Advances in High Energy Physics

Volume 2015 (2015), Article ID 430606, 9 pages

http://dx.doi.org/10.1155/2015/430606

## The Unified Hydrodynamics and the Pseudorapidity Distributions in Heavy Ion Collisions at BNL-RHIC and CERN-LHC Energies

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

Received 23 June 2014; Accepted 29 September 2014

Academic Editor: Fu-Hu Liu

Copyright © 2015 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

The charged particles produced in nucleus-nucleus collisions are divided into two parts. One is from the hot and dense matter created in collisions. The other is from leading particles. The hot and dense matter is assumed to expand according to unified hydrodynamics and freezes out into charged particles from a space-like hypersurface with a fixed proper time of . The leading particles are conventionally taken as the particles which inherit the quantum numbers of colliding nucleons and carry off most of incident energy. The rapidity distributions of the charged particles from these two parts are formulated analytically, and a comparison is made between the theoretical results and the experimental measurements performed in Au-Au and Pb-Pb collisions at the respective BNL-RHIC and CERN-LHC energies. The theoretical results are well consistent with experimental data.

#### 1. Introduction

Since the elementary work of Landau in 1953 [1], relativistic hydrodynamics has been applied to calculate a large number of variables developed in the context of nucleon or nucleus collisions at high energy. In particular, owing to the successful description of elliptic flow measured at the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) [2] and recently at the Large Hadron Collider (LHC) at CERN [3], the hydrodynamic research has entered into a more active phase. It has now been widely accepted as one of the best approaches for understanding the space-time evolution of the matter created in such collisions [4–10].

Although at present there are powerful numerical approaches to deal with certain hydrodynamic problems, this will require a very large scale of calculation and skillful sophisticated techniques to avoid instabilities. On the contrary, analytical solutions, since their simple and transparent forms, usually providing us with an invaluable insight into the characteristics of matter created in collisions, are always our pursuit of the goal though usually at the price of some ideal assumptions.

Due to the tremendous complexity of hydrodynamic equations, the progress in finding exact solutions is not going well. Up till now, most of this work mainly involves dimensional flows with simple equation of state [11–19]. The dimensional hydrodynamics is less developed, and no general exact solutions are known so far.

An important application of dimensional hydrodynamics is the analysis of the pseudorapidity distributions of the charged particles produced in nucleon or nucleus collisions. In this paper, by taking into account the effect of leading particles, we will discuss such distributions in the context of unified hydrodynamics [14]. The main points of this model are listed in Section 2. Its solution is then exploited in Section 3 to formulate the rapidity distributions of the charged particles frozen out from fluid at a space-like hypersurface with a fixed proper time . In Section 4, the theoretical results are compared with the experimental data performed in nucleus-nucleus collisions at BNL-RHIC and CERN-LHC energies. Section 5 is about conclusions.

#### 2. A Brief Description of the Unified Hydrodynamics

The expansion of a perfect fluid follows the equations:where is the time, is the longitudinal coordinate along beam direction, andis the energy-momentum tensor, and , , , and are, respectively, the metric tensor, 4-velocity, energy density, and pressure of fluid. For a constant speed of sound, and are related by the equation of state:where is the speed of sound. Investigations have shown that changes very slowly with energy and centrality [20–23]. For a given incident energy, it can be well taken as a constant.

Using (2) and (3), and noticing the light-cone components of the 4-velocitywhere is the ordinary rapidity of fluid, (1) readswhere and are the compact notation of partial derivatives with respect to light-cone coordinates , is the proper time, and is the space-time rapidity of fluid. The solutions of above equations are

The key ingredient of unified hydrodynamics is that it generalizes the relation between and by where are a priori arbitrary function obeying equationwhere is a constant. In case ofEquation (7) reduces to , returning to the boost-invariant picture of Hwa-Bjorken. Otherwise, (7) describes the nonboost-invariant geometry of Landau. Accordingly, (7) unifies the Hwa-Bjorken and Landau hydrodynamics together. It paves a way between these two models.

Substituting (7) into (6), we havewhere and is an arbitrary constant. From the above equations, we can get the entropy density of fluid:where, by definitionand in terms of , we have from (7) and (8)where .

#### 3. The Rapidity Distributions of the Charged Particles Frozen out from Fluid

By using (11), we can obtain the rapidity distributions of the charged particles frozen out from fluid or hot and dense matter created in collisions. To this end, we first evaluate the entropy distributions of the fluid on a space-like hypersurface with a fixed proper time , from which the fluid will freeze out into the charged particles. Such distributions take the formwhere is the 4-dimensional unit vector normal to the hypersurfaceAlso , and is the space-like infinitesimal length element along hypersurface

Considering a hypersurfacewhere is a constant, we haveThe light-cone components of unit vector can be expressed by asThen, the expression on the right-hand side of (15) is

Furthermore, known from (14),In terms of and , can be written asThus, (22) becomesSubstituting it into (21) and then into (15), we have

In the above equation, the right-hand side is evaluated at the hypersurface with proper time . Known from (18), this hypersurface can be taken asComparing the above equation with (19), we getInserting it together with (11) into (25) and noticing the proportional relation between entropy and the number of the charged particles, we havewhere , independent of rapidity , is an overall normalization constant. is the impact parameter, and is the center-of-mass energy per pair of nucleons.

#### 4. Comparison with Experimental Measurements and the Rapidity Distributions of Leading Particles

From (28), we can get the rapidity distributions of the charged particles generated in collisions. Figure 1 presents such distributions for , , , , , and resulting from 5% most central Au-Au collisions at GeV. The symbols are the experimental measurements given by BRAHMS Collaboration at BNL-RHIC [24–26]. The solid curves are the theoretical predictions from (28). From this figure, we can see that (28) coincides well with the experimental data of all the charged particles with the exception of . For proton , an obvious discrepancy appears in the rapidity interval of ~3.0. This might be caused by the effects of leading particles.