Particle ratio fluctuations and isobaric ensemble for chemical freeze-out scenario

Particle ratio fluctuations and isobaric ensemble for chemical freeze-out scenario

Licinio L. S. Portugal

Instituto de Física, Universidade Federal do Rio de Janeiro, Cx.P. 68528, CEP 21945-970, Rio de Janeiro, RJ, Brazil

ABSTRACT

In this work, we study the behavior of event-by-event distribution of particle ratios in Gran Canonical and Isobaric ensemble.We show that the experimental value of the width in the K/p ratio distribution is much wider than that given by a unique grand canonical ensemble or Isobaric ensembe and maybe a mixture is required.

1 Introduction

The main purpose of the relativistic heavy ion program at CERN and RHIC is to study the properties of matter in hadronic interactions at very high energy density. Among many novel phenomena expected, the observation of the quark gluon plasma (QGP) predicted by the QCD is one of the main goals. For these studies, a crucial factor is to know whether or not the system reaches thermal equilibrium at some stage of the collision. In this respect, thermal models are extensively used to analyze the final hadronic abundances [1]. In these models, it is assumed that there exists a stage where the chemical abundances of hadrons (including resonances) are suddenly frozen out, keeping the exact memory of the last instant of the chemical equilibrium. Such stage is referred to as chemical freeze-out and the observed hadronic abundances are completely determined by the temperature T and baryonic chemical potential m_B of the system at this point.

The above picture is equivalent to say that the hadronic gas at the chemical freeze-out is described by a grand canonical ensemble (GC). The fact is that the observed particle ratios are remarkably well reproduced adjusting just two parameters, T and m_B. Such fits have been done for 4p NA49 data and for RHIC Au+Au data in a rapidity window of rapidity. Although the resulting fits are impressive[1], it is very difficult to imagine that final hadrons are in a global chemical equilibrium (rapidity distributions of particles, especially those of hyperons, are far from constant). It may well be possible that to have a nice fit only for the chemical abundances does not imply that T and m_B obtained correspond to those for a real chemical equilibrium. They could be just some effective parameters to characterize certain average of statistical ensembles corresponding to different physical conditions for chemical freeze-out.

In order to clarify the above mentioned aspect, we should go one step further than the mean values of chemical abundances, and study the behavior of less inclusive quantities such as event-by-event fluctuations and correlations of chemical abundances among different hadrons. In this paper, we analyze the event-by-event fluctuations of K/p ratio in the thermal model. The distribution of K/p ratio is measured for N A49 experiment. We show that the pure statistical fluctuation given by a unique GC ensemble of resonance gas is much smaller than the experimental data. This imply that the observed particle abundances can not be reproduced in terms of one single GC ensemble, but we need surely a mixture of many statistical ensembles. To explain the observed width of the distribution of K/p ratio[2], we have to fluctuate, for example, the chemical potential m_B. For nucleus-nucleus collisions, the mixture of different m_B is rather natural because of the baryons in the incident nuclei and fluctuating initial conditions[3].

When we consider the fluctuation of the ratio K/p, it is important to see if there exists any correlation among K and p. For a GC ensemble of ideal hadronic gas, there exists no correlation for the production of K and p. For an infinite system these assumptions really do not matter, but when one consider a domain of particle production with finite volume (fireball) it is not clear at all whether a GC ensemble represents the best scenario. In particular for kaons, the strangeness conservation imposes some effects. Analysis based on the canonical ensemble has been done by Koch [4]. Another possible approach is the use of isobaric ensemble (ISO). This ensemble describe a system with fixed pressure allowing volume fluctuations.

When we deal only with the average values of chemical abundances, there will be no difference between the GC and ISO ensembles, especially if the total number of the particles in the system is large enough. However, as we discuss later, the multiplicity distribution and correlations among particles are completely different between them. To understand more clearly, let us consider the K/p ratio. For ISO ensemble, the pressure is fixed so that for a fixed event large multiplicity of pions corresponds to a large volume event. Therefore kaons are also abundant in such events and vice-versa. That is, K and p multiplicities have positive correlations in ISO ensembles. For GC ensembles, if there exists no excluded volume correlation, the K and p multiplicities have no correlation among them. Furthermore, since the chemical freeze-out should be a local process, it seems natural that its mechanism is governed by the intensive parameters. Statistically speaking, the observed chemical freeze-out state may correspond to the state of maximum entropy under a fixed pressure, and not under a fixed volume. It is worthwhile to investigate if there exist differences between these two scenarios in the particle abundances, especially in the distribution of K/p ratio.

2 Grand Canonical Ensemble

Let us consider first the case of GC ensemble of hadronic resonances. For simplicity, we consider ideal Boltzmann gas of hadronic resonances, neglecting the quantum statistics and the effect of excluded volume. Except for pions, for temperatures we are interested, the Boltzmann gas approximation is quite well. Furthermore, since we consider just the ratio of particle multiplicities, the effect of excluded volume is not very important. Under these approximations, the probability P(N₁,...N_h) of having N₁ particles of the hadron 1, N₂ particles of the hadron 2, and so on, till h-th hadron species is given as just the product of Poisson distributions for each hadronic species,

where i refers to the hadron species, V is the volume and

with b = 1/T and m_i is the chemical potential of the i – th hadron. It is given by

with B_i, S_i and are baryon number, strangeness and 3-rd component of isospin of the i – th hadron. The strangeness and isospin chemical potentials m_S and m₃ are determined using the charge and strangeness neutrality. m_i and g_i are the mass and statistical factor of the hadron, and K₂ is the modified Bessel function.

The observed particle abundances are not directly given by the thermal multiplicities. We should take into account the feeding stream from the resonance decays. The multiplicity of a stable (observable) hadron species, say a, in a single collision event is then given by

where N_{i ® a} is the number of a hadron produced by the decay chain of the parent hadron i. If we consider the decay chain has no correlation with the thermal production mechanism, we may safely substitute N_{i ® a} (integers) in terms of their average values F_{i ® a} (not necessarily integers). We constructed the table for all F_{i ® a}'s following the decay chain of the i – th hadron using the values given in the particle data table [5]. The distribution of the particle ratio, r_ab is then calculated to be

where áOñ denotes the average over collisional events of an observable O. In practice, we introduce a finite bin size D for the variable r_a/b and calculate the probability of finding an event within a given bin specified by [r_ab – D/2,r_ab + D/2] . The average number of a – th hadron is

where in the case of GC, we have

áNñ_i = f_iV.

In Fig. 1, we show the result of Monte Carlo evaluation of P(r_K/p) for the Pb+Pb central collision.

Thermal parameters {T,m_B} are determined to give the best c²-fit to the particle ratios of stable hadrons as done in Ref. [1]. The value of c² is around 20, which is comparable to that of [1]. This means that the average particle ratios are very well reproduced. However, as we see, the calculated distribution is very much narrower than the experimental value. The data [2] refers to a narrow rapidity window, and the corrections from detector acceptance are not included we may not consider the value shown here as definitive but the difference between the value from a single GC ensemble seems to be too large. It may be possible that the contamination of kaons from the neighboring rapidity windows causes such a broadening of the distribution [4]. Therefore the observed large width may be, is due to the mixture of many GC with different thermodynamical parameters, {T,m_B} . For Pb+Pb collisions at SPS, the baryonic chemical potential m_B varies more than 200MeV from the central to fragmentation region. In Fig. 1, we also show how the distribution of K/p ratio changes with the chemical potential for fixed temperature. This results indicates that, the observed width of the K/p ratio distribution requires a mixture of GC ensemble of appreciably different parameters.

3 Isobaric Ensemble

As mentioned in Introduction, it maybe possible that the chemical freeze-out mechanism is governed by a pressure value than a fixed volume as constraint. The isobaric partition function is given as

where Z_GC(b,m,V) is the partition function of the GC ensemble. For a Boltzmann ideal gas, we can perform the integral easily to get

where we have introduced the excluded volume v₀ here as in [6].

It can be shown that the probability function corresponding to Eq.(1) becomes

where

The single particle multiplicity distribution is then given by

where:

which is a geometric distribution.

Different from the GC case, the convolution of geometric distributions is not a geometric distribution, but so-called negative binomial distribution. Suppose that the final state of nuclear collision is formed of k independent 'fireballs', described by the same isobaric ensemble specified T and P. Let be the number of i – th hadron contained in the q – th "fireball". Then the total number of i – th hadron is

In this case, the final probability distribution of having a set of hadrons, { N₁,...,N_h} , is given as the convolution of Eq.(09),

where the summations are taken over all satisfying the constraints Eq.(12). Correspondingly, the single particle distribution Eq.(11) is modified as[7]

The average multiplicity of i – th hadron is then

The particle ratio and its distribution are calculated by evaluating Eq.5) by the Monte Carlo method, generating {N_i} which obey the probability distribution Eq.(13). For this purpose, use of the conditional probabilities for the ISO ensemble is found to be efficient. The probability of having N_i particles for i – th hadron after having already N₁ particles for the hadron 1, N₂ particles for the hadron 2, and so on till N_i–1 is

where Z_i = N_j and = h_i/(1 – å_{j > i} h_j). From this, we can evaluate Eq.(5) very efficiently. In Fig.2, we show the calculated distribution of K/p ratio for the ISO ensemble for several clusters. The ISO ensemble gives a larger width compared to the GC ensemble, especially for smaller numbers of clusters. This can be understood in terms of the positive correlation among K and p multiplicities in the ISO ensemble, as mentioned before. On the other hand, for large number of clusters, the convoluted distribution Eq. (13) tends to the GC ensemble. This is a natural consequence of the statistical central limit theorem. However, different from the GC ensemble, for a given set of T an P, the value of chemical potential changes to keep the baryon number conservation so that the average values of K/p value changes for different number of clusters in the system.

4 Discussion and Perspectives

In this work, we studied the behavior of event-by-event distribution of particle ratios in different statistical ensembles. We have shown that the experimental value of the width in the K/p ratio distribution is much wider than that given by a unique grand canonical ensemble. If the observed width is due to the mixture of GC ensembles corresponding to different values of chemical potentials, a very large variation of chemical potential is required to reproduce the data. On the other hand, the isobaric ensemble seems to be more adequate for the description of the chemical freeze-out scenario. It already furnishes a wider distribution of K/p ratio due to the correlations in particle multiplicities. On the other hand, if the final state of the system is composed of many clusters, convolution of the ISO distributions from these clusters tends to that of the GC distribution. However, in this case, the fluctuation of number of clusters with fixed (T,P) can also contribute to the width of the K/p distribution. A further investigation in this line is in progress.

Acknowledgments

This work has been developed in collaboration with Drs. A.G. Grunfeld, C.E. Aguiar and T. Kodama and is supported by CNPq, CAPES, FAPERJ and FAPESP

References

[1] See for example, P. Braun-Munzinger et al., nucl-th/0304013.

[3] C.E. Aguiar, Y. Hama, T. Kodama, T. Osada, AIP Conf.Proc.631:686-694,2003

[4] A. Majumder and V. Koch, nucl-th/0305047.

[5] Review of Particle Physics, Phys. Rev. D66 (2002).

Received on 15 August, 2003.

• [2] S.V. Afanasiev, et. al., Phys. Rev. Lett. 86, 1965 (2001).
• [6] G. D. Yen, M. Gorenstein, W. Greiner and S. N. Yang, Phys. Rev. C56, 2210 (1997).
• [7] M. Gorenstein, Sov. J. Nucl. Phys. 31, 845 (1980).

Publication Dates

• Publication in this collection
  11 May 2004
• Date of issue
  Mar 2004

History

• Received
  15 Aug 2003
• Accepted
  15 Aug 2003