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

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.


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  c = 180 MeV or energy density  c = 2 GeV/fm 3 .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 √ NN = 200 and 130 GeV is evaluated to be much higher than  c [3].Further studies have shown that QGP might be indeed generated in these collisions [4][5][6][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][9][10][11][12].
In the past decade, a number of bulk observables about charged particles, such as the Fourier harmonic coefficients V  of azimuth-angle distributions [13,14], rapidity or pseudorapidity distributions [15][16][17][18], and transverse momentum distributions [19][20][21][22][23][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 .
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][27][28][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 √ NN = 200 and 130 GeV [16][17][18][19][20]. Section 4 is about conclusions.

A Brief Introduction to the Model
The main content of the theoretical model [25] is as follows.
(1) In the process of expansions, the energy and momentum of fluid are conserved.Hence, the movement of fluid follows the continuity equation where  ] = ( 0 ,  1 ) = (, ),  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  ] =  ] = diag(1, −1) is the metric tensor.  = ( 0 ,  1 ) = (cosh  L , sinh  L ) is the 4-velocity of fluid with rapidity  L . and  in (2) are the energy density and pressure of fluid, related by the equation of state where , , and  s are the temperature, entropy density, and the sound speed of fluid, respectively.
(2) In order to solve (1), Khalatnikov potential  is introduced which makes the coordinate base of (, ) transform to that of (,  L ) via relations where  0 is the initial temperature of fluid and  = ln( 0 /).
Equation ( 1) is translated to the so-called telegraphy equation (3) Along with the expansions of matter created in collisions, its temperature becomes lower and lower.As the temperature drops from initial  0 to critical  c , 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  s .In QGP,  s =  0 = 1/ √ 3, which is the sound speed of a massless perfect fluid, being the maximum of  s .In the hadronic state,  s =  h <  0 .At the point of phase transition; that is, as  =  c ,  s is discontinuous.
(4) The solutions of (5) for the sectors of QGP and hadrons are, respectively [25], where  0 is a constant determined by tuning the theoretical results to experimental data. 0 is the 0th-order modified Bessel function of the first kind, and It is evident that if  h =  0 , then  h =  0 , () =   0  ,  h =  0 , and thus  h =  0 .At the point of phase transition,  =  c ,  =  c , () =   0  c , and

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 socalled chemical freeze-out temperature  FO , 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. T = √ 2 +  2 T is the transverse mass of produced charged particle with rest mass . B in ( 9) is the baryochemical potential.For Fermi charged particles,  = 1 in the denominator of ( 9), and for Bosons,  = −1.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  FO and transverse mass  T resulting from the thermalization of fluid.
From ( 4) and ( 8), it can be shown that where where  1 is the 1st-order modified Bessel function of the first kind.
The integral interval of  L in ( 9) is [− h ,  h ].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.
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 ( + +  − )/2 in Figure 2 therein, which is, however, only a theoretical prediction with no experimental comparisons.
In calculations,  c in (7) takes the well-recognized value of  c = 180 MeV. h takes the value of  h = 0.35 from the investigations of [32,[47][48][49].The chemical freeze-out temperature  FO in ( 9)- (12) takes  FO = 160 MeV from the studies of [23], which also show that the baryochemical potential  B in ( 9) is about equal to 20 and 27 MeV in Au-Au collisions at √ NN = 200 and 130 GeV, respectively.The initial temperature  0 in (7) takes  0 = 700 and 550 GeV in central Au-Au collisions at respective energy of √ NN = 200 and 130 GeV [32,[50][51][52][53].This allows us to determine the constant  0 in (9) for different kinds of charged particles as Advances in High Energy Physics listed in Tables 1 and 2. Keeping  0 unchanged, we can get  0 in different centrality cuts as shown in Tables 1 and 2. As we might expect,  0 decreases with increasing centrality cuts.
The fitted results of  0 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  FO for different charged particles, the integral part of ( 9) is the same for particles and antiparticles.Hence,  0 should be proportional to the abundance of corresponding particles.

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  0 , the sound speed  h in hadronic phase, the phase transition temperature  c , the chemical freeze-out temperature  FO , the baryochemical potential  B , and the initial temperature  0 .With the exception of  0 , the other five coefficients take the values either from the well-known theoretical results or from experimental measurements.As for  0 , it takes the values from other researches for the most central collisions.For the rest centrality cuts,  0 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 √ NN = 200 GeV.For the transverse momentum distributions measured in Au-Au collisions at √ NN = 200 and 130 GeV, the theoretical results match up well with experimental data for the transverse momentum up to about  T = 1.1 GeV/c.

Figure 4 :
Figure 4: The transverse momentum distributions of  + and  − in Au-Au collisions at √ NN = 200 GeV for  T up to  T = 2.0 GeV/c.The centrality cuts are the same as those in Figure2.The solid dots are the experimental data[19].The solid curves are the hydrodynamic results of (9).

Table 1 :
The  2 /NDF and the fitted initial temperature  0 and constant  0 at 68.3% confidence level in different centrality Au-Au collisions at √ NN = 200 GeV.

Table 2 :
The  2 /NDF and the fitted initial temperature  0 and constant  0 at 68.3% confidence level in different centrality Au-Au collisions at √ NN = 130 GeV.