Advances in High Energy Physics

Volume 2018, Article ID 9285759, 9 pages

https://doi.org/10.1155/2018/9285759

## Energy Dependence of Particle Ratios in High Energy Nucleus-Nucleus Collisions: A USTFM Approach

Department of Physics, Jamia Millia Islamia (Central University), New Delhi, India

Correspondence should be addressed to Rameez Ahmad Parra; moc.liamg@arrapzeemar

Received 7 February 2018; Accepted 1 April 2018; Published 13 May 2018

Academic Editor: Chun-Sheng Jia

Copyright © 2018 Inam-ul Bashir et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We study the identified particle ratios produced at mid-rapidity () in heavy-ion collisions, along with their correlations with the collision energy. We employ our earlier proposed unified statistical thermal freeze-out model (USTFM), which incorporates the effects of both longitudinal and transverse hydrodynamic flow in the hot hadronic system. A fair agreement seen between the experimental data and our model results confirms that the particle production in these collisions is of statistical nature. The variation of the chemical freeze-out temperature and the baryon chemical potential with respect to collision energies is studied. The chemical freeze-out temperature is found to be almost constant beyond the RHIC energy and is found to be close to the QCD predicted phase-transition temperature suggesting that the chemical freeze-out occurs soon after the hadronization takes place. The vanishing value of chemical potential at LHC indicates very high degree of nuclear transparency in the collision.

#### 1. Introduction

Relative hadron yields and their correlations are observable which can provide information on the nature, composition, and size of the medium from which they originate in high energy heavy-ion collisions where a strongly interacting nuclear matter at high energy density and temperatures is formed. Within the framework of the statistical model, it is assumed that a hot and dense fireball is formed over an extended region for a brief period of time (~a few fm/c) after the initial collision which undergoes collective expansion leading to a decrease in its temperature and finally to the hadronization. After the hadronization of the hot fireball, the hadrons keep interacting with each other and the particle number changing (inelastic) reaction processes continue to take place till the temperature drops to a certain value where a given reaction process almost comes to a stop. Those particle number changing reaction processes (e.g., strangeness exchange process) stop earlier for which the threshold energy is larger. The temperature at which the particle number changing process for a given hadron almost stops is called the “chemical freeze-out” temperature of that hadronic specie. However, the (elastic) rescattering still takes place and continues to build up the collective (hydrodynamic) expansion. Consequently, the matter becomes dilute and the mean free path for the elastic reaction processes of given hadronic species becomes comparable with the system size. At this stage the scattering processes stop and the given hadron decouples from the rest of the system. This is called the “kinetic or thermal freeze-out” after which the hadron’s energy/momentum spectrum is frozen in time [1]. As the inelastic cross sections are only a small fraction of the total cross section at lower (thermal) energies, the inelastic processes stop well before the* elastic* ones. Thus chemical freeze-out precedes kinetic or thermal freeze-out [2].

Statistical thermal models have successfully reproduced the essential features of particle production in heavy-ion collisions [3] as well as in many types of elementary collisions [4–7] at LHC energies suggesting a statistical nature of particle production in these collisions. Systematic studies of particle yields using experimental results at different beam energies have revealed a clear underlying freeze-out pattern for particle yields in heavy-ion collisions [8, 9]. The success of the statistical (thermal) models in describing the ratios of hadron yields produced in heavy-ion collisions is remarkable. The agreement of the particle ratios with simple predictions of the statistical models is a key argument for the thermalization of the system formed in heavy-ion collisions. Measurements of antiparticle to particle ratios in these collisions give information on the net baryon density or baryon chemical potential achieved and are thus of interest in characterizing the environment created in these collisions. It has also been suggested that the measurement of strange antibaryon to baryon ratios could help distinguish between a hadron gas and deconfined plasma of quarks and gluons [10]. For a boost invariant system at mid-rapidity for the RHIC and LHC energies, the particle yields change only by a few percentages in the rapidity window . The particles ratios detected at mid-rapidity are the integrated yield from various parts of the fireball.

In this paper, we attempt to reproduce the particle ratios and to study their correlations and the energy dependence in the hadron gas (HG) scenario by using our phenomenological boost invariant unified statistical thermal freeze-out model (USTFM) [1, 13–17] which assumes that at freeze-out all the hadrons in the hadron gas resulting from a high energy nuclear collision follow an equilibrium distribution. The local particle phase space densities have the form of the Fermi-Dirac or Bose-Einstein statistical distributions.

#### 2. Model Description

The nuclear matter created in high energy heavy-ion collisions is assumed to form an ideal gas that can be described by Grand Canonical Ensemble. The density of the particle can then be written aswhere is the energy, is the momentum of the particle specie, is the spin degeneracy factor, is the chemical potential of the particle species , and is the temperature. The (+) sign is for fermions and (−) sign is for bosons. For high temperatures and energies, the Bose-Einstein or Fermi-Dirac statistics can be replaced with the Boltzmann statistics by dropping the ±1 term. The chemical freeze-out relates to the equilibrium between different flavors. If the hadron gas reaches chemical equilibrium, the particle abundance is described by chemical potentials and temperatures. The information of the chemical freeze-out can be extracted from particle ratios in the measurement. Relative particle production can be studied by particle ratios of the integrated yields. If we neglect the decay contributions and consider only the primordial yield, the antiparticle to particle ratios are found to be controlled only by their respective fugacities. That is,Ratios of particles with the same mass, but different quark content, such as and , are sensitive to the balance between matter and antimatter, characterized by the baryon chemical potential . As strange quarks are created during the collision and are not transported from the incoming nuclei, strangeness production is expected to be a good estimator of the degree of equilibration of the produced fireball [18]. The ratio in accordance with (2) can be written as follows:The other particle ratios of thermal yields (i.e., without feed-down contributions from the heavier resonances) can be correlated accordingly with the ratio as follows:Incidentally, the above given relations of the mid-rapidity equal mass particle ratios, emitted from a hadronic fireball maintaining a high degree of thermal and chemical equilibration, hold even when the resonance decay contributions are included [18]. In (8), the ratio is absent because of the complete strange quark content of Omega mesons.

In our model [1, 13–17], it is assumed that the rapidity axis is populated with hot hadronic regions moving along the beam axis with monotonically increasing rapidity . This essentially emerges from the situation where the colliding nuclei exhibit transparency effects. Hence the regions away from the mid-region also consist of the constituent partons of the colliding nucleons, which suffer less rapidity loss due to partial nuclear transparency. Due to this, these regions have an excess of quarks over the antiquarks and hence maintain larger baryon chemical potentials on either side of the mid-region in a symmetric manner. For this reason, a quadratic-type rapidity-dependent chemical potential has been considered in our model as follows:where the model parameter defines the chemical potential at mid-rapidity and the parameter gives the variation of the chemical potential along the rapidity axis. We focus on the mid-rapidity data (), for which a bulk of published hadrons yields is available. We have also employed the strangeness conservation criteria in a way such that the total strangeness in the fireball is zero.

We have tabulated above the different values of chemical potentials obtained in our previous papers [1, 13–17] for different center-of-mass energies, as shown in Table 1, by using our unified statistical thermal freeze-out model. For the sake of comparison, we have also mentioned the values of obtained at different SPS and RHIC energies by STAR collaboration [11] and by ALICE at LHC [12].