Advances in High Energy Physics

Volume 2018, Article ID 7682325, 9 pages

https://doi.org/10.1155/2018/7682325

## A Description of the Transverse Momentum Distributions of Charged Particles Produced in Heavy Ion Collisions at RHIC and LHC 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 5 November 2017; Revised 27 January 2018; Accepted 6 March 2018; Published 8 April 2018

Academic Editor: Bhartendu K. Singh

Copyright © 2018 Jia-Qi Hui 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

By assuming the existence of memory effects and long-range interactions, nonextensive statistics together with relativistic hydrodynamics including phase transition are used to discuss the transverse momentum distributions of charged particles produced in heavy ion collisions. It is shown that the combined contributions from nonextensive statistics and hydrodynamics can give a good description of the experimental data in Au+Au collisions at GeV and in Pb+Pb collisions at TeV for and in the whole measured transverse momentum region and for in the region of GeV/c. This is different from our previous work using the conventional statistics plus hydrodynamics, where the describable region is only limited in GeV/c.

#### 1. Introduction

The primary goals of the experimental programs performed in high energy heavy ion collisions are to find the deconfined nuclear matter, namely, the quark-gluon plasma (QGP), which is believed to have filled in the early universe several microseconds after the Big Bang. Therefore, studying the properties of QGP is important for both particle physics and cosmology. In the past decade, a number of bulk observables about charged particles, such as the Fourier coefficients of azimuth-angle distributions [1, 2], transverse momentum spectra [3–8], and pseudorapidity distributions [9–12], have been extensively studied in nuclear collisions at both RHIC and LHC energies. These investigations have shown that the matter created in these collisions 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 [13–31]. Therefore, the space-time evolution of sQGP can be described in the scope of relativistic hydrodynamics, which connects the static aspects of sQGP properties and the dynamical aspects of heavy ion collisions [32].

In our previous work [33], by considering the effects of thermalization, we once used a hydrodynamic model incorporating phase transition in analyzing the transverse momentum distributions of identified charged particles produced in heavy ion collisions. In that model, the quanta of hot and dense matter are supposed to obey the standard statistical distributions and the experimental measurements in Au+Au collisions at and 130 GeV can be well matched up in the region of GeV/c. Known from the investigations in [34, 35], the memory effects and long-range interactions might appear in the hot and dense matter. This guarantees the reasonableness of nonextensive statistics in describing the thermal motions of quanta of hot and dense matter, since this new statistics is just suitable for the thermodynamic system including memory effects and long-range interactions. Hence, in this paper, on the basis of hydrodynamics taking phase transition into consideration, we will use nonextensive statistics instead of conventional statistics to simulate the transverse collective flow of the matter created in collisions.

The nonextensive statistics are also known as Tsallis nonextensive thermostatistics, which were first proposed by Tsallis in 1988 in his pioneering work [36]. This statistical theory overcomes the shortcomings of the conventional statistics in many physical problems with long-range interactions, memory effects, or fractal space-time constrains [34]. It has a wide range of applications in high energy physics [37–39], astrophysical self-gravitating systems [40], cosmology [41], the solar neutrino problem [42], many-body theory, dynamical linear response theory, and variational methods [43].

In Section 2, a brief description is given on the employed hydrodynamics, presenting its analytical solutions. The solutions are then used in Section 3 to formulate the transverse momentum distributions of charged particles produced in heavy ion collisions in the light of nonextensive statistics and Cooper-Frye prescription. The last section is about conclusions.

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

The key points of the hydrodynamic model [18] used in the present paper are as follows.

The expansions of fluid follow the continuity equationwhere andis the energy-momentum tensor of perfect fluid, where is the metric tensor and is the four-velocity of fluid with rapidity . The energy density and the pressure of fluid are related by the sound speed via the equation of statewhere is the temperature and is the entropy density of fluid.

Projecting (1) to the direction of giveswhich is the continuity equation for entropy conservation. Projecting (1) to the direction Lorentz perpendicular to leads to the following equation:which means the existence of scalar function satisfyingFrom and Legendre transformation, Khalatnikov potential is introduced bywhich makes the coordinates of transform towhere is the initial temperature of fluid and . In terms of , (4) can be rewritten as the so-called telegraphy equation:

With the expansions of created matter, its temperature becomes lower and lower. When the temperature reduces from initial temperature to the critical temperature , the matter transforms from the sQGP state to the hadronic state. The produced hadrons are initially in the violent and frequent collisions, which are mainly inelastic. Hence, the abundance of identified hadrons is constantly changing. Furthermore, the mean free paths of these primary hadrons are very short. In sQGP, . The evolution of them still satisfies (9) with only difference being the values of . In the hadronic state, . At the point of phase transition, is discontinuous.

The solutions of (9) for the sQGP and hadronic state are, respectively [18],where is a constant determined by fitting the theoretical results with experimental data, is the 0th-order modified Bessel function, and

#### 3. The Transverse Momentum Distributions of Charged Particles Produced in Heavy Ion Collisions

##### 3.1. The Energy of Quantum of Produced Matter

In the nonextensive statistics, there are two basic postulations [36, 40].

(a) The entropy of a statistical system possesses the form ofwhere is the probability of a given microstate among different ones and is a real parameter.

(b) The mean value of an observable is given bywhere is the value of an observable in the microstate .

From the above two postulations, the average occupational number of the quantum in the state with temperature can be written in a simple analytical form [44]:where is the energy of quantum and is its baryochemical potential. for fermions, and for bosons. In the limit of , it reduces to the conventional Fermi-Dirac or Bose-Einstein distributions. Hence, the value of reflects the discrepancies of nonextensive statistics from the conventional ones. Known from (15), the average energy of quantum in the state with temperature readswhere is the rapidity of quantum and is its transverse mass with rest mass and transverse momentum .

##### 3.2. The Rapidity Distributions of Charged Particles in the State of Fluid

In terms of Khalatnikov potential , the rapidity distributions of charged particles in the state of fluid can be written as [45]whereis the area of overlap region of collisions, is impact parameter, and is the radius of colliding nucleus. From (8), the expression in the round brackets in (17) becomes

##### 3.3. The Transverse Momentum Distributions of Charged Particles Produced in Heavy Ion Collisions

Along with the expansions of hadronic matter, its temperature becomes even lower. As the temperature drops to kinetic freeze-out temperature , the inelastic collisions among hadronic matter stop. The yields of identified hadrons remain unchanged, becoming the measured results. According to Cooper-Frye scheme [45], the invariant multiplicity distributions of charged particles take the form [18, 45, 46]where and the integrand takes values at the moment of .

Substituting in (19) by of (11), it becomeswherewhere and is the 1st-order modified Bessel function.

By using (17) and (20)–(22), we can obtain the transverse momentum distributions of identified charged particles as shown in Figures 1 and 2.