arXiv:nucl-th/9702016v1 7 Feb 1997 o n es arncmte,hayincollisions heavy-ion matter, hadronic dense and hot words key 11 N.Y. Brook, Stony Brook, Stony USA at York New of University State Physics of Department Li US 94720, Guoqiang CA Berkeley, Laboratory, National Berkeley Lawrence Division Science Nuclear Koch USA Volker 77843, Texas Station, College University, A&M Department Texas Physics and Institute Cyclotron Ko Ming Che MEDIUM NUCLEAR THE IN HADRONS OF PROPERTIES 3 2 1 e-mail:[email protected] e-mail:[email protected] e-mail:[email protected] osqec n eut ilb reviewed. be nu experime will the the effect results and in-medium and phi), proposed consequence (pion omega, each (rho, for bosons Also, mesons particul Goldstone vector cleon. In the the on eta), condensate. effects and chiral kaon medium the discuss of shall change we the f to disentang interactions due to many-body those conventional given restora to is partial due attention the effects to medium Special relation symmetry. its chiral and of medium nuclear the in hsrve sdvtdt h icsino arnpropertie hadron of discussion the to devoted is review This hrlsmer,Glsoebsn,vco eos baryons, mesons, vector bosons, Goldstone symmetry, Chiral : 2 3 1 Abstract LBNL-39866 ein- le tion rom ntal ar, s - , A 794, 2 KO, KOCH & LI
Contents
1 INTRODUCTION 2
2 GOLDSTONE BOSONS 5 2.1 The pion ...... 6 2.2 Kaons ...... 10 2.3 Etas ...... 18
3 VECTOR MESONS 18 3.1 The rho meson ...... 20 3.2 The omega meson ...... 29 3.3 The phi meson ...... 29
4 BARYONS 31
5 RESULTS FROM LATTICE QCD CALCULATIONS 32
6 SUMMARY 35
1 INTRODUCTION
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, < qq¯ >= 0. It is believed and supported by lattice QCD calculations [1] that6 at temperatures around 150MeV, QCD undergoes a phase transition to a chirally restored phase, characterized by the vanishing of the order parameter, the chiral condensate < qq¯ >. This is supported by results obtained within chiral perturbation theory [2, 3]. Effective chiral models predict that a similar transition also takes place at finite nuclear density. The only way to create macroscopic, strongly interacting systems at finite temperature and/or density in the laboratory is by colliding heavy nuclei at high energies. Experiments carried out at various bombarding energies, ranging from 1 AGeV (BEVALAC, SIS) to 200 AGeV (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 EFFECTS 3
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 finite density. At nuclear density the quark condensate is estimated to be reduced by about 30% [2, 3, 4, 5] so that effects 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. Confinement 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 < qq¯ >. 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 effects 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 affected by final-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 effects. 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 effects such as the Pauli principle are at work, which, as we shall discuss, lead to considerable modifica- tions of hadronic properties in some cases. How these effects are related to the partial restoration of chiral symmetry is a new and unsolved ques- 4 KO, KOCH & LI
tion in nuclear many-body physics. One example is the effective 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 finite 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-field description one also arrives at a reduced effective 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 finite 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 effects 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 difficult 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 data. Compu- 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 effects have been found to be important at least at energies up to 2 AGeV. At higher energies there seems to be a chance that one can separate the reaction in an initial hard scattering phase and a final 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 different observables. IN-MEDIUM EFFECTS 5
This review is devoted to the discussion of in-medium effects 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 deconfinement in the Quark Gluon Plasma, namely the suppression of the J/Ψ. This idea, which has been first 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/Ψ is not bound anymore. As a result, if such a Quark Gluon Plasma is formed in a relativistic heavy ion collision, the abundance of J/Ψ should be considerably reduced. Of course also this signal suffers from more conventional backgrounds, namely the dissociation of the J/Ψ due 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 effects on hadrons. Whenever possible, we will try to disentangle conventional in- medium effects from those we believe are due to new physics, namely the change of the chiral condensate. We first 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 effective 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 field 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.
2 GOLDSTONE BOSONS
This first part is devoted to in-medium effects of Goldstone bosons, specifically 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 6 KO, 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 finite density as we shall discuss below.
2.1 The 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 finite 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 finite 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 finds to first order in the nuclear density ρ
2 −1 ∆m = 4π(b0)eff ρ, (b0)eff 0.024m . (1) π − ≃− π At nuclear matter density the shift of the pion mass is ∆m π 8%, (2) mπ ∼ which is small as expected from low energy theorems based on chi- ral symmetry. Theoretically, the value of (b0)eff is well understood as the combined contribution from a single scattering (impulse approxi- mation) involving the small s-wave isoscalar πN scattering length b0 = IN-MEDIUM EFFECTS 7
−1 0.010(3)mπ and the contribution from a (density dependent) corre- lation− or re-scattering term, which is dominated by the comparatively −1 large isovector-s-wave scattering length b1 = 0.091(2)m , i.e., − π mπ 2 2 1 (b0)eff = b0 [1 + (b0 +2b1)] < >, (3) − MN r where the correlation length < 1/r > 3pf /(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, b0 and b1, are controlled by chiral symmetry and its explicit breaking [33, 37]. Pions at finite momenta, on the other hand, interact very strongly with nuclear matter through the p-wave interaction, which is dominated by the P33 delta-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
2 2 ω(k,ρ)= mπ + k + Π(ω, k), (4) where the pion self-energy is given by k2χ(ω, k) Π(ω, k)= , (5) 1 g′χ(ω, k) − with g′ 0.6 the Migdal parameter which accounts for short-range correlations,≈ also known as the Ericson-Ericson – Lorentz-Lorenz effect [41]. The pion susceptibility χ is given by
2 8 f ∆ ω χ(ω, k) πN R exp 2k2/b2 ρ, (6) ≈ 9 m ω2 ω2 − π − R where fπN∆ 2 is the pion-nucleon-delta coupling constant, b 7mπ ≈ k2 ≈ is the range of the form factor, and ω + m∆ m . R ≈ 2m∆ − N 8 KO, KOCH & LI
The pion dispersion relation obtained in this model is shown in Fig. 1. The pion branch in the lower part of the figure is seen to become softened, while the delta-hole branch in the upper part of the figure is stiffened. Naturally this model oversimplifies 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 significantly reduced [42, 43, 44, 45, 46, 47]. However, up to momenta of about 2mπ the 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].
Figure 1: Pion dispersion relation in the nuclear medium. The normal nuclear matter density is denoted by ρ0.
In the context of relativistic heavy ion collisions interest in the in- medium modified pion dispersion relation got sparked by the work of IN-MEDIUM EFFECTS 9
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 effect 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 finds [39] that as a result of the attractive interaction the yield of pions at low pt is enhanced by about a factor of 2. This low pt enhancement is indeed seen in experimental data for π− 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 refined calculations are needed, and they are presently being carried out [47].
Finite temperature
The pionic properties at finite temperature instead of finite 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 the ρ-resonance. This again results in a softened dispersion relation which, however, is not as dramatic as that obtained at finite density [60, 61, 62, 63]. Also the phenomenological consequences are similar. The modified dispersion relation results in an enhancement of dileptons from the pion annihila- tion by a factor of about two at invariant masses around 300 400 MeV [64, 65, 66]. Unfortunately, in the same mass range other channels− dom- inate the dilepton spectrum (see discussion in section 3.1) so that this enhancement cannot be easily observed in experiment [65]. The mod- ified 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, 10 KO, KOCH & LI
is too fast for the attractive pion interaction to affect the pion spectrum at low transverse momenta [62]. This is different at the lower energies around 1 GeV. 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 effects for the pions at finite density as well as finite temperature are dominated by p-wave reso- nances, the ∆(1230) and the ρ(770), respectively. Both are not related directly to chiral symmetry and its restoration but rather to what we call here many-body effects, 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).
2.2 Kaons Contrary to the pion, the kaon is not such a good Goldstone boson. Effects 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 of 1 GeV. In addition, since the kaon carries strangeness, its behavior in non-strange, isospin symmetric matter will be different from that of the pion. Rather interesting phenomenological consequences arise from this difference such as a possible condensation of antikaons in neutron star matter [69, 70, 71, 72].
Chiral Lagrangian
This difference can best be exemplified by studying the leading order effective SU(3)L SU(3)R Lagrangian obtained in heavy baryon chiral perturbation theory× [72, 73]
2 2 f µ + f + 0 = Tr ∂ U∂ U + r Tr M (U + U 2) L 4 µ 2 q − µ µ +Tr Bi¯ vµ B +2D Tr BS¯ Aµ,B Dµ { } +2F Tr BS¯ [Aµ,B]. (7)
In the above formula, we have U = exp(2iπ/f) with π and f being the pseudoscalar meson octet and their decay constant, respectively; B is 2 the baryon octet; vµ is the four velocity of the heavy baryon (v = 1); 1 µ and Sµ stands for the spin operator Sµ = 4 γ5[vν γν ,γµ] and vµS = 0; IN-MEDIUM EFFECTS 11
Mq is the quark mass matrix; and r, D as well as F are empirically determined constants. Furthermore, = ∂ B + [V ,B] (8) DµB µ µ 1 i V = (ξ∂ ξ† + ξ†∂ ξ), A = (ξ∂ ξ† ξ†∂ ξ), (9) µ 2 µ µ µ 2 µ − µ with ξ2 = U. To leading order in the chiral counting the explicit symmetry break- + ing term Tr Mq(U + U 2) gives rise to the masses of the Goldstone bosons. The∼ interesting difference− between the behavior of pions and kaons in matter arises from the term involving the vector current Vµ, µ i.e. Tr Biv¯ µ B. In case of the pion, this term is identical to the well-known Weinberg-TomozawaD term [37]
1 ¯ µ δ W T = − 2 (N~τγ N) (~π (∂µ~π)) . (10) L 4fπ · × It contributes only to the isovector s-wave scattering amplitude and, therefore, does not contribute to the pion optical potential in isospin symmetric nuclear matter. This is different in case of the kaon, where we have i ↔ ↔ δ = − 3(Nγ¯ µN)(K ∂ K) + (N~τγ¯ µN)(K~τ ∂ K) . (11) LW T 8f 2 µ µ The first 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 difference between pions and kaons, namely the strength of the explicit symmetry breaking. The next to leading order effective kaon-nucleon Lagrangian can be written as [72]
ΣKN C =2 = NN¯ (KK¯ )+ N~τN¯ K~τK¯ Lν f 2 f 2 · D˜ D˜ ′ + NN¯ (∂ K∂¯ K)+ N~τN¯ ∂ K~τ∂¯ K . (12) f 2 t t f 2 · t t 1 The value of the kaon-nucleon sigma-term ΣKN = 2 (mq + ms) N uu¯ + ss¯ N depends on the strangeness content of the nucleon, y =2 Nh ss¯| N | i h | | i 12 KO, KOCH & LI
/ N uu¯ + dd¯ N 0.1 0.2. Using the light quark mass ratio m /m h | | i ≈ − s ≈ 29, one obtains 370 < ΣKN < 405MeV. The additional parameters, C, D,˜ D˜ ′ are then fixed by comparing with K+-nucleon scattering data [72]. This so determined effective 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 finds a repulsive amplitude [74]. This discrepancy has been attributed to the existence of the Λ(1405) which is located below the K−N-threshold. From the analysis of K−p Σπ reactions it is known that this resonance couples strongly to the I→=0 K−p state, and, there- fore, leads to repulsion in the K−p amplitude.
The Role of the Λ(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, effects which are due to the proximity of a resonance, such as the Λ(1405), will only show up in terms of rather high order in the chiral counting. Thus, it is not too surprising that the first 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 explicit Λ(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 state Λ(1405) [77]. This picture of the Λ(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 the Λ(1405) would be significantly 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 the Λ(1405) in matter, the K− optical potential changes IN-MEDIUM EFFECTS 13
sign from repulsive to attractive at a density of about 1/4 of nuclear matter density (see Fig. 2) in agreement with a recent analysis of K− atoms [79]. These findings have been confirmed in ref. [80]. This in- medium change of the Λ(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 the Λ(1405) is mostly a genuine three quark state, the Pauli blocking should not affect its properties. Therefore, it would be very interesting to confirm 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 the Λ(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-field potentials are the subthreshold production [83, 84, 85] as well as kaon flow [86, 87]. Given an attractive/repulsive mean-field potential for the kaons it is clear that the subthreshold production is enhanced/reduced. In case of the kaon flow an attractive interaction between kaons and nucleons aligns the kaon flow with that of the nucleons whereas a repulsion leads to an anti-alignment (anti-flow). However, both observables are also extremely sensitive to the overall reaction dynamics, in particular to the properties of the nuclear mean field and to reabsorption processes especially in case of the antikaons. Therefore, transport calculations are required in order to consistently incorporate all these effects. 14 KO, KOCH & LI
6.0 ρ = 0 ρ ρ = 0.5 0 ρ = ρ 4.0 0
2.0 Im f(I=0) [fm]
0.0 1.38 1.48
Ec.m. [GeV]
0.0 Re(U) Im(U)
[GeV] −0.1 opt U
−0.2 0.0 1.0 2.0 ρ ρ / 0
Figure 2: (a) Imaginary part of the I = 0 K−-proton scattering am- plitude for different densities. (b) real and Imaginary part of the K− optical potential. IN-MEDIUM EFFECTS 15
In Fig. 3 we show the result obtained from such calculations for the K+ subthreshold production and flow 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 fields (dotted curves) as well as with a repulsive mean field 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-field potential is included. Although the subthreshold production of kaons can also be explained without any kaon mean field [90, 91], the assumption used in these calculations that a lambda particle has the same mean-field 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 fit 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 field, so the same observables can be used [95, 96]. However, the measurement and interpretation of K− observables is more difficult since it is produced less abundantly. Also reabsorption effects due to the reaction K−N Λπ are strong, which further complicates the analysis. Nevertheless, the→ effect of the attractive mean field has been shown to be significant as illustrated in Fig. 4, where results from transport cal- culations with (solid curves) or without (dotted curves) attractive mean field 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-field potential. For the K− flow, there only exist very preliminary data from the FOPI collaboration [98], which seem to show that the K−’s have a positive flow rather an antiflow, thus again consistent with an attractive antikaon mean-field 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 effects comes from the Weinberg- Tomozawa vector type interaction, which contributes because contrary 16 KO, KOCH & LI
Figure 3: Kaon yield (left panel) and flow (right panel) in heavy ion collisions. The solid and dotted curves are results from transport model calculations with and without kaon mean-field potential, respectively. The data are from refs. [88, 89]. IN-MEDIUM EFFECTS 17
Figure 4: Antikaon yield (left panel) and flow (right panel) in heavy ion collisions. The solid and dotted curves are results from transport model calculations with and without kaon mean-field potential, respectively. The data are from ref. [97]. 18 KO, 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 flow 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.
2.3 Etas The properties of etas are not only determined by consideration of chiral symmetry but also by the explicit breaking of the UA(1) axial symmetry due to the axial anomaly in QCD. As a result, the singlet eta, which would be a Goldstone boson if UA(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 η and η′. The mixing is such that the η′ is mostly singlet and thus heavy, and the η is mostly octet and therefore has roughly the mass of the kaons. If, as has been speculated [99, 100], the UA(1) symmetry is restored at high temperature due to the instanton effects, 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 the η/π0 ratio to be not more than 20 % larger for central than for peripheral collisions [104]. This imposes severe constraints on the changes of the η properties in hadronic matter. We note that the restoration of chiral symmetry is important in ′ the η η sector, because without the UA(1) breaking both would be Goldstone− bosons of an extended U(3) U(3) symmetry. However, it is still being debated what the effects precisely× are.
3 VECTOR MESONS
Of all particles it is probably the ρ-meson which has received the most attention in regards of in-medium corrections. This is mainly due to the fact that the ρ is directly observable in the dilepton invariant mass spec- trum. Also, since the ρ carries the quantum numbers of the conserved vector current, its properties are related to chiral symmetry and can, IN-MEDIUM EFFECTS 19
as we shall discuss, be investigated using effective chiral models as well as current algebra and QCD sum rules. That possible changes of the ρ can be observed in the dilepton spectrum measured in heavy ion colli- sions has been first demonstrated in ref. [105] and then studied in more detail in [106, 107]. The in-medium properties of the a1 are closely related to that of the ρ since they are chiral partners and their mass difference 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 a1 in hadronic matter. The ω-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 the ω mass as one approaches chiral restoration. Observationally, the ω can probably be studied best due to its rather small width and its decay channel into dileptons. Finally, there is the φ meson. In an extended SU(3) chiral symmetry, the φ and the ω are a superposition of the singlet and octet states with nearly perfect mixing, i.e. the φ contains only strange quarks whereas the ω is made only out of light quarks. Both QCD sum rules [109] and effective chiral models [110] predict a lowering of the φ-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 the ρ-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 finds some support from lattice QCD calculations in that the gluons condensate drops by about 50% at the chiral phase transition [116]. To what extent this is also reflected in changes in hadron masses in not clear at the moment 20 KO, KOCH & LI
(see section 5), 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 finite current quark masses, which are presumably generated at a much higher (Higgs) scale.
3.1 The rho meson
Since the ρ is a vector meson, it couples directly to the isovector current which then results in the direct decay of the ρ into virtual photons, i.e. dileptons. Consequently, properties of the ρ meson can be investigated by studying two-point correlation functions of the the isovector currents, i.e.,
iqx 4 Πµν (q)= i e TJµ(x)Jν (0) ρd x. (13) Z h i
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 field identity [117] holds, namely that the current operator is proportional to the ρ-meson field. 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 the ρ-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 the ρ meson can be addressed theoretically by evaluating the current-current correlator in the hadronic environment. The current-current correlator can be evaluated either in effective 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 EFFECTS 21
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 a1 meson in the nuclear medium through the axial vector correlation function. Then, once chiral symmetry is restored, there should be no observable difference 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 identified with the degeneracy of the ρ and a1 mesons 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 different scenarios, for which the vector and axial vector correlator are identical (see Fig. 5).
CV,A CV,A CV,A
mρ m= mρ m a1 M 1 Ma M (1) (2) (3)
Figure 5: Several possibilities for the vector and axial-vector spectral functions in the chirally restored phase.
1. The masses of ρ and a1 are 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 the ρ and the mass of the a1. 22 KO, KOCH & LI
3. Both spectral functions could be smeared over the entire mass range. Because of thermal broadening of the mesons and the onset of deconfinement, the structure of the spectral function may be washed out, and it becomes meaningless to talk about mesonic states.
Finite temperature
At low temperatures, where the heat bath consists of pions only, one can employ current algebra as well as PCAC to obtain an essentially model- independent result for the properties of the ρ. Using this technique Dey, Eletzky and Ioffe [121] could show that to leading order in the temperature, T 2, the mass of the ρ-meson does not change. Instead one finds an admixture of the axial-vector correlator, i.e. that governed by the a1-meson. Specifically to this order the vector correlator is given by C (q,T )=(1 ǫ) C (q,T = 0)+ ǫ C (q,T = 0)+ (T 4), (14) V − V A O 2 2 with ǫ = T /(6fπ). Here CV , CA stand for the vector-isovector and axial-vector-isovector correlation functions, respectively. One should note, that to the same order the chiral condensate is reduced [59], 2 < qq¯ >T T 4 =1 2 + (T ). (15) < qq¯ >0 − 8fπ O Therefore, to leading order a drop in the chiral condensate does not affect the mass of the ρ-meson but rather reduces its coupling to the vector current and in particular induces an admixture of the a1-meson. This finding is at variance with the Brown-Rho scaling hypothesis. The reason for this difference is not yet understood. The admixture of the axial correlator is directly related with the onset of chiral restoration. If chiral symmetry is restored, the vector and axial-vector correlators should be identical. The result of Dey et al. suggests that this is achieved by a mixing of the two instead of a degeneracy of the ρ and a1 masses. If the mixing is complete, i.e. ǫ = 1/2, then the extrapolation of the low temperature result (14) would give a critical temperature of Tc = √3fπ 164 MeV , which is surprisingly close to the value given by recent lattice≃ calculations. Corrections to the order T 4 involve physics beyond chiral symmetry. As nicely discussed in ref. [122], to this order the contributions can be separated into two distinct contributions. The first arises if a pion from the heat bath couples via a derivative coupling to the current IN-MEDIUM EFFECTS 23
under consideration. In this case the contribution is proportional to the invariant pion density 3 d q −ω/T 2 nπ = e T , (16) Z 2ω(2π)3 ∼ and the square of the pion momentum q2 T 2. A typical example would be for instance the self-energy correction∼ to the ρ meson from the standard two-pion loop diagram involving the p-wave ππρ coupling. The other contribution comes from interactions of two pions from the heat bath with the current, without derivative couplings. This is pro- portional to the square of the invariant pion density and thus T 4 as well. These pure density contributions again can be evaluated in∼ a model independent fashion using current algebra techniques and, as before, do not change the mass of the ρ-meson, but change the coupling to the current and induce a mixing with the axial-vector correlator. Actually, it can be shown [122] that to all orders in the pion density these pure density effects do not change the ρ mass but induce mixing and reduce the coupling. At the same time, however, these contributions reduce the chiral condensate. The contributions due to the finite pion momenta have been esti- mated in [122] to give a downward shift of both the ρ and a1 mass of about δm 10%. (17) m ≃ for temperatures of 150 200 MeV. We should point out, however, that in this analysis∼ those− changes in the masses arise from Lorentz- nonscalar condensates in the operator product expansion. Therefore, they are not directly related to the change of the Lorentz-scalar chiral- condensate. Effective chiral models also have been used to explore the order T 4 corrections. Ref. [123] reports that the mass of the ρ drops whereas that of the a1 increases to this order. This is somewhat at variance with the findings of [124] where no drop in the mass of the ρ but a decrease in the a1 mass has been found. In this calculation, however, the terms leading to the order T 4 changes have not been explicitly identified but rather a calculation to one loop order has been carried out. Presently, the difference in these results is not understood. Also the difference between the analysis of [122] and the effective chiral models is not resolved yet. The analysis of [122] uses dispersion relations to relate phenomenological space-like photon-pion amplitudes with the time-like ones needed to 24 KO, KOCH & LI
calculate the mass shift. The effective Lagrangian methods, on the other hand, rely on pion-pion scattering data as well as measured decay width in order to fix their model parameters. One would think that both methods should give a reasonably handle on the leading order momentum-dependent couplings. At higher temperatures as well as at finite density vacuum effects due to the virtual pair correction or the nucleon-antinucleon polariza- tion could become important and they tend to reduce vector meson masses [125]. Because of the large mass of the nucleon, these corrections however, do not affect the leading temperature result T 2 discussed previously. Also this approach assumes that the physical∼ vacuum con- sists of nucleon-antinucleon rather than quark-antiquark fluctuations. Whether this is the correct picture is, however, not yet resolved.
Finite density
Since the density effect on the chiral condensate is much stronger than that of the temperature, one expects the same for the rho meson mass. But the situation is more complicated at finite density as one cannot make use of current algebra arguments and thus model-independent result as the one discussed previously are not available at this time. Present model calculations, however, disagree even on the sign of the mass shift. One class of models [126, 127, 128, 129, 130] considers the ρ as a pion-pion resonance and calculates its in-medium modifications due to those for the pions as discussed in section 2.1. These calculations typically show an increased width of the rho since it now can also decay into pion-nucleon-hole or pion-delta-hole states. At the same time this leads to an increased strength below the rho meson mass. The mass of the rho meson, defined as the position where the real part of the correlation function goes through zero, is shifted only very little. Most calculations give an upward shift but also a small downward shift has been reported [130, 131]. This difference seems to depend on the specific choice of the cutoff functions for the vertices involved [132], and thus is model dependent. However, the imaginary part of the correlation function, the relevant quantity for the dilepton measurements, hardly depends on a small upward or downward shift of the rho. The important and apparently model independent feature is the increased strength at low invariant masses due to the additional decay-channels available in nuclear matter. Calculation using QCD-sum rules [133] predict a rather strong de- crease of the ρ-mass with density ( 20% at nuclear matter density), ∼ IN-MEDIUM EFFECTS 25
which is similar to the much discussed Brown-Rho scaling [8]. Here, as in the finite temperature case, the driving term is the four quark condensate which is assumed to factorize 16 (¯qγ λaq)(¯qγµλaq) qq¯ 2. (18) h µ iρ ≈ − 9 h iρ To which extent this factorization is correct at finite density is not clear. Also, when it comes to the parameterization of the phenomenological spectral distribution, these calculations usually assume the standard pole plus continuum form with the addition of a so called Landau damp- ing contribution at q2 = 0 [133]. The additional strength below the mass of the rho as predicted by the previously discussed models is usu- ally ignored. An attempt, however, has been made to see if a spectral distribution obtained from the effective models described above does saturate the QCD sum rules [134]. These authors found that they could only saturate the sum rule if they assumed an additional mass shift of the rho-meson peak downwards by 140 MeV. However, in this calcu- lation only leading order density corrections∼ to the quark condensates whereas infinity order density effects have been taken into account in order to calculate the spectrum. Another comparison with the QCD sum rules has recently been carried out in ref. [131] where a reasonable saturation of the sum-rule is reported using a similar model for the in medium correlation function. There are also attempts to use photon-nucleon data in order to esti- mate possible mass shifts of the ρ and ω mesons. In ref. [135] a simple pion and sigma exchange model is used in order to fit data for photopro- duction of ρ- and ω-mesons. Assuming vector dominance, this model is then used to calculate the self-energy of these vector mesons in nuclear matter. The authors find a downward shift of the ρ of about 18% at nu- clear matter density, in rough agreement with the prediction from QCD sum rules. Quite to the contrary, ref. [136] using photoabsorption data and dispersion relations find an upwards shift of 10 MeV or 50 MeV for the longitudinal and transverse part of the ρ, respectively. This result, however, is derived for a ρ-meson which is not at rest in the nuclear matter frame, and, therefore a direct comparison of the two predictions is not possible. Very recently Friman and Pirner have pointed out that the ρ-meson couples very strongly with the N ∗(1720) resonance [31, 137]. The cou- pling is of p-wave nature and, similarly to the pion coupling to delta- hole states, the ρ may mix with N ∗-nucleon-hole states resulting in a modified dispersion relation for the ρ-meson in nuclear matter. Since 26 KO, KOCH & LI
the coupling is p-wave, only ρ-mesons with finite momentum are mod- ified and shifted to lower masses. This momentum dependence of the low mass enhancement in the dilepton spectrum is an unique prediction which can be tested in experiment.
Experimental results
First measurement of dileptons in heavy ion collisions have been carried out by the DLS collaboration [53, 54] at the BEVALAC. The first pub- lished data based on a limited data set could be well reproduced using state of the art transport models [106, 138, 139] without any additional in-medium corrections. However, a recent reanalysis [140] including the full data set seems to show a considerable increase over the originally published data. It remains to be seen if these new data can also be understood without any in-medium corrections to hadronic properties. Recently, low mass dilepton data taken at the CERN SPS have been -1 -1 ) ) 2 CERES S-Au 200 GeV/u 2 HELIOS-3 S-W 200 GeV/u Koch-Song Koch-Song Li-Ko-Brown -5 Li-Ko-Brown -5 Cassing-Ehehalt-Ko 10 Cassing-Ehehalt-Kralik 10 Srivastava-Sinha-Gale Srivastava-Sinha-Gale Baier-Redlich UQMD-Frankfurt ) (100 MeV/c η /d
ch -6 -6
10 /charged (50 MeV/c 10 µµ dm) / (dN η
/d -7 -7 p-W data hadron decays ee 10 10 hadron decays after freezeout after freezeout N 2
(d 0 0.25 0.5 0.75 1 0 0.5 1 2 [ 2] mee (GeV/c ) mµµ GeV/c
Figure 6: Dilepton invariant mass spectrum from S+Au and S+W colli- sions at 200 GeV/nucleon without dropping vector meson masses. The experimental statistical errors are shown as bars and the systematic errors are marked independently by brackets. (The figure is from ref. [141]) published by the CERES collaboration [142]. Their measurements for p+Be and p+Au could be well understood within a hadronic cocktail, which takes into account the measured particle yields from p+p exper- iments and their decay channels into dileptons. In case of the heavy IN-MEDIUM EFFECTS 27
ion collision, S+Pb, however, the hadronic cocktail considerably under- predicted the measured data in particular in the invariant mass region of 300MeV Minv 500 MeV. A similar enhancement has also been reported by≤ the HELIOS-3≤ collaboration [143]. -1 -1 ) ) 2 CERES S-Au 200 GeV/u 2 HELIOS-3 S-W 200 GeV/u
Li-Ko-Brown -5 Li-Ko-Brown -5 Cassing-Ehehalt-Ko Cassing-Ehehalt-Kralik 10 10 ) (100 MeV/c η /d
ch -6 -6
10 /charged (50 MeV/c 10 µµ dm) / (dN η
/d -7 -7 p-W data hadron decays ee 10 10 hadron decays after freezeout after freezeout N 2
(d 0 0.25 0.5 0.75 1 0 0.5 1 2 2 mee (GeV/c ) mµµ (GeV/c )
Figure 7: Same as Fig. 6 with dropping vector meson masses. (The figure is from ref. [141])
Of course it is well-known that in a heavy ion collisions at SPS- energies a hadronic fireball consisting predominantly of pions is created. These pions can pairwise annihilate into dileptons giving rise to an ad- ditional source, which has not been included into the hadronic cocktail. While the pion annihilation contributes to the desired mass range, many detailed calculations [65, 144, 145], which have taken this channel into account, still underestimate the data by about a factor of three as shown Fig. 6. Also, additional hadronic processes, such as πρ πe+e−, have been taken into account [146, 147]. However, ‘conservative’→ calculations could at best reach the lower end of the sum of statistical and systematic error bars of the CERES data. Furthermore, in-medium corrections to the pion annihilation process together with corrections to the ρ-meson spectral distribution have been considered using effective chiral mod- els [64, 148]. But, these in-medium corrections, while enhancing the contribution of the pion annihilation channel somewhat, are too small to reproduce the central data points. Only models which allow for a dropping of the ρ-meson mass give enough yield in the low mass region [144, 149, 150, 151] as shown in Fig. 7. In these models, the change 28 KO, KOCH & LI
of the rho meson mass is obtained from either the Brown-Rho scaling [149, 150, 151] or the QCD sum rules [144]. Furthermore, the vector dominance is assumed to be suppressed in the sense that the pion elec- tromagetic from factor, which in free space is dominated by the rho me- son and is proportional to the square of the rho meson mass, is reduced as a result of the dropping rho meson mass. At finite temperature, such a suppression has been shown to exist in both the hidden gauge theory [152] and the perturbtive QCD [153], where the pion electromagnetic form factor is found to decrease at the order T 2. This effect is related to the mixing between the vector-isovector and axial-vector-isovector correlators at finite temperature we discussed earlier. In most studies of dilepton production without a dropping rho meson mass, this effect has not been included. Although the temperature reached in heavy ion collisions at SPS energies is high and the ratio of pions to baryons in the final state is about 5 to 1, the authors in [144, 149, 150, 151] found that they had to rely on the nuclear density rather than on the temperature in order to reproduce the data. There are also preliminary data from the Pb+Au collisions at 150 GeV/nucleon [141], which also seem to be consistent with the dropping in-medium rho meson scenario as well. Unfortunately, the experimental errors in this case are even larger than in the S+Au data and thus these new, preliminary data do not further discriminate between the dropping rho-mass and more conventional scenarios. Thus, additional data with reduced error bars are needed before firm conclusions about in-medium changes of the rho meson mass can be drawn. Although the data from the HELIOS-3 collaboration shown in the right panels of Figs. 6 and 7 do have very small errors, these are only the statistical ones while the systematic errors are not known. To summarize this section on the rho meson, our present theoreti- cal understanding for possible in-medium changes at low temperatures and vanishing baryon density indicates that one expects no (to leading order) and a small (to next to leading order) changes of the ρ-mass. At the same time to leading order in the temperature, the chiral con- densate is reduced. Therefore, a direct connection between changes in the chiral condensate and the mass of the ρ meson has not been estab- lished as assumed in the Brown-Rho scaling. However, it may well be that the latter is only valid at higher temperature near the chiral phase transition. At finite density additional effects such as nuclear many- body excitations come into play, which make the situation much more complicated. Nevertheless, the observed enhancement of low mass dilep- tons from CERN-SPS heavy ion collisions can best be described by a IN-MEDIUM EFFECTS 29
dropping rho meson mass in dense matter. Finite density and zero tem- perature is another area where possible changes of the vector mesons can be measured in experiment. Photon, proton and pion induced dilepton production from nuclei will soon be measured at CEBAF [154] and at GSI [155]. In principle these measurements should be able to determine the entire spectral distribution. If the predictions of the Brown-Rho scaling and the QCD sum rules are correct, mass shifts of the order of 100 MeV should occur, which would be visible in these experiments. Furthermore, by choosing appropriate kinematics, the properties of the ρ meson at rest as well as at finite momentum with respect to the nuclear rest frame can be measured.
3.2 The omega meson The omega meson couples to the isoscalar part of the electromagnetic current. In QCD sum rules it is also dominated by the four quark con- densate. If the latter is reduced in medium, then the omega meson mass also decreases. Assuming factorization for the four quark con- densate, Hatsuda and Lee [133] showed that the change of the omega meson mass in dense nuclear matter would be similar to that of the rho meson mass. The dropping omega meson mass in medium can also be obtained by considering the nucleon-antinucleon polarization in medium [125, 156, 157] as in the case of the rho meson. On the other hand, us- ing effective chiral Lagrangians, one finds at finite temperature an even smaller change in the mass of the omega as compared to that of the rho [124]. Because of its small decay width, which is about an order of magnitude smaller than that of rho meson, most dileptons from omega decay in heavy ion collisions are emitted after freeze out, where the medium effects are negligible. However, with appropriate kinematics an omega can be produced at rest in nuclei in reactions induced by the photon, proton and pion. By measuring the decay of the omega into dileptons allows one to determine its properties in the nuclear matter. Such experiments will be carried out at CEBAF [154] and GSI [155].
3.3 The phi meson For the phi meson, the situation is less ambiguous than the rho meson as both effective chiral Lagrangian and QCD sum-rule studies predict that its mass decreases in medium. Based on the hidden gauge theory, Song [110] finds that at a temperature of T = 200 MeV the phi meson mass is reduced by about 20 MeV. The main contribution is from the thermal 30 KO, KOCH & LI
kaon loop. In QCD sum rules, the reduction is much more appreciable, i.e., about 200 MeV at the same temperature [109]. The latter is due to the significant decrease of the strange quark condensate at finite temperature as a result of the abundant strange hadrons in hot matter. Because of the relative large strange quark mass compared to the up and down quark masses, the phi meson mass in QCD sum rules is mainly determined by the strange quark condensate instead of the four quark condensate as in the case of rho meson mass. However, the temperature dependence of the strange quark condensate in [109] is determined from a non-interacting gas model, so effects due to interactions, which may be important at high temperatures, are not included. In [110], only the lowest kaon loop has been included, so the change of kaon properties in medium is neglected. As shown in [158], this would reduce the phi meson mass if the kaon mass becomes small in medium. Also, vacuum effects in medium due to lambda-antilambda polarization has also been shown to reduce the phi meson in-medium mass [159]. QCD sum rules have also been used in studying phi meson mass at finite density [133], and it is found to decrease by about 25 MeV at normal nuclear matter density. Current experimental data on phi meson mass from measuring the KK¯ invariance mass spectra in heavy ion collisions at AGS energies do not show a change of the phi meson mass [160]. This is not surprising as these kaon-antikaon pairs are from the decay of phi mesons at freeze out when their properties are the same as in free space. If a phi meson decays in medium, the resulting kaon and antikaon would interact with nucleons, so their invariant mass is modified and can no longer be used to reconstruct the phi meson. However, future experiments on measuring dileptons from photon-nucleus reactions at CEBAF [154] and heavy ion collisions at GSI [155] will provide useful information on the phi meson properties in dense nuclear matter. For heavy ion collisions at RHIC energies, matter with low-baryon chemical potential is expected to be formed in central rapidities. Based on the QCD-sum rule prediction for the mass shift of the phi meson it has been suggested that a low mass phi peak at 880 MeV besides the normal one at 1.02 GeV appears in the dilepton∼ spectrum if a first-order phase transition or a slow cross-over between the quark-gluon plasma and the hadronic matter occurs in the collisions [161, 162]. The low-mass phi peak is due to the nonnegligible duration time for the system to stay near the transition temperature compared with the lifetime of a phi meson in vacuum, so the contribution to dileptons from phi meson decays in the mixed phase becomes comparable to that from their decays at freeze IN-MEDIUM EFFECTS 31
out. Without the formation of the quark-gluon plasma, the low-mass phi peak is reduced to a shoulder in the dilepton spectrum. Thus, one can use this double phi peaks in the dilepton spectrum as a signature for identifying the quark-gluon plasma to hadronic matter phase transition in ultrarelativistic heavy ion collisions.
4 BARYONS
As far as the in medium properties of the baryons are concerned most is known about the nucleon. But also the properties of the hyperons such as the Λ can be determined in the medium by studying hypernuclei. Also the Λ(1405), which we have discussed in connection with the K− optical potential, is another example for in-medium effects of baryons. In the following we will limit ourself on a brief discussion of how the nucleon properties can be viewed in the context of chiral symmetry. As mentioned briefly in the introduction, medium effects on a nu- cleon due to many-body interactions have long been studied, leading to an effective mass, which is generally reduced as a result of the finite range of the nucleon interaction and higher order effects. Its relation to chiral symmetry restoration is best seen through the QCD sum rules [2, 3, 163, 164, 165, 166, 167]. Using in-medium condensates, it has been shown that the change of scalar quark condensate in medium leads to an attractive scalar potential which reduces the nucleon mass, while the change of vector quark condensate leads to a repulsive vector potential which shifts its energy. The nucleon scalar and vector self-energies in this study are given, respectively, by
2 2 8π 8π ΣπN ΣS 2 ( qq¯ ρ qq¯ 0) 2 ρN , ≈ −MB h i − h i ≈−MB mu + md 2 2 64π + 32π ΣV 2 q q ρ = 2 ρN , (19) ≈ 3MB h i MB where the Borel mass M is an arbitrary parameter. With M m , B B ≈ N and mu + md 11 MeV, these self-energies have magnitude of a few hundred MeV≈ at normal nuclear density. These values are similar to those determined from both the Walecka model [168] and the Dirac- Brueckner-Hartree-Fock (DBHF) approach [169] based on the meson- exchange nucleon-nucleon interaction. Experimental evidences for these strong scalar and vector potentials have been inferred from the proton- nucleus scattering at intermediate energies [170] via the Dirac phe- 32 KO, KOCH & LI
nomenology [171, 172, 173, 174] in which the Dirac equation with scalar and vector potentials is solved. The importance of medium effects on the nucleon mass can be seen from the significant decrease of the Q-value for the reaction NN NNNN¯ in nuclear medium as a result of the attractive scalar poten-→ tial. This effect has been included in a number of studies based on transport models. In Fig. 8, theoretical results from these calculations for the antiproton differential cross section in Ni+Ni collisions at 1.85 GeV/nucleon [175, 176, 177] and C+Cu collisions at 3.65 GeV/nucleon [178] are compared with the experimental data from GSI [179] and Dubna [180]. In [175, 176, 177], the relativistic transport model has been used with the antiproton mean-field potential obtained from the Walecka model. The latter has a value in the range of -150 to -250 MeV at normal nuclear matter density. The results of ref. [178] are based on the nonrelativistic Quantum Molecular Dynamics with the produced nucleon and antinucleon masses taken from the Nambu Jona-Lasinio model. Within this framework, these studies thus show that− in order to describe the antiproton data from heavy-ion collisions at subthreshold energies it is necessary to include the reduction of both nucleon and antinucleon masses in nuclear medium. Even at AGS energies, which are above the antiproton production threshold in NN interaction, the medium effects on antiproton may still be important [181]. Indeed, a recent study using the Relativistic Quantum Molecular Dynamics shows that medium modifications of the antiproton properties are important for a quantitative description of the experimental data [182]. We note, however, a better understanding of antiproton annihilation is needed, as only about 5-10% of produced antiprotons can survive, in order to determine more precisely the antiproton in-medium mass from heavy ion collisions.
5 RESULTS FROM LATTICE QCD CAL- CULATIONS
Another source of information about possible in medium changes are lattice QCD-calculations. For a review see e.g. [1]. Presently lattice calculations can only explore systems at finite temperature but van- ishing baryon chemical potential. Therefore, their results are not di- rectly applicable to present heavy ion experiments, where system at finite baryon chemical potential are created. But future experiments at RHIC and LHC might succeed in generating a baryon free region. But IN-MEDIUM EFFECTS 33
Figure 8: Antiproton momentum spectra from Ni+Ni collisions at 1.85 GeV/nucleon, and C+Cu collisions at 3.85 GeV/nucleon. The left, mid- dle, and right panels are from Refs. [175], [177], and [178], respectively. The solid and dotted curves are from transport model calculations with free and dropping antiproton mass, respectively. The experimental data from ref. [179] for Ni+Ni collisions and from ref. [180] for C+Cu colli- sions are shown by solid circles. 34 KO, KOCH & LI
aside from the experimental aspects, lattice results provide an impor- tant additional source where our model understanding can be tested. Lattice calculation are usually carried out in Euclidean space, i.e. in a space with imaginary time. As a consequence plane waves in Minkowski- space translate into decaying exponentials in Euclidean space. At zero temperature, masses of the low lying hadrons are extracted from two point functions, which carry the quantum numbers of the hadron under consideration.
3 2 C(τ)= d x
Mscreen = nπT, (21) where n is the number of quarks of a given hadron, i.e. n = 2 for mesons and n = 3 for baryons. Only for the pion and sigma meson, lat- IN-MEDIUM EFFECTS 35
tice calculation found a significant deviation from this simple behavior, which is due to strong residual interactions in these channel. All other hadrons considered so far, such as the ρ and a1 as well as the nucleon, exhibit the above screening mass [185, 186]. These trivial contributions from independent quarks are absent, however, in the time-like corre- lator. There has been one attempt to extract meson masses from the time-like correlators in four flavor QCD [187]. In this case it has been found that the mass of the ρ-meson does not change significantly below Tc. Above the critical temperature, on the other hand, the correlation function appeared to be consistent with one of noninteracting quarks, indicating the onset of deconfinement. There have been attempts to extract the mass of the scalar σ meson as well as that of the pion from so called susceptibilities [188]. These are nothing else than the total four-volume integral over the two-point functions. If the two-point function is dominated by one hadronic pole, then this integral should be inversely proportional to the square of the mass of the lightest hadron under consideration. Using this method, the degeneracy of the pion and sigma mass close to Tc has been demon- strated. Finally, let us note that a scenario where all hadron masses scale linearly with the chiral condensate is consistent with strong rise in the energy and entropy density around the phase transition as observed in Lattice calculations [116].
6 SUMMARY
In this review, hadron properties, particularly their masses, in the nu- clear medium have been discussed. Both conventional many-body inter- actions and genuine vacuum effects due to chiral symmetry restoration have effects on the hadron in-medium properties. For Goldstone bosons, which are directly linked to the spontaneously broken chiral symmetry, we have discussed the pion, kaons, and etas. For the pion, its mass is only slightly shifted in the medium due to the small s-wave interactions as a result of chiral symmetry. On the other hand, the strong attractive p-wave interactions due to the ∆(1230) and the ρ(770) lead to a softening of the pion dispersion relation in medium. These effects result from many-body interactions rather from chiral symmetry. Phenomenologically, the small change in the pion in- medium mass does not seem to have any observable effects. However, the effects due to the softened pion in-medium dispersion relation may 36 KO, KOCH & LI
be detectable through the enhanced low transverse momentum pions in heavy ion collisions. For kaons, medium effects due to the Weinberg-Tomozawa vector type interaction, which are negligible for pions in asymmetric nuclear matter, are important as they have only one light quark. However, because of the large explicit symmetry breaking due to the finite strange quark mass higher order corrections in the chiral expansion are non- negligible, leading to an appreciable attractive scalar interaction for both the kaon and the antikaon in the medium. Available experimental data on both subthreshold kaon production and kaon flow are consistent with the presence of this attractive scalar interaction. For etas, their properties are more related to the explicit breaking of the UA(1) axial symmetry in QCD. If the UA(1) symmetry is restored in medium as indicated by the instanton liquid model, then their masses are expected to decrease as well. Heavy ion experiments at the CERN- SPS, however, rule out a vast enhancement of the final-state eta yield. In the case of vector mesons, we have discussed the rho, omega, and phi. At finite temperature, all model calculations agree with the cur- rent algebra result that the mass of the rho does not change to order T 2. To higher order in the temperature and at finite density QCD sum rules predict a dropping of the masses in the medium. Predictions from chiral models, on the other hand, tend to predict an increase of the rho mass with temperature. However, presently available dilepton data from CERN SPS heavy ion experiments are best described assuming a dropping rho mass in dense matter but the errors in the present exper- imental data are too large to definitely exclude some more conventional explanations. For baryons, particularly the nucleon, the change of their properties in the nuclear medium has been well-known in studies based on conven- tional many-body theory. On the other hand, the nucleon mass is also found to be reduced in the medium as a result of the scalar attractive potential related to the quark condensate. A clear separation of the vac- uum effects due to the condensate from those of many-body interactions is a topic of current interest and has not yet been resolved. A dropping nucleon effective mass in medium seems to be required to explain the large enhancement of antiproton production in heavy ion collisions at subthreshold energies. In principle lattice QCD calculations could help answer quite a few of these questions, although they are restricted to systems at finite tem- perature and vanishing baryon density. However, with the presently available computing power reliable quantitative predictions about in- IN-MEDIUM EFFECTS 37
medium properties of hadrons are still not available. The study of in-medium properties of hadrons is a very active field, both theoretically and experimentally. While many questions are still open and require additional measurements and more careful calcula- tions, it is undisputed that the study of in medium properties of hadrons provides us with an unique opportunity to further our understanding about the long range, nonperturbative aspects of QCD.
Acknowledgments: We are grateful to many colleagues for helpful discussions over the years. Also, we would like to thank G. Boyd, G. E. Brown, A. Drees, B. Friman, C. Gaarde, F. Klingl, R. Rapp, C. Song, T. Ullrich, and W. Weise for the useful information and discussions during the preparation of this review. The work of CMK was supported in part by the National Science Foundation under Grant No. PHY-9509266. V.K. was supported the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of Nuclear Physics, and by the Office of Basic Energy Sciences, Division of Nuclear Sciences, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098. GQL was supported by the Department of Energy under Contract No. DE-FG02-88Er40388.
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PROPERTIES OF HADRONS IN
THE NUCLEAR MEDIUM
Che Ming Ko
Cyclotron Institute and Physics Department
Texas A M University College Station Texas USA
Volker Koch
Nuclear Science Division
Lawrence Berkeley National Lab oratory Berkeley CA USA
Guoqiang Li
Department of Physics
State University of New York at Stony Bro ok Stony Bro ok N Y
USA
key words Chiral symmetry Goldstone b osons vector mesons baryons
hot and dense hadronic matter heavy ion collisions
Abstract
This review is devoted to the discussion of hadron prop erties
in the nuclear medium and its relation to the partial restoration
of chiral symmetry Sp ecial attention is given to disentangle in
medium e ects due to conventional many bo dy interactions from
those due to the change of the chiral condensate In particular
we shall discuss medium e ects on the Goldstone b osons pion
kaon and eta the vector mesons rho omega phi and the nu
cleon Also for each prop osed in medium e ect the exp erimental
consequence and results will b e reviewed
Contents
INTRODUCTION
GOLDSTONE BOSONS
The pion
Kaons
Etas
KO KOCH LI
VECTOR MESONS
The rho meson
The omega meson
The phi meson
BARYONS
RESULTS FROM LATTICE QCD CALCULATIONS
SUMMARY
INTRODUCTION
The atomic nucleus provides a unique lab oratory to study the long range
and bulk prop erties of QCD Whereas QCD is well tested in the p er
turbative regime rather little is known ab out its prop erties in the long
range nonp erturbative region One of the central nonp erturbative prop
erties of QCD is the sp ontaneous breaking chiral symmetry in the ground
state resulting in a nonvanishing scalar quark condensate q q
It is b elieved and supp orted by lattice QCD calculations that at
temp eratures around MeV QCD undergo es a phase transition to
a chirally restored phase characterized by the vanishing of the order
parameter the chiral condensate q q This is supp orted by results
obtained within chiral p erturbation theory E ective chiral mo dels
predict that a similar transition also takes place at nite nuclear density
The only way to create macroscopic strongly interacting systems at
nite temp erature and or density in the lab oratory is by colliding heavy
nuclei at high energies Exp eriments carried out at various b ombarding
energies ranging from AGeV BEVALAC SIS to AGeV SPS
have established that one can generate systems of large density but mo d
erate temp eratures SIS BEVALAC systems of b oth large density and
temp erature AGS as well as systems of low density and high temp er
atures SPS Therefore a large region of the QCD phase diagram can
b e investigated through the variation of the b ombarding energy But
in addition the atomic nucleus itself represents a system at zero tem
p erature and nite density At nuclear density the quark condensate is
estimated to b e reduced by ab out so that e ects due to
the change of the chiral order parameter may b e measurable in reactions induced by a pion proton or photon on the nucleus
IN MEDIUM EFFECTS
Calculations within the instanton liquid mo del as well as results
from phenomenological mo dels for hadrons suggest that the prop er
ties of the light hadrons such as masses and couplings are controlled
by chiral symmetry and its sp ontaneous 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 prop erties should de
p end on the value of the chiral condensate q q Consequently we
should exp ect that the prop erties 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 This
has since stimulated extensive theoretical and exp erimental studies on
hadron in medium prop erties
By studying medium e ects on hadronic prop erties one can directly
test our understanding of those non p erturbative asp ects of QCD which
are resp onsible for the light hadronic states The b est way to investigate
the change of hadronic prop erties in exp eriment 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 b e seen di
rectly in the dilepton invariant mass sp ectrum In addition as we shall
discuss the measurement of subthreshold particle pro duction such as
kaons and antiprotons 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 bo dy 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 bo dy e ects such as the Pauli principle
are at work which as we shall discuss lead to considerable mo di ca
tions of hadronic prop erties in some cases How these e ects are related
to the partial restoration of chiral symmetry is a new and unsolved ques
tion in nuclear many bo dy physics One example is the e ective mass of
a nucleon in the medium which was introduced long time ago
to mo del the momentum dep endence of the nuclear force which is due
to its nite range or to mo del its energy dep endence which results
from higher order particle 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 this is due to so called virtual pair corrections and may b e
KO KOCH LI
related to chiral symmetry restoration as suggested by recent studies
Another environment which is somewhat cleaner from the theorists
p oint of view is a system at nite temp erature and vanishing baryon
chemical p otential At low temp eratures such a system can b e system
atically explored within the framework of chiral p erturbation theory
and essentially mo del indep endent statements ab out the e ects of chi
ral restoration on hadronic masses and couplings may thus b e given In
the high temp erature regime eventually lattice QCD calculations should
b e able to tell us ab out the prop erties of hadrons close to the chiral
transition temp erature Unfortunately such a system is very di cult to
create in the lab oratory
Since relativistic heavy ion collisions are very complex pro cesses
one has to resort to careful mo deling in order to extract the desired
in medium correction from the available exp erimental data Compu
tationally the description of these collisions is b est carried out in the
framework of transp ort theory since nonequi
librium e ects have b een found to b e imp ortant at least at energies up
to AGeV At higher energies there seems to b e a chance that one can
separate the reaction in an initial hard scattering phase and a nal equi
librated phase It app ears that most observables are dominated by the
second stage of the reaction so that one can work with the assumption
of lo cal thermal equilibrium However as compared to hydrodynamical
calculations also here the transp ort 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 b e successfully treated in the transp ort approach
Thus it app ears p ossible that one can explore the entire range of exp er
iments within one and the same theoretical framework which of course
has the advantage that one reduces the ambiguities of the mo del to a
large extent by exploring di erent observables
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 p ossible in medium ef
fect which is related to the decon nement in the Quark Gluon Plasma
namely the suppression of the J This idea which has b een rst pro
p osed by Matsui and Satz is based on the observation that due to
the screening of the color interaction in the Quark Gluon Plasma the
J is not b ound anymore As a result if such a Quark Gluon Plasma
IN MEDIUM EFFECTS
is formed in a relativistic heavy ion collision the abundance of J
should b e considerably reduced Of course also this signal su ers from
more conventional backgrounds namely the disso ciation of the J due
to hadronic collisions To which extent present data can b e
understo o d in a purely hadronic scenario is extensively debated at the
moment We refer the reader to the literature for further details
In this review we will concentrate on prop osed in medium e ects on
hadrons Whenever p ossible we will try to disentangle conventional in
medium e ects from those we b elieve are due to new physics namely
the change of the chiral condensate We rst will discuss medium ef
fects on the Goldstone b osons such as the pion kaon and eta Then we
will concentrate on the vector mesons which have the advantage that
p ossible changes in their mass can b e directly observed in the dilep
ton sp ectrum 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 prop erties As
will b ecome clear from our discussion progress in this eld requires the
input from exp eriment Many question cannot b e settled from theoret
ical consideration alone Therefore we will always try to emphasize the
observational asp ects for each prop osed in medium correction
GOLDSTONE BOSONS
This rst part is devoted to in medium e ects of Goldstone b osons
sp eci cally the pion kaon and eta The fact that these particles are
Goldstone b osons means that their prop erties are directly linked to the
sp ontaneous breakdown of chiral symmetry Therefore rather reliable
predictions ab out their prop erties can b e made using chiral symmetry
arguments and techniques such as chiral p erturbation theory Contrary
to naive exp ectations the prop erties of Goldstone b osons are rather
robust with resp ect to changes of the chiral condensate On second
thought however this is not so surprising b ecause as long as chiral
symmetry is sp ontaneously broken there will b e Goldstone b osons 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 asso ciated 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
KO KOCH LI
we shall discuss b elow
2.1 The pion
Of all hadrons the pion is probably the one where in medium corrections
are b est known and understo o d From the theoretical p oint of view
pion prop erties can b e well determined b ecause the pion is such a go o d
Goldstone b oson The explicit symmetry breaking terms in this case are
small as they are asso ciated with the current masses of the light quarks
Therefore the prop erties of the pion at nite density and temp erature
can b e calculated using for instance chiral p erturbation theory But
more imp ortantly there exists a considerable amount of exp erimental
data ranging from pionic atoms to pion nucleus exp eriments to charge
exchange reactions These data all address the pion prop erties 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 p otential This has
b een measured to a very high precision in pionic atoms and one
nds to rst order in the nuclear density
m b b m
e e
At nuclear matter density the shift of the pion mass is
m
m
which is small as exp ected from low energy theorems based on chi
ral symmetry Theoretically the value of b is well understo o d as
e
the combined contribution from a single scattering impulse approxi
mation involving the small s wave isoscalar N scattering length b
m and the contribution from a density dep endent corre
lation or re scattering term which is dominated by the comparatively
large isovector s wave scattering length b m i e
m
b b b b
e
M r
N
where the correlation length r p is essentially due to
f
the Pauli exclusion principle The weak and slightly repulsive s wave
IN MEDIUM EFFECTS
pion p otential has also b een found in calculations based on chiral p er
turbation theory as well as the phenomenological Nambu Jona
Lasinio mo del This is actually not to o surprising b ecause b oth am
plitudes b and b are controlled by chiral symmetry and its explicit
breaking
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 delta 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 sp ectrum develops several
branches which to leading order corresp ond to the pion particle hole
and delta hole excitations Because of the attractive p wave interaction
the disp ersion relation of the pionic branch b ecomes considerably softer
than that of a free pion A simple mo del which has b een used in prac
tical calculations is the so called delta hole mo del which
concentrates on the stronger pion nucleon delta interaction ignoring the
nucleon hole excitations In this mo del the pion disp ersion relation in
nuclear medium can b e written as
k m k k
where the pion self energy is given by
k k
k
g k
with g the Migdal parameter which accounts for short range
correlations also known as the Ericson Ericson Lorentz Lorenz e ect
The pion susceptibility is given by
f
N R
k exp k b
m
R
where f is the pion nucleon delta coupling constant b m
N
2
k
is the range of the form factor and m m
R N
m
The pion disp ersion relation obtained in this mo del is shown in Fig
The pion branch in the lower part of the gure is seen to b ecome
softened while the delta hole branch in the upp er part of the gure is
sti ened Naturally this mo del 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 However up to momenta
of ab out m the pionic branch remains pretty narrow and considerably
KO KOCH LI
softer than that of a free pion The disp ersion relation of the pion can
b e measured directly using H e t charge exchange reactions In
these exp eriments the pro duction of coherent pion have b een inferred
from angular correlations b etween the outgoing pion and the transferred
momentum The corresp onding energy transfer is smaller than that of
free pions indicating the attraction in the pion branch discussed ab ove
More detailed measurements of this type are currently b eing analyzed
Figure Pion disp ersion relation in the nuclear medium The normal
nuclear matter density is denoted by
In the context of relativistic heavy ion collisions interest in the in
medium mo di ed pion disp ersion relation got sparked by the work of
Gale and Kapusta Since the density of states is prop ortional to
the inverse of the group velocity of these collective pion mo des which
can actually vanish dep ending on the strength of the interaction see
Fig they argued that a softened disp ersion relation would lead to
a strong enhancement of the dilepton yield This happ ens at invariant
mass close to twice the pion mass b ecause the number of pions with
low energy will b e considerably increased However as p ointed out by
Korpa and Pratt and further explored in more detail in the
IN MEDIUM EFFECTS
initially prop osed strong enhancement is reduced considerably once cor
rections due to gauge invariance are prop erly taken into account Other
places where the e ect of the pion disp ersion relation is exp ected to
play a role are the inelastic nucleon nucleon cross section and the
shap e of the pion sp ectrum In the latter case one nds
that as a result of the attractive interaction the yield of pions at low p
t
is enhanced by ab out a factor of This low p enhancement is indeed
t
seen in exp erimental data for pro duction from the BEVALAC
and in more recent data from the TAPS collab oration which mea
sures neutral pions Certainly in order to b e conclusive more re ned
calculations are needed and they are presently b eing carried out
Finite temperature
The pionic prop erties at nite temp erature 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
And analogous to the case of nuclear matter there is a strong at
tractive p wave interaction which is now dominated by the resonance
This again results in a softened disp ersion relation which however is
not as dramatic as that obtained at nite density Also
the phenomenological consequences are similar The mo di ed disp ersion
relation results in an enhancement of dileptons from the pion annihila
tion by a factor of ab out two at invariant masses around MeV
Unfortunately in the same mass range other channels dom
inate the dilepton sp ectrum see discussion in section so that this
enhancement cannot b e easily observed in exp eriment The mo d
i ed disp ersion relation has also b een invoked in order to explain the
enhancement of low transverse momentum pions observed at CERN
SPS heavy ion collisions However at these high energies
the expansion velocity of the system which is mostly made out of pions
is to o fast for the attractive pion interaction to a ect the pion sp ectrum
at low transverse momenta This is di erent at the lower energies
around GeV There the dominant part of the system and the source
for the pion p otential are the nucleons which move considerably slower
than pions Therefore pions have a chance to leave the p otential well
b efore it has disapp eared 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 temp erature are dominated by p wave reso
KO KOCH LI
nances the and the resp ectively Both are not related
directly to chiral symmetry and its restoration but rather to what we
call here many bo dy e ects which of course do es not make them less
interesting The s wave interaction is small b ecause the pion is such a
go o d Goldstone Boson probably to o small to have any phenomenolog
ical consequence at least for heavy ion collisions
2.2 Kaons
Contrary to the pion the kaon is not such a go o d Goldstone b oson
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 of GeV In addition since the kaon carries
strangeness its b ehavior in non strange isospin symmetric matter will
b e di erent from that of the pion Rather interesting phenomenological
consequences arise from this di erence such as a p ossible condensation
of antikaons in neutron star matter
Chiral Lagrangian
This di erence can b est b e exempli ed by studying the leading order
e ective SU SU Lagrangian obtained in heavy baryon chiral
L R
p erturbation theory
f f
L Tr U U r Tr M U U
q
Tr B i v D B D Tr BS fA B g
F Tr BS A B
In the ab ove formula we have U exp i f with and f b eing the
pseudoscalar meson o ctet and their decay constant resp ectively B is
the baryon o ctet v is the four velocity of the heavy baryon v
v and v S and S stands for the spin op erator S
M is the quark mass matrix and r D as well as F are empirically
q
determined constants Furthermore
D B B V B
i
y y y y
A V
with U
To leading order in the chiral counting the explicit symmetry break
ing term Tr M U U gives rise to the masses of the Goldstone q
IN MEDIUM EFFECTS
b osons The interesting di erence b etween the b ehavior of pions and
kaons in matter arises from the term involving the vector current V
i e Tr B iv D B In case of the pion this term is identical to the
well known Weinberg Tomozawa term
N N L
WT
f
It contributes only to the isovector s wave scattering amplitude and
therefore do es not contribute to the pion optical p otential in isospin
symmetric nuclear matter This is di erent in case of the kaon where
we have
i
L N N K K N N K K
WT
f
The rst term contributes to the isoscalar s wave amplitude and there
fore gives rise to an attractive or repulsive optical p otential 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 to o
repulsive as compared with exp eriment 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
b etween pions and kaons namely the strength of the explicit symmetry
breaking The next to leading order e ective kaon nucleon Lagrangian
can b e written as
C
KN
K K NN KK N N L
f f
D D
K K K K NN N N
t t t t
f f
The value of the kaon nucleon sigma term m m hN juu
KN q s
ss jN i dep ends on the strangeness content of the nucleon y hN jss jN i
hN juu ddjN i Using the light quark mass ratio m m
s
one obtains MeV The additional parameters
KN
C D D are then xed by comparing with K nucleon scattering data
This so determined e ective Lagrangian can then b e used to pre
dict the K nucleon scattering amplitudes and one obtains an attrac
tive isoscalar s wave scattering length in contradiction with exp eriments
where one nds a repulsive amplitude This discrepancy has b een
attributed to the existence of the which is lo cated b elow the
K N threshold From the analysis of K p reactions it is known
that this resonance couples strongly to the I K p state and there
fore leads to repulsion in the K p amplitude
KO KOCH LI
The Role of the
Already in the sixties there have b een attempts to understand the
as a b ound 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
b ecause a b ound state is formed This concept is familiar to the nuclear
physicist from the deuteron which is b ound b ecause of the attractive
interaction b etween proton and neutron The existence of the deuteron
then leads to a repulsive scattering length in spite of the attractive
interaction b etween neutron and proton Chiral p erturbation theory is
based on a systematic expansion of the S matrix elements in p owers of
momenta and therefore e ects which are due to the proximity of a
resonance such as the will only show up in terms of rather
high order in the chiral counting Thus it is not to o 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 p erturbation
calculation has b een extended to either include an explicit state
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 b ound state
This picture of the as a K p b ound state has recently re
ceived some considerable interest in the context of in medium correc
tions In ref it has b een p ointed out that in this picture as a result
of the Pauli blocking of the proton inside this b ound state the prop erties
of the would b e signi cantly changed in the nuclear environ
ment With increasing density its mass increases and the strength of
the resonance is reduced see Fig Because of this shifting and dis
app earance of the in matter the K optical p otential changes
sign from repulsive to attractive at a density of ab out of nuclear
matter density see Fig in agreement with a recent analysis of K
atoms These ndings have b een con rmed in ref This in
medium change of the due to the Pauli blo cking can only o ccur
if a large fraction of its wave function is indeed that of a K proton
b ound 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 p otentials But certainly if the is mostly a genuine
three quark state the Pauli blo cking should not a ect its prop erties
Therefore it would b e very interesting to con rm the mass shift of the
for instance by a measurement of the missing mass sp ectrum
of kaons in the reaction p K at CEBAF Thus the
IN MEDIUM EFFECTS
6.0 ρ = 0 ρ ρ = 0.5 0 ρ = ρ 4.0 0
2.0 Im f(I=0) [fm]
0.0 1.38 1.48
Ec.m. [GeV]
0.0 Re(U) Im(U)
[GeV] −0.1 opt U
−0.2 0.0 1.0 2.0 ρ ρ
/ 0
Figure a Imaginary part of the I K proton scattering am
plitude for di erent densities b real and Imaginary part of the K
optical p otential
atomic nucleus provides a unique lab oratory to investigate the prop er
ties of elementary particles From our discussion it is clear that this
in medium change of the is not related to the restoration of
chiral symmetry
Experimental results
Phenomenologically the attractive optical p otential for the K in nu
clear matter is of particular interest b ecause it can lead to a p ossible
kaon condensation in neutron stars This would limit the max
imum mass of neutron stars to ab out one and a half solar masses and
give rise to sp eculations ab out many small black holes in our galaxy
Kaonic atoms of course only prob e the very low density b ehavior of the kaon optical p otential and therefore an extrap olation to the large
KO KOCH LI
densities relevant for neutron stars is rather uncertain Additional in
formation ab out the kaon optical p otential can b e obtained from heavy
ion collision exp eriments where densities of more than twice nuclear
matter density are reached
Observables that are sensitive to the kaon mean eld p otentials are
the subthreshold pro duction as well as kaon ow
Given an attractive repulsive mean eld p otential for the kaons it is
clear that the subthreshold pro duction is enhanced reduced In case
of the kaon ow an attractive interaction b etween kaons and nucleons
aligns the kaon ow with that of the nucleons whereas a repulsion leads
to an anti alignment anti ow However b oth observables are also
extremely sensitive to the overall reaction dynamics in particular to
the prop erties of the nuclear mean eld and to reabsorption pro cesses
esp ecially in case of the antikaons Therefore transp ort calculations are
required in order to consistently incorp orate all these e ects
In Fig we show the result obtained from such calculations for
the K subthreshold pro duction and ow together with exp erimental
data solid circles from the KaoS collab oration and from the FOPI
collab oration 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 exp erimental data are nicely
repro duced when the kaon mean eld p otential is included Although
the subthreshold pro duction of kaons can also b e explained without any
kaon mean eld the assumption used in these calculations that
a lambda particle has the same mean eld p otential as a nucleon is
not consistent with the phenomenology of hypernuclei Since it is
undisputed that these observables are sensitive to the in medium kaon
p otential a systematic investigation including all observables should
eventually reveal more accurately the strength of the kaon p otential in
dense matter Additional evidence for a repulsive K p otential comes
nucleus exp eriments There the measured cross sections and from K
angular distributions can b e pretty well understo o d within a simple
impulse approximation Actually it seems that an additional
repulsion is required to obtain an optimal t to the data which has
b een suggested as a p ossible evidence for a swelling of the nucleon size
or a lowering of the omega meson mass in the nuclear medium
As for the K b oth chiral p erturbation theory and dynamical mo d
els of the K nucleon interaction indicate the existence of an attrac
tive mean eld so the same observables can b e used However
the measurement and interpretation of K observables is more di cult
IN MEDIUM EFFECTS
Figure Kaon yield left panel and ow right panel in heavy ion
collisions The solid and dotted curves are results from transp ort mo del
calculations with and without kaon mean eld p otential resp ectively
The data are from refs
KO KOCH LI
since it is pro duced less abundantly Also reabsorption e ects due to the
reaction K N are strong which further complicates the analysis
Nevertheless the e ect of the attractive mean eld has b een shown to
b e signi cant as illustrated in Fig where results from transp ort cal
culations with solid curves or without dotted curves attractive mean
eld for the antikaons are shown It is seen that the data solid
circles on subthreshold K pro duction supp ort the existence of an
attractive antikaon mean eld p otential For the K ow there only
exist very preliminary data from the FOPI collab oration which
seem to show that the K s have a p ositive ow rather an anti ow
thus again consistent with an attractive antikaon mean eld p otential
Let us conclude this section on kaons by p ointing out that the prop
erties of kaons in matter are qualitatively describ ed in chiral p erturba
tion theory Although some of the e ects comes from the Weinberg
Tomozawa vector type interaction which contributes b ecause contrary
to the pion the kaon has only one light quark in order to explain the ex
p erimental data on kaon subthreshold pro duction 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
2.3 Etas
The prop erties of etas are not only determined by consideration of chiral
symmetry but also by the explicit breaking of the U axial symmetry
A
due to the axial anomaly in QCD As a result the singlet eta which
would b e a Goldstone b oson if U were not explicitly broken b ecomes
A
heavy Furthermore b ecause of SU symmetry breaking the strange
quark mass is considerably heavier than that of up and down quark
the o ctet eta and singlet eta mix leading to the observed particles
and The mixing is such that the is mostly singlet and thus
heavy and the is mostly o ctet and therefore has roughly the mass of
the kaons If as has b een sp eculated the U symmetry
A
is restored at high temp erature due to the instanton e ects one would
exp ect considerable reduction in the masses of the etas as well as in their
mixing A dropping eta meson in medium mass is exp ected
to provide a p ossible explanation for the observed enhancement of low
transverse momentum etas in SIS heavy ion exp eriments at subthreshold
energies However an analysis of photon sp ectra from heavy ion
collisions at SPS energies puts an upp er limit on the ratio to b e not more than larger for central than for p eripheral collisions
IN MEDIUM EFFECTS
Figure Antikaon yield left panel and ow right panel in heavy ion
collisions The solid and dotted curves are results from transp ort mo del
calculations with and without kaon mean eld p otential resp ectively
The data are from ref
KO KOCH LI
This imp oses severe constraints on the changes of the prop erties in
hadronic matter
We note that the restoration of chiral symmetry is imp ortant in
the sector b ecause without the U breaking b oth would b e
A
Goldstone b osons of an extended U U symmetry However it is
still b eing debated what the e ects precisely are
VECTOR MESONS
Of all particles it is probably the meson which has received the most
attention in regards of in medium corrections This is mainly due to the
fact that the is directly observable in the dilepton invariant mass sp ec
trum Also since the carries the quantum numbers of the conserved
vector current its prop erties are related to chiral symmetry and can
as we shall discuss b e investigated using e ective chiral mo dels as well
as current algebra and QCD sum rules That p ossible changes of the
can b e observed in the dilepton sp ectrum measured in heavy ion colli
sions has b een rst demonstrated in ref and then studied in more
detail in The in medium prop erties of the a are closely
related to that of the since they are chiral partners and their mass
di erence in vacuum is due to the sp ontaneous breaking of chiral sym
metry Unfortunately there is no direct metho d to measure the
changes of the a in hadronic matter The meson on the other hand
is a chiral singlet and the relation of its prop erties to chiral symmetry
is thus not so direct However calculations based on QCD sum rules
predict also changes in the mass as one approaches chiral restoration
Observationally the can probably b e studied b est due to its rather
small width and its decay channel into dileptons Finally there is the
meson In an extended SU chiral symmetry the and the are a
sup erp osition of the singlet and o ctet states with nearly p erfect mixing
i e the contains only strange quarks whereas the is made only out
of light quarks Both QCD sum rules and e ective chiral mo dels
predict a lowering of the 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 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 b eing worked on the basic argument of Brown
and Rho is as follows Hadron masses such as that of the meson
IN MEDIUM EFFECTS
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 which is prop ortional to the
Gluon condensate Thus one could imagine that with the disapp ear
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 prop osed by Pisarski But the
conjecture of Brown and Rho go es even further They assume that the
gluon condensate can b e separated into a hard and soft part the lat
ter of which scales with the quark condensate and is also resp onsible
for the masses of the light hadrons This picture nds some supp ort
from lattice QCD calculations in that the gluons condensate drops by
ab out at the chiral phase transition To what extent this is
also re ected in changes in hadron masses in not clear at the moment
see section although one should mention that the rise in the en
tropy density close to the critical temp erature can b e explained if one
assumes the hadron masses to scale with the quark condensate
Another asp ect 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
exp ected to b e suppressed In their picture Goldstone b osons are not
sub ject 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
3.1 The rho meson
Since the is a vector meson it couples directly to the isovector current
which then results in the direct decay of the into virtual photons i e
dileptons Consequently prop erties of the meson can b e investigated
by studying two point correlation functions of the the isovector currents
i e
Z
iq x
q i e hTJ x J i d x
The masses of the rho meson and its excitations corresp ond to
the p ositions of the p oles of this correlation function This is b est seen
if one assumes that the current eld identity holds namely that
the current op erator is prop ortional to the meson eld In this case
the ab ove correlation function is identical up to a constant to the
KO KOCH LI
meson propagator The imaginary part of this correlation function
is also directly prop ortional to the electron p ositron annihilation cross
section where the meson is nicely seen In addition at higher
center of mass energies one sees a continuum in the electron p ositron
annihilation cross section which corresp onds to the excited states of the
rho mesons as well as to the onset of p erturbative QCD pro cesses
In medium changes of the meson can b e addressed theoretically by
evaluating the current current correlator in the hadronic environment
The current current correlator can b e evaluated either in e ective chi
ral mo dels or using current algebra arguments or directly in QCD In
the latter one evaluates the correlator in the deeply Euclidean region
q using the Wilson expansion where all the long distance
physics is expressed in terms of vacuum exp ectation values of quark and
gluon op erators the so called condensates Disp ersion relations are then
used to relate the correlator in the Euclidean region to that for p osi
tive q where the hadronic eigenstates are lo cated One then assumes
a certain shap e for the phenomenological sp ectral functions typically
a delta function which represents the b ound state and a continuum
which represents the p erturbative regime These so called QCD sum
rules therefore relate the observable hadronic sp ectrum with the QCD
vacuum condensates for a review of the QCD sum rule techniques see
e g These relations can then either b e used to determine the
condensates from measured hadronic sp ectra or to make predictions
ab out changes of the hadronic sp ectrum due to in medium changes of
the condensates
Similarly one can study the prop erties of a meson in the nuclear
medium through the axial vector correlation function Then once chiral
symmetry is restored there should b e no observable di erence b etween
left handed and right handed or equivalently vector and axial vector
currents Consequently the vector and axial vector correlators should
b e identical Often this identity of the correlators is identi ed with the
degeneracy of the and a mesons in a chirally symmetric world This
however is not the only p ossibility as was p ointed out by Kapusta and
Shuryak There are at least three qualitatively di erent scenarios
for which the vector and axial vector correlator are identical see Fig
The masses of and a are the same The value of the common
mass however do es not follow from chiral symmetry arguments
alone
The mixing of the sp ectral functions i e b oth the vector and
IN MEDIUM EFFECTS
CV,A CV,A CV,A
mρ m= mρ m a1 M 1 Ma M
(1) (2) (3)
Figure Several p ossibilities for the vector and axial vector sp ectral
functions in the chirally restored phase
axial vector sp ectral functions have p eaks of similar strength at
b oth the mass of the and the mass of the a
Both sp ectral functions could b e smeared over the entire mass
range Because of thermal broadening of the mesons and the onset
of decon nement the structure of the sp ectral function may b e
washed out and it b ecomes meaningless to talk ab out mesonic
states
Finite temperature
At low temp eratures where the heat bath consists of pions only one can
employ current algebra as well as PCAC to obtain an essentially mo del
indep endent result for the prop erties of the Using this technique
Dey Eletzky and Io e could show that to leading order in the
temp erature T the mass of the meson do es not change Instead one
nds an admixture of the axial vector correlator i e that governed by
the a meson Sp eci cally to this order the vector correlator is given
by
C q T C q T C q T O T
V V A
with T f Here C C stand for the vector isovector and
V A
axial vector isovector correlation functions resp ectively One should
note that to the same order the chiral condensate is reduced
q q T
T
O T
q q f
Therefore to leading order a drop in the chiral condensate do es not
a ect the mass of the meson but rather reduces its coupling to the
vector current and in particular induces an admixture of the a meson
KO KOCH LI
This nding is at variance with the Brown Rho scaling hypothesis The
reason for this di erence is not yet understo o d The admixture of the
axial correlator is directly related with the onset of chiral restoration
If chiral symmetry is restored the vector and axial vector correlators
should b e identical The result of Dey et al suggests that this is
achieved by a mixing of the two instead of a degeneracy of the and a
masses If the mixing is complete i e then the extrap olation
of the low temp erature result would give a critical temp erature of
p
f M eV which is surprisingly close to the value given T
c
by recent lattice calculations
Corrections to the order T involve physics b eyond chiral symmetry
As nicely discussed in ref to this order the contributions can
b e separated into two distinct contributions The rst arises if a pion
from the heat bath couples via a derivative coupling to the current
under consideration In this case the contribution is prop ortional to the
invariant pion density
Z
d q
T
n e T
and the square of the pion momentum q T A typical example
would b e for instance the self energy correction to the meson from
the standard two pion lo op diagram involving the p wave coupling
The other contribution comes from interactions of two pions from the
heat bath with the current without derivative couplings This is pro
p ortional to the square of the invariant pion density and thus T as
well These pure density contributions again can b e evaluated in a mo del
indep endent fashion using current algebra techniques and as b efore do
not change the mass of the meson but change the coupling to the
current and induce a mixing with the axial vector correlator Actually
it can b e shown that to all orders in the pion density these pure
density e ects do not change the mass but induce mixing and reduce
the coupling At the same time however these contributions reduce the
chiral condensate
The contributions due to the nite pion momenta have b een esti
mated in to give a downward shift of b oth the and a mass of
ab out
m
m
for temp eratures of MeV We should p oint out however
that in this analysis those changes in the masses arise from Lorentz
nonscalar condensates in the op erator pro duct expansion Therefore
IN MEDIUM EFFECTS
they are not directly related to the change of the Lorentz scalar chiral
condensate
E ective chiral mo dels also have b een used to explore the order T
corrections Ref rep orts that the mass of the drops whereas that
of the a increases to this order This is somewhat at variance with the
ndings of where no drop in the mass of the but a decrease in the
a mass has b een found In this calculation however the terms leading
to the order T changes have not b een explicitly identi ed but rather
a calculation to one lo op order has b een carried out Presently the
di erence in these results is not understo o d Also the di erence b etween
the analysis of and the e ective chiral mo dels is not resolved yet
The analysis of uses disp ersion relations to relate phenomenological
space like photon pion amplitudes with the time like ones needed to
calculate the mass shift The e ective Lagrangian metho ds on the
other hand rely on pion pion scattering data as well as measured decay
width in order to x their mo del parameters One would think that
b oth metho ds should give a reasonably handle on the leading order
momentum dependent couplings
At higher temp eratures as well as at nite density vacuum e ects
due to the virtual pair correction or the nucleon antinucleon p olariza
tion could b ecome imp ortant and they tend to reduce vector meson
masses Because of the large mass of the nucleon these corrections
however do not a ect the leading temp erature result T discussed
previously Also this approach assumes that the physical vacuum con
sists of nucleon antinucleon rather than quark antiquark uctuations
Whether this is the correct picture is however not yet resolved
Finite density
Since the density e ect on the chiral condensate is much stronger than
that of the temp erature one exp ects the same for the rho meson mass
But the situation is more complicated at nite density as one cannot
make use of current algebra arguments and thus mo del indep endent
result as the one discussed previously are not available at this time
Present mo del calculations however disagree even on the sign of the
mass shift One class of mo dels considers the
as a pion pion resonance and calculates its in medium mo di cations
due to those for the pions as discussed in section These calculations
typically show an increased width of the rho since it now can also decay
into pion nucleon hole or pion delta hole states At the same time this
leads to an increased strength b elow the rho meson mass The mass
KO KOCH LI
of the rho meson de ned as the p osition where the real part of the
correlation function go es through zero is shifted only very little Most
calculations give an upward shift but also a small downward shift has
b een rep orted This di erence seems to dep end on the sp eci c
choice of the cuto functions for the vertices involved and thus
is mo del dep endent However the imaginary part of the correlation
function the relevant quantity for the dilepton measurements hardly
dep ends on a small upward or downward shift of the rho The imp ortant
and apparently mo del indep endent feature is the increased strength at
low invariant masses due to the additional decay channels available in
nuclear matter
Calculation using QCD sum rules predict a rather strong de
crease of the mass with density at nuclear matter density
which is similar to the much discussed Brown Rho scaling Here
as in the nite temp erature case the driving term is the four quark
condensate which is assumed to factorize
a a
h q q q q i hq q i
To which extent this factorization is correct at nite density is not clear
Also when it comes to the parameterization of the phenomenological
sp ectral distribution these calculations usually assume the standard
p ole plus continuum form with the addition of a so called Landau damp
ing contribution at q The additional strength b elow the
mass of the rho as predicted by the previously discussed mo dels is usu
ally ignored An attempt however has b een made to see if a sp ectral
distribution obtained from the e ective mo dels describ ed ab ove do es
saturate the QCD sum rules These authors found that they could
only saturate the sum rule if they assumed an additional mass shift of
the rho meson p eak downwards by MeV However in this calcu
lation only leading order density corrections to the quark condensates
whereas in nity order density e ects have b een taken into account in
order to calculate the sp ectrum Another comparison with the QCD
sum rules has recently b een carried out in ref where a reasonable
saturation of the sum rule is rep orted using a similar mo del for the in
medium correlation function
There are also attempts to use photon nucleon data in order to esti
mate p ossible mass shifts of the and mesons In ref a simple
pion and sigma exchange mo del is used in order to t data for photopro
duction of and mesons Assuming vector dominance this mo del is then used to calculate the self energy of these vector mesons in nuclear
IN MEDIUM EFFECTS
matter The authors nd a downward shift of the of ab out at nu
clear matter density in rough agreement with the prediction from QCD
sum rules Quite to the contrary ref using photoabsorption data
and disp ersion relations nd an upwards shift of MeV or MeV for
the longitudinal and transverse part of the resp ectively This result
however is derived for a meson which is not at rest in the nuclear
matter frame and therefore a direct comparison of the two predictions
is not p ossible
Very recently Friman and Pirner have p ointed out that the meson
couples very strongly with the N resonance The cou
pling is of p wave nature and similarly to the pion coupling to delta
hole states the may mix with N nucleon hole states resulting in a
mo di ed disp ersion relation for the meson in nuclear matter Since
the coupling is p wave only mesons with nite momentum are mo d
i ed and shifted to lower masses This momentum dep endence of the
low mass enhancement in the dilepton sp ectrum is an unique prediction
which can b e tested in exp eriment
Experimental results
First measurement of dileptons in heavy ion collisions have b een carried
out by the DLS collab oration at the BEVALAC The rst pub
lished data based on a limited data set could b e well repro duced using
state of the art transp ort mo dels without any additional
in medium corrections However a recent reanalysis including the
full data set seems to show a considerable increase over the originally
published data It remains to b e seen if these new data can also b e
understo o d without any in medium corrections to hadronic prop erties
Recently low mass dilepton data taken at the CERN SPS have b een
published by the CERES collab oration Their measurements for
p Be and p Au could b e well understo o d within a hadronic co cktail
which takes into account the measured particle yields from p p exp er
iments and their decay channels into dileptons In case of the heavy
ion collision S Pb however the hadronic co cktail considerably under
predicted the measured data in particular in the invariant mass region
of MeV M MeV A similar enhancement has also b een
inv
rep orted by the HELIOS collab oration
Of course it is well known that in a heavy ion collisions at SPS
energies a hadronic reball consisting predominantly of pions is created
These pions can pairwise annihilate into dileptons giving rise to an ad ditional source which has not b een included into the hadronic co cktail
KO KOCH LI -1 -1 ) ) 2 CERES S-Au 200 GeV/u 2 HELIOS-3 S-W 200 GeV/u Koch-Song Koch-Song Li-Ko-Brown -5 Li-Ko-Brown -5 Cassing-Ehehalt-Ko 10 Cassing-Ehehalt-Kralik 10 Srivastava-Sinha-Gale Srivastava-Sinha-Gale Baier-Redlich UQMD-Frankfurt ) (100 MeV/c η /d
ch -6 -6
10 /charged (50 MeV/c 10 µµ dm) / (dN η
/d -7 -7 p-W data hadron decays ee 10 10 hadron decays after freezeout after freezeout N 2
(d 0 0.25 0.5 0.75 1 0 0.5 1 2 [ 2]
mee (GeV/c ) mµµ GeV/c
Figure Dilepton invariant mass sp ectrum from S Au and S W colli
sions at GeV nucleon without dropping vector meson masses The
exp erimental statistical errors are shown as bars and the systematic
errors are marked indep endently by brackets The gure is from ref -1 -1 ) ) 2 CERES S-Au 200 GeV/u 2 HELIOS-3 S-W 200 GeV/u
Li-Ko-Brown -5 Li-Ko-Brown -5 Cassing-Ehehalt-Ko Cassing-Ehehalt-Kralik 10 10 ) (100 MeV/c η /d
ch -6 -6
10 /charged (50 MeV/c 10 µµ dm) / (dN η
/d -7 -7 p-W data hadron decays ee 10 10 hadron decays after freezeout after freezeout N 2
(d 0 0.25 0.5 0.75 1 0 0.5 1 2 2
mee (GeV/c ) mµµ (GeV/c )
Figure Same as Fig with dropping vector meson masses The
gure is from ref
IN MEDIUM EFFECTS
While the pion annihilation contributes to the desired mass range many
detailed calculations which have taken this channel into
account still underestimate the data by ab out a factor of three as shown
Fig Also additional hadronic pro cesses such as e e have
b een taken into account However conservative calculations
could at b est reach the lower end of the sum of statistical and systematic
error bars of the CERES data Furthermore in medium corrections to
the pion annihilation pro cess together with corrections to the meson
sp ectral distribution have b een considered using e ective chiral mo d
els But these in medium corrections while enhancing the
contribution of the pion annihilation channel somewhat are to o small
to repro duce the central data p oints Only mo dels which allow for a
dropping of the meson mass give enough yield in the low mass region
as shown in Fig In these mo dels the change
of the rho meson mass is obtained from either the Brown Rho scaling
or the QCD sum rules Furthermore the vector
dominance is assumed to b e suppressed in the sense that the pion elec
tromagetic from factor which in free space is dominated by the rho me
son and is prop ortional to the square of the rho meson mass is reduced
as a result of the dropping rho meson mass At nite temp erature such
a suppression has b een shown to exist in b oth the hidden gauge theory
and the p erturbtive QCD where the pion electromagnetic
form factor is found to decrease at the order T This e ect is related
to the mixing b etween the vector isovector and axial vector isovector
correlators at nite temp erature we discussed earlier In most studies
of dilepton pro duction without a dropping rho meson mass this e ect
has not b een included Although the temp erature reached in heavy ion
collisions at SPS energies is high and the ratio of pions to baryons in the
nal state is ab out to the authors in found that
they had to rely on the nuclear density rather than on the temp erature
in order to repro duce the data
There are also preliminary data from the Pb Au collisions at
GeV nucleon which also seem to b e consistent with the dropping
in medium rho meson scenario as well Unfortunately the exp erimental
errors in this case are even larger than in the S Au data and thus these
new preliminary data do not further discriminate b etween the dropping
rho mass and more conventional scenarios Thus additional data with
reduced error bars are needed b efore rm conclusions ab out in medium
changes of the rho meson mass can b e drawn Although the data from
the HELIOS collab oration shown in the right panels of Figs and
do have very small errors these are only the statistical ones while the
KO KOCH LI
systematic errors are not known
To summarize this section on the rho meson our present theoreti
cal understanding for p ossible in medium changes at low temp eratures
and vanishing baryon density indicates that one exp ects no to leading
order and a small to next to leading order changes of the mass
At the same time to leading order in the temp erature the chiral con
densate is reduced Therefore a direct connection b etween changes in
the chiral condensate and the mass of the meson has not b een estab
lished as assumed in the Brown Rho scaling However it may well b e
that the latter is only valid at higher temp erature near the chiral phase
transition At nite density additional e ects such as nuclear many
b o dy excitations come into play which make the situation much more
complicated Nevertheless the observed enhancement of low mass dilep
tons from CERN SPS heavy ion collisions can b est b e describ ed by a
dropping rho meson mass in dense matter Finite density and zero tem
p erature is another area where p ossible changes of the vector mesons can
b e measured in exp eriment Photon proton and pion induced dilepton
pro duction from nuclei will so on b e measured at CEBAF and at
GSI In principle these measurements should b e able to determine
the entire sp ectral distribution If the predictions of the Brown Rho
scaling and the QCD sum rules are correct mass shifts of the order of
MeV should o ccur which would b e visible in these exp eriments
Furthermore by choosing appropriate kinematics the prop erties of the
meson at rest as well as at nite momentum with resp ect to the nuclear
rest frame can b e measured
3.2 The omega meson
The omega meson couples to the isoscalar part of the electromagnetic
current In QCD sum rules it is also dominated by the four quark con
densate If the latter is reduced in medium then the omega meson
mass also decreases Assuming factorization for the four quark con
densate Hatsuda and Lee showed that the change of the omega
meson mass in dense nuclear matter would b e similar to that of the rho
meson mass The dropping omega meson mass in medium can also b e
obtained by considering the nucleon antinucleon p olarization in medium
as in the case of the rho meson On the other hand us
ing e ective chiral Lagrangians one nds at nite temp erature an even
smaller change in the mass of the omega as compared to that of the
rho Because of its small decay width which is ab out an order of magnitude smaller than that of rho meson most dileptons from omega
IN MEDIUM EFFECTS
decay in heavy ion collisions are emitted after freeze out where the
medium e ects are negligible However with appropriate kinematics an
omega can b e pro duced at rest in nuclei in reactions induced by the
photon proton and pion By measuring the decay of the omega into
dileptons allows one to determine its prop erties in the nuclear matter
Such exp eriments will b e carried out at CEBAF and GSI
3.3 The phi meson
For the phi meson the situation is less ambiguous than the rho meson as
b oth e ective chiral Lagrangian and QCD sum rule studies predict that
its mass decreases in medium Based on the hidden gauge theory Song
nds that at a temp erature of T MeV the phi meson mass is
reduced by ab out MeV The main contribution is from the thermal
kaon lo op In QCD sum rules the reduction is much more appreciable
i e ab out MeV at the same temp erature The latter is due
to the signi cant decrease of the strange quark condensate at nite
temp erature as a result of the abundant strange hadrons in hot matter
Because of the relative large strange quark mass compared to the up and
down quark masses the phi meson mass in QCD sum rules is mainly
determined by the strange quark condensate instead of the four quark
condensate as in the case of rho meson mass However the temp erature
dep endence of the strange quark condensate in is determined from
a non interacting gas mo del so e ects due to interactions which may
b e imp ortant at high temp eratures are not included In only the
lowest kaon lo op has b een included so the change of kaon prop erties
in medium is neglected As shown in this would reduce the phi
meson mass if the kaon mass b ecomes small in medium Also vacuum
e ects in medium due to lambda antilambda p olarization has also b een
shown to reduce the phi meson in medium mass
QCD sum rules have also b een used in studying phi meson mass at
nite density and it is found to decrease by ab out MeV at
normal nuclear matter density
Current exp erimental data on phi meson mass from measuring the
K K invariance mass sp ectra in heavy ion collisions at AGS energies do
not show a change of the phi meson mass This is not surprising
as these kaon antikaon pairs are from the decay of phi mesons at freeze
out when their prop erties are the same as in free space If a phi meson
decays in medium the resulting kaon and antikaon would interact with
nucleons so their invariant mass is mo di ed and can no longer b e used to
reconstruct the phi meson However future exp eriments on measuring
KO KOCH LI
dileptons from photon nucleus reactions at CEBAF and heavy ion
collisions at GSI will provide useful information on the phi meson
prop erties in dense nuclear matter For heavy ion collisions at RHIC
energies matter with low baryon chemical p otential is exp ected to b e
formed in central rapidities Based on the QCD sum rule prediction
for the mass shift of the phi meson it has b een suggested that a low
mass phi p eak at MeV b esides the normal one at GeV
app ears in the dilepton sp ectrum if a rst order phase transition or
a slow cross over b etween the quark gluon plasma and the hadronic
matter o ccurs in the collisions The low mass phi p eak is
due to the nonnegligible duration time for the system to stay near the
transition temp erature compared with the lifetime of a phi meson in
vacuum so the contribution to dileptons from phi meson decays in the
mixed phase b ecomes comparable to that from their decays at freeze
out Without the formation of the quark gluon plasma the low mass
phi p eak is reduced to a shoulder in the dilepton sp ectrum Thus one
can use this double phi p eaks in the dilepton sp ectrum as a signature for
identifying the quark gluon plasma to hadronic matter phase transition
in ultrarelativistic heavy ion collisions
BARYONS
As far as the in medium prop erties of the baryons are concerned most is
known ab out the nucleon But also the prop erties of the hyperons such
as the can b e determined in the medium by studying hypernuclei
Also the which we have discussed in connection with the K
optical p otential is another example for in medium e ects of baryons
In the following we will limit ourself on a brief discussion of how the
nucleon prop erties can b e viewed in the context of chiral symmetry
As mentioned brie y in the introduction medium e ects on a nu
cleon due to many bo dy interactions have long b een studied leading
to an e ective mass which is generally reduced as a result of the nite
range of the nucleon interaction and higher order e ects Its relation
to chiral symmetry restoration is b est seen through the QCD sum rules
Using in medium condensates it has b een
shown that the change of scalar quark condensate in medium leads to
an attractive scalar p otential which reduces the nucleon mass while the
change of vector quark condensate leads to a repulsive vector p otential
which shifts its energy The nucleon scalar and vector self energies in
IN MEDIUM EFFECTS
this study are given resp ectively by
N
hq q i hq q i
S N
M M m m
u d
B B
hq q i
N V
M M
B B
where the Borel mass M is an arbitrary parameter With M m
B B N
and m m MeV these self energies have magnitude of a few
u d
hundred MeV at normal nuclear density These values are similar to
those determined from b oth the Walecka mo del and the Dirac
Brueckner Hartree Fock DBHF approach based on the meson
exchange nucleon nucleon interaction Exp erimental evidences for these
strong scalar and vector p otentials have b een inferred from the proton
nucleus scattering at intermediate energies via the Dirac phe
nomenology in which the Dirac equation with scalar
and vector p otentials is solved The imp ortance of medium e ects on
the nucleon mass can b e seen from the signi cant decrease of the Q
value for the reaction NN NNN N in nuclear medium as a result of
the attractive scalar p otential This e ect has b een included in a num
b er of studies based on transp ort mo dels In Fig theoretical results
from these calculations for the antiproton di erential cross section in
Ni Ni collisions at GeV nucleon and C Cu col
lisions at GeV nucleon are compared with the exp erimental
data from GSI and Dubna In the relativistic
transp ort mo del has b een used with the antiproton mean eld p otential
obtained from the Walecka mo del The latter has a value in the range
of to MeV at normal nuclear matter density The results of
ref are based on the nonrelativistic Quantum Molecular Dynam
ics with the pro duced nucleon and antinucleon masses taken from the
Nambu Jona Lasinio mo del Within this framework these studies thus
show that in order to describ e the antiproton data from heavy ion colli
sions at subthreshold energies it is necessary to include the reduction of
b oth nucleon and antinucleon masses in nuclear medium Even at AGS
energies which are ab ove the antiproton pro duction threshold in NN
interaction the medium e ects on antiproton may still b e imp ortant
Indeed a recent study using the Relativistic Quantum Molecular
Dynamics shows that medium mo di cations of the antiproton prop er
ties are imp ortant for a quantitative description of the exp erimental
data We note however a b etter understanding of antiproton an
nihilation is needed as only ab out of pro duced antiprotons can
KO KOCH LI
Figure Antiproton momentum sp ectra from Ni Ni collisions at
GeV nucleon and C Cu collisions at GeV nucleon The left mid
dle and right panels are from Refs and resp ectively
The solid and dotted curves are from transp ort mo del calculations with
free and dropping antiproton mass resp ectively The exp erimental data
from ref for Ni Ni collisions and from ref for C Cu colli sions are shown by solid circles
IN MEDIUM EFFECTS
survive in order to determine more precisely the antiproton in medium
mass from heavy ion collisions
RESULTS FROM LATTICE QCD CAL
CULATIONS
Another source of information ab out p ossible in medium changes are
lattice QCD calculations For a review see e g Presently lattice
calculations can only explore systems at nite temp erature but van
ishing baryon chemical p otential Therefore their results are not di
rectly applicable to present heavy ion exp eriments where system at
nite baryon chemical p otential are created But future exp eriments at
RHIC and LHC might succeed in generating a baryon free region But
aside from the exp erimental asp ects lattice results provide an imp or
tant additional source where our mo del understanding can b e tested
Lattice calculation are usually carried out in Euclidean space i e in a
space with imaginary time As a consequence plane waves in Minkowski
space translate into decaying exp onentials in Euclidean space At zero
temp erature masses of the low lying hadrons are extracted from two
p oint functions which carry the quantum numbers of the hadron under
consideration
Z
X
exp m C d x J x J
i
i
i
Here the sum go es over all hadronic states which carry the quantum
numbers of the op erator J At large imaginary times only the state
with the lowest mass survives and its mass can b e determined from
the exp onential slop e Zero temp erature lattice calculation by now re
pro duce the hadronic sp ectrum to a remarkable accuracy If one
wants to extract masses at nite temp erature however things b ecome
more complicated The reason is that nite temp erature on the lattice
is equivalent to requiring p erio dic or anti perio dic b oundary conditions
in the imaginary time direction for b osons or fermions resp ectively
Therefore one cannot study the correlation functions at arbitrary large
but is restricted to T where T is the temp erature under
max
consideration As a result the lowest lying states cannot b e pro jected
out so easily This has led p eople to study the correlation functions as
a function of the spatial distance where no restriction to the spatial
extent exist to extract so called screening masses At zero temp erature
KO KOCH LI
the Euclidean time and space direction are equivalent and the screen
ing masses are identical to the actual hadron masses By analyzing
the correlation function in the spatial direction one essentially measures
the range of a virtual particle emission As already demonstrated by
Yukawa who deduced the mass of the pion from the range of the nu
clear interaction this can b e used to extract particle masses Screening
masses however are not very useful if one wants to study hadronic
masses at high temp erature close to T At these temp eratures the
c
spatial correlators are dominated by the trivial contribution from free
indep endent quarks and the resulting screening masses simply turn out
to b e
M n T
screen
where n is the number of quarks of a given hadron i e n for
mesons and n for baryons Only for the pion and sigma meson lat
tice calculation found a signi cant deviation from this simple b ehavior
which is due to strong residual interactions in these channel All other
hadrons considered so far such as the and a as well as the nucleon
exhibit the ab ove screening mass These trivial contributions
from indep endent quarks are absent however in the time like corre
lator There has b een one attempt to extract meson masses from the
time like correlators in four avor QCD In this case it has b een
found that the mass of the meson do es not change signi cantly b elow
T Ab ove the critical temp erature on the other hand the correlation
c
function app eared to b e consistent with one of noninteracting quarks
indicating the onset of decon nement
There have b een attempts to extract the mass of the scalar meson
as well as that of the pion from so called susceptibilities These
are nothing else than the total four volume integral over the two point
functions If the two point function is dominated by one hadronic p ole
then this integral should b e inversely prop ortional to the square of the
mass of the lightest hadron under consideration Using this metho d
the degeneracy of the pion and sigma mass close to T has b een demon
c
strated
Finally let us note that a scenario where all hadron masses scale
linearly with the chiral condensate is consistent with strong rise in the
energy and entropy density around the phase transition as observed in Lattice calculations
IN MEDIUM EFFECTS
SUMMARY
In this review hadron prop erties particularly their masses in the nu
clear medium have b een discussed Both conventional many bo dy inter
actions and genuine vacuum e ects due to chiral symmetry restoration
have e ects on the hadron in medium prop erties
For Goldstone b osons which are directly linked to the sp ontaneously
broken chiral symmetry we have discussed the pion kaons and etas
For the pion its mass is only slightly shifted in the medium due to
the small s wave interactions as a result of chiral symmetry On the
other hand the strong attractive p wave interactions due to the
and the lead to a softening of the pion disp ersion relation in
medium These e ects result from many bo dy interactions rather from
chiral symmetry Phenomenologically the small change in the pion in
medium mass do es not seem to have any observable e ects However
the e ects due to the softened pion in medium disp ersion relation may
b e detectable through the enhanced low transverse momentum pions in
heavy ion collisions
For kaons medium e ects due to the Weinberg Tomozawa vector
type interaction which are negligible for pions in asymmetric nuclear
matter are imp ortant as they have only one light quark However
b ecause of the large explicit symmetry breaking due to the nite strange
quark mass higher order corrections in the chiral expansion are non
negligible leading to an appreciable attractive scalar interaction for b oth
the kaon and the antikaon in the medium Available exp erimental data
on b oth subthreshold kaon pro duction and kaon ow are consistent with
the presence of this attractive scalar interaction
For etas their prop erties are more related to the explicit breaking of
the U axial symmetry in QCD If the U symmetry is restored in
A A
medium as indicated by the instanton liquid mo del then their masses
are exp ected to decrease as well Heavy ion exp eriments at the CERN
SPS however rule out a vast enhancement of the nal state eta yield
In the case of vector mesons we have discussed the rho omega and
phi At nite temp erature all mo del calculations agree with the cur
rent algebra result that the mass of the rho do es not change to order
T To higher order in the temp erature and at nite density QCD sum
rules predict a dropping of the masses in the medium Predictions from
chiral mo dels on the other hand tend to predict an increase of the
rho mass with temp erature However presently available dilepton data
from CERN SPS heavy ion exp eriments are b est describ ed assuming a dropping rho mass in dense matter but the errors in the present exp er
KO KOCH LI
imental data are to o large to de nitely exclude some more conventional
explanations
For baryons particularly the nucleon the change of their prop erties
in the nuclear medium has b een well known in studies based on conven
tional many bo dy theory On the other hand the nucleon mass is also
found to b e reduced in the medium as a result of the scalar attractive
p otential related to the quark condensate A clear separation of the vac
uum e ects due to the condensate from those of many bo dy interactions
is a topic of current interest and has not yet b een resolved A dropping
nucleon e ective mass in medium seems to b e required to explain the
large enhancement of antiproton pro duction in heavy ion collisions at
subthreshold energies
In principle lattice QCD calculations could help answer quite a few
of these questions although they are restricted to systems at nite tem
p erature and vanishing baryon density However with the presently
available computing p ower reliable quantitative predictions ab out in
medium prop erties of hadrons are still not available
The study of in medium prop erties of hadrons is a very active eld
b oth theoretically and exp erimentally While many questions are still
op en and require additional measurements and more careful calcula
tions it is undisputed that the study of in medium prop erties of hadrons
provides us with an unique opp ortunity to further our understanding
ab out the long range nonp erturbative asp ects of QCD
Acknowledgments We are grateful to many colleagues for helpful
discussions over the years Also we would like to thank G Boyd G E
Brown A Drees B Friman C Gaarde F Klingl R Rapp C Song T
Ullrich and W Weise for the useful information and discussions during
the preparation of this review The work of CMK was supp orted in part
by the National Science Foundation under Grant No PHY
V K was supp orted the Director O ce of Energy Research O ce of
High Energy and Nuclear Physics Division of Nuclear Physics and by
the O ce of Basic Energy Sciences Division of Nuclear Sciences of the
U S Department of Energy under Contract No DE AC SF
GQL was supp orted by the Department of Energy under Contract No
DE FG Er
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