origin, are suppressed. A solution to these problems is to work with two sets of links in the fermion action. In the relevant dimension-four operators, one works with the untouched

176

links generated via Monte Carlo methods, while the smeared fat links are introduced only in the higher dimension irrelevant operators. The effect this has on decay constants is under investigation and will be reported elsewhere. In this paper we present the first results of simulations of the spectrum of light mesons and baryons at light quark masses using this variation of the clover action. In particular, we will start with the standard clover action and replace the links in the irrelevant operators with APE smeared 10 , or fat links. We shall refer to this action as the Fat-Link Irrelevant Clover (FLIC) action 11 . 2. Gauge Action The simulations are performed using a tree-level (9(a 2 )-Symanzik-improved 12 gauge action on a 163 x 32 lattice at 0 = 4.60, providing a lattice spacing a = 0.125(2) fm determined from the string tension with yfa = 440 MeV. A total of 50 configurations are used in this analysis, and the error analysis is performed by a third-order, single-eliminationjackknife, with the \ 2 P e r degree of freedom (Nj^p) obtained via covariance matrix fits. Further details of this simulations may be found in Ref. [11]. 3. Fat-Link Irrelevant Fermion Action Fat links 6 ' 9 are created by averaging or smearing links on the lattice with their nearest neighbours in a gauge covariant manner (APE smearing). The smearing procedure 10 replaces a link, U^x), with a sum of the link and a times its staples 4

+Ul(x - ua)Ufi(x - va)Uu{x - ua + fia) ,

(2)

followed by projection back to SU(3). We select the unitary matrix J/J L which maximizes Heti(U^U',!),

(3)

by iterating over the three diagonal SU(2) subgroups of SU(3). We repeat this procedure of smearing followed immediately by projection n times. We create our fat links by setting a = 0.7 and comparing n — 4 and 12 smearing sweeps. The mean-field improved FLIC action now becomes S & = Sw - ^

J

^x)a^F^(x)

,

(4)

where F^ is constructed using fat links, and where the mean-field improved Fat-Link Irrelevant Wilson action is

177 Table 1. The value of the mean link for different numbers of smearing sweeps, n. n

KL)4

"o

0 4 12

0.88894473 0.99658530 0.99927343

0.62445197 0.98641100 0.99709689

(5) X

In x,p — V

-*

FL U0

ip(x+n)-

4>{X +fl)+

ip(x - p.) «o

wo

„FLF L u;0

nX •/*)

(6)

with K = l/(2m + 8r). We take the standard value r = 1. The 7-matrices are hermitian and a^ = [7^, 7„]/(2i). As reported in Table 1, the mean-field improvement parameter for the fat links is very close to 1. Hence, the mean-field improved coefficient for Csw is expected to be adequate a . In addition, actions with many irrelevant operators (e.g. the D234 action) can now be handled with confidence as treelevel knowledge of the improvement coefficients should be sufficient. Another advantage is that one can now use highly improved definitions of F^ (involving terms up to UQ2), which give impressive near-integer results for the topological charge 13 . In particular, we employ an 0(a4) improved definition13 of F^ in which the standard clover-sum of four l x l Wilson loops lying in the fi, v plane is combined with 2 x 2 and 3 x 3 Wilson loop clovers. Work by DeForcrand et a/.14 suggests that 7 cooling sweeps are required to approach topological charge within 1% of integer value. This is approximately 15 16 APE smearing sweeps at a = 0.7. However, achieving integer topological charge is not necessary for the purposes of studying hadron masses, as has been well established. To reach integer topological charge, even with improved definitions of the topological charge operator, requires significant smoothing and associated loss of short-distance information. Instead, we regard this as an upper limit on the number of smearing sweeps. "Our experience with topological charge operators suggests that it is advantageous to include «o factors, even as they approach 1.

178

Using unimproved gauge fields and an unimproved topological charge operator, Bonnet et al.7 found that the topological charge settles down after about 10 sweeps of APE smearing at a — 0.7. Consequently, we create fat links with APE smearing parameters n = 12 and a = 0.7. This corresponds to ~ 2.5 times the smearing used in Refs. [6,9]. Further investigation reveals that improved gauge fields with a small lattice spacing (a = 0.125 fm) are smooth after only 4 sweeps. Hence, we perform calculations with 4 sweeps of smearing at a = 0.7 and consider n = 12 as a second reference. Table 1 lists the values of UQ L for n = 0, 4 and 12 smearing sweeps. We also compare our results with the standard Mean-Field Improved Clover (MFIC) action. We mean-field improve as defined in Eqs. (4) and (6) but with thin links throughout. The standard Wilson-loop definition of F^ is used. A fixed boundary condition is used for the fermions by setting Ut{x, nt) = 0 and UfL(x, nt) = 0

Vx ,

(7)

in the hopping terms of the fermion action. The fermion source is centred at the space-time location (x,y,z,t) — (1,1,1,3), which allows for two steps backward in time without loss of signal. Gauge-invariant gaussian smearing 16 in the spatial dimensions is applied at the source to increase the overlap of the interpolating operators with the ground states. 4. R e s u l t s Hadron masses are extracted from the Euclidean time dependence of the calculated two-point correlation functions. The effective masses are given by M(t + 1/2) = log[G(<)] - Iog[G(t + 1)] •

(8)

The critical value of f\l y Kg ) IS determined by linearly extrapolating m\ as a function of mq to zero. We used five values of quark mass and the strange quark mass was taken to be the second heaviest quark mass. Effective masses in Eq. (8) are calculated as a function of time and various time-fitting intervals are tested with a covariance matrix to obtain X V - ^ D F Good values of X2/NUF are obtained for many different time-fitting intervals as long as one fits after time slice 8. All fits for this action ("FLIC4") are therefore performed on time slices 9 through 14. For the Wilson action and the FLIC action with n - 12 ("FLIC12") the fitting regimes used are 9-13 and 9-14, respectively. The behaviour of the p, nucleon and A masses as a function of squared pion mass is shown in Fig. 1 for the various actions. The first feature to note is the excellent agreement between the FLIC4 and FLIC12 actions. On the other hand, the Wilson action appears to lie somewhat low in comparison. It is also

179 i

i

1

1

* ** ** * ** « * A * * *| ** N * ** * * * • FLIC12 P *> & *

IO.

• 0

_ji

i

0.0

0.2

0.4 m/

0.6 (GeV8)

-

~

FLIC4 Wilson i

0.8

1.0

Figure 1. Masses of the nucleon, A and p meson versus m\ for the FLIC4, FLIC12 and Wilson actions.

reassuring that all actions give the correct mass ordering in the spectrum. The value of the squared pion mass at m^/mp = 0.7 is plotted on the abscissa for the three actions as a reference point. This point is chosen in order to allow comparison of different results by interpolating them to a common value of m^/mp = 0.7, rather than extrapolating them to smaller quark masses, which is subject to larger systematic and statistical uncertainties. The scaling behaviour of the different actions is illustrated in Fig. 2. The present results for the Wilson action agree with those of Ref. [4]. The first feature to observe in Fig. 2 is that actions with fat-link irrelevant operators perform extremely well. For both the vector meson and the nucleon, the FLIC actions perform significantly better than the mean-field improved clover action. It is also clear that the FLIC4 action performs systematically better than the FLIC12. This suggests that 12 smearing sweeps removes too much shortdistance information from the gauge-field configurations. On the other hand, 4 sweeps of smearing combined with our 0(a4) improved F^„ provides excellent results, without the fine tuning of Csw m the NP improvement program. Notice that for the p meson, a linear extrapolation of previous mean-field improved clover results in Fig. 2 passes through our mean-field improved clover result at a2a ~ 0.08 which lies systematically low relative to the FLIC actions. However, a linear extrapolation does not pass through the continuum limit result, thus confirming the presence of significant 0(a) errors in the mean-field improved clover fermion action. While there are no NP-improved clover plus improved gluon simulation results at a2
180

b

>

a

NP Clover

•» FLIC4, 200conf |» i Opr jTilgon I ,

0.00

0.05

,

* ,

Np

,

Clover+imp glue I , , , , I

0.10 a2 a

0.15

Figure 2. Nucleon and vector meson masses for the Wilson, NP-improved and FLIC actions obtained by interpolating our results of Fig. 1 to mTjmp = 0.7. Results from the present simulations are indicated by the solid points. The fat links are constructed with n = 4 (solid squares) and n = 12 (stars) smearing sweeps at a = 0.7.

obtained with a NP-improved clover fermion action. Having determined FLIC4 is the preferred action, we have increased the number of configurations to 200 for this action. As expected, the error bars are halved and the central values for the FLIC4 points move to the upper end of the error bars on the 50 configuration result, further supporting the promise of excellent scaling. Finally, in order to search for exceptional configurations by pushing the bare quark mass down, we would like our preferred action to be efficient when inverting the fermion matrix. In Fig. 3 we compare the convergence rates of the different actions by plotting the number of stabilized biconjugate gradient 17 iterations required to invert the fermion matrix as a function of mn/mp. For any particular value of mv/mp, the FLIC actions converge faster than both the Wilson and mean-field improved clover fermion actions. Also, the slopes of the FLIC lines are smaller in magnitude than those for Wilson and meanfield improved clover actions, which provides great promise for performing cost effective simulations at quark masses closer to the physical values. Problems with exceptional configurations have prevented such simulations in the past. The ease with which one can invert the fermion matrix using FLIC fermions leads us to attempt simulations of three lighter quark masses corresponding

181 450

1

1

-

400

1

- - - FLIC12 ---FLIC4 ---Wilson - - - MFIC

N

\

350

\

-

x

M300

\

-

.2 250 xx xx

£200
x

-

X N

xx

-

\

^ X Xx x

0.6

1

0.7 m / m

-

\

\ XS

xx 1

\

X

-

0.5

\ \

\

«.«.xx NNX

\ s

1

0.8

N

^

" 0.9

Figure 3. Average number of stabilized biconjugate gradient iterations for the Wilson, FLIC and mean-field improved clover (MFIC) actions plotted against mn /mp. The fat links are constructed with n = 4 (solid squares) and n = 12 (stars) smearing sweeps at a = 0.7.

to mn/mp = 0.53, 0.45, 0.36. Previous attempts with Wilson-style fermion actions have only succeeded in getting down 18 to m„/mp = 0.47. Figure 4 shows the behaviour of the TV, A, E, S masses as a function of m£ for all eight masses considered. The first two of these lighter quark masses show no sign of exceptional configurations, however at the lightest mass tested, the pion correlator exhibited divergences on 1% of the configurations. These two exceptional configurations were omitted in producing Fig. 4. At the lightest quark mass, the Compton wavelength of the pion is A^ = l/mn ~ 0.66 fm. The general rule for studying hadrons at light quark masses is to ensure Xw < ( l / 4 x length of spatial dimensions). The physical length of the spatial dimensions of our lattice is ~ 2 fm which is < 4 x A^-. Hence it is important to investigate the impact of finite volume effects on the exceptional configuration problem. Simulations on a 20 3 x 40 lattice with a physical length ~ 2.7 fm are now under way using the precision 5-loop improved from of FM„ complemented by two additional APE smearing sweeps. 5. Conclusions We have examined the hadron mass spectrum using a novel Fat Link Irrelevant Clover (FLIC) fermion action, in which only the irrelevant, higher-dimension operators involve smeared links. One of the main conclusions of this work is that the use of fat links in the irrelevant operators provides excellent results. Fat links promise improved scaling behaviour over mean-field improvement. This technique also solves a significant problem with 0(a) nonperturbative

182

1.8

i

1

1

1.7 1.6 ~ O 1.5 H* 3 I-* g 1.3

1.0 0.0

s 9

i

i 5

s

i

§

5

z i3

-

5

.

$ i

A*

1.1 -•/v} -

s 1 s

s

a>

a 1.2

1

$ 1 i

0.2

0.4 TO '

i

0.6 (GeV2)

' 0.8

1.0

Figure 4. Masses of the nucleon, A, E and H versus m\ for the FLIC4 fermion action.

improvement on mean field-improved gluon configurations. Simulations are possible and the results are competitive with nonperturbative-improved clover results on plaquette-action gluon configurations. We have found that minimal smearing holds the promise of better scaling behaviour. Our results suggest that too much smearing removes relevant information from the gauge fields, leading to poorer performance. Fermion matrix inversion for FLIC actions is more efficient and results show no sign of exceptional configuration problems down to mn/mp = 0.45. However we encounter divergences in the pion correlator at m^/rrip = 0.36 on 1% of the configurations analysed on this particular lattice. This work paves the way for promising future studies. It will be of great interest to consider different lattice spacings to further test the scaling of the fat-link actions. Current work is under way to further explore the exceptional configuration problem where a precision field-strength tensor and additional smearing hold promise. A study of the spectrum of excited hadrons using the fat-link clover actions is currently in progress 19 . Acknowledgements This work was supported by the Australian Research Council. We would also like to thank the National Computing Facility for Lattice Gauge Theories for the use of the Orion Supercomputer. W.M. and F.X.L. were partially supported by the U.S. Department of Energy contract DE-AC05-84ER40150, under which the Southeastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility (Jefferson Lab).

183

References 1. K.G. Wilson, in New Phenomena in Subnuclear Physics, Part A, A. Zichichi (ed.), Plenum Press, New York, p. 69, 1975. 2. B. Sheikholeslami and R. Wohlert, Nucl. Phys. B259, 572 (1985). 3. M. Luscher et al., Nucl. Phys. B478, 365 (1996) [hep-lat/9605038]; M. Luscher et ai, Nucl. Phys. B491, 323 (1997) 323 [hep-lat/9609035]. 4. R.G. Edwards, U.M. Heller and T.R. Klassen, Phys. Rev. Lett. 80, 3448 (1998). [hep-lat/9711052]; see also R.D. Kenway, Nucl. Phys. Proc. Suppl. 73, 16 (1999) [hep-lat/9810054] for a review. 5. W. Bardeen et al., Phys. Rev. D57, 1633 (1998) [hep-lat/9705008]; W. Bardeen et al., Phys. Rev. D57, 3890 (1998). 6. T. DeGrand et al. (MILC Collaboration), [hep-lat/9807002]. 7. F.D. Bonnet et al., Phys. Rev. D62, 094509 (2000) [hep-lat/0001018]. 8. M.C. Chu et al., Phys. Rev. D49, 6039 (1994) [hep-lat/9312071]. 9. T. DeGrand (MILC collaboration), Phys. Rev. D60, 094501 (1999) [heplat/9903006]. 10. M. Falcioni et al., Nucl. Phys. B251, 624 (1985); M. Albanese et al., Phys. Lett. B192, 163 (1987). 11. J.M. Zanotti et al. Phys. Rev. D60, 074507 (2002) [hep-lat/0110216]; Nucl.Phys.Proc.Suppl. 109, 101 (2002) [hep-lat/0201004]. 12. K. Symanzik, Nucl. Phys. B226, 187 (1983). 13. S. Bilson-Thompson et al., Nucl.Phys.Proc.Suppl 109 116 (2002) [heplat/0112034]; hep-lat/0203008. 14. P. de Forcrand et al., Nucl. Phys. B499, 409 (1997) [hep-lat/9701012]; P. de Forcrand et ai, [hep-lat/9802017]. 15. F.D. Bonnet et al., [hep-lat/0106023]. 16. S. Gusken, Nucl. Phys. Proc. Suppl. 17, 361 (1990). 17. A. Frommer et al., Int. J. Mod. Phys. C 5 , 1073 (1994) [hep-lat/9404013]. 18. M.D. Morte et al., hep-lat/0111048. 19. W. Melnitchouk et al., Nucl. Phys. Proc. Suppl 109, 116 (2002), hep-lat/0201005; hep-lat/0202022.

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5. Nuclear and Nucleon Structure Functions

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SMALL-X N U C L E A R EFFECTS IN PARTON D I S T R I B U T I O N S

V. GUZEY Special Research Centre for the Subatomic Structure of Matter, University of Adelaide, Adelaide, 5005, Australia E-mail: [email protected]

We summarize the status of nuclear parton distribution functions in nuclei at low x and show t h a t they are not constrained well enough by the available data. Measurements of Drell-Yan dimuon pair production on nuclei at the JHF is singled out as a good candidate to significantly increase information about antishadowing and shadowing of valence and antiquark parton distributions in nuclei. A brief description of the leading twist approach to nuclear shadowing is given and outstanding theoretical problems are reviewed.

1. Parton Distributions in Nuclei It is now very well established that the distributions of partons (quarks and gluons) in nuclei differ from those in free nucleons. Experimental information on nuclear parton distribution functions (nPDFs) is obtained from the following fixed nuclear target experiments: inclusive deep inelastic scattering (DIS) of leptons, Drell-Yan dimuon production by hadron beams, and lepto- and hadroproduction of vector mesons. Each of the afore mentioned processes is sensitive to a particular type or combination of nPDFs. Inclusive DIS, l + A—tl' + X, measures the nuclear structure function F£, which, to leading order, takes the form F2A(x,Q*)=Yje](f?(x,Q2)

+ fi(x,Q*)).

(1)

i

Here f* is the quark nPDF; f£ is that of the antiquark. Inclusive DIS constrains best the combination f£ + f£. It is important to note that the fixedtarget kinematics correlates the values of x and Q 2 entering Eq. (1). For instance, for the NMC experiment 1 , the condition Q2 > 1 GeV 2 means that x > 5 x 1 0 - 3 . Hence the data set eligible for the perturbative QCD analysis becomes reduced, and the reliable extraction of nPDF, especially at low x, becomes more difficult. Once F£ is measured, the gluon distribution in nuclei, gA(x,Q2), can be

187

188

determined indirectly and approximately through scaling violations of FA :

^^?f

2 l s W

>.

(2)

However, the accuracy and extent in x and Q2 variables of the available data do not offer enough statistics to pin down gA{x, Q2) reliably enough. Alternatively, gA{x,Q2) can be studied directly via the "proton-gluon fusion" mechanism : 7 *-|V

">•//* + A

(3)

which dominates lepto- and hadroproduction of vector mesons (J/^f, W, T) at high energies. Another important source of information on nPDFs is Drell-Yan high-mass dimuon production, p + A —> fi+^~ + X. To leading order, the differential cross section of this process reads

£k - C £ E - J ( ' . < ' ' > * <»>>+tt-itfc))'

(4)

i

where y/s is the centre-of-mass energy; x\ and x-i are the Bjorken variables of the projectile and target, respectively. In the kinematics, where X\ ^> x%, the first term dominates, and Eq. (4) gives a measure of the distribution of antiquarks in nuclei. How well do we know nPDFs after almost 20 years of experiments? The dispersion of predictions of modern models for nPDFs indicates that nPDFs are not constrained sufficiently. Figures 1 and 2 demonstrate predictions for the ratios uA/(AuN) and gA/(AgN) at Q = 1.5 GeV as a function of x for 208 the heavy nucleus of Pb. The curves are results of Frankfurt et al.2 (solid curve labelled FGMS), Eskola et al.3 (dotted curve labelled ESK98), Li and Wang 4 (dotted curve labelled HIJING), and Kumano et al.5 (dash-dotted curve labelled HKM). While the solid curve is a theoretical prediction made within the framework of a leading twist approach (see Sec. 3), the other three curves are fits to the available data with several simplifying assumptions. There is no region in x, where the four results would be consistent with each other. 2. Drell-Yan Dimuon Production on Nuclei at JHF The Japan Hadron Facility (JHF) offers a unique possibility to perform very precise measurements of Drell-Yan high-mass dimuon production on nuclei, and thus to determine the nuclear sea and valence quark distributions 6 . Let us briefly discuss relevant kinematics of the reaction (see Eq. (4)). To leading order, Drell-Yan dimuon production can be envisioned as annihilation

189

£•••••

-

I

**f 0.9

0.8

0.7

/

. . /

J ^ 0.6

"

/

y

FGMS ESK98 HIJING HKM

Q-1.5 GeV 208

0.5

Pb

'/ i

Figure 1.

The ratio of the antiquark n P D F , uA/(AuN),

i

within four different models.

of a quark (antiquark) from the beam by an antiquark (quark) from the target into a virtual photon with subsequent production of a pair of oppositely charged muons. The Bjorken variable x\ refers to the beam (proton), while xi refers to the nuclear target. These variables are related by x\X2 = Q2/s, where Q2 is the virtuality of the photon. Q2 is equal to the invariant mass squared of the detected dimuon pair, and, since one wants to stay away from the n+(i~ resonances, Q2 is large, Q2 > 16 GeV 2 . Drell-Yan dimuon production on nuclei was studied by the Fermilab E772 experiment with 800 GeV/c protons 7 . In this case, 0.05 < x 2 < 0.3 and the second term in Eq. (4) can be safely neglected. Thus, the experiment gave a direct measure of the antiquark nPDFs in several nuclei with the conclusion that antiquarks are not modified in nuclei for 0.1 < x < 0.3. This finding imposes powerful constraints on models of nuclear shadowing and antishadowing for antiquarks in nuclei: the scenario of Kumano et al. is ruled out. Also, the E772 result rules out models of the EMC effect with significant pion excess. JHF will have two advantages over the E772 experiment. First, the beam energy is lower, Ep = 50 GeV, which results in an increase by factor 16 of the

190

FGMS ESK98 Q=1.5GeV

HIJING HKM

208pb

Figure 2.

The ratio of the gluon n P D F , gA/(Ag

), within four different models.

differential cross section in Eq. (4) because of the factor 1/s. Second, a much higher flux will give another factor 16 gain. The values of xi will be higher, #2 > 015 — 0.2, which means that both terms in Eq. (4) should be retained. This will allow one to study both antiquark and valence quark nPDFs. One should note that the depletion of nPDFs at low x (nuclear shadowing), is followed by some enhancement (antishadowing) at 0.05 < x < 0.2 (see Fig. 2). Thus, in the available domain of X2, JHF will be able to determine the pattern of antishadowing with much better accuracy than any previous experiment. Among other exciting possibilities one should also mention measurements of hotly debated parton energy loss, using Drell-Yan dimuon production 6 . Besides very high statistics, the values of x-i values are higher at JHF than at Fermilab (Fermilab experiment 8 E866), which would enable a much cleaner analysis of the data.

191

3. Leading Twist Nuclear Shadowing and Antishadowing of Singlet P D F s In Figs. 1 and 2 we presented our predictions for the singlet (antiquark and gluon) nPDFs. Inspecting Fig. 2 one notes that the ratio gA/(AgN) < 1 for 4 10~ < x < 0.03, which means that the gluon distribution is depleted, or shadowed, in nuclei. For 0.03 < x < 0.2, gA/(AgN) > 1, which represents an enhancement, which is termed antishadowing. Both effects are characteristic to small x and are believed to arise from the coherent (simultaneous) interaction of the projectile with several nucleons in the target. In what follows we shall give a brief account of the derivation of our leading twist result. For more details, we refer the reader to the original publication 2 . Our leading twist approach to nuclear shadowing in DIS on nuclei is based on the space-time picture of hadron-deuteron scattering developed by Gribov 9 . It was observed that at high energies the nuclear shadowing correction to the total cross section, which arises from the simultaneous interaction of the projectile with both nucleons, is dominated by the excitation of diffractive intermediate states. Ultimately this enables one to relate nuclear shadowing in hadron-deuteron scattering to diffraction in hadron-nucleon scattering. These ideas can be generalized to lepton DIS on any nucleus 10 . Moreover, using the QCD factorization theorems for inclusive and hard diffractive processes, nuclear shadowing can be formulated for each nuclear parton distribution, fj/A, separately. Introducing the shadowing correction Sfj/A as Sfj/A = fj/A - Afj/N, we obtain Sfj/A =

A i

\

^lQxRe

/•OO

(1-itf

P dxrfflrl(P,Q'>,xr,0)

/»00

/

pA{b,z^) pA{b,*,)<,<•'•""<•'

JX

xe

-{A/2)(l-iV)aift

//*

dzpA(z)

(5)

Here j denotes the parton species (flavour of the quark or the gluon); 77 is the ratio of the imaginary to real part of the diffractive scattering amplitude; 6 and z\ and z-i are the transverse and longitudinal coordinates of any two nucleons of the target nucleus; xo is a cut-off parameter (xo = 0 . 1 for antiquarks and # 0 = 0.03 for gluons); PA is the nuclear density normalized to unity. The exponential factor takes into account the suppression due to the interaction with three and more nucleons. The essence of the leading twist approach is that Sfj/A is proportional to the diffractive parton distribution of the nucleon, ff/N, which is a function of the usual diffractive variables (3 = x/xp, xp and t (note

192

that t — 0). Since fP,N obeys the leading twist DGLAP evolution equations, so does Sfj/A- So a slow, logarithmic, or leading-twist Q2-dependence of nuclear shadowing is expected. One of the characteristic features of Eq. (5) is that nuclear shadowing for gluons is more sizable than for the sea quarks. Indeed, as observed at HERA, DIS diffraction is dominated by the gluon diffractive distribution, i.e. fq/N -^ fq'/N- Hence, &9A > SuA. This trend can be seen in Figs. 1 and 2. The cut-off parameter xo cannot be determined in our approach from the first principles and should be inferred from the comparison to the data. This parameter defines the transition point from shadowing to antishadowing. As discussed above, antishadowing for the sea quarks was not observed by the E772 experiment so that xo can be taken arbitrary large (xo = 0.1 in our case). The analysis of scaling violations of FA indicated that gluons in nuclei might be enhanced 11 for 0.03 < x < 0.2. Thus, in our analysis we choose XQ = 0.03 for gluons and complement Eq. (5) by modelling the enhancement of the gluon nPDF in the interval 0.03 < x < 0.2 by requiring that the fraction of the total momentum carried by gluons is the same in nuclei and the proton: /•0.2

/

/-0.2

dxxgA(x,Q2)/A=

dxxgN{x,Q2).

(6)

Finally, we note that the leading twist approach is applicable for the evaluation of nuclear shadowing of the antiquark and gluon nPDFs. Indeed, high energy diffraction is believed to be dominated by the 2-channel exchange with vacuum quantum numbers, the Pomeron exchange. Exactly this type of exchange drives singlet parton distributions (antiquarks and gluons) at low x. 4. Shadowing and Antishadowing of Valence n P D F s The situation with nuclear shadowing of the valence nPDFs is less transparent. The coherent theoretical picture of nuclear shadowing is absent, and the available fixed-target data on FA are not sensitive to the valence distributions in nuclei at low x. From the theoretical point of view, nuclear shadowing for valence nPDFs could be formulated in a form similar to Eq. (5) with the important modification that the Pomeron exchange is replaced by the interference between the Pomeron and Reggeon (non-vacuum quantum numbers) exchanges. However, the experimental information on such type of interference is absent. It is interesting to note that high precision measurements of Drell-Yan dimuon productions at JHF for x 2 > 0.15 — 0.2 should significantly clarify the situation with the valence quarks. Indeed, the baryon charge sum rule, for

193 instance, A

L dxut,(x,Q )/A 2

= 2,

(7)

combined with the measured expected enhancement for x > 0.05, would give a tight constraint on the net contribution of the low-a: shadowing region t o the sum rule in Eq. (7). Finally, we cannot help b u t reflect on outstanding theoretical challenges in the theory of nuclear shadowing and antishadowing. First, while the origin of nuclear shadowing is understood, the dynamics of antishadowing is unknown: Eq. (5) does not include antishadowing in n P D F s . Ideally one would like t o have a theory, which would include b o t h effects on equal footing, and would also automatically satisfy the baryon charge and m o m e n t u m sum rules. Second, a theory of small-a; nuclear effects for the valence and non-singlet n P D F s is lacking. Once available, one can immediately analyse low-a; behaviour of such quantities as non-singlet combinations of nuclear structure functions, ^ 2 H e - -F2 H . o r polarized n P D F s . 5. Conclusions and Discussion T h e J a p a n Hadron facility (JHF) can significantly contribute to studies of lowx nuclear phenomena, shadowing and antishadowing, using Drell-Yan dimuon production on nuclear targets. Such measurements should be precise enough t o study the upper-a; end of nuclear shadowing and the whole range of antishadowing of antiquark and valence n P D F s . This will help understand the presently unknown dynamics of antishadowing and shadowing for the valence n P D F s and will have implications for small-a; DIS on polarized nuclear targets. As experiments complimentary to the m a i n physics program of J H F , we suggest to study Drell-Yan dimuon production on nuclei induced by pion and kaon beams, which should allow one t o eliminate the resonance background and decrease the mass of the measured dimuon pair and, thus, Q2 of the probed partons. T h i s will enable to take d a t a in previously unattainable kinematical region, which will facilitate a comparison to the inclusive d a t a generally taken at lower Q2. One can also suggest to measure inelastic J/\I? production on nuclei. W i t h the luminosity of J H F , one should be able to extract the gluon n P D F in the antishadowing region with unprecedented accuracy. Acknowledgments This work was supported in part the Australian Research Council and the University of Adelaide.

194

References 1. NMC Collab., P. Amaudruz et al., Nucl. Phys. B441, 3 (1995); M. Arneodo et al., Nucl. Phys. B441, 12 (1995). 2. L. Frankfurt, V. Guzey, M. McDermott, and M. Strikman, JHEP 202, 27 (2002). 3. K. J. Eskola, V. J. Kolhinen, and P. V. Ruuskanen, Nucl. Phys. B535, 351 (1998). 4. Shi-yuan Li and X. N. Wang, Phys. Lett. B527, 85 (2002). 5. M. Hirai, S. Kumano, and M. Miyama, Phys. Rev. D64, 034003 (2001). 6. M. Asakawa et al., Expression of interest for Nuclear/Hadron Physics Experiments at the 50-GeVProton Synchrotron, KEK Report 2000-11, October 2000. 7. D. M. Aide et al., Phys. Rev. Lett. 64, 2479 (1990). 8. M. A. Vasiliev et al., Phys. Rev. Lett. 83, 2304 (1999). 9. V. N. Gribov, Sov. Phys. JETP 29, 483 (1969). 10. L. Frankfurt and M. Strikman, Eur. Phys. J. A 5 , 293 (1999). 11. T. Gousset and H. J. Pirner, Phys. Lett. B375, 349 (1996).

THE N U T E V ANOMALY A N D S Y M M E T R Y B R E A K I N G IN T H E PARTON D I S T R I B U T I O N F U N C T I O N S

F. G. CAO AND A. I. SIGNAL Institute

of Fundamental Sciences, Massey University Palmerston North, 5301, New Zealand E-mail: [email protected]; [email protected]

The NuTeV collaboration recently reported a measurement of sin 8^ which is about 3 standard deviations above the standard model prediction. Quarkantiquark symmetry breaking in the strange sea and charge symmetry breaking in the parton distribution functions may affect the NuTeV result. We present theoretical calculations of these symmetry breakings a using meson cloud model. It was found that the corrections from these symmetry breakings would not be significant enough to explain the NuTeV anomaly.

1. The NuTeV Result The NuTeV collaboration recently measured 1 the electroweak parameter sin 2 6w using neutrino (anti-neutrino) -nucleon scattering. In order to reduce the uncertainties associated with charm production, the Paschos-Wolfenstein ratio 2 was measured, P- _

are

a

NC

~ aNC

"cc

a

_

1

0;„2fl

M\

z

cc

ne

where cr^c and
195

196

In this paper we will study explanation (2), particularly the corrections from possible symmetry breaking in the parton distribution functions (PDFs). The Paschos-Wolfenstein relation (Eq. (1)) is obtained under the assumption of parton distribution functions of the nucleon being strange quark-antiquark symmetric (s — s) and charge symmetric (dp — un and up = dn). There have been experimental analyses 7,8 ' 9 and several theoretical investigations 10,11,12 ' 13 on the possible breaking of s-s symmetry and charge symmetry in the PDFs since the discovery and establishment of flavour symmetry breaking in the nucleon sea 14 . Taking into account these symmetry breakings, the PaschosWolfenstein relation, Eq. (1), becomes4 R~ — — — sin 2 0w 2 (Suv) - (Sdv) - (Ss) 2 (9LU ~9RJ+ ((u) + (d))/2

^Id

~ 92Rd)

(2)

where (q) — f dxxq(x) and (Sq) — f dxxSq(x) are the second moments of the corresponding distributions, and giq and gRq (q — u, d) are the Z couplings. 8dv — dvv — u" and Suv = up — d™ 'measure' charge symmetry breaking (CSB) in the PDFs, and Ss — sp -sp — sn - sn 'measures' quark-antiquark symmetry breaking in the strange sea. Eq. (2) clearly shows that s-s symmetry breaking and CSB in the PDFs will affect the extraction of sin2 Ow in the NuTeV measurement. It is the second moments of the symmetry breakings that enter into the Paschos-Wolfenstein relation, and a value {Suv) — {Sdv) — (6s) m —0.0038 is needed4 to make the NuTeV result to be consistent with the standard model prediction. We investigated this breaking 11,13 using a meson cloud model which can provide a natural explanation of the flavour symmetry breaking in the nucleon (d > u for the proton) 14 . 2. s-s Symmetry Breaking In the meson cloud model (MCM) 15 the nucleon can be viewed as a bare nucleon surrounded by a mesonic cloud. The physical nucleon state can be expressed in terms of a series of baryon and meson components, |P>phys. = | p > t o r e + | J V 7 r ) + | A j r } + | A ( E ) / i 0 + . . . .

(3)

The pion is needed for chiral symmetry in quark models and plays an important role in understanding of the nucleon structure at lower energy. A crucial observation was made by Sullivan16 in the early 70's about the importance of these baryon-meson components in high energy processes, for example, about the contributions to the nucleon structure functions measured in deep inelastic scattering. If the lifetime of the baryon-meson Fock states is much longer

197 than the strong interaction time between the hard photon and nucleon, then the interaction between the hard photon and the baryons and mesons in the expansion of Eq. (3) may become important. The contribution to the PDFs of the nucleon from such kind of processes can be written as convolution of the fluctuation function, which describes the fluctuation N —¥ BM, with parton distributions of the baryon and meson. Then the flavour asymmetry d > u emerges naturally since the probability of finding the Fock state \TIK+) is much larger than that of | A + + T T _ ) . For the strange content of the nucleon sea, the important baryon-meson components are AA' and EA'. The non-perturbative contributions to the strange and anti-strange distributions in the proton can be written as 1 1 dy

fBK{y)sBC-)

Jx

y

(4) y

*"(*) = fdJ-fKB{y)sK{X-),

(5)

Jx y y where /sjf(y) = / K B ( 1 — y) (y and 1 — y being the longitudinal momentum fractions of the baryon and meson) are the fluctuation functions, and s (B = A, E) and sK are the s and s distributions in the A (E) baryons and K meson, respectively. The fluctuation functions are calculated 11 from an effective meson-baryon-nucleon interaction Lagrangian using time-ordered perturbation theory in the infinite momentum frame, _ 9NBK P (»)-l6^-y0

*MCM,.A

fBK

dk

l G%K(y,kl) {ymN-mB)2 , ( 1 _ y) {m% - m%KY y

+ kl '

(6)

where gNBK is the effective coupling constant; mBK — (m2B + k]_)/y+ (m2K + &j_)/(l—y) is the invariant mass squared of the BK Fock state; and GBK{V^ k\) is a phenomenological vertex form factor for which we adopt an exponential form GBK{y, fc±) = exp

N-mBK(y,kl) 2A?

(7)

with Ac being a cut-off parameter. From Eqs. (4) and (5) we know that the non-perturbative contributions to s and s distributions in the nucleon are different, and the difference s — s depends on both the fluctuation functions (/BK ano> IKB) and the parton distributions in the baryon and meson (sB and sK). Due to the A (S) baryons being heavier than the K meson, fBx(y) is harder than fxB(y), which suggests giv y gN -m £ n e j a r g e x r e g i o n . On the other hand, the s distribution of the K meson (sK(x)) is generally believed to be harder than the s distribution of

198

the baryon (sB(x)) as the baryon contains one more valence quark than the meson, which implies sN < sN in the large x region. The final prediction of the s-s asymmetry will depend on these two competing effects. We employed the following two prescriptions for the strange and antistrange distributions in the A (£) baryons and K meson : (1) Use SU(3) symmetry for the parton distribution functions of the baryons, i.e. sA = s s = \uN where the MRST98 17 parameterization for uN is adopted, and GRS98 18 parameterization for s , which is obtained by connecting sK to the valence quark distribution in the pionic meson based on the constituent quark model; (2) Calculate the strange distributions in the bag model in order to take into account the SU(3) symmetry breaking effect in the PDFs of the baryons, sA ^ s s ^ \uN, which could be important 19 . The numerical results are given in Fig. 1. It can be seen that the two prescrip0.0015

--

0.001 0.0005 0

-0.0005 0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 1. x(s — s) versus x. The solid and dotted curves are the results using prescription (1) and (2) for sB and SK, respectively.

tions give very different predictions for the s-s asymmetry. The calculations also depend on the cut-off parameter Ac introduced in Eq. (7). The baryon and meson production data can impose some constrains on this parameter and a value of about 1.1 GeV is preferred 20 . We studied the dependence on this parameter by allowing Ac to change from 0.8 GeV to 1.5 GeV. The probabilities of finding the AA' (£A') Fock states in the proton wavefunction change from 0.18% (0.09%) to 5.4% (3.5%). Consequently the independence and the magnitude of the s-s asymmetry change dramatically (see Fig. 2). The Ac-dependence of (Ss) is plotted in Fig. 3. (6s) depends on Ac much less dramatically than the difference x(s — s) because of the cancellation

199 SU3

0

0.1

0.2

0.3

Bag

0.4

0.5

0.1

0.6

0.2

0.3

0.4

0.5

0.6

Figure 2. x(s — s) versus x. The solid, dashed and dotted curves are the results with A c = 1.5, 1.08 and 0.8 GeV, respectively.

0.8

0.9

1.1

1.3

1.5

Figure 3. (Ss) versus A c . T h e dotted and solid curves are the results using prescription (1) and (2) for s and s , respectively.

between the contributions in the small x and large x regions (see Fig. 2). (Ss) changes from being positive to negative with the increase of the cut-off Ac and has a maximum at about Ac = 1.1 GeV. So the theoretical estimates for (Ss) lie in the range from -0.0005 to 0.0001, which is about an order of magnitude smaller than that is needed to explain the NuTeV anomaly. 3. C h a r g e S y m m e t r y B r e a k i n g It has been generally believed that charge symmetry is highly respected in the nucleon system. However some theoretical calculations 12 have suggested that the charge symmetry breaking (CSB) in the valence quark distributions may be as large as 2% - 10%, which is rather large compared to the low-energy

200

results. Any unexpected large CSB will greatly affect our understanding of non-perturbative dynamics and hadronic structure, and also the extraction of sin 2 6w from neutrino scattering. CSB in both the valence and sea quark distributions comes from nonperturbative dynamics. Thereby the meson cloud model (MCM), describing the non-perturbative structure of the nucleon, can provide a natural explanation of CSB in the valence and sea quark distributions of the nucleon 13 . The fluctuations we consider include TV —>• Nir and N —»• An. The baryons (mesons) in the respective virtual Fock states of the proton and neutron may carry different charges. If we neglect the mass differences between these baryons (mesons), the fluctuation functions for the proton and neutron will be the same, and the contributions to the parton distributions of the nucleon from these fluctuations will be charge symmetric. The electromagnetic interaction induces mass differences3, among these baryons (mesons) 21 , trip — f^n = —1-3 MeV, mAo - m A + = 1.3 MeV,

mn± — m„o = 4.6 MeV, mA- - mAo = 3.9 MeV.

(8)

Due to these mass difference the probabilities of corresponding fluctuations for the proton and neutron may be different, thereby the contributions to the PDFs of the proton and neutron from these fluctuations may be different. The numerical result for CSB in the valence quark sector is given in Fig. 4. We find that xSdv and xSuv have similar shapes and both are negative, which °T •S § -0.0002a* u xSdv f

-0.0004-

>

-0.0006-

x5uv

u -0.0008 -• 0

0.1

0.2

0.3

0.4

0.5

X

Figure 4.

Charge symmetry breaking in the valence sector.

is quite different from the quark model prediction 12 for x6dv being positive for a

Unlike the other mass splittings, the splitting between charged and uncharged pion masses is not a violation of charge symmetry - it is a violation of isospin symmetry.

201

most values of x. Our numerical results are about 10% of the quark model estimates 12 . We did not find any significant large-x enhancement of the ratio Rmin = 8dv/d%, which is predicted in the quark model calculations 12 . The smallness of any CSB effect as x -^ 1 is natural in the MCM, as all the fluctuation functions go to zero as y —>• 1, and hence there is no non-perturbative contribution to the parton distributions at large x. We estimate (Su)—{Sd) is about —0.0003, which is much smaller than required to explain the NuTeV anomaly. Considering both s-s symmetry breaking and charge symmetry breaking we have (Su) — (Sd) — (5s) ca 0.0002 0.0004, which is an of magnitude order smaller than needed to explain the NuTeV discrepancy. 4. Summary The NuTeV measurement of sin2 6W being 0.2277 ±0.0013(stat.)±0.0009{sy si.) is about 3 standard deviations above the standard model prediction. Possible symmetry breaking in the parton distribution functions of the nucleon, strangeantistrange symmetry breaking and charge symmetry breaking may affect the interpretation of NuTeV result. We reported theoretical calculations for these symmetry breakings using a meson cloud model, which could naturally explain flavour symmetry breaking in the nucleon sea (d > u). It was found that the corrections from the two symmetry breakings would not be significant enough to alter the NuTeV result. Moreover there is no established experimental evidences for or against quark-antiqu ark symmetry and charge symmetry in the parton distribution functions of the nucleon. More studies are needed to explain the NuTeV anomaly. Acknowledgments This work was partially supported by the Science and Technology Postdoctoral Fellowship of the Foundation for Research Science and Technology, and the Marsden Fund of the Royal Society of New Zealand. References 1. G. P. Zeller et al, NuTeV Collaboration, Phys. Rev. Lett. 88, 091802 (2002). 2. 3. 4. 5. 6.

E. A. Paschos and L. Wolfenstein, Phys. Rev. D7, 91 (1973). G. A. Miller and A. W. Thomas, hep-ex/0204007. S. Davidson et al., hep-ph/0112302. G. P. Zeller et al., NuTeV Collaboration, hep-ex/0203004. E. Ma and D. P. Roy, Phys. Rev. D65, 075021 (2002); S. Barshay and G. KreyerhofT, Phys. Lett. B535, 201 (2002); C. Giunti and M. Laveder, hepph/0202152.

202 7. 8. 9. 10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21.

A. O. Bazarko et al., CCFR Collaboration, Z. Phys. C65, 189 (1995). M. Goncharov et al., NuTeV Collaboration, Phys. Rev. D64, 112006 (2001). V. Barone, C. Pascaud and F. Zomer, Eur. Phys. J. C12, 243 (2000). A. I. Signal and A. W. Thomas, Phys. Lett. B191, 205 (1987); S. J. Brodsky and B. Q. Ma, Phys. Lett. B381, 317 (1996). F.-G. Cao and A. I. Signal, Phys. Rev. D60, 074021 (1999). For a recent review see J. T. Londergan and A. W. Thomas, in Progress in Particle and Nuclear Physics, Volume 41, P. 49, ed. A. Faessler (Elsevier Science, Amsterdam, 1998); E. Sather, Phys. Lett. B274, 433 (1992); E. Rodionov, A. W. Thomas and J. T. Londergan, Mod. Phys. Lett. A9, 1799 (1994). F.-G. Cao and A. I. Signal, Phys. Rev. C62, 015203 (2000). E. A. Hawker et al., E866/NuSea Collaboration, Phys. Rev. Lett. 80, 3715 (1998); For a recent review see J.-C. Peng and G. T. Garvey, hep-ph/9912370. A. W. Thomas, Phys. Lett. B126, 97 (1983). J. D. Sullivan, Phys. Rev. D5, 1732 (1972). A. D. Martin, R. G. Roberts, W. J. Stirling and R. S. Thome, Eur. Phys. J. C5, 463 (1998). M. Gliick, E. Reya and M. Stratmann, Eur. Phys. J. C2, 159 (1998). C. Boros and A. W. Thomas, Phys. Rev. D60, 074017 (1999). H. Holtmann, A. Szczurek and J. Speth, Nucl. Phys. A569, 631 (1996). D. E. Groom et al., Particle Data Group, Eur. Phys. J. C15, 1 (2000).

N U C L E O N S AS RELATIVISTIC T H R E E - Q U A R K STATES

M. OETTEL Max-Planck~Institut fur Metallforschung Heisenbergstr. 3, 70569 Stuttgart, Germany E-mail: [email protected]

A covariant nucleon model is formulated which uses dressed quarks compatible with recent lattice d a t a and Dyson-Schwinger results. Two—quark correlations are modelled as a series of quark loops in the scalar and axial vector channel. Faddeev equations for nucleon and delta are solved and nucleon form factors are calculated in a fully covariant and gauge-invariant scheme. The insufficience of a pure valence quark description becomes apparent in results for the electric form factor of the neutron and the ratio HGE/GM for the proton.

1. Introduction to the Model Virtually all intricacies and complications of QCD are present in the analysis of the simplest baryonic bound state, the nucleon. Spectroscopy suggests that it is mainly composed of three confined constituent quarks of mass ~ 0.33 GeV. Probing its structure in deep inelastic scattering (DIS) reveals a completely different picture: Depending on the energy of the incoming photon, the proton appears to be a complicated mixture of a three-quark valence core supplemented with higher qq and gluonic components. To be precise, such a Fock-state picture is rigorously valid only in the light-cone gauge, and the quark states should correspond to the nearly massless current quarks from the QCD Lagrangian. From both points of view, the nucleon is a complicated relativistic bound state which calls for a covariant description since already the simplest processes which are described by the various nucleon form factors involve considerable momentum transfers. It has become common wisdom that massive constituent quarks are generated by the spontaneous breaking of the approximate chiral symmetry of QCD. Assuming that the gluon propagator and the quark-gluon vertex possess enough integrated strength, Dyson-Schwinger (DS) equation studies 1 ' 2 have predicted a running quark mass which is about the constituent quark mass in the infrared and drops to the current mass in the ultraviolet. This has been confirmed on the lattice 3 ' 4 . Although all these results have been obtained

203

204

0

02

0.4

0.6

1

0.8

1.2

1.4

1.6

1.8

rfm

Figure 1. Lattice results for the static three-quark potential. The d a t a points for the potential (little dashes with error bars) are compared to static, string-like potentials of Y and A type between the three quarks. (Adapted from Ref. [8].)

in Landau gauge and not in the light-cone gauge, we may assume that a good part (though by no means all) of the complicated Fock state structure of the proton is buried in the structure of the constituent quark. Covariant bound states wave functions may be obtained by use of the covariant Bethe-Salpeter (BS) equation. The properties of pseudoscalar mesons are described successfully in combined DS/BS-studies 2 which emphasize their dual role as both q — q bound states and Goldstone bosons. Along these lines the nucleon's bound state amplitude can be obtained by solving a relativistic Faddeev equation if irreducible three-quark correlations are neglected. This is a strong assumption which has been tested on the lattice only for static quarks, see Fig. 1. The Faddeev equation needs as input the full solution for the q — q scattering kernel (whose determination is itself almost an unsurmountable task). Truncating the interaction between the two quarks to lowest order (one effective gluon which generates the constituent quark mass) one finds diquark poles with scalar (0 + ) diquarks (RS 0.7 — 0.8 GeV) and axial vector (1 + ) diquarks (m 0.9 — 1.0 GeV) having the lowest masses 2,5 . Although the poles may not survive in higher orders 6 the dominance of scalar and axial vector correlations in the q — q matrix has also be seen on the lattice (in Landau gauge), see Fig. 2. Despite these signals for pronounced q — q correlations, no evidence for compact (i.e. pointlike) diquarks has been seen in DIS or other processes.

205 40

i

••

p(o>)

I 1

35 30

.

25

'

1

-i

1

1

scalar (303) diquark axial vector (613) diquark

! "mass" peak K = 0.147

20 15

.

10

•

5

K

(a) 0

0.5

1

/ 1.5

continuum ••-.,. 2.5

0)[GeV] 3.5

Figure 2. Lattice results for the spectral function in the colour antitriplet (3) q — q channel. There is an almost perfect mass peak in the scalar diquark channel while for the axial vector diquark the mass peak and the continuum are of equal importance. The spectral functions of other diquarks is of no significance. (Adapted from Ref. [7].)

Putting all these motivations together, we formulate a covariant model which incorporates 9 • quark propagators with a running mass function in accordance with DS and lattice results. A parameterization in terms of entire functions is used which has been fitted to a number of low-energy meson observables 10 . Although the general shape agrees with recent lattice data a (see Fig. 3) the latter suggest a somewhat broader mass function. The relevance of this for the electromagnetic form factors will be discussed. • separable q — q correlations truncated to the 0 + and 1 + channels, see Fig. 3. Their parameterization involves diquark vertex functions x> which are described by their dominant Dirac structure, and a scalar function F, which assigns a width wo+[i+] to the diquarks. The diquark propagator is obtained as a sum of quark polarization loops. An external probe (e.g. a photon) will resolve the quark substructure of the diquarks by coupling to the quarks in the loop. • strict covariance in solving the resulting Faddeev equations. Thanks to the separability assumption the Faddeev equations reduce to an a

N o t e t h a t the fit has been obtained some years earlier than lattice d a t a became available.

206

A —

" Z 2€>=CC

Lattice, m ^ = 91 MeV u/d quark parametrizatkm m K . . = 5MeV

a=/0+,l+}

= A,, */"'"

/=Po+(75C)F(»o+)

Figure 3. Left panel: Parametrized quark mass function compared to lattice d a t a from Ref. [3]. Right panel: The separability assumption for the q — q t matrix.

effective diquark-quark Bethe-Salpeter equation where the attractive interaction is effected through the quark exchange between the quark and diquark, see Fig. 4. With the masses of the nucleon and delta as input parameters, we can vary only the width of the scalar diquark which in turn determines the diquark masses through the zeros of the inverse diquark propagators. The rich structure of the BS wave functions and the numerical procedure to solve the BS equation can be found in Refs. [11,12] respectively. Exemplary width parameters for which a solution can be found together with the resulting diquark masses are given in Table 1. We see that for a scalar diquark width of ~ 0.3 fm the diquark masses are in agreement with the position of the lattice peaks. Smaller widths give approximately equal masses for the scalar and axial vector diquark while larger widths tend to increase the mass difference. We remark that the width of the nucleon BS wave function (which can be interpreted as a quark-diquark separation) is of about 0.4 fm. On the one hand, this indicates no strong diquark clustering (for which there is no evidence anyway). On the other hand the diquark width should not exceed the quark-diquark width for

a,i={0, f } Figure 4. The Bethe-Salpeter equation for the effective baryon-quark-diquark vertex function »o+[i+l.

207

our diquark-quark picture to make sense. Table 1. Parameter sets which give a solution to the BS equation with the physical mass of nucleon and delta. Set I II III

w0+ wl+ -fm0.34 0.25 0.28 0.17 0.25 0.14

m0+ 0.75 0.80 0.86

TOj+ Mff -GeV0.92 0.89 0.94 0.87

M&

1.32

2. Electromagnetic Form Factors The so-called gauging method 13 gives the correct, gauge invariant prescription how to treat processes where an external photon couples to a bound state whose wave function in turn is described by the solution of an integral equation. It basically consists of coupling the photon to all momentum dependent terms (propagators, vertices) in the kernel of the integral equation. Consequently we have impulse approximation diagrams where the external photon couples to the spectator quark and the quarks within the diquark. Inspecting the kernel of the BS equation, we find furthermore diagrams where the photon couples to the exchange quark and directly to the quark-diquark vertices (seagull graphs). The most important diagram, though, is the impulse approximation quark diagram, and its central element is the quark-photon vertex. It consists of a longitudinal and a transverse part,

r£ = r£ L + r £ T )

(l)

where the longitudinal part T^ L (the Ball-Chiu vertex), which is fixed by the Ward-Takahashi identity, is entirely determined by the form of the quark propagator13. The remaining transverse part might receive dynamical contributions of which the p — w meson poles in the q — q vector channel are presumably the most important ones. The most thorough study of these contributions has been made in Ref. [14] where the quark-photon vertex is analysed in the DS/BS framework. It was shown/that the transverse vertex contributes about one half to the pion charge radius and that it is essential for reproducing the b

T o be precise, the Ball-Chiu vertex contains a fixed transverse part since it should be free of kinematical singularities.

208

Figure 5. Electric form factor of the proton (left panel) and neutron (right panel). T h e small dotted curve in the right panel refers to a calculation from Ref. [15], assuming an electromagnetically compact (pointlike) diquark.

experimental results for the pion form factor. Following the analysis of Ref. [14] we write for the transverse vertex r

I * r f = ^ Q ^ ^ ^

W = *-P).

(2)

Here the BS wave function of the vector meson ^ is modelled by its two dominant structures. It is properly normalized and reproduces the experimental decay constant fp. The exponential describes an off-shell damping of the vector meson propagator, and we fit the constant a to the pion form factor, for details see Ref. [15]. The resulting electric form factors for proton and neutron are plotted in Fig. 5. We find for all sets that the transverse part of the quark-photon vertex contributes about 25% to the proton electric charge radius. Not surprisingly the form factor becomes softer with increasing diquark size which needs to be above 0.3 fm to bring the theoretical curve close to the data. Turning to the neutron, we see that all sets give a positive GE (a relativistic effect due to lower components in the BS wave function) but miss the experimental low-Q 2 behaviour completely. For comparison we have given results from a calculation with pointlike diquarks which are roughly compatible with the data. It appears that the latter assumption mimics non-valence effects that may arise if e.g. a pion-cloud is added to the nucleon. A pion-cloud would not only influence neutron's GE but also induce a mass shift for the nucleon between 16 200 and 300 MeV and somewhat less for the delta. We therefore solved the BS equations for higher core masses of the nucleon and delta and found quantitatively little difference to the old form factor results. Especially

209

•

Jones etal.PRL 84(2000) 1398

0.1 0I

0

Figure 6.

,

1 0.5

Proton's ratio HGE/GM

,

1 1

.

1 1.5

,

1 2

in comparison with the experimental d a t a .

neutron's GE remains quenched. Thus, a covariant calculation of pionic effects would clearly be desirable. For realistic diquark sizes, proton's ratio /J,GE/GM is underestimated in our calculations. This discrepancy can be traced back to the structure of the Ball-Chiu vertex. After some reshuffling of its tensor structure 9 one can isolate a transverse term which is proportional to the difference in the quark mass function for the outgoing and the incoming quark, i.e. ~ [M(k2) — M(p2)]/[k2 — p 2 ] . This term (which is absent in c o n s t a n t - m a s s constituent models) produces quite large negative contributions to GE for intermediate Q2 and therefore quenches the ratio. To investigate this in more detail, we replaced the fit of the running quark mass in Fig. 3 by a fit to the chiral extrapolation of latest lattice d a t a 4 , see the left panel of Fig. 7. Replacing the quark mass in the Ball-Chiu vertex by (a) the new fit and (b) by a constant we find results for (IGE/G/M which are depicted in the right panel of Fig. 7. Surprisingly enough the constant constituent mass does the best j o b but also the new fit to the lattice d a t a causes some noticeable change compared to the results with the original meson fit. Since the running quark mass is now well established and should be an element of any serious nucleon model we therefore conclude t h a t the observed ratio is most likely a consequence of a subtle interplay of various contributions such as a realistic quark propagator, vector mesons in the q u a r k - p h o t o n vertex and possibly also pions from the cloud. Further work which analyses these contributions in much more detail t h a n could be done here is clearly required. To summarize, we have investigated a covariant nucleon model involving three constituent quarks whose propagators capture the essential features of lattice simulations and DS calculations. Three-quark irreducible interactions

210

p[GeV]

o*[GeV*]

Figure 7. Left panel: lattice d a t a and the two fits to the quark mass function. Right panel: the ratio I*GE/GM for calculations with different quark mass functions in the Ball-Chiu vertex. The parameter for the scalar diquark width is here w0+ = 0.30 fm which results in the diquark masses rn0+ = 0.77 GeV and mj+ = 0.91 GeV.

were discarded and the problem was reduced to an effective q u a r k - d i q u a r k problem. T h e form factor results, especially for neutron's GE, point to the necessity of incorporating meson cloud contributions. T h e observed ratio [IGE/GM eludes a simple interpretation and is possibly a consequence of a number of mechanisms.

Acknowledgments T h e a u t h o r wants to thank the Special Research Centre for the Subatomic Structure of M a t t e r (CSSM) in Adelaide where most of this study was conducted. He is also grateful to the Alexander-von-Humboldt foundation which supported this research by a Feodor-Lynen grant. Special thanks go t o Reinhard Alkofer with whom the work was done and to Tony T h o m a s for m a n y insightful remarks.

References 1. R. Alkofer and L. von Smekal, Phys. Rep. 353, 281 (2001). 2. C. D. Roberts and S. M. Schmidt, Prog. Part. Nucl. Phys. 45, SI (2000). 3. J. Skullerud, D. B. Leinweber and A. G. Williams, Phys. Rev. D64, 074508 (2001). 4. P. O. Bowman, U. M. Heller and A. G. Williams, hep-lat/0203001. 5. P. Maris, nucl-th/0204020. 6. A. Bender, W. Detmold, C. D. Roberts and A. W. Thomas, nucl-th/0202082. 7. C. Alexandrou, P. De Forcrand and A. Tsapalis, Phys. Rev. D65, 054503 (2002). 8. I. Wetzorke and F. Karsch, hep-lat/0008008. 9. M. Oettel and R. Alkofer, hep-ph/0204178. 10. C. J. Burden, C. D. Roberts and M. J. Thomson, Phys. Lett. B371, 163 (1996).

211 11. M. Oettel, G. Hellstern, R. Alkofer and H. Reinhardt, Phys. Rev. C58, 2459 (1998). 12. M. Oettel, L. von Smekal and R. Alkofer, Comp. Phys. Comm. 144, 63 (2002). 13. A. N. Kvinikhidze and B. Blankleider, Phys. Rev. C60, 044003 (1999). 14. P. Maris and P. C. Tandy, Phys. Rev. C 6 1 , 045202 (2000). 15. M. Oettel, R. Alkofer and L. von Smekal, Eur. Phys. J. A 8 , 553 (2000). 16. M. Oettel and A. W. Thomas, nucl-th/0203073.

(POLARIZED) H A D R O P R O D U C T I O N OF O P E N C H A R M AT THE JHF IN NLO QCD

I. B O J A K CSSM, University of Adelaide Adelaide, SA 5005, Australia E-mail: [email protected]

We present the complete next-to-leading order QCD corrections to (polarized) hadroproduction of heavy flavors and investigate how they can be studied experimentally in (polarized) pp collisions at the JHF in order to constrain the (polarized) gluon density. It is demonstrated that the dependence on the unphysical renormalization and factorization scales is strongly reduced beyond the leading order. We also briefly discuss how the high luminosity of the JHF can be used to control the remaining theoretical uncertainties.

1. Introduction Although we have gained precise information concerning the total quark spin contribution to the nucleon spin in the last decade, the spin-dependent gluon density Ag remains elusive, see Fig. 1. Hence current and future experiments focus strongly on the issue of constraining Ag. The JHF could play a prominent role in these efforts using production of open charm. The gluon participates are dominantly in heavy flavor pair creation in longitudinally polarized pp collisions. Figure 1 also shows that even the unpolarized gluon distribution has considerable uncertainties at large x, and hence could be pinned down by a corresponding unpolarized measurement. In leading order (LO), heavy flavor pair production in hadron-hadron collisions proceeds through two parton-parton subprocesses, gg^QQ

and

qq^QQ

.

(1)

Gluon-gluon fusion is by far the most dominant mechanism for charm and bottom production in the unpolarized case in all experimentally relevant regions of the phase space 1,2 ' 3,4 . This should hold true in the polarized case unless Ag is very small. However, it is necessary to include next-to-leading order (NLO) QCD corrections for a reliable description. The LO results depend strongly on the arbitrary factorization and renormalization scales. Furthermore, in the unpolarized case the NLO corrections are known to be large 1,2 . Finally, new

212

213

1.2 1 0.8 0.6

xAG(x)

NLO

5 GeV2 x G ( x ) (GRV)

0.4 0.2 0 10

1 0 —x-

^T

1 0 "*

1

x Figure 1. Uncertainty in Ag (shaded la error bands) and in g at large a; (hatched). This is a copy of Fig. 5 of t h e analysis of Bliimlein and Bottcher 7 (BB).

processes with a single light quark in the initial state contribute for the first time in NLO. Unpolarized NLO results 1 ' 2,3,4 and polarized LO expressions 5 ' 6 have been available before, but the complete NLO results are presented here for the first time.

2. Technical F r a m e w o r k The 0{oP$) NLO QCD corrections to heavy flavor production consist of the one-loop virtual and the real "2 —• 3" corrections, the latter include the new production mechanism
214

gluons: |M| 2 (A 1) A 2 ) = | M | 2 + A 1 A 2 A|M| 2 .

(2) 1,2,3,4

The unpolarized \M\ in Eq. (2) can be compared to the literature . This is an important consistency check for the correctness of our new A|M| 2 . We treat the Levi-Civita e-tensor and 75 in n-dimensions by applying the HVBM prescription 8 . However, this leads to e-dimensional scalar products appearing alongside the usual n-dimensional ones. These contributions can be accounted for by an appropriately modified phase space formula 9,10 . The spin-dependent parton densities and cross sections are denned by Af(x^F)

= f+(x,fiF)-ft(x,fiF)

(3)

Acr = ![*(+,+)-*(+,-)] •

(4)

Here /+ (ft) gives the probability to find a parton / = q,q,g at a scale HF with momentum fraction x and helicity + (—) in a proton with helicity +. Similarly, cr(+, +) is the hadronic ( denotes both the unpolarized quantity and the polarized analogue A^. The virtual cross sections are obtained from the interference between the virtual and Born amplitudes and contain tensor loop integrals, which are Passarino-Veltman-reduced to scalar integrals 11 . The required scalar integrals can be found in 1,10 . Phase space integrations for the real 2 -» 3 gluon bremsstrahlung corrections require extensive partial fractioning to transform them to a standard form 1,9,10 . A sufficient set of four- and n-dimensional integrals can again be found in 1 ' 10 . The final color-averaged results for gluon-gluon fusion can be decomposed according to color structure as \Mgg | 2 ~ [(2C F ) 2 Mq E D + C2AM0q + M K Q + 2CFMRF + CAMQL\

,

(5)

with Casimir operators CF = (N^ — l)/2Nc and CA = Nc, Nc denoting the number of colors. The choice in Eq. (5) ensures that the "Abelian" MQED is identical to the QED part 9 of 75 -» QQ with the usual factor l/(2Nc) for replacing a photon by a gluon. Similar color decompositions are made for the other subprocesses. 3. Numerical and Phenomenological Results The total partonic subprocess cross sections &ij, i,j — q,q,g can be expressed in terms of LO and NLO functions /,-•' and f>-', /> •', respectively, which

215

0.4

|

Tt|

Alff+3.7<

0.35 0.3

f

(0) + 3

7.f

(l)

f (0)

0.25 0.2 0.15 0.1 0.05

le-05

0.0001

0.001

0.01

100

0.1 1 r|=s/(4 m2) -1

1000

10000

Figure 2. (m 2 j'a\)agg in NLO (MS) and LO as a function of rj according to Eq. (6), with fip = HR = "i for simplicity and Airas = 3.7 as appropriate for charm production.

depend only on a single scaling variable n = s/(4m 2 ) — 1:

)

! W m + *™* W+f. I)3 ln42 + A^»ln41} 2 3 ?(i)

m

87r

(6)

HF\\

where s is the available partonic center of mass (c.m.s.) energy squared, m is the heavy quark mass, as = a s (pfj), and /30 = (HCU — 2nj/)/3. The last term in Eq. (6) vanishes for the standard choice (ip = MR- In Fig. 2 we show {tn21a2s)agg in LO and NLO for fiF — fj,R — m as a, function of r] in the MS scheme. The NLO corrections are significant in the entire n range. In LO 'gg

~r

0" "at" threshold, but in NLO & ""i^""*") """ -" ^ ~ ~ ~

,

8m2

^

2(AT£-I)

[(2CF)2 - C\ + §] ^

due to a "Coulomb singularity". Logarithms from soft gluon emissions also contribute significantly even at the lowest 7] shown. At high energies gluon exchanges in the t-channel drive the unpolarized NLO result to a plateau value 1 , whereas the polarized NLO cross section vanishes. The large deviations between NLO and LO arise from Feynman diagram topologies that occur for the first time at NLO. Figure 3 shows the decomposition of the polarized NLO (MS) coefficient function A / ^ into contributions with different color structure as defined in

216 0.04

0.02

Af^QED)

-0.02

Af^COQ) Af^CKQ) -0.04 50-Afg^CQL) le-06

le-05 0.0001 0.001

0.01 0.1

1

10

100

1000

T|=s/(4 n T ) - 1 Figure 3. NLO (MS) coefficient function Afgg0) color decomposed according to Eq. (5). T h e quark loop (QL) contribution is enhanced by a factor 50. Also included are A / J g and fgg'.

Eq. (5). Notable are the large cancellations between the QED and the first nonAbelian (OQ) contributions in the threshold region t] —> 0. For completeness we also present here the coefficient functions Afgg' and fgg in Eq. (6), which arise from the mass factorization procedure. The RF contribution just moved into the last term in Eq. (6) and is not shown. The physical total cross section is obtained by convoluting the partonic cross sections in Eq. (6) with the parton densities a(S,m2)

= 5Z /

dxi

dx2fi(xi,iiF)fj{x2,VF)vij{s,m2)

(7)

where 5 is the hadronic cm. energy squared, s = X1X2S, and x m i n = Am2 IS. The differential heavy (anti-)quark inclusive distributions like d2a/dpT(lpL are obtained similarly by convoluting with our double differential partonic cross sections. In Fig. 4 we plot xix2Ag(xi,fij)Ag(x2,Hf)Aagg(xiX2S)/(a2/m2) with fij = \iT — m, 4iras = 3.7, and GRSV'OO val. helicity densities 12 . This shows the gluon-gluon fusion contribution in Eq. (7), with £1X2 included to display the right volume with logarithmic axes. We see that the z-region xia;2 — Zmin = 4m2/S is dominant. Furthermore, a peak on that line is situ-

217

Logxi ^0.75. 0.25

0

o .008 0 .006 0 004 0. 002 0 -0.25 Logx2 Figure 4. The gluon-gluon fusion contribution xiX2&g(xi,iif)Ag(x2,Hf)&<7gg(xiZ2S)/ (a2Jm2) for Eq. (7) with nf = tir = m, 4wa, = 3.7, and GRSVOO val. helicity densities 1 2 .

ated at xx ~ x%. Thus at VS ~ 10 GeV the JHF will probe x ~ ^/xmin ~ 0.3. The situation is similar for the unpolarized case. Note that in photoproduction, the region x ~ xmin is dominant. Thus the COMPASS 13 measurement will probe smaller x ~ 0.08 even if using the same cm. energy as the JHF. Experiments will usually study the longitudinal spin asymmetry defined by A(S,m2)

=

Aa(S, ro2)
(8)

The advantage is that one does not need to determine the absolute normalization of the cross section a(S, m 2 ). In Fig. 5 we demonstrate that the NLO results for the total charm spin asymmetry are indeed more robust under scale variations than LO estimates. We vary HF and mc independently of each other. The range for / i | = / r o 2 is given by 1 < / < 4.5, with the conventional choice pit = HF. The range for the charm mass m c = cGeV is 1.2 < c < 1.6. The JHF beam energy of Eb = 50 GeV and the GRSVOO val. helicity densities 12 have been used. We show A(f,c)/A(f = 2.5,c = 1.4) - 1, i.e. the deviation in percent of the asymmetry according to Eq. (8) with respect to a reference asymmetry taken at fixed nF = fi2R = 2.5m2 with rac = 1.4 GeV. To better guide the eye, contour lines in steps of 30% are plotted at the base of each plot. Here we also indicate m c = 1.4 GeV by a thin solid line. The NLO result in the right part of Fig. 5 is considerably "flatter" than the LO result in the

218

Ajp(f,cyAjp(2.5,1.4)-l

Eb=50 GeV f=|if/nig, mc=c GeV, Mr=Mf Figure 5. Deviation [in %] of the total charm spin asymmetry in LO (left) and NLO (right) from a reference choice ("0-pin" marker, see text) as a function of fip and mc with (J.R = fip. The LO result is multiplied by a factor (-1). Corresponding contour lines in steps of 30% are given at the base of each plot.

left part with respect to the variations. Variations of the charm mass cause the major uncertainty of about ±30% in the NLO predictions. In LO we find unacceptably large theoretical uncertainties. We predict the charm production asymmetry at the JHF in Fig. 6 using a range of recent 12 ' 14,15 and older 16 ' 17 helicity densities. These sets mainly differ in their assumptions about Ag. For calculating the required unpolarized a, we have used the associated unpolarized parton distributions specified in the references. For consistency, mc is also taken as in these fits. All results are obtained for the choice pip — p?R = 2.5(m2 + P y ) . Figure 6 shows that open charm production at the JHF can be very useful in pinning down Ag. The estimated statistical error of such a measurement, 8A = p i ,—, 1± with luminosity C = 7.66 fb _ 1 in 120 days, charm detection efficiency ec = 1 0 - 3 , polarizations for beam and target P^Pt = 1/2, and dilution factor fu = 0.15, is significantly smaller than the total spread of the predictions. Gluons of small size, e.g. the oscillating A<7 of GS C (see Ref. [17]), cannot be detected.

219

0.8 pTM Figure 6. The NLO charm asymmetry A for a JHF beam energy Eb — 50 GeV as a function of p™'n with px > p™ ln . Also shown is an estimate for the statistical error (see text).

In Fig. 7 we show the dependence of the asymmetry prediction on the beam energy for the GRSV'OO val. helicity densities, including the theoretical error from HF and [MR variations. At small p™in the sensitivity to the differences in A<7 is largest, see Fig. 6, but here also any uncertainties in the beam energy can have large effects. Hence it is crucial that the beam energy will be known with good precision. The theoretical uncertainties of polarized Aa and unpolarized cross sections a are much larger than those of the asymmetries A, since they cancel to a large extent in the ratio. Figure 8 shows the huge uncertainty in
220 ~

I

T"

T

I

I

I

[

I

1

1

T"

2

10

r

2
m2%^4m2

5

c

* ?

c

&S 0 E i-

Al

-5

GRSV'OO va

i-

E =40 GeV

< -10

b

E =50 GeV D

-15

E =60 GeV b

-j -20 0.0

i

i

i_

0.5

1.0 p™n [GeV]

1.5

2.0

Figure 7. The NLO charm asymmetry A in percent at JHF beam energies Eb = 40,50,60 GeV with the GRSV'OO val. densities 12 as a function of p^in. The bands show the uncertainty due to varying mc < (fj-r,^f) < 2m c .

4. Summary Determining the gluon helicity density Ag is one of the most pressing problems in spin physics today. The JHF could contribute significantly to pinning down Ag in the large x region using polarized production of open charm. The NLO QCD corrections20 presented here will make this feasible by reducing the theoretical uncertainties to a viable level. However, it would be highly advantageous if the JHF limited those uncertainties further by first determining the unpolarized cross section. Since polarized beam and target are planned for a later stage of the JHF, this is in accordance with the current experimental plans at the JHF anyway.

Acknowledgments I thank M. Stratmann for fruitful collaboration. This work has been supported by the Australian Research Council.

221 r

l_J^I

1000

1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l

_o c

i

-

T~ 600 E tQ.

c
c

C

C

m2%2^4m2 \

'•

i

I I I '

m 2 ^u 2 ^4m 2

\ \

,—. 800

1 I l-I-p-l

T

\ \

hr-^

™f

GRV'98

-

: m

\

'-

= 1 3 GeV

c -

";

.cT 400 b 200 0 0.0

0.2

0.4

0.6 0.8 p™n [GeV]

1.0

1.2

1.4

Figure 8. The unpolarized NLO charm cross section a at the JHF for different mc = 1.3,1.4,1.5 GeV with the GRV'98 densities 18 . The bands show the uncertainty due to varying m c < (^ r ,/i/) < 2m c .

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

W. Beenakker et al., Phys. Rev. D40, 54 (1989). W. Beenakker et al, Nucl. Phys. B351, 507 (1991). S. Dawson, R.K. Ellis, and P. Nason, Nucl. Phys. B327, 49 (1988). S. Dawson, R.K. Ellis, and P. Nason, Nucl. Phys. B303, 607 (1988). A.P. Contogouris, S. Papadopoulos, and B. Kamal, Phys. Lett. B246, 523 (1990). M. Karliner and R.W. Robinett, Phys. Lett. B324, 209 (1994). J. Bliimlein and H. Bottcher, arXiv:hep-ph/0203155. G. 't Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972); P. Breitenlohner and D. Maison, Comm. Math. Phys. 52, 11 (1977). I. Bojak and M. Stratmann, Phys. Lett. B433, 411 (1998); Nucl. Phys. B540, 345 (1999); Erratum, ibid. B569, 694 (2000). I. Bojak, Ph.D. thesis, Univ. of Dortmund, April 2000, arXiv:hep-ph/0005120. G. Passarino and M. Veltman, Nucl. Phys. B160, 151 (1979); W. Beenakker, Ph.D. thesis, Univ. of Leiden, 1989. M. Gliick et al., Phys. Rev. D63, 094005 (2001). COMPASS Collaboration, G. Baum et al., report CERN/SPSLC 96-14,30. AA Collaboration, Y. Goto et al., Phys. Rev. D62, 034017 (2000). D. de Florian and R. Sassot, Phys. Rev. D62, 094025 (2000). M. Gliick et al., Phys. Rev. D53, 4775 (1996). T. Gehrmann and W.J. Stirling, Phys. Rev. D53, 6100 (1996). M. Gliick, E. Reya and A. Vogt, Eur. Phys. J. C 5 , 461 (1998). J. Pumplin et al., arXiv:hep-ph/0201195. I. Bojak and M. Stratmann, arXiv:hep-ph/0112276 and in preparation.

COVARIANT LIGHT-FRONT D Y N A M I C S A N D ITS APPLICATION TO THE MESON WAVE F U N C T I O N S

O. M. A. L E I T N E R 1 ' 2 A N D A . W . T H O M A S 1 Department of Physics and Mathematical Physics, and Special Research Centre for the Subatomic Structure of Matter, University of Adelaide, Adelaide 5005, Australia E-mail: oleitnerQphysics.adelaide.edu.au ; [email protected] J.-F. M A T H I O T 2 2 Laboratoire de Physique Corpusculaire, Universite Blaise Pascal, CNRS/IN2PS, 24 avenue des Landais, 63177 Aubiere Cedex, France E-mail: [email protected]

In the framework of covariant formulation of light-front dynamics, we determine phenomenologically various meson wave functions. We focus on the pseudo-scalar particles B, D, w and K, as well as on the vector mesons J/ip, p and w. All these relativistic wave functions are defined on the light-front plane w-x = 0, in four dimensional space. Their relativistic structure is fully determined. The parameters are well constrained by several experimental d a t a sets: decay constants, transition form factors, electromagnetic form factors and charge radii.

1. Covariant Light-Front Dynamics In CLFD 1 , the state vector which describes the physical bound state is defined on the light-front plane given by the equation ui-x = 0, where u> is an unspecified light-like four vector (w2 = 0). It defines the position of the light-front plane. This approach is a generalization of the standard Light-Front Dynamics (LFD). The latter can easily be recovered with a special choice of the lightfront orientation, u> = ( 1 , 0 , 0 , - 1 ) . The description of relativistic systems in LFD provides several advantages compared with the standard formulation. The main properties are the following1: • The formalism does not involve vacuum fluctuation contributions. Therefore, the state vector describing the physical bound state contains a definite number of particles, described by Fock state components. • The Fock components of the state vector satisfy a three dimensional equation.

222

223

• The relativistic wave function and off-shell amplitude depend on the orientation of the light-front plane which is fully parametrized in terms of the four vector u>. The state vector describing a meson of momentum p, denned on a lightfront plane characterized by w, is given by (retaining the two-body component only):

M L = J\v) = (^)3/2j xf

W f r + fc2 -

p

-

UT)

*/xAfflia
e x p f t r ^ f r • p)dr

( 2 )

d3k\ r ) , / 8 ^ ^

d3k-> (2ir),/a ^

• W

where £/. = y/\s? + m 2 . We emphasize that the bound state wave function (see Fig. 1) is always an off-energy-shell object (r ^ 0 because of its binding energy). The parameter r is entirely determined by the on-mass-shell condition for the individual constituents.

>

q,
Figure 1.

(p)

Two-body relativistic wave function.

The two-body wave function $(/:i, k^,p,UT) can be parametrized in terms of several sets of variables. In order to make a close connection to the nonrelativistic case, it will be convenient to introduce the pair of variables 1 (k, n). The relativistic momentum k corresponds, in the frame where ki + k2 = 0, to the usual relative momentum between the two particles. The unit vector, n, corresponds in this frame to the spatial direction of w. The second set of variables which we shall use in the following is the usual light front coordinates ( I , R J _ ) . All details can be found in Ref. [1]. The two-body component $ is the solution of a three dimensional equation, Eq. (2). For simplicity, let us consider the case of a two-body scalar

224

system composed of two scalar particles. The wave function is scalar, and can be parametrized in terms of $(x, R-jJ- The homogeneous equation, which <3> should satisfy, is given by :

R]_ + n x(l — x)

M2)^(X,H2L)

(2

=

^ f » ( « ' , B f l A C ( « ' , R ' ± ; x , R x , M 2 ) £ * ^ , (2)

where m is the constituent quark mass (for two identical particles) and M is the bound state mass. In this form, this equation is similar to the Weinberg equation. The kernel, K, depends on the dynamics of the system. In our case, we shall use a kernel with one-gluon exchange.

2. Pseudo-Scalar Mesons The explicit covariance of our approach allows us to write down the general structure of the two-body bound state. For a pseudo-scalar particle composed of an antiquark and a quark of mass mi and m,2, respectively, it has the form : 1 *PS =

ih

75"(*i) ,

^"(*2)

(3)

where u(k^) and v(k\) are the usual Dirac spinors, and A\, Ai are the two scalar components of the wave function. The mass mr is defined by mr = 2 m ' m a and ^

'

J

'

mi+m2

is chosen here just for convenience. A\ and Ai will be expressed in terms of ^i and 52 which are Gaussian wave functions written as gt = 4.7T2a,-/J,- exp(—/3,k2), with oti and /?,- two parameters to be determined from experimental data. 2.1.

Normalization

In the spirit of the constituent quark model, the state vector is decomposed in Fock components, and only the two-body component is retained. Thus one gets, for a state of zero total angular momentum, the following normalization condition 1 : N

* =l = l

E ^AiA^A^ ,

(4)

A1A2

where D is an invariant phase space element given by, 1 d 3 ki _ 1 d 3 k _ 1 d 2 R x da; ~ (2?r)3 (1 - x)2ekl ~ (2?r)3 ek ~ (2ir)3 2x(l - x) '

(

'

225

With the pseudo-scalar wave function written in Eq. (3), the expression for the normalization is therefore : 1 • / .

R j + (m2x + mi (1 - z)) 2 m2x(l — x) ' +4AlA2

2.2. Decay

| " i ( ! - « ) + "*»

_

D.(Q)

Constant

According to the usual definition, the semi-leptonic decay amplitude is TM = (0\Jfi\PS) where JM is the 7^75 current. Since our formulation is explicitly covariant, we can decompose T^ in terms of all momenta available in our system, i.e. the incoming meson momentum p^ and u^. We have therefore : T^ = fps Pp + Bun

,

(7)

where fps is the physical decay constant. Using the diagrammatical rules of CLFD, we can calculate T^, and then the decay constant is given by :

! n<'-"-'> + "1">U + : te (i-, M ,

-W.K

/PS = 2N/6/

2.3. Electromagnetic

Form

D.

(8)

Factor

The electromagnetic form factor is one of the most useful tools to understand the internal structure of a bound state. Moreover, from the electromagnetic form factor at low Q2, it is possible to determine its charge radius. This physical observable is therefore very powerful in order to constrain the phenomenological structure of the wave function. By definition, the general physical electromagnetic amplitude of a spinless system 1 is : JP = (P + P')P FPS (Q 2 ) + ^B(Q2)

,

(9)

where Fps(Q2) is the physical form factor. In any exact calculation, B(Q2) should be zero, while it is non-zero for any approximate calculation. It should therefore be removed since it is a non-physical contribution. We choose for convenience u> • q = 0. This implies automatically that the form factors Fps (Q2) and B(Q2) depend on Q2 = — q2 only since, for homogeneity arguments, they can only depend on u> • p/ui • p' = 1. The physical electromagnetic form factor, Fps{Q2), can be obtained by contracting both sides in Eq. (9) with u>p. One then has :

F

?^=Bp-

^

226

After applying the diagrammatical rules of CLFD, one gets for the electromagnetic form factor : 2

f r ( T O l (l-,) + m 1

JD L

A imii]

+ 2(A1A'2 + A[A2)

2

^

~

Ri-,Rx.A

+ x m

) r

x)

+ m2X)

+ 4z(l - x)A2A'2] D . (11) mr J From this form factor, we can extract two major pieces of information : the first one is the charge radius of the bound state is defined by : (rls)

-~

2 = -^FPS{Q ) dQ 2

_

„ ,

Q2=0

(12)

and the second one is the behaviour of the electromagnetic form factor at high Q2 (asymptotic form). It is now well accepted that the asymptotic behaviour of the pion form factor, Fn(Q2) ~ l/Q2, is fully determined by the one-gluon exchange mechanism. This mechanism can either be considered explicitly in the hard scattering amplitude, or incorporated in the relativistic wave function of the meson. Here we adopt the second strategy. At asymptotically large Q2, the form factor is dominated by the contribution from the relativistic component A% in Eq. (3). The high momentum tail of the wave function is generated by the one-gluon exchange kernel. 2.4. Pion Transition

Form

Factor

The quantum numbers of the TT transition amplitude, -K -> 7*7, are similar to the deuteron electro-disintegration amplitude near threshold, as detailed in Ref. [1]. Thus, the exact physical amplitude has the form : F„ = ^epllvXq»PxF^

,

(13)

where P — p+p' and q — p'—p. In any approximate calculation, the amplitude Fpp has to depend on u. It should thus be decomposed in terms of all possible tensor structures compatible with the quantum numbers of the transition, as we did above for the semi-leptonic decay constant and electromagnetic form factor. Therefore, we can extract the physical form factor F*"7 by the following contraction : 2Q2(Wp)

e^p^xq^xFup

.

(14)

At leading order, the transition form factor is given by two relevant diagrams (other diagrams which should be taken into account in leading order either

227

correspond to vacuum diagrams or are equal to zero for u • q = 0) and the total amplitude reads :

x\Ax+

AD s ) - 2 R j . - A + xQ 2 R , .A 1 Rj^A 2A2x(l -x) + A2 "- 2 (1 - x) , ( 1 5 ) ~Q

where eq is the quark electric charge. Two physical constraints can be extracted from the transition form factor. The first is the axial anomaly which is defined at Q2 = 0 GeV . It gives the transition form factor when both photons are on their mass-shell. One should have : FW1(Q2 = 0) = -\-

.

(16)

The second one is the asymptotic behaviour : at high Q 2 , the transition form factor behaves like l/Q2. 3. Vector Mesons 3.1.

Formalism

We shall now concentrate on the structure of vector mesons, Jp = 1~. The wave function is decomposed in terms of all possible, independent spin structures. One gets 1 : d ( * i - * J . P . W T ) = V ^ e £ ( p ) ^{k2)

^ v^ih)

,

(17)

with (f)^ = Q(i_6, which depend on two invariant scalar variables. In terms of the variables k and n, the wave function takes the form 1 :

^20i(k,n)

= V^42V>A(k,n)w0l ,

(18)

with V(k,n) = T{fi-\fi). The relation between ipx and if) is the same as the relation between the spherical function Y]*(n) and n. The coefficients of the spin structures in ^(k, n) are scalar functions of two independent invariants, which we can choose to be k 2 and k - n , since these variables are only rotated by a Lorentz boost 1 . In the non-relativistic limit, only two components remain, f\ and f2, and they only depend on k 2 . We shall neglect in first approximation the tensor component, f2, so one is left with the

228

non-relativistic wave function / i = ^ ^ ^ ( k 2 ) . should be normalized 2 and one gets, N2=l

The wave function ^ ^ ( k 2 )

= m f \<j>NR(k2)\2D .

(19)

JD

3.2. Decay

Constant

The leptonic vector decay diagram is similar to the semi-leptonic one. The decay amplitude M^ to produce a photon with polarisation e^ from a vector state of polarisation ep can be decomposed in terms of all possible tensor structures 2 . Therefore we can write M.>ip as :

M"> = Fa\> + ^ - < + P-

+ Dal" ,

(20)

where the tensors 2 a f describe the kinematics. By simple algebraic 2 manipulation , we can isolate the physical amplitude F as a function of Mfip, and one obtains : F=^{I1-

2/ 2 + h + h) , where U = M^af

.

(21)

On the other hand, we can calculate M^p from the CLFD diagrammatical rules and thus extract the physical amplitude, F, which can be written as :

L

'-2V6m3/2

1i -- 2"^(I^ + 5*) Vet m)

^"(k2) D .

(22)

In this approach, the non-relativistic wave function ^ ^ ( k 2 ) is parametrized as a Gaussian which includes a and 8, the two parameters determined from experimental data sets. After averaging over the polarization states of the vector meson, one gets for the decay width : r (

^e+e-) = ^ a

2

£

2

| F | \

(23)

where Eq is a factor arising from the electric charges of the constituent quarks, M is the mass of the vector meson and ae is the electromagnetic fine-structure constant. 3.3. Radiative

Corrections

to the Wave

Function

Since we determine the parameters of the wave function from the leptonic decay width only, it is more appropriate here to correct the decay amplitude directly for radiative corrections. In the non-relativistic limit, this gives rise to the well known correction : ^NR

rr

(i - ^ )

,

(24)

229 which implies a huge correction, even for the J/^ (50% correction). However it has been shown in Ref. [3] that the relativistic calculation of these radiative corrections leads to a result that is better founded physically. The kinematical relativistic corrections induced by the finite relative momentum between the quark-antiquark pair, as detailed above, gives rise to a first reduction. The dynamical correction induced by the one-gluon exchange contribution to the amplitude leads to a reduction of the correction 1 | " s by 70%. The latter reduction is indeed rather important since the radiative correction is known to be of the order of 15% for the J/ty. This gives us somewhat more confidence in the expansion in as.

4. Results We used all available physical constraints to determine the meson wave functions. In regard to pseudo-scalar particles, we studied heavy (B and D) and light (TT and K) particles. For vector particles, we also studied both heavy {J/ip) and light (p and w) particles . The parameters included for the phenomenological description of these mesons have been extracted from physical constraints such as decay constants, electromagnetic form factors, charge radii, transition form factors and the normalization. For B and D particles, two constraints were used : the decay constant and the normalization condition. Their distribution amplitude is shown Fig. 2 and emphasizes their internal structure (heavy quark b or d). As output, the average transverse momentum is A / ( R ^ ) — 0.556 GeV for B and >/(S^y = 0.494 GeV for D. For the pion, the electromagnetic form factor at low Q2 is in total agreement with all experimental data sets. From this, we found the pion charge radius squared equal to 0.417 fm2, which is within experimental uncertainties. At high Q2, the electromagnetic form factor is in good agreement with the latest experimental data sets, because of the inclusion of the one-gluon exchange. For the transition form factor, we found an asymptotic limit in agreement with experimental data. For the kaon, the lack of data does not allow us to extract accurate results, but we obtained a nice agreement with a previous study 4 . At low Q2, the electromagnetic form factor is compatible with different studies and with experimental data sets even though they are not sufficiently numerous and precise. Moreover, we found a good approximation for the kaon charge radius squared since one gets 0.388 fm . This result is still in agreement with the experimental value. At high Q2, we found the same behaviour for this one as for the pion.

230

l 0.8

—

B

....

D

— •

K

—- Jt

*(x)

0.6 0.4

0.2 \--'

"0

I

0.2

••.-••• I

0.4

0.6

0.8

1

x Figure 2. Pseudo-scalar distribution amplitude. Full line, dotted line, dashed line and dot-dashed line represent the results for B, D, K and TT, respectively.

The distribution amplitudes for the pion and kaon are also shown in Fig. 2. As numerical outputs, the value of the transverse momentum is V^Rj.} = 0.320 GeV for the pion and y/(R?±) = 0.340 GeV for the kaon. Considering vector particles, we studied three particles : J/tp,p and w. Since our approach has been phenomenological, we only concentrated on the first two components of the wave function. That is the reason why we just used the normalization and decay constant. We found results in total agreement with the quark model. The distribution amplitudes for the vector mesons are shown in Fig. 3. As usually observed, the distribution amplitudes for p and w are very close because of the similar internal quark structure. In our approach, the radiative corrections have been taken into account only for J/ip and explain the main features of the distribution amplitude. The p and u> structure (light quarks) does not require the same treatment, but they should be analysed in the same way in the future. As output, the average transverse momentum has been calculated, and one gets \ / ( R l ) = 0.517,0.459,0.434 GeV, for the J/ip, p and u, respectively. This study has shown that CLFD is a very powerful framework which takes into account the dynamical spin structure of the wave functions of pseudoscalar and vector particles. We stress that both wave functions and transition form factors are described in a relativistic formalism. Through this work, we found similar results, in total consistency with soft QCD 4 (for n and K). Moreover, the results obtained also reproduce the heavy quark effective theory 5 (for B and D). All particles can be derived by following this formalism and by using experimental data sets as constraints.

231 1

1

0.6 *(x) 0.4 ~ 0.8

0.2 n

1

—

'

JA|/

i

/

CO

._. p

/ \ i

'

1

'

-

\

L *:"•'•"••-•.T-.-.Oi SL'

// //

'•' I /

- / .y

i 0.2

A

\

\

i 0.4

1

0.6

,

')

\ \ " 0.8

Figure 3. Vector meson distribution amplitude. Full line, dotted line and dot-dashed line represent the results for J/ip, w and p, respectively.

Acknowledgement s This work was supported in part by the Australian Research Council and the University of Adelaide.

References 1. J. Carbonell, B. Desplanques, V.A. Karmanov and J.-F. Mathiot, Phys. Rep. 300, 215 (1998). 2. S. Louise, J.-J. Dugne and J.-F. Mathiot, Phys. Rev. Lett. 472, 357 (2000). 3. F. Bissey, J.-J. Dugne and J.-F. Mathiot, hep-ph/0112079. 4. P. Maris and P.C. Tandy, Phys. Rev. C62, 055204 (2000). 5. H.-Y. Cheng, C.-Y. Cheung, C.-W. Hwang and W.-M. Zhang, Phys. Rev. D57, 5598 (1998).

N U C L E O N S T R U C T U R E F U N C T I O N S AT FINITE D E N S I T Y IN THE NJL MODEL

H. M I N E O Department of Physics, National Taiwan University 1 Roosevelt, Section 4t Taipei, Taiwan E-mail: [email protected] W.BENTZ Department of Physics, Tokai University Hiratsuka-shi, Kanagawa 259-1207, Japan E-mail: [email protected] A.W. T H O M A S Special Research Centre for the Subatomic Structure of Matter, and Department of Physics and Mathematical Physics, Adelaide University Adelaide, SA 5005, Australia E-mail: athomas©physics.adelaide.edu.au N. ISHII The Institute of Physical and Chemical Research (RIKEN) 2-1, Hirosawa, Wako-shi, Saitama 351-0198, Japan E-mail: [email protected] K. YAZAKI Department

of Physics, Tokyo Woman's Christian Suginami-ku, Tokyo 167-8585, Japan E-mail: [email protected]

University

In this work we discuss the EMC effect of the nucleon structure functions in nuclear matter, using a simple approximation to the relativistic Faddeev description of the nucleon in the framework of the Nambu-Jona-Lasinio model. We adopt a stable nuclear matter equation of state, calculated in the Nambu-Jona-Lasinio model, which incorporates confinement effects phenomenologically so as to avoid unphysical thresholds for the decay into quarks. We will compare our results for the BMC ratio in nuclear matter within the Nambu-Jona-Lasinio model with the parametrized fits to the experimental data.

232

233

1. Introduction The investigation of the nucleon structure functions is a very active field of current research 1,2,3 . On the theoretical side, it is not yet possible to directly use QCD for their description, although it is now possible to calculate the first few moments in lattice QCD and discuss the extrapolation to the chiral limit 4 . For a direct description of structure functions, however, effective quark theories, like bag models 5 , soliton models 6 and Faddeev descriptions 7 , play important roles. In this work we calculate the quark light-cone (LC) momentum distributions in the nucleon7 and discuss the medium effects on the nucleon structure functions, that is, the EMC effect in nuclear matter 8 . Our approach is based on the relativistic Faddeev equation 9 in the NJL model 10 , assuming a simple "static approximation" 11 for the Faddeev kernel. In our approach, the nucleon is described as a bound state of a quark and a scalar diquark (Jp = 0 + , T — 0). Axial vector diquarks (Jp = 1 + ,T = 1) have been investigated in Ref. [12], and it was shown that their effects should be subordinate to the scalar diquark ones. Pion cloud effects can be taken into account perturbatively, as has been done in Ref. [7,12] for the free nucleon structure functions. In this work, however, axial vector diquark and pion cloud effects will not yet be included. In a recent work 13 , a stable nuclear matter equation of state has been obtained in the NJL model by combining the quark-scalar diquark description of the nucleon with the mean field (Hartree) description of the nuclear matter ground state, using a 'hybrid model'. For the stability of nuclear matter it was essential to incorporate confinement effects phenomenologically so as to avoid unphysical decay thresholds. This has been done by introducing an infrared (IR) cut-off, in addition to the usual ultraviolet (UV) one, in the proper time regularization scheme 14 . We will present our results for the EMC ratio in nuclear matter based on this equation of state. 2. Calculations 2.1.

Model

The Lagrangian of the flavour SU(2) NJL model has the form C = V> (i$ — m) ifr + Ci, where m is the current quark mass and Li is a chiral invariant 4-Fermi interaction. By using Fierz transformations, any given £ / can be decomposed into various qq and qq channel interactions. The ratio r, = G3/Gn of the effective coupling constant in the scalar diquark channel (Gs) to the one in the pionic qq channel (Gn) depends on the assumed form of £ / , but in this work we will use rs as a parameter reflecting different forms of the interaction Lagrangian. We fix the constituent quark mass in the vacuum, which is related

234

to the current quark mass via the gap equation, as M = 400 MeV. The UV cut-off A is fixed by the pion decay constant (/* = 93 MeV), and Gn is fixed by reproducing the pion mass at zero density (mn = 140 MeV) as the pole of the qq t-matrix in the ladder approximation to the Bethe-Salpeter equation. Then the ratio rs is determined so as to reproduce the nucleon mass at zero density {MM = 940 MeV) as the pole of the quark-scalar diquark t-matrix in the ladder approximation to the relativistic Faddeev equation 7 . This automatically determines also the value of the scalar diquark mass Ms as the pole of the t-matrix in the scalar qq channel. In this work we neglect the momentum dependence of the propagator of the exchanged quark in the Faddeev kernel, that is, we work in the "static approximation". In this approximation the Faddeev equation can be solved almost analytically. Based on this quark-scalar diquark picture of the single nucleon, a nuclear matter equation of state has been constructed in Ref. [13] using the mean field approximation for the nuclear matter ground state. The essential difference to chiral point-like nucleon models, like the linear 100 MeV, the results are very similar 13 . The resulting equation of state gives the density dependence of the effective masses in the medium, M*, M* and MN, which are less than those at zero density, while the pion mass in the medium, m*, is somewhat larger than mn, since in this mean field approach only the (slightly) repulsive S-wave interaction between the pion and the nucleon is taken into account. The values of these masses are listed in Table 1 for the case of zero density, p = 0.16 f m - 3 (normal nuclear matter density) and p = 0.21 f m - 3 (saturation point of the nuclear matter equation of state in this model a ) .

a

Since in this model there is only one additional parameter (the strength of the 4-Fermi vector interaction Gu) as we go from the zero density to the finite density case, one cannot fit the empirical saturation point. In Ref. [13] the value of Gw has been adjusted so t h a t the binding energy curve passes through the empirical saturation point.

235 Table 1. The effective masses in the medium with density p p, m i " 3 Af*, MeV M*, MeV M*N, MeV ml, MeV

2.2. Quark LC Momentum

0 400 576 940 140

0.16 308 413 707 159

0.21 282 368 649 167

Distributions

The quark and antiquark light-cone (LC) momentum distributions in the proton can be obtained from the function 15 ?» = \ j

^ • > - « - ( p | T ( ? , ( 0 ) 7 + l M * - ) ) \P),

(1)

with q(x) — q(x) and q(x) = — q~(—x) for 0 < x < 1. Here x is the fraction of the proton's LC momentum component p_ = (po - P3)/V% carried by the quark (or antiquark) with flavour q — u,d, and \p) denotes the proton state with momentum p. The valence and sea quark distributions are then given by qv(x) — q(x) — q(x) and qs(x) = q(x). As explained in 15 , the evaluation of the distribution in Eq. (1) can be reduced to a straightforward Feynman diagram calculation by noting that it is nothing but the Fourier transform of the quark two-point function in the proton traced with 7 + , where the component k- of the quark LC momentum is fixed as k- — xp_ and the other quark momentum components (fc+,kj_) are integrated out. Since the proper time regularization scheme is defined in Euclidean space, it is very difficult to calculate the quark LC distributions directly from Eq. (1). Instead of direct calculation, we first calculate the moments and then construct the distribution functions in terms of its moments 15 , which can be done analytically within our model. The resulting expression becomes q(x) = ^[Tq(x

- ie) -Tq(x

+ ie)]

(2)

oo

r,(*)=x;j?/*n. where J£ is the nth moment of q(x) defined as J£ = £1dxxn-1q(x). Since the moments are given by the expectation values of local quark operators, they can be calculated also in Euclidean regularization schemes. The equation of state of Ref. [13] is also used to calculate the nucleon LC momentum distribution in nuclear matter. This function, /N/A^A), where yA is A times the fraction of the LC momentum of the whole system carried by

236

the nucleon (0 < yA < A), was discussed in detail in Ref. [16]. The quark LC momentum distributions in the medium are then obtained by applying the standard convolution formalism3: %/A(XA)

= I

dyA J

dzS(xA - yAz)q*(z)f^/A(yA),

(3)

where the asterisk indicates that the in-medium masses are used, and the variables are defined as xA=A

O2 k+ - 2 — =A-T %PA • g

(0 < xA < A)

p\

yA=A-?^= A^r (0 1. Since in general the distributions near x = 1 are very sensitive to the regularization procedure, the discontinuity at x = 1 must be considered as an artifact of the proper time cut-off scheme. (It does not appear, for example, in the regularization scheme used in Ref. [7].) This behaviour will affect the result for the EMC ratio at large x, as will shortly be seen.

237

xu. (x)

•

/

^ O l = 0.16 GeV2

\

Q2 = 4.0 GeV2

/

\

-/--<

•

\

/x f"-'\

/ /

/

/

emp i r i oa I

\ %N

\

\.

Figure 1. The valence u-quark distribution at zero density. The result after the Q2 evolution is compared with the empirical distribution of Ref. [2] at Q2 = 4.0 GeV 2 . Here the variable x is the same as the Bjorken variable Xff.

xd,(x) 0.7

L

/—s^

0.6

/

0.5

/

/

\

^

Do = 0.16 GeV2

\

Q' = 4.0 Gev\

0.4

0J

0.2

•

/

•

•// j

0.1

•

/

/

/

"

-

.

.

\

t %,

empirical

\

\

" \ ,N

* *-*

\

^^

Figure 2. The valence d-quark distribution at zero density. The result after the Q2 evolution is compared with the empirical distribution of Ref. [2] at Q2 = 4.0 GeV 2 . Here the variable x is the same as the Bjorken variable x^j.

Since our aim is to calculate the EMC ratio for isospin symmetric nuclear matter, we show in Fig. 3 the isoscalar quark distributions at zero and finite density, and also the result after performing the convolution with the nucleon momentum distribution in nuclear matter according to Eq. (3). We see that

238

the isoscalar quark distribution in the nucleon at finite density is shifted towards smaller x, i.e. it is softer for a bound nucleon in comparison to a free nucleon. (This can be related to the increase of the radius of the distribution of baryon charge, which in our calculation amounts to 3 ~ 4%.) After the convolution with the nucleon momentum distribution, the isoscalar quark distribution becomes a smooth function with a small non-zero support also for x > 1. Nevertheless, for x < 1 it always stays below the zero density distribution at large x.

0

0.2

0.4

0.6

OS

1

1.2

X

Figure 3. Isoscalar quark distributions at zero density (solid line) and finite density (dashed line). T h e isoscalar quark distribution after convolution with the nucleon momentum distribution in nuclear matter is shown by the dotted line. Here x corresponds to the Bjorken variable Xff for the zero density quark distributions, the variable z (see Eq. (3)) for the finite density quark distributions, and XA for the quark distributions after convolution with the nucleon momentum distribution.

In Fig.4 we plot the EMC ratio F^(x)/F2>(x) in comparison to the parametrized fits to the experimental data 1 8 . Here F*(x) is the nuclear matter structure function per nucleon and F ^ x ) is the isoscalar nucleon structure function at zero density. In the intermediate x region, where the quark distributions are softened by medium effects, our calculated ratio can reproduce the trend of the data. In the large x region, however, our results continue to decrease, which is due to the artificial end-point behaviour of the quark distributions at zero density due to the regularization procedure, see Fig. 3. In the small x region our calculation gives an enhancement of the EMC ratio, which can be understood from the baryon number sum rule, but for very small x the calculated ratio does not show the empirically observed suppression, which is

239 related to shadowing. F£(X)/F2°(X)

0

0.2

0.4

0.6

0£

1

X

Figure 4. The ratio of nuclear to deuteron structure functions. The parametrized fits to the experimental d a t a are taken from Ref. [18]. The variable x here corresponds to the Bjorken variable x^.

4.

Conclusion

In this work we calculated the spin independent nucleon structure functions at zero and finite density in the NJL model based on the relativistic Faddeev approach. One advantage of our calculation is that we treated the free nucleon, nuclear matter, and the structure functions at zero and finite density in one consistent framework. In particular, we wish to emphasize that the vector potential has been consistently included not only for the nuclear matter equation of state, but also for the description of the nucleon and quark momentum distributions. This has the important consequence that the baryon number and momentum sum rules for the nucleon and quark momentum distributions are satisfied automatically. Our model can explain the softening of the valence quark distributions within the nucleon bound in nuclear matter, corresponding to an increase of the radius associated with the distribution of the baryon number. The observed suppression of the EMC ratio for intermediate values of x can therefore be reproduced qualitatively. However, for large x our quark distributions show a discontinuity which is an artifact of the cut-off procedure (proper time scheme), and therefore the EMC ratio does not increase as x —> 1, although the Fermi

240 motion of nucleons has been properly taken into account. It is possible to remedy this situation by improving the higher m o m e n t s . Also, the overall shape of the quark distributions in the free nucleon is expected to improve if one includes the pion cloud effects 7 , 1 2 . These points are now under investigation and will be discussed in detail elsewhere 1 9 . Acknowledgements T h e authors t h a n k M. Miyama and S. K u m a n o for the Q2 evolution code of Ref. [17]. This work was supported by the Australian Research Council, Adelaide University, and the Grant in Aid for Scientific Research of the Japanese Ministry of Education, Culture, Sports, Science and Technology, Project No. C2-13640298. References 1. J. Gomez et al., Phys. Rev. D49, 4348 (1994); M. Arneodo et al., Nucl. Phys. B483, 3 (1997); D. F. Geesaman, K. Saito and A. W. Thomas, Ann. Rev. Nucl. Part. Sci. 45, 337 (1995). 2. A.D. Martin, R.G. Roberts, W.J. Stirling and R.S. Thone, Eur. Phys. J. C14, 133 (2000). 3. A.W. Thomas and W. Weise, The Structure of the Nucleon, Wiley-VCH, 2001. 4. W. Detmold, W. Melnitchouk, J. W. Negele, D. B. Renner and A. W. Thomas, Phys. Rev. Lett. 87, 172001 (2001). W. Detmold, W. Melnitchouk and A. W. Thomas, Eur. Phys. J. C13, 1 (2001). 5. A. W. Schreiber, A. I. Signal and A. W. Thomas, Phys. Rev. D44, 2653 (1991). 6. M. Wakamatsu and T. Kubota, Phys. Rev. D57, 5755 (1998). 7. H. Mineo, W. Bentz and K. Yazaki, Phys. Rev. C60, 065201 (1999). 8. European Muon Collab., Phys. Lett. B123, 275 (1983); Nucl. Phys. B293, 740 (1983); A. Bodek et al. (SLAC), Phys. Rev. Lett. 50, 1431 (1983); R. G. Arnold et al. (SLAC), Phys. Rev. Lett. 52, 727 (1984). 9. N. Ishii, W. Bentz and K. Yazaki, Nucl. Phys. A578, 617 (1995). 10. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1960); 124, 246 (1961). 11. A. Buck, R. Alkofer and H. Reinhardt, Phys. Lett. B286, 29 (1992). 12. H. Mineo, W. Bentz, N. Ishii, and K. Yazaki, Nucl. Phys. A703, 785 (2002). 13. W. Bentz and A. W. Thomas, Nucl. Phys. A696, 138 (2001). 14. D. Ebert, T. Feldmann and H. Reinhardt, Phys. Lett. B388, 154 (1996). G. Hellstern, R. Alkofer and H. Reinhardt, Nucl. Phys. A625, 697 (1997). 15. R. L. Jaffe, 1985 Los Alamos School on Relativistic Dynamics and Quark Nuclear Physics, eds. M.B. Johnson and A. Pickleseimer, Wiley, New York, 1985; R. L. Jaffe, Nucl. Phys. B229, 205 (1983). 16. G.A. Miller and J.R. Smith, Phys. Rev. C65, 015211 (2002). 17. M. Miyama and S. Kumano, Comp. Phys. Commun. 94, 185 (1996). 18. I. Sick and D. Day, Phys. Lett. B274, 16 (1992). 19. H. Mineo, W. Bentz, N. Ishii, A.W. Thomas and K. Yazaki, to be published.

VIOLATION OF SUM RULES FOR TWIST-3 PARTON D I S T R I B U T I O N S IN QCD

M. B U R K A R D T Dept. of Physics, New Mexico State University, Las Cruces, NM 88003, E-mail: burkardtQnmsu.edu

USA

Y. K O I K E Dept. of Physics, Niigata University, Niigata 950-2181, E-mail: [email protected]

Japan

Sum rules for twist-3 distributions are re-examined. Integral relations between twist-3 and twist-2 parton distributions suggest the possibility for a <5-function at x = 0. We confirm and clarify this result by constructing hi and hL (the quark-gluon interaction dependent part of hi) explicitly from their moments for a one-loop dressed massive quark. The physics of these results is illustrated by calculating hi(x,Q2) using light-front time-ordered pQCD to O(cts) on a quark target.

1. Introduction Ongoing experiments with polarized beams and/or targets conducted at RHIC, HERMES and COMPASS etc. are providing us with important information on the spin distribution carried by quarks and gluons in the nucleon. They are also enabling us to extract information on the higher twist distributions which represent the effect of quark-gluon correlations. In particular, the twist3 distributions gr{x,Q2) and IIL(X,Q2) are unique in that they appear as a leading contribution in some spin asymmetries. For example, gr can be measured in the transversely polarized lepton-nucleon deep inelastic scattering and hi appears in the longitudinal-transverse spin asymmetry in the polarized nucleon-nucleon Drell-Yan process 1 . The purpose of this paper is to reexamine the validity of the sum rules for these twist-3 distributions. A complete list of twist-3 quark distributions is given by the light-cone correlation functions in a hadron with momentum P, spin S and mass M:

J

^eiX'(PSmOh"^(Xn)\Q,\PS) = 2 [gi(x, Q2)pf(S • n) + gT(x, Q2)S"± + M2g3(x, Q2)n»(S • n)] ,

241

(1)

242

/

^eiXx(PS\i>{Q)a^il5xl;{\n)\Q,\PS) +hL(x, Q2)M{ptinv

/

= 2 [^(x,Q2)(Stf

- p"n".)(5 • n) + h3(x, Q2)M(S^nv

^eiX*(PSMO)^n)\Qi

\PS) = 2Me(x, Q2).

- S1_p»)/M - Sin'1)]

(2) (3)

The light-like vectors p and n are introduced by the relation p2 — n2 = 0, n+ = p~ = 0, P" = p" + ^-n" and S" = (5-n)p" + (5-p)n' 1 +S^. The variable a; G [—1,1] represents the parton's light-cone momentum fraction. Anti-quark distributions
(ps\mrl5m\Q>\ps) = 2J

dx [9l(x,Q2)p^S-n)

(4) + gT(x,Q2)Sl

+

M2g3(x,Q2)n"(S-n)].

From rotational invariance, it follows that the left hand side of Eq. (4) is proportional to the spin vector 5^ and thus gi,T,3(x
dx9l(x,Q2)

=J

dxgx(x,Q2)

= 2j

dxgT(x,Q2), dxg3(x,Q2).

(5) (6)

The same argument for Eq. (2) leads to the sum rule relations for h\tLi3(x, Q2): j f

dxh1(x,Q2)=f dxh1(x,Q2)

dxhL(x,Q2), =2 f

dxh3{x,Q2).

(7) (8)

The sum rule in Eq. (5) is known as Burkhardt-Cottingham sum rule 2 and Eq. (7) was first derived in Refs. [3,4]. Since the twist-4 distributions g3, h3 are unlikely to be measured experimentally, the sum rules involving those functions, Eqs. (6) and (8), are practically useless and will not be addressed in the subsequent discussions. As clear from the above derivation, these sum rules are mere consequences of rotational invariance and there is no doubt in their validity in a mathematical sense. However, if one tries to confirm those sum rules by experiment, great care is required to perform the integral including x — 0. In DIS, x is identified as the Bjorken variable XB — Q2/2P • q and

243

x = 0 corresponds to P • q -> oo, which can never be achieved in a rigorous sense. Accordingly, if fiL(x,Q2) has a contribution proportional to S(x) and h\(x, Q2) does not, experimental measurement would claim a violation of the sum rule. In this paper we re-examine the sum rule involving the first moment of the twist-3 distribution h^{x, Q2). In particular we argue that hi(x, Q2) has a rf(x)-singularity at x — 0. Starting from the general QCD-based decomposition of fiL, we show that it contains a function h™ which has a <$(x)-singularity. In Sec. 3, we construct hi for a massive quark from the moments of h\ at the oneloop level and show that h\ also has a <J(a;)-singularity, which together with the singularity in /i™ gives rise to a <J(a;)-singularity in h^ itself. In Sec. 4, we perform an explicit light-cone calculation of hi in the one-loop level to confirm the result of the previous sections. Details can be found in Ref. [5]. 2. <S(a;)-Functions in HL(X, Q2) The OPE analysis of the correlation function in Eq. (2) allows us to decompose hi(x, Q2) into a contribution expressed in terms of twist-2 distributions and the rest which we call h\(x,Q2). Since the scale dependence of each distribution is inessential in the following discussion, we shall omit it in this section for simplicity. Introducing the notation for the moments on [—1,1], Mn[hi] = f_1 dx xnhi{x), this decomposition is given in terms of the moment relation 1 : Mn[hi] = -^-Mn[h1]

+ ^-1^-Mn-1[g1}

+ Mn[hl],

M0[hL} = M 0 [/ii],

(n>l)

(9) (10)

with the conditions M„[fti] = Mi[Ai] = 0.

(11)

By inverting the moment relation, one finds hL(x) = hYw(x)

2x[

Jx

+ h>E(x) + hl(x)

dy±M2 + y

J_y

y2

(12)

kiM 2x.o. ^\!!Mr f\jjM\ dy +hi{x) M [ x

M [ x

{x>0) (13)

Jx

7-i

T

+ h3L(x),

(x<0)

where the first and second terms in Eq. (12) denote the corresponding terms in (13). In this notation the sum rule in Eq. (10) and the condition of Eq. (11)

244

imply3, M0[h]>] = 0.

(14)

Naively integrating Eq. (13) over a; for a; > 0, while dropping all surface terms 6 one arrives at fQ dxhi(x) — JQ dxhi(x) + JQ dxh3L{x) and likewise for f_xdxhi(x). Together with Eq. (11), this yields f_1dxhi = f_1dxhi. However, this procedure may be wrong due to the very singular behaviour of the functions involved near x — 0. Investigating this issue will be the main purpose of this paper. We first address the potential singularity at x = 0 in the integral expression for h™(x) in Eq. (13). In order to regulate the region near x — 0, we first multiply h™(x) by x&, integrate from 0 to 1 and let /3 —• 0. This yields lldxhtix) = %

}zpj\xx^9l{y)

= ^l(0+),

(15)

while multiplying Eq. (13) by \x\& and integration from —1 to 0 yields J°~ dxhT(x)

= -£L\ijnpJ0~

dx\xf~lgi(y)

= - ^ i ( O - ) , (16)

where we have assumed that gx (0±) is finite. Adding these results we have j°^dxh'E(x)

+ lo dxhZ(x)=^(gi(0+)-gi(0-)).

(17)

Equation (14) and the fact that, in general, limx^o gi(%) — gi{—x) ^ 0, imply b TYI

h?(x) = h?(x)reg

- ^L ( 5 l ( 0 + ) - 5 1 ( 0 - ) ) 6(x),

(18)

where h™(x)reg is defined by the integral in Eq. (13) at x > 0 and x < 0 and is regular at x = 0. Equation (18) indicates that hi has a S(x) term unless h\{x) has a S(x) term and it cancels the above singularity in h'£{x). Equation (18) demonstrates that the functions constituting h^x) are more singular near x — 0 than previously assumed and great care needs to be taken when replacing integrals over nonzero values of x by integrals that involve the origin. In particular, if HL{X) itself contains a S(x) term, then Eq. (10) implies f JQ+ a

(M*)~M*))+ /

(hL(x)-h1(x))^0,

(19)

J-i

More precisely, the original OPE tells us M0[h\ + h^] = 0. But as long as 31 (0±) is finite, which we will assume, this is equivalent to the stronger relations, Eqs. (11) and (14). b For example, dressing a quark at O(as) yields fli(0+) 5^ 0 and gi(0—) = 31 (0+) = 0.

245

and, since h\{x) is singularity free at x — 0: o-

,.1

,.1

(x). (20) 'o+ dxhi(x) + I dxhi(x) = I dxhi(i •l J-i 3 would obviously fail. Accordingly, an attempt to verifyJo+ the "/it-sum rule" However, in order to see whether the S(x) identified in Eq. (18) eventually survives in /IL(X), we have to investigate the behaviour of hL(x) at x = 0. To this end we will explicitly construct hi{x) for a massive quark to 0(as). /

3. hj,(x, Q2) from the Moment Relations In this section we will construct IIL(X,Q2) for a massive quark (mass mq) to O(as) from the one-loop calculation of Mn[h\\: hL(x,Q2)^hL°\x)

+ ^CFln^hL1)(x),

(21)

where the scale Q2 is introduced as an ultraviolet cutoff and the Cp — 4/3 is the colour factor. hL ' ' m ( ' \x) are denned similarly. g[°'(x) — h\'(x) = S(l — x) gives hL '(x) = S(l - x), as it should. One-loop calculations for g\{x) and h\(x) for a quark yield the well known splitting functions 7,8 :

'"'""" = p ^ + 1 4 " - >

<22> <23'

*S"w = i r ^ i : +1 4 * 1 - *>•

Inserting these equations into the defining equation in Eq. (13), one obtains h^1\x)=3x h

™{1){x)

=

+ Ax\n^-

(i^x]7 - 4 * l n i i r

(24) _3+3

* _ \S{x)-

(25)

In the first line of Eq. (25), the term (3a:-§<5(l — x)) comes from the self-energy correction, i.e. from expanding M = mq 1 + f f C F f l n ^ -\&{x) — -^g\(G+)S(x)

in h™^\x)

accounts for the second term on the right

hand side of Eq. (18). We also note that hLVW (

in Eq. (12), and

does not have any singularity

at x = 0 and satisfies f„ dx hL ' (x) = /„ dx h\ (x) as it should. The structure function hL' (x) can be constructed if we know the purely twist-3 part /iLv (x) at the one-loop level. One-loop renormalization of hL was completed in 9 and the mixing matrix for the local operators contributing to

246

the moments of hi,(x,Q2) quark distribution 5

was presented. We obtain for the moment of the

f1 dx xnhf\x) = -J_ - J L + I, y_!

L

n+l

n+ 2

(26) K

2

for n > 2. From this result, together with Eq. (11), we can construct h^ as hf\x)

= 3 - 6x + l-8{\ -x)~

\s(x).

'

'(x) (27)

We emphasize that the — l/2S(x) in Eq. (27) is necessary to reproduce the n = 0 moment of h3LW(x). From Eqs. (24), (25) and (27), one obtains

M * . Q2) = <*(! - *) +2n

2

n

^ 2%°F m .[1-*]+

. + ls(l-z)-S(x) 2

•

(28)

We remark that the above calculation indicates that the S(x) term appears not only in h™ but also in h\. Furthermore, they do not cancel but add up to give rise to — S(x) in hi{x,Q2) itself. In the next section we will confirm Eq. (28) through a direct calculation of hi(x, Q2) for a quark. 4. Light-Cone Calculation of hz,(x,

Q2)

In order to illustrate the physical origin of the S(x) terms in fiL(x) and to develop a more convenient procedure for calculating such terms, we now evaluate /IL(X) using the time-ordered light-front (LF) perturbation theory. The method has been outlined in Ref. [10] and we will restrict ourselves here to the essential steps only. There are two equivalent ways to perform time-ordered LF perturbation theory: one can either work with the LF Hamiltonian of QCD and perform old-fashioned perturbation theory 10 , or one can start from Feynman perturbation theory and integrate over the LF-energy k~ first. In the following, we will use the latter approach for the one-loop calculation of hi(x). In LF gauge, A+ — 0, parton distributions can be expressed in terms of LF momentum densities (&+-densities). Therefore, one finds for a parton distribution, characterized by the Dirac matrix T at O(as) and for 0 < x < 1 /

H4k 1 1 79^4 7" 7 ^ 7 r7"«(p) (zir) k — m a + ie k — m„ + is

xS (x- — j where Dfll/(q) = ~rfe

D^lp-k),

(29)

g^ - fy""*"*19" is the gauge field propagator in LF

gauge, and n^ is a light-like vector such that n • A = A+ ~ (^1° + A3) /y/2 for

247

any four vector A*. The k integrals in expressions like Eq. (29) are performed using Cauchy's theorem, yielding for 0 < k+ < p+

I

dk2T

(p _

m 2 + i£y

(30)

(p - k)2 + ie

1 1 (2fc+)2 2(p+ - k+)

1

kx-oo

1 1

where we used k+ = xp+. In order to integrate all terms in Eq. (29) over k~, Cauchy's theorem is used to replace any factors of k~ in the numerator of Eq. (29) containing k~ by their on-shell value at the pole of the gluon propagator k

^

k

~P

2(p+-*+)'

(31)

In the following we will focus on the UV divergent contributions to the parton distribution only. This helps to keep the necessary algebra at a reasonable level. We find for 0 < x < 1 to 0(as)

M-,«a) = ^ l n ^

i r

J^-,

where the usual +-prescription for >1_1^ applies at x = 1, i.e. n 1 ^

(32) = j ^ for

x < 1 and f0 t f a n g i = 0. Furthermore, /IL(X) = 0 for x < 0, since antiquarks do not occur in the O(as) dressing of a quark. In addition to Eq. (32), there is also an explicit S(x — 1) contribution at x = 1. These are familiar from twist-2 distributions, where they reflect the fact that the probability to find the quark as a bare quark is less than one due to the dressing with gluons. For higher twist distributions, the wave function renormalization contribution is T^-CF In Qj^5(x — 1). The same wave function renormalization also contributes at twist-3. However, for all higher twist distributions there is an additional source for S(x — 1) terms which has, in parton language, more the appearance of a vertex correction, but which arises in fact from the gauge-piece of selfenergies connected to the vertex by an 'instantaneous fermion propagator' 3-f. For gr{x, Q2) these have been calculated in Ref. [10] where they give an additional contribution — ff-Cf In §pS(x - 1), i.e. the total contribution at x = 1 for gr(x, Q2) was found to be ff-Cf \n^^S(x

- 1). We found the same

2 c

S(x - 1) terms also for /iz,(x, Q ) . Combining the S(x - 1) piece with Eq. (32) c

I n LF gauge, different components of the fermion field acquire different wave function

248

we thus find for 0 < x < 1 hL(x,Q2)

= 8(x-l)

+ ^CFln^

f r ^ - ^ y - + U(x - 1)1 •

(33)

Comparing this result with the well known result 8 for hi

h1(x,Q2) = S(x-l)

+ ^\n^CF\j-^r

+ h(x-l)},

(34)

%CF

(35)

one realizes that lim /

dx [hL(x,Q2)

- hi{x,Q2]

= ~\n

± 0,

i.e. if one excludes the possibly problematic region x = 0, then the /i^-sum rule 3 is violated already for a quark dressed with gluons at O(as). In the above calculation, we carefully avoided the point x — 0. For most values of k+, the denominator in Eq. (29) contains three powers of k~ when k~ —» oo. However, when k+ = 0, k2 — m2 becomes independent of k~ and the denominator in Eq. (29) contains only one power of k~. Therefore, for those terms in the numerator which are linear in k~, the A;~-integral diverges linearly. Although this happens only for a point of measure zero (namely at k+ = 0), a linear divergence is indicative of a singularity of HL(X, Q2) at that point d . To investigate the k+ & 0 singularity in these terms further, we consider k

/(*+, kj.) = Jdk-n9

,^

9

,_

,*, , ^

(36)

(k2 — m2 + is)2 (p — k)2 + is

=

dk — (k2 — m2 + is)2 \{p — k)2 + is] /'

Jcan.yk

, k x ) + /sjn.(«

j^x),

where the 'canonical' piece fcan. is obtained by substituting for k~ its on energy-shell value k~ — p~ — 2tp+-k+\ [^ e v a -l u e a * the pole at (p — k)2 = 0 , Eq. (31)]. For k+ = xp+ ^ 0, it is only this canonical piece which contributes. To see this, we note that k~ — k~ = — 2r -T-1-H, and therefore Jsinyk

dk~

i KJ_ •

/

k~ 1 2 (k — m + ie) (p — k)2 + is 2

2(p+ - *+) /

2

dk'

(k2

m2 + ie)2

(37)

renormalization. However, since all twist-3 parton distributions involve one LC-good and one LC-bad component, one finds the same wave function renormalization for all three twist-3 distributions. d N o t e that the divergence at fc+ = p~^ is only logarithmic.

249 Obviously 1 2 , / dk

(2fc+fc -_ k 2~

_ m 2 + < e ) 2 = 0 for Ar+ ^ 0 because one can always m 2 + k 2 — ie

q avoid enclosing the pole a t k~ = by closing the contour in the 2k+— appropriate half-plane of the complex A;~-plane. However, on the other hand /d2**(*£-kiim;+,-0' = ki+^f a n d t h e r e f o r e

fsin{k+,^)

=

-Li^ll.

(38 )

Upon collecting all terms oc k~ in the numerator of Eq. (29), and applying Eq. (38) t o those terms we find for the terms in hi(x, Q2) t h a t are singular at x~0 Q2) = - ^

hL,sin{x,

In Q?CFS(x).

Together with Eq. (33), this gives our final result for / I L , u p to 0(as), also for x = 0

hL(x,Q') = 6(x-l)

+

as^,

,

-lCF\n

Z7T

Q ?7l~

L

-iw + H ^ + ^ - 1 »

(39) valid

.(40)

As expected, hi from Eq. (40) does now satisfy the hi-sum rule, provided of course the origin is included in the integration. This result is important for several reasons. First of all, it confirms our result for hi(x,Q2) as determined from the m o m e n t relations. Secondly, it provides us with a method for calculating these S(x) terms and thus enabling us to address the issue of validity of the naive sum rules more systematically. And finally, it shows t h a t there is a close relationship between these S(x) t e r m s and the infamous zero-modes in LF field theory 1 3 . T h e result of Ref. [11], where canonical Hamiltonian light-cone perturbation theory is used to calculate hi(x) for x ^ 0, agrees with ours which provides an independent check of the formalism and the algebra. However, the canonical light-cone perturbation theory used in Ref. [11] is not adequate for studying the point x — 0. From the s m o o t h behaviour of hi{x) near x = 0 the a u t h o r s of Ref. [11] conclude t h a t the sum rule for the parton distribution hi{x) is violated to O(as)- Our explicit calculation for hi[x) not only proves t h a t the sum rule for hi(x) is not violated to this order if the point x — 0 is properly included, but also shows t h a t it is incorrect to draw conclusions from s m o o t h behaviour near x — 0 about the behaviour at x = 0. 5.

Summary

We have investigated the twist-3 distribution h^x) and found t h a t the s u m rule for its lowest m o m e n t is violated if the point x = 0 is not properly included.

250

For a massive quark, to O(as) we found AL (*, Q J ) = 8(x - 1) +

T^-CF

2n

In

mf

9

1

[1 - ar]+

2

1

{41)

At 0(0:5), fti(a;, Q2) does not satisfy its sum rule if one excludes the origin from the region of integration (which normally happens in experimental attempts to verify a sum rule). Of course, QCD is a strongly interacting theory and parton distribution functions in QCD are nonperturbative observables. Nevertheless, if one can show that a sum rule fails already in perturbation theory, then this is usually a very strong indication that the sum rule also fails nonperturbatively (while the converse is often not the case!). From the QCD equations of motion, we were able to show nonperturbatively that the difference between fiL(x,Q2) and h\(x,Q2) contains a S(x) term [hL{*,Q2)

- hl(x,Q2)]singular

= - ^

M 0 + , Q 2 ) -
Since gi(0+,Q2) - gi{0-,Q2) = \imx-+0gi(x,Q2) - g\(x.Q2) seems to be 6 nonzero (it may even diverge ), one can thus conclude that either hi(x,Q2) 3 2 or h L(x, Q ) or both do contain such a singular term. We checked the validity of this relation to O(as) and found that, to this order, both h\ and /i£ contain a term oc 6(x). We also verified that even though the sum rule for 1IL(X) is violated if x = 0 is not included, it is still satisfied to O(as) if the contribution from x = 0 (the S(x) term) is included. Acknowledgments M.B. was supported by a grant from the DOE (FG03-95ER40965). Y.K. is supported by the Grant-in-Aid for Scientific Research (No. 12640260) of the Ministry of Education, Culture, Sports, Science and Technology (Japan). We are also grateful to JSPS for the Invitation Fellowship for Research in Japan (S-00209) which made it possible to materialize this work. References 1. R.L. Jaffe and X. Ji, Nucl. Phys. B375, 527 (1992). 2. H. Burkhardt and W.N. Cottingham, Ann. of Phys. 56, 453 (1970). 3. M. Burkardt, Proc. to Spin 92, Eds. Hasegawa et al., (Universal Academy Press, Tokyo, 1993); R.D. Tangerman and P.J. Muilders, hep-ph/9408305. 4. M. Burkardt, Phys. Rev. D52, 3841 (1995). 5. M. Burkardt and Y. Koike, to appear in Nucl. Phys. B, hep-ph/0111343. e

I n the next-to-leading order QCD for a quark, \imx-,.o 9l(x) divergent 1 4 .

— 9l(x)

>s logarithmically

251 6. D. Boer, Ph.D. Thesis, Vrije Univ., Amsterdam 1998. 7. V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15, 438 (1972); Yu. L. Dokshitser, Sov. Phys. JETP, 46, 641 (1977); G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977). 8. X. Artru and M. Mekhfi, Z. Phys. C45, 669 (1990). 9. Y. Koike and K. Tanaka, Phys. Rev. D 5 1 , 6125 (1995). 10. A. Harindranath and W.-M. Zhang, Phys. Lett. B408, 347 (1997). 11. R. Kundu and A. Metz, Phys. Rev. D65, 014009 (2002). 12. S.-J. Chang and T.-M. Yan, Phys. Rev. D7, 1147 (1972). 13. M. Burkardt, Advances Nucl. Phys. 23, 1 (1996). 14. W. Vogelsang, Phys. Rev. D54, 2023 (1996).

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6. Hadronic Properties of Nuclei

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PROPERTIES OF H A D R O N S IN N U C L E A R M A T T E R

F.C.KHANNA Physics Department

University

of Alberta Edmonton Alberta, T6G 2J1 Canada * and Center for the Subatomic Structure of Matter, University of Adelaide, Adelaide, 5000, Australia E-mail: khannaQphys.ualberta.ca D.U.MATRASULOV Physics Department University of Alberta Edmonton Alberta, T6G 2J1 Canada [email protected]. ca

Nucleon properties change in nuclear matter at normal and high densities as well as with increasing momentum transfer (q2). It is known that with nuclear matter at high temperature, the mass and coupling constant of mesons are changed. The Japan Hadron Facility will provide an opportunity to increase the nuclear density, temperature and q2 of nuclear matter simultaneously. This provides a unique opportunity to study all these variations at once. Furthermore, with a proton beam of 50 GeV incident on a target, there will be production of mesons and hadrons with one and two charm and beauty quarks. This will provide an opportunity to change the flavour of quarks in hadrons and then study the interaction and decay of these hadrons in nuclear matter.

1. Introduction The Japan Hadron Facility1 (JHF) will produce a proton beam of 50 GeV. Such a beam of protons incident on finite nuclei will produce local regions of high density, 2-3 times the nuclear matter density, and of high temperature, a few tens of MeV. Furthermore, in interaction with nucleons it will produce a very large variety of mesons, hyperons and hyperons with heavy flavor like charm and beauty. The phase space alone suggests that there will be copious production of charm mesons and nucleons. As in the case of hyperons 2 with one or two strange quarks, the charm hyperons (Ac) and beauty hyperons (At) will be absorbed in nuclear fragments or in target nuclei. This will initiate the first study of nuclei with charm and beauty baryons in them and will * permanent address

255

256

provide an opportunity for the study of the behaviour and motion of heavy quarks in nuclear matter. There is also the possibility that the baryons with two heavy quarks are produced. Certainly baryons with two charm quarks are more likely than with two beauty quarks. Spectroscopy of such baryons would be most interesting and the motion of such baryon in nuclear matter would be very exciting. JHF is thus expected to open a window to new phenomena in nuclear systems. Measurements to isolate these phenomena and attribute them to heavy quarks will require ingenuity and clever experimentation. But it will be well worth it to understand these new phenomena with new unusual probes. 2. Density Dependence of Nucleon and Meson Properties It is well-known that nucleon and meson properties like masses, coupling constants and vertex form factors depend on density and momentum transfered (q2). We have proposed a medium modified Skyrme-like Lagrangian which includes the distortion of the nonlinear meson field by the medium. A scalarisoscalar sigma meson is added that is identified with the quantum dilaton. The medium effect on the pion is introduced through the self-energy operator while its effect on the dilaton field is limited to renormalization of the gluon condensate. Table 1. Change of masses of 7r, a, N and F with density, (all quantities are given in MeV). P/PO 0 0.5 1.0

TOir

139.0 144.90 144.06

mN

ma

550.1 513.8 493.8

1413 1217 11.57

1 a—t-7T7r

251.2 88.7 34.6

In Table 1 we show the change of masses of 7r, a and N in the medium with density p, po is the nuclear matter density. Both a and N masses decrease. The IV-^Tr is also shown. It indicates that a becomes a sharp resonance at nuclear matter density while it is a very broad resonance in the free state. Table 2. P/PO 0.5 1.0

Change of various coupling constants with density.

3"A/9 A 0.96 0.92

9*NN/9*NN

0.91 0.80

9^NN/9
AJ/A, 0.70 0.56

A;/A C T 0.90 0.84

In Table 2 we show various quantities normalised to the free space value.

257 T h e axial vector coupling constant decreases towards the value of nuclear m a t ter (0.8). This value is realised without the dilaton field. T h e coupling constants of 7T and a decrease by 20% while the decrease of A,r in medium is quite drastic, approaching a factor of 2. Finally, the 7TJVJV and
258

Hamiltonian etc. and ii) a pure finite temperature vacuum state of this new Hilbert spaces defined by using Bogoliubov transformations. The propagators are 2 x 2 matrices. In TFD it is possible to use Wick's theorem and the Feynman diagrammatic approach as in the case of zero temperature field theory. The free meson-nucleon vertex function is independent of temperature. The temperature dependence is calculated at one-loop level and the results are limited to terms linear in the fermi- and bose distribution function. The result may be written as TA(t,T)

= TA(t) + i^AAB(t,T),

(1)

B

with /

dAk —^TB(k2)(a

+ M)TA{t){b +

x [Go(a)Go(b)D$(k2)

M)TB{k2)

+ 2Go(a)GT(b)DB(k2)},

(2)

t = q2 = q2o-q2 = (p-p')2, a = p'-k, b=p-k, 1 Go GHa) = 2wiS{a2 ^ = a2-M2M2 + ~ M2)^(fl) + isis ' _1 D k2 D °( ) = 152 iTTTTT T(*2) = ~^iS(k2 - m2)NB(k) k - M2 + is '

(3)

where

(4) (5)

and NF(a) - 0(ao)nf{a)

+ 0(-a0)nf{a),

(6)

with nF(a) = [ ^ ( M - ^ )

+

i]-i

nB(fc) = [ e ^ 0 l - l ] - 1

f /,

n / (a) = [e (l°

o|+ l)

1

' + l]- .

(7) (8)

The coupling constants (monopole) at zero temperature and the vertex form factors are taken from the BONN-A potential. The results indicate some very interesting features when variation of coupling constants with temperature and density and the corresponding change of vertex form factors with q2 are considered. They may be summarised as follows : i) There is a critical temperature (Tc) where the coupling constants for different mesons show a very sharp change. Tc is different for different mesons: 77 = 360 MeV, T° = 95 MeV, T/ = 200 MeV and T? = 175 MeV. The most notable point is that the w—meson coupling constant increases while the

259 other coupling constants decrease to zero at some temperature. It is worth pointing out that g„NN, goes to zero at the lower temperature suggesting that the strong attraction provided by the cr-meson disappears. What impact this has on the overall stability of the nucleus needs to be carefully considered. The sharp increase of gWNN arises due to the isospin factors for p- and w-meson. This implies that the overall repulsion increases making it much more difficult to keep a stable nucleus. It is possible that these variations depend on the choice of the vertex form factors. ii) The vertex form factors as a function of q2 decrease more rapidly as the temperature is incresed, thus approaching zero at much smaller values of q2. iii) The change of the coupling constant and the vertex form factors is not as drastic with the changing density as it is with a change of temperature. For example, the density dependence of Tc for different mesons is shown in Table 1 and the change is small. iv) The usual procedure of finding the critical temperature of transition from hadronic state to the quark-gluon plasma is based on lattice QCD. In such calculations it is usual to look for critical temperature for hadronization of the quark-gluon system and it is conceivable that such a transition temperature is unique. In the present study, starting with a hadronic system as the temperature is raised the role of different mesons in maintaining an equiliblium state of nuclear matter is disturbed by a different response of various mesons. It appears that the attractive parts of the interaction are depleted while the repulsive parts are enhanced. This would suggest that the nucleon system will approach hard-sphere gas and thus would be much harder to compress it to high densities. What may be the magnitude of such a density is not established in these calculations. v) Since the experiments at RHIC and other heavy ion facilities start with heavy ions, colliding beam or fixed target, it is imperative to understand more carefully the approach to the critical temperature as the nuclei are heated and compressed at the same time. Will the transition temperature for deconfinement of hadrons and the transition temperature for hadronization of quark-gluon plasma be the same? A resolution will be possible if we had a clear and an unambigous signal for the formation or presence of a quark-gluon plasma. 4. Production of New Particles The Japan Hadron Facility with a proton beam of 50 GeV incident on a nuclear target will produce new exotic particles profusely, in particular the charm and beauty hadrons. Certainly there will be many more charm hadrons than beauty hadrons. Similar to the case of hypernuclei with one hyperon (A in particular)

260 there will be hypernuclei with A c and Af,. These particles will propagate in nuclei. Interactions of A c or At, with nucleons, their propagation and finally their decay will give valuable information on the c- and b - quarks in a sea of u- and d-quarks. No doubt the production of unusual hyperons with an s-quark will be higher and thus will open a new era in our understanding of A in nuclear m a t t e r , its decay and propagation. T h e variety of new physics may be p u t in several different categories : i) Large scale production of n, K and other mesons at such energies. T h e cross sections are not measured very well so far. ii) Production of oniums, charmoniums and beautyoniums, will be large and will provide an opportunity to study their decay in detail. T h e system has been studied extensively within Schrodinger and Bethe-Salpeter approaches 7 . T h e spectrum is understood reasonably well b u t the decays are not known or understood very well. Though it is possible t h a t separating such events among the massive debris may be a difficult task. iii) Mesons with one heavy and one light quarks, mesons like cq, bq, cq and bq where q is a light quark («, d or s) will be produced in large numbers and will then decay. T h e spectrum of such mesons has been studied within Schrodinger, Dirac and Bete-Salpeter equations 7 , 8 , 9 . T h e decay of such mesons will again give us valuable information about heavy quarks. iv) Baryons with two heavy quarks. It is possible to produce hyperons with strangeness and these particles are absorbed by nuclei before decaying. These are accompanied by production of two /^-mesons. At J H F there will be large flux of such particles in the outgoing particles of various kind. However in addition there are baryons, i.e. hyperons with two c- or 6-quarks. Such particles, as the strangeness baryons will have interactions with the nuclei, they will propagate and eventually decay. Observation and study of such particles would give first evidence for c — c interaction and interaction of such baryon with nuclear m a t t e r . Study of such individual baryons with two heavy quarks (s, c or b) and one light quark have been done with various approximations. Studies with Schrodinger and Dirac approach 8 ' 9 using a two-center Coulomb plus linear confinement potential have been carried out. These give spectrum of the hadrons. Bethe-Salpeter approach 1 0 using the two heavy quarks as a diquark and light quark motion in this potential have been carried out. In this approach, since the wave functions of the effective two particle system are obtained, it is possible to study transition rates also. No studies have been done where these heavy hadrons are inserted in nuclear m a t t e r . How does the presence of finite density of ordinary hadrons inhibit the weak decay of these hadrons? W h a t is the interaction of nucleons with these hadrons? Is

261

it attractive? The JHF machine with its capability to produce many of these exotic particles will eventually open the door to a vast array of new physics that would eventually produce a clear and better understanding of interaction among quarks and the corresponding interaction among hadrons with one or two heavy quarks in them. Such studies at JHF will truly open the door to "charm"ing and " beauty"ful nuclei and to a wide array of new physics. 5. Conclusions To summarise, we have indicated that increasing the density and increasing the temperature have a complementary though somewhat different effect on the production of mesons, nucleons and their interactions. But the changes are sufficiently profound that they cannot be ignored in any realistic analysis of the results from JHF which is expected to produce densities up to few times of nuclear matter density and create temperatures at high as tens of MeV. The production of a wide variety of hadrons with heavy quarks (c and b) and strange quarks opens a new era in the study of nuclear physics. The physics of hypernuclei will be extended to include Ac and Aj, in addition to E c , S c , fic, £{,, Eb and fi{, as possible baryons in nuclear matter. The study of their interaction with nucleons would open a new chapter in the study of nuclear matter. JHF will be a unique facility that will allow us to change temperature, density, momentum transfer and the type of constituent quarks that make up a large nucleus. That is a great deal more than it has been possible to do so far. Acknowledgements The authors wishes to thank their collaborators, the Tashkent group, M. M. Musakhanov, A. Rakhimov and U. Yakhshiev for valuable contributions towards several of the conclusions reached in this talk. The research is supported in part by Natural Science and Engineering Research Council of Canada. Finally, the author (FCK) wishes to thank Prof. Tony Thomas for hospitality at CSSM where this talk was written up. References 1. S. Sawada, in this proceedings; T. Nagae, in this proceedings. 2. A. Gal, Adavances in nuclear physics (Eds. M. Baranger and E.Vogt) Vol. 8 (Plenum Press, N. Y. 1975); B. Povh, Rep. Prog. Phys. 39, 824 (1976). 3. A. M. Rakhimov, M. M. Musakhanov, F. C. Khanna and U. T. Yakhshiev, Phys. Rev. C58, 1738 (1998); A. M. Rakhimov, F. C. Khanna, U. T. Yakhshiev and M. M. Musakhanov, Nucl. Phys. A643, 383 (1998); M. Rho, Ann. Rev. Nucl. Sci. 34,

262 531 (1984); G. E. Brown and M. Rho, Phys. Rep. 269, 333 (1996); K. Saito, K. Tsushima and A. W. Thomas, Phys. Rev. C55, 2637 (1997); M. K. Bannerjee and J. A. Tjon, Phys. Rev. C56, 497 (1997) The literature on this topic is extensive. 4. A. M. Rakhimov, U. T. Yakhshiev and F. C. Khanna, Phys.Rev. C61, 024907 (2000); C. A. Dominguez, C. van Gend and M. Loewe, Phys. Lett. B429, 64 (1998); Y.-J. Zhang, S. Gao and R.-K. Su, Phys. Rev. C56, 3336 (1997); A. M. Rakhimov and F. C. Khanna, Phys. Rev. C64, 064907 (2001). 5. Y. Takahashi and H. Umezawa, Collect. Phenom. 2 55 (1975); This article is reproduced in Int. J. Mod. Phys. A10 1755 (1996); A. Das, Finite temperture field theory (World Scientific, 1977); H. Umezawa, Advanced field theory : Micro, Macro and Thermal physics (AIP, New York, 1994). 6. R. Machleidt, Adv. Nucl.Phys. 19, 189 (1989). 7. X. Q. Zhu, F. C. Khanna, R. Gourishankar and R. Teshima Phys. Rev. D47, 1155 (1993); S. Godfrey and N. Isgur, Phys. Rev.T>23, 189 (1985). 8. D. U. Matrasulov, M. M. Musakhanov and T. Morii, Phys. Rev. C61, 045204 (2000); V. V. Kiselev, A. K. Likhoded and A. I. Onishchenko, Phys. Rev. D 6 0 , 014007 (1999); S. S. Gershtein, V. V. Kiselev, A. K. Likhoded and A. I. Onishchenko, Phys. Rev. D62, 054021 (2000). 9. D. U. Matrasulov, F. C. Khanna, K. Y. Rakhimov and K. T. Butanov to appear in Eur. J. Phys. A. 10. S.-P. Tong et al Phys. Rev. D62, 054024 (2000).

RELATIVISTIC MEAN-FIELD THEORY W I T H P I O N IN FINITE NUCLEI

H. TOKI Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka, 567-0047, Japan, and RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0105, Japan E-mail: [email protected]

K. IKEDA AND S. SUGIMOTO RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0105, Japan E-mail: [email protected]; [email protected]

First we review the present relativistic mean-field theory from the view point of the missing pion contribution. Then we discuss the role played by pions in hadrons and light nuclei. We introduce the interesting experimental d a t a on pionic states taken at RCNP. These d a t a seem to suggest the occurrence of surface pion condensation in finite nuclei. We show the results of the relativistic mean-field theory with the pion. We find that the pion mean-field becomes finite with a reasonable parameter choice. A qualitative discussion is presented on the consequence of a finite pion mean-field on Gamow-Teller transitions and spin response functions as well as other quantities.

1. Relativistic Mean-Field Theory We start with the relativistic mean-field theory. The relativistic mean-field (RMF) theory is quantitatively very successful. This is mainly because RMF naturally includes the strong three-body repulsion, which is responsible for saturation, and a strong spin-orbit force, which is responsible for magic numbers, if expressed in the non-relativistic framework. However, this theory does not include pions, which should be the most important ingredient in hadron physics. In fact, the Lagrangian of the relativistic mean-field (RMF) theory is L = i^da

-M-9o
+ i duad»a - \mW

- g^pP"

- ±g2a3 - \g3
+ i C 3 ( W / i W «) 2 - \G%G^

+ \rn\W

263

- e7/J

(1

\H,„H>"'

- \F,VF^

~

Ts)

A»ty

+ \ m l ^ ^ ,

(1)

264 where the field tensors H, G and F for the vector fields are defined through H^v = d„w„ - d„u,i

F^

= d»Av

- dvA„

,

(2)

and other symbols have their usual meanings. Here, a denotes the scalar meson, u> the vector meson and p the isovector-vector meson. A denotes the photon. This Lagrangian apparently does not include the pion. T h e pion term should be there, but this term does not contribute in the R M F approximation due to the conservation of parity and isospin of single particle states. There are 7 parameters in the R M F Lagrangian. Under the mean-field approximation we can construct coupled differential equations for nucleons, mesons and photon field, which could be easily solved numerically. By adjusting the parameters to the existing d a t a on binding energies and radii of proton magic nuclei, these parameters are fixed by Sugahara and Toki 1 . These parameter sets were named T M l , T M 2 and T M A , which have non-linear sigma and omega self-coupling terms. T h e T M l parameter set is used for the nuclei with A > 40 and T M 2 for A < 40. T h e T M A parameter set has a smooth mass dependence in order to describe nuclei in the entire mass region. We found good descriptions of binding energies and radii (the binding energies are shown in Fig. 1). Good quality of the parameter sets has been demonstrated by calculating all the even-even mass nuclei in the entire mass region 2 . In addition, we have calculated giant resonances, equation of state of nuclear m a t t e r and superheavy elements. We are very much satisfied with the performance of the R M F theory with the T M parameter sets. However, if we look at hadron physics, chiral symmetry and its spontaneous breaking are the essential ingredients of the successful description of the experimental d a t a .

2. T e n s o r F o r c e i n Light N u c l e i T h e pion exchange interaction is discussed in nuclear physics in terms of the tensor force. Let us see this connection first. T h e pion exchange force in a non-relativistic form can be written as 5=i • qa2 • q = #i • a2q2 + -S12 ,

(3)

where Su is called the tensor force. We can then see by direct calculations t h a t the tensor force is much larger than the central spin-spin force. Therefore, we can say t h a t the pion exchange interaction is the tensor force. Knowing

265 this, let us see what we know about the contribution of the tensor force in light nuclei. There is a variational calculation on the a particle 3 . According to this calculation, about half of the potential energy is caused by the tensor force. In addition, the wave function of the a particle contains about 10% of the 2p-2h components, which consist of 2 particles in the p orbit and 2 holes in the s orbit, even though the energy difference between the two orbits is at least 20 MeV. This large admixture is caused by the strong tensor force coming from the pion exchange. We further remind ourselves of the findings of fewbody systems, which are treated rigorously without the restriction of model space with a realistic nucleon-nucleon interaction, which has explicitly a shortrange repulsive core and a tensor force4,5. The recent variational calculations of the Argonne group for the up to A=8 systems, with the use of a realistic

9.5

8.5

%

"

< Ul 6.5

5.5

-

45

i i i i i 1 i i i i I i i i i I i i i i I i i i i I i i ii

0

50

100

150

200

250

300

Figure 1. Binding energy per particle as a function of the mass number for proton magic number nuclei. The experimental data are shown by the dots, while the TMA results are shown by the solid curves and the NL1 results by the dashed curves.

266

two-body interaction, provide very good description of the light nuclei5. In addition, these calculations show the dominant role of the pion, which amounts to 70-80% in two-body interaction part in the 3
3. R C N P (p,n) Spin Experiments There are two important experiments on pionic excitations performed at RCNP using the (p,n) reactions by Sakai group 7 ' 8 . They are the zero-degree spectra in the (p,n) reactions and the large momentum transfer (p,n) reactions at E ~ 300 MeV. We start with the zero-degree (p,n) reactions. We take 90 Zr as an example 7 . Taking a simple shell model, we expect two states being populated by the Gamow-Teller operator : #7/2<7^/2 particle-hole state and <79/2<7^/2 state. Due to a repulsive interaction between these two states, we expect these two states mix and the higher state carries most of the strength, which is called the giant Gamow-Teller state. The experimental data are obtained at forward angles with the incident energy of 300 MeV and are analysed in terms of multipole components. The large fraction of the GT strength is then found to be shifted further up to the higher states than the giant GT state by 30 to 40 percent. This shift of the GT strength to higher states was studied theoretically as the coupling of the GT state to 2p-2h states due to the strong tensor force. With the coupling of highly excited states, the perturbative calculations are able to provide the large strengths in the continuum. However, the change of the GT strengths seems too large considering the calculations are done perturbatively. The GT strength seems to be exhausted by nucleon degrees freedom. This fact suggests that the coupling of the GT states to delta isobars is very small. In terms of g' for delta-hole coupling in the spin-isospin channel, g'A < 0.2. This fact indicates that the delta-hole states should contribute largely to pion condensation and its precritical phenomenon. This possibility is rejected by the experimentally missing precritical phenomenon 9,10 . Relativity arguments are suggested to reduce the pionic collectivity, but it seems not enough 11 . The precritical phenomenon is expected also in the spin response functions 12 . Hence, we discuss now the second experiment of the Sakai group. It is the (p,n) reaction with the measurement of polarizations in the initial and final channels at large momentum transfer. In addition, the isovector

267

transfer is identified as unity by the use of the (p,n) reaction. This spin measurement is able to provide the response functions in the pion and the rho meson channels 8 . In the standard model of spin correlations, we use the interactions in the pion (longitudinal) and rho meson (transverse) channels as

P

F

m2

q2 + m2.

P

f

Vp = —^(g' - Cp-^r~)#i xq-a2x mJ q2 + my

qn • f2 .

(5)

These interactions provide strong attraction in the pion channel and strong repulsion in the rho meson channel at high momentum region due to small pion mass, m^, and large rho meson mass, mp. Naturally then we expect that the pion response is softened from the non-interacting case and the rho meson response is hardened. This tells us that the ratios of the spin longitudinal (TT) and the spin transverse (p) responses are larger than unity at smaller excitation energies. The experimental data show surprisingly enough that the ratios are not larger than unity 8 . This fact is difficult to understand in the present framework. The relativistic description of the response functions is not enough to turn around the ratios unless we use different g' between the pion and the rho meson channels 13 . This experimental data indicate that there is a serious problem in the present theoretical framework in the pion channel (spin-isospin channel). 4. Surface Pion Condensation We propose to take the pion terms seriously now, which should be present in the Lagrangian 14 . For simplicity, we write explicitly only the sigma and pion terms in the Lagrangian density L = ^[ii^dn -M-g„(T-g«i<>-i,idliTa'Ka}i>

+ Lmeson

•

(6)

We then assume that the expectation value of the pion field is finite. We write the equation of motion for nucleons and pions as [i^d^ -M-gaa-

g*VnaWTa]ijj

= 0,

(7)

and [V2 - m2]7ra = -gMHlTat/j).

(8)

The sigma meson and others follow the same equations of motion as in the standard case. These equations tell us the reason why we have not included

268 the pion mean-field until now. T h e source term of the pion Klein-Gordon equation is non-vanishing only when the parity and the isospin are mixed in the single particle state. This violation of the parity and isospin is caused by the pion term in the above Dirac equation for nucleons. Hence, the single particle state can be expressed as 1>njm(x)

= Y,

W

"tKjm,t •

(9)

K,t

Here 4>K,jmit denotes the eigenfunctions of nucleons without the pion mean-field term. T h e sum over K implies parity mixing and the sum over t isospin mixing. We call a non-vanishing pion mean-field surface pion condensation, since the pion source t e r m involves derivatives of the mean-field.

5. N u m e r i c a l R e s u l t s w i t h F i n i t e P i o n F i e l d In this section, we present our numerical results. We take the T M 1 parameter set of Ref. [1] for all the parameters except for the pion-nucleon coupling. For the latter we take the value for the Bonn A potential 1 6 , which corresponds to <7?r = fn/irin and fn ~ 1. We stress again t h a t we use all the terms given in the Lagrangian, including the non-linear terms. This means t h a t the saturation property is guaranteed and the bulk part of the nucleus tends to have the saturation density. Since we are especially interested in the occurrence of finite pion mean-field and want to see its effect under the simplest condition, we neglect the Coulomb term. We calculate such N = Z closed shell nuclei as 12 C , 1 6 0 , 4 0 C a , 5 6 Ni, 8 0 Zr, 1 0 0 Sn and 1 6 4 P b . Next we discuss the results with and without the pion mean-field. W i t h the present choice of the parameters, we find the actual occurrence of the finite pion mean-field for all the nuclei studied here. This is not a trivial result t h a t the pion mean-field becomes finite. As we will show later, the pion mean-field become finite only when the pion-nucleon coupling constant is larger t h a n the critical value. There we will see the dependence of the pion mean-field on the pion-nucleon coupling strength together with the critical strength for the onset of the pion mean-field. T h e rho meson mean-field vanishes for N = Z nuclei. T h e case with the finite pion mean-field provides total binding energies larger t h a n those obtained without pion mean-field. T h e pion term reduces the total energy, while all the other terms such as kinetic energy, the combined sigma and omega energy do not favour a finite pion mean-field. This means t h a t a single particle state is brought up to the next major shell by the pionic correlation and, therefore, the kinetic energy becomes larger. However, the net energy is smaller for the case with finite pion mean-field.

269

We show the mass number dependence of the pion energy per nucleon in Fig. 2. We note that the kinetic energy and the sigma and omega energies per particle are almost constant (independent of the mass number), and hence they are volume-like. We see, on the other hand, a peculiar behaviour in the pion energy. The magnitudes of the pion energy are clearly separated into two groups. One group is large and the common feature is jj-closed shell nuclei; the magic number nuclei due to a larger spin-orbit partner (j-upper) being filled. The other group is smaller and consists of LS-closed shell nuclei. The pion energy per nucleon for jj-closed shell nuclei decreases monotonically with the mass number. The rate of the decrease is stronger than A-1/3. This means that the pion mean-field energy behaves in proportion to the nuclear surface or even stronger than that. Hence, we name this phenomenon surface pion condensation. We also mention that the separation of the LS-closed and jj-closed shell

I

s

10

^

^ - ^ ^ * N i

^

J-J L-S •

O •

•

•

:

•

20A" 1/3 •

>

o

0 40

100

Ca

Sn

O

J

164

\

I . I !

1

>

•

'

-

'

'

Pb -

"f>Zr . * .

1

10

. 100

A Figure 2. The pion energy per nucleon as a function of the mass number on the log-log plot. There are two groups: first is the jj-closed shell nuclei denoted by open circle and the second is the LS-closed shell nuclei denoted by filled circles. The pion energy per nucleon for the jj-closed shell nuclei decreases monotonically and follows more steeply t h a n A~ '3, which is shown by solid line.

270

nuclei into two groups of the pion energy is related to the spin-orbit splitting. The spin-orbit partner having larger total spin near the Fermi-surface does not contribute to the total energy in the case of surface pion condensation in the LS-closed shell nuclei 14 .

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

f2/f02 Figure 3. The pion energy per nucleon as a function of the pion-nucleon coupling constant squared. In this systematic study, we have fixed the set of the filled single particle states with the same spins as in the closed shell configurations.

In Fig. 3 we present our results of the finite pion mean-field by plotting the pion energy per nucleon for various nuclei as a function of the pion nucleon coupling constant (f„) in Fig. 3. If the pion nucleon coupling constant is decreased by 5 %, the onset of the finite pion mean-field disappears except for 12 C. This fact indicates that the symmetry breaking mean-field is fragile and various effects as delta isobar contributions, the rho meson tensor coupling effects, the effects of the Fock (exchange) term and several others would influence the occurrence of the finite pion mean-field in finite nuclei. Quantitatively, this aspect will be worked out in the subsequent paper 15 . Next we discuss examples of the qualitative consequence of finite pion meanfield. First, we concentrate on the Gamow-Teller (GT) transitions. Without pion condensation, there exist no transitions for LS-closed shell nuclei such as 1 6 0 , 4 0 Ca and only two transitions, for example, for 90 Zr. However, the

271

mixing of parity in the intrinsic state allows transitions of 2p-2h states. This makes the spectrum of the GT transitions with some GT strengths to lie above the two dominant peaks 7 in 90 Zr. Hence, naturally we have strengths in the region of the higher excitation energy in the mean-field theory with the pion, as the experiment demands. Second, the longitudinal spin response functions, which are caused by the pionic correlations, should be largely modified due to pion condensation. Since the large pionic strength is used up to construct the nuclear ground state, the pionic fluctuation should be reduced largely. This should make the spin response in the pion channel weak. This remains to be demonstrated in the future work. Until now the phenomena involving high momentum components have been separated into two groups: nuclear correlations induced by the repulsive core and the tensor force. Surface pion condensation provides us with the possibility to describe the part of the correlation effect induced by the tensor force. This fact indicates that we are now able to separate the short range correlation phenomena into two classes: those associated with the pion and those associated with as the repulsive core. In fact, the surface pion condensation automatically provides a large amount of the 2p-2h excitations in the nuclear ground state. There should be many other consequences of surface pion condensation in nuclear phenomena. The pairing correlations and the spin-orbit couplings are all surface phenomena, and surface pion condensation would couple with these correlations and provide rich phenomena. 6. Conclusion We have discussed the possible occurrence of the finite pion mean field in finite nuclei by introducing the pion field in the relativistic mean-field (RMF) theory. We have extended the RMF theory by introducing the parity-mixed single particle basis to accommodate the finite pion mean-field. We have taken the TM1 parameter set in the RMF theory and introduce the pion field in the pseudovector coupling with the nucleon. Making use of the pion-nucleon coupling constant in free space, we have made calculations for N=Z closed shell nuclei and demonstrated the actual occurrence of finite pion mean-field. We have demonstrated that the potential energy associated with the pion behaves as proportional to or even stronger than the nuclear surface. Hence, we name the onset of the finite pion mean-field as surface pion condensation. We qualitatively discussed the consequence of surface pion condensation on the Gamow-Teller strengths, the spin response functions and the short range correlations. The large difference of the pion energies in jj-closed shell and LSclosed shell nuclei has been found. This may be connected to the mechanism of spin-orbit splitting due to the tensor force discussed long time ago by Takagi

272 et al. and Terasawa . We would like to stress t h a t we are at the initial stage of our investigation of the role of the pion in finite nuclei. We have to perform various studies in order to establish the mean-field theory with the inclusion of the pion, which was pursued in this paper for medium and heavy nuclei. We have to definitely introduce the rho meson tensor coupling term, which acts against the pion finite mean-field. We should include also the delta isobar pion coupling terms, which favour a finite pion mean-field. We have to study carefully the effect of the short range repulsions, the so-called g' term in the non-relativistic framework. We would also like to work out the exchange terms as well as the Fock terms in the relativistic many-body theory. We then have to work out the parameter search for the coupling constants in the Lagrangian. We should further perform the parity projection. Further studies are necessary in order to establish the occurrence of surface pion condensation. We are at the gate of exploring both theoretically and experimentally the phenomena caused by the most essential boson, the pion, in nuclear physics. At the J a p a n Hadron Facility ( J H F ) , we expect copious d a t a on physics with strangeness. We expect flavour SU(3) symmetry in various observables. We shall extend the concept of surface pion condensation to hypernuclei, in which the role of the pion is expected to be extremely i m p o r t a n t . Acknowledgments This is an invited talk presented by H. Toki in International Workshop on Physics at J H F held through March 14-21, 2002 in Adelaide. H . T . is grateful to Prof. T h o m a s for the organization of this workshop and providing exciting discussions on the physics to be studied at JHF. References 1. Y. Sugahara and H. Toki, Nucl. Phys. A579, 557 (1994). 2. D. Hirata, K. Sumiyoshi, I. Tanihata, Y. Sugahara, T. Tachibana and H. Toki, Nucl. Phys. A616, 438c (1997). 3. Y. Akaishi, Cluster Models and Other Topics (World Scientific, Singapore, 1986), p. 259. 4. J. Carlson and R. Schiavilla, Rev. Mod. Phys. 70, 743 (1998). 5. R.B. Wiringa, S.C. Pieper, J. Carlson, and V.R. Pandharipande, Phys. Rev. C62, 014001 (2000); S.C. Pieper and R.B. Wiringa, Annu. Rev. Nucl. Part. Sci. 51, 53 (2001). 6. Y. Suzuki and K. Varga, Stochastic Variational Approach to QuantumMechanical Few-Body Problems (Springer-Verlag, Heidelberg, 1998), Chap. 11. 7. T. Wakasa et al., Phys. Rev. C55, 2909 (1997). 8. T. Wakasa et al., Phys. Rev. C59, 3177 (1999).

273 9. 10. 11. 12. 13. 14. 15. 16. 17.

H. Toki and W. Weise, Phys. Rev. Lett. 42, 1034 (1979). E. Oset, H. Toki and W. Weise, Phys. Rep. 83, 281 (1982). H. Toki and I. Tanihata, Phys. Rev. C59, 1196 (1999). E. Shiino, Y. Saito, M. Ichimura and H. Toki, Phys. Rev. C34, 1004 (1986). K. Yoshida and H. Toki, Nucl. Phys. A648, 75 (1999). H. Toki, S. Sugimoto and K. Ikeda, RCNP preprint (2001), nucl-th/0110017. S. Sugimoto, H. Toki, and K. Ikeda, in preparation. R. Brockmann and R. Machleidt, Phys. Rev. C42, 1965 (1990). S. Takagi, W. Watari, and M. Yasuno, Prog. Theor. Phys. 22, 549 (1959); T. Terasawa, Prog. Theor. Phys. 23, 87 (1960); A. Arima and T. Terasawa, Prog. Theor. Phys. 23, 115 (1960).

EQUATION OF STATE OF Q U A R K - N U C L E A R M A T T E R

G. K R E I N A N D V. E . V I Z C A R R A Instituto de Fisica Teorica, Universidade Est a dual Paulista Rua Pamplona, 145, 01405-900 Sao Paulo, SP, Brazil E-mail: gkreinQijt.unesp.br; [email protected]

Quark-nuclear matter is a many-body system containing hadrons and deconfined quarks. Starting from a microscopic quark-meson coupling Hamiltonian with a density dependent quark-quark interaction, an effective quark-hadron Hamiltonian is constructed via a mapping procedure. The mapping is implemented with a unitary operator such that composites are redescribed by elementary particle field operators that satisfy canonical commutation relations in an extended Fock space. Application of the unitary operator to the microscopic Hamiltonian leads to effective, hermitian operators that have a clear physical interpretation. At sufficiently high densities, the effective Hamiltonian contains interactions that lead to quark deconfinement. The equation of state of quark-nuclear matter is obtained using standard many-body techniques with the effective quark-hadron Hamiltonian. At low densities, the model is equivalent to quark-meson coupling model with confined quarks. Beyond a critical density, when quarks start to deconfine, the equation of state predicted for quark-nuclear matter is softer than the quarkmeson coupling equation of state with confined quarks.

1. Introduction One of the most exciting open questions in the study of high density hadronic matter is the identification of the appropriate degrees of freedom to describe the different matter phases. For systems with matter densities several orders of magnitude larger than the nuclear saturation density, one expects a phase of deconfined matter composed of quarks and gluons whose properties very likely can be described by perturbative QCD. For ground state nuclei, there is a large body of experimental evidence that their gross properties can be described more economically employing hadronic degrees of freedom, rather than quarks and gluons. On the other hand, for matter at densities not asymptotically higher than the saturation density, like the ones in dense stars and produced in high-energy nuclear collisions, the situation seems to be very complicated, since hadrons and deconfined quarks and gluons can be simultaneously present in the system. Presently, it is not possible to employ QCD directly to study such systems and the use of effective, tractable models are essential for making

274

275

progress in the field. One attractive model to study the different phases of hadronic matter in terms of explicit quark-gluon degrees of freedom is the quark-meson coupling (QMC) model, originally proposed by Guichon and subsequently improved by Saito and Thomas 1 . For a list of references on further improvements of the model and recent work, see Ref. [2]. In the QMC model, matter at low density is described as a system of nonoverlapping MIT bags interacting through effective scalar and vector meson degrees of freedom. The effective mesonic degrees of freedom couple directly to the quarks in the interior of the baryons. At very high density and/or temperature, when one expects that baryons and mesons dissolve, the entire system of quarks and gluons becomes confined within a single, big MIT bag. In a regime of very high density, the description of hadronic matter in terms of nonoverlapping bags should of course break down, since once the relative distance between two bags becomes much smaller than the diameter of a bag, the individual bags loose their identity. The density at which this starts to happen is presently unknown within QCD. In the present communication we introduce a generalization of the QMC model that allows to include quark deconfinement at high density. Our starting point is a relativistic quark potential model 3 . From the model quark Hamiltonian, we construct a unitarily equivalent Hamiltonian that contains quark and hadron degrees of freedom. Starting from the Fock space representation of single-hadron states, a unitary transformation is constructed such that the composite-hadron field operators are redescribed in terms of elementary particle field operators in an extended Fock space. When the unitary transformation is applied to the quark Hamiltonian, effective, hermitian Hamiltonians with a clear physical interpretation are obtained 4 . In particular, one of such effective Hamiltonians describes the deconfinement of quarks from the interior of hadrons. The equation of state of quark-nuclear matter (QNM) can be calculated using standard many-body techniques with the quark-hadron Hamiltonian. We will show that at low densities, the model is equivalent to the QMC model and, beyond a critical density, when quarks start to deconfine, the equation of state predicted for QNM is softer than the QMC equation of state.

2. QMC Model with Confined Quarks The nucleons are bound states of three constituent quarks. Constant scalar (o) meson fields couple to the constituent quarks in the interior of the nucleons. Each constituent quark satisfies a Dirac equation of the

276

form -iS • V + 0°m* + 1/2(1 + p)V(rj\

ip(r) = Elr/>(r),

(1)

where . -,nq-gla0,

E*q = s* - g * w 0 ,

m„ q

V(r)=
(2)

The only difference from the model of Toki et al.3 is the form of the potential, while theirs is a harmonic oscillator, ours is a linearly rising one. For a linearly rising potential, the Dirac equation cannot be solved analytically. We use the saddle point variational principle (SPVP) 5 to obtain an approximated solution. Since the Dirac Hamiltonian does not have a lower bound for the energy, the traditional nonrelativistic variational method cannot be employed. The SPVP amounts to minimizing (maximizing) the energy expectation value with respect to the variational parameters corresponding to the upper (lower) component of the Dirac wave function. We use as an ansatz for the Dirac wave function5 *(r)=(.-U(.r)M)*, \i
(3)

J

with u(r) = Ne-x2r2'2,

v{r) = if/\ff-Vu(r),

(4)

where N is a normalization constant, and A and 7 are the variational parameters. The parameters are found by minimizing the energy eigenvalue e with respect to A and maximizing it with respect to 7. We note here that for a harmonic oscillator potential, the SPVP with this ansatz leads to the exact solution. Following the traditional path in the QMC model, we initially fix the parameters of the model in vacuum. The nucleon mass in vacuum (
(5)

where the last term above is used to take into account the era. energy of the three-quark state and other short-distance effects are not taken into account by the confining potential, such as gluon exchanges. £0 is fitted to obtain MJV = 939 MeV. The value of the string tension is taken to be a = 0.203 GeV 2 and mq = 313 MeV. With these parameters, the SPVP leads to A = 2.38 frn - 1 and 7 = 0.346. The value required for £0 to fit the nucleon mass is 4.67 MeV fm. Next, we proceed to obtain the energy of nuclear matter. Nuclear matter in the QMC model is modelled as a system of nucleons treated in the mean field approximation, in which the quarks remain confined within the nucleons. The nucleon mass is now obtained as above, but now including the mean fields

277

coupled to the quarks. The energy density of symmetrical nuclear matter is given by the traditional expression in the QMC model E

, fkF =

V

d3

P r,. / N n a + 7MEEN(P) ^C^oPB

J 2

1

2 2

1

+ ^maa0--rnuw0,

2

2

(6)

2

where E'fc = \ / p + Mjy ; />s is the baryon density; ma and raw are the masses of the mesonic excitations. The next step consists in determining the mean fields. The vector mean field, from its equation of motion, is simply given in terms of pg, and the scalar field is obtained by minimizing E with respect to = Bt |0>,

Bi = ^ v l W ^ t ^ t ^

(7)

where the q^'s are constituent-quark creation operators and ip£ 1 ' i2 ' i3 is the Fock space nucleon amplitude. For independent quarks, this is simply a product of three single-quark wave functions. The convention of summing over repeated indices is used throughly. The quark creation and annihilation operators satisfy usual canonical anticommutation relations {*ypi»i»

= SaP .

(9)

278

In the abbreviated notation we are using, the Hamiltonian corresponding to Eq. (1) can be written as Hq = Tq + Vqq = T (fi) q\qii + -Vqq <jxv; ap) q^Upq^

(10)

where Vqq is a confining potential. Using the quark anticommutation relations of Eq. (8) and the normalization condition of Eq. (9), one can shown that the nucleon operators, Ba and Ba, satisfy the following anticommutation relations {Ba,Bl} = Safi-Aa0,

{Ba,B0}=O,

(11)

where 9

Hap — o *

v^

a

9j/ 3 9^ 3

2

a

p

iV3qV2q^iii3-

\1*)

In addition, one has

{«/..*£} = y/ln^'qlql,

ii^Ba] = o.

(13)

The term Aap is responsible for the noncanonical nature of the baryon anticommutator. This term and the nonzero value of the first commutator in Eq. (13) are manifestations of the composite nature of the baryons and the kinematical dependence of the quark and nucleon operators. This fact complicates enormously the mathematical treatment of many-body systems in which deconfined qur.rks and nucleons are simultaneously present. The mapping formalism of Ref. [4], known as the Fock-Tani (FT) representation 6 , is a way to circumvent such complications. We will shortly review this formalism in the context of the present model. For further details, and applications for more general models, the reader is referred to Ref. [4]. The basic idea is to extend the original Fock space by introducing fictitious, or ideal nucleons that satisfy canonical anticommutation relations. The unitary operator is constructed in the extended Fock space such that | a ) = 5 t | 0 ) - ^ t / - 1 | a ) = |a) = 6t,|0),

(14)

where the ideal baryon operators 6^ and ba satisfy, by definition, the canonical anticommutation relations {6a, *£} = *«/»,

{ba, W = 0.

(15)

The state |0) is the vacuum of both q and 6 degrees of freedom in the new representation. In addition, in the new representation, the quark operators gt and q are kinematically independent of the ba and ba

to..*-} = {«,..*!,} = o.

(16)

279 The unitary operator U can be constructed as a power series in the bound state amplitude \?. The rational for this is clear : in situations the quarks remain confined in the interior of the nucleons, the term A a/ 3 plays no role and it can be taken to be zero and the unitary operator becomes trivial 6 ' 4 . This is the situation for low densities, when the internal structures of the nucleons do not overlap significantly in the system. As the density of the system increases, the quark structures of different nucleons start to overlap. An expansion in powers of VP's offers a power counting procedure to construct the unitary operator. The effective Hamiltonian is constructed by applying the unitary operator to the microscopic quark Hamiltonian of Eq. (10), -ffe// = U~1HqU. The zeroth-order U is trivial and not interesting. The first-order U brings interesting effects. At this order, we denote the effective Hamiltonian by H^J,, where the superscript (1) means that U has been evaluated up to the first order in $ . H^J. can be written as H{e)) = Tq + Hb + Vqq + Vqb + ---.

(17)

The • • • refer to terms not relevant for our discussion here; Hb = Tb + Vbb, where Tb is a single-nucleon energy and Vbb is an effective nucleon-nucleon interaction without a quark exchange. This term leads to the normal QMC model, in which the many-body system is described by nonoverlapping nucleons - no quark-exchange. In particular, it can describe Fock terms in the QMC model 7 . The term Vqq contains two-quark and three-quark interactions. It can be shown that if ip is a bound-state eigenstate of the original quark Hamiltonian, Vqq collapses to V

qq = 2 V11^V'^ °P)
(18)

It is not difficult to show that this interaction leads to a quark Hamiltonian that has a positive semidefinite spectrum. That is, after the transformation, the resulting Hamiltonian involving only quark operators is unable to bind three quarks to form a nucleon, it describes only states in the continuum. The term Vqb is given by

Vbq = ^=[H(nm;ap)^y^ - H(wap)^pT3A(w2Mpvri)]ql1qlaqlabf)+h.c.,

(19)

where h.c. denotes hermitian conjugation and A((XVT; crp\) = J^a ^aVT^*apX is known as the bound-state kernel. If $ is a stationary state of the microscopic quark Hamiltonian, one obtains Vbq = 0,

(20)

280

since A(pvT;
Y,Q>lK) = ZB,

$>],?„> = (1 - Z)B,

(21)

\i

a

where B is, as in the previous section, the total baryon number. In the mean field approximation - or independent-particle approximation - and for sufficiently small Vbq, Z can be estimated by the perturbative formula 1+6

V

^H0-E,Vbq

b ,

(22)

where Ho is Tq + Tb, and P = 1 — |6)(6| is a projection operator. In order to evaluate Eq. (22), we postulate a density dependence of the confining interaction in the form8 V{r)=are-"2r\

(23)

where p is a prescribed function of ps- We use a simple formula for p, such that it is zero for baryon densities below three times the normal nuclear matter density po, and for higher densities it increases linearly with the density as

281

p = PB/%PO — 1- In Fig- (1) we show the potential of Eq. (23) for zero density, and 5 and 10 times the saturation density of normal nuclear matter.

1.0

1

'

1

yi

- 1 —

1

-

-

0.75 --

PBIPQ

=

-

=5

.-•—

>

~ ^ \

0.5 --

\

N

// 0.25 — //'

-•

x

PB/PO = 1

0.0 0.0

0.5

—

10 -,.!., 1.0

i

1

1.5

•

2.0

r(fm) Figure 1. The confining potential in vacuum (solid line) and in matter for two different baryon densities.

The energy density of the system can be written as

V

— ^ g E*N{p) + 3g%u0 PB + ^mlal

J0

+12

fk?

- -m2uuj2

d3k

(24)

l <55^<"-

where E*(k) = \/k2 + m*2, and the Fermi momenta kF and kqF are related to the nucleon density and quark density as Pb

ZPB = 2 ( ^ ) 2 / 3 T T 2 ,

Pq

= (1 - Z)pB = 2(kqF)2/7T2 .

(25)

At this point, it is important to notice that since p only starts to operate for densities larger than three times the normal density, the coupling constants g% and g% are the same as before. Of course, for higher densities, there is a somewhat complicated self-consistency problem to be solved, since Z is density dependent. Therefore, in the process of obtaining a, the iterative problem becomes more complicated.

282 1.2

1

1

1

1

'

1

'

\/

/ // / /

/ s / s s^' // / s / s / / s/ / s /y

1.1 -

'

s ss

ss

—

//

/s//

^

1.0 _

0.9

/

1

1

""

,

4

1

.

1

,

10

6 PB/PO

Figure 2. Equation of state of quark nuclear matter. The solid line is for matter composed of nucleons only and the dashed line is for matter composed by nucleons and quarks.

In order to proceed, we need Z as a function of pg. It can be calculated numerically with our ansatz wave function given above. The calculation, however, involves multidimensional integrals that must be done using a Monte Carlo integrator. For our purposes here, in this initial investigation we make some approximations. Initially we neglect the lower component of the Dirac spinor. This does not seem to be too drastic an approximation, since 7 in Eq. (4) is a small quantity. In this approximation, one obtains \F(h 1 + / d 3 M 3 M 3 fc 3 |$ P (A;i,&2,&3) AE(p,k k ,k ) / u 2 3 with F(q) given by F{q) = fd3kAV(k)e-k2'x2[e^x2

- l) ,

(26)

(27)

where $ p (&i, Ar2, ^3) is the three-quark wave function of the nucleon with cm. momentum p (see Ref. [4]), AE(p, k\, fo, ^3) is the difference between of the energies of the three unbound quarks and of the three quarks bound in the potential, and AV(k) is the Fourier transform of AV(r), where AV(r) =
-l).

(28)

This clearly shows that once p, = 0, i.e. the potential is density independent, one regains the original QMC model.

283 Now, Eqs. (26) and (27) still require a lot of numerical work. We simplify t h e m further by making two additional approximations. T h e first one consists in neglecting the m o m e n t u m dependence of the energy denominator, and the second one is to use an average value for q2 in F(q2). Both approximations taken together seem not to be a bad approximation, since the energy denominator under the integral is dominated by low m o m e n t a . Now the problem consists in a single one dimensional integral t h a t can easily be performed with a Gaussian integration m e t h o d . In Fig. 2 we present the results for the energy per baryon number, E/B, as a function of the ratio PB/POT h e solid line in this figure is the result for the Q M C model of the previous section. T h e dashed line shows t h a t the deconfming of quarks leads to softening of the equation of state. It would be interesting to investigate the consequences of this softening for neutron star phenomenology. Soft equations of state seem to be required to explain recent observational d a t a on compact stellar objects. 4.

Conclusions

We have generalized the Q M C model to include quark deconfinement in m a t ter. T h e model is based on an effective quark-hadron Hamiltonian obtained via a m a p p i n g procedure from a relativistic microscopic quark Hamiltonian with a density dependent quark-quark interaction. T h e equation of state of QNM was obtained using the effective quark-hadron Hamiltonian. It was found t h a t beyond a critical density, when quarks start to deconfine, the equation of state predicted for QNM is softer t h a n the usual Q M C equation of state. Implications of this equation of state for the phenomenology of compact stellar objects were pointed out. Acknowledgements This work partially supported by the Brazilian agencies C N P q and FAPESP. References 1. P. A. M. Guichon, Phys. Lett. B200, 235 (1988); K. Saito and A.W. Thomas, Phys. Lett. B327, 9 (1994). 2. P.K. Panda, M.E. Bracco, M. Chiapparini, E. Conte and G.Krein, Phys. Rev. C (in press), nucl-th/0205051. 3. H. Toki, U. Meyer, A. Faessler, and R. Brokmann, Phys. Rev. C58, 3749 (1998). 4. D. Hadjimichef, G. Krein, S. Szpigel and J.S. da Veiga, Phys. Lett. B367, 317 (1996); Ann. Phys. (NY), 268, 105 (1998). 5. J. Franklin and R.L. Intemann, Phys. Rev. Lett. 54, 2068 (1985); J.D. Talman, Phys. Rev. Lett. 57, 1091 (1986).

284 6. M.D. Girardeau, Phys. Rev. Lett. 27, 1416 (1971); J. Math. Phys. 16, 1901 (1975). 7. G. Krein, A.W. Thomas and K. Tsushima, Nucl. Phys. A650, 313 (1999). 8. W. Alberico, P. Czerski and M. Nardi, Eur. Phys. J. A4, 195 (1999).

EQUATIONS OF STATE FOR N U C L E A R MATTER A N D Q U A R K MATTER IN THE NJL MODEL

W. BENTZ AND T. HORIKAWA Department of Physics, Tokai University Hiratsuka-shi, Kanagawa 259-1207, Japan E-mail: [email protected]; [email protected] N. ISHII The Institute of Physical and Chemical Research (RIKEN) 2-1, Hirosawa, Wako-shi, Saitama 351-0198, Japan E-mail: [email protected] A.W. THOMAS Special Research Centre for the Subatomic Structure of Matter, and Department of Physics and Mathematical Physics, Adelaide University Adelaide, SA 5005, Australia E-mail: [email protected]

Using the Nambu-Jona-Lasinio model as an effective quark theory, we construct the nuclear matter and quark matter equations of state at zero temperature. The nuclear matter equation of state is based on the quark- diquark description of the single nucleon, and the quark matter equation of state includes the effects of scalar diquark condensation. The phase transition from nuclear matter to quark matter is discussed in this framework.

1. Introduction In recent works1 the successful quark-diquark description of the single nucleon, which is based on the relativistic Faddeev method 2,3 , has been extended to describe the equations of state (EOS) of nuclear matter (NM) in the NambuJona-Lasinio (NJL) model 4 . The essential ingredient to obtain a saturated EOS was the introduction of an infrared cut-off (in addition to the usual ultraviolet one) 5 in order to avoid unphysical thresholds for the decay of hadrons into free quarks. The resulting NM EOS has many interesting features, for example a natural suppression of the famous "Z-graph contributions" and a very mild density dependence of the sigma meson mass which is essential for saturation.

285

286

In this work we extend this successful description of matter at normal densities to the high density region. Since one expects a phase transition from NM to quark matter (QM) at densities of several times the normal nuclear matter density, we first construct the EOS of QM in the NJL model. Since our description of the single nucleon is based on the interactions in the scalar diquark channel, it is natural to consider the possibility of diquark condensation in QM 6 . We will discuss the static phase transition from NM to QM, in particular the role played by the pairing interaction. Throughout this work we will limit ourselves to flavour SU(2) and isospin symmetric matter. 2. Model and Nuclear Matter EOS The Lagrangian of the flavour SU(2) NJL model has the form C — ip (i$ — m) ip + Cj, where m is the current quark mass and JCI a chirally symmetric 4-Fermi interaction. Any 4-Fermi interaction can be decomposed into various qq (meson) and qq (diquark) channels 2 , and for our purposes we need the interactions in the 0 + , 0~ and l~ meson (m) channels, as well as the 0 + diquark (d) channel : Ci,m = GV ((^V>)2 - (^(75r)V) 2 ) - Gu ( V W ) 2

(1)

Ci,d = Gs (?( 75 C) r 2 /?V) (V>T ( C - S ) r2PA^) -

(2)

Here flA — \J"&/1 XA (A — 2,5,7) are the colour 3 matrices, and C = 27270We consider the ratio r s = Gs/Gw as a free parameter, reflecting the form of the original interaction Lagrangian. Separating the scalar and vector mean field parts from the Lagrangian in Eq. (1) and adding them to the free part, we obtain the mean field Lagrangian : CMF =^{i$-M-

2G W 7"",.) V> -

( M

~

m )

+ Guu^,

(3)

where M = m — 2Gn(p\ipip\p) is the constituent quark mass and u11 — (/9|V'7MV,|p)- (The "physical value" of M, i.e. the solution of the gap equation, will be denoted by M* for finite density, and by Mo for zero density.) Here \p) refers to the ground state under consideration characterized by the baryon density p, i.e. the vacuum (p = 0), NM or QM. To construct the nucleon as quark-scalar diquark state, we use Eq. (3) as the one-body part together with the residual interaction in Eq. (2) to calculate the quark-diquark t-matrix (T). If the total system is at rest it has a pole at € N{P) = 6Ga,w° ± \fp1 + Mjy, where the nucleon mass Mjv is a function of the constituent quark mass MN(M). For the present finite density calculation, we will restrict ourselves to the "static approximation" 7 to the relativistic Faddeev

287

equation for T, where the momentum dependence of the quark exchange kernel is neglected. Based on the quark-diquark description of the single nucleon, the EOS of NM in the mean field approximation has been derived in Ref. [1] by using a kind of "hybrid assumption" for the NM ground state, similar in spirit to the model of Guichon and collaborators 8 . The resulting effective potential V{n), where fj. is the chemical potential for the baryon number, differs from the familiar expression in chiral point-nucleon models 9 essentially only by the fact that MN{M) is a non-linear function of the scalar potential $ = Mo — M. (The expression for the energy density can be found in Ref. [1].) Like most other models based on a linear realization of chiral symmetry, the vacuum part of V(fi) is characterized by the "Mexican hat" shape as a function of M. Its curvature decreases rapidly as one moves away from the vacuum state (M = Mo) towards smaller M, which implies a decreasing effective a meson mass and therefore an increasing attraction between nucleons with increasing $ (or p). This leads to an instability of the nuclear matter ground state in point-like nucleon models like the linear cr-model9. In our composite nucleon model, however, the function M^{M) has a finite curvature ("scalar polarizability" 10 ), which can lead to a repulsive contribution to the sigma meson mass and stabilization of the system. However, a sufficiently strong repulsive contribution arises only if confinement effects are incorporated phenomenologically so as to avoid the unphysical thresholds. The simplest method is to introduce an infrared cut-off (A//?) in the framework of the proper time regularization scheme 5 . By choosing A/.R > 0.1 GeV one obtains a saturated NM EOS characterized by an effective cr-meson mass which is almost independent of the density 1 . The stabilization of the system due to the scalar polarizability is demonstrated in Fig. 1, which shows the nucleon mass as a function of $ = Mo — M and the resulting binding energy per nucleon as a function of the density a .

3. Quark Matter EOS and Phase Transition To construct the EOS of QM, we allow for the possibility that a nonzero scalar diquark condensate develops in one particular direction in colour space, say A — 3, due to the interaction in Eq. (2). The mean field Lagrangian is then

a

T h e parameters m, Auv, G„ and r3 are determined by reproducing m* — 0.14 GeV, f„ = 0.093 GeV, Mo = 0.4 GeV, and the nucleon mass in the vacuum Mjvo = 0.94 GeV. ru. = Gu/G„ is fixed so that the binding energy curve passes through the empirical saturation point. In the proper time scheme with \ I R = 0.2 GeV, this leads to m = 0.0169 GeV, Auv = 0.6385 GeV, G„ = 19.60 G e V - 2 , rs = 0.508 and rw = 0.369.

288

Figure 1. Nucleon mass as a function of scalar potential (left) and binding energy per nucleon as function of density (right) for the cases hm = 0.2 GeV and AJR = 0.

given by Eq. (3) supplemented by SCMF = - \

tyiskfaCvif

- ^C^^i^A^)

,

(4)

where the real field A is defined by A = -G.iptfinfoCvif

- i>TC-lT2i^M\p).

(5)

We then introduce a chemical potential for the quark number (/iq — p/3) into CMF and apply the Hartree approximation. For QM at rest the energy spectrum of the quarks is given by eq(p) — \iq — ±J{EV ± (i*)2 + A 2 , where Ep = s/p2 + M2, A = sfzj2&, and //* = fiq - 2Gwu°. The effective potential naturally separates into the vacuum quark loop term, mean field terms due to M, w° and A, the Fermi motion part of the quarks, and a contribution which arises from the change of the quark energy spectrum and occupation numbers due to the finite gap. In the numerical calculations we use the proper time regularization scheme with the same values of m, G^, Auv and AIR = 0.2 GeV as in the NM case b . However, we will allow for variations of rs and ru in order to show the dependencies. Before discussing the phase transition, we compare in Fig. 2 the effective quark mass M* in QM for the case without pairing (r, = 0 in QM) to the one in NM. From this figure one can expect that, if there is a transition from NM to QM at several times the normal NM density, QM will already be in the b

W e note that the choice AIR = 0, which might be favourable since in QM the quark decay thresholds would be welcome in contrast to the NM case, leads to results for the EOS which are qualitatively similar to AIR = 0.2 GeV, except for the behaviour of A at small fj, see Ref. [11].

289 0.4

0.3

0.1

" 0.0

0.2

0.4

_ 0.6

p[fm- 3 ]

0.8

1.0

Figure 2. Effective quark mass in nuclear matter (NM) and quark matter (QM) as functions of the density.

"chirally restored phase" (M* ~ 0). 4. Phase Transition The solid line in Fig. 3 shows the P-fi plot for NM, where the density increases along the line starting from the point n = MN0 = 940 MeV. (The vacuum solution P = 0 exists in the range 0 < fi < MN0, but is not drawn in the figure.) In the low density region there is the familiar gas-liquid phase transition, see the insert in Fig. 3. The EOS for QM without the effects of diquark condensation is shown by the dashed lines for the cases rw = Gu/Gn = 0.369 (same as in NM) and rw = 0. They start at // = 3M 0 = 1.2 GeV. In the former case the "chiral phase transition" is of second order and therefore not directly visible on this plot, while for the latter case it is of first order and similar to the gas-liquid phase transition in nuclear matter 12 . The lower dotted line shows the case when M* is set equal to zero by hand for rw = 0. It corresponds to the massless quark gas EOS with the choice BllA = BlJjL = 0.181 GeV for the bag constant. For comparison we also show the EOS for a massless quark gas with B1/4 = B^4IT = 0.145 GeV by the upper dotted line. Figure 3 shows that there is no phase transition from NM to QM if one uses the same strength of the vector interaction in QM as determined from the saturation properties of NM. It turns out that this situation does not change

290

'

4 3 2 1 0 200 — - 1 . -2 9 0 920 930 940 950

. . | . . , . | y . . ,

-

B='Bm 0

•*

/

y/

QM

y

100 quai

/

A-

NU

>

*

//y

•k gas

y

y ,-•.

1000

y

/•

/

*

y s

..

/

/

t r„=o

1 . . . . 1200

r„=0.369

-•

,'

,-'

-

y :

y'

1 . . . .

1600

/4MeV]

Figure 3. P — \i plots for nuclear matter (NM) and quark matter (QM). The insert shows the behaviour of the solid line in the low density region. For the explanation of the lines see text.

even for a finite gap in QM, i.e. NM is the ground state for all densities. Since this situation seems unrealistic, one might conclude that the use of a vector interaction in the high density QM phase is physically unreasonable. From Fig. 3 we also see that, for the case rw = 0 in QM, there is still no phase transition, but this situation changes if the effect of the gap is included, as will be discussed below. In the following we will set rw = 0 in QM and investigate the effect of the diquark condensation. Figure 4 shows the P—fi plots for several values of the pairing strength r, in QM in comparison to the NM case. We see that diquark condensation favours the NM —> QM phase transition, but r„ in QM should not be larger than ~ 0.3: For rs = 0.3 the NM phase would be stable only up to p = 0.45 fm~ 3 , and for rs = 0.4 the QM phase would be the ground state for all densities. Also, the gap becomes unreasonably large for the case r s > 0.3, see Fig. 5. This figure shows that the trend towards a chiral phase transition in QM persists also for finite gap, although the corresponding density region is shifted upwards, and that the gaps are typically of the order of 100-300 MeV, which is large

291 300

200 I

6

*t-i

> S

10

°

PL.

0

800

1000

1200

1400

1600

fj. [MeV]

Figure 4. P — /x plots for nuclear matter (NM) and quark matter (QM) for various pairing strengths rs.

compared to the results obtained in the 3-momentum sharp cut-off scheme0. From this we conclude that the strength rs — 0.508 determined by reproducing the experimental nucleon mass in the pure quark-scalar diquark model is too strong to be used as the pairing strength in QM. We have to note, however, that rs = 0.508 is probably an overestimate, since pion cloud 13 and/or axial vector diquark 14 contributions will reduce the value of rs needed to reproduce the nucleon mass. In order to discuss the NM -» QM phase transition including diquark condensation, we use rs = 0.25 in QM as an example. The path of the system in the P-\i plane (as shown in Fig. 4) is the following: (i) Vacuum (P = 0) up to the saturation point of NM, (ii) NM up to the transition point, and (iii) QM. The corresponding p — p. and M* — p (or A* — p) plots are shown in Fig. 6, where for any fixed p only the quantities in the stable phase (VAC, NM or QM) are indicated. The density jump during the NM - • QM phase transition is quite large. The P-p and S-p plots are shown in Fig. 7, where the dashed lines indicate the mixed phases. c

T h e momenta which give the most important contributions to the gap equation are p ~ fi*, and in the density region of interest this is very similar to the UV cut-off. A sharp 3momentum cut-off therefore eliminates a part of the peak around p ~ fi*, while the proper time scheme corresponds to a smooth cut-off function.

292 ' ' ' ' 1' ' ' ' 1' ' ' ' .

....

i

•

•

•

•

i

•

•

•

•

r

•

.,.,....

OH ( r , - O . Z )

ini

OH ( r . - 0 . 3 ) .

0.3

fi i

^\v N\\J \••;

0.2

a

---

QM (r,-O.Z) -

QM (r.-Q.4)

0.3

-

-

-

"

"

...'

"

0.2

"<

V\ ....!..,. 0.2

•

>

QM (r,-0.4)

v-\^^

0.0

-

on (r>»o.a) '

\Si

0.1

.

_..-••'"

0.4

V-,

-

. r: : n S f f i " 5 -^»»?»^ [ t a -r 0.8

-""" :

0.1

. /. . 1 . . . . 0.0

1.0

0.2

1. . . .

1 . 0.0

0.4

P I'm 3 J

1.0

Figure 5. Effective quark mass (left) and gap (right) in quark matter as functions of the density for various rs. 1' ' ' ' 1 '

...,....,....,

,,,,.,,,, /on

r.(0JI)-0.2S

-

r,(qu)-0.20 M*(VAC)

G

'

400

!

0.7i

-

"

a

300

"VftM)

V 200

-

M"(M(P" "

NU 100

VAC

,

1 • . . . 1 . . . . 1 . . . . 1

1300

Ktw 1400

BOO

000

1000

1100 1100

1200

1300

1400

/* [MeV]

Figure 6. Density (left) and M*, A* (right) of the stable phases (vacuum, nuclear m a t t e r or quark matter) as functions of the chemical potential; r s = 0 . 2 5 is used in QM. 1 " "

' 1 ' ' ' ' 1 ' ' ' 1' ' -

/-

r.(Qll)-0.2S

-

/<W

NM->QM

-

1500

1000

1 ' '

,.| . -

r.(QK)-0.26

-

-

,.'''H«->QII

y

-

600

•

AM

• VAC->IOI 0.00

0.26

y

. 1 . . 0-60

. . 1 . . . . 1 . 0.75 _ , 1.00 3

P ['Hi" ]

,..!.. 1.25

,.-'VAC->mi 1.60

0.00

0.21

0.60

r0«__n

1.00

1.S0

1.S0

p [fm 3 ]

Figure 7. Pressure (left) and energy density (right) of the various phases as functions of the density; r s = 0 . 2 5 is used in QM.

5. Summary In this work we used the NJL model as an effective quark theory to describe the single nucleon, NM and QM including the effects of diquark condensation.

293 We focused on the possibility of a phase transition NM -> QM. Our results can be summarized as follows: (i) T h e strength of the vector interaction in QM has to be reduced drastically as compared to the NM case, where it can be related to the saturation properties, in order to make the phase transition possible. (ii) Diquark condensation favours the phase transition, but reasonable values for the strength of the pairing interaction are only about half of those derived from the pure quark-scalar diquark model of the single nucleon. (iii) For any reasonable scenario, the quark phase is already in the chiral symmetric, colour broken phase (M* ~ 0, A* > 0) at the transition density.

Acknowledgements This work was supported by the Australian Research Council, Adelaide University, and the Grant in Aid for Scientific Research of the Japanese Ministry of Education, Culture, Sports, Science and Technology, Project No. C2-13640298. References 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14.

W. Bentz and A.W. Thomas, Nucl. Phys. A696, 138 (2001). N. Ishii, W. Bentz and K. Yazaki, Nucl. Phys. A578, 617 (1995). U. Zuckert, R. Alkofer, H. Weigel and H. Reinhardt, Phys. Rev. 55, 2030 (1997). Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1960); 124, 246 (1961). G. Hellstern, R. Alkofer and H. Reinhardt, Nucl. Phys. A625, 697 (1997); D. Ebert, T. Feldmann and H. Reinhardt, Phys. Lett. B388, 154 (1996). M. Afford, K. Rajagopal, and F. Wilczek, Phys. Lett. B422, 247 (1998); O. Kiriyama, S. Yasui and H. Toki, Int. J. Mod. Phys. E10, 501 (2001). A. Buck, R. Alkofer and H. Reinhardt, Phys. Lett. B286, 29 (1992). RA.M. Guichon, Phys. Lett. B200, 235 (1988); P.A.M. Guichon, K. Saito, E. Rodionov and A.W. Thomas, Nucl. Phys. A 6 0 1 , 349 (1996). W. Bentz, L.G. Liu and A. Arima, Ann. Phys. 188, 61 (1988). M.C. Birse, Phys. Rev. C51, R1083 1995); S.J. Wallace, F. Gross and J.A. Tjon, Phys. Rev. Lett. 74, 228 (1995). W. Bentz, T. Horikawa, N. Ishii and A.W. Thomas, to be published. M. Buballa, Nucl. Phys. A611, 393 (1996). N. Ishii, Phys. Lett. B 431, 1 (1998). H. Mineo, W. Bentz, N. Ishii and K. Yazaki, Nucl. Phys. A703, 785 (2002).

COLOUR S U P E R C O N D U C T I V I T Y IN D E N S E QCD A N D S T R U C T U R E OF C O O P E R PAIRS

H. ABUKI AND T. HATSUDA Department of Physics, University of Tokyo, Tokyo 113-0033, E-mail: [email protected]. ac.jp; hatsuda&phys.s.u-tokyo.

Japan ac.jp

K. I T A K U R A RIKEN

BNL Research Center, BNL, Upton, NY 11973, USA E-mail: [email protected]

Two-flavour colour superconductivity is examined over a wide range of baryon densities within a single model. To study the structural change of Cooper pairs, quark correlations in colour superconductor is calculated both in momentum space and in coordinate space. At extremely high baryon density (~ O(10 1 0 po))i our model becomes equivalent to the usual perturbative QCD treatment and the gap is shown to have a sharp peak near the Fermi surface due to the weak-coupling nature of QCD. On the other hand, the gap is a smooth function of the momentum at lower densities (~ O(10po)) due to strong colour magnetic and electric interactions. The size of the Cooper pair is shown to become comparable to the averaged interquark distance at low densities, which indicates that a crossover from a BCS state to Bose-Einstein condensate of tightly bound Cooper pairs may take place at low density.

1. Introduction Because of asymptotic freedom and Debye screening in QCD, deconfined quark matter is expected to be realized for baryon densities much larger than the normal nuclear matter density 1 . This weak-coupling fermionic matter is, however, qualitatively different from a perturbed free Fermi gas system, if the temperature is low enough. This is because any attractive quark-quark interaction in the cold quark matter causes an instability of the Fermi surface due to the formation of Cooper pairs and leads to the colour superconducting phase 2 ' 3,4 . Current understanding of colour superconductivity has been based on two different theoretical approaches. One is an analysis of Schwinger-Dyson equations with a perturbative one-gluon exchange, which is valid in the weak coupling limit at asymptotically high densities 5 . The dominant contribution to the formation of Cooper pairs comes from collinear scattering through the long range magnetic gluon exchange 6 . In such a weak-coupling regime, formation

294

295

of Cooper pairs takes place only in a small region near the Fermi surface. The second approach is the mean-field approximation with the QCD-inspired 4fermion model introduced to study lower density regions 3 ' 7 . In this approach, the magnitude of the gap becomes as large as 100 MeV and is almost constant in the vicinity of the Fermi surface. The main purpose of this contribution is to discuss the superconducting gap over a wide range of baryon densities within a single model and to make a bridge between high and low density regimes 8 . To this aim, we make an extensive analysis on the structural change in spatial-momentum dependence of a Cooper pair in 2-flavour colour superconductivity at zero temperature. The momentum dependence of the gap, diffuseness of the Fermi surface, the quark-quark correlations in the superconductor are the characteristic quantities reflecting the departure from the weak-coupling picture. In particular, we show that the spatial size of Cooper pairs which is calculated from the quark-quark correlations, indicates a clear deviation from the weak-coupling BCS theory. A more complete and detailed analysis is given in Ref. [8]. 2. Gap Equation with Spatial Momentum Dependence In the standard Nambu-Gor'kov formalism for a two component Dirac spinor \p = ( ^ , ^ ' ) , the quark self-energy E(fcp) with Minkowski 4-momentum A;p satisfies J4

where Da£v is the gluon propagator in medium, S(qp) the full quark propagator, and T£ the quark-gluon vertex. We study the gap function in the flavour antisymmetric, colour anti-symmetric and Jp — 0 + channel (the most attractive channel within the one-gluon exchange model) 5 : A(*„) = (A 2 r 2 C 7 5) (A+(* p )A+(fc) + A_(fcp)A_(fe)) ,

(2)

where r 2 is the Pauli matrix acting on the flavour space, A2 a colour anti-symmetric Gell-Mann matrix, and C the charge conjugation operator. A±(fc) = ( l i f e - a ) / 2 is the projector on positive (+) and negative ( - ) energy quarks. For the vertex in Eq. (1), we use T£ = diag( 7 ^A a /2, -( 7 M A a /2) f ). For g2 in Eq. (1), we use a momentum dependent coupling g2{q,k) in the "improved ladder approximation" 9 (ft, = (11JVC - 2iV»/3 = 29/3):

296

where p2 plays a role of a phenomenological infrared regulator which prevents the coupling constant from being too large at low momentum q, k ~ AQCDAt high momentum, g2 shows the same logarithmic behaviour as the usual running coupling with A identified with AQCD- The quark propagator in the improved ladder approximation in vacuum is known to have a high momentum behaviour consistent with that expected from the renormalization group and the operator product expansion. We adopt A=400 MeV and p%=1.5 A2 which are determined to reproduce the low energy meson properties for the Nj = 2 vacuum 10 . The gluon propagator in Eq. (1) in the Landau gauge reads,

/?„„(*/>; *o < |*|) = -.2^.,ffin/ii,i - wf~r> 2

k +iM \k0\/\k\

fc

(4)

+mJ3

where Pj^ are the transverse and longitudinal projectors. The longitudinal part of the propagator (the electric part) has static screening by the Debye mass rrijy = (Nj /2-K2)g2fj,2, while the transverse part (the magnetic part) has dynamical screening by Landau damping M2 = (7r/4)m23. This form is a quasi-static approximation of the full gluon propagator in the sense that only the leading frequency dependence is considered u . 2.1. Full Gap

Equation

The gap equation with the approximations shown above is obtained from the 2-1 element of the Schwinger-Dyson (matrix) equation, Eq. (1), which reads A±(*) = [°°

Jo

dqV±{q,k;e+e±)-=^±M=

q

'

fe

V^+(?)2

+ |A+(?)l2

A + [ dqV^Sq^-^-^t) , ~^ q Jo ^ * V^-(ff) 2 + |A_(g)|2

(5) K)

Here we used a simplified notation: A±(efc ,k) -* A±(fc) with E±(q) = q =F fi and ejr being the quasi-particle energy as a solution of (e*) 2 = E"±{q) + A\(e^, q). The explicit form of V± is given in Ref. [8]. At extremely high density, the Cooper pairing is expected to take place only near the Fermi surface due to the weak coupling property. In this case, we can safely neglect the antiquark-pole contribution for calculating A + . Furthermore, one may replace the momentum dependent coupling constant in Eq. (3) by that on the Fermi surface g2(fi,fi). On the other hand, at low densities, sizable diffusion of the Fermi surface occurs and the weak-coupling approximation is not justified, and we need to solve the coupled gap equations, Eq. (5), numerically.

297 2.2. Occupation Length

Number,

Correlation

Function

and

Coherence

To clarify the structural change of the colour superconductor from high to low densities, it is useful to examine the following physical quantities : (I) The quark and antiquark occupation number which is related to the diagonal (1-1) element of the quark propagator, (V>j (t,y)ii>i{t,x))supeT, in the NambuGor'kov formalism;

ni 2(

' " " i ('" T S P T E w ) • 4<,) = 9("""• "S-(,) = ° ' ( 6 )

where the superscripts (1,2 and 3) stand for colour indices. Since the third axis in the colour space is chosen to break colour symmetry, quarks with the third colour remains ungapped. (II) The q-q and q-q correlation functions in momentum space ±{q) and in coordinate space J (2TT)3 ^ W ' where N is a normalization constant determined by f d3r\ip+(r)\2 = 1.

(7) ^'

(III) The coherence length £c characterizing the typical size of a Cooper pair: It is defined simply as the root mean square radius of ip+ (r): Jd\

^

r2|y+(r)|'

f
f0°° dk fc2 \dy+(k)/dk\2

tfdkVtf+ik)?

'

U

It can be shown8 that the quark correlation
where the Pippard length is given by £p = (irA+(fi))~1. In a typical type-I superconductor in metals, the Pippard length is of the semi-macroscopic order £p ~ 10~ 4 cm, whereas the inverse Fermi momentum is of microscopic order kp1 ~ 10~ 8 cm. Besides, there is another scale uijy, the Debye cutoff, which limits the range of attractive interactions to be just around the Fermi surface \k —fcp|< WQ. The inverse of the Debye cutoff is in between the two scales above: w^ 1 ~ 1 0 - 6 cm. Therefore there is a clear scale hierarchy, A < C U D C kp. On the other hand, since there is no intrinsic scale WD in QCD, scale hierarchy at extremely high density simply reads A ~ pe~c'3
298

At lower densities, however, such scale separation becomes questionable for coupling constants which are not small. 3. M o m e n t u m Dependent Gap from High to Low Densities In Fig. 1(a), we show the gap A+(k) as a solution of the full gap equation for a wide range of densities. Since the actual position of the Fermi surface moves as we vary the density, we use k/fi as a horizontal axis in the figure which helps us to understand the change of global behaviour. The figure shows that the sharp peak at high density gradually gets broadened and simultaneously the magnitude of the gap increases as we decrease the density. The characteristic features at high density are: (i) There is a sharp peak at the Fermi surface, and (ii) The gap decays rapidly but is nonzero for momentum far away from the Fermi surface. The property (i) is similar to the standard BCS superconductivity but (ii) is not, due to the absence of an intrinsic ultraviolet Debye-cutoff of the gluonic interaction in QCD. As for the magnitude of the gap at high density, if one estimates the several contributions to the kernel V± in the gap equation, Eq. (5), separately one finds that the colour-electric interaction enhances the gap considerably. This may be understood as follows. In coordinate space, the Debye-screened electric interaction is a Yukawa potential. Such a short-range interaction alone can form only loosely bound Cooper pairs and a very small gap. However, if the magnetic and electric interactions coexist, small size Cooper pairs are formed primarily by the long-range magnetic interaction. Then, even the short-range electric interaction becomes effective to generate further attraction between the quarks. The characteristic features at low density are : (iii) The sharp peak at the Fermi surface disappears, and (iv) All the contributions neglected in the weak-coupling limit are actually not negligible. A close look at the effects of each contribution to the gap leads us to the conclusion that the colour superconductivity at low density is not a phenomenon just around the Fermi surface. This is confirmed by computing the occupation number following Eq. (6). The result shows that the Fermi surface is substantially diffused at low density. In Fig. 1(b), the gap at the Fermi surface A + (/t) is shown as a function of the chemical potential. It decreases monotonically as fi increases as is shown on the figure, but increases at much higher density ft > 106 MeV. An analytic solution based on the weak coupling approximation (includes only the magnetic gluons, fixed coupling, and ignores the antiquark pole contribution) is also shown in Fig. 1(b). The magnitude of the analytic solution is normalized to the numerical solution at the highest density (i = 212A ~ 1.6 x 106 MeV. At high density, the /x-dependence of the numerical result is in good agreement with an

299

0

1

2

3

,„»

10«

10»

10«

Figure 1. (a) A+ (A;) as a function of &/M for various densities n = 2™ A with n = 1,2,3,12. (A =400MeV). All the calculations are done with the momentum dependent vertex g2(q, k) and with the antiquark-pole contribution, (b) Chemical potential dependence of the gap A+ (k = //) in the full calculation compared with the analytic result which is normalized at the highest density n = 2 A.

analytic form with a parametric dependence A+(//) oc g~5(iexp(—Sir2/\/2g) with <72 = g2(fi,fi). On the other hand, the difference between the two curves at low density implies the failure of the weak-coupling approximation.

4. Correlation Function and Coherence Length One of the advantages of treating the momentum dependent gap is that we are able to calculate the correlation function which physically corresponds to the "wavefunction" of the Cooper pair. Such correlations have first been studied in Ref. [12] in the context of colour superconductivity (but with a much simpler model than Eq. (5), and only at lower density). Figure 2(a) shows the coherence length £c of a quark Cooper pair defined as the root mean square radius of the correlation function [see Eq. (8)]. The size of the Cooper pair becomes smaller as we go to lower densities. This tendency is understood by the behaviour of the Pippard length £p = l/jrA+(/i) (which gives a rough estimate of the coherence length) together with the behaviour of A+(/i) shown in Fig. 1(b). Also, the Cooper pair becomes smaller as density increases beyond fi = 2 12 A. However, it does not necessarily imply the existence of tightly bound Cooper pairs. In fact, the size of a Cooper pair makes sense only in comparison to the typical length scale of the system, namely the averaged inter-quark

300

10

10

10 /j[MeV]

10 fi[MeV]

Figure 2. (a): Density dependence of the coherence length, (b): Ratio of the coherence length £ c and the average inter quark distance dq as a function of the chemical potential.

distance dq for free quarks defined as dq =

7T2\1/31

(10)

As we go to higher densities, the ratio £c/dq increases monotonically as shown in Fig. 2(b). Namely, loosely bound Cooper pairs similar to the BCS superconductivity in metals are formed at extremely high densities. (Recall that the typical ratio in superconductivity is kp/A ~ 104.) At the lowest density in Fig. 2, the size of the Cooper pair is less than 4 fm and the ratio £c/dq is less than 10. This is not similar to the usual BCS system. The transition from £c/dq S> 1 to £c/dq ~ 1 as \i decreases is analogous to the crossover from the BCS-type superconductor to the BoseEinstein condensation (BEC) of tightly bound Cooper pairs 13 . Our result here suggests that quark matter possibly realized in the core of neutron stars may be rather like the BEC of tightly bound Cooper pairs. For better understanding of the internal structure of the quark Cooper pair, let us consider the correlation function in coordinate space. Figure 3 shows the spatial correlation of a Cooper pair at various chemical potentials normalized

as/d^+OOI^l. At high density, most of the quarks participating in forming a Cooper pair have the Fermi momentumfcp= \i giving a sharp peak in the momentum space correlation. In the coordinate space, this corresponds to an oscillatory distribution with a wavelength A = l//i without much structure near the origin. (The oscillation is also evident from the factor sin(pr) in the approximate correlation function in Eq. (9).) At lower densities, accumulation of the correlation

301

o X

Figure 3. The quark-quark correlation function
near the origin in coordinate space is much more prominent in Fig. 3. This implies a localized Cooper pair composed of quarks with various momentum. 5. Summary and Discussion We have studied the spatial-momentum dependence of a superconducting gap and the structure of Cooper pairs in two-flavour colour superconductivity, using a single model in a very wide region of density. A nontrivial momentum dependence of the gap manifests itself at low densities, where relatively large QCD coupling allows the Cooper pairing to take place in a wide region around the Fermi surface. Note that our results can be easily extended to the Nj = 3 case. Our results imply that quark matter which might exist in the core of neutron stars or in the quark stars could be rather different from that expected from the weak-coupling BCS picture and could be more like a BEC of tightly bound Cooper pairs. A study of the finite T phase transition of this strong coupling system is currently under investigation. Acknowledgments T.H. would like to thank A. W. Thomas, A. G. Williams, D. Leinweber and the members of the local organizing committee for giving him the opportunity to give a talk at the Joint CSSM/JHF Workshop on Physics at Japan Hadron Facility (March 14-21, Adelaide, 2002). This research was supported in part by the National Science Foundation under Grant No. PHY99-07949.

302

References 1. J. C. Collins and M. J. Perry, Phys. Rev. Lett. 34, 1353 (1975). 2. D. Bailin and A. Love, Phys. Rep. 107, 325 (1984). M. Iwasaki and T. Iwado, Phys. Lett. B 350, 163 (1995). 3. M. Alford, K. Rajagopal, and F. Wilczek, Phys. Lett. B422, 247 (1998). R. Rapp, T. Schafer, E. V. Shuryak, and M. Velkovsky, Phys. Rev. Lett. 81, 53 (1998). 4. For a recent review, see K. Rajagopal and F. Wilczek, hep-ph/0011333. 5. T. Schafer and F. Wilczek, Phys. Rev. D60, 114033 (1999); D. K. Hong, V. A. Miransky, I. A. Shovkovy and L.C.R. Wijewardhana, Phys. Rev. D 6 1 , 056001 (2000), Erratum-ibid. D62, 059903 (2000); R. D. Pisarski and D. H. Rischke, Phys. Rev. D60, 094013 (1999); D 6 1 , 074017 (2000). 6. D. T. Son, Phys. Rev. D 59, 094019 (1999). 7. J. Berges and K. Rajagopal, Nucl. Phys. B538, 215 (1999). G. W. Carter and D. Diakonov, Phys. Rev. D60, 016004 (1999). 8. H. Abuki, T. Hatsuda and K. Itakura, Phys. Rev. D65, 074014 (2002). 9. K. Higashijima, Phys. Rev. D29, 1228 (1984); Prog. Theor. Phys. Suppl. 104, 1 (1991). V. A. Miransky, Sov. J. Nucl. Phys. 38, 280 (1983). 10. See e.g., T. Ikeda, Prog. Theor. Phys. 107, 403 (2002); hep-ph/0107105 and references therein. 11. K. Iida and G. Baym, Phys. Rev. D63, 074018 (2001). 12. M. Matsuzaki, Phys. Rev. D62, 017501 (2000). 13. P. Nozieres and S. Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985). See also, E. Babaev, Int. J. Mod. Phys. A16, 1175 (2001) and references therein.

H A D R O N PROPERTIES IN N U C L E A R M E D I U M A N D THEIR IMPACTS ON OBSERVABLES

K. TSUSHIMA Department

of Physics and Astronomy, University Athens, GA 30602, USA E-mail: [email protected]

of Georgia

The effect of changes in hadron properties in a nuclear medium on physical observables is discussed. Highlighted results are: (1) hypernuclei, (2) meson-nuclear bound states, (3) K-meson production in heavy ion collisions, and (4) J/
1. Hadrons in Nuclear Medium: Treatment in QMC Here we discuss hadrons in a nuclear medium in the quark-meson coupling (QMC) model 1 . The model has been extended and successfully applied to many problems 2 ' 3 ' 4,5 ' 6 ' 7 ' 8 . A detailed description of the Lagrangian density, the mean-field equations of motion, and the treatment of finite nuclei are given in Refs. [2,3]. As examples, we show in Fig. 1 results for the 4 0 Ca nucleus in QMC 3 , which is based on the quark structure of nucleon or nucleus. *°CCL single

particle

prattma

energies

nmrttrona

c- -so

.« -30

f £ —*o -SO -60 0

1

2

3

4

5

Exp.

QHD

QMC

Exp.

QHO

QUO

J

6

r(fm)

Figure 1. Charge density distribution (the left panel) and energy spectrum (the right panel) for the 4 0 C a nucleus, compared with the experimental d a t a and predictions using Quantum Hadrodynamics (QHD, also labelled by "Walecka-Serot") 1 0 .

The Dirac equations for quarks and antiquarks in hadron bags (q = u,u, d or d, hereafter) neglecting the Coulomb force, are given by (|as| < bag radius) 2 ' 3 ' 4 ' 5 ' 6 ' 7 ' 8 :

il • dx - (m, - V§) =F 7° (v* + \v^j

\M*)J

303

(i)

304

[*7 • dx - rnS:C] ipSiC{x) (or ipj,e(x)) = 0.

(3)

The mean-field potentials for a bag in nuclear matter are defined by V§=g%a, v 2= 9%u and V^=gjb, with g%, g* and gj the corresponding quark-meson coupling constants. The bag radius in medium for a hadron ft, R*h, will be determined through the stability condition for the mass of the hadron against the variation of the bag radius 1,2 (see Eq. (4)). The hadron masses in a nuclear medium m*h (free mass will be denoted by m/j), are calculated by v

j=q,q,Q,Q

njWj-ZH

4

3

dml

= 0,

(4)

Rh—R^

where Q* = SI* = [x2g + ( ^ m ^ ) 2 ] 1 / 2 (q = u,d), with m* = mq-g$a, 2 1 2 fig = ft^- = [x^ + ( ^ m Q ) ] / ( Q = s, c), and X 9I Q being the bag eigenfrequencies. B is the bag constant, nq(riq) and TIQ^-Q) are the lowest mode quark (antiquark) numbers for the quark flavours q and Q in the hadron ft, respectively, and the ZH parametrize the sum of the centre-of-mass and gluon fluctuation effects (assumed to be independent of density). The parameters are determined in free space to reproduce the corresponding masses. We chose the values (mg,ms,mc) = (5,250,1300) MeV for the current quark masses, and -R/v = 0.8 fm for the bag radius of the nucleon in free space. The quark-meson coupling constants, g%, g% and g^, are adjusted to fit the nuclear saturation energy and density of symmetric nuclear matter as well as the bulk symmetry energy 2 . However, in the studies of the kaon system, we found that it was phenomenologically necessary to increase the strength of the vector coupling to the non-strange quarks in the K+ (by a factor of 1.42, i.e. gqKu = 1.42#2,) in order to reproduce the empirically extracted A' + -nucleus interaction 5 . We assume this also for the D and D mesons 7,8,9 . The scalar (Vj1) and vector (Vvh) potentials felt by the hadrons ft, in nuclear matter, are given by Vsh = ml - mh, Vvh = (n, - nq)V* - hV« (V« -> 1.42Vtf for K,~K,

D,D),(S)

where I3 is the third component of isospin projection of the hadron ft. In Fig. 2 we show effective masses and mean field potentials for various hadrons in symmetric nuclear matter. Several comments on the results shown in Fig. 2 are in order : (1) Physical 7/ and 7/ mesons are treated including a mixing angle, 8p — — 10°, as (77,77') = (r]scos9p — rjisindp, rig sin 6p +rji cosOp), where 771 = (l/\/3)(«w + dd+ ss) and 778 = (1/V / 6)(MU + dd — 2ss).

305

Figure 2. Effective mass ratios and mean field potentials for hadrons in nuclear m a t t e r (po = 0.15 f m - 3 ) . " H " stands for the H particle with an input mjj — 2m\.

(2) The H particle is treated as a 6-quark bag, uuddss, with ra# = 2mA(3) The scalar potential for the hadron h, Vsh, shows a universal quark number scaling rule, Vsh/VSN ~ (nq + n ? ) / 3 , where V ^ is the scalar potential for the nucleon. (See Eq. (5).) (4) The scalar potential for the <j> meson arises entirely from the (j> — u mixing in QMC, and is tiny. (5) Reduction of the effective mass of A+ implies the role of the light quarks in partial restoration of chiral symmetry in a heavy-light quark system, and merits further studies. In following sections we will discuss how these changes in hadron properties in a nuclear medium impact physical observables.

306

2. Hypernuclei When a hyperon, Y, is embedded in a nucleus, the potential felt by the hyperon at a given position of the nucleus can be calculated self-consistently4. Simultaneously one can also get energy levels for the hyperon Y in a shell-model calculation 4 . Because the treatment is based on the quark degrees of freedom, we need to consider the possibility of a Pauli blocking effect at the quark level, and also the effect of E-A channel coupling for the S- and A-hypernuclei. We included these effects in a phenomenological way 4 . Nevertheless, we can study all A-, £-, S-hypernuclei in a systematic way, due to the quark degrees of freedom, because the hyperon-meson coupling constants in nuclear medium are automatically determined 4 . In Table 1 we present QMC predictions for the energy levels of hyperons in various hypernuclei. Results for other heavier mass hypernuclei are given in Ref. [4]. Table 1. Energy levels for hyperons Y (in MeV), for ^O, ^ C a and ^?Ca hypernuclei, calculated in QMC. Experimental d a t a are taken from Ref. [11]. ™0 (Expt.) -12.5

X'o -14.1

J?-°

-5.1 -5.0

-17.2 -8.7 -8.0

"Ca

*LCa -23.5 -17.1 -16.5 -10.6 -9.3 -9.7

14/2

-19.5 -12.3 -12.3 -4.7 -3.5 -4.6 49 Ca -21.0 -13.9 -13.8 -6.5 -5.4 -6.4

1/7/2

—

lSl/2 1P3/2 lPl/2 lSl/2 1P3/2 lPl/2

-2.5 (lp) ^°Ca (Expt.) -20.0 -12.0 (lp)

14/2 2s

l/2

1^3/2

— 1*1/2 1P3/2 lPl/2

14/2 2«l/2

*?_Ca -19.3 -11.4 -10.9 -5.8 -6.7 -5.2 -1.2

17

O -9.6 -3.2 -2.6 41 Cn -13.4 -8.3 -7.7 -2.6 -1.2 -1.9

17

O -3.3

41

— — fa

-9.9 -3.4 -3.4

tlCa

17

O -4.5

— — i^Ca

-17.0 -11.2 -11.3 -5.5 -5.4 -5.6

-8.1 -3.3 -3.4

Pa

i 9 _Ca

itt.Ca

-14.6 -9.4 -8.9 -3.8 -2.6 -3.1

-11.5 -7.5 -7.0 -2.0 -1.2

-14.7 -8.7 -8.8 -3.8 -4.6 -3.8

-12.0 -7.4 -7.4 -2.1 -1.1 -2.2

—

—

—

—

49

Ca

-4.1

i7_o

— — — — — 49

—

— — —

3. Meson-Nucleus Bound States Next, we discuss the meson-nuclear bound states. We have solved the KleinGordon equation for mesons j (j = w, 77,rf, D, D) with almost zero momenta, using the potentials calculated in QMC 6 , 7 : [V 2 + Ef

- fhf(r)]


E* = Ej + mj - iTj/2

(6)

307

m*{r) = m){r) - %- [(mj - m*(r)) 7 j + Tj) = m*(r) -

'-Tfr),

(7)

where Ej is the complex valued, total energy of the meson, and we included the widths of the mesons in a nucleus assuming a specific form using */j, which are treated as phenomenological parameters. We calculate the single-particle energies for the values j u = 0.2, and 7, = 0.5, which are expected to correspond best to experiments 6 , while for the 77', D and U, the widths H = 0 are assumed 7 . For a comparison we present also results for the w calculated using the potentials obtained in QHD 12 . Results are given in Tables 2 and 3. Table 2. Calculated u-, T)- and ^'-nuclear bound state energies (in MeV), E-j = Re(E* - m}) (j = w, r;, ri'), in QMC 6 and those for the w in QHD with cr-w mixing effect 1 2 . The complexeigenenergiesare given by, E* = Ej+m,j —iVj/2. (* not calculated) IT, = 0.5

(QMC)

lw = 0.2

(QMC)

•yu = 0.2

(QHD)

*V

fcjtjj

14.5

rw

hiuj

?He

Is

!'B

Is

-24.5

22.8

Is

-38.8 -17.8

28.5 23.1

fMg le

O

fCa ?°Zr 208 p

b

Table 3. state Is lp 2s

(QMC) r„

Er, -10.7

24.7

-97.4

r„ 33.5

-80.8

28.8

-129

38.5

-99.7

31.1

-144

39.8

29.4 24.8 30.6

-121 -80.7 -134

37.8 33.2 38.7

IP 2s Is

—

—

* * * * *

-32.6

26.7

-41.3

-78.5 -42.8 -93.4

lp Is

-7.72 -46.0

18.3 31.7

-22.8 -51.8

-64.7 -111

27.8 33.1

-103 -148

35.5 40.1

lp 2s Is

-26.8 -4.61

26.8 17.7

-52.9

33.2

-38.5 -21.9 -56.0

-90.8 -65.5 -117

31.0 28.9 33.4

-129 -99.8 -154

38.3 35.6 40.6

IP 2s

-40.0 -21.7

30.5 26.1

-56.3

33.2

-105 -86.4 -118

32.3 30.7

Is

-47.7 -35.4 -57.5

33.1

-143 -123 -157

39.8 38.0 40.8

lp 2s

-48.3 -35.9

31.8 29.6

-52.6 -44.9

-111 -100

32.5 31.7

-151 -139

40.5 39.5

-55.6

D~, D° and D° bound state energies (in MeV). The widths are all set to zero,

D-(1.42VJ) -10.6 -10.2 -7.7

D~(V2) -35.2 -32.1 -30.0

£ > - ( V j , no Coulomb) -11.2 -10.0 -6.6

£>°(1.42VJ) unbound unbound unbound

£>°(VJ) -25.4 -23.1 -19.7

D°{V2) -96.2 -93.0 -88.5

Our results suggest that w, r) and rj mesons should be bound in all the nuclei considered. Furthermore, the D~ meson should be bound in 2 0 8 Pb in any case, assisted by the Coulomb force .

308

4. i£-Meson Production in Heavy Ion Collisions Here we focus on the kaon production reactions, -KN —¥ AA", in nuclear matter. We calculate the in-medium reaction amplitudes, taking into account the scalar and vector potentials for incident, final and intermediate mesons and baryons. The processes considered in the calculation, which were already established in studies for free space 13 , are shown in Fig. 3.

Figure 3.

Processes included for the TTN —> AA' reactions.

(GeV)

(GeV)

Figure 4. Energy dependence of the total cross sections. The solid, dashed, dotted lines in the left panel correspond to nuclear densities pg = (0,po,3po)- The solind and dotted lines in the right panel correspond to pg = 0, and the result averaged over the nuclear density distribution in Au+Au collisions at 2A GeV 1 4 . The dots are d a t a in free space 1 5 , pg = 0.

Results are shown in Fig. 4. (See caption for explanations.) From the results we conclude that if one accounts for the in-medium modification of the production amplitude correctly, it is possible to understand A"+ production data in heavy ion collisions at SIS (Darmstadt, Germany) energies, even if the K+-meson feels the theoretically expected, repulsive mean field potential. The apparent failure to explain the K+ production data, if one includes the purely kinematic effects of the in-medium modification of the A' + -meson and hadrons, appears to be a consequence of the omission of these effects in the reaction amplitudes.

309

5. J/$? Dissociation in Nuclear Matter There is a great deal of interest in possible signals of Quark-Gluon Plasma (QGP) formation, and J / $ suppression is a promising candidate as suggested by Matsui and Satz 16 . Our interest here is how much the J/ty absorption cross sections in hadronic dissociation processes will be modified, if the inmedium hadron potentials are included, which has never been addressed in QGP analyses. We consider the reactions involving the J/$!, shown in Fig. 5. Recent cal: D * ID ID* ID D D ID ID ID

ID

Figure 5.

ID

ID

Processes included for J/Vt dissociation.

culations for the processes in free space 17 indicate a much lower cross sections than necessary to explain the data for the J / \ t suppression. However, this situation changes when the in-medium potentials of the charmed (and also p) mesons are taken into account, as shown in the left panel of Fig. 6. Clearly, the J/\P absorption cross sections are substantially o> r + J / .

\ |, V-. \\ j ^ 20

Pb + Pb * 1996 • 1 9 9 6 - M i n . Bios • 1998- Min. Bias

%*^.

10

E, (GeV)

Figure 6.

Energy dependence of the total cross section.

enhanced not only because of the downward shift of the reaction threshold, but also because of the in-medium effect on the reaction amplitude. In order to compare our results with the NA38/NA50 data 1 8 on J / $ suppression in Pb+Pb collisions, we have adopted the heavy ion model proposed in Ref. [19] with the ET model from Ref. [20]. Our result is shown in the right panel of Fig. 6 by the solid line, using the density dependent, thermally

310

averaged cross section 8 . The dashed line shows the result reported in Ref. [20] using the phenomenological constant cross section, 4.5 mb. Both lines clearly reproduce the data 1 8 quite well, including most recent results from NA50 on the ratio of J/vp over Drell-Yan cross sections, as a function of the transverse energy ET up to about 100 GeV. It is important to note that if one neglects the in-medium modification of the J/9 absorption cross section, the large value of the J/9 dissociation cross section of 4.5 mb used in Ref. [20], could not be justified by microscopic theoretical calculations, and thus the NA50 data 1 8 could not be described. 6. pp —• ppu and pp —¥ pp Reactions at Near Thresholds Here we report results for the vector meson production reactions near threshold, pp —> ppv (v = w, 4>)21'22. Description of the model and parameters determined by other reactions are given in Ref. [23]. The vector meson production amplitude, M, is depicted in Fig. 7. Our main results for vector meson production are : (1) Tensor to vector coupling ratio, KV = fVNN/gvNN, for the w (KW), where a typical NN interaction model 24 sets KU = 0; (2) <j)NN coupling constant, g<j>NN, in connection with ss component in the nucleon wave function and the OZI rule violation.

-t-

(i-

=~2>

—[—-

o>

o

o>

Figure 7. Amplitudes, M , for the pp -»• ppv (v = u, ) reactions considered in the study. TMNI ISI and FSI stand for, meson-nucleon (MN) T-matrix, initial state interaction and final state interaction, respectively.

Results for angular distributions are shown in Fig. 8, together with the experimental data for the u in Ref. [25] and <£in Ref. [26]. The total contribution comes from following two sources. First, the nucleonic source : the vector meson is produced from the vNN (meson-nucleon-nucleon) vertex. Second, the mesonic one : the vector meson is produced from the vpir (mesonic vertex),

311

where the n and p are attached to two different protons. The u meson production is dominated equally by the nucleonic and mesonic mechanisms, while for the <j>, the mesonic one is strongly dominant. Thus, production mechanisms for these two vector mesons are different. For the u meson, a large value pp ->pp$: TJab = 2.85 GeV(K=-2.0, KH=ll90MeV) PP —» ppta:

Q—173 MeV(K^~1000

MeV) A.^I20M«V

A.=2O40M«V

IJTHH-^fTiViai- r]~ih*+h4T*M

A.o2240MeV

s

piaiE,

-1.2 V2200MeV

j*tf^ l„=-2.0 A.*2220MeV

-^a±mi^ti. cos(e)

oo«(0)

°»(9)

C0S(6)

Figure 8. Angular distributions for pp -¥ ppui at excess energy Q = 173 MeV (the left panel), and for pp -»• ppcj> at excess energy Q = 83 MeV (the right panel). They are normalized to the total cross sections, ) = 190 nb in Ref. [26], respectively. The (solid, dash-dotted, dotted) lines correspond to the (total, nucleonic, mesonic) contributions, respectively.

KU = —2.0 with g^NN = (—9.0)2 = 81.0 fits best the data 2 5 , and turned out to be insensitive to the value of gWNN21'22, where in the Bonn NN potential model 24 KW = 0 with g%NN ~ 24(4TT) ~ 301.6 is used. As for the <j> meson, several parameter sets can possibly reproduce the data 2 6 . Surprisingly, for K4, = —2.0, a large absolute value |<70AW| = I — 2.0| can also reproduce the data. This implies a large violation of the OZI rule. However, this turned out to give a distinguish ably different energy dependence for the total cross section at near threshold 22 . Thus, we can distinguish if the energy dependence of the total cross section is measured at near threshold. 7. Conclusion In conclusion, there is strong motivation to perform experiments to study the changes in hadron properties in nuclear medium or the partial restoration of chiral symmetry in nuclear medium. Acknowledgments The author would like to thank, D.H. Lu, K. Nakayama, K. Saito, A. Sibirtsev and A.W. Thomas for exciting collaborations, and for warm hospitality at CSSM during the workshop. Special thanks go to Prof. A.W. Thomas for the support enabling me to attend the workshop. This work was partially

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List of Participants

Prof. Iraj A F N A N SoCPES Flinders University G P O box 2100 Adelaide, SA, 5001 AUSTRALIA [email protected]

Dr. Chris ALLTON University of Wales Swansea-UK University of Queensland Department of Mathematics St. Lucia, Brisbane, 4072 AUSTRALIA c.allton@swan. ac. uk

Wesley A R M O U R 9 Beaufort Court Cadle Swansea Wales, SA5 4PH U.K. [email protected]

Jonathan A S H L E Y CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

Dr. Wolfgang B E N T Z Dept. of Physics, School of Science Tokai University, Japan 1117 Kitakaname, Hiratsuka-shi Kanagawa 259-1292 JAPAN [email protected]

Sundance B I L S O N - T H O M P S O N CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

Dr. Ingo B O J A K CSSM University of Adelaide 10 Pulteney Street, Level 4 Adelaide, SA, 5000 AUSTRALIA [email protected]

Prof. Matthias B U R K A R D T Department of Physics New Mexico State Univ Las Cruces NM 88003 U.S.A. [email protected]

313

314 Dr. Fu-Guang C A O Institute of Fundamental Sciences Massey University Private Bag 11-222 Palmerston North NEW ZEALAND [email protected]

Prof. Abraham C H I A N WISER/NITP University of Adelaide 10 Pulteney Street, Level 4 Adelaide, SA, 5000 AUSTRALIA [email protected]

Ian C L O E T CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

Benjamin C R O U C H CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

Dr. Will D E T M O L D CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

Matthew G A R B U T T Theory Group School of Physics The University of Melbourne VIC,3010 AUSTRALIA [email protected]

Dr. Xin-Heng G U O CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

Dr. Vadim G U Z E Y CSSM University of Adelaide 10 Pulteney Street, Level 4 Adelaide, SA, 5000 AUSTRALIA [email protected]

Dr. Johann H A I D E N B A U E R Forschungszentrum Juelich IKP D-52425 Juelich GERMANY [email protected]

Prof. Tetsuo H A T S U D A University of Tokyo Physics Department Tokyo 113-0033 JAPAN [email protected]

315 John H E D D I T C H CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

Prof. Atsushi H O S A K A RCNP Osaka University Ibaraki 567-0047 JAPAN [email protected]

Dr. Alexander KALLONIATIS CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

Waseem K A M L E H CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

Prof. Faqir K H A N N A Physics Department Theoretical Physics Institute University of Alberta Edmonton,Alberta,T6G 2J1 CANADA [email protected]

Dr. Ayse KIZILERSU CSSM University of Adelaide 10 Pulteney Street, Level 4 Adelaide, SA, 5000 AUSTRALIA akiziler ©physics. adelaide.edu.au

Prof. Gastao K R E I N Instituto de Fisica Teorica Rua Pamplona, 145 01405-900 Sao Paulo, SP BRAZIL [email protected]

Ben L A S S C O C K CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA 5000 AUSTRALIA [email protected]

Dr. Derek L E I N W E B E R CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

Dr. Olivier L E I T N E R CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

316 Dr. Max LOHE University of Adelaide Department of Phys. and Math. Phys. Adelaide, SA, 5005 AUSTRALIA [email protected]

Prof. Bruce M C K E L L A R School of Physics University of Melbourne VIC,3010 AUSTRALIA [email protected]

Hirobumi M I N E O Department of Physics University of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033 JAPAN [email protected]

Prof. Tomofumi N A G A E KEK 1-1 Oho Tsukuba 305-0801 JAPAN [email protected]

Dr. Martin O E T T E L Max-Planck-Institut fiir Metallforschung Heisenbergstr. 3, 70569 Stuttgart GERMANY oettel@mf. mpg.de

Prof. Makoto O K A Department of Physics Tokyo Institute of Technology Oh-Okayama, Meguro, Tokyo 152-8551 JAPAN [email protected]

Dr. Shin'ya S A W A D A KEK 1-1 Oho Tsukuba Ibaraki 305-0801 JAPAN [email protected]

Kai S C H M I D T - H O B E R G CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

Dr. Andreas S C H R E I B E R CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

Mark S T A N F O R D CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

317 Prof. Anthony T H O M A S CSSM University of Adelaide 10 Pulteney Street, Level 4 Adelaide, SA, 5000 AUSTRALIA [email protected]

Prof. Hiroshi T O K I RCNP, Osaka University MihogaokalO-l Ibaraki Osaka 567-0047 JAPAN [email protected]

Dr. Stuart T O V E Y Research Centre for HEP School of Physics University of Melbourne V I C , 3010 AUSTRALIA [email protected]

Dr. Kazuo T S U S H I M A Department of Physics and Astronomy University of Georgia Athens GA 30602 USA tsushima@physast. uga. edu

Dr. Raymond V O L K A S School of Physics The University of Melbourne Melbourne VIC,3010 AUSTRALIA [email protected]

A/Prof. Anthony W I L L I A M S CSSM University of Adelaide 10 Pulteney Street, Level 4 Adelaide, SA, 5000 AUSTRALIA [email protected]

Ross Y O U N G CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]. au

James Z A N O T T I CSSM University of Adelaide 10 Pulteney Street, Level 13 Adelaide, SA, 5000 AUSTRALIA [email protected]

Dr. Jioanbo Z H A N G CSSM University of Adelaide 10 Pulteney Street, Level 4 Adelaide, SA, 5000 AUSTRALIA [email protected]

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