The Novel Scaling of Tsallis Parameters from the Transverse Momentum Spectra of Charged Particles in Heavy-Ion Collisions

The transverse momentum $(p_T)$ spectra of charged particles measured in Au + Au collisions from the beam energy scan (BES) program, Cu + Cu collisions at $\sqrt{s_{NN}}=62.4$, 200 GeV at the RHIC and Pb + Pb, Xe + Xe collisions at the LHC are investigated in the framework of Tsallis thermodynamics. The theory can describe the experimental data well for all the collision systems, energies and centralities investigated. The collision energy and centrality dependence of the Tsallis distribution parameters, i.e., the temperature $T$ and the nonextensive parameter $q$, for the \mbox{A + A} collisions are also studied and discussed. A novel scaling between the temperature divided by the natural logarithm of collision energy $(T/\ln\sqrt{s})$ and the nonextensive parameter $q$ is presented.


I. INTRODUCTION
In experiments, the transverse momentum (p T ) spectra of particles produced in the high energy heavy-ion collisions can be measured and provide information about the collision system. They are used to constrain and improve the transport models designed to study the heavy-ion collisions, as well as to gain deep insights into the collision processes and to extract the kinetic freeze-out information, such as the kinetic freeze-out temperature and volume, of the collision system. Abundant data of the transverse momentum spectra of particles have been measured by different experimental collaborations at the RHIC and LHC in the past two decades, ranging from p+p collisions to Pb + Pb collisions at different energies and centralities [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].
Recently, the Tsallis distribution has attracted lots of attention due to its successful applications to the particle transverse momentum spectra and the pseudorapidity distribution of charged particles produced in high energy heavyion collisions [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. In the present study, we apply the Tsallis distribution to systematically analyze the transverse momentum spectra of charged particles produced in Au + Au collisions from the beam energy scan (BES) program, i.e., collision energies ranging from √ s N N = 7.7 to 200 GeV, Cu + Cu collisions at √ s N N = 62.4 and 200 GeV, Pb + Pb collisions at √ s N N = 2.76 and 5.02 TeV and Xe + Xe collisions at √ s N N = 5.44 TeV. We also study the collision energy and centrality dependence of Tsallis distribution parameters, i.e., the temperature T and nonextensive parameter q, and search for some novel phenomena. The remainder of this paper is organized as follows. In Section II, we briefly introduce the Tsallis distribution for the transverse momentum spectra of charged particles. In Section III, we show the results along with the experimental data generated in different heavy-ion collision systems at the RHIC and LHC. Furthermore, a novel scaling between the ratio T / ln √ s and q is found for all the collision systems investigated. In Section IV, a brief conclusion is drawn.

II. FORMULA OF THE TRANSVERSE MOMENTUM DISTRIBUTION
As a generalization of the Boltzmann-Gibbs distribution in classical thermodynamics, the Tsallis distribution was proposed several decades ago [54]. The transverse momentum distribution of particles derived from the Tsallis distribution can be written as [40][41][42][43] where g is particle state degeneracy, m T = m 2 0 + p 2 T is the transverse mass and m 0 is the particle rest mass, y is the rapidity and µ is the chemical potential. V is the volume, T is the temperature and q is the entropic factor which arXiv:2206.12202v1 [hep-ph] 24 Jun 2022 measures the nonadditivity of the entropy [54,55]. When q = 1, Equation (1) recovers the Boltzmann distribution. For the collision energies explored in this work, the multiplicities of charged particles and their antiparticles are approximately equal, therefore we take µ = 0. The experimental data are taken in the mid-rapidity, y ≈ 0; thus, Equation (1) can be rewritten as In order to be consistent with the experimental data, we convert the rapidity to pseudorapidity in Equation (2). The relation between rapidity and pseudorapidity in the mid-rapidity is As it is well known, most of the charged particles produced in high energy heavy-ion collisions are π + (π − ), K + (K − ) and p(p). Therefore, we can sum over the Tsallis distributions of these particles at the mid-rapidity to fit the transverse momentum distribution of charged particles produced in heavy-ion collisions. Then, we obtain [40] where i = π, K, p. The degeneracy factor g i are g π = g K = 2 and g p = 4 to take into account the spin and the contribution from antiparticles. We noted multiplicity differences between protons and anti-protons at low collision energies but these are only a small portion of the total charged particles. Equation (4) will be used to extract the parameters T and q by fitting the experimental transverse momentum distribution of charged particles.

III. RESULTS
In Figures 1 and 2  for the fits are listed in the Appendix A. The Tsallis distribution can well reproduce the experimental data of all the collision systems and centralities investigated. Noting that we have applied the cut p T < 8 GeV/c to the transverse momentum spectra of charged particles in Pb + Pb and Xe + Xe collisions at LHC [40]. To show the agreement between the data and the Tsallis distribution in linear scale, the ratios of data/fit are shown in the bottom panels of Figures 1 and 2. The discrepancies are within 30% for the low collision energies and it reduces to 15% for the high collision energies. The centrality dependence of the fitting quality is not obvious at the RHIC. However, the Tsallis distribution can fit the transverse momentum spectrum better at the peripheral collisions than at the central collisions for the collision systems at LHC. This can be attributed to the medium effects for central collisions, because it is empirically known that the Tsallis distribution can fit well the transverse momentum spectra produced in p + p collisions. The peripheral collisions are more like the p + p collisions than the central collisions [39,44]. We analyze the collision energy and centrality dependence of the Tsallis distribution parameters, i.e., the temperature T and the nonextensive parameter q. In Figure 3, the results of T and q versus collision energy from the RHIC to LHC for the most central collisions are shown. We use √ s to denote √ s N N in units of GeV, i.e., ; thus, it is a dimensionless variable that is suitable used in an expression like ln √ s. For the temperature T , it is observed that a linear decrease from √ s N N = 7.7 GeV to a certain collision energy and then it is approximately constant for the higher collision energies, which is the asymptotic value connected to the Hagedorn temperature, i.e., the pion mass. The lines are drawn to guide the eyes. Unlike the temperature T , the parameter q shows a linear monotonic increasing dependence on the collision energy in the whole energy region investigated. A linear fit gives q = 0.0116 ln √ s + 1.00116 shown in the Figure 3b. This indicates that the higher the collision energy is, the less the collision system reaches thermal equilibrium during the evolution and the temperature fluctuation is larger [55]. The parameters from the other centralities showing similar behaviors have been observed.
In Figure 4, the results of the centrality dependence of T and q for A + A collisions at different collision energies are presented. A nice parabolic fit can be performed for temperature T and centrality, see the solid lines in Figure  4a,b. The corresponding parameters are listed in Table I. Unlike the temperature T , the dependence between q and the centrality is linear as shown by the linear fits in Figure 4c,d. The fit parameters are listed in Table I as well. The observation is that the temperature T decreases from central to peripheral collisions for all collision systems and energies as expected. This trend is universal and is consistent with the average momentum dependence of centrality [56]. The system can survive for a longer time in central collisions than in peripheral collisions since more nucleons are affected and more particles are produced in the collisions. Therefore more collisions can occur resulting in the higher temperatures for central collisions. The nonextensive parameter q increases from central to peripheral collisions, having the opposite trend respect to the temperature T versus centrality. Similarly to the discussion above, it indicates that the peripheral collisions are further away from thermal equilibrium and have larger temperature fluctuations than the central collisions for the same collision system and energy. In Figure 5, the results of the novel scaling discovered between the temperature divided by the natural logarithm of collision energy (T / ln √ s) and nonextensive parameter q for all the A + A collision systems and centralities investigated is shown. It is clear that all the data points are scaled into one curve. We are able to fit it with the function indicated in the legend of Figure 5. This observed strong scaling indicates that the parameters of Tsallis distribution obtained from the charged particle transverse momentum spectra are not independent of each other but are anticorrelated. It also suggests that further fundamental characteristics of the nonextensive statistics are yet to be studied. The emergence of this scaling maybe attributed to hydrodynamical scaling [57] but further investigations are needed. A u + A u 7 . 7 G e V A u + A u 1 1 . 5 G e V A u + A u 1 4 . 5 G e V A u + A u 1 9 . 6 G e V A u + A u 2 7 G e V A u + A u 3 9 G e V A u + A u 6 2 . 4 G e V A u + A u 1 3 0 G e V A u + A u 2 0 0 G e V C u + C u 6 2 . 4 G e V C u + C u 2 0 0 G e V P b + P b 2 7 6 0 G e V P b + P b 5 0 2 0 G e V X e + X e 5 4 4 0 G e V

IV. CONCLUSIONS
In this paper, the transverse momentum spectra of charged particles produced at different collision systems, energies and centralities at the RHIC and LHC have been studied by using Tsallis distribution. The results show that Tsallis distribution can fit well all the charged particle spectra in a wide range of p T investigated. In the centrality and collision energy dependence analysis, it is found that the temperature T linearly decreases with ln √ s to a certain collision energy and becomes almost a constant for a selected centrality while the nonextensive parameter q linearly increases with ln √ s. For a fixed collision energy, the temperature decreases from central to peripheral collisions and can be fitted with a parabolic function and the parameter q linearly increases from central to peripheral collisions indicating that the peripheral collisions are further away from the thermal equilibrium and have larger temperature fluctuations referring to central collisions. Furthermore, we have found a novel scaling between the temperature divided by the natural logarithm of the collision energy T / ln √ s and the nonextensive parameter q. More works related to nonextensive statistics may be stimulated by this finding.