A description of pseudorapidity distributions of charged particles produced in Au+Au collisions at RHIC energies

The charged particles produced in heavy ion collisions consist of two parts: One is from the freeze-out of hot and dense matter formed in collisions. The other is from 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.


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 . 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 .
The best approach for describing the space-time evolution of fluid-like sQGP is the relativistic hydrodynamics, which was first put forward by L. D. Landau in his pioneering work in 1953 [34].
However, since the partial differential equations of relativistic hydrodynamics are highly nonlinear, it is a formidable task to solve them analytically. This is the reason why, from the time of Landau until now, the exact solutions of relativistic hydrodynamics are mainly limited to 1+1 expansion for a perfect fluid with simple equation of state. To solve the equations of high-dimensional expansions especially for situations incorporating the effect of viscosities or pressure anisotropies, one has to resort to the means of computer simulations.
One of the most important applications of 1+1 dimensional hydrodynamics is the analysis of the pseudorapidity distributions of charged particles in high energy physics. In this paper, combing the effect of leading particles, we will discuss such distributions in the framework of hydrodynamic model including phase transition [10]. In section 2, a brief introduction is given to the theoretical model, presenting its exact solutions. The solutions are then used in section 3 to formulate the pseudorapidity distributions of charged particles resulted from the freeze-out of sQGP. Together with the contribution from leading particles, the results are then compared with the experimental observations performed by PHOBOS Collaboration at RHIC in Au+Au collisions at NN s =200 and 19.6 GeV [5], respectively. The last section 4 is traditionally about conclusions.

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].
(1) The movement of fluid follows the continuity equation , t is the time and z is the longitudinal coordinate along beam direction.
T  is the energy-momentum tensor, which, for a perfect fluid, takes the form where   is the 4-velocity of fluid, F y is its rapidity.  and p in Eq. (2) is needed, where s c is the sound speed of fluid, which takes different values in sQGP and in hadronic phase.
(2) Project Eq. (1) to the direction of u  and the direction perpendicular to u  , respectively.
This leads to equations Eq. (6) is the continuity equation for entropy conservation. Eq. (7) means the existence of a scalar function  satisfying relations cosh , sinh .
From  and Legendre transformation, Khalatnikov potential  is introduced via relation cosh sinh In terms of  , the variables t and z can be expressed as 0 0 cosh sinh , sinh cosh , where 0 T is the initial temperature of fluid and (3) Along with the expansions of matter created in collisions, it becomes cooler and cooler.
As its temperature drops from the initial 0 T to the critical c T , phase transition occurs.
where 0 q is a constant determined by tuning the theoretical results to experimental data. 0 I is the 0th order modified Bessel function of the first kind, and In the sector of hadrons, the solution of Eq. (11) is [10] where

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 [35]   where   A b is the area of overlap region of collisions, being the function of impact parameter b or centrality cuts. Inserting Eq. (10) into above equation, the part in the round brackets is the transverse mass of produced charged particle with rest mass m . The integral interval of F y in Eq. (18) is where 1 I is the 1st order modified Bessel function of the first kind.
(2) The invariant multiplicity distributions of leading particles It is believed that the leading particles are formed outside the nucleus, that is, outside the colliding region [36,37]. The generation of leading particles is therefore free from fluid evolution.
Hence, their rapidity distributions are beyond the scope of hydrodynamic description and should be treated separately.
In our previous work [24][25][26], we once argued that the rapidity distributions of leading particles take the Gaussian form where   The investigations have shown that [38], for certain rapidity, the invariant multiplicity distributions of leading particles possess the form where a is a constant. Then, as a function of rapidity, the invariant multiplicity distributions of leading particles can be written as which is normalized to lead N .    2  2  2  T  T   2  2  2  T  T   cosh  sinh  1 ln  2 cosh sinh is in order.
Substituting Eqs. (18) and (24) into Eq. (25) and carrying out the integration of T p , we can get the pseudorapidity distributions of charged particles produced in high energy heavy ion collisions. The solid curves are the sums of the four types of curves. The 2 NDF  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.   Table 1. It can be seen that 0 T decreases slowly with increasing centralities especially in the first four cuts. Table 1 Table 1. The meanings of different types of curves are the same as those in Figure 1. It can be seen that, in the absence of leading particles, the hydrodynamics alone can give a good description to the experimental observations. This is different from Figure 1. Where, leading particles are essential in fitting experimental data. This difference might be caused by the different transparencies of nuclei in different energies. As the analyses given in Ref. [43], in central Au+Au collisions at NN 200 GeV s  , the leading particles locate at about 0 2.91 y  . This position is far away from the mid-rapidity region. Where, relative to the low yields of charged particles frozen out from sQGP, the effect of leading particles is evident which should be considered separately. On the contrary, in case of Au+Au collisions at NN 19.6 GeV s  , 0 1.28 y  . This position is so close to the mid-rapidity region that the effect of leading particles is hidden by the large yields of charged particles generated from the freeze-out of sQGP. Therefore, there is no need to consider the contribution of leading particles separately.
In drawing Figure 2, 0 T takes the values as those listed in Table 1.

Conclusions
By taking into consideration the effect of leading particles, the hydrodynamic model incorporating the phase transition is used to analyze the pseudorapidity distributions of charged particles produced in Au+Au collisions at RHIC energies.