Centrality dependency of proton, deuteron, and triton’s temperatures in Au+Au collisions at 200 GeV

The transverse momentum (pT) spectra of protons (p), deuterons (d), and tritons (t) in 200 GeV gold–gold (Au + Au) collisions at RHIC are examined across a range of centrality bins using the Levy Tsallis (TS) statistical model. The model's predictions closely match the experimental results from PHENIX (p) and STAR (d and t) Collaborations. Kinetic freeze-out temperatures of hadrons are obtained from particle spectra. The results showed that the kinetic freeze-out temperature decreases as collisions move from center to the periphery. This work found mass-dependent kinetic freeze-out temperatures, heavier particles arrive to the freeze-out phase before lighter ones. Comparison with same data fitted by blast wave function with Tsallis statistics (BWTS) showed that T0 values are increasing from central to peripheral collisions, while in case of TS function (current analysis) it decreases. This behavior puts a question mark on the reliability of using such functions for temperature extraction.


The method and formalism
Particles are created in high-energy collisions through both soft and hard processes.These processes can be modeled using a variety of methods, including the standard distribution 39,40 , the blast wave model using Boltzmann Gibbs statistics [5][6][7] , and the Hagedorn thermal model 21 .The list also contains Levy Tsallis function 11 and blast wave model using Tsallis statistics 25 .Effective temperature (T eff ), initial temperature (T i ), thermal or kinetic freeze-out temperature (T 0 ), thermal freeze-out volume (V) of the interacting system, and transverse flow velocity (β T ) of the final-state particles are just a few of the parameters that can be understood by analyzing the p T spectra of the particles 11 .To extract these details, we use a fitting strategy that makes use of several models and distributions.For this reason, we use Tsallis statistics in the current study.
Use of the Levy Tsallis function 11 and its results comparison to blast wave model using Tsallis statistics is the main topic of this work.Levy Tsallis function is given by; dp T dy , T 0 is the freeze-out temperature of the hadrons, N is the normalization constant, n is the entropy index and m T and m 0 are the transverse mass and the rest mass of the hadron, respectively.It works both at low and high p T regions of the p T spectra of hadrons.
Both Levy Tsallis function and Blast wave model using Tsallis statistics have been extensively used to get the kinetic freeze-out temperatures of hadrons for decades 8,11,25 .It is used in 25 in which Au+Au collisions are studied for the production of protons, deuterons, and tritons.According to 8 , the blast wave fit using Tsallis statistics is used to derive the probability density function.

This gives
where C is normalization constant, which leads the integral in Eq. ( 2), to be normalized to 1, g is the degeneracy factor which is different for different particles based on Here m 0 is rest mass of the particle, Φ is the azimuthal angle, r is the radial coordinate, R is the maximum r, q is the measure of degree of deviation of the system from an equilibrium state, ρ = tanh −1 {β(r)} is the boost angle.Here {β(r)} = βS(r/R) no is a self-similar flow profile in which β S is the flow velocity on the surface, as a mean of β(r), β(r) = (2/R 2 ) r 0 rβ(r)dr = 2βs/(n 0 + 2) = 2βs/3, and n 0 = 1, furthermore, the index −1/(q -1) in Eq. (1) can be replaced by −q/(q -1), because q is close to 1.This replacing results in a small and negligible divergence in the Tsallis distribution.

Ethical approval
The authors affirm that they have adhered to all ethical standards with regard to the topic of this study.
It is clear from Fig. 1a-c that Eq. (1) accurately passes through the transverse momentum spectra of all the three hadrons obtained in Au+Au collisions at √ s NN = 200 GeV.To highlight parameters trends, Fig. 2 shows how the kinetic freeze-out temperature (T 0 ) varies with respect to centrality. Figure 2a contains results from Eq. (1) which is current study, while Fig. 2b has results from Eq. ( 2) studied in reference 25 .It is noteworthy that, in current analysis, T 0 is significantly higher in central collisions and gradually decreases with diminishing centrality.This denotes a change from center to periphery collisions and a longer lifetime for the fireball.However in case of Fig. 2b we see opposite trends.T 0 seems to be significantly lower in central collisions and steadily increases with lessening centrality.Such behavior is also reported in following references: decreasing [19][20][21][22] and increasing [23][24][25] .This trend puts a big question mark on the credibility of use of such functions for temperature extraction.However we can expect such behavior if we consider the physical meaning of both fit functions.We have used Tsallis statistics, so both include possible deviation of the data from the equilibrium Boltzmann distribution.But only BWTS takes into account the radial flow of the particles.Nothing is referring to the radial flow in the TS formula.Since we know that the transverse momentum distribution's slope is affected by the temperature and the radial flow.Radial flow pushes the particles towards higher transverse momentum, making the spectrum less steep.So, if the radial flow is present in experimental data, but the fitted function does not include the radial flow, then the extracted slope parameter will be affected.If the intuition from the Boltzmann distribution can be transferred to the Tsallis distribution, then the parameter T 0 gets artificially larger because the radial flow is not considered.And this is what we indeed see in this research article.Also it can be observed that T 0 has a mass-dependent pattern as well, with tritons having the highest value, followed by deuterons, then protons.This means that heavier particles arrive to the freeze-out phase before their lighter counter parts do.T 0 versus mass-dependency has same behavior in both cases.
Table 1 also reveals that T 0 values obtained via Eq.(1) are greater than those obtained via Eq.( 2).Greatest value for triton emission in central collisions is 642 ± 91 MeV in case of TS fittings.It is very much greater than 160 ± 20 MeV obtained in case of BWTS equation for most peripheral collisions.
Figure 3 shows the same information in a different way.Data from both Tsallis fitting models have been put on single place for each hadron.Figure 3      Decrease in slopes of hadrons is also particle mass dependent.Greater the mass, greater is the reduction slope.
The values of T 0 in the TS approach are, in most cases, higher (sometimes much) than the typical temperatures of chemical freeze-out obtained in the Statistical Hadronization Model approaches and the phase transition temperatures from the Quark-Gluon Plasma to the hadronic matter, calculated with Lattice QCD.This is hard to be expected.On the other hand, the temperatures of the kinetic freeze-out in the BWTS approach are in the same range or lower than the chemical freeze-out and the phase transition temperatures, which is much more intuitive.
In fact, one can investigate this, using Fig. 4, which demonstrates that when particle mass increases, the parameter T 0 rises.The temperature, T 0 , is essentially noticeably greater in central collisions and decreases toward the periphery.
With the Boltzmann statistics, when radial flow is not included in fits to transverse momentum spectra, there is a relation (Eq. 3) between the slope of the spectrum T 0 , average transverse flow velocity β T and the kinetic freeze-out temperature T kin : The formula is rough already with the Boltzmann statistics, so transferring it to the Tsallis one perhaps does not cause big additional error.Therefore, it is worth applying it to the data shown in Fig. 4 and comparing the results to the ones from BWTS 25 in a slightly fairer way.Fitting of Eq. ( 3) gives the values of free parameters listed in Table 3.
(3) Table 3.Values of variable β T obtained from fit Eq. ( 3) and average β T obtained using BWTS 25 .  3and average β T obtained using BWTS 25 given in Table 3, reveals that β T decreases with centrality in both cases.However values of β T obtained using Eq. ( 3) and TS equation are much higher as compared to those obtained via BWTS equation.It is important that their dependency on centrality is now of the same type i.e. both are decreasing.So we can say that both fitting functions, TS and BWTS may be used to extract temperature of hadrons.
Figure 5a shows relationship between N and centrality, whereas Fig. 5b shows the relationship between n and centrality.It is obvious that a drop in centrality causes a drop in N and n values.This can be explained by the system's internal interactions, which become more intense in central collisions.This effect causes a spike in central collision part.

Conclusion
In order to extract the kinetic freeze-out temperature the study of transverse momentum spectra of protons (p), deuterons (d), and tritons (t) has been done via using the Levy Tsallis statistics.Levy Tsallis equation accurately passed through the p T spectra of all three hadrons obtained in Au+Au collisions at √ s NN = 200 GeV.Switching from central to peripheral collisions: the kinetic freeze-out temperature (T 0 ) decreases, denotes a change from center to periphery collisions and a longer lifetime of the fireball.Comparison with same data, but different model fitting reveals that T 0 values are increasing from central to peripheral collisions when we use BWTS function, while in case of TS function it gives counter results.Slopes of hadrons' temperatures in case of TS fittings are negative while positive in case of BWTS function.This trend puts a big question mark on the credibility of using such functions for temperature extraction.After careful analysis we reached to the conclusion that even though there is a discrepancy between different functions concerning temperature, still they can reproduce, for example, p T distributions.And since dependency of β T on centrality reveals decreasing behavior in cases of both func- tions, so we can say that both fitting functions, TS and BWTS may be used to extract temperature of hadrons.
The decrease in T 0 values for hadrons is greater in the TS case than the increase in values of T 0 in the BWTS case.The decrease in slopes of hadrons is particle mass dependent.The greater the mass, greater the slope reduction.Kinetic freeze-out temperature depends on mass of the particles.It increases with increasing particle mass.It also means that heavier particles arrive to the freeze-out phase before their lighter counterparts do.Drop in centrality causes a drop in N and n values.
shows a trend of T 0 = aC + b, with C-centrality, a, b-free parameters.Here a is the slope of the curve, which shows how fast or slow variation in T 0 takes place, as we go from central to peripheral collisions.Values of variables a and b are given in Table2.Slopes of hadrons in the case of TS fittings are negative while positive in the case of BWTS function.TS slopes are much steep as compared to those of BWTS

Figure 3 .
Figure 3.Comparison of centrality dependency of T 0 for protons (a), deuterons (b) and tritons (c) obtained via fittings of Levy Tsallis function (Eq. 1) (present work) and blast wave function with Tsallis statistics (Eq.2) 25 .Red solid line is the linear function (T 0 = aC + b) fit curve.

Figure 4 .
Figure 4. T o Dependency on masses of the hadrons.

Table 2 .
Values Decrease in T 0 values for hadrons is greater in TS case than increase in values of T 0 in BWTS case.
of variables a and b extracted from linear fit equation, T 0 = aC + b.