# A Description of Transverse Momentum Distributions inp+p Collisions at RHIC and LHC Energies

Authors: Jia-Qi Hui; Zhi-Jin Jiang
Journal: Advances in High Energy Physics
 (2019)

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## Abstract

It has long been debated whether the hydrodynamics is suitable for the smaller colliding systems such asp+p collisions. In this paper, by assuming the existence of longitudinal collective motion and long-range interactions in the hot and dense matter created in p+p collisions, the relativistic hydrodynamics incorporating with the nonextensive statistics is used to analyze the transverse momentum distributions of the particles. The investigations of the present paper show that the hybrid model can give a good description of the currently available experimental data obtained in p+p collisions at RHIC and LHC energies, except for p and p¯ produced in the range of pT>3.0 GeV/c at s=200 GeV.


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## Body

## 1. Introduction

In the past decade, the experimental results of heavy ion collisions at both RHIC and LHC energies have been extensively studied. These studies have shown that the strongly coupled quark-gluon plasma (sQGP) might be created in these collisions [1–9], which exhibits a clear collective behavior almost like a perfect fluid with very low viscosity [10–28]. Therefore, the evolution of sQGP can be described in the scope of relativistic hydrodynamics. However, unlike heavy ion collisions, p+p collisions are a relatively smaller system with lower multiplicity, larger viscosity, and larger fluctuation [29]. The reasonableness of applying relativistic hydrodynamics in depicting the evolution of sQGP created in p+p collisions has undergone an endless debate.In this paper, by supposing the existence of collective flow inp+p colliding direction, the relativistic hydrodynamics including phase transition is introduced to describe the longitudinal expansion of sQGP. Besides the collective flow, the thermal motion also exists in sQGP. The evolution of sQGP is therefore the superposition of collective flow and thermal motion. Known from the investigations of [30, 31], the long-range interactions and memory effects might appear in sQGP. This guarantees the reasonableness of nonextensive statistics in describing the thermodynamic aspects of sQGP. Hence, in this paper, we will use the nonextensive statistics instead of conventional statistics to characterize the thermal motion of the matter created in p+p collisions.The nonextensive statistics,i.e., Tsallis nonextensive thermostatistics, is the generalization of conventional Boltzmann-Gibbs statistics, which is proposed by C. Tsallis in his pioneer work of [32]. This statistical theory overcomes the inabilities of the conventional statistical mechanics by assuming the existence of long-range interactions, long-range microscopic memory, or fractal space-time constraints in the thermodynamic system. It has a wide range of applications in cosmology [33], phase shift analyses for the pion-nucleus scattering [34], dynamical linear response theory, and variational methods [35]. It has achieved a great success in solving many physical problems, such as the solar neutrino problems [36], many-body problems, the problems in astrophysical self-gravitating systems [37], and the transverse momentum spectra [38–40].The article is organized as follows. In Section2, a brief description is given about the employed hydrodynamics, presenting its analytical solutions. The solutions are then used in Section 3 to formulate the transverse momentum distributions of the particles produced in p+p collisions in the light of Cooper-Frye prescription. The last Section 4 is about conclusions.


## 2. A Brief Introduction to the Hydrodynamic Model

The main content of the relativistic hydrodynamic model [15, 41] used in this paper is as follows.The expansion of fluid obeys the continuity equation(1)∂μTμν=0,μ,ν=0,1,where(2)Tμν=ε+puμuν-pgμνis the energy-momentum tensor of fluid and gμν=diag⁡1,-1 is the metric tensor. The four-velocity of fluid uμ=u0,u1=cosh⁡yF,sinh⁡yF, where yF is the rapidity of fluid. ε and p in Equation (2) are the energy density and pressure of fluid, respectively, which are related by the sound speed cs of fluidvia the equation of state(3)dpdε=sdTTds=cs2,where T and s are the temperature and entropy density of fluid, respectively.The projection of Equation (1) to the direction of uμ leads to the continuity equation for entropy conservation(4)∂νsuν=0.The projection of Equation (1) to the direction perpendicular to uμ gives equation (5)∂Tsinh⁡yF∂t+∂Tcosh⁡yF∂z=0,which means the existence of a scalar function ϕ satisfying(6)∂ϕ∂t=Tcosh⁡yF,∂ϕ∂z=-Tsinh⁡yF.By using ϕ and Legendre transformation, Khalatnikov potential χ can be introducedvia relation(7)χ=ϕ-tTcosh⁡yF+zTsinh⁡yF,which changes the coordinate base of t,z to that of (ω,yF)(8)t=eωT0∂χ∂ωcosh⁡yF+∂χ∂yFsinh⁡yF,(9)z=eωT0∂χ∂ωsinh⁡yF+∂χ∂yFcosh⁡yF,where T0 is the initial temperature of sQGP, and ω=-ln⁡T∕T0. In terms of χ, Equation (4) can be rewritten as the so-called equation of telegraphy(10)∂2χ∂ω2-2β∂χ∂ω-1cs2∂2χ∂yF2=0,β=1-cs22cs2.With the expansion of created matter, its temperature becomes lower and lower. When the temperature drops from the initial temperatureT0 to the critical temperature Tc, phase transition occurs. This will modify the value of sound speed of fluid. In sQGP, cs=c0=1∕3, which is the sound speed of a massless perfect fluid, being the maximum of cs. In the hadronic state, 0<cs=ch≤c0. At the point of phase transition, cs is discontinuous.The solutions of Equation (10) for sQGP and hadronic state are, respectively [15],(11)χ0ω,yF=Q0c02eβ0ωI0β0ω2-c02yF2,(12)χhω,yF=Q0c02SωI0λω,yF,where I0 is the 0th order modified Bessel function, and(13)β0=1-c022c02=1,Sω=eβhω-ωc+β0ωc,λω,yF=βhchyh2ω-yF2,where βh=1-ch2∕2ch2, ωc=-ln⁡Tc∕T0, and yhω=ω-ωc∕ch+ωc∕c0. The Q0 in Equations (11) and (12) is a free parameter determined by fitting the theoretical results with experimental data.


## 3. The Transverse Momentum Distributions of the Particles Produced inp+p Collisions

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

The nonextensive statistics is based on the following two postulations [32, 36].(a) The entropy of a statistical system possesses the form of(14)sq=1q-11-∑i=1Ωpiq,where pi is the probability of a given microstate among Ω ones and q is a fixed real parameter. The defined entropy has the usual properties of positivity, equiprobability, and irreversibility, and, in the limit of q→1, it reduces to the conventional Boltzmann-Gibbs entropy(15)s=-∑ipiln⁡pi.(b) The mean value of an observableO is defined as(16)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 quantum in the state with temperatureT can be written in a simple analytical form [42](17)n-q=11+q-1E-μB/T1∕q-1+δ.Here, as usual, E is the energy of quantum, and μB is its baryochemical potential. For baryons δ=1 and for mesons δ=-1. In the limit of q→1, it reduces to the conventional Fermi-Dirac or Bose-Einstein distributions. Hence, the value of q in the nonextensive statistics represents the degree of deviation from the conventional statistics. Known from Equation (17), the average energy of quantum in the state with temperature T reads(18)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 Transverse Momentum Distributions of the Particles Produced inp+p Collisions

With the expansion of hadronic matter, its temperature becomes even lower. As the temperature drops to the so-called kinetic freeze-out temperatureTf, the inelastic collisions among hadronic matter stop. The yields of produced particles remain unchanged, becoming the measured results. According to Cooper-Frye scheme [43], the invariant multiplicity distributions of produced particles take the form [15, 43](19)ⅆ2N2πpTⅆyⅆpT=A2π3∫-yhωfyhωfcoshydzdyF-sinhydtdyFE-qT=TfⅆyF,where A is the area of overlap region of collisions, ωf=-ln⁡Tf∕T0, and the integrand takes values at the moment of T=Tf. The meaning of Equation (19) is evident. The part of integrand in the round brackets is proportional to the rapidity density of fluid [43]. Hence, Equation (19) is the convolution of rapidity of fluid with the energy of the particles in the state with temperature T. From Equations (8) and (9)(20)coshy⁡ⅆzⅆyF-sinh⁡yⅆtⅆyF=1Tcs2∂∂ωχ+∂χ∂ωcosh⁡y-yF-1T∂∂yFχ+∂χ∂ωsinh⁡y-yF.Substituting χ in Equation (20) by the χh of Equation (12) and taking the values at the moment of T=Tf, it becomes(21)cosh⁡yⅆzⅆyF-sinh⁡yⅆtⅆyFT=Tf=Q0c0Tfβhch2SωfBωf,yFsinh⁡y-yF+Cωf,yFcosh⁡y-yF,where(22)Bωf,yF=βhyFλωf,yFβhchyhωfλωf,yFI0λωf,yF+βh+1βh-2βhchyhωfλ2ωf,yFI1λωf,yF,(23)Cωf,yF=βh+1βh+βhchyhωf2λ2ωf,yFI0λωf,yF+1λωf,yFyhωfch+1-2βhchyhωf2λ2ωf,yFI1λωf,yF,where λωf,yF=βhchyh2(ωf)-yF2, I1 is the 1st order modified Bessel function.By using Equations (19) and (21)-(23), we can obtain the transverse momentum distributions of produced particles as shown in Figures 1, 2, 3, and 4.Figure 1
The transverse momentum distributions of strange particlesy<0.5 produced in p+p collisions at s= 200 GeV. The solid dots, circles, and solid triangles represent the experimental data of the STAR Collaboration [44]. The solid curves are the results calculated from Equation (19).Figure 2
The transverse momentum distributions ofπ+, π-, K+, K-, p, and p¯ produced in p+p collisions at s= 200 GeV at midrapidity. The solid dots, solid triangles, solid squares, circles, triangles, and squares represent the experimental data of the PHENIX Collaboration [45]. The solid curves are the results calculated from Equation (19).Figure 3
The transverse momentum distributions ofp and p¯ produced in p+p collisions at s= 200 GeV in the whole measured pT range. The solid squares and squares represent the experimental data of the PHENIX Collaboration [45]. The solid curves are the results calculated from Equation (19).Figure 4
The transverse momentum distributions of the identified charged particles (y< 1) produced in p+p collisions at s= 0.9, 2.76, and 7 TeV (from top to bottom). The solid dots, solid triangles, solid squares, circles, triangles, and squares represent the experimental data of the CMS Collaboration [46]. The solid curves are the results calculated from Equation (19).Figure1 shows the transverse momentum spectra of KS0, K+, K-, Λ, Λ-, Ξ-, Ξ-+, and Ω-+Ω-+ produced in p+p collisions at s=200 GeV. The solid dots, circles, and solid triangles represent the experimental data of the STAR Collaboration [44]. The solid curves are the results calculated from Equation (19). The values of free parameters q, Q0, and χ2/NDF are listed in Table 1. It can be seen that the present model can give a good description of the transverse momentum distributions of strange particles. Since strangeness enhancement is originally proposed as a signature of sQGP produced in nuclear collisions, this proves the reasonableness of hypothesis given at the beginning of this paper that sQGP might appear in p+p collisions.Table 1
The values ofq, Q0, and χ2/NDF obtained from the analyses of STAR data [44] in p+p collisions at s= 200 GeV.
Parameters K S 0 K +/K- Λ/Λ- Ξ -/Ξ-+ Ω - + Ω - +  q 1.083±0.002 1.083±0.005 1.062±0.001 1.075±0.003 1.068±0.007 1.083±0.006 1.062±0.001 1.075±0.003 Q 0 0.084±0.005 0.087±0.012 0.379±0.026 0.026±0.006 0.016±0.010 0.086±0.012 0.337±0.023 0.026±0.005 χ 2/NDF 0.68 0.32/0.39 0.47/0.90 0.47/0.64 0.02Figure2 presents the transverse momentum spectra of π+, π-, K+, K-, p, and p¯ produced in p+p collisions at s= 200 GeV. The solid dots, solid triangles, solid squares, circles, triangles, and squares represent the experimental data of the PHENIX Collaboration [45]. The solid curves are the results calculated from Equation (19). The values of free parameters q, Q0, and χ2/NDF are summarized in Table 2. The theoretical model can give a good description of the experimental data for π+, π-, K+, K- in the whole measured transverse momentum range, and for p and p¯ in the range of pT≤ 3.0 GeV/c. In the range of pT> 3.0 GeV/c, the deviation appears as shown in Figure 3, which shows the transverse momentum distributions of p and p¯ in the whole measured pT range.Table 2
The values ofq, Q0, and χ2/NDF obtained from the analyses of PHENIX data [45] in p+p collisions at s= 200 GeV.
Parameters π +/π- K +/K- p/p-  q 1.075±0.003 1.080±0.003 1.060±0.002 1.075±0.003 1.080±0.003 1.060±0.001 Q 0 10.439±0.015 3.699±0.005 13.099±1.551 10.342±0.015 3.602±0.006 10.479±1.221 χ 2/NDF 5.25/3.28 1.57/1.01 0.47/0.17Figure4 shows the transverse momentum spectra of π+, π-, K+, K-, p, and p¯ produced in p+p collisions at s= 0.9, 2.76, and 7 TeV. The solid dots, solid triangles, solid squares, circles, triangles, and squares represent the experimental data of the CMS Collaboration [46]. The solid curves are the results calculated from Equation (19). The values of free parameters q, Q0, and χ2/NDF are summarized in Table 3.Table 3
The values ofq, Q0, and χ2/NDF obtained from the analyses of CMS data [46] in p+p collisions at LHC energies.
s Parameters π +/π- K +/K- p/p-  0.9 TeV q 1.064±0.002 1.090±0.002 1.071±0.001 1.064±0.003 1.090±0.002 1.071±0.001 Q 0 0.161±0.004 0.045±0.001 0.152±0.005 0.159±0.004 0.044±0.002 0.145±0.003 χ 2/NDF 8.50/10.85 0.37/0.47 1.10/1.84  2.76 TeV q 1.078±0.002 1.100±0.002 1.088±0.001 1.078±0.002 1.100±0.002 1.088±0.001 Q 0 0.051±0.001 0.015±0.0004 0.037±0.001 0.050±0.001 0.014±0.0004 0.036±0.001 χ 2/NDF 8.90/8.25 0.30/0.62 1.75/1.35  7 TeV q 1.084±0.002 1.120±0.003 1.105±0.001 1.084±0.003 1.120±0.002 1.105±0.001 Q 0 0.004±0.00004 0.001±0.00002 0.002±0.00004 0.004±0.00008 0.001±0.00002 0.002±0.00002 χ 2/NDF 7.28/7.69 0.23/0.35 1.13/1.43In calculations, the sound speed in hadronic state takes the value ofch= 0.35 [47, 48]. The critical temperature takes the value of Tc= 0.16 GeV [49]. For s= 200 GeV, the initial temperature takes the value of T0= 0.35 GeV [50], the kinetic freeze-out temperature takes the values of Tf= 0.12 GeV for strange particles and pions, and, for protons, Tf=0.13 GeV from the investigation of [51], which also shows that the baryochemical potential takes the value of μB=0.01 GeV. For s= 0.9, 2.76, and 7 TeV, referring to [50], the initial temperatures are estimated to be T0= 0.4, 0.6, and 1.5 GeV, respectively. The kinetic freeze-out temperature takes the values of Tf=0.12 GeV for pions and kaons, and, for protons, Tf=0.13 GeV. The baryochemical potential takes the value of μB=0 [51].The parametersQ0 and T0 have the same effects. They all affect the amplitudes of the theoretical curves. They are different from parameter q which affects the slopes of the theoretical curves. From the above analysis we can see that the value of the parameter q increases with the increase of the CMS beam energy. However, the values of q do not seem completely consistent with the CMS and the RHIC beam energies.


## 3.1. The Energy of Quantum of Produced Matter

The nonextensive statistics is based on the following two postulations [32, 36].(a) The entropy of a statistical system possesses the form of(14)sq=1q-11-∑i=1Ωpiq,where pi is the probability of a given microstate among Ω ones and q is a fixed real parameter. The defined entropy has the usual properties of positivity, equiprobability, and irreversibility, and, in the limit of q→1, it reduces to the conventional Boltzmann-Gibbs entropy(15)s=-∑ipiln⁡pi.(b) The mean value of an observableO is defined as(16)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 quantum in the state with temperatureT can be written in a simple analytical form [42](17)n-q=11+q-1E-μB/T1∕q-1+δ.Here, as usual, E is the energy of quantum, and μB is its baryochemical potential. For baryons δ=1 and for mesons δ=-1. In the limit of q→1, it reduces to the conventional Fermi-Dirac or Bose-Einstein distributions. Hence, the value of q in the nonextensive statistics represents the degree of deviation from the conventional statistics. Known from Equation (17), the average energy of quantum in the state with temperature T reads(18)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 Transverse Momentum Distributions of the Particles Produced inp+p Collisions

With the expansion of hadronic matter, its temperature becomes even lower. As the temperature drops to the so-called kinetic freeze-out temperatureTf, the inelastic collisions among hadronic matter stop. The yields of produced particles remain unchanged, becoming the measured results. According to Cooper-Frye scheme [43], the invariant multiplicity distributions of produced particles take the form [15, 43](19)ⅆ2N2πpTⅆyⅆpT=A2π3∫-yhωfyhωfcoshydzdyF-sinhydtdyFE-qT=TfⅆyF,where A is the area of overlap region of collisions, ωf=-ln⁡Tf∕T0, and the integrand takes values at the moment of T=Tf. The meaning of Equation (19) is evident. The part of integrand in the round brackets is proportional to the rapidity density of fluid [43]. Hence, Equation (19) is the convolution of rapidity of fluid with the energy of the particles in the state with temperature T. From Equations (8) and (9)(20)coshy⁡ⅆzⅆyF-sinh⁡yⅆtⅆyF=1Tcs2∂∂ωχ+∂χ∂ωcosh⁡y-yF-1T∂∂yFχ+∂χ∂ωsinh⁡y-yF.Substituting χ in Equation (20) by the χh of Equation (12) and taking the values at the moment of T=Tf, it becomes(21)cosh⁡yⅆzⅆyF-sinh⁡yⅆtⅆyFT=Tf=Q0c0Tfβhch2SωfBωf,yFsinh⁡y-yF+Cωf,yFcosh⁡y-yF,where(22)Bωf,yF=βhyFλωf,yFβhchyhωfλωf,yFI0λωf,yF+βh+1βh-2βhchyhωfλ2ωf,yFI1λωf,yF,(23)Cωf,yF=βh+1βh+βhchyhωf2λ2ωf,yFI0λωf,yF+1λωf,yFyhωfch+1-2βhchyhωf2λ2ωf,yFI1λωf,yF,where λωf,yF=βhchyh2(ωf)-yF2, I1 is the 1st order modified Bessel function.By using Equations (19) and (21)-(23), we can obtain the transverse momentum distributions of produced particles as shown in Figures 1, 2, 3, and 4.Figure 1
The transverse momentum distributions of strange particlesy<0.5 produced in p+p collisions at s= 200 GeV. The solid dots, circles, and solid triangles represent the experimental data of the STAR Collaboration [44]. The solid curves are the results calculated from Equation (19).Figure 2
The transverse momentum distributions ofπ+, π-, K+, K-, p, and p¯ produced in p+p collisions at s= 200 GeV at midrapidity. The solid dots, solid triangles, solid squares, circles, triangles, and squares represent the experimental data of the PHENIX Collaboration [45]. The solid curves are the results calculated from Equation (19).Figure 3
The transverse momentum distributions ofp and p¯ produced in p+p collisions at s= 200 GeV in the whole measured pT range. The solid squares and squares represent the experimental data of the PHENIX Collaboration [45]. The solid curves are the results calculated from Equation (19).Figure 4
The transverse momentum distributions of the identified charged particles (y< 1) produced in p+p collisions at s= 0.9, 2.76, and 7 TeV (from top to bottom). The solid dots, solid triangles, solid squares, circles, triangles, and squares represent the experimental data of the CMS Collaboration [46]. The solid curves are the results calculated from Equation (19).Figure1 shows the transverse momentum spectra of KS0, K+, K-, Λ, Λ-, Ξ-, Ξ-+, and Ω-+Ω-+ produced in p+p collisions at s=200 GeV. The solid dots, circles, and solid triangles represent the experimental data of the STAR Collaboration [44]. The solid curves are the results calculated from Equation (19). The values of free parameters q, Q0, and χ2/NDF are listed in Table 1. It can be seen that the present model can give a good description of the transverse momentum distributions of strange particles. Since strangeness enhancement is originally proposed as a signature of sQGP produced in nuclear collisions, this proves the reasonableness of hypothesis given at the beginning of this paper that sQGP might appear in p+p collisions.Table 1
The values ofq, Q0, and χ2/NDF obtained from the analyses of STAR data [44] in p+p collisions at s= 200 GeV.
Parameters K S 0 K +/K- Λ/Λ- Ξ -/Ξ-+ Ω - + Ω - +  q 1.083±0.002 1.083±0.005 1.062±0.001 1.075±0.003 1.068±0.007 1.083±0.006 1.062±0.001 1.075±0.003 Q 0 0.084±0.005 0.087±0.012 0.379±0.026 0.026±0.006 0.016±0.010 0.086±0.012 0.337±0.023 0.026±0.005 χ 2/NDF 0.68 0.32/0.39 0.47/0.90 0.47/0.64 0.02Figure2 presents the transverse momentum spectra of π+, π-, K+, K-, p, and p¯ produced in p+p collisions at s= 200 GeV. The solid dots, solid triangles, solid squares, circles, triangles, and squares represent the experimental data of the PHENIX Collaboration [45]. The solid curves are the results calculated from Equation (19). The values of free parameters q, Q0, and χ2/NDF are summarized in Table 2. The theoretical model can give a good description of the experimental data for π+, π-, K+, K- in the whole measured transverse momentum range, and for p and p¯ in the range of pT≤ 3.0 GeV/c. In the range of pT> 3.0 GeV/c, the deviation appears as shown in Figure 3, which shows the transverse momentum distributions of p and p¯ in the whole measured pT range.Table 2
The values ofq, Q0, and χ2/NDF obtained from the analyses of PHENIX data [45] in p+p collisions at s= 200 GeV.
Parameters π +/π- K +/K- p/p-  q 1.075±0.003 1.080±0.003 1.060±0.002 1.075±0.003 1.080±0.003 1.060±0.001 Q 0 10.439±0.015 3.699±0.005 13.099±1.551 10.342±0.015 3.602±0.006 10.479±1.221 χ 2/NDF 5.25/3.28 1.57/1.01 0.47/0.17Figure4 shows the transverse momentum spectra of π+, π-, K+, K-, p, and p¯ produced in p+p collisions at s= 0.9, 2.76, and 7 TeV. The solid dots, solid triangles, solid squares, circles, triangles, and squares represent the experimental data of the CMS Collaboration [46]. The solid curves are the results calculated from Equation (19). The values of free parameters q, Q0, and χ2/NDF are summarized in Table 3.Table 3
The values ofq, Q0, and χ2/NDF obtained from the analyses of CMS data [46] in p+p collisions at LHC energies.
s Parameters π +/π- K +/K- p/p-  0.9 TeV q 1.064±0.002 1.090±0.002 1.071±0.001 1.064±0.003 1.090±0.002 1.071±0.001 Q 0 0.161±0.004 0.045±0.001 0.152±0.005 0.159±0.004 0.044±0.002 0.145±0.003 χ 2/NDF 8.50/10.85 0.37/0.47 1.10/1.84  2.76 TeV q 1.078±0.002 1.100±0.002 1.088±0.001 1.078±0.002 1.100±0.002 1.088±0.001 Q 0 0.051±0.001 0.015±0.0004 0.037±0.001 0.050±0.001 0.014±0.0004 0.036±0.001 χ 2/NDF 8.90/8.25 0.30/0.62 1.75/1.35  7 TeV q 1.084±0.002 1.120±0.003 1.105±0.001 1.084±0.003 1.120±0.002 1.105±0.001 Q 0 0.004±0.00004 0.001±0.00002 0.002±0.00004 0.004±0.00008 0.001±0.00002 0.002±0.00002 χ 2/NDF 7.28/7.69 0.23/0.35 1.13/1.43In calculations, the sound speed in hadronic state takes the value ofch= 0.35 [47, 48]. The critical temperature takes the value of Tc= 0.16 GeV [49]. For s= 200 GeV, the initial temperature takes the value of T0= 0.35 GeV [50], the kinetic freeze-out temperature takes the values of Tf= 0.12 GeV for strange particles and pions, and, for protons, Tf=0.13 GeV from the investigation of [51], which also shows that the baryochemical potential takes the value of μB=0.01 GeV. For s= 0.9, 2.76, and 7 TeV, referring to [50], the initial temperatures are estimated to be T0= 0.4, 0.6, and 1.5 GeV, respectively. The kinetic freeze-out temperature takes the values of Tf=0.12 GeV for pions and kaons, and, for protons, Tf=0.13 GeV. The baryochemical potential takes the value of μB=0 [51].The parametersQ0 and T0 have the same effects. They all affect the amplitudes of the theoretical curves. They are different from parameter q which affects the slopes of the theoretical curves. From the above analysis we can see that the value of the parameter q increases with the increase of the CMS beam energy. However, the values of q do not seem completely consistent with the CMS and the RHIC beam energies.


## 4. Conclusions

By assuming the existence of longitudinal collective motion and long-range interactions in sQGP produced inp+p collisions, the relativistic hydrodynamics including phase transition together with the nonextensive statistics is used to discuss the transverse momentum distributions of the particles produced in p+p collisions at s= 0.2, 0.9, 2.76, and 7 TeV.The theoretical model used in this paper contains rich information about the transport coefficients of fluid, such as the sound speedc0 in sQGP, the sound speed ch in hadronic state, the initial temperature T0, the critical temperature Tc, the kinetic freeze-out temperature Tf, and the baryochemical potential μB. Except for T0, the other five parameters take the values either from the widely accepted theoretical results or from experimental measurements. As for T0, there are no acknowledged values so far. In this paper, T0 takes the values from other studies. The investigations of the present paper show the conclusions as follows.(a) The theoretical model can give a good description of the currently available experimental data collected inp+p collisions at RHIC and LHC energies with the only exception of p and p¯ measured in the range of pT> 3.0 GeV/c at s= 200 GeV, which might be caused by the hard scattering process [52]. To improve the fitting conditions, the results of perturbative QCD should be taken into account.(b) The fitted values ofq are close to 1. This means that the deviation between nonextensive statistics and conventional statistics is small, while it is this small difference that plays an essential role in fitting the experimental data.



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