Advances in High Energy Physics

Volume 2018 (2018), Article ID 1369098, 8 pages

https://doi.org/10.1155/2018/1369098

## A Description of Pseudorapidity Distributions of Charged Particles Produced 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 Z. J. Jiang

Received 15 October 2017; Revised 10 December 2017; Accepted 27 December 2017; Published 24 January 2018

Academic Editor: Fu-Hu Liu

Copyright © 2018 Z. J. 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

In heavy ion collisions, charged particles come from two parts: the hot and dense matter and the leading particles. In this paper, the hot and dense matter is assumed to expand according to the hydrodynamic model including phase transition and decouples into particles via the prescription of Cooper-Frye. The leading particles are as usual supposed to have Gaussian rapidity distributions with the number equaling that of participants. The investigations of this paper show that, unlike low energy situations, the leading particles are essential in describing the pseudorapidity distributions of charged particles produced in high energy heavy ion collisions. This might be due to the different transparencies of nuclei at different energies.

#### 1. Introduction

The BNL Relativistic Heavy Ion Collider (RHIC) accelerates nuclei up to the center-of-mass energies from a dozen GeV to 200 GeV per nucleon. In the past decade, the measurements from such collisions have triggered an extensive research for the properties of matter at extreme conditions of very high temperature and energy densities [1–33]. One of the most important achievements from such research is the discovery that the matter created in nucleus-nucleus collisions at RHIC energies is in the state of strongly coupled quark-gluon plasma (sQGP) exhibiting a clear collective behavior nearly like a perfect fluid with very low viscosity [10–33].

The best approach for describing the space-time evolution of fluid-like sQGP is the relativistic hydrodynamics. However, owing to the formidable complexities of hydrodynamic equations, the most analytical work so far is mainly limited to 1+1 expansion for a perfect fluid with simple equation of state, which can be found as an important application in the analysis of the pseudorapidity distributions of charged particles in high energy physics. In this paper, combining the effects of leading particles, we will discuss such distributions in the framework of 1+1 hydrodynamic model including phase transition [10].

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

Here, for the purpose of completeness and applications, we will list the key ingredients of the hydrodynamic model [10].

The expansion of fluid is subject to the conservation of energy and momentum. This is reflected in continuity equationwhere is the energy-momentum tensor. For a perfect fluidwhereis the 4-velocity of fluid and is its rapidity. and in (2) are the energy density and pressure of fluid, which meet the thermodynamical relationswhere and are the temperature and entropy density of fluid, respectively. To close (1), another relation, namely, the equation of stateis needed, where is the sound speed of fluid, which takes different values in sQGP and in hadronic phase.

Project (1) to the direction of and the direction perpendicular to , respectively. This leads to equationsEquation (6) is the continuity equation for entropy conservation. Equation (7) means the existence of a scalar function satisfying relationsFrom and Legendre transformation, Khalatnikov potential is introduced via relationIn terms of , the variables and can be expressed aswhere is the initial temperature of fluid and . Through the above equations, the coordinate base of is transformed to that of , and (6) is translated into the so-called telegraphy equation

Along with the expansions of matter created in collisions, it becomes cooler and cooler. As its temperature drops from the initial to the critical , phase transition occurs. The matter transforms from sQGP 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 abundances of identified hadrons are 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 (11) with only difference being the value of . In sQGP, , 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 solution of (11) for the sector of sQGP is [10]where is a constant determined by tuning the theoretical results to experimental data and is the 0th-order modified Bessel function of the first kind, and

In the sector of hadrons, the solution of (11) is [10]where

#### 3. The Pseudorapidity Distributions of Charged Particles

*(1**) The Invariant Multiplicity Distributions of Charged Particles Frozen Out from sQGP*. From Khalatnikov potential , the rapidity distributions of charged particles frozen out from fluid-like sQGP read [34]where is the area of overlap region of collisions, being the function of impact parameter or centrality cuts. Inserting (10) into the above equation, the part in the round brackets becomes

With the expansions of hadronic matter, it continues becoming cooler. According to the prescription of Cooper-Frye [34], as the temperature drops to the freeze-out temperature , the inelastic collisions among hadrons cease. The yields of identified hadrons remain unchanged becoming the measured results in experiments. The invariant multiplicity distributions of charged particles equal [10, 15, 34]where is the transverse mass of produced charged particle with rest mass . in (18) is the baryochemical potential. For Fermi charged particles, in the denominator of (18), and for Bosons, . The meaning of (18) is evident. It is the convolution of with the energy of the charged particles in the state with temperature .

The integral interval of in (18) is . The integrand is evaluated with . At this moment, the fluid freezes out into the charged particles. Replacing in (17) by of (14), it becomeswherewhere is the 1st-order modified Bessel function of the first kind.

*(2**) The Invariant Multiplicity Distributions of Leading Particles*. Investigations have shown that the leading particles are formed outside the overlap region of collisions [35, 36]. The generation and movement of them are therefore beyond the scope of hydrodynamic description and should be treated separately.

In our previous work [24–26], we once argued that the rapidity distributions of leading particles take the Gaussian formwhere and are, respectively, the central position and width of distributions. It can be expected that should increase with increasing energies and centrality cuts, while should not, at least not apparently, depend on the energies, centrality cuts, and even colliding systems. The specific values of them can be determined by comparing the theoretical results with experimental data. in (21) represents the number of leading particles, which, for an identical nucleus-nucleus collision, equals half of the number of participants.

The investigations have shown that [37], for certain rapidity, the invariant multiplicity distributions of leading particles possess the formwhere is a constant. Then, as a function of rapidity, the invariant multiplicity distributions of leading particles can be written aswhich is normalized to .

*(3**) The Pseudorapidity Distributions of Charged Particles*. Writing invariant multiplicity distributions in terms of pseudorapidity, we havewhereTo fulfill the transformation of (24), another relationis in order.

Substituting (18) and (23) into (24) and carrying out the integration of , we can get the pseudorapidity distributions of charged particles produced in high energy heavy ion collisions. Figures 1 and 2 show such distributions in Au+Au collisions at and 62.4 GeV, respectively. The solid dots in the figures are the experimental measurements [5]. The dashed, dashed-dotted, and dotted curves are, respectively, the contribution from pions, kaons, and protons got from the hydrodynamic result of (18). The dotted-star curves are the components of leading particles obtained from (23). The solid curves are the sums of the four types of curves. for each curve is listed in Table 1. It can be seen that the combined contribution from both hydrodynamics and leading particles matches up well with experimental data.