Advances in High Energy Physics

Volume 2017, Article ID 6896524, 8 pages

https://doi.org/10.1155/2017/6896524

## The Rapidity Distributions and the Thermalization Induced Transverse Momentum Distributions in Au-Au Collisions at RHIC Energies

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Correspondence should be addressed to Zhi-Jin Jiang; moc.361@562jzj

Received 23 January 2017; Revised 28 April 2017; Accepted 9 May 2017; Published 4 July 2017

Academic Editor: Sally Seidel

Copyright © 2017 Zhi-Jin Jiang 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

It is widely believed that the quark-gluon plasma (QGP) might be formed in the current heavy ion collisions. It is also widely recognized that the relativistic hydrodynamics is one of the best tools for describing the process of expansion and hadronization of QGP. In this paper, by taking into account the effects of thermalization, a hydrodynamic model including phase transition from QGP state to hadronic state is used to analyze the rapidity and transverse momentum distributions of identified charged particles produced in heavy ion collisions. A comparison is made between the theoretical results and experimental data. The theoretical model gives a good description of the corresponding measurements made in Au-Au collisions at RHIC energies.

#### 1. Introduction

The primary goal of experimental program performed at Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) and at Large Hadron Collider (LHC) at CERN is to create a hot and dense matter consisting of partonic degrees of freedom, usually called the quark-gluon plasma (QGP), which is believed to have filled in the early universe several microseconds after the big bang. The calculations of Lattice Quantum Chromodynamics (LQCD) have predicted [1] that such matter may exist in the environment with critical temperature of about MeV or energy density GeV/fm^{3}. By means of the Bjorken estimation [2] and the measurements of PHENIX Collaboration at RHIC, the spatial energy density in central Au-Au collisions at and 130 GeV is evaluated to be much higher than [3]. Further studies have shown that QGP might be indeed generated in these collisions [4–7]. In fact, it has long been argued that QGP might even have come into being in collisions at the energies of Intersecting Storage Rings (ISR) and Super Proton Synchrotron (SPS) at CERN [8–12].

In the past decade, a number of bulk observables about charged particles, such as the Fourier harmonic coefficients of azimuth-angle distributions [13, 14], rapidity or pseudorapidity distributions [15–18], and transverse momentum distributions [19–24], have experienced a series of extensive investigations in heavy ion collisions at both RHIC and LHC energies. These investigations have provided us with a compelling evidence that the matter created in heavy ion collisions exhibits a clear collective behavior, expanding nearly like a perfect fluid with very low viscosity. This sets up the prominent position of relativistic hydrodynamics in analyzing the properties of bulk observables in heavy ion collisions [25–46].

Apart from collective movement, the quanta of produced matter also have the components of thermal motion stemmed from the thermalization of fluid. The evolution of produced matter is then the superposition of these two parts. To clarify the role of thermalization in the expansions of the produced matter is the major subject of this paper. To this end, we may as usual in analytical treatment ignore the collective flow in the transverse directions. The transverse movement of the produced matter is therefore only induced by the thermalization.

The collective movement of produced matter in the longitudinal direction can be solved analytically. There are a few schemes in dealing with such exact calculations. In this paper, the hydrodynamic model proposed by Suzuki is employed [25]. Besides the analyticity, the other typical feature of this model is that it, like some other models [26–29], incorporates the effects of phase transition into solutions. This coincides with the current experimental observations as mentioned above. Hence, the employed model might be more in line with the realistic situations. In addition, the model is related to the initial temperature of QGP, the sound speed in both partonic and hadronic media, the baryochemical potential, and the critical temperature of phase transition. This work may therefore help us understand various transport coefficients of expanding system.

In Section 2, a brief introduction is given to the theoretical model [25], presenting its analytical formulations. The solutions are then used in Section 3 to formulate the invariant multiplicity distributions of charged particles produced in heavy ion collisions which are in turn compared with the experimental measurements carried out by BRAHMS and PHENIX Collaboration in Au-Au collisions at RHIC energies of and 130 GeV [16–20]. Section 4 is about conclusions.

#### 2. A Brief Introduction to the Model

The main content of the theoretical model [25] is as follows.

() In the process of expansions, the energy and momentum of fluid are conserved. Hence, the movement of fluid follows the continuity equationwhere , is the time, and is the longitudinal coordinate along beam direction. is the energy-momentum tensor, which, for a perfect fluid, takes the formwhere is the metric tensor. is the 4-velocity of fluid with rapidity . and in (2) are the energy density and pressure of fluid, related by the equation of statewhere , , and are the temperature, entropy density, and the sound speed of fluid, respectively.

() In order to solve (1), Khalatnikov potential is introduced which makes the coordinate base of transform to that of * via* relationswhere is the initial temperature of fluid and . Equation (1) is translated to the so-called telegraphy equation

() Along with the expansions of matter created in collisions, its temperature becomes lower and lower. As the temperature drops from initial to critical , phase transition occurs. The matter transforms from QGP state to hadronic state. The produced hadrons are initially in the violent and frequent collisions. The major part of these collisions is inelastic. Hence, the abundance of an identified hadron is in changing. Furthermore, the mean free paths of these primary hadrons are very short. The movement of them is still like that of a fluid meeting (5) with only difference being the value of . In QGP, , which is the sound speed of a massless perfect fluid, being the maximum of . In the hadronic state, . At the point of phase transition; that is, as , is discontinuous.

() The solutions of (5) for the sectors of QGP and hadrons are, respectively [25],where is a constant determined by tuning the theoretical results to experimental data. is the 0th-order modified Bessel function of the first kind, andIt is evident that if , then , , , and thus . At the point of phase transition, , , , and . Then. That is, the potential is discontinuous at point of .

#### 3. The Rapidity Distributions and the Thermalization Induced Transverse Momentum Distributions of Identified Charged Particles

With the expansions of hadronic matter, its temperature continues becoming lower. According to the prescription of Cooper-Frye [31], as the temperature drops to the so-called chemical freeze-out temperature , the inelastic collisions among hadrons cease. The abundance of an identified hadron maintains unchanged becoming the measured results in experiments. The invariant multiplicity distributions of charged particles are given by [25, 31, 32]where is the area of overlap region of collisions, being the function of impact parameter or centrality cuts. is the transverse mass of produced charged particle with rest mass . in (9) is the baryochemical potential. For Fermi charged particles, in the denominator of (9), and for Bosons, . That is, Fermi and Bose charged particles follow the Fermi-Dirac and Bose-Einstein distributions, respectively. The meaning of (9) is evident. The part of integrand in the round brackets is proportional to the rapidity density of fluid resulting from the collective movement along longitudinal direction [31]. The rest part is the energy of the charged particles in the state with temperature and transverse mass resulting from the thermalization of fluid.

From (4) and (8), it can be shown thatwherewhere is the 1st-order modified Bessel function of the first kind.

The integral interval of in (9) is . By applying (9)–(12), together with the definitions in (7), we can get the rapidity distributions and the thermalization induced transverse momentum distributions of identified charged particles as shown in Figures 1, 2, and 3.