Study of proton, deuteron and triton at 54.4 GeV

Transverse momentum spectra of proton, deuteron and triton in gold-gold (Au-Au) collisions at 54.4 GeV are analyzed in different centrality bins by the blast wave model with Tsallis statistics. The model results are approximately in agreement with the experimental data measured by STAR Collaboration in special transverse momentum ranges. We extracted the kinetic freeze out temperature, transverse flow velocity and freeze out volume from the transverse momentum spectra of the particles. It is observed that the kinetic freeze out temperature is increasing from central to peripheral collisions. However the transverse flow velocity and freeze out volume decrease from central to peripheral collisions. The present work reveals the mass dependent kinetic freeze out and volume differential freeze out scenario in collisions at STAR Collaboration. In addition, the parameter q characterizes the degree of non-equilibrium of the produced system, and it increase from central to peripheral collisions and increase.


Introduction
The two important stages in the evolution system are chemical and kinetic freeze out. The degree of excitation of the interacting system at the two stages are different from each other. The chemical and kinetic freeze out temperatures are used to describe the different excitation degree of two stages. In general, the ratios of different kinds of particles are no longer change at the stage of chemical freeze out. The chemical freeze out temperature can be obtained from different particle ratios in the framework of thermal model [1][2][3]. The transverse momentum spectra of different particles are no longer changed at the stage of kinetic freeze out and thermal/kinetic freeze out temperature can be obtained from the transverse momentum spectra according to hydrodynamical model [4].
It is important to point out that the transverse momentum spectra even though in a narrow p T range, but it contains both the contribution of thermal motion and transverse flow velocity. The random thermal motion reflect the excitation and the transverse flow velocity reflects the degree of expansion of interacting system. In order to extract the kinetic freeze out temperature (T 0 ), we have excluded the contribution of transverse flow velocity (β T ), that is to disengage the random thermal motion and transverse flow velocity. There are various methods to disengage the two issues. The methods include but are not limited to blast wave fit with boltzmann Gibbs statics [5][6][7], blast wave model with Tsallis statistics [8][9][10], and alternative methods [11][12][13][14][15][16][17].
The dependence of T 0 and β T on centrality is very complex situation. There are two schools of thought. (1) T 0 increase decrease from central to peripheral collisions [18][19][20][21] (2) T 0 increase from central to peripheral collision [22][23]. Both have their own explanations. Larger T 0 in the central collisions explain higher degree of excitation of the system due to more violent collisions, while smaller T 0 in the central collisions indicates longer liver fireball in the central collisions. It is very important to find out which collision system contains larger T 0 . Furthermore, there are several opinions about the freeze out of particles which include single, double or multiple kinetic freeze out. It is also very important to dig our the correct freeze out scenario.
In the present work, we will analyze the p T spectra of proton, deuteron and triton and will extract T 0 and β T . Deuteron and triton are light nuclei. The fundamental mechanism, for light nuclei production in relativistic heavy ion collision is not well understood [24][25][26]. Coalescence of anti-(nucleons) is possible approach [27][28][29][30][31]. Because of small binding energies (d with 2.2 MeV and t with 8.8 MeV), the light nuclei cannot persists when the temperature is much higher than their binding energy.
The typical kinetic freeze out temperature is around 100 MeV for light hadrons, so they might disintegrate and be formed again by final state coalescence after nucleons are decoupled from the hot and dense system. Hence the study of the light nuclei can be useful in the extraction of information of nucleons distribution at the freeze out [27,30,32].
Before going to the formalism, we would point out the concept of volume is important in high energy collisions. The volume occupied by the ejectiles when the mutual interactions become negligible, and the only force they feel is the columbic repulsive force, is called the kinetic freeze-out volume (V ). Various freeze-out volumes occurs at various freeze-out stages, but we are only focusing on the kinetic freeze-out volume V in the present work. The information about the information of the co-existence of phase-transition, and is important in the extraction of multiplicity, micro-canonical heat capacity and it's negative branch or shape of the caloric curves under the thermal constraints can be obtained V .
The remainder of the paper consists of method and formalism in section 2, followed by the results and discussion in section 3. In section 4, we summarized our main observations and conclusions.

The method and formalism
In high energy collisions there are two types particles production process. (1) soft process ad (2) hard process. For soft process, there are various methods which includes but are not limited to blast wave model with boltzmann Gibbs statistics [5][6][7], blast wave model with Tsallis statistics [8][9][10], Hagedorn thermal model [20] and Standard distribution [33,34] etc. We are interested in blast wave model with Tsallis statistics. According to [8], the blast wave fit with Tsallis statistics results in the probability density function be where C denotes the normalization constant that leads the integral in Eq. (1) to be normalized to 1, g is the degeneracy factor which is different for different particles based on g n =2S n +1, m T = p 2 T + m 2 0 is the transverse mass, m 0 is denotes rest mass of the particle, φ shows the azimuthal angle, r is the radial coordinate, R is the maximum r, q represents the measure of degree of deviation of the system from an equilibrium state, ρ = tanh −1 [β(r)] is the boost angle, β(r) = β S (r/R) n0 is a self-similar flow profile, β S represents the flow velocity on the surface,as mean of (1) can be substituted by −q/(q − 1) due to the reason that q is being close to 1. This substitution results in a small and negligible divergence in the Tsallis distribution.
In case of a not too wide p T range, the above Eqn. can be used to describe the p T spectra and we can extract T 0 and β T . But if we use the wide p T spectra, then the contribution of hard scattering process can be considered. According to quantum chromodynamics (QCD) calculus [35][36][37], the contribution of hard process is parameterized to be an inverse power law.
which is the Hagedorn function [38,39], A is the normalization constant while p 0 and n are the free parameters. The superposition of soft and hard scattering process can be used if the p T spectra is distributed in a wide range. If Eqn. (1) describes the contribution of soft process, then the contribution of hard process can be described by Eqn. (2). To describe the spectrum in a wide p T range, one can superpose the two-component superposition like this where k denotes shows the contribution fraction of soft excitation and (1 − k) shows hard scattering process, f S denotes the soft process which contributes in the low p T region and f H is the hard process which contributes in a whole p T region. The two contributions overlap each other in the low p T region. We may also use the usual step function to superpose the two functions. According to Hagedorn model [38] where A 1 and A 2 are the normalization constants that synthesize A 1 f S (p 1 )=A 1 f H (p 1 ) and θ(x) is the usual step function. The symbols represent the experimental data measured by the STAR Collaborations [40] and the curves are our fitting results by using the blast-wave model with Tsallis statistics. Each panel is followed by its corresponding data/fit. The related parameters, χ 2 and degree of freedom (dof) are listed in Table 1. One can see that Eq.

Results and discussion
(1) fits well the data in Au-Au collisions at 54.4 GeV at the RHIC.
To show the trend of the extracted parameters, Fig.  2 shows the dependence of kinetic freeze out temperature on centrality. One can see that T 0 in central collisions is smaller and it is increasing with decrease of centrality which indicates a decrease of lifetime of fireball from central collisions to peripheral collisions. Furthermore, T 0 is observed to be mass dependent, as it larger from triton, followed by deuteron and then proton which means heavy particles freeze out early than lighter particles. Fig. 3 shows the centrality dependence of transverse flow velocity. β T is observed to decrease with the decrease of centrality due to the reason that in central collisions the system undergoes more violent collisions and the system expands very rapidly. In addition, β T is observed smaller for heavy particles. Fig. 4 is same as fig. 3 but shows the centrality dependent freeze out volume. The freeze out volume decrease with decrease of centrality due to decreasing the number of participant nucleons. There are large number of binary collisions due to the re-scattering of partons in central collisions and therefore the system with more participants reaches quickly to equilibrium state. Furthermore, the volume differential scenario is observed and heavy particles is observed to have less freeze out volume and this shows the early freeze out heavier particles. The different freeze out of different particles exhibits different freeze out surfaces different particles. Fig. 5 and fig. 6 are the same. Fig. 5 shows the correlation of T 0 and β T and fig. 6 show the correlation of T 0 and V . one can see that both T 0 and β T and T 0 and V exhibits a two dimensional anti-correlation band. Larger the T 0 , smaller the β and V . Fig. 7 shows the dependence of q on centrality. The parameter q is smaller in central collisions but as we go from central to peripheral collisions it is going to increase. The parameter is varying with mass of the particle, larger q is observed for light particles. The interacting system stays at equilibrium because q is being very close to 1.
In Fig 8 the parameter N 0 decrease with the mass of the particle. N 0 basically shows the multiplicity and it is larger in central collision which decrease towards periphery Further discussion 3.2 The study of p T spectra of the particles may give some fruitful information about effective temperature (T ef f ), initial temperature (T i ), thermal/kinetic freezeout temperature (T 0 ), thermal freezeout volume (V ) of the interacting system, and transverse flow velocity (β T ) of the final sate particles. We use the fitting method to extract these information by using different models and distributions. In the present work, the blast wave model with Tsallis statistics is used.
The structure of transverse momentum (p T ) spectra of charged particles generated in high energy heavy ion collisions is very complex. It is not enough to use only one probability density function to describe the p T spectra, though this function can be of various forms. Particularly, in case when the maximum p T reaches to 100 GeV at LHC collisions [41]. Several p T regions are observed by the model analysis [42], including the first region p T < 4-6 Gev/c, 4-6 Gev/c<p T <17-20 GeV/c and the third region with p T >17-20 GeV/c. The boundaries of different p T regions at the RHIC are slightly lower. It is expected that different p T regions correspond to different interacting mechanisms. Even for the same p T region, there are different explanations due to different model methods and microcosmic pictures.
According to [42], different whole features of frag- , the first p T region has the minimum number of strings and maximum number of hadrons. In some cases there may be the contribution region (p T < 0.2-0.3 GeV/c) of very soft process which is due to resonant production of charged particles e:g pions, and this region is considered as the fourth p T region. Different components in a unified superposition can describe the four p T regions. We have two methods in order to structure the unified superposition. First method is the common method of overlapping of the contribution regions of various components, however the second

Au-Au collisions
Demonstrates the dependence of T 0 on centrality method is the Hagedorn model [38] which exclude this overlapping. If the contribution of hard component in the first method is be neglected in low p T region due to its small value, the first method can be changed into second method. Indeed the contribution to T 0 and β T is less for the hard component. If the spectra in low p T region is analyzed to extract only T 0 and β T , then we can give up the second part of eqs. (3) and (4). That is, f S (p T ) can be used directly from eqs. (1) which also includes the contribution of very-soft component that comes from resonance decays if available in the data. In the present work, the contribution of hard component in low p T region if available is included in the extraction of T 0 and β T which may cause a slight increase in T 0 and/or β T but the relative increase can be neglected due to small values [43]. In the present work we only use eqn. (1), which means that the fraction of hard component is zero in low p T region. But we also show eqs. (3) and (4) to show a method for further analysis if necessary.

Conclusions
The main observations and conclusions are summarized here.
a) The transverse momentum spectra of proton (p), deuteron (d) and triton (t) are analyzed by the blast wave model with Tsallis statistics and the bulk properties in terms of the kinetic freeze out temperature, transverse flow velocity and kinetic freeze out volume are extracted.
b) The kinetic freeze out temperature (T 0 ) is ob- served to increase from central to peripheral collisions. However the transverse flow velocity and freeze out volume is decreasing from central to peripheral collision. c) The entropy index (q) increasing with from centrality while the parameter N 0 is decreasing with centrality.
d) The kinetic freeze out temperature, transverse flow velocity and freeze out volume decrease with the increasing mass of the particle. Therefore mass differential kinetic freeze out scenario and volume differential freeze out scenario is observed. e) both the entropy index (q) and the parameter N 0 decreasing the mass of particle. )

Data availability
The data used to support the findings of this study are included within the article and are cited at relevant places within the text as references.

Compliance with Ethical Standards
The authors declare that they are in compliance with ethical standards regarding the content of this paper.