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

Authors: Zhi-Jin Jiang; Jia-Qi Hui; Yu Zhang
Journal: Advances in High Energy Physics
 (2017)

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## 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.


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## Body

## 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 Tc=180 MeV or energy density εc=2 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 sNN=200 and 130 GeV is evaluated to be much higher than εc [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 p-p(p-) 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 coefficientsvn 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 Section2, 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 sNN=200 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.(1) In the process of expansions, the energy and momentum of fluid are conserved. Hence, the movement of fluid follows the continuity equation(1)∂Tμν∂xν=0,μ,ν=0,1,where xν=(x0,x1)=(t,z), t is the time, and z is the longitudinal coordinate along beam direction. Tμν is the energy-momentum tensor, which, for a perfect fluid, takes the form(2)Tμν=ε+puμuν-pgμν,where gμν=gμν=diag⁡1,-1 is the metric tensor. uμ=u0,u1=cosh⁡yL,sinh⁡yL is the 4-velocity of fluid with rapidity yL. ε and p in (2) are the energy density and pressure of fluid, related by the equation of state(3)dpdε=sdTTds=cs2,where T, s, and cs are the temperature, entropy density, and the sound speed of fluid, respectively.(2) In order to solve (1), Khalatnikov potential χ is introduced which makes the coordinate base of t,z transform to that of ω,yLvia relations(4)t=eωT0∂χ∂ωcosh⁡yL+∂χ∂yLsinh⁡yL,z=eωT0∂χ∂ωsinh⁡yL+∂χ∂yLcosh⁡yL,where T0 is the initial temperature of fluid and ω=ln⁡T0/T. Equation (1) is translated to the so-called telegraphy equation(5)∂2χ∂ω2-2β∂χ∂ω-1cs2∂2χ∂yL2=0,β=1-cs22cs2.(3) Along with the expansions of matter created in collisions, its temperature becomes lower and lower. As the temperature drops from initial T0 to critical Tc, 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 cs. In QGP, cs=c0=1/3, which is the sound speed of a massless perfect fluid, being the maximum of cs. In the hadronic state, cs=ch<c0. At the point of phase transition; that is, as T=Tc, cs is discontinuous.(4) The solutions of (5) for the sectors of QGP and hadrons are, respectively [25],(6)χ0ω,yL=Q0c02eβ0ωI0β0c0y02ω-yL2,χhω,yL=Q0c02BωI0λω,yL,where Q0 is a constant determined by tuning the theoretical results to experimental data. I0 is the 0th-order modified Bessel function of the first kind, and(7)β0=1-c022c02=1,y0ω=ωc0,Bω=eβhω-ωc+β0ωc,λω,yL=βhchyh2ω-yL2,βh=1-ch22ch2,ωc=ln⁡T0Tc,yhω=ω-ωcch+ωcc0.It is evident that if ch=c0, then βh=β0, Bω=eβ0ω, yh=y0, and thus χh=χ0. At the point of phase transition, T=Tc, ω=ωc, Bω=eβ0ωc, and yh=y0=ωc/c0. Then(8)χ0ω,yLT=Tc=Q0c02eβ0ωcI0β0ωc2-c02yL2,χhω,yLT=Tc=Q0c02eβ0ωcI0βhchc0ωc2-c02yL2,χhω,yLT=Tc≠χ0ω,yLT=Tc. That is, the potential χ is discontinuous at point of T=Tc.


## 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 TFO, 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](9)d2N2πpTdydpT=Q0c02Ab2π3∫cosh⁡ydzdyL-sinh⁡ydtdyLT=TFO×mTcosh⁡y-yLexp⁡mTcosh⁡y-yL-μB/TFO+δdyL,where Ab is the area of overlap region of collisions, being the function of impact parameter b or centrality cuts. mT=m2+pT2 is the transverse mass of produced charged particle with rest mass m. μB in (9) is the baryochemical potential. For Fermi charged particles, δ=1 in the denominator of (9), and for Bosons, δ=-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 TFO and transverse mass mT resulting from the thermalization of fluid.From (4) and (8), it can be shown that(10)cosh⁡ydzdyL-sinh⁡ydtdyLT=TFO=1TFOβhch2BωFOSωFO,yLsinh⁡y-yL+CωFO,yLcosh⁡y-yL,where(11)SωFO,yL=βhyLλωFO,yLβhchyhωFOλωFO,yLI0λωFO,yL+βh+1βh-2βhchyhωFOλ2ωFO,yLI1λωFO,yL,(12)CωFO,yL=βh+1βh+βhchyhωFO2λ2ωFO,yLI0λωFO,yL+1λωFO,yLyhωFOch+1-2βhchyhωFO2λ2ωFO,yLI1λωFO,yL,where I1 is the 1st-order modified Bessel function of the first kind.The integral interval ofyL in (9) is -yh,yh. 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 ofπ±, K±, and pp- in central Au-Au collisions at sNN=200 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 atsNN=200 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%  (×104), 10–20%  (×103), 20–40%  (×102), 40–60%  (×101), and 60–92%  (×100), respectively.Figure 3
The centrality dependences of transverse momentum distributions of identified charged particles produced at midrapidity ofy≤1 in Au-Au collisions at sNN=130 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.Figure1 shows the rapidity distributions of identified charged particles in central Au-Au collisions at sNN=200 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 p, the theoretical predictions are in good agreement with experimental data. For proton p, the experimental data show an evident uplift in the rapidity interval between y=2.0 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 π++π-/2 in Figure 2 therein, which is, however, only a theoretical prediction with no experimental comparisons.Figures2 and 3 are for the centrality dependences of thermalization induced transverse momentum distributions of π±, K±, and pp- in Au-Au collisions at sNN=200 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%  (×104), 10–20%  (×103), 20–40%  (×102), 40–60%  (×101), and 60–92%  (×100), respectively, counted from top to bottom in Figure 2. The centrality cuts in Figure 3 are 0–5%  (×5), 5–15%  (×2), 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 χ2/NDF 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 pT=1.1 GeV/c.Table 1
Theχ2/NDF and the fitted initial temperature T0 and constant Q0 at 68.3% confidence level in different centrality Au-Au collisions at sNN=200 GeV.
Centrality cuts (%)
0–10
10–20
20–40
40–60
60–92
χ
2
/
N
D
F
π+
0.92
1.52
0.75
2.13
1.44
π-
1.05
1.02
1.07
0.95
0.98
K+
1.02
0.90
1.48
1.26
1.30
K-
1.15
2.25
2.20
2.31
1.25
p
3.20
3.64
3.61
3.26
3.25
p-
1.34
1.30
1.65
2.52
2.33
T
0
M
e
V
700
±
24
618
±
31
534
±
31
450
±
35
332
±
27
Q
0
π
+
/
π
-: 2.21±0.17/2.21±0.15;
K
+
/
K
-: 1.04±0.16/0.98±0.19;
p
/
p
-: 2.15±0.24/1.61±0.20.Table 2
Theχ2/NDF and the fitted initial temperature T0 and constant Q0 at 68.3% confidence level in different centrality Au-Au collisions at sNN=130 GeV.
Centrality cuts (%)
0–5
5–15
15–30
30–60
60–92
χ
2
/
N
D
F
π+
0.85
0.80
0.72
0.11
2.19
π-
0.85
0.80
0.72
0.11
2.19
K+
3.34
0.82
0.87
0.72
3.77
K-
3.36
1.23
1.25
0.93
3.20
p
4.05
3.05
2.96
2.99
0.90
p-
4.17
4.02
3.03
3.02
0.88
T
0 (MeV)
550
±
40
542
±
36
502
±
28
403
±
22
260
±
34
Q
0
π
+
/
π
-: 3.50±0.13/3.50±0.13;
K
+
/
K
-: 2.32±0.18/2.15±0.21;
p
/
p
-: 9.83±0.33/7.05±0.38.In the region ofpT>1.1 GeV/c, the deviation appears as shown in Figure 4 which shows the transverse momentum distributions for π+ and π- in Au-Au collisions at sNN=200 GeV for pT up to pT=2.0 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 pT<2.0 GeV/c [20–22]. Furthermore, to describe the experimental data in the region of pT≥2.0 GeV/c, the contributions of blast-wave and perturbative QCD [24] should be introduced.Figure 4
The transverse momentum distributions ofπ+ and π- in Au-Au collisions at sNN=200 GeV for pT up to pT=2.0 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,Tc in (7) takes the well-recognized value of Tc=180 MeV. ch takes the value of ch=0.35 from the investigations of [32, 47–49]. The chemical freeze-out temperature TFO in (9)–(12) takes TFO=160 MeV from the studies of [23], which also show that the baryochemical potential μB in (9) is about equal to 20 and 27 MeV in Au-Au collisions at sNN=200 and 130 GeV, respectively. The initial temperature T0 in (7) takes T0=700 and 550 GeV in central Au-Au collisions at respective energy of sNN=200 and 130 GeV [32, 50–53]. This allows us to determine the constant Q0 in (9) for different kinds of charged particles as listed in Tables 1 and 2. Keeping Q0 unchanged, we can get T0 in different centrality cuts as shown in Tables 1 and 2. As we might expect, T0 decreases with increasing centrality cuts.The fitted results ofQ0 in Tables 1 and 2 give the ratios(13)Q0π+Q0π-=1,Q0K+Q0K-=1.06,Q0pQ0p-=1.34200GeV,Q0π+Q0π-=1,Q0K+Q0K-=1.08,Q0pQ0p-=1.39130GeV.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 TFO for different charged particles, the integral part of (9) is the same for particles and antiparticles. Hence, Q0 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 QGPc0, the sound speed ch in hadronic phase, the phase transition temperature Tc, the chemical freeze-out temperature TFO, the baryochemical potential μB, and the initial temperature T0. With the exception of T0, the other five coefficients take the values either from the well-known theoretical results or from experimental measurements. As for T0, it takes the values from other researches for the most central collisions. For the rest centrality cuts, T0 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 atsNN=200 GeV. For the transverse momentum distributions measured in Au-Au collisions at sNN=200 and 130 GeV, the theoretical results match up well with experimental data for the transverse momentum up to about pT=1.1 GeV/c.



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