Separate freeze-out of strange particles and the quark-hadron phase transition

The scenario of the independent chemical freeze-outs for strange and nonstrange particles is discussed. Within such a scenario an apparent in-equilibrium of strangeness is naturally explained by a separation of chemical freeze-out of strange hadrons from the one of non-strange hadrons, which, nevertheless, are connected by the conservation laws of entropy, baryonic charge and third isospin projection. An interplay between the separate freeze-out of strangeness and its residual non-equilibrium is studied within an elaborate version of the hadron resonance gas model. The developed model enables us to perform a high-quality fit of the hadron multiplicity ratios measured at AGS, SPS and RHIC with an overall fit quality χ2/do f = 0.93. A special attention is paid to a description of the Strangeness Horn and to the well-known problem of selective suppression of Λ− and Ξ hyperons. It is remarkable that for all collision energies the strangeness suppression factor γs is about 1 within the error bars. The only exception is found in the vicinity of the center-of-mass collision energy 7.6 GeV, at which a residual enhancement of strangeness of about 20 % is observed.


Introduction
Evolution of strongly interacting system created in high energy central heavy ion collisions includes several stages governed by different physical degrees of freedom. Switching between them is expected to have a character of a phase transition. For an unambiguous identification of these transitions of QCD we need the reliable signals which can be measured experimentally. However, an interpretation of such signals is highly non-trivial, since in the region of major interest the lattice formulation of QCD a e-mail: bugaev@fias.uni-frankfurt.de EPJ Web of Conferences 182, 02057 (2018) https://doi.org/10.1051/epjconf/201818202057 ICNFP 2017 EPJ Web of Conferences is inapplicable due to high baryonic charge densities. The problems is enhanced because most of such signals are affected by the last stage of nucleus-nucleus collision, at which the strongly interacting matter exists in the form of a hadron gas. Therefore, one of the most important theoretical tasks is to separate or to account for an influence of the hadron gas expansion from true signals of phase transformation. Evidently, to solve this task one needs a reliable theoretical description of hadron gas properties emitted from the hypersurface of chemical freeze-out [1][2][3][4]. During last four years such an approach known as the hadron-resonance gas model (HRGM) [1][2][3][4] was worked out. Up to now, the HRGM is the most successful approach to describe the yields of produced hadrons. However, we have to point out that the obsolete versions of this model which have only one or two independent hadronic hard core radii were not able to give a high quality description of available experimental data. The true progress in obtaining an accurate description of the data was recently achieved by the multicomponent version of the HRGM (MHRGM) [3]. Thus, having only two additional (global) fitting parameters the high quality description of the most problematic ratios of hadrons was achieved [3,4]. In the works [3,4] the ratios K + /π + and Λ/π − were described with the quality χ 2 K + /π + /do f = 7.7 14 and χ 2 Λ/π − /do f = 16 12 which is a perfect description compared to the results χ 2 K + /π + /do f = 21.8 14 and χ 2 Λ/π − /do f = 79 12 obtained in Ref. [5]. Unfortunately, even this advanced model failed to reproduce the yields of strange particles measured at several collision energies. Therefore, one cannot use such a model to reliably study the irregularities associated to strange hadrons at chemical freeze-out. It is clear that without accurate fit of the data one cannot draw any reliable conclusion about such irregularities. Therefore, in order to improve the description of strange particle yields in Ref. [6] it was assumed that strange hadrons can deviate from the full chemical equilibrium. The degree of such a deviation from chemical equilibrium is controlled by the γ s factor [6], which modifies the one-particle thermal density φ i of strange hadron of sort i according to the rule Here s i denotes the total number of strange valence quarks and antiquarks in a given hadron. If γ s = 1, then there is no deviation of strange particles from chemical equilibrium. Their production is enhanced, if γ s > 1, and is suppressed, if γ s < 1. Being treated as a free parameter, the γ s factor, indeed, sizably improves the description of particle yields within the MHRGM and provides an overall fit with χ 2 /do f = 1.15 in case of four independent hard-core radii of hadrons [8] and with χ 2 /do f = 0.95, if the number of these radii is five [4,9]. However, a physical reason lying behind the deviation of strange charge from chemical equilibrium remains unclear. In order to elucidate this reason here we consider several scenaria of two independent freeze-outs, which correspond to nonstrange and strange hadrons, which, nevertheless, are connected by the conservation laws. Due to the latter feature this approach significantly improves the quality of experimental data description.
In the next section we briefly formulate the MHRGM and introduce the concept of separate freezeout of strange particles (SFO) from the freeze-out of non-strange hadrons (FO). Analysis of experimental data is presented in Section 3. The question about the source of residual in-equilibrium of strange particles is also discussed in this section. Finally, our conclusions are given in Section 4.

Separate freeze-out of strange particles
The MHRGM model treats hadron gas as a multicomponent mixture of hard spheres with the hard core radii R i with i = 1, 2, 3, .... The grand canonical pressure p of such a mixture is a sum of hadronic partial pressures p i defined as a solution of the following nonlinear system Here T denotes the temperature, while µ i = µ B Q B i + µ I3 Q I3 i + µ S Q S i is the chemical potential of a given hadron sort, which is expressed in terms of the baryonic µ B , the third isospin projection µ I3 and strange µ S chemical potentials associated with the corresponding charges Q B i , Q I3 i and Q S i of this hadron, respectively. A symmetric matrix b i j = 4 3 π(R i + R j ) 3 is composed from the second virial coefficients of different hadron species. The one particle thermal density φ i of a hadron with the spin-isospin degeneracy g i and the mean mass m i is defined as the momentum integral of the Boltzmann exponential, which is averaged over the normalized Breit-Wigner mass distribution f BW with the threshold m T h i and the width Γ i where the normalization fact is defined as . If all partial pressures p i are known, then the thermal densities of hadrons are calculated by a differentiation of p with respect to its chemical potential, i.e. n i = ∂p ∂µ i . Contrary to the conventional approach, the present model allows one to freeze-out the multiplicities of strange and non-strange hadrons independently from each other. As a result, the thermodynamic parameters of FO and SFO are different in general case. These ten parameters are the temperatures T A , the baryonic µ B A , the strange µ S A and the third isospin projection µ I3 A chemical potentials defined for A ∈ {FO, S FO} as well as the volumes of the FO and SFO hypersurfaces, respectively, V FO and V S FO . However, a hydrodynamic evolution of the system between these two hypersurfaces of heavy ion collisions implies the conservation of entropy. The other conserved quantities are the baryonic and electric charges. Therefore, we assume that where A denote the total densities of entropy, baryonic charge and the third isospin projection at FO or SFO. These equations allow us to exclude three parameters of SFO, i.e. V S FO , µ B S FO and µ I3 S FO , from the list of independent ones. Two more parameters, namely the strange chemical potentials µ S FO and µ S S FO , are excluded by the requirement of zero total strangeness at FO and SFO. Thus, only five parameters remains independent within our approach. They are T FO , T S FO , µ B FO , µ I3 FO and V FO . In other words, in the present model the number of independent fitting parameters for each collision energy is exactly the same as in the HRGM with the γ s factor, but instead of the latter one, the parameter T S FO is used.
The strong decays of hadronic resonances modify their multiplicities, while the weak ones can be safely neglected in the considered range of collision energies. Thus, the total number of produced hadrons of sort i is calculated as Here the branching ratio Br( j → i) defines a probability that the hadron of sort j decays into a hadron of sort i and Br(i → i) = 1 is introduced for the sake of convenience. In order to get rid of effective volume of the system V FO at FO we fitted the particle number ratios R i j ≡ The masses of hadrons, their spin-isospin degeneracies, the widths, as well as the branching ratios and thresholds of strong decays, are taken from the particle tables used by the thermodynamic code THERMUS [7]. The hard core radii of hadrons were fixed to the best fit values from Refs. [3,8].
The fit procedure to find out the parameters T FO , T S FO , µ B FO and µ I3 FO at each collision energy searches for a global minimum of the quantity where summation is performed over all independent pairs of particle sorts i and j, while the superscript indices "theor" and "exp" denote, respectively, theoretically calculated and experimentally measured values of the corresponding ratio R i j which has the experimental errors δR exp i j . , the dashed and short dashed curves correspond to parameterizations from Ref. [31]. Middle panel. Dependence of the γ s factor on the collision energy from the combined fit (see text). Right panel. Collision energy dependence of the ratio K + /π + . Horizontal bars and empty diamonds represent the results obtained for the combined fit and for the SFO fit, respectively. The solid curve represents the parameterization of baryonic chemical potential and temperature taken from Ref. [31].
conclude that traditionally problematic ratios which involve the strange particles experience the most significant influence due to separate SFO. This effect is stronger at the intermediate collision energies.
For example, at √ s NN = 6.3 AGeV an introduction of SFO makes the ratios K − /K + , Λ/Λ and Λ/π − much closer to their experimental values (see the left panel of Fig. 1). It is worth to stress, that the description of another problematic ratio p/p is also greatly improved within the MHRGM with separate SFO. This means that the present approach relaxes a connection between strange and nonstrange baryons and significantly simplifies their simultaneous description. As it is seen from the middle panel of Fig. 1 the separate SFO approach sizably improves a description of the K − /π + , K + /π + Λ/π − , p/p and Λ/Λ ratios at √ s NN = 7.6 AGeV. Since, the highest sensitivity of the particle yield ratios on the SFO parameters is achieved at √ s NN from 6.3 to 17 AGeV, then there is nothing unexpected that separation of the curves of two freeze-outs is the most pronounced in this region. Such a conclusion can be drawn from the right panel of Fig. 1, where the curves of SFO and FO are shown in the plane of baryonic chemical potential and temperature. This figure shows that these curves are rather close to each other. Although the difference between T S FO ( √ s NN ) and T FO ( does not exceed 20 MeV, it is sufficient for a systematic improvement of the data description reflected in the overall value of χ 2 /do f = 58.5/55 1.06 [8].
It is remarkable that introduction of separate SFO reduces the χ 2 /do f value by about 10% from initial value 1.16 obtained in Ref. [3]. At the same time the MHRGM without SFO and with the free γ s parameter allows one to describe the data with overall value of χ 2 /do f = 1.15 which is only slightly less than the result of Ref. [3]. This means that the scenario of separate SFO is favored by experimental data. Moreover, the concept of separate SFO naturally explains an apparent chemical in-equilibrium of strange hadrons used to heal the inconsistency between the experimental and theoretical particle ratios. Indeed, if the SFO is ignored, then the multiplicities of strange hadrons are assumed to be controlled by the temperature and chemical potentials of FO which, however, are not the equilibrium parameters of these particles and can not provide the high quality description of their yields. This effect looks like a deviation of strangeness from chemical equilibrium. Thus, a poor description of yields of strange hadrons obtained in the earlier versions of HRGM can not be