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Jia-Qi Hui, Zhi-Jin Jiang, "A Description of Transverse Momentum Distributions in Collisions at RHIC and LHC Energies", Advances in High Energy Physics, vol. 2019, Article ID 8372416, 9 pages, 2019. https://doi.org/10.1155/2019/8372416
A Description of Transverse Momentum Distributions in Collisions at RHIC and LHC Energies
It has long been debated whether the hydrodynamics is suitable for the smaller colliding systems such as collisions. In this paper, by assuming the existence of longitudinal collective motion and long-range interactions in the hot and dense matter created in 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 collisions at RHIC and LHC energies, except for and produced in the range of GeV/c at GeV.
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, collisions are a relatively smaller system with lower multiplicity, larger viscosity, and larger fluctuation . The reasonableness of applying relativistic hydrodynamics in depicting the evolution of sQGP created in collisions has undergone an endless debate.
In this paper, by supposing the existence of collective flow in 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 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 . 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 , phase shift analyses for the pion-nucleus scattering , dynamical linear response theory, and variational methods . It has achieved a great success in solving many physical problems, such as the solar neutrino problems , many-body problems, the problems in astrophysical self-gravitating systems , and the transverse momentum spectra [38–40].
The article is organized as follows. In Section 2, 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 collisions in the light of Cooper-Frye prescription. The last Section 4 is about conclusions.
2. A Brief Introduction to the Hydrodynamic Model
The expansion of fluid obeys the continuity equationwhereis the energy-momentum tensor of fluid and is the metric tensor. The four-velocity of fluid , where is the rapidity of fluid. and in Equation (2) are the energy density and pressure of fluid, respectively, which are related by the sound speed of fluid via the equation of statewhere and are the temperature and entropy density of fluid, respectively.
The projection of Equation (1) to the direction of leads to the continuity equation for entropy conservationThe projection of Equation (1) to the direction perpendicular to gives equation which means the existence of a scalar function satisfyingBy using and Legendre transformation, Khalatnikov potential can be introduced via relationwhich changes the coordinate base of to that of where is the initial temperature of sQGP, and . In terms of , Equation (4) can be rewritten as the so-called equation of telegraphy
With the expansion of created matter, its temperature becomes lower and lower. When the temperature drops from the initial temperature to the critical temperature , phase transition occurs. This will modify the value of sound speed of fluid. In sQGP, , which is the sound speed of a massless perfect fluid, being the maximum of . In the hadronic state, . At the point of phase transition, is discontinuous.
The solutions of Equation (10) for sQGP and hadronic state are, respectively ,where is the 0th order modified Bessel function, andwhere , , and . The 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 in Collisions
3.1. The Energy of Quantum of Produced Matter
(a) The entropy of a statistical system possesses the form ofwhere is the probability of a given microstate among ones and is a fixed real parameter. The defined entropy has the usual properties of positivity, equiprobability, and irreversibility, and, in the limit of , it reduces to the conventional Boltzmann-Gibbs entropy
(b) The mean value of an observable is defined aswhere is the value of an observable in the microstate .
From the above two postulations, the average occupational number of quantum in the state with temperature can be written in a simple analytical form Here, as usual, is the energy of quantum, and is its baryochemical potential. For baryons and for mesons . In the limit of , it reduces to the conventional Fermi-Dirac or Bose-Einstein distributions. Hence, the value of 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 readswhere is the rapidity of quantum and is its transverse mass with rest mass and transverse momentum .
3.2. The Transverse Momentum Distributions of the Particles Produced in Collisions
With the expansion of hadronic matter, its temperature becomes even lower. As the temperature drops to the so-called kinetic freeze-out temperature , the inelastic collisions among hadronic matter stop. The yields of produced particles remain unchanged, becoming the measured results. According to Cooper-Frye scheme , the invariant multiplicity distributions of produced particles take the form [15, 43]where is the area of overlap region of collisions, , and the integrand takes values at the moment of . The meaning of Equation (19) is evident. The part of integrand in the round brackets is proportional to the rapidity density of fluid . Hence, Equation (19) is the convolution of rapidity of fluid with the energy of the particles in the state with temperature . From Equations (8) and (9)Substituting in Equation (20) by the of Equation (12) and taking the values at the moment of , it becomeswherewhere , is the 1st order modified Bessel function.
Figure 1 shows the transverse momentum spectra of , , , , , , , and produced in collisions at GeV. The solid dots, circles, and solid triangles represent the experimental data of the STAR Collaboration . The solid curves are the results calculated from Equation (19). The values of free parameters , , and /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 collisions.
Figure 2 presents the transverse momentum spectra of , , , , , and produced in collisions at 200 GeV. The solid dots, solid triangles, solid squares, circles, triangles, and squares represent the experimental data of the PHENIX Collaboration . The solid curves are the results calculated from Equation (19). The values of free parameters , , and /NDF are summarized in Table 2. The theoretical model can give a good description of the experimental data for , , , in the whole measured transverse momentum range, and for and in the range of 3.0 GeV/c. In the range of 3.0 GeV/c, the deviation appears as shown in Figure 3, which shows the transverse momentum distributions of and in the whole measured range.
Figure 4 shows the transverse momentum spectra of , , , , , and produced in collisions at 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 . The solid curves are the results calculated from Equation (19). The values of free parameters , , and /NDF are summarized in Table 3.
In calculations, the sound speed in hadronic state takes the value of 0.35 [47, 48]. The critical temperature takes the value of 0.16 GeV . For 200 GeV, the initial temperature takes the value of 0.35 GeV , the kinetic freeze-out temperature takes the values of 0.12 GeV for strange particles and pions, and, for protons, 0.13 GeV from the investigation of , which also shows that the baryochemical potential takes the value of 0.01 GeV. For 0.9, 2.76, and 7 TeV, referring to , the initial temperatures are estimated to be 0.4, 0.6, and 1.5 GeV, respectively. The kinetic freeze-out temperature takes the values of 0.12 GeV for pions and kaons, and, for protons, 0.13 GeV. The baryochemical potential takes the value of .
The parameters and have the same effects. They all affect the amplitudes of the theoretical curves. They are different from parameter which affects the slopes of the theoretical curves. From the above analysis we can see that the value of the parameter increases with the increase of the CMS beam energy. However, the values of do not seem completely consistent with the CMS and the RHIC beam energies.
By assuming the existence of longitudinal collective motion and long-range interactions in sQGP produced in 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 collisions at 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 speed in sQGP, the sound speed in hadronic state, the initial temperature , the critical temperature , the kinetic freeze-out temperature , and the baryochemical potential . Except for , the other five parameters take the values either from the widely accepted theoretical results or from experimental measurements. As for , there are no acknowledged values so far. In this paper, 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 in collisions at RHIC and LHC energies with the only exception of and measured in the range of 3.0 GeV/c at 200 GeV, which might be caused by the hard scattering process . To improve the fitting conditions, the results of perturbative QCD should be taken into account.
(b) The fitted values of 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.
The experimental data used to support the findings of this study have been deposited in https://doi.org/10.1103/PhysRevC.75.064901; https://doi.org/10.1103/PhysRevC.83.064903; and https://doi.org/10.1140/epjc/s10052-012-2164-1.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work is supported by the Shanghai Key Lab of Modern Optical System.
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Copyright © 2019 Jia-Qi Hui and Zhi-Jin Jiang. 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.