Advances in High Energy Physics

Volume 2015, Article ID 641906, 10 pages

http://dx.doi.org/10.1155/2015/641906

## On Distributions of Emission Sources and Speed-of-Sound in Proton-Proton (Proton-Antiproton) Collisions

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

Received 17 July 2015; Revised 5 November 2015; Accepted 19 November 2015

Academic Editor: Luca Stanco

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

The revised (three-source) Landau hydrodynamic model is used in this paper to study the (pseudo)rapidity distributions of charged particles produced in proton-proton and proton-antiproton collisions at high energies. The central source is assumed to contribute with a Gaussian function which covers the rapidity distribution region as wide as possible. The target and projectile sources are assumed to emit isotropically particles in their respective rest frames. The model calculations obtained with a Monte Carlo method are fitted to the experimental data over an energy range from 0.2 to 13 TeV. The values of the squared speed-of-sound parameter in different collisions are then extracted from the width of the rapidity distributions.

#### 1. Introduction

In hadron-hadron, hadron-nucleus, and nucleus-nucleus (heavy ion) collisions at high energies, the final-state particles could be produced by multiple emission sources (which we denote also by fireballs) within the interacting system. This scenario can be tested by means of an analysis of the kinematic distributions of final-state particles. Proton-proton (proton-antiproton) collisions are usually used as a reference for the measurements in nucleus-nucleus collisions, in which the quark-gluon plasma (QGP) is expected to be formed [1]. From proton-proton collisions to nucleus-nucleus collisions, the distributions of emission sources may be similar to each other due to the small influences of nuclear effects such as the spectators and stopping power. The distributions of emission sources in high energy proton-proton (proton-antiproton) collisions can provide information on particle production in longitudinal rapidity and transverse momentum spaces.

In the study of distributions of emission sources, a principal question is whether there is a central source at midrapidity. If yes, what is the dependence of the relative contribution to particle production of the central source on collision energy? If no, what is the dependence of the width of the gap between two neighbouring sources around midrapidity on collision energy? In experiments at available accelerators and colliders, do the distributions of emission sources change with energy? Generally, common phenomenological models are useful in answering these questions. We focus on a few phenomenological models such as the three-fireball model [2–7], the three-source relativistic diffusion model [8–11], the multisource thermal model [12–14], the model with two Tsallis (or Boltzmann-Gibbs) clusters of fireballs [15–17], and the Landau hydrodynamic model [18–21] which results in a Gaussian shape for the rapidity distribution [21, 22] and a few revised versions such as works of Gao and Liu [23], Wong [24], Jiang et al. [25–30], Beuf et al. [31], and Bialas et al. [32].

The three-fireball model [2–7] assumes the nucleon to be an extended object and to consist of valence quarks, sea quarks, and gluons. The interacting system formed in high energy proton-proton collisions can be divided into three fireballs: a central fireball located at midrapidity, a target fireball located in the backward (target) rapidity region, and a projectile fireball located in the forward (projectile) rapidity region. The three-source relativistic diffusion model [8–11] was also proposed to have three fireballs or sources: a central source located at midrapidity and arising from interactions between low-momentum gluons in both target and projectile, a target-like source located in the backward rapidity region and arising from interactions between valence quarks in the target and low-momentum gluons in the projectile, and a projectile-like source located in the forward rapidity region and arising from interactions between low-momentum gluons in the target and valence quarks in the projectile.

The multisource thermal model [12–14] is the successor of the thermalized cylinder model and the two-cylinder model. The thermalized cylinder model was proposed to have a (central) thermalized cylinder at midrapidity and two leading nucleon sources in target and projectile rapidity regions, respectively. The two-cylinder model uses two (target/projectile) cylinders and two leading (target/projectile) nucleon sources. In the case of the two cylinders having a gap between them, there is no source at midrapidity. In most cases, there are sources at midrapidity due to the two cylinders overlapping each other or having no gap between them. In the model with two Tsallis clusters of fireballs [15], the rapidity distributions are described by using a superposition of two Tsallis fireballs along the rapidity axis. In the energy range from 0.5 to 7 TeV, the model result [15] shows that there is a gap between the two clusters, and there is no source at midrapidity. The Tsallis clusters can be replaced by others such as the Boltzmann-Gibbs clusters [16, 17].

The Landau hydrodynamic model [18–21] uses only a central source at midrapidity. The rapidity distribution obtained in the model is simply a Gaussian function [21, 22] in which the width is related to the speed-of-sound. A few revised versions were proposed in literature [22–32] to give better descriptions for experimental data, some of which include the contributions of leading and nonleading nucleons. We focus on a simple revised version [23] in which a central source at midrapidity, a target source in the backward rapidity region, and a projectile source in the forward rapidity region are used. In the simple revised version, the central source is assumed to contribute with a Gaussian function which covers the rapidity distribution region as wide as possible. The target and projectile sources are assumed to emit isotropically particles in their respective rest frames.

From the above introduction, we see that most models were proposed to have a central source at midrapidity. In particular, under the assumption that the emitting sources are thermalized fireballs, the width of the rapidity distribution obtained by the (revised) Landau hydrodynamic model can be used to extract the squared speed-of-sound parameter, which is related to the mean free path of the strongly interacting particles in the sources in the central, forward, and backward rapidity regions. We use the simple revised Landau hydrodynamic model [23] to describe the (pseudo)rapidity distribution. Because the widths of rapidity distributions obtained from the central source and from the target (or projectile) source are different, we expect to extract different values of the squared speed-of-sound parameter for the central and target (or projectile) sources.

In view of the wider application of the Landau hydrodynamic model [18–22] and its various revisions [22, 24–32], we have used the simple revised (three-source) Landau hydrodynamic model to study the pseudorapidity and rapidity distributions of charged particles produced in symmetric and asymmetric nuclear collisions at high energies in our recent work [23]. As a follow-up paper of [23], this work focuses on proton-proton (and proton-antiproton) collisions and presents a more detailed description of the model and its implementation. The definition of kinetic quantities is the same as in [23] according to universal representations, in order to provide a consistent picture.

In this paper, we use the simple revised (three-source) Landau hydrodynamic model to study the pseudorapidity and rapidity distributions of charged particles produced in proton-proton (proton-antiproton) collisions at high energies. In Section 2, a description of the model and calculation method is presented. Both pseudorapidity and rapidity distributions are obtained separately [33]. In Section 3, the pseudorapidity distributions are compared with the experimental data of proton-proton (proton-antiproton) collisions over an energy range from 0.2 to 13 TeV [34–42]. The values of the squared speed-of-sound parameter are then extracted from the width of the rapidity distributions. In Section 4, we summarize our main observations and conclusions.

#### 2. The Revised Landau Hydrodynamic Model and Calculation Method

The first 1+1-dimensional hydrodynamic model was proposed by Landau many years ago [18]. Later, a complex analytical solution was obtained by Khalatnikov [19]. The rapidity distribution of charged particles obtained from the complex analytical solution by Belenkij and Landau is [20]where is the logarithmic Lorentz contraction factor, denotes the center-of-mass energy per pair of nucleons, and denotes the rest mass of a proton. A later study [21] showed that the rapidity distribution of charged particles follows a Gaussian functionin the case of .

The second 1+1-dimensional hydrodynamic model and its analytical solution were obtained by Hwa [43] in the limit of . A plateau structure in rapidity distribution was obtained, which departs from available experimental results. Based on Hwa’s work, Bjorken obtained the energy density of particles in high energy collisions [44]. By taking into account the contributions of leading particles and based on a theory of unified description of Hwa-Bjorken and Landau relativistic hydrodynamics, pseudorapidity distributions of charged particles in agreement with the experimental results were obtained in a recent work [23]. For the central source, the unified hydrodynamic model describes the rapidity distribution of charged particles to be [22]where is the normalization constant, is the squared speed-of-sound, , denotes the initial temperature, and denotes the kinetic freeze-out temperature.

Equation (3) is similar to a Gaussian distribution [22]where is the normalization constant, is the distribution width, and is the midrapidity. In symmetric collisions, in the center-of-mass reference frame. In our simple revised version of the Landau hydrodynamic model [23], should be large enough to cover the rapidity region as wide as possible. The relation between and can be given by [18, 24, 45–49]Then, is expressed by using to be

To perform the calculation for (pseudo)rapidity distribution as accurately as possible, we need also the transverse momentum () distribution. Here we use the simplest Boltzmann distribution [50] where is the normalization constant, denotes the Boltzmann constant, is the effective temperature which is larger than the kinetic freeze-out temperature , and denotes the rest mass of the considered particle. The chemical potential and the distinction for fermions and bosons are not included due to small effects on transverse momentum distribution at high energy. Equation (7) means that we assume thermal emission of the final-state particles.

We introduce the transverse momentum distribution in the model so that we can obtain rapidity and pseudorapidity separately. In fact, to convert between rapidity and pseudorapidity distributions, additional limit is needed. The Boltzmann distribution describes the most fraction of the transverse momentum distribution. Equation (7) is not the sole choice for the transverse momentum distribution. In fact, we can also use other choices such as the Tsallis distribution [51, 52], the Tsallis form of standard distribution [53, 54], and the Erlang distribution [55]. In the study of rapidity or pseudorapidity distribution, the form of (7) and the value of parameter in it are not sensitive factors. Instead, the distributions of emission sources influence largely the rapidity or pseudorapidity distribution.

In the simple revised version of the Landau hydrodynamic model, we use in fact three sources: a central () source described by (4) and (7), a target () source described by an isotropic emission picture and (7), and a projectile () source equal to the target source. The central source describes the contributions of all produced particles, partly nonleading nucleons, and all leading nucleons, while the target and projectile sources describe the contributions of partly nonleading nucleons. In the rapidity space, the central source stays at midrapidity and the target (projectile) source stays at rapidity () in the backward (forward) target (projectile) region.

We used a Monte Carlo method to perform the calculation for the central source. Let denote random numbers in . According to (4) and (7), we have and , respectively. The longitudinal momentum is , the momentum , and the pseudorapidity

In the calculation for the target and projectile sources, particles are assumed to be emitted isotropically in their respective rest frames. In the Monte Carlo method, the emission angle is as follows: where (or ) in the case of the first term in the above equation being larger than 0 (or smaller than 0) and denotes a random number in . This isotropic emission results approximately in a Gaussian pseudorapidity distribution with the width of 0.91–0.92 [56], which can be compared with the theory of unified description of Hwa-Bjorken and Landau relativistic hydrodynamics [22], in which the rapidity of particle is assumed to obey a Gaussian distribution with the width of 0.85. In the present work, we do not need to study further the width corresponding to the target and projectile sources due to the fixed value of 0.91.

The transverse momentum of the particle produced in the target or projectile source has the same expression as those in the central source. The longitudinal momentum , the momentum , and the energy in the rest frame of the considered source can be extracted. In the laboratory or center-of-mass reference frame, the rapidity is given by The longitudinal momentum , the momentum , and the pseudorapidity have the same expressions as those for the central source.

In the above discussions, the rapidity distribution, , and the pseudorapidity distribution, , can be obtained separately, where denotes the multiplicity of charged particles.

#### 3. Comparisons with Experimental Data and Discussion

The pseudorapidity distributions, , of charged particles produced in proton-proton () collisions at 0.2, 0.41, 0.9, 2.36, 7, and 13 TeV are presented in Figure 1, where is simplified to for proton-proton and proton-antiproton collisions. The squares represent the experimental data measured by the PHOBOS [34] ((a), (b)), ALICE [35, 36] ((c), (d), (g)), and CMS [37, 38] ((e), (f), (h)) Collaborations, whereas the curves represent our model results calculated by means of the Monte Carlo method. The types of collisions, inelastic interactions (INEL) or nonsingle diffractive interactions (NSD), are marked in the panels. We take for the central source and for the target and projectile sources, respectively. The former is estimated from an average of the weighted masses and yields of , , , and [57]; the latter is the mass of a proton due to protons being the charged particles in the target (projectile) source.