Advances in High Energy Physics
Volume 2017 (2017), Article ID 6896524, 8 pages
https://doi.org/10.1155/2017/6896524

Research Article

The Rapidity Distributions and the Thermalization Induced Transverse Momentum Distributions in Au-Au Collisions at RHIC Energies

Zhi-Jin Jiang, Jia-Qi Hui, and Yu Zhang

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

Correspondence should be addressed to Zhi-Jin Jiang; jzj265@163.com

Received 23 January 2017; Revised 28 April 2017; Accepted 9 May 2017; Published 4 July 2017

Academic Editor: Sally Seidel

Copyright © 2017 Zhi-Jin 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 SCOAP³.

Abstract.

It is widely believed that the quark-gluon plasma (QGP) might be formed in the current heavy ion collisions. It is also widely recognized that the relativistic hydrodynamics is one of the best tools for describing the process of expansion and hadronization of QGP. In this paper, by taking into account the effects of thermalization, a hydrodynamic model including phase transition from QGP state to hadronic state is used to analyze the rapidity and transverse momentum distributions of identified charged particles produced in heavy ion collisions. A comparison is made between the theoretical results and experimental data. The theoretical model gives a good description of the corresponding measurements made in Au-Au collisions at RHIC energies.

1. Introduction

The primary goal of experimental program performed at Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) and at Large Hadron Collider (LHC) at CERN is to create a hot and dense matter consisting of partonic degrees of freedom, usually called the quark-gluon plasma (QGP), which is believed to have filled in the early universe several microseconds after the big bang. The calculations of Lattice Quantum Chromodynamics (LQCD) have predicted [1] that such matter may exist in the environment with critical temperature of about  MeV or energy density  GeV/fm³. By means of the Bjorken estimation [2] and the measurements of PHENIX Collaboration at RHIC, the spatial energy density in central Au-Au collisions at and 130 GeV is evaluated to be much higher than [3]. Further studies have shown that QGP might be indeed generated in these collisions [4–7]. In fact, it has long been argued that QGP might even have come into being in collisions at the energies of Intersecting Storage Rings (ISR) and Super Proton Synchrotron (SPS) at CERN [8–12].

In the past decade, a number of bulk observables about charged particles, such as the Fourier harmonic coefficients of azimuth-angle distributions [13, 14], rapidity or pseudorapidity distributions [15–18], and transverse momentum distributions [19–24], have experienced a series of extensive investigations in heavy ion collisions at both RHIC and LHC energies. These investigations have provided us with a compelling evidence that the matter created in heavy ion collisions exhibits a clear collective behavior, expanding nearly like a perfect fluid with very low viscosity. This sets up the prominent position of relativistic hydrodynamics in analyzing the properties of bulk observables in heavy ion collisions [25–46].

Apart from collective movement, the quanta of produced matter also have the components of thermal motion stemmed from the thermalization of fluid. The evolution of produced matter is then the superposition of these two parts. To clarify the role of thermalization in the expansions of the produced matter is the major subject of this paper. To this end, we may as usual in analytical treatment ignore the collective flow in the transverse directions. The transverse movement of the produced matter is therefore only induced by the thermalization.

The collective movement of produced matter in the longitudinal direction can be solved analytically. There are a few schemes in dealing with such exact calculations. In this paper, the hydrodynamic model proposed by Suzuki is employed [25]. Besides the analyticity, the other typical feature of this model is that it, like some other models [26–29], incorporates the effects of phase transition into solutions. This coincides with the current experimental observations as mentioned above. Hence, the employed model might be more in line with the realistic situations. In addition, the model is related to the initial temperature of QGP, the sound speed in both partonic and hadronic media, the baryochemical potential, and the critical temperature of phase transition. This work may therefore help us understand various transport coefficients of expanding system.

In Section 2, a brief introduction is given to the theoretical model [25], presenting its analytical formulations. The solutions are then used in Section 3 to formulate the invariant multiplicity distributions of charged particles produced in heavy ion collisions which are in turn compared with the experimental measurements carried out by BRAHMS and PHENIX Collaboration in Au-Au collisions at RHIC energies of and 130 GeV [16–20]. Section 4 is about conclusions.

2. A Brief Introduction to the Model

The main content of the theoretical model [25] is as follows.

() In the process of expansions, the energy and momentum of fluid are conserved. Hence, the movement of fluid follows the continuity equation

where , is the time, and is the longitudinal coordinate along beam direction. is the energy-momentum tensor, which, for a perfect fluid, takes the form

where is the metric tensor. is the 4-velocity of fluid with rapidity . and in (2) are the energy density and pressure of fluid, related by the equation of state

where , , and are the temperature, entropy density, and the sound speed of fluid, respectively.

() In order to solve (1), Khalatnikov potential is introduced which makes the coordinate base of transform to that of via relations

where is the initial temperature of fluid and . Equation (1) is translated to the so-called telegraphy equation

() Along with the expansions of matter created in collisions, its temperature becomes lower and lower. As the temperature drops from initial to critical , phase transition occurs. The matter transforms from QGP state to hadronic state. The produced hadrons are initially in the violent and frequent collisions. The major part of these collisions is inelastic. Hence, the abundance of an identified hadron is in changing. Furthermore, the mean free paths of these primary hadrons are very short. The movement of them is still like that of a fluid meeting (5) with only difference being the value of . In QGP, , which is the sound speed of a massless perfect fluid, being the maximum of . In the hadronic state, . At the point of phase transition; that is, as , is discontinuous.

() The solutions of (5) for the sectors of QGP and hadrons are, respectively [25],

where is a constant determined by tuning the theoretical results to experimental data. is the 0th-order modified Bessel function of the first kind, and

It is evident that if , then , , , and thus . At the point of phase transition, , , , and . Then

. That is, the potential is discontinuous at point of .

3. The Rapidity Distributions and the Thermalization Induced Transverse Momentum Distributions of Identified Charged Particles

With the expansions of hadronic matter, its temperature continues becoming lower. According to the prescription of Cooper-Frye [31], as the temperature drops to the so-called chemical freeze-out temperature , the inelastic collisions among hadrons cease. The abundance of an identified hadron maintains unchanged becoming the measured results in experiments. The invariant multiplicity distributions of charged particles are given by [25, 31, 32]

where is the area of overlap region of collisions, being the function of impact parameter or centrality cuts. is the transverse mass of produced charged particle with rest mass . in (9) is the baryochemical potential. For Fermi charged particles, in the denominator of (9), and for Bosons, . That is, Fermi and Bose charged particles follow the Fermi-Dirac and Bose-Einstein distributions, respectively. The meaning of (9) is evident. The part of integrand in the round brackets is proportional to the rapidity density of fluid resulting from the collective movement along longitudinal direction [31]. The rest part is the energy of the charged particles in the state with temperature and transverse mass resulting from the thermalization of fluid.

From (4) and (8), it can be shown that

where

where is the 1st-order modified Bessel function of the first kind.

The integral interval of in (9) is . By applying (9)–(12), together with the definitions in (7), we can get the rapidity distributions and the thermalization induced transverse momentum distributions of identified charged particles as shown in Figures 1, 2, and 3.

Figure 1: The rapidity distributions of , , and in central Au-Au collisions at  GeV. The scattered dots are the experimental measurements [16–18]. The solid curves are the results from (9).

Figure 2: The centrality dependences of transverse momentum distributions of identified charged particles produced in Au-Au collisions at  GeV. The solid dots are the experimental data [19]. The solid curves are the hydrodynamic results of (9). The centrality cuts counted from top to bottom in each panel are 0–10%  , 10–20%  , 20–40%  , 40–60%  , and 60–92%  , respectively.

Figure 3: The centrality dependences of transverse momentum distributions of identified charged particles produced at midrapidity of in Au-Au collisions at  GeV. The solid dots are the experimental data [20]. The solid curves are the hydrodynamic results of (9). The centrality cuts counted from top to bottom in each panel are 0–5% (×5), 5–15% (×2), 15–30%, 30–60%, and 60–92%, respectively.

Figure 1 shows the rapidity distributions of identified charged particles in central Au-Au collisions at  GeV. The solid dots are the experimental measurements [16–18]. The solid curves are the results from (9). It can be seen that, except for proton , the theoretical predictions are in good agreement with experimental data. For proton , the experimental data show an evident uplift in the rapidity interval between and 3.0. This might be resulting from parts of leading particles [33, 34], which are free from the description of hydrodynamics.

In [25], by approximating the Bose-Einstein to Maxwell-Boltzmann distribution for the Bose charged particles, the author also presented a rapidity distribution of in Figure 2 therein, which is, however, only a theoretical prediction with no experimental comparisons.

Figures 2 and 3 are for the centrality dependences of thermalization induced transverse momentum distributions of , , and in Au-Au collisions at and 130 GeV, respectively. The left columns are for positive charged particles. The right columns are for negative ones. The centrality cuts are 0–10%  , 10–20%  , 20–40%  , 40–60%  , and 60–92%  , respectively, counted from top to bottom in Figure 2. The centrality cuts in Figure 3 are 0–5%  , 5–15%  , 15–30%, 30–60%, and 60–92%, respectively, counted from top to bottom. The solid dots are the experimental measurements [19, 20]. The solid curves are the hydrodynamic predictions of (9). The corresponding values of are listed in Tables 1 and 2. It can be seen that the theoretical results are in good accordance with experimental data for transverse momentum up to about  GeV/c.

Table 1: The and the fitted initial temperature and constant at 68.3% confidence level in different centrality Au-Au collisions at  GeV.

Centrality cuts (%)
0–10
10–20
20–40
40–60
60–92

0.92
1.52
0.75
2.13
1.44

1.05
1.02
1.07
0.95
0.98

1.02
0.90
1.48
1.26
1.30

1.15
2.25
2.20
2.31
1.25

3.20
3.64
3.61
3.26
3.25

1.34
1.30
1.65
2.52
2.33

: ;

: ;

:

Table 2: The and the fitted initial temperature and constant at 68.3% confidence level in different centrality Au-Au collisions at  GeV.

Centrality cuts (%)
0–5
5–15
15–30
30–60
60–92

0.85
0.80
0.72
0.11
2.19

0.85
0.80
0.72
0.11
2.19

3.34
0.82
0.87
0.72
3.77

3.36
1.23
1.25
0.93
3.20

4.05
3.05
2.96
2.99
0.90

4.17
4.02
3.03
3.02
0.88

(MeV)

: ;

: ;

:

In the region of  GeV/c, the deviation appears as shown in Figure 4 which shows the transverse momentum distributions for and in Au-Au collisions at  GeV for up to  GeV/c. To overcome this deviation, some other mechanisms should be introduced. The existing investigations have shown that if the transverse collective flow is taken into account, the describable region is extended to about  GeV/c [20–22]. Furthermore, to describe the experimental data in the region of  GeV/c, the contributions of blast-wave and perturbative QCD [24] should be introduced.

Figure 4: The transverse momentum distributions of and in Au-Au collisions at  GeV for up to  GeV/c. The centrality cuts are the same as those in Figure 2. The solid dots are the experimental data [19]. The solid curves are the hydrodynamic results of (9).

In calculations, in (7) takes the well-recognized value of  MeV. takes the value of from the investigations of [32, 47–49]. The chemical freeze-out temperature in (9)–(12) takes  MeV from the studies of [23], which also show that the baryochemical potential in (9) is about equal to 20 and 27 MeV in Au-Au collisions at and 130 GeV, respectively. The initial temperature in (7) takes and 550 GeV in central Au-Au collisions at respective energy of and 130 GeV [32, 50–53]. This allows us to determine the constant in (9) for different kinds of charged particles as listed in Tables 1 and 2. Keeping unchanged, we can get in different centrality cuts as shown in Tables 1 and 2. As we might expect, decreases with increasing centrality cuts.

The fitted results of in Tables 1 and 2 give the ratios

These ratios are well consistent with the relative abundances of particles and antiparticles presented in [23]. This consistence may be attributed to the fact that, in case of adopting a common for different charged particles, the integral part of (9) is the same for particles and antiparticles. Hence, should be proportional to the abundance of corresponding particles.

4. Conclusions

In order to see the importance of thermalization in the expansions of the hot and dense matter created in high energy heavy ion collisions, the transverse collective flow is as usual not taken into account. The expansion of matter in the transverse directions is therefore only induced by the thermalization of fluid. Multiplied with longitudinal collective flow, we can get the invariant multiplicity distributions of charged particles.

The model contains rich information about the transport coefficients of fluid, such as the sound speed in QGP , the sound speed in hadronic phase, the phase transition temperature , the chemical freeze-out temperature , the baryochemical potential , and the initial temperature . With the exception of , the other five coefficients take the values either from the well-known theoretical results or from experimental measurements. As for , it takes the values from other researches for the most central collisions. For the rest centrality cuts, is determined by comparing the theoretical results with experimental data.

From the coincidences between the theoretical curves and experimental data points, we can see that the theoretical model can give a good description of rapidity distributions measured in central Au-Au collisions at  GeV. For the transverse momentum distributions measured in Au-Au collisions at and 130 GeV, the theoretical results match up well with experimental data for the transverse momentum up to about  GeV/c.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by the Shanghai Key Lab of Modern Optical System.

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