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PHYSICS LETTERS B

ELSEVIER Physics Letters B 596 (2004) 61-69 ■

www.elsevier.com/locate/physletb

Chemical freeze-out and the QCD phase transition temperature

P. Braun-Munzingera, J. Stachelb, Christof Wetterichc

a GSIDarmstadt, Germany b Physikalisches Institut, Universität Heidelberg, Germany c Institut für Theoretische Physik, Universität Heidelberg, Germany

Received 3 November 2003; received in revised form 8 March 2004; accepted 27 May 2004

Available online 2 July 2004

Editor: J.-P. Blaizot

Abstract

We argue that hadron multiplicities in central high energy nucleus-nucleus collisions are established very close to the phase boundary between hadronic and quark matter. In the hadronic picture this can be described by multi-particle collisions whose importance is strongly enhanced due to the high particle density in the phase transition region. As a consequence of the rapid fall-off of the multi-particle scattering rates the experimentally determined chemical freeze-out temperature is a good measure of the phase transition temperature. © 2004 Elsevier B.V. All rights reserved.

PACS: 25.75.-q

The yield of (multi-)strange hadrons produced in central high energy nucleus-nucleus collisions was proposed two decades ago [1] as a signature of quark-gluon plasma (QGP) formation. Hadron yields observed in such collisions at AGS, SPS, and RHIC energies are found to be described with high precision within a hadro-chemical equilibrium approach [2-10], governed by a chemical freeze-out temperature Tch, baryo-chemical potential \x and the fireball volume Vch. A recent review can be found in [11].

E-mail address: p.braun-munzinger@gsi.de (P. Braun-Munzinger).

Importantly, the data at SPS and RHIC energy comprise multi-strange hadrons including the Q and Q. Their yields agree with the chemical equilibrium calculation and are strongly enhanced as compared to observations inpp collisions. The time needed to achieve this equilibrium for Q baryons via two-body collisions was estimated [1] to be much longer than reasonable lifetimes of the fireball. The observations were thus interpreted as a sign that the system had reached a par-tonic phase prior to hadron production [12-14].

In this Letter we argue that the chemical freeze-out temperature Tch is actually very close to the critical temperature Tc of the QCD phase transition. This observation has afar reaching consequence: since Tch has

0370-2693/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.05.081

been measured for different values of x the approximate association of Tc with Tch implies that we have experimental knowledge of part of the critical line in the QCD-phase diagram.

Let us first sketch our overall picture and detail our arguments subsequently. Hadro-chemical equilibration is achieved during or at the end of the phase transition. In particular, the number of strange quarks may be established in the plasma phase and/or hadroniza-tion of the QGP. During the very early stages of the hadronic phase the relative numbers of strange baryons and mesons K, K*,A,S,E,Q are then realized at the thermal equilibrium values according to the Bose-Einstein or Fermi distribution

"J = {% f p2ip{exp[{Ejw-iij)/T]±ir1-0

Here Ej = Mj + p2 and Mj proportional to the vacuum mass of the hadron j, ¡j is the effective chemical potential, and gj counts degrees of freedom (for details see [11]). The high accuracy of the distribution (1) in reproducing the data suggests that Tch plays effectively the role of a universal temperature, governing simultaneously the chemical and kinetic distributions.

In the hadronic picture the production of multistrange hadrons can be described by multi-hadron strangeness exchange reactions.1 The multi-hadron scattering is substantial, however, only in the immediate vicinity of the critical temperature Tc. As T decreases the multi-particle rates drop very rapidly with a high power of the particle density. Below some temperature Tch very close to Tc only two-particle interactions and decays remain as relevant processes for a change in the relative particle numbers. These are too slow in order to equilibrate the distributions or to catch up with the decreasing temperature—chemical freeze-out occurs for Tch ^ Tc. We proceed to discuss

1 Production of multi-strange baryons by multi-particle collisions has also been considered by [15]. Their argument focuses on anti-hyperon production at high baryon density, where indeed relatively short equilibration times are obtained. The authors conclude that their approach should not be applicable for RHIC energies, unless the hadronic phase has a rather long lifetime. Furthermore this approach does not take account of the expected rapid change of density near a phase transition which is central for our argument.

the three main points of this scenario in the following in more detail.

(A) The QCD phase transition corresponds to a change in the effective degrees of freedom (from hadrons to quarks and gluons) in a narrow temperature interval.2 For both the hadronic and quark-gluon phases sufficiently far away from Tc the dominant processes in thermal equilibrium are two-particle scattering and decays. This is consistent with an effective (pseudo-)particle description. Close to Tc, however, collective phenomena play an important role. (As an example, near Tc the a -resonance may behave like a particle with mass almost degenerate with the pions.) If Tch is close to Tc the chemical equilibration may be described in several equivalent pictures: multi-hadron scattering, time evolving classical fields or hadronization. We emphasize, however, that no picture should contradict a hadronic description. In the end, the chemical equilibrium distribution has to be established by fast processes involving hadrons (not quarks and gluons).

Let us approach the phase transition (or a rapid crossover) from the hadronic phase. For T sufficiently below Tc not much happens on the level of microscopic scattering processes between hadrons. The main effect of an increase of the temperature is an increase of the density. Near Tc, however, the density is so high that new dynamics can be associated with collective excitations. It is precisely the behavior of these collective excitations that triggers the transition. On the level of individual hadrons the propagation and scattering of collective excitations is expressed in the form of multi-hadron scattering. Since the collective dynamics becomes dominant only near Tc the same holds for the multi-hadron scattering.

We next argue that the temperature range where multi-hadron processes can dominate is actually very narrow (typically a few MeV). This is due to (i) a rapid increase of the particle densities as a function of T and (ii) a very steep dependence of multi-particle scattering rates on the density of the incoming particles. Evidence for rapid energy density changes near Tc comes from recent lattice QCD results [16]. This is shown in Fig. 1, where the observed rapid rise in en-

2 Our central argument will make no distinction between a true phase transition and a rapid "crossover".

14 12 10 8 6 4 2

if 2+1 LQCD-

Slat. Model wo. excl. Vol. .

^ Stat. Model

T (GeV)

Fig. 1. Comparison of the temperature dependence of energy density in lattice calculations with 2 light and 1 heavier quark flavor [16] with results from the hadronic gas model of [2,4] with and without repulsive interactions.

ergy density beyond the simple T4 dependence reflects the large increase in degrees of freedom. The statistical model of a hadron resonance gas [2,4] exhibits only a moderate increase beyond the T4 behavior (see Fig. 1), determined by an interplay between the relevant number of hadronic states and the increased importance of repulsive interactions modeled by an excluded volume correction.3

The rate of an individual multi-particle scattering process with nm incoming particles and average density n grows as n (T)nin. In addition, also the phase space increases with temperature. In the temperature region of multi-particle scattering dominance one expects that many processes (with different nin) become of similar strength, thus increasing again the total rate. For the purpose of demonstration we model the temperature dependence of a typical rate (say for scatterings which change the number of Q -baryons) by r ~ n(T)Y, with y ^ 1. Within a narrow temperature interval AT = 5 MeV this rate changes by a substantial factor (1 + ^ffy with (i = dlnw/dlnr » 1. This is demonstrated in Fig. 2.At Tc, the multi-particle scattering rates are substantial. For T < Tc, however,

3 Neglecting the repulsive interactions the energy density of a hadronic gas diverges, reflecting the Hagedorn temperature (see dotted line in Fig. 1 and [17]).

Fig. 2. Time tq = nq/rq needed to bring q baryons into chemical equilibrium via multi-particle collisions. We display the dependence on the pion density nn and on t. The temperature scale is obtained as discussed in the text. The arrow indicates the chemical freeze-out temperature Tch = 176 MeV.

they drop so rapidly that only a small temperature interval around Tc is left where the multi-particle scattering processes can dominate. The evaluation of these multi-particle rates is detailed in point (C) below.

(B) Let us next turn to the issue of chemical equilibrium and argue that two-particle scattering is insufficient to achieve or sustain it. We focus in the following on the analysis of data at RHIC energies and comment at the end on the applicability of our considerations at lower energies. The high accuracy of the statistical model predictions in reproducing experimental particle ratios varying over several orders of magnitude [7,11] should be taken as an indication that hadro-chemical equilibration has occurred in the system. Further, chemical equilibration according to Eq. (1) requires two crucial ingredients. (i) The particle number changing reaction rates must be sufficiently high such that the particle distributions can adapt to a given T. (ii) At freeze-out the masses of the different hadrons must be proportional to theirvacuum masses. From (ii) we conclude immediately that chemical equilibration and freeze-out occur in the hadronic phase, Tch < Tc. The particle distribution in the quark-gluon plasma has no memory of the individual hadronic vacuum

masses—for example, the relative density of strange particles (at the same given f¿ and T) would be determined by the strange quark mass m5 rather than by the individual masses of K, K*,A,S,3,n.

To demonstrate that Tch is close to Tc we exclude two possible alternative scenarios with Tch substantially smaller than Tc: first, we show that an extended period of "hadronic chemical equilibrium" with Tch < T < Tc is quite unlikely. Second, we generalize this argument to demonstrate that "late equilibration" at a temperature close to Tch but significantly smaller than Tc is also disfavored.

The first argument is conceptually simple since the condition for chemical equilibrium at a temperature T substantially smaller than Tc can be formulated by using the equilibrium rates and distributions. For our purpose complete thermalization of all quantities is not necessary—the "prethermalization" [18] of some rough quantities like relative particle densities and approximate momentum distributions is sufficient,such that it makes sense to speak about temperature and chemical potential (in the sense of Eq. (1)) and to compute rates in thermal equilibrium.

We first need the relevant time scale to which the two-particle scattering rates have to be compared. In equilibrium this is given by the inverse rate of decrease of temperature tt . For this purpose we assume, guided by recent results from two-pion correlation measurements [19-22], two possible scenarios for the evolution of the fireball. In both cases we use the observation that the density decreases by only 30% between chemical and thermal freeze-out and our knowledge of Tch = 176 MeV. From the two-pion correlation data we obtain, for a central rapidity slice, the transverse and longitudinal radii at thermal freeze-out of 5.75 and 7.0 fm, a longitudinal expansion velocity j\\ = 1, and a transverse expansion velocity = 0.5. The thermal freeze-out radii are to be understood as widths of Gaussian distributions and give a volume of Vf = 3650 fm3. Scenario (1) assumes that the shape of the density distributions is the same (i.e., Gaussian) at thermal and chemical freeze-out and that accordingly an increase in density by 30% is equivalent to a 30% decrease in volume corresponding to Vch = 2600 fm3. In scenario (2) we use an initial volume of 1450 fm3 corresponding to a flat distribution over one unit of rapidity. With the assumption of isentropic expansion at the above velocities we find the longitudinal and

transverse radius parameters at Tch as well as the duration of the expansion and the thermal freeze-out temperature Tf. For scenarios (1) and (2) the duration of the expansion in the hadronic phase is rf = 0.9 and 2.3 fm and the thermal freeze-out temperature is Tf = 158 and 132 MeV. Consequently, in the hadronic phase near Tch the rate of decrease in temperature may be estimated as | T/T | = t-1 = (13 ± 1)%/fm = (7.7 ± 0.6 fm)-1. Note that these time scales are entirely consistent with the duration of pion emission, also obtained from the two-pion correlation data, of less than 2 fm. In both scenarios the lifetime of the fireball is rather short, leaving little room for an extended period of "hadronic cooking" at temperatures significantly below Tc.

As a simple example, a decrease in temperature by AT = 5 MeV reduces the equilibrium ratio of Q-baryons over kaons, nQ/nK, by a factor FQK = 1.13 (using thermal model densities of [7]). Adaptation of the particle distribution to the changing temperature requires FQK = exp(aAt) = exp(atT AT / Tch) with a = dln(nQ/nK)/dt. Let us define the rate of change of individual particle number densities as

rj = ^=nj+njV/V.

(The last term accounts for the decrease in particle number densities due to the volume change.) Maintaining chemical equilibrium needs

Tq tk lnFqk Tch 1.10 - 0.55 0.55

The numerical evaluation of the two terms in the difference agrees well with the direct evaluation of the term involving FQK in Eq. (3). For the difference in the rate of relative density change of Q -baryons to protons we obtain similarly a value of (1.10 -0.90)/fm = 0.20/fm. Both rates are evaluated here at Tch. Eq. (3) can be obeyed either by the destruction of Q -baryons or the production of kaons. Since Q -baryons decay weakly the effect of decays is completely negligible. A typical two-particle scattering Q + K ^ 3 + n (or Q + n ^ 3 + K) yields at most rQ/nQ = n^(va) = 0.018/fm. Forthis estimate we used a (large) strangeness exchange cross section of a = 10 mb and a relative velocity of v = 0.6. Similarly, for a typical kaon production process n + n ^

K + K one would obtain rK/nK = 0.18/fm, using the measured cross section of 3 mb [23]. Clearly both numbers are much too small to maintain equilibrium close to Tch. Analogous arguments can be made for the Q/p ratio. The reaction n + + S- ^ K- + p contributes to the rate of relative density change of protons a value of 0.018/fm. One would need about 50 reactions with similar cross section (12 mb, [24]) to keep the proton density in equilibrium. For the Q baryons there is obviously no way to achieve this. We conclude that two-particle reactions and decays are not fast enough to maintain chemical equilibrium for multi-strange baryons in the hadronic phase near Tch. This finding is also supported by studies using cascade codes [25] and rate equations [26].

We next turn to the production rates for situations where thermal equilibrium has not yet been realized, e.g., during hadronization. The production rates should be consistent between different pictures for such processes and permit a hadronic description, at least towards the end of the chemical equilibration process. Therefore our picture does not require a detailed understanding of hadronization. Although our estimates of rj/nj have used thermal distributions for the incoming particles our arguments can be extended to non-thermal situations: there is no reason why the rates should be much larger or the available times longer. In the final approach to chemical equilibrium (needed in order to achieve the high accuracy of the observed thermal description of the data) the densities of incoming particles must already be close to equilibrium. Also only very rough features of the momentum distribution of the incoming particles are needed for an estimate of the magnitude of rj/nj. Therefore, the two-particle scattering is also too slow to achieve chemical equilibration in the production of multistrange hadrons. In particular, this applies to a possible picture of chemical equili-brationby hadronization: hadronization at T much below Tc would not lead to equilibrium abundances since there is not enough time to produce the multistrange baryons with rates corresponding to T. (Otherwise the hadronization picture would be in clear discrepancy with an equivalent hadronic picture for which the rates are dominated by two particle scattering.) This closes our argument: at Tch either multi-particle interactions must be important or the cross section must be dramatically larger than in the vacuum. Both

possibilities are conceivable only for Tch very close to Tc.

(C) The chemical equilibration of hadrons should be accessible to a hadronic description, at least in a rough sense. Consistency of the hadronic picture for equilibration requires that multi-hadron processes changing the numbers of Q,S, etc., must be fast enough in order to build up the observed particle numbers at Tch. (We do not assume in this part thermal equilibrium with detailed balance of individual rates.) Keeping in mind the considerable quantitative uncertainties we now proceed to evaluate rates for scattering processes involving more than two incoming particles and demonstrate the importance of such processes near Tc. For an understanding of multi-particle interactions we write for the rate of scattering events per volume with nin ingoing and nout outgoing particles

r(nin,nout) =n(T)nin\M\2$ (4)

The rate is proportional to nm powers of the particle densities of the incoming particles that we denote for short by n(T). For the outgoing particles $ is the Lorentz invariant phase space factor which we evaluate case-by-case with the program given by [27], and which needs to be weighted (see below) with the thermal probability to find a particular cm energy in the initial state. The magnitude of the squared transition amplitude4 \M2\ is evaluated using measured cross sections. We assume constant \M2\ independent of temperature and density.

Inspection of measured cross section systematics shows that production cross sections of strange particles are at most a few mb, and usually much smaller, as is reflected in the known strangeness suppression factor in hadronic collisions. Strangeness exchange reactions may reach cross sections in the 20 mb range. As cases in point we evaluate, using Eq. (4), in the

4 We use a normalization adapted to incoming fermions and outgoing bosons. For each outgoing fermion the additional factor 2E in the phase space integrals is absorbed here in the squared amplitude. For each incoming boson there is an additional factor 2M in |.M|2 which is essentially canceled by an additional factor 1/(2Ei) in r.

following the rate rQ for Q production through the reaction 2n + 3K ^ NNQ and similar rates for E and A production. For this case,

rQ = nl(uK/nny\M\2$.

The densities used to evaluate this expression are taken from the thermal model predictions which describe the measured yields at RHIC [7] using the excluded volume correction. At T = 176 MeV these are: nK = 0.174/fm3 and nK/nn = 0.172. Leaving out the excluded volume correction would increase the densities by approximately a factor of 2. Note that nn and nK stand here for "generic pion and kaon densities". Indeed, the incoming particles can include all sorts of resonances like p, etc. Therefore nn and nK are not the thermal value of individual pion and kaon densities but rather comprise effectively all non-strange and strange degrees of freedom. The 2n 3K reaction is likely to be an important channel for Q production as it involves particles with the largest densities (pions, kaons) and the required amount of strangeness. Similarly we obtain 3 's from 3n2K and ^'s from 4nK reactions.

The numerical rate evaluation needs as input both the matrix element and phase space factor. The phase factor <f> depends on sfs and needs to be weighted by the probability f(s) that the five-meson-scattering occurs at a given value of sfs. For this purpose we assume thermal momentum distributions for the kaons and pions in the entrance channel. The function f is evaluated numerically by a Monte Carlo program. Its results were cross-checked for massless particles against an analytic evaluation [28]. The phase space factor is then obtained by folding $ with f in the energy range from threshold to infinity.

For an estimate of the matrix element we note that the cross section for p + p ^ 5n has been measured. Close to threshold it takes a value of about 40 mb and is falling exponentially with cm energy sfs according to a5lT = (871 mb)exp( —y/i 1.95/GeV) [1]. For the Q + N ^ 2n + 3K reaction the threshold is 2.61 GeV and the peak of f(s) occurs at 3.25 GeV. To evaluate the corresponding matrix element we assume that the cross section at sfs = 3.25 GeV is 6.4 mb, close to the pp ^ 5n cross section at the same energy above threshold. From the known cross section and phase space the matrix element |M|2 can be extracted by the usual formula. Using our normalization conven-

tion this yields |M|2 = 9.5 x 109/GeV8. The final result for Q production through this channel is then tq = 1.39 x 10-4/(fm4) at T = 176 MeV and scales with the 5th power of the pion density. Furthermore, / scales approximately [28] as (sfs )23/2 exp - sfs/'/'. leading to a further increase of rQ with temperature (and hence density).

We have alternatively evaluated the rate for Q production in a semi-classical approach, in which the standard two-body rate equation r = n1n2(av) is generalized to multi-particle collisions. Inspired by the approach taken in cascade codes a reaction takes

place if nm particles approach within a volume V = _^

4tt/3 v/(>rjnc| /tt )'. For the inelastic cross section we take a typical value of a;nel = 40 mb. A particular exit channel (such as QN) is obtained by multiplying with a probability Px. For the reaction under consideration this yields

ra = 8Pxnl (nK/n„ )3J{cr£/7c3){v).

Using for typical relative velocities (v) = 0.6 and Px = 0.10 yields tq = 1.4 x 10-4/fm4, indicating that a similar result as above is obtained with reasonable parameters.

The meaning of this result is as follows: for a density nn = 0.174/fm3, as used above, the final Q density of 3.0 x 10-4/fm3 can be built up within a characteristic time tq = nQ/rQ = 2.2 fm. Since the Q yield scales in this approach as n5n, and taking into account the temperature dependence of f, already an increase of nn to 0.2/fm3 reduces this time by more than a factor of three. The time tq is depicted in Fig. 2 as function of the pion density. We have also added a temperature scale in this figure. This scale is obtained as follows: we take the variation of energy density with temperature from the lattice results (see Fig. 1) and assume that the density scales5 as e/T. Fixing the overall scale by nn = 0.174/fm3 at T = 176 MeV as above this determines the T-dependence of nn near Tc. In consequence, close to Tc,the Q equilibration time scales approximately as tq a T-60.

5 Part of the increase in e is due to the increase in the number of

effective degrees of freedom. Similarly, more relevant channels for

five particle scattering open up. As argued above nn should be inter-

preted as an average weighted density n of all particles contributing

to five particle scattering with Q-production.

We note that the above time of 2.2 fm is a reasonable time for hadronization and phase change, considering the necessary decrease by about a factor of 3 in the degrees of freedom and the concomitant volume increase. A decrease of the pion density by 1/3 only will increase this time to about 27 fm; this time is even much longer than the total lifetime of the fireball as measured by two-pion correlations [19] while we should consider here only the time between begin of hadronization and chemical freeze-out.

We have checked the numerics of our approach by noting that, at equilibrium, detailed balance can be used to evaluate the 2n 3K ^ Qn rate in the reverse direction.6 Since both sides scale very differently with (overall) density, setting both rates equal determines the equilibrium density (prior to resonance decays) for each temperature. The so-determined equilibrium density is to within 25% equal to that computed independently with our thermal model, lending strong support to our calculations.

Our main results are fairly insensitive to the details of the calculation. As can be seen by inspecting Fig. 1 the energy (and consequently particle) density is very rapidly increasing with temperature near T = 176 MeV due to the phase transition. Density values of 20% different from those used in our calculations are reached through temperature changes of less than 3 MeV Possible uncertainties in our rate estimates of even a factor of 2.5 would be compensated by such an increase or decrease in density, indicating the stability of our estimate near Tc . Furthermore, our rate estimates above are more likely overestimates because of the use of a comparatively large Q N ^ 3K 2n cross section. The resulting larger densities needed for chemical equilibration are, however, easily reached as in our picture the phase transition is passed through from above.

We have, exactly along the lines for Q production, evaluated the equilibration times for E and Л production with values of te = 0.71 fm/c and тЛ = 0.66 fm/c, indicating that all strange baryons have similar equilibration times with similar density and temperature dependencies.The corresponding time for protons and antiprotons is typically shorter. We conclude that for all particle species the multi-particle

6 We thank C. Greiner and I. Shovkowy for pointing this out.

rates are sufficient to produce the equilibrium abundances.

It is an interesting question if for some particle species like the pion-proton-antiproton system the hadronic multi-particle rates could be sufficient to maintain (restricted) chemical equilibrium even for some temperature range below Tc. In this case the measured chemical temperature for the proton to antiproton ratio should reflect a lower T < Tch, which is not the case observationally. This may be taken as an experimental indication that an extended period of hadronic equilibrium for T < Tc is not realized. It may favor the idea that the relevant prethermalization process could be associated to hadronization.

We note that thermal models have also been used [29] to describe hadron production in e+e- and hadron-hadron collisions, leading to temperature parameters close to 170 MeV. Indeed,this suggests that hadronization itself can be seen as a prethermalization process. However, to account for the strangeness un-dersaturation in such collisions, multi-strange baryons can only be reproduced by introducing a strangeness suppression factor of about 0.5, leading to a factor of 8 suppression of Q baryons. In contrast to heavy ion collisions, tq exceeds here the available time. In the hadronic picture this is due to the "absence" of sufficient multi-particle scattering since the system is not in a high density phase due to a phase transition.

Has the critical temperature of the QCD-phase transition been fixed by observation? The answer to this question needs a quantitative estimate of the difference AT = Tc - Tch. An accurate determination is difficult since it involves the detailed understanding of equilibration/chemical freeze-out. From Fig. 2 we conclude that a temperature decrease of AT = 5 MeV below the critical temperature lowers the five meson contribution to rQ by more than a factor of 10. This factor is even larger if scattering processes with more than five incoming mesons dominate at Tch. Unless strangeness exchanging rates are implausibly high at Tc, such a sharp drop should make sufficient Q production impossible and we conclude AT < 5 MeV.

The accuracy of the experimental determination of Tc could be limited by a possible temperature dependence of the hadron masses. Indeed, the hadronic yields determine the ratio Tch/M rather than the absolute value of Tch. A large uncertainty in M(T) would reflect in a large uncertainty in Tch and we

want to limit the size of this effect. Using chiral symmetry arguments one may surmise that the masses of hadrons except for pions and kaons are proportional to the value of the order parameter a responsible for chiral symmetry breaking. As a result, they depend on T and x even in the hadronic phase [30]

Mj(T) = hj(T, x)a(T, x).

Neglecting the T-and x-dependence of the dimen-sionless couplings hj these masses scale proportional to a(T, x). However, not all hadron masses scale proportional a(T, x)—the pions and kaons scale differently. Therefore, too large mass changes are not easily consistent with the universality of chemical freeze-out. We notice that the "observed temperature" Tobs = 176 MeV is fitted to the vacuum masses. The true freeze-out temperature Tch therefore obeys

a(Tch,x) a(0,0)

Tch Tobs

Since a is expected to decrease for T > 0 and \x\ > 0 (see, e.g., [30]) we infer that the true freeze-out temperature Tch ^ Tc is lower than Tobs. Already a moderate relative change of a by 10% lowers Tch by 18 MeV [31]. Keeping the rather vague character of our "error estimate" in mind we infer the critical temperature for small x

Tc = 176+58 MeV. (10)

A continuous crossover, second order phase transition or extremely weak first order phase transition make a ratio a(Tch)/a (0) close to one rather unlikely. The observed thermal distribution of the hadron yields strongly suggests, however, that the temperature dependence of the effective hadron masses should not be too substantial. This even could be interpreted as an experimental indication that the QCD-phase transition is of first order! An accurate determination of the temperature dependence of hadron masses M(T)/M(0) by lattice simulations could greatly reduce the (lower) error in our estimate of Tc.

In summary, hadronic many body collisions near Tc can consistently account for chemical equilibration at RHIC energies and lead to Tch = Tc to within an accuracy of a few MeV Any hadronic equilibrium scenario with Tch substantially smaller than Tc would require that either multi-particle interactions dominate even much below Tc or that the two-particle cross sections

are bigger than in the vacuum by a large factor. Both of the latter hypotheses seem unlikely in view of the rapid density decrease.

It was proposed [13,14,32] that the observed hadron abundances arise from a direct production of strange (and non-strange) particles by hadronization. How this happens microscopically is unclear. Nevertheless, in order to escape our argument that Tch = Tc one would have to argue that no (even rough) hadronic picture for this process exists at all—this is unlikely since the abundances are determined by hadronic properties (masses) with high precision. Second, one may question if the "chemical temperature" extracted from the abundances is a universal temperature which also governs the local kinetic aspects and can be associated with the critical temperature of a phase transition in equilibrium. Indeed, in a prethermalization process, different equilibrium properties are realized at different time scales. Nevertheless, all experience shows that kinetic equilibration occurs before chemical equilibration. It seems quite hard to imagine that chemical equilibrium abundances are realized at a time when the rough features of kinematic distributions (like relation between particle density and average kinetic energy per degree of freedom) are not yet close to their equilibrium values. Defining the kinetic temperature by the average kinetic energy one expects that the chemical and kinetic temperatures coincide at chemical freeze-out (with a typical precision at the percent level). More specific properties, like detailed balance of hadronic processes, may be realized later or never.

The critical temperature determined from RHIC for Tch « Tc coincides well with lattice estimates [16] for x = 0. The same arguments as discussed here for RHIC energy also hold for SPS energies: it is likely that also there the phase transition drives the particle densities and ensures chemical equilibration. The values of Tch and x collected in [11] would thus trace out the phase boundary for these energies. Whether the phase transition also plays a role a lower beam energies is currently an open question.

Acknowledgements

We thank J. Knoll for helpful discussions concerning phase space integrals. P.B.M. and J.S. acknowl-

edge the hospitality of the INT Seattle, where part of this work was performed.

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