Particle ratios at RHIC: Effective hadron masses and chemical freeze-out

The measured particle ratios in central heavy-ion collisions at RHIC-BNL are investigated within a chemical and thermal equilibrium chiral SU(3) \sigma-\omega approach. The commonly adopted noninteracting gas calculations yield temperatures close to or above the critical temperature for the chiral phase transition, but without taking into account any interactions. Contrary, the chiral SU(3) model predicts temperature and density dependent effective hadron masses and effective chemical potentials in the medium and a transition to a chirally restored phase at high temperatures or chemical potentials. Three different parametrizations of the model, which show different types of phase transition behaviour, are investigated. We show that if a chiral phase transition occured in those collisions, ''freezing'' of the relative hadron abundances in the symmetric phase is excluded by the data. Therefore, either very rapid chemical equilibration must occur in the broken phase, or the measured hadron ratios are the outcome of the dynamical symmetry breaking. Furthermore, the extracted chemical freeze-out parameters differ considerably from those obtained in simple noninteracting gas calculations. In particular, the three models yield up to 35 MeV lower temperatures than the free gas approximation. The in-medium masses turn out differ up to 150 MeV from their vacuum values.

Recently hadron abundances and particle ratios have been measured in heavy-ion collisions from SIS, AGS, SPS to RHIC energies. These data have revived the interest in the extraction of temperatures and chemical potentials from thermal equilibrium "chemical" model analyses. The experimentally determined hadron ratios can be fitted well with straightforward noninteracting gas model calculations [4,6,9,10,11,12,13], if a sudden breakup of a thermalized source is assumed and once the subsequent feeding of the various channels by the strongly decaying resonances is taken into account. From the χ 2 freeze-out fits one has constructed a quite narrow band of freeze-out values in the T − µ B plane (see e.g. [12,13]).
The extracted freeze-out parameters are fairly close to the phase transition curve for SPS and RHIC energies. However, when we are indeed so close to the phase transition or to a crossover as suggested by the data for T and µ B , we can not afford to neglect the very in-medium effects we are after -and which, after all, do produce the phase transition. Thus, since noninteracting gas models neglect any kind of possible in-medium modifications they can not yield information about the phase transition.
Therefore, we will employ below a relativistic selfconsistent chiral model of hadrons and hadron matter developed in [14,15,16]. This model can be used as a thermodynamically consistent effective theory or as a toy model, which embodies the restoration of chiral symmetry at high temperatures or densities. Therefore the model predicts temperature and density dependent hadronic masses and effective chemical potentials, which have already been proposed and considered in [5,14,17,18,19,20]. Thus, using the chiral SU(3) model we can investigate, whether the freeze-out in fact takes place close to the phase transition boundary (if it exists) and if the extracted T, µ B parameters are strongly model dependent.
Depending on the chosen parameters and degrees of freedom different scenarios for the chiral phase change are predicted by the model: Strong or weak first order phase transition or a crossover. The transitions take place around T c = 155 MeV [14,21], which is in qualitative agreement with lattice predictions [22] for the critical temperature for the onset of a deconfined phase which coincides with that of a chirally restored phase [23].

MODEL DESCRIPTION
The chiral SU(3) model is presented in detail in [14,16]. We will briefly introduce the model here: We consider a relativistic field theoretical model of baryons and mesons built on chiral symmetry and broken scale invariance. The general form of the Lagrangean looks as follows: L kin is the kinetic energy term, L BW includes the interaction terms of the different baryons with the various spin-0 and spin-1 mesons (see [16] for details). The baryon masses are generated by both, the nonstrange σ (< qq >) and the strange ζ (< ss >) scalar condensate.
L VP contains the interaction terms of vector mesons with pseudoscalar mesons. L vec generates the masses of the spin-1 mesons through interactions with spin-0 fields, and L 0 gives the meson-meson interaction terms which induce the spontaneous breaking of chiral symmetry.
It also includes a scale-invariance breaking logarithmic potential. Finally, L SB introduces an explicit symmetry breaking of the U(1) A , the SU(3) V , and the chiral symmetry. All these terms have been discussed in detail in [14,16].
The hadronic matter properties at finite density and temperature are studied in the meanfield approximation [24]. Then the Lagrangean (1) becomes where m i is the effective mass of the hadron species i. σ and ζ correspond to the scalar condensates, ω and φ represent the non-strange and the strange vector field respectively, and χ is the scalar-isoscalar dilaton field, which mimics the effects of the gluon condensate [25]. Only the scalar (L BX ) and the vector meson terms (L BV ) contribute to the baryonmeson interaction, since for all other mesons the expectation value vanishes in the mean-field approximation. The grand canonical potential Ω per volume V as a function of chemical potential µ and temperature T can be written as: with the baryons (top sign) and mesons (bottom sign). The vacuum energy V vac (the potential at ρ B = 0, T = 0) has been subtracted in order to get a vanishing vacuum energy.
γ i denote the hadronic spin-isospin degeneracy factors. The single particle energies are The mesonic fields are determined by extremizing Ω V (µ, T = 0). The density of particle i can be calculated by differentiating Ω with respect to the corresponding chemical potential µ i . This yields: All other thermodynamic quantities can also be obtained from the grand canonical potential.
In the present calculation the lowest lying baryonic octet and decuplet and the lowest lying mesonic nonets are coupled to the relativistic mean fields. Depending on the coupling of the baryon resonances (the decuplet) to the field equations, the model shows a first order phase transition or a crossover (for details see [21]). We will use three different parameter sets: Parameter set CI treats the members of the baryon decuplet as free particles, which yields a crossover behaviour. Parameter sets CII and CIII include also the (anti)-baryon decuplet as sources for the meson field equations. They differ by an additional explicit symmetry breaking for the baryon resonances along the hypercharge direction, as described in [16] for the baryon octet. This is included in CII and not used in CIII. This leads to a weak first order phase transition at µ = 0 for CII and two first order phase transitions for CIII, which can be viewed as one strong first order phase transition. Heavier resonances up to m = 2 GeV are always included as free particles. The resulting baryon masses for CI and CIII are shown in fig. 1. We observe a continous decrease of the baryon masses for CI starting at T ≈ 150 MeV. In contrast, CIII shows two jumps around T = 155 MeV. The critical energy densities, the entropy densities and the transition temperatures for µ q = µ s = 0   Since the chiral SU(3) model predicts density and temperature dependent hadronic masses and effective potentials, in contrast to noninteracting models, the resulting particle ratios and therefore the deduced freeze-out temperatures and baryon chemical potentials are expected to change [26]. Hence in the following, we identify combinations of temperatures and chemical potentials that fit the observed particle ratios in the chiral model. In all calculations the value of the strange chemical potential µ S is chosen such that the net strangeness f s = 0. We are looking for minima of χ 2 with Here r exp i is the experimental ratio, r model i is the ratio calculated in the model and σ i represents the error in the experimental data points. We use the same ratios as in [6]:p/p,Λ/Λ, We rather focus on the principal question whether an interacting chiral SU(3) approach with m * = m vac can at all describe the particle yields at RHIC. Fine tuning of the χ 2 by adjustment of the weak decay scheme is not our intention. Even though it has been shown [29] that χ 2 values may be improved by including weak decays.
To compare the quality of the fits obtained in the chiral model with those obtained from the noninteracting gas approach, we set all masses and chemical potentials contained in the chiral model to their vacuum values and again use the same UrQMD feeding procedure as for the interacting model. This yields the ideal gas denoted ig F F M . We find that the resulting ideal gas ratios are not identical but comparable to those obtained in the literature [6,26,29,30]. The differences should only result from a different treatment of weak interactions and from the uncertainty in the decay scheme of high mass resonances.

RESULTS FOR AU + AU COLLISIONS AT RHIC
First, we find that a reasonable fit of the measured particle ratios at RHIC is possible in all three phase transition scenarios of the chiral model and the ideal gas case with comparable quality.
Second, the resulting freeze-out values depend on the model employed, i.e. crossover, weak first order, strong first order or free thermal gas.
Third, a reasonable description of the data is impossible above T c in the models showing a first order phase transition. This shows that no direct freeze-out from the restored phase is observed.    The resulting best-fit particle ratios, The fact that the freeze-out appears right at the phase boundary or at crossover implies that there are large in-medium corrections, in particular for the effective masses, a phenomenon observed already in [17].  These results, together with the steep χ 2 contours from Fig. 2, suggest that the relative particle abundances "freeze" shortly after the spontaneous breaking of chiral symmetry.
The success of our fit suggests extremely rapid chemical equilibration (through abundancechanging reactions) in the state with broken symmetry. Fig. 2 shows that the chemical composition of the hadronic system has to change substantially within a small temperature interval, just before freeze-out, even for the crossover transition (i.e. parameter set CI); for reference, we have indicated the dynamical path in the T − µ B plane corresponding to the expansion of a perfect fluid (i.e. with constant entropy per net baryon [34]). While 2 → n reactions are perhaps too slow to explain such rapid chemical equilibration [35,36], m → n processes with several particles in the initial state may be important as well [37,38,39,40].
Alternatively, the appearence of chemical equilibrium right after the phase transition (or the crossover) to the state of broken chiral symmetry might just be the outcome of the dynamical symmetry breaking process itself [41], with statistical occupation of the various hadronic channels according to phase space [42,43,44,45]. If so, number-changing reactions in the broken phase need not proceed at a high rate. To test this picture experimentally, it might be useful to consider central collisions of small ions like protons or deuterons, at similar energy and particle densities in the central region as for central Au+Au. For systems of transverse extent comparable to the correlation lengths of the chiral condensates, the dynamical symmetry breaking process should be different from that in large systems (for example, the mean field approximation should not apply). The correlation lengths ξ σ,ζ are given by and accordingly for ξ ζ . We evaluate the curvature of the thermodynamical potential at the global minimum and for T , µ B , µ S at the freeze-out point. For parameter sets CI, CII, CIII we obtain ξ σ = 0.37 fm, 0.41 fm, 0.40 fm, respectively. For the correlation length of the strange condensate we obtain ξ ζ = 0.20 fm in all three cases. The correlation lengths are not very much smaller than, say, the radius of a proton. Thus, even if the freeze-out point for high-energy pp collisions happens to be close to that for Au+Au collisions at RHIC energies, the transition from the symmetry restored to the broken phase might be different.
Finally, we also note that the correlation lengths obtained from our effective potential are not larger than the thermal correlation length 1/T at freeze-out, and so corrections beyond the mean-field approximation employed here should be analyzed in the future.