Modified excluded volume hadron resonance gas model with Lorentz contraction

Abstract

In this work we discuss a modified version of Excluded Volume Hadron Resonance Gas model and also study the effect of Lorentz contraction of the excluded volume on scaled pressure and susceptibilities of conserved charges. We find that the Lorentz contraction, coupled with the variety of excluded volume parameters reproduces the lattice QCD data quite satisfactorily.

Introduction

Studies of strongly interacting matter at high temperatures and/or high densities have been of great interest for some decades. In the early universe, a few microseconds after the Big Bang, strongly interacting matter is expected to have existed in the colour charge deconfined quark-gluon phase [1]. On the other hand, dense strongly interacting matter can be found inside neutron stars [2]. There are several ongoing and up-coming experiments with ultra-relativistic heavy ion collision which are recreating such phases of strongly interacting matter. Among the experimental programmes, Large Hadron Collider (LHC) at CERN, Geneva and Relativistic Heavy Ion Collider (RHIC) at Brookhaven, New York, have already enriched lot of our understanding in this direction. The ongoing experiments in these facilities as well as the upcoming facilities at GSI, Darmstadt and at JINR, Dubna, are expected to further expand our knowledge about the various facets of strongly interacting matter at high temperature and density.

The experimental investigations have been very well supported by significant advancements of the theoretical approaches. One of the main objectives of these explorations is to understand the thermodynamic phases as well as the phase diagram of strong interactions at high temperatures and high densities. At high temperature and low density, the phase boundary between hadronic and quark-gluon matter is found to be a crossover [3], [4], while at low temperature and high density there is possibly a first order phase transition [5], [6], [7], [8], [9], [10]. Thereby the possibility of observing signatures of a critical end point (CEP) [11], [12], [13], [14] has become an extremely active field of research.

Though quantum chromodynamics (QCD) is the theory of strong interactions, the traditional perturbative methods of field theory is inadequate because the strong interaction coupling may not be small for the temperatures and densities concerned. Lattice QCD (LQCD) is the most important non-perturbative tool that can describe strongly interacting matter at high temperatures [3], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. However, the Monte Carlo techniques of LQCD cannot be applied to a system with finite baryon chemical potential ${\mu }_B$, as the fermion determinant becomes complex. However, the Taylor expansion of thermodynamic quantities around ${\mu }_B=0$, for a given temperature T, can be used until ${\mu }_B$ is close to a phase boundary. For this reason, people build effective models to study properties of strongly interacting matter in non-perturbative domain. Some examples of such models are Polyakov Loop Extended Nambu-Jona-Lasinio (PNJL) Model [30], [31], [32], [33], [34], [35], [36], Hadron Resonance Gas (HRG) Model [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], Polyakov-Quark-Meson (PQM) model [47], [48], [49], [50], Chiral Perturbation Theory [51], etc., which successfully describe some aspects of strongly interacting matter.

HRG model is used to describe a system of dilute gas consisting of hadrons and is based on Dashen, Ma, Bernstein theorem [52]. In this model attractive interactions among hadrons are taken care of by considering the unstable resonance particles as stable particles. HRG model successfully describes the experimental data of a system at freeze out [53]. However, repulsive interaction among the particles in the hadronic system is also important [54]. This is taken into account in the modified version of HRG model, namely, Excluded Volume Hadron Resonance Gas (EVHRG) model where repulsive interaction comes into play due to finite excluded hardcore volume of the particles [55], [56], [57], [58], [59], [60], [61].

Recently, several works have been done on EVHRG using different sizes for different hadrons. In Ref. [62], the authors have studied the effect of excluded volume on the equation of state. In particular, they have looked in to the pressure and the trace of the energy momentum tensor and compared their results with the lattice data. They have found that the best fits are obtained when the excluded volume is inversely proportional to the mass of the particle. Recently there is a renewed interest in studying multiplicity data [63], [64]. However, a more prudent approach has been followed in Ref. [65] where HRG model has been studied in the light of both multiplicity and fluctuation data. The authors have treated temperature and chemical potential as parameters and tried to fit those by fitting the data using non-interacting HRG model. Fluctuations of baryon number and strangeness within HRG model with repulsive mean field approach with the effect of missing resonances have been studied in Ref. [66]. HRG model with parity-doubled baryons having temperature dependent mass was used to calculate charge susceptibilities in Ref. [67] and it was found that mass reduction at high temperatures overshoots the lattice data. The contribution of heavy resonances through exponential Hagedorn mass spectrum to fluctuations of conserved charges was discussed in Ref. [68]. In Ref. [69], a comparison between EVHRG hardcore repulsion and interaction based on S-matrix in HRG framework was considered and it was found that for a πNΔ system, there is good agreement between the two approaches with excluded volume radius $R=0.3\text{fm}$ for pions and nucleons. In Ref. [70], the authors have shown that the mid-rapidity data for hadron yield ratios at AGS, SPS and two highest RHIC energies, are best fit when $R_{\pi }=0,R_K=0.35\text{fm},R_{mesons}=0.35\text{fm},R_{baryon}=0.5\text{fm}$. In Ref. [71], a choice of 2nd virial coefficient for nucleons $=3.42{\text{fm}}^3$ was found to generate the ground state nuclear properties well. Some other choices can be found in Refs. [72], [73], [74]. This indicates that there is no strict consensus about the value of excluded volume of hadrons.

Fluctuations of conserved charges are useful indicators of phase transition between hadronic and quark gluon plasma phase. Existence of CEP can also be indicated by divergent fluctuations. One can calculate charge susceptibilities which are related to fluctuations via fluctuation-dissipation theorem. If net baryon number of the system is small then transition from hadronic to QGP phase is continuous and fluctuations are expected not to show singular behaviour. On the other hand Lattice QCD calculation shows that at small chemical potentials, susceptibilities show rapid increment near the crossover region. Higher order susceptibilities are considered to be more sensitive to phase transition. Fluctuation of conserved charges has been studied in Refs. [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89].

This paper is organised as follows. In the next section we briefly discuss the Hadron Resonance Gas (HRG) model and its interacting version Excluded Volume Hadron Resonance Gas (EVHRG) model briefly. We also introduce a modified version of EVHRG model, namely, MEVHRG model and its extended version Lorentz contracted MEVHRG model or LMEVHRG model. In section 3 we discuss our results of pressure and fluctuation of conserved charges. We also compare our results with the recent LQCD data. Finally, in section 4 we conclude our findings.