Properties of Hadrons in the Nuclear Medium

a r X i v :n u c l -t h /9702016v 1 7 F e

b 1997LBNL-39866PROPERTIES OF HADRONS IN THE NUCLEAR MEDIUM Che Ming Ko 1Cyclotron Institute and Physics Department Texas A&M University,College Station,Texas 77843,USA Volker Koch 2Nuclear Science Division Lawrence Berkeley National Laboratory,Berkeley,CA 94720,USA Guoqiang Li 3Department of Physics State University of New York at Stony Brook,Stony Brook,N.Y.11794,USA key words :Chiral symmetry,Goldstone bosons,vector mesons,baryons,hot and dense hadroni

c matter,heavy-ion collisions

2KO,KOCH&LI

Contents

1INTRODUCTION2 2GOLDSTONE BOSONS5

2.1The pion (6)

2.2Kaons (10)

2.3Etas (18)

3VECTOR MESONS18

3.1The rho meson (20)

3.2The omega meson (29)

3.3The phi meson (29)

4BARYONS31 5RESULTS FROM LATTICE QCD CALCULATIONS32 6SUMMARY35

1INTRODUCTION

The atomic nucleus provides a unique laboratory to study the long range and bulk properties of QCD.Whereas QCD is well tested in the per-turbative regime rather little is known about its properties in the long range,nonperturbative region.One of the central nonperturbative prop-erties of QCD is the spontaneous breaking chiral symmetry in the ground state resulting in a nonvanishing scalar quark condensate,<ˉq q>=0. It is believed and supported by lattice QCD calculations[1]that at temperatures around150MeV,QCD undergoes a phase transition to a chirally restored phase,characterized by the vanishing of the order parameter,the chiral condensate<ˉq q>.This is supported by results obtained within chiral perturbation theory[2,3].E?ective chiral models predict that a similar transition also takes place at?nite nuclear density.

The only way to create macroscopic,strongly interacting systems at ?nite temperature and/or density in the laboratory is by colliding heavy nuclei at high energies.Experiments carried out at various bombarding energies,ranging from1AGeV(BEVALAC,SIS)to200AGeV(SPS), have established that one can generate systems of large density but mod-erate temperatures(SIS,BEVALAC),systems of both large density and

IN-MEDIUM EFFECTS3 temperature(AGS)as well as systems of low density and high temper-atures(SPS).Therefore,a large region of the QCD phase diagram can be investigated through the variation of the bombarding energy.But in addition the atomic nucleus itself represents a system at zero tem-perature and?nite density.At nuclear density the quark condensate is estimated to be reduced by about30%[2,3,4,5]so that e?ects due to the change of the chiral order parameter may be measurable in reactions induced by a pion,proton or photon on the nucleus.

Calculations within the instanton liquid model[6]as well as results from phenomenological models for hadrons[7]suggest that the proper-ties of the light hadrons,such as masses and couplings,are controlled by chiral symmetry and its spontaneous as well as explicit breaking. Con?nement seems to play a lesser role.If this is the correct picture of the low energy excitation of QCD,hadronic properties should de-pend on the value of the chiral condensate<ˉq q>.Consequently,we should expect that the properties of hadrons change considerably in the nuclear environment,where the chiral condensate is reduced.Indeed, based on the restoration of scale invariance of QCD,Brown and Rho have argued that masses of nonstrange hadrons would scale with the quark condensate and thus decrease in the nuclear medium[8].This has since stimulated extensive theoretical and experimental studies on hadron in-medium properties.

By studying medium e?ects on hadronic properties one can directly test our understanding of those non-perturbative aspects of QCD,which are responsible for the light hadronic states.The best way to investigate the change of hadronic properties in experiment is to study the pro-duction of particles,preferably photons and dileptons,as they are not a?ected by?nal-state interactions.Furthermore,since vector mesons decay directly into dileptons,a change of their mass can be seen di-rectly in the dilepton invariant mass spectrum.In addition,as we shall discuss,the measurement of subthreshold particle production such as kaons[9]and antiprotons[10]may also reveal some rather interesting in-medium e?ects.

Of course a nucleus or a hadronic system created in relativistic heavy ion collisions are strongly interacting.Therefore,many-body excitations can carry the same quantum numbers as the hadrons under considera-tion and thus can mix with the hadronic states.In addition,in the nu-clear environment‘simple’many-body e?ects such as the Pauli principle are at work,which,as we shall discuss,lead to considerable modi?ca-tions of hadronic properties in some cases.How these e?ects are related to the partial restoration of chiral symmetry is a new and unsolved ques-

4KO,KOCH&LI tion in nuclear many-body physics.One example is the e?ective mass of a nucleon in the medium,which was introduced long time ago[11,12] to model the momentum dependence of the nuclear force,which is due to its?nite range,or to model its energy dependence,which results from higher order(2-particle-1-hole etc.)corrections to the nucleon self energy.On the other hand,in the relativistic mean-?eld description one also arrives at a reduced e?ective mass of the nucleon.Accord-ing to[13],this is due to so-called virtual pair corrections and may be related to chiral symmetry restoration as suggested by recent studies [3,14,15,16].

Another environment,which is somewhat‘cleaner’from the theorists point of view,is a system at?nite temperature and vanishing baryon chemical potential.At low temperatures such a system can be system-atically explored within the framework of chiral perturbation theory, and essentially model-independent statements about the e?ects of chi-ral restoration on hadronic masses and couplings may thus be given.In the high temperature regime eventually lattice QCD calculations should be able to tell us about the properties of hadrons close to the chiral transition temperature.Unfortunately,such a system is very di?cult to create in the laboratory.

Since relativistic heavy ion collisions are very complex processes, one has to resort to careful modeling in order to extract the desired in-medium correction from the available experimental 645ac6fb941ea76e58fa04a3pu-tationally the description of these collisions is best carried out in the framework of transport theory[17,18,19,20,21,22,23]since nonequi-librium e?ects have been found to be important at least at energies up to2AGeV.At higher energies there seems to be a chance that one can separate the reaction in an initial hard scattering phase and a?nal equi-librated phase.It appears that most observables are dominated by the second stage of the reaction so that one can work with the assumption of local thermal equilibrium.However,as compared to hydrodynamical calculations,also here the transport approach has some advantages since the freeze out conditions are determined from the calculation and are not needed as input parameter.As far as a photon or a proton induced reaction on nuclei is concerned,one may resort to standard reaction theory.But recent calculations seem to indicate that these reaction can also be successfully treated in the transport approach[24,25,26]. Thus it appears possible that one can explore the entire range of exper-iments within one and the same theoretical framework which of course has the advantage that one reduces the ambiguities of the model to a large extent by exploring di?erent observables.

IN-MEDIUM EFFECTS5 This review is devoted to the discussion of in-medium e?ects in the hadronic phase and its relation to the partial restoration of chiral sym-metry.We,therefore,will not discuss another possible in-medium ef-fect which is related to the decon?nement in the Quark Gluon Plasma, namely the suppression of the J/Ψ.This idea,which has been?rst pro-posed by Matsui and Satz[27]is based on the observation that due to the screening of the color interaction in the Quark Gluon Plasma,the J/Ψis not bound anymore.As a result,if such a Quark Gluon Plasma is formed in a relativistic heavy ion collision,the abundance of J/Ψshould be considerably reduced.Of course also this signal su?ers from more conventional backgrounds,namely the dissociation of the J/Ψdue to hadronic collisions[28,29,30].To which extent present data can be understood in a purely hadronic scenario is extensively debated at the moment[31].We refer the reader to the literature for further details [32].

In this review we will concentrate on proposed in-medium e?ects on hadrons.Whenever possible,we will try to disentangle conventional in-medium e?ects from those we believe are due to new physics,namely the change of the chiral condensate.We?rst will discuss medium ef-fects on the Goldstone bosons,such as the pion,kaon and eta.Then we will concentrate on the vector mesons,which have the advantage that possible changes in their mass can be directly observed in the dilep-ton spectrum.We further discuss the e?ective mass of a nucleon in the nuclear medium.Finally,we will close by summarizing the current status of Lattice QCD calculations concerning hadronic properties.As will become clear from our discussion,progress in this?eld requires the input from experiment.Many question cannot be settled from theoret-ical consideration alone.Therefore,we will always try to emphasize the observational aspects for each proposed in-medium correction.

2GOLDSTONE BOSONS

This?rst part is devoted to in-medium e?ects of Goldstone bosons, speci?cally the pion,kaon and eta.The fact that these particles are Goldstone bosons means that their properties are directly linked to the spontaneous breakdown of chiral symmetry.Therefore,rather reliable predictions about their properties can be made using chiral symmetry arguments and techniques,such as chiral perturbation theory.Contrary to naive expectations,the properties of Goldstone bosons are rather robust with respect to changes of the chiral condensate.On second

6KO,KOCH&LI thought,however,this is not so surprising,because as long as chiral symmetry is spontaneously broken there will be Goldstone bosons.The actual nonvanishing value of their mass is due to the explicit symmetry breaking,which–at least in case of the pion–are small.Changes in their mass,therefore,are associated with the sub-leading explicit symmetry breaking terms and are thus small.We should stress,however, that in case of the kaon the symmetry breaking terms are considerably larger,leading to sizeable corrections to their mass at?nite density as we shall discuss below.

2.1The pion

Of all hadrons,the pion is probably the one where in-medium corrections are best known and understood.From the theoretical point of view, pion properties can be well determined because the pion is such a‘good’Goldstone boson.The explicit symmetry breaking terms in this case are small,as they are associated with the current masses of the light quarks. Therefore,the properties of the pion at?nite density and temperature can be calculated using for instance chiral perturbation theory.But more importantly,there exists a considerable amount of experimental data ranging from pionic atoms to pion-nucleus experiments to charge-exchange reactions.These data all address the pion properties at?nite density which we will discuss now.

Finite density

The mass of a pion in symmetric nuclear matter is directly related to the real part of the isoscalar-s-wave pion optical potential.This has been measured to a very high precision in pionic atoms[33],and one ?nds to?rst order in the nuclear densityρ

?m2π=?4π(b0)e?ρ,(b0)e???0.024m?1π.(1) At nuclear matter density the shift of the pion mass is

?mπ

IN-MEDIUM EFFECTS 7?0.010(3)m ?1πand the contribution from a (density dependent)corre-lation or re-scattering term,which is dominated by the comparatively large isovector-s-wave scattering length b 1=?0.091(2)m ?1π,i.e.,

(b 0)e?=b 0?[1+m π

r >,(3)

where the correlation length <1/r >?3p f /(2π)is essentially due to the Pauli exclusion principle.The weak and slightly repulsive s-wave pion potential has also been found in calculations based on chiral per-turbation theory [34,35]as well as the phenomenological Nambu -Jona-Lasinio model [36].This is actually not too surprising because both am-plitudes,b 0and b 1,are controlled by chiral symmetry and its explicit breaking [33,37].

Pions at ?nite momenta,on the other hand,interact very strongly with nuclear matter through the p-wave interaction,which is dominated by the P 33delta-resonance.This leads to a strong mixing of the pion with nuclear excitations such as a particle-hole and a delta-hole exci-tation.As a result the pion-like excitation spectrum develops several branches,which to leading order correspond to the pion,particle-hole,and delta-hole excitations.Because of the attractive p-wave interaction the dispersion relation of the pionic branch becomes considerably softer than that of a free pion.A simple model,which has been used in prac-tical calculations [38,39]is the so-called delta-hole model [33,40]which concentrates on the stronger pion-nucleon-delta interaction ignoring the nucleon-hole excitations.In this model,the pion dispersion relation in nuclear medium can be written as

ω(k ,ρ)=m 2π+k 2+Π(ω,k ),(4)where the pion self-energy is given by

Π(ω,k )=k 2χ(ω,k )

9 f πN ?

ω2?ω2R exp ?2k 2/b 2

ρ,(6)

where f πN ?

≈2is the pion-nucleon-delta coupling constant,b ≈7m πis the range of the form factor,and ωR ≈k 2

8KO,KOCH&LI The pion dispersion relation obtained in this model is shown in Fig.

1.The pion branch in the lower part of the?gure is seen to become softened,while the delta-hole branch in the upper part of the?gure is sti?ened.Naturally this model oversimpli?es things and once couplings to nucleon-hole excitations and corrections to the width of the delta are consistently taken into account,the strength of the delta-hole branch is signi?cantly reduced[42,43,44,45,46,47].However,up to momenta of about2mπthe pionic branch remains pretty narrow and considerably softer than that of a free pion.The dispersion relation of the pion can be measured directly using(3He,t)charge exchange reactions[48].In these experiments the production of coherent pion have been inferred from angular correlations between the outgoing pion and the transferred momentum.The corresponding energy transfer is smaller than that of free pions indicating the attraction in the pion branch discussed above. More detailed measurements of this type are currently being analyzed [49].

Figure1:Pion dispersion relation in the nuclear medium.The normal nuclear matter density is denoted byρ0.

In the context of relativistic heavy ion collisions interest in the in-medium modi?ed pion dispersion relation got sparked by the work of

IN-MEDIUM EFFECTS9 Gale and Kapusta[50].Since the density of states is proportional to the inverse of the group velocity of these collective pion modes,which can actually vanish depending on the strength of the interaction(see Fig.1),they argued that a softened dispersion relation would lead to a strong enhancement of the dilepton yield.This happens at invariant mass close to twice the pion mass because the number of pions with low energy will be considerably increased.However,as pointed out by Korpa and Pratt[51]and further explored in more detail in[52],the initially proposed strong enhancement is reduced considerably once cor-rections due to gauge invariance are properly taken into account.Other places,where the e?ect of the pion dispersion relation is expected to play a role,are the inelastic nucleon-nucleon cross section[38]and the shape of the pion spectrum[39,55,56].In the latter case one?nds[39] that as a result of the attractive interaction the yield of pions at low p t is enhanced by about a factor of2.This low p t enhancement is indeed seen in experimental data forπ?production from the BEVALAC[57] and in more recent data from the TAPS collaboration[58],which mea-sures neutral pions.Certainly,in order to be conclusive,more re?ned calculations are needed,and they are presently being carried out[47]. Finite temperature

The pionic properties at?nite temperature instead of?nite density are qualitatively very similar,although the pion now interacts mostly with other pions instead of nucleons.Again,due to chiral symmetry,the s-wave interaction among pions is small and slightly repulsive,leading to a small mass shift of the pion[59].

And analogous to the case of nuclear matter,there is a strong at-tractive p-wave interaction,which is now dominated by theρ-resonance. This again results in a softened dispersion relation which,however,is not as dramatic as that obtained at?nite density[60,61,62,63].Also the phenomenological consequences are similar.The modi?ed dispersion relation results in an enhancement of dileptons from the pion annihila-tion by a factor of about two at invariant masses around300?400MeV [64,65,66].Unfortunately,in the same mass range other channels dom-inate the dilepton spectrum(see discussion in section3.1)so that this enhancement cannot be easily observed in experiment[65].The mod-i?ed dispersion relation has also been invoked in order to explain the enhancement of low transverse momentum pions observed at CERN-SPS heavy ion collisions[61,67,68].However,at these high energies the expansion velocity of the system,which is mostly made out of pions,

10KO,KOCH&LI is too fast for the attractive pion interaction to a?ect the pion spectrum at low transverse momenta[62].This is di?erent at the lower energies around1GeV.There,the dominant part of the system and the source for the pion potential are the nucleons,which move considerably slower than pions.Therefore,pions have a chance to leave the potential well before it has disappeared as a result of the expansion.

To summarize this section on pions,the e?ects for the pions at?nite density as well as?nite temperature are dominated by p-wave reso-nances,the?(1230)and theρ(770),respectively.Both are not related directly to chiral symmetry and its restoration but rather to what we call here many-body e?ects,which of course does not make them less interesting.The s-wave interaction is small because the pion is such a ‘good’Goldstone Boson;probably too small to have any phenomenolog-ical consequence(at least for heavy ion collisions).

2.2Kaons

Contrary to the pion,the kaon is not such a good Goldstone boson. E?ects of the explicit chiral symmetry breaking are considerably bigger, as one can see from the mass of the kaon,which is already half of the typical hadronic mass scale of1GeV.In addition,since the kaon carries strangeness,its behavior in non-strange,isospin symmetric matter will be di?erent from that of the pion.Rather interesting phenomenological consequences arise from this di?erence such as a possible condensation of antikaons in neutron star matter[69,70,71,72].

Chiral Lagrangian

This di?erence can best be exempli?ed by studying the leading order e?ective SU(3)L×SU(3)R Lagrangian obtained in heavy baryon chiral perturbation theory[72,73]

L0=f22r Tr M q(U+U+?2)

+TrˉBi vμDμB+2D TrˉBSμ{Aμ,B}

+2F TrˉBSμ[Aμ,B].(7) In the above formula,we have U=exp(2iπ/f)withπand f being the pseudoscalar meson octet and their decay constant,respectively;B is the baryon octet;vμis the four velocity of the heavy baryon(v2=1); and Sμstands for the spin operator Sμ=1

IN-MEDIUM EFFECTS 11M q is the quark mass matrix;and r ,D as well as F are empirically determined constants.Furthermore,

D μB =?μB +[V μ,B ](8)V μ=12(ξ?μξ??ξ??μξ),(9)with ξ2=U .

To leading order in the chiral counting the explicit symmetry break-ing term ～Tr M q (U +U +

?2)gives rise to the masses of the Goldstone bosons.The interesting di?erence between the behavior of pions and kaons in

matter arises from the term involving the vector current V μ,

i.e.Tr ˉBiv μD μB .In case of the pion,this term is identical to the

well-known Weinberg-Tomozawa term [37]

δL W T =?1

8f 2 3(ˉNγμN )(K ??μK )+(ˉN τγμN )(K τ?

?μK ) .(11)

The ?rst term contributes to the isoscalar s-wave amplitude and,there-fore,gives rise to an attractive or repulsive optical potential for K ?and K +in symmetric nuclear matter.It turns out,however,that this lead-ing order Lagrangian leads to an s-wave scattering length which is too repulsive as compared with experiment.Therefore,terms next to lead-ing order in the chiral expansion are needed.Some of these involve the kaon-nucleon sigma term and thus are sensitive to the second di?erence between pions and kaons,namely the strength of the explicit symmetry breaking.The next to leading order e?ective kaon-nucleon Lagrangian can be written as [72]

L ν=2=ΣKN

f 2 ˉN τN · ˉK τK

+?

D f 2 ˉN τN · ?t ˉK τ?t K .(12)

The value of the kaon-nucleon sigma-term ΣKN =1

12KO,KOCH&LI / N|ˉu u+ˉdd|N ≈0.1?645ac6fb941ea76e58fa04a3ing the light quark mass ratio m s/m≈29,one obtains370<ΣKN<405MeV.The additional parameters, C,?D,?D′are then?xed by comparing with K+-nucleon scattering data [72].This so determined e?ective Lagrangian can then be used to pre-dict the K?-nucleon scattering amplitudes,and one obtains an attrac-tive isoscalar s-wave scattering length in contradiction with experiments, where one?nds a repulsive amplitude[74].This discrepancy has been attributed to the existence of theΛ(1405)which is located below the K?N-threshold.From the analysis of K?p→Σπreactions it is known that this resonance couples strongly to the I=0K?p state,and,there-fore,leads to repulsion in the K?p amplitude.

The Role of theΛ(1405)

Already in the sixties[75]there have been attempts to understand the Λ(1405)as a bound state of the proton and the K?.In this picture, the underlying K?-proton interaction is indeed attractive as predicted by the chiral Lagrangians but the scattering amplitude is repulsive only because a bound state is formed.This concept is familiar to the nuclear physicist from the deuteron,which is bound,because of the attractive interaction between proton and neutron.The existence of the deuteron then leads to a repulsive scattering length in spite of the attractive interaction between neutron and proton.Chiral perturbation theory is based on a systematic expansion of the S-matrix elements in powers of momenta and,therefore,e?ects which are due to the proximity of a resonance,such as theΛ(1405),will only show up in terms of rather high order in the chiral counting.Thus,it is not too surprising that the?rst two orders of the chiral expansion predict the wrong sign of the K?N amplitude.To circumvent this problem,the chiral perturbation calculation has been extended to either include an explicitΛ(1405)state [76]or to use the interaction obtained from the leading order chiral Lagrangian as a kernel for a Lippman-Schwinger type calculation,which is then solved to generate a bound stateΛ(1405)[77].

This picture of theΛ(1405)as a K?p bound state has recently re-ceived some considerable interest in the context of in-medium correc-tions.In ref.[78]it has been pointed out that in this picture as a result of the Pauli-blocking of the proton inside this bound state,the properties of theΛ(1405)would be signi?cantly changed in the nuclear environ-ment.With increasing density,its mass increases and the strength of the resonance is reduced(see Fig.2).Because of this shifting and‘dis-appearance’of theΛ(1405)in matter,the K?optical potential changes

IN-MEDIUM EFFECTS13 sign from repulsive to attractive at a density of about1/4of nuclear matter density(see Fig.2)in agreement with a recent analysis of K?atoms[79].These?ndings have been con?rmed in ref.[80].This in-medium change of theΛ(1405)due to the Pauli blocking can only occur if a large fraction of its wave function is indeed that of a K?-proton bound state.Of course one could probably allow for a small admixture of a genuine three quark state without changing the results for the mea-sured kaon potentials.But certainly,if theΛ(1405)is mostly a genuine three quark state,the Pauli blocking should not a?ect its properties. Therefore,it would be very interesting to con?rm the mass shift of the Λ(1405)for instance by a measurement of the missing mass spectrum of kaons in the reaction p+γ→Λ(1405)+K+at CEBAF.Thus,the atomic nucleus provides a unique laboratory to investigate the proper-ties of elementary particles.From our discussion it is clear that this in-medium change of theΛ(1405)is not related to the restoration of chiral symmetry.

Experimental results

Phenomenologically,the attractive optical potential for the K?in nu-clear matter is of particular interest because it can lead to a possible kaon condensation in neutron stars[72,81].This would limit the max-imum mass of neutron stars to about one and a half solar masses and give rise to speculations about many small black holes in our galaxy[82]. Kaonic atoms,of course,only probe the very low density behavior of the kaon optical potential and,therefore,an extrapolation to the large densities relevant for neutron stars is rather uncertain.Additional in-formation about the kaon optical potential can be obtained from heavy ion collision experiments,where densities of more than twice nuclear matter density are reached.

Observables that are sensitive to the kaon mean-?eld potentials are the subthreshold production[83,84,85]as well as kaon?ow[86,87]. Given an attractive/repulsive mean-?eld potential for the kaons it is clear that the subthreshold production is enhanced/reduced.In case of the kaon?ow an attractive interaction between kaons and nucleons aligns the kaon?ow with that of the nucleons whereas a repulsion leads to an anti-alignment(anti-?ow).However,both observables are also extremely sensitive to the overall reaction dynamics,in particular to the properties of the nuclear mean?eld and to reabsorption processes especially in case of the antikaons.Therefore,transport calculations are required in order to consistently incorporate all these e?ects.

14KO,KOCH &LI

0.0 1.0

2.0

ρ / ρ0?0.2?0.1

0.0

U o p t [G e V ]Re(U)Im(U)1.38 1.48

E c.m. [GeV]

0.02.0

4.06.0

I m f (I =0) [f m ]ρ = 0ρ = 0.5 ρ0ρ = ρ0Figure 2:(a)Imaginary part of the I =0K ?-proton scattering am-plitude for di?erent densities.(b)real and Imaginary part of the K ?optical potential.

IN-MEDIUM EFFECTS15 In Fig.3we show the result obtained from such calculations for the K+subthreshold production and?ow together with experimental data(solid circles)from the KaoS collaboration[88]and from the FOPI collaboration[89]at GSI.Results are shown for calculations without any mean?elds(dotted curves)as well as with a repulsive mean?eld obtained from the chiral Lagrangian(solid curves),which is consistent with a simple impulse approximation.The experimental data are nicely reproduced when the kaon mean-?eld potential is included.Although the subthreshold production of kaons can also be explained without any kaon mean?eld[90,91],the assumption used in these calculations that a lambda particle has the same mean-?eld potential as a nucleon is not consistent with the phenomenology of hypernuclei[92].Since it is undisputed that these observables are sensitive to the in-medium kaon potential,a systematic investigation including all observables should eventually reveal more accurately the strength of the kaon potential in dense matter.Additional evidence for a repulsive K+potential comes from K+-nucleus experiments.There the measured cross sections and angular distributions can be pretty well understood within a simple impulse approximation.Actually it seems that an additional(15%) repulsion is required to obtain an optimal?t to the data[93],which has been suggested as a possible evidence for a swelling of the nucleon size or a lowering of the omega meson mass in the nuclear medium[94].

As for the K?,both chiral perturbation theory and dynamical mod-els of the K?-nucleon interaction[78]indicate the existence of an attrac-tive mean?eld,so the same observables can be used[95,96].However, the measurement and interpretation of K?observables is more di?cult since it is produced less abundantly.Also reabsorption e?ects due to the reaction K?N→Λπare strong,which further complicates the analysis. Nevertheless,the e?ect of the attractive mean?eld has been shown to be signi?cant as illustrated in Fig.4,where results from transport cal-culations with(solid curves)or without(dotted curves)attractive mean ?eld for the antikaons[95]are shown.It is seen that the data(solid circles)on subthreshold K?production[97]support the existence of an attractive antikaon mean-?eld potential.For the K??ow,there only exist very preliminary data from the FOPI collaboration[98],which seem to show that the K?’s have a positive?ow rather an anti?ow, thus again consistent with an attractive antikaon mean-?eld potential.

Let us conclude this section on kaons by pointing out that the prop-erties of kaons in matter are qualitatively described in chiral perturba-tion theory.Although some of the e?ects comes from the Weinberg-Tomozawa vector type interaction,which contributes because contrary

16KO,KOCH&LI

Figure3:Kaon yield(left panel)and?ow(right panel)in heavy ion collisions.The solid and dotted curves are results from transport model calculations with and without kaon mean-?eld potential,respectively. The data are from refs.[88,89].

IN-MEDIUM EFFECTS17

Figure4:Antikaon yield(left panel)and?ow(right panel)in heavy ion collisions.The solid and dotted curves are results from transport model calculations with and without kaon mean-?eld potential,respectively. The data are from ref.[97].

18KO,KOCH&LI to the pion the kaon has only one light quark,in order to explain the ex-perimental data on kaon subthreshold production and?ow in heavy ion collisions an additional attractive scalar interaction is required,which is related to the explicit breaking of chiral symmetry and higher order corrections in the chiral expansion.

2.3Etas

The properties of etas are not only determined by consideration of chiral symmetry but also by the explicit breaking of the U A(1)axial symmetry due to the axial anomaly in QCD.As a result,the singlet eta,which would be a Goldstone boson if U A(1)were not explicitly broken,becomes heavy.Furthermore,because of SU(3)symmetry breaking–the strange quark mass is considerably heavier than that of up and down quark –the octet eta and singlet eta mix,leading to the observed particles ηandη′.The mixing is such that theη′is mostly singlet and thus heavy,and theηis mostly octet and therefore has roughly the mass of the kaons.If,as has been speculated[99,100],the U A(1)symmetry is restored at high temperature due to the instanton e?ects,one would expect considerable reduction in the masses of the etas as well as in their mixing[101,102].A dropping eta meson in-medium mass is expected to provide a possible explanation for the observed enhancement of low transverse momentum etas in SIS heavy ion experiments at subthreshold energies[103].However,an analysis of photon spectra from heavy ion collisions at SPS-energies puts an upper limit on theη/π0ratio to be not more than20%larger for central than for peripheral collisions[104]. This imposes severe constraints on the changes of theηproperties in hadronic matter.

We note that the restoration of chiral symmetry is important in theη?η′sector,because without the U A(1)breaking both would be Goldstone bosons of an extended U(3)×U(3)symmetry.However,it is still being debated what the e?ects precisely are.

3VECTOR MESONS

Of all particles it is probably theρ-meson which has received the most attention in regards of in-medium corrections.This is mainly due to the fact that theρis directly observable in the dilepton invariant mass spec-trum.Also,since theρcarries the quantum numbers of the conserved vector current,its properties are related to chiral symmetry and can,

IN-MEDIUM EFFECTS19 as we shall discuss,be investigated using e?ective chiral models as well as current algebra and QCD sum rules.That possible changes of theρcan be observed in the dilepton spectrum measured in heavy ion colli-sions has been?rst demonstrated in ref.[105]and then studied in more detail in[106,107].The in-medium properties of the a1are closely related to that of theρsince they are chiral partners and their mass di?erence in vacuum is due to the spontaneous breaking of chiral sym-metry[108].Unfortunately,there is no direct method to measure the changes of the a1in hadronic matter.Theω-meson on the other hand is a chiral singlet,and the relation of its properties to chiral symmetry is thus not so direct.However,calculations based on QCD sum rules predict also changes in theωmass as one approaches chiral restoration. Observationally,theωcan probably be studied best due to its rather small width and its decay channel into dileptons.Finally,there is the φmeson.In an extended SU(3)chiral symmetry,theφand theωare a superposition of the singlet and octet states with nearly perfect mixing, i.e.theφcontains only strange quarks whereas theωis made only out of light quarks.Both QCD sum rules[109]and e?ective chiral models [110]predict a lowering of theφ-meson mass in medium.

The question on whether or not masses of light hadrons change in the medium has received considerable interest as a result of the conjecture of Brown and Rho[8],which asserts that the masses of all mesons,with the exception of the Goldstone Bosons,should scale with the quark con-densate.While the detailed theoretical foundations of this conjecture are still being worked on[111,112,113]the basic argument of Brown and Rho is as follows.Hadron masses,such as that of theρ-meson, violate scale invariance,which is a symmetry of the classical QCD La-grangian.In QCD scale invariance is broken on the quantum level by the so-called trace anomaly(see e.g.[114]),which is proportional to the Gluon condensate.Thus one could imagine that with the disappear-ance of the gluon condensate,i.e.the bag pressure,scale invariance is restored,which on the hadronic level implies that hadron masses have to vanish.Therefore,one could argue that hadron masses should scale with the bag-pressure as originally proposed by Pisarski[115].But the conjecture of Brown and Rho goes even further.They assume that the gluon condensate can be separated into a hard and soft part,the lat-ter of which scales with the quark condensate and is also responsible for the masses of the light hadrons.This picture?nds some support from lattice QCD calculations in that the gluons condensate drops by about50%at the chiral phase transition[116].To what extent this is also re?ected in changes in hadron masses in not clear at the moment

20KO,KOCH&LI (see section5),although one should mention that the rise in the en-tropy density close to the critical temperature can be explained if one assumes the hadron masses to scale with the quark condensate[116]. Another aspect of the Brown and Rho scaling is that once the scaling hadron masses are introduced in the chiral Lagrangian,only tree-level diagrams are needed as the contribution from higher order diagrams is expected to be suppressed.In their picture,Goldstone bosons are not subject to this scaling,since they receive their mass from the explicit chiral symmetry breaking due to?nite current quark masses,which are presumably generated at a much higher(Higgs)scale.

3.1The rho meson

Since theρis a vector meson,it couples directly to the isovector current which then results in the direct decay of theρinto virtual photons,i.e. dileptons.Consequently,properties of theρmeson can be investigated by studying two-point correlation functions of the the isovector currents, i.e.,

Πμν(q)=i e iqx T Jμ(x)Jν(0) ρd4x.(13)

The masses of the rho meson and its excitations(ρ′...)correspond to the positions of the poles of this correlation function.This is best seen if one assumes that the current?eld identity[117]holds,namely that the current operator is proportional to theρ-meson?eld.In this case the above correlation function is identical,up to a constant,to the ρ-meson propagator.The imaginary part of this correlation function is also directly proportional to the electron-positron annihilation cross section[118],where theρ-meson is nicely seen.In addition,at higher center-of-mass energies,one sees a continuum in the electron-positron annihilation cross section,which corresponds to the excited states of the rho mesons as well as to the onset of perturbative QCD processes.

In-medium changes of theρmeson can be addressed theoretically by evaluating the current-current correlator in the hadronic environment. The current-current correlator can be evaluated either in e?ective chi-ral models,or using current algebra arguments,or directly in QCD.In the latter,one evaluates the correlator in the deeply Euclidean region (q2→?∞)using the Wilson expansion,where all the long distance physics is expressed in terms of vacuum expectation values of quark and gluon operators,the so-called condensates.Dispersion relations are then

IN-MEDIUM EFFECTS21 used to relate the correlator in the Euclidean region to that for posi-tive q2,where the hadronic eigenstates are located.One then assumes a certain shape for the phenomenological spectral functions,typically a delta function,which represents the bound state,and a continuum, which represents the perturbative regime.These so called QCD sum rules,therefore,relate the observable hadronic spectrum with the QCD vacuum condensates(for a review of the QCD sum-rule techniques see e.g.[119]).These relations can then either be used to determine the condensates from measured hadronic spectra,or,to make predictions about changes of the hadronic spectrum due to in-medium changes of the condensates.

Similarly,one can study the properties of a1meson in the nuclear medium through the axial vector correlation function.Then,once chiral symmetry is restored,there should be no observable di?erence between left-handed and right-handed or equivalently vector and axial vector currents.Consequently,the vector and axial vector correlators should be identical.Often,this identity of the correlators is identi?ed with the degeneracy of theρand a1mesons in a chirally symmetric world.This, however,is not the only possibility,as was pointed out by Kapusta and Shuryak[120].There are at least three qualitatively di?erent scenarios, for which the vector and axial vector correlator are identical(see Fig.

5).

C C

C

(2)

(1)(3) Figure5:Several possibilities for the vector and axial-vector spectral functions in the chirally restored phase.

1.The masses ofρand a1are the same.The value of the common

mass,however,does not follow from chiral symmetry arguments alone.

2.The mixing of the spectral functions,i.e,both the vector and

axial-vector spectral functions have peaks of similar strength at both the mass of theρand the mass of the a1.

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