A description of the transverse momentum distributions of charged particles produced in heavy ion collisions at RHIC and LHC energies

By assuming the existing of memory effects and long-range interactions in the hot and dense matter produced in high energy heavy ion collisions, the nonextensive statistics together with the relativistic hydrodynamics including phase transition is 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 to the experimental data in Au+Au collisions at sqrt(s_NN )= 200 GeV and in Pb+Pb collisions at sqrt(s_NN) )= 2.76 TeV for pi^(+ -) , K^(+ -) in the whole measured transverse momentum region, and for p(p-bar) in the region of p_T<= 2.0 GeV/c. This is different from our previous work, where, by using the conventional statistics plus hydrodynamics, the describable region is only limited in p_T<= 1.1 GeV/c.


Introduction
The primary goals of 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 coefficients of azimuth-angle distributions [1,2], transverse momentum spectra [3][4][5][6][7][8] and pseudorapidity

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 where = ( 0 , 1 ) = ( , ), and where, is the temperature, and is the entropy density of fluid.
Project Eq. (1) to the direction of giving which is the continuity equation for entropy conservation. Project Eq. (1) to the direction perpendicular to leading to equation which means the existence of scalar function satisfying From and Legendre transformation, Khalatnikov potential is introduced by which makes the coordinates of ( , ) transform to where 0 is the initial temperature of fluid, and = ln( 0 ∕ ). In terms of , Eq. (4) can be rewritten as the so-called telegraphy equation With the expansions of created matter, its temperature becomes lower and lower. When the temperature reduces from initial temperature 0 to critical temperature , 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 in changing. Furthermore, the mean free paths of these primary hadrons are very short. The evolution of them still satisfies Eq. (10) with only difference being the values of . In sQGP, = 0 = 1 ∕ √3 . In the hadronic state, 0 < = ℎ ≤ 0 . At the point of phase transition, is discontinuous.
The solutions of Eq. (10) for the sQGP and hadronic state are respectively [18], where 0 is a constant determined by fitting the theoretical results with experimental data, 0 is the 0th order modified Bessel function, and

collisions (1) The energy of quantum of produced matter
In the nonextensive statistics, there are two basic postulations [36,39] (a) The entropy of a statistical system possesses the form of where is the probability of a given microstate among different ones and is a real parameter.
(b) The mean value of an observable is given by where is the value of an observable in the microstate .
From above two postulations, the average occupational number of quantum in the state with temperature can be written in a simple analytical form [41] where, is the energy of quantum, and is its baryochemical potential. = 1 for fermions, and = −1 for Bosons. In the limit of → 1, it reduces to the conventional Fermi-Dirac or Bose-Einstein distributions. Hence, the value of reflects the discrepancies of nonextensive statistics from conventional one. Known from Eq. (15), the average energy of quantum in the state with temperature reads where, is the rapidity of quantum, = √ 2 + 2 is its transverse mass with rest mass and transverse momentum .

(2) The rapidity distributions of charged particles in the state of fluid
In terms of Khalatnikov potential , the rapidity distributions of charged particles in the state of fluid can be written as [42] ⅆ where ( ) = 2 2 arccos 2 − √ 2 − ( 2 ) 2 is the area of overlap region of collisions, is impact parameter, and is the radius of colliding nucleus. From Eqs. (8) and (9), the expression in the round brackets in Eq. (17) becomes

(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 temperature , the inelastic collisions among hadronic matter stop. The yields of identified hadrons keep unchanged becoming the measured results.
According to Cooper-Frye scheme [42], the invariant multiplicity distributions of charged particles take the form as [18,42,43] where where 1 is the 1st order modified Bessel function.    In analyses, the sound speed in hadronic state takes the value of ℎ = 0.35 [43,[44][45][46]. The critical temperature takes the value of = 0.18 GeV [47].
These ratios are in good agreement with the relative abundances of particles and antiparticles given in Ref. [3]. This consistency may be due to the fact that the integrand of Eq. (19) is the same for particles and antiparticles in case that takes a common constant for these two kinds of particles. Therefore, 0 might be proportional to the abundance of corresponding particles. The 0 in Pb+Pb collisions is also independent of centrality cuts for ± and ± . While for ( ̅ ), 0 increases with centrality cuts from semicentral to peripheral collisions. The 0 listed in table 2 gives the ratios 0 ( − ) 0 ( + ) = 0 ( − ) 0 ( + ) = 0 ( ̅ ) 0 ( ) = 1.
This is consistent with the above stated fact that, in Pb+Pb collisions at √ = 2.76 TeV, the yield of charged particles is equal to that of antiparticles.

Conclusions
By introducing nonextensive statistics, we employ the relativistic hydrodynamics including phase transition to discuss the transverse momentum distributions of charged particles produced in (2) The fitted is close to 1. It 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 .
(3) 0 is independent of centrality cuts for different charged particles in Au+Au collisions at