Advances in High Energy Physics

Volume 2015, Article ID 184713, 23 pages

http://dx.doi.org/10.1155/2015/184713

## On Pseudorapidity Distribution and Speed of Sound in High Energy Heavy Ion Collisions Based on a New Revised Landau Hydrodynamic Model

Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China

Received 3 April 2015; Revised 12 June 2015; Accepted 18 June 2015

Academic Editor: Sally Seidel

Copyright © 2015 Li-Na Gao and Fu-Hu Liu. 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 propose a new revised Landau hydrodynamic model to study systematically the pseudorapidity distributions of charged particles produced in heavy ion collisions over an energy range from a few GeV to a few TeV per nucleon pair. The interacting system is divided into three sources, namely, the central, target, and projectile sources, respectively. The large central source is described by the Landau hydrodynamic model and further revised by the contributions of the small target/projectile sources. The modeling results are in agreement with the available experimental data at relativistic heavy ion collider, large hadron collider, and other energies for different centralities. The value of square speed of sound parameter in different collisions has been extracted by us from the widths of rapidity distributions. Our results show that, in heavy ion collisions at energies of the two colliders, the central source undergoes a phase transition from hadronic gas to quark-gluon plasma liquid phase; meanwhile, the target/projectile sources remain in the state of hadronic gas. The present work confirms that the quark-gluon plasma is of liquid type rather than being of a gas type.

#### 1. Introduction

In fields of particle physics and nuclear physics, heavy ion (nucleus-nucleus) collisions at high energies are a very important research subject. Many charged and neutral particles are produced in final state of the collisions and can be measured in experiments. To study the behavior of the particles could help us to understand the processes of interacting system in the collisions. The pseudorapidity distributions of charged particles are an important quantity which can be measured in the early stage of the measurements. According to ecumenical textbooks, the pseudorapidity is simply defined as , where is the emission angle of the considered particle.

The pseudorapidity distributions can be used to study stopping and penetrating powers of the target and projectile nuclei, positions and contribution ratios of different emission sources, contribution ratios of leading nucleons, square speed of sound, and other related topics. Many models have been introduced in order to study the pseudorapidity distributions, transverse momentum distributions, azimuthal correlations, and other distributions and correlations in relativistic heavy ion collisions. These theoretical models can be classified mainly into two classes: (i) thermal and statistical model and (ii) transport and dynamical model. Among them, the three-fireball model [1–6], the three-source relativistic diffusion model [7–12], and the Landau hydrodynamic model [13–21] are of great interest for us and will be used in the present work.

A lot of experimental data on nuclear collisions at high energies has been published in literature. The relativistic heavy ion collider (RHIC) has performed gold-gold (Au-Au), copper-copper (Cu-Cu), deuteron-gold (-Au), and other collisions at various GeV energies [22–25]. Also, the large hadron collider (LHC) has performed lead-lead (Pb-Pb), proton-lead (-Pb), and other collisions at TeV energies [26]. These collider experiments show rich and exciting results on the pseudorapidity distributions and other distributions. In fixed target experiments as well (such as in nuclear emulsion experiments at high energies), proton to gold induced emulsion (-Em and Au-Em) collisions have presented pseudorapidity distributions with abundant structures [27–29]. More other ions such as helium, carbon, oxygen, neon, silicon, sulphur, and krypton have also been used [30, 31].

In this paper, we propose a new revision on the Landau hydrodynamic model based on the three-source picture to describe systematically the pseudorapidity distributions of charged particles produced in Au-Au, Cu-Cu, Pb-Pb, -Au, Au-Em, and -Em collisions at high energies. The central source with a large enough contribution is described by the Landau hydrodynamic model. The small contributions of the target and projectile sources are considered as a revision on the Landau hydrodynamic model for the central source. Based on the descriptions of pseudorapidity distributions, the values of square speed of sound parameter are extracted from the widths of rapidity distributions.

#### 2. The Model and Calculation Method

Enlightened by the three-fireball model [1–6] and the three-source relativistic diffusion model [7–12], we classify the particle emission sources into three types: a central source (C), a target source (T), and a projectile source (P). Generally, the central source’s center stays at the midrapidity, and its contribution covers a wide enough region in the whole rapidity distribution. The central source includes the contributions of all produced particles and most nonleading nucleons. The target source stays in the left side of the central source’s center and covers an appropriate region. It includes the contributions of all leading and a few nonleading target nucleons. The projectile source stays in the right side of the central source’s center and covers an appropriate region. It includes the contributions of all leading and a few nonleading projectile nucleons. As revisions of the central source, the distribution ranges of particles produced from the target and projectile sources are covered by those from the central source.

The central source can be described by the Landau hydrodynamic model [13–21]. The target and projectile sources are assumed to emit isotropically particles in their respective rest frames. The three sources may have different contribution ratios which are regarded as free parameters in the model. The central source can produce pions, kaons, nucleons, and other particles. The target and projectile sources emit only nucleons due to the two sources consisting of leading and nonleading nucleons. In fact, the target/projectile sources are a complementarity and revision of the central source. Generally, spectator nucleons appear in the very backward or forward rapidity region in noncentral collisions; their contributions should be subtracted in our analyses. Most experimental distributions do not contain the contributions of spectator nucleons due to relative central collisions and narrow region of measurement. Our treatment is in fact a new revision of the Landau hydrodynamic model [13–21].

In center-of-mass reference frame and for symmetric collisions such as Au-Au and Pb-Pb collisions, the central source stays at the peak position at , the target source stays at a peak position in the range of , and the projectile source stays at a peak position in the range of . In an actual calculation for symmetric collisions, and are regarded as free parameters. The distributions contributed by the target/projectile sources revise (in fact increase) somewhere the probabilities underestimated by the central source. In the case of considering asymmetric collisions such as -Au collisions, the central source stays at a peak position at , and the situations for the target/projectile sources are similar to those in symmetric collisions. In an actual calculation for asymmetric collisions, , , and are regarded as free parameters, where is close to the peak position. In the fixed target experiments in laboratory reference system, for both symmetric and asymmetric collisions. Obviously, we have in any case.

In center-of-mass or laboratory reference frame, according to the Landau hydrodynamic model [13–21], particles produced in the central source can be described by a Gaussian rapidity () distribution with a width of [15–21]. We have where denotes the multiplicity, is the normalization constant, and should be large enough to cover a wide enough rapidity region. At the same time, the transverse momentum () distribution is assumed to obey the simplest form of Boltzmann distribution [32] where is the normalization, denotes the rest mass of the considered particle, denotes the Boltzmann constant, and denotes the source temperature. In the simplest form (see (2)), the chemical potential and the distinction for fermions and bosons are not included due to small effects at high energy.

Based on (1) and (2), we can get a series of values of pseudorapidity for the particles produced from the central source by using the Monte Carlo method. Then, we can get the pseudorapidity distribution contributed by the central source by the statistical method. Now we describe the process which calculates one of values of pseudorapidity. Let denote random numbers in . The Gaussian rapidity distribution (see (1)) results in being A given satisfies The longitudinal momentum can be given by The momentum is Then, we have the pseudorapidity Repeating the above process (see (3)–(7)) for many times, a series of values of pseudorapidity for the particles produced in the central source can be obtained.

The situation for the target/projectile sources is somewhere different from that for the central source. We now describe the process which calculates one of the values of pseudorapidity for the particles produced from the target or projectile source. Let denote random number in . In the rest frame of the target or projectile source, particles are assumed to be emitted isotropically. The Monte Carlo method gives the emission angle where (or ) in the case of the first item in the above equation being greater than 0 (or less than 0). This isotropic emission results in a Gaussian pseudorapidity distribution with the width of 0.91–0.92 [27]. In this study, we have not studied further the width corresponding to the target and projectile sources. The transverse momentum has the same expression as (2) and (4). The longitudinal momentum , the momentum , and the energy are written by respectively. In laboratory or center-of-mass reference frame, the rapidity of the particle produced in the target or projectile source can be given by The longitudinal momentum , the momentum , and the pseudorapidity have the same expressions as (5)–(7), respectively. Repeating the above process for many times, we can obtain a series of values of for the particles produced in the target or projectile source.

If necessary, distribution for particles produced in the central source can be obtained by the statistical method. At the same time, distribution for particles produced in the target or projectile source can be obtained by the statistical method, too. To compare with experimental distribution, we have to use the statistical method to collect lots of for particles produced in the central, target, and projectile sources. In the statistical process, the contribution ratios of the three sources have to be considered naturally. A reasonable and accordant comparison with experimental data can determine a set of parameters. Particularly, the width of rapidity distribution for particles produced in the central source is the most interesting parameter, although other parameters can be obtained together.

The relation between the square speed of sound and the rapidity distribution width can be given by [15, 16, 18, 20, 21]where denotes the mass of a proton. Then, is expressed by using asThe square speed of sound parameter can be obtained from the above equation. It should be noted that not only the central source but also the target/projectile sources discussed in the present work are formed in the participants but not in the spectators which mainly appeared in noncentral or peripheral collisions. The relation between the square speed of sound and rapidity distribution width can be applied for the groups of particles produced from the three sources.

We would like to point out that, as the previous version, people apply the hydrodynamic model for hadron production that was originally developed by Landau [13] and Belen’kji and Landau [14]. Subsequently other researchers have extended the model [15–20], always based on an expanding central source for the produced particles in the rapidity/pseudorapidity space. The variance of this Gaussian source in Landau’s original work is in the pseudorapidity space and later works in the rapidity space. Generally, the variance of this Gaussian source is related analytically to the logarithm of the center-of-mass energy and the speed of sound. Hence the speed of sound can in principle be inferred from a comparison of calculated (pseudo)rapidity distributions with data given this particular model. We think that the application of Gaussian source in the rapidity space is a revision of that in the pseudorapidity space. Therefore, the Gaussian rapidity distribution is used in the present work.

In addition, according to the Landau hydrodynamic model [13–21], in hadron-nucleus and nucleus-nucleus collisions, the number and energy of primary hadrons, which form an ideal liquid, fluctuate from event to event. Therefore, in already formed ideal liquid in laboratory system, the (pseudo)rapidity of its center-of-mass also fluctuates on the (pseudo)rapidity scale. The total inclusive distribution is then the sum of Gaussian distributions with different centers and widths, and the shape of the inclusive distributions differs from the form of a Gaussian distribution, in the laboratory system. These situations are changed in the center-of-mass system, where the mentioned centers are the same, and the mentioned widths fluctuate slightly from event to event within a given centrality range. Because our calculation is performed in the center-of-mass system and for the given centrality ranges in most cases, we will not take into account the fluctuations of the center and width of Gaussian distribution for the central source from event to event for the purpose of convenience. Even if for a few minibias samples and emulsion experiments in the laboratory system, our treatment gives an average for different events. For the target and projectile sources in their respective rest frames, we will not take into account the fluctuations of the center either, and the width is fixed due to isotropic emission and given temperature.

In the calculation, we do not need either to consider the Jacobian transformation between the rapidity and pseudorapidity spaces explicitly but instead to calculate pseudorapidity distribution for produced charged hadrons directly in the Monte Carlo approach, in which we assume Gaussian rapidity distribution and Boltzmann transverse momentum distribution for the central sources and isotropic emission and Boltzmann transverse momentum distribution for the target and projectile sources in their respective rest frames. At the same time, we assume Landau’s prescription for the speed of sound to be valid for the three sources. Based on the description of experimental pseudorapidity distribution, speed of sound parameters are then determined for the three sources from their respective widths of rapidity distributions.

#### 3. Comparisons with Experimental Data

The pseudorapidity distributions, , of charged particles produced in Au-Au collisions for different centralities at , 62.4, 130, and 200 GeV are presented in Figures 1–4, respectively. In the four figures, the circles represent the experimental data measured by the PHOBOS collaboration [22], and the curves are our results calculated by the Monte Carlo method, where the spectator contributions in Figures 1(e)–1(j) are not considered. In the calculation, we take GeV/ for the particles produced in the central source and GeV/ for the particles produced in the target and projectile sources, respectively. The former one is estimated by us from an average weighing the masses and yields of , , , and [33]. The later one is the mass of a proton. The temperature is taken to be 0.156 GeV [34]. The values of peak position and contribution ratio of the target source, the rapidity distribution width of the central source, and per degree of freedom () obtained in the fitting are given in Table 1, where the contributions of spectators in Figures 1(e)–1(j) are not counted in the description, which result in larger if we consider them in the calculation of , and there is no spectator to be expected in other figures. The last two columns in Table 1 will be discussed later. To estimate the values of parameters, the least-squared fitting method is used. For the symmetric collisions, the peak position and contribution ratio of the projectile source, and the peak position and contribution ratio of the central source, can be given by , , , and , respectively.