# A Description of the Transverse Momentum Distributions of Charged Particles Produced in Heavy Ion Collisions at RHIC and LHC Energies

Authors: Jia-Qi Hui; Zhi-Jin Jiang; Dong-Fang Xu
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

By assuming the existence of memory effects and long-range interactions, nonextensive statistics together with relativistic hydrodynamics including phase transition are used to discuss the transverse momentum distributions of charged particles produced in heavy ion collisions. It is shown that the combined contributions from nonextensive statistics and hydrodynamics can give a good description of the experimental data in Au+Au collisions atsNN=200 GeV and in Pb+Pb collisions at sNN=2.76 TeV for π± and K± in the whole measured transverse momentum region and for pp- in the region of pT≤2.0 GeV/c. This is different from our previous work using the conventional statistics plus hydrodynamics, where the describable region is only limited in pT≤1.1 GeV/c.


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## Body

## 1. Introduction

The primary goals of the experimental programs performed in high energy heavy ion collisions are to find the deconfined nuclear matter, namely, the quark-gluon plasma (QGP), which is believed to have filled in the early universe several microseconds after the Big Bang. Therefore, studying the properties of QGP is important for both particle physics and cosmology. In the past decade, a number of bulk observables about charged particles, such as the Fourier coefficientsvn of azimuth-angle distributions [1, 2], transverse momentum spectra [3–8], and pseudorapidity distributions [9–12], have been extensively studied in nuclear collisions at both RHIC and LHC energies. These investigations have shown that the matter created in these collisions 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 [13–31]. Therefore, the space-time evolution of sQGP can be described in the scope of relativistic hydrodynamics, which connects the static aspects of sQGP properties and the dynamical aspects of heavy ion collisions [32].In our previous work [33], by considering the effects of thermalization, we once used a hydrodynamic model incorporating phase transition in analyzing the transverse momentum distributions of identified charged particles produced in heavy ion collisions. In that model, the quanta of hot and dense matter are supposed to obey the standard statistical distributions and the experimental measurements in Au+Au collisions at sNN=200 and 130 GeV can be well matched up in the region of pT≤1.1 GeV/c. Known from the investigations in [34, 35], the memory effects and long-range interactions might appear in the hot and dense matter. This guarantees the reasonableness of nonextensive statistics in describing the thermal motions of quanta of hot and dense matter, since this new statistics is just suitable for the thermodynamic system including memory effects and long-range interactions. Hence, in this paper, on the basis of hydrodynamics taking phase transition into consideration, we will use nonextensive statistics instead of conventional statistics to simulate the transverse collective flow of the matter created in collisions.The nonextensive statistics are also known as Tsallis nonextensive thermostatistics, which were first proposed by Tsallis in 1988 in his pioneering work [36]. This statistical theory overcomes the shortcomings of the conventional statistics in many physical problems with long-range interactions, memory effects, or fractal space-time constrains [34]. It has a wide range of applications in high energy physics [37–39], astrophysical self-gravitating systems [40], cosmology [41], the solar neutrino problem [42], many-body theory, dynamical linear response theory, and variational methods [43].In Section2, a brief description is given on the employed hydrodynamics, presenting its analytical solutions. The solutions are then used in Section 3 to formulate the transverse momentum distributions of charged particles produced in heavy ion collisions in the light of nonextensive statistics and Cooper-Frye prescription. The last section is about conclusions.


## 2. A Brief Introduction to the Hydrodynamic Model

The key points of the hydrodynamic model [18] used in the present paper are as follows.The expansions of fluid follow the continuity equation(1)∂Tμv∂xv=0,μ,v=0,1,where xv=x0,x1=t,z and(2)Tμv=ε+puμuv-pgμvis the energy-momentum tensor of perfect fluid, where gμv=diag⁡1,-1 is the metric tensor and uμ=u0,u1=cosh⁡yF,sinh⁡yF is the four-velocity of fluid with rapidity yF. The energy density ε and the pressure p of fluid are related by the sound speed cs via the equation of state(3)ⅆpⅆε=sdTTds=cs2,where T is the temperature and s is the entropy density of fluid.Projecting (1) to the direction of uμ gives(4)∂suv∂xv=0,which is the continuity equation for entropy conservation. Projecting (1) to the direction Lorentz perpendicular to uμ leads to the following equation:(5)∂Tsinh⁡yF∂t+∂Tcosh⁡yF∂z=0,which means the existence of scalar function ϕ satisfying(6)∂ϕ∂t=Tcosh⁡yF,∂ϕ∂z=-Tsinh⁡yF.From ϕ and Legendre transformation, Khalatnikov potential χ is introduced by(7)χ=ϕ-tTcosh⁡yF+zTsinh⁡yF,which makes the coordinates of t,z transform to(8)t=eωT0∂χ∂ωcosh⁡yF+∂χ∂yFsinh⁡yF,z=eωT0∂χ∂ωsinh⁡yF+∂χ∂yFcosh⁡yF,where T0 is the initial temperature of fluid and ω=ln⁡T0/T. In terms of χ, (4) can be rewritten as the so-called telegraphy equation:(9)∂2χ∂ω2-2β∂χ∂ω-1cs2∂2χ∂yF2=0,β=1-cs22cs2.With the expansions of created matter, its temperature becomes lower and lower. When the temperature reduces from initial temperatureT0 to the critical temperature Tc, the matter transforms from the sQGP state to the hadronic state. The produced hadrons are initially in the violent and frequent collisions, which are mainly inelastic. Hence, the abundance of identified hadrons is constantly changing. Furthermore, the mean free paths of these primary hadrons are very short. In sQGP, cs=c0=1/3. The evolution of them still satisfies (9) with only difference being the values of cs. In the hadronic state, 0<cs=ch≤c0. At the point of phase transition, cs is discontinuous.The solutions of (9) for the sQGP and hadronic state are, respectively [18],(10)χ0ω,yF=Q0c02eβ0ωI0β0ω2-c02yF2,(11)χhω,yF=Q0c02eβhω-ωc+β0ωcI0βhchyh2ω-yF2,where Q0 is a constant determined by fitting the theoretical results with experimental data, I0 is the 0th-order modified Bessel function, and(12)β0=1-c022c02=1,βh=1-ch22ch2,ωc=ln⁡T0Tc,yhω=ω-ωcch+ωcc0.


## 3. The Transverse Momentum Distributions of Charged Particles Produced in Heavy Ion Collisions

### 3.1. The Energy of Quantum of Produced Matter

In the nonextensive statistics, there are two basic postulations [36, 40].(a) The entropy of a statistical system possesses the form of(13)sq=1q-1∑i=1Ωpi1-piq-1,where pi is the probability of a given microstate among Ω different ones and q is a real parameter.(b) The mean value of an observableO is given by(14)O-q=∑i=1ΩpiqOi,where Oi is the value of an observable O in the microstate i.From the above two postulations, the average occupational number of the quantum in the state with temperatureT can be written in a simple analytical form [44]:(15)n-q=11+q-1E-μB/T1∕q-1+δ,where E is the energy of quantum and μB is its baryochemical potential. δ=1 for fermions, and δ=-1 for bosons. In the limit of q→1, it reduces to the conventional Fermi-Dirac or Bose-Einstein distributions. Hence, the value of q reflects the discrepancies of nonextensive statistics from the conventional ones. Known from (15), the average energy of quantum in the state with temperature T reads(16)E-q=mTcosh⁡y-yF1+q-1mTcosh⁡y-yF-μB/T1/q-1+δ,where y is the rapidity of quantum and mT=pT2+m2 is its transverse mass with rest mass m and transverse momentum pT.


### 3.2. The Rapidity Distributions of Charged Particles in the State of Fluid

In terms of Khalatnikov potentialχ, the rapidity distributions of charged particles in the state of fluid can be written as [45](17)ⅆNⅆyF=Q0c02Abcosh⁡yⅆzⅆyF-sinh⁡yⅆtⅆyF,where(18)Ab=2r2arccos⁡b2r-br2-b22is the area of overlap region of collisions, b is impact parameter, and r is the radius of colliding nucleus. From (8), the expression in the round brackets in (17) becomes(19)cosh⁡yⅆzⅆyF-sinh⁡yⅆtⅆyF=1Tcs2∂∂ωχ+∂χ∂ωcosh⁡y-yF-1T∂∂yFχ+∂χ∂ωsinh⁡y-yF.


### 3.3. The Transverse Momentum Distributions of Charged Particles Produced in Heavy Ion Collisions

Along with the expansions of hadronic matter, its temperature becomes even lower. As the temperature drops to kinetic freeze-out temperatureTf, the inelastic collisions among hadronic matter stop. The yields of identified hadrons remain unchanged, becoming the measured results. According to Cooper-Frye scheme [45], the invariant multiplicity distributions of charged particles take the form [18, 45, 46](20)ⅆ2N2πpTⅆyⅆpT=12π3∫-yhωfyhωfⅆNⅆyFE-qT=TfⅆyF,where ωf=ln⁡T0/Tf and the integrand takes values at the moment of T=Tf.Substitutingχ in (19) by χh of (11), it becomes(21)coshy⁡ⅆzⅆyF-sinh⁡yⅆtⅆyFT=Tf=1Tfβhch2eβhω-ωc+β0ωcBωf,yFsinh⁡y-yF+Cωf,yFcosh⁡y-yF,where(22)Bωf,yF=βhyFλωf,yFβhchyhωfλωf,yFI0λωf,yF+βh+1βh-2βhchyhωfλ2ωf,yFI1λωf,yF,Cωf,yF=βh+1βh+βhchyhωf2λ2ωf,yFI0λωf,yF+1λωf,yFyhωfch+1-2βhchyhωf2λ2ωf,yFI1λωf,yF,where λωf,yF=βhchyh2(ωf)-yF2 and I1 is the 1st-order modified Bessel function.By using (17) and (20)–(22), we can obtain the transverse momentum distributions of identified charged particles as shown in Figures 1 and 2.Figure 1
The invariant yields ofπ±, K±, and pp- as a function of pT in Au+Au collisions at sNN=200 GeV. The solid dots represent the experimental data of the PHENIX Collaboration [3]. The solid curves are the results calculated from (20). 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.
(a) (b)Figure 2
The invariant yields ofπ±, K±, and pp- as a function of pT in Pb+Pb collisions at sNN=2.76 TeV. The circles and squares represent the experimental data of the ALICE Collaboration [4]. The solid curves are the results calculated from (20). The centrality cuts counted from top to bottom in each panel are 0–5% (×29), 5–10% (×28), 10–20% (×27), 20–30% (×26), 30–40% (×25), 40–50% (×24), 50–60% (×23), 60–70% (×22), 70–80% (×21), and 80–90% (×20), respectively.
(a) (b)Figure1 shows the invariant yields of π±, K±, and pp- as a function of pT in Au+Au collisions at sNN=200 GeV. Figure 2 shows the same distributions in Pb+Pb collisions at sNN=2.76 TeV. The solid dots represent the experimental data [3, 4]. The solid curves are the results calculated from (20). It can be seen that the theoretical results are in good agreement with the experimental data for π± and K± in the whole measured pT region. For pp-, the theoretical model works well in the region of pT≤2.0 GeV/c. Beyond this region, the deviation appears as shown in Figure 3, which presents the fittings for pp- in the most peripheral collisions for pT up to about 4 GeV/c.Figure 3
The invariant yields ofpp- as a function of pT in 60–92% Au+Au collisions at sNN=200 GeV (a) and in 80–90% Pb+Pb collisions at sNN= 2.76 TeV (b). The meanings of solid dots, circles, squares, and solid curves are the same as those in Figures 1 and 2.
(a) (b)In analyses, the sound speed in hadronic state takes the value ofch=0.35 [46–49]. The critical temperature takes the value of Tc=0.16 GeV [50]. The kinetic freeze-out temperature Tf takes the values of 0.12 GeV for pions and kaons and 0.13 GeV for protons, respectively, from the investigations of [8], which also shows that the average value of baryochemical potential μB is about 0.019 GeV in Au+Au collisions. For Pb+Pb collisions, μB takes the value of μB=0 [8]. The initial temperature in central Au+Au and Pb+Pb collisions takes the values of T0=0.35 GeV and T0=0.6 GeV, respectively [51].Tables1 and 2 list the values of T0, q, Q0, and χ2/NDF in different centrality cuts. It can be seen that T0 decreases with increasing centrality cuts. The value of q, affecting the slopes of curves, is slightly larger than 1 for different kinds of charged particles. It is almost irrelevant to centrality cuts, while it increases with increasing beam energies and the masses of charged particles.Table 1
The values ofT0, q, and Q0 in different centrality Au+Au collisions at sNN= 200 GeV.
Centrality cuts T 0 (GeV) q ( π / K / p ) Q 0 ( π + / π - ) Q 0 ( K + / K - ) Q 0 ( p / p - ) χ 2 / N D F 0–10% 0.35 0 ± 0.001 1.0808 ± 0.00001 0.7953 ± 0.0081 0.7952±0.0078 0.3552 ± 0.0043 0.3315±0.0040 0.4490 ± 0.0050 0.3228±0.0040 25/23 0.350 ± 0.001 1.0880 ± 0.00004 27/33 0.350 ± 0.001 1.1012 ± 0.0004 18/14  10–20% 0.349 ± 0.002 1.0822 ± 0.0002 0.7953 ± 0.0112 0.7965±0.0110 0.3545 ± 0.0043 0.3307±0.0041 0.4494 ± 0.0052 0.3259±0.0040 20/17 0.349 ± 0.001 1.0892 ± 0.0002 21/21 0.349 ± 0.001 1.1012 ± 0.0004 17/13  20–40% 0.320 ± 0.001 1.0838 ± 0.0002 0.7953 ± 0.0098 0.8010±0.0060 0.3514 ± 0.0030 0.3276±0.0030 0.4470 ± 0.0036 0.3274±0.0029 24/17 0.320 ± 0.001 1.0904 ± 0.0003 21/18 0.320 ± 0.001 1.1012 ± 0.0002 31/32  40–60% 0.278 ± 0.001 1.0838 ± 0.0005 0.7953 ± 0.0067  0.7985±0.0062 0.3260 ± 0.0031 0.3053±0.0031 0.5494 ± 0.0048 0.4093±0.0038 19/14 0.278 ± 0.001 1.0905 ± 0.0005 22/10 0.278 ± 0.001 1.0912 ± 0.0002 19/27  60–92% 0.210 ± 0.001 1.0854 ± 0.0002 0.7953 ± 0.0100 0.8040±0.0067 0.3134 ± 0.0033 0.2943±0.0032 0.9299 ± 0.0090 0.7029±0.0066 26/32 0.210 ± 0.001 1.0908 ± 0.0005 28/25 0.210 ± 0.001 1.0744 ± 0.0002 3.6/9.2Table 2
The values ofT0, q, and Q0 in different centrality Pb+Pb collisions at sNN= 2.76 TeV.
Centrality cuts T 0 (GeV) q ( π / K / p ) Q 0 ( π + / π - ) Q 0 ( K + / K - ) Q 0 ( p / p - ) χ 2 / N D F 0–5% 0.60 ± 0.02 1.104 ± 0.001 0.20 ± 0.02 0.05 0 ± 0.003 0.0069 ± 0.0005 1.88/2.00 0.60 ± 0.01 1.130 ± 0.003 6.50/5.42 0.6 0 ± 0.01 1.260 ± 0.005 2.35/1.50  5–10% 0.59 ± 0.02 1.104 ± 0.001 0.20 ± 0.03 0.051 ± 0.004 0.0073 ± 0.0005 2.46/2.56 0.59 ± 0.02 1.13 0 ± 0.005 6.15/5.22 0.59 ± 0.01 1.260 ± 0.004 2.75/2.15  10–20% 0.58 ± 0.02 1.104 ± 0.001 0.20 ± 0.02 0.048 ± 0.003 0.0068 ± 0.0005 2.21/2.14 0.58 ± 0.01 1.13 0 ± 0.004 5.65/4.85 0.58 ± 0.01 1.260 ± 0.004 3.43/2.79  20–30% 0.54 ± 0.01 1.104 ± 0.001 0.20 ± 0.02 0.048 ± 0.004 0.0069 ± 0.0005 2.19/2.00 0.54 ± 0.01 1.13 0 ± 0.004 4.55/3.46 0.54 ± 0.01 1.260 ± 0.004 4.60/4.08  30–40% 0.50 ± 0.02 1.104 ± 0.002 0.20 ± 0.02 0.047 ± 0.004 0.0156 ± 0.0011 2.69/2.41 0.5 0 ± 0.01 1.13 0 ± 0.003 3.29/2.54 0.50 ± 0.01 1.200 ± 0.004 2.27/1.87  40–50% 0.45 ± 0.01 1.103 ± 0.001 0.20 ± 0.02 0.046 ± 0.004 0.0221 ± 0.0015 2.68/2.49 0.45 ± 0.01 1.13 0 ± 0.003 2.35/1.91 0.45 ± 0.01 1.180 ± 0.003 1.89/2.00  50–60% 0.4 0 ± 0.01 1.103 ± 0.001 0.20 ± 0.01 0.048 ± 0.004 0.0215 ± 0.0016 5.66/5.12 0.40 ± 0.01 1.125 ± 0.003 1.31/1.22 0.40 ± 0.01 1.180 ± 0.002 3.97/4.86  60–70% 0.34 ± 0.01 1.100 ± 0.001 0.20 ± 0.02 0.050 ± 0.005 0.0560 ± 0.0043 5.20/4.66 0.34 ± 0.01 1.12 0 ± 0.003 0.58/1.05 0.34 ± 0.01 1.130 ± 0.003 0.91/1.25  70–80% 0.28 ± 0.01 1.100 ± 0.001 0.20 ± 0.02 0.052 ± 0.005 0.0686 ± 0.0054 8.21/7.86 0.28 ± 0.01 1.115 ± 0.004 0.22/0.27 0.28 ± 0.01 1.120 ± 0.002 1.01/1.43  80–90% 0.21 ± 0.01 1.099 ± 0.001 0.20 ± 0.02 0.056 ± 0.006 0.1250 ± 0.0117 6.42/5.98 0.21 ± 0.01 1.115 ± 0.002 0.86/1.19 0.21 ± 0.01 1.100 ± 0.003 0.40/1.42The parameterQ0 has the same effects as T0 on curves. They all affect the heights of the curves. The fitted Q0 in Tables 1 and 2 gives the ratios(23)
Q
0
π
-
Q
0
π
+
=
1
,
Q
0
K
-
Q
0
K
+
=
0.93
,
Q
0
p
-
Q
0
p
=
0.72
A
u
+
A
u
200
G
e
V
,
Q
0
π
-
Q
0
π
+
=
Q
0
K
-
Q
0
K
+
=
Q
0
p
-
Q
0
p
=
1
P
b
+
P
b
2.76
T
e
V
.These ratios are in good agreement with the relative abundance of particles and antiparticles given in [3, 4]. This agreement may be due to the fact that the integrand of (20) is the same for particles and antiparticles in case that Tf takes a common constant for these two kinds of particles. Hence, Q0 might be proportional to the abundance of corresponding particles.


## 3.1. The Energy of Quantum of Produced Matter

In the nonextensive statistics, there are two basic postulations [36, 40].(a) The entropy of a statistical system possesses the form of(13)sq=1q-1∑i=1Ωpi1-piq-1,where pi is the probability of a given microstate among Ω different ones and q is a real parameter.(b) The mean value of an observableO is given by(14)O-q=∑i=1ΩpiqOi,where Oi is the value of an observable O in the microstate i.From the above two postulations, the average occupational number of the quantum in the state with temperatureT can be written in a simple analytical form [44]:(15)n-q=11+q-1E-μB/T1∕q-1+δ,where E is the energy of quantum and μB is its baryochemical potential. δ=1 for fermions, and δ=-1 for bosons. In the limit of q→1, it reduces to the conventional Fermi-Dirac or Bose-Einstein distributions. Hence, the value of q reflects the discrepancies of nonextensive statistics from the conventional ones. Known from (15), the average energy of quantum in the state with temperature T reads(16)E-q=mTcosh⁡y-yF1+q-1mTcosh⁡y-yF-μB/T1/q-1+δ,where y is the rapidity of quantum and mT=pT2+m2 is its transverse mass with rest mass m and transverse momentum pT.


## 3.2. The Rapidity Distributions of Charged Particles in the State of Fluid

In terms of Khalatnikov potentialχ, the rapidity distributions of charged particles in the state of fluid can be written as [45](17)ⅆNⅆyF=Q0c02Abcosh⁡yⅆzⅆyF-sinh⁡yⅆtⅆyF,where(18)Ab=2r2arccos⁡b2r-br2-b22is the area of overlap region of collisions, b is impact parameter, and r is the radius of colliding nucleus. From (8), the expression in the round brackets in (17) becomes(19)cosh⁡yⅆzⅆyF-sinh⁡yⅆtⅆyF=1Tcs2∂∂ωχ+∂χ∂ωcosh⁡y-yF-1T∂∂yFχ+∂χ∂ωsinh⁡y-yF.


## 3.3. The Transverse Momentum Distributions of Charged Particles Produced in Heavy Ion Collisions

Along with the expansions of hadronic matter, its temperature becomes even lower. As the temperature drops to kinetic freeze-out temperatureTf, the inelastic collisions among hadronic matter stop. The yields of identified hadrons remain unchanged, becoming the measured results. According to Cooper-Frye scheme [45], the invariant multiplicity distributions of charged particles take the form [18, 45, 46](20)ⅆ2N2πpTⅆyⅆpT=12π3∫-yhωfyhωfⅆNⅆyFE-qT=TfⅆyF,where ωf=ln⁡T0/Tf and the integrand takes values at the moment of T=Tf.Substitutingχ in (19) by χh of (11), it becomes(21)coshy⁡ⅆzⅆyF-sinh⁡yⅆtⅆyFT=Tf=1Tfβhch2eβhω-ωc+β0ωcBωf,yFsinh⁡y-yF+Cωf,yFcosh⁡y-yF,where(22)Bωf,yF=βhyFλωf,yFβhchyhωfλωf,yFI0λωf,yF+βh+1βh-2βhchyhωfλ2ωf,yFI1λωf,yF,Cωf,yF=βh+1βh+βhchyhωf2λ2ωf,yFI0λωf,yF+1λωf,yFyhωfch+1-2βhchyhωf2λ2ωf,yFI1λωf,yF,where λωf,yF=βhchyh2(ωf)-yF2 and I1 is the 1st-order modified Bessel function.By using (17) and (20)–(22), we can obtain the transverse momentum distributions of identified charged particles as shown in Figures 1 and 2.Figure 1
The invariant yields ofπ±, K±, and pp- as a function of pT in Au+Au collisions at sNN=200 GeV. The solid dots represent the experimental data of the PHENIX Collaboration [3]. The solid curves are the results calculated from (20). 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.
(a) (b)Figure 2
The invariant yields ofπ±, K±, and pp- as a function of pT in Pb+Pb collisions at sNN=2.76 TeV. The circles and squares represent the experimental data of the ALICE Collaboration [4]. The solid curves are the results calculated from (20). The centrality cuts counted from top to bottom in each panel are 0–5% (×29), 5–10% (×28), 10–20% (×27), 20–30% (×26), 30–40% (×25), 40–50% (×24), 50–60% (×23), 60–70% (×22), 70–80% (×21), and 80–90% (×20), respectively.
(a) (b)Figure1 shows the invariant yields of π±, K±, and pp- as a function of pT in Au+Au collisions at sNN=200 GeV. Figure 2 shows the same distributions in Pb+Pb collisions at sNN=2.76 TeV. The solid dots represent the experimental data [3, 4]. The solid curves are the results calculated from (20). It can be seen that the theoretical results are in good agreement with the experimental data for π± and K± in the whole measured pT region. For pp-, the theoretical model works well in the region of pT≤2.0 GeV/c. Beyond this region, the deviation appears as shown in Figure 3, which presents the fittings for pp- in the most peripheral collisions for pT up to about 4 GeV/c.Figure 3
The invariant yields ofpp- as a function of pT in 60–92% Au+Au collisions at sNN=200 GeV (a) and in 80–90% Pb+Pb collisions at sNN= 2.76 TeV (b). The meanings of solid dots, circles, squares, and solid curves are the same as those in Figures 1 and 2.
(a) (b)In analyses, the sound speed in hadronic state takes the value ofch=0.35 [46–49]. The critical temperature takes the value of Tc=0.16 GeV [50]. The kinetic freeze-out temperature Tf takes the values of 0.12 GeV for pions and kaons and 0.13 GeV for protons, respectively, from the investigations of [8], which also shows that the average value of baryochemical potential μB is about 0.019 GeV in Au+Au collisions. For Pb+Pb collisions, μB takes the value of μB=0 [8]. The initial temperature in central Au+Au and Pb+Pb collisions takes the values of T0=0.35 GeV and T0=0.6 GeV, respectively [51].Tables1 and 2 list the values of T0, q, Q0, and χ2/NDF in different centrality cuts. It can be seen that T0 decreases with increasing centrality cuts. The value of q, affecting the slopes of curves, is slightly larger than 1 for different kinds of charged particles. It is almost irrelevant to centrality cuts, while it increases with increasing beam energies and the masses of charged particles.Table 1
The values ofT0, q, and Q0 in different centrality Au+Au collisions at sNN= 200 GeV.
Centrality cuts T 0 (GeV) q ( π / K / p ) Q 0 ( π + / π - ) Q 0 ( K + / K - ) Q 0 ( p / p - ) χ 2 / N D F 0–10% 0.35 0 ± 0.001 1.0808 ± 0.00001 0.7953 ± 0.0081 0.7952±0.0078 0.3552 ± 0.0043 0.3315±0.0040 0.4490 ± 0.0050 0.3228±0.0040 25/23 0.350 ± 0.001 1.0880 ± 0.00004 27/33 0.350 ± 0.001 1.1012 ± 0.0004 18/14  10–20% 0.349 ± 0.002 1.0822 ± 0.0002 0.7953 ± 0.0112 0.7965±0.0110 0.3545 ± 0.0043 0.3307±0.0041 0.4494 ± 0.0052 0.3259±0.0040 20/17 0.349 ± 0.001 1.0892 ± 0.0002 21/21 0.349 ± 0.001 1.1012 ± 0.0004 17/13  20–40% 0.320 ± 0.001 1.0838 ± 0.0002 0.7953 ± 0.0098 0.8010±0.0060 0.3514 ± 0.0030 0.3276±0.0030 0.4470 ± 0.0036 0.3274±0.0029 24/17 0.320 ± 0.001 1.0904 ± 0.0003 21/18 0.320 ± 0.001 1.1012 ± 0.0002 31/32  40–60% 0.278 ± 0.001 1.0838 ± 0.0005 0.7953 ± 0.0067  0.7985±0.0062 0.3260 ± 0.0031 0.3053±0.0031 0.5494 ± 0.0048 0.4093±0.0038 19/14 0.278 ± 0.001 1.0905 ± 0.0005 22/10 0.278 ± 0.001 1.0912 ± 0.0002 19/27  60–92% 0.210 ± 0.001 1.0854 ± 0.0002 0.7953 ± 0.0100 0.8040±0.0067 0.3134 ± 0.0033 0.2943±0.0032 0.9299 ± 0.0090 0.7029±0.0066 26/32 0.210 ± 0.001 1.0908 ± 0.0005 28/25 0.210 ± 0.001 1.0744 ± 0.0002 3.6/9.2Table 2
The values ofT0, q, and Q0 in different centrality Pb+Pb collisions at sNN= 2.76 TeV.
Centrality cuts T 0 (GeV) q ( π / K / p ) Q 0 ( π + / π - ) Q 0 ( K + / K - ) Q 0 ( p / p - ) χ 2 / N D F 0–5% 0.60 ± 0.02 1.104 ± 0.001 0.20 ± 0.02 0.05 0 ± 0.003 0.0069 ± 0.0005 1.88/2.00 0.60 ± 0.01 1.130 ± 0.003 6.50/5.42 0.6 0 ± 0.01 1.260 ± 0.005 2.35/1.50  5–10% 0.59 ± 0.02 1.104 ± 0.001 0.20 ± 0.03 0.051 ± 0.004 0.0073 ± 0.0005 2.46/2.56 0.59 ± 0.02 1.13 0 ± 0.005 6.15/5.22 0.59 ± 0.01 1.260 ± 0.004 2.75/2.15  10–20% 0.58 ± 0.02 1.104 ± 0.001 0.20 ± 0.02 0.048 ± 0.003 0.0068 ± 0.0005 2.21/2.14 0.58 ± 0.01 1.13 0 ± 0.004 5.65/4.85 0.58 ± 0.01 1.260 ± 0.004 3.43/2.79  20–30% 0.54 ± 0.01 1.104 ± 0.001 0.20 ± 0.02 0.048 ± 0.004 0.0069 ± 0.0005 2.19/2.00 0.54 ± 0.01 1.13 0 ± 0.004 4.55/3.46 0.54 ± 0.01 1.260 ± 0.004 4.60/4.08  30–40% 0.50 ± 0.02 1.104 ± 0.002 0.20 ± 0.02 0.047 ± 0.004 0.0156 ± 0.0011 2.69/2.41 0.5 0 ± 0.01 1.13 0 ± 0.003 3.29/2.54 0.50 ± 0.01 1.200 ± 0.004 2.27/1.87  40–50% 0.45 ± 0.01 1.103 ± 0.001 0.20 ± 0.02 0.046 ± 0.004 0.0221 ± 0.0015 2.68/2.49 0.45 ± 0.01 1.13 0 ± 0.003 2.35/1.91 0.45 ± 0.01 1.180 ± 0.003 1.89/2.00  50–60% 0.4 0 ± 0.01 1.103 ± 0.001 0.20 ± 0.01 0.048 ± 0.004 0.0215 ± 0.0016 5.66/5.12 0.40 ± 0.01 1.125 ± 0.003 1.31/1.22 0.40 ± 0.01 1.180 ± 0.002 3.97/4.86  60–70% 0.34 ± 0.01 1.100 ± 0.001 0.20 ± 0.02 0.050 ± 0.005 0.0560 ± 0.0043 5.20/4.66 0.34 ± 0.01 1.12 0 ± 0.003 0.58/1.05 0.34 ± 0.01 1.130 ± 0.003 0.91/1.25  70–80% 0.28 ± 0.01 1.100 ± 0.001 0.20 ± 0.02 0.052 ± 0.005 0.0686 ± 0.0054 8.21/7.86 0.28 ± 0.01 1.115 ± 0.004 0.22/0.27 0.28 ± 0.01 1.120 ± 0.002 1.01/1.43  80–90% 0.21 ± 0.01 1.099 ± 0.001 0.20 ± 0.02 0.056 ± 0.006 0.1250 ± 0.0117 6.42/5.98 0.21 ± 0.01 1.115 ± 0.002 0.86/1.19 0.21 ± 0.01 1.100 ± 0.003 0.40/1.42The parameterQ0 has the same effects as T0 on curves. They all affect the heights of the curves. The fitted Q0 in Tables 1 and 2 gives the ratios(23)
Q
0
π
-
Q
0
π
+
=
1
,
Q
0
K
-
Q
0
K
+
=
0.93
,
Q
0
p
-
Q
0
p
=
0.72
A
u
+
A
u
200
G
e
V
,
Q
0
π
-
Q
0
π
+
=
Q
0
K
-
Q
0
K
+
=
Q
0
p
-
Q
0
p
=
1
P
b
+
P
b
2.76
T
e
V
.These ratios are in good agreement with the relative abundance of particles and antiparticles given in [3, 4]. This agreement may be due to the fact that the integrand of (20) is the same for particles and antiparticles in case that Tf takes a common constant for these two kinds of particles. Hence, Q0 might be proportional to the abundance of corresponding particles.


## 4. Conclusions

By introducing the nonextensive statistics, we employ the relativistic hydrodynamics including phase transition to discuss the transverse momentum distributions of charged particles produced in heavy ion collisions. The model contains rich information about the transport coefficients of fluid, such as the initial temperatureT0, the critical temperature Tc, the kinetic freeze-out temperature Tf, the baryochemical potential μB, the sound speed in sQGP state c0, and the sound speed in hadronic state ch. Except for T0, the other five parameters take the values either from widely accepted theoretical results or from experimental measurements. As for T0, there are no widely acknowledged values so far. In this paper, T0 takes the values referring to other studies for the most central collisions, and, for the rest of centrality cuts, T0 is determined by tuning the theoretical results to experimental data. The present investigations show the conclusions as follows:( 1 ) The theoretical model can give a good description of the experimental data in Au+Au collisions at sNN=200 GeV and in Pb+Pb collisions at sNN=2.76 TeV for π± and K± in the whole measured transverse momentum region and for pp- in the region of pT≤2.0 GeV/c.( 2 ) The fitted q is close to 1. This might mean that the difference between nonextensive statistics and conventional statistics is small. However, it is this small difference that plays an essential role in extending the fitting region of pT.( 3 ) The fitted q increases with beam energies and the masses of charged particles. This might mean that the nonextensive statistics are more suitable for these conditions.( 4 ) The model cannot describe the experimental measurements for pp- in the region of pT≥2.0 GeV/c for the both kinds of collisions. This might be caused by the hard scattering process [52]. To improve the fitting conditions, the results from perturbative QCD should be taken into account.



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