Kinetic freeze-out temperature and transverse flow velocity in Au-Au collisions at RHIC-BES energies

Based on the data-driven analysis, the mid-rapidity transverse momentum spectra of charged hadrons produced in central and peripheral gold-gold (Au-Au) collisions from the Beam Energy Scan (BES) program at the relativistic Heavy Ion Collider (RHIC) are fitted by the blast-wave model with Boltzmann-Gibbs statistics. The model result are in agreement with the experimental data measured by the STAR Collaboration at the RHIC-BES energies. We observe that the kinetic freeze-out temperature, transverse flow velocity, mean transverse momentum, and initial temperature increase with the collision energy and with the event centrality.


Introduction
One of the most fundamental questions in nuclear matter is to determine the phase structure of the strongly-interacting quantum chromodynamics (QCD) matter [1,2,3]. The yield ratios, transverse momentum (p T ) spectra and other data for various identified particles produced in proton-proton (pp), proton-nucleus (pA) and nucleus-nucleus (AA) collisions at high energies are important observable quantities for determining the phase structure. The experimental facilities, for example the Relativistic Heavy-Ion Collider (RHIC) and the Large Hadron Collider (LHC) provide excellent tools to study the properties of Quark-Gluon Plasma (QGP) [4,5,6].
The phase diagram of the QCD matter is usually expressed in terms of the chemical freeze-out temperature (T ch ) and the baryon chemical potential (µ B ) [7,8]. Besides, other quantities such as the kinetic freeze-out temperature (T kin or T 0 ) and transverse flow velocity (β T ) are useful to understand the phase diagram [9]. To search for the possible critical energy in the phase transition from hadronic matter to QGP in high energy collisions, the STAR Collaboration has been performing the Beam Energy Scan (BES) program [10,11,12,13] at the RHIC. Besides, other experiments at similar or lower energies at other accelerators are scheduled [14,15].
Generally, the processes of high energy collisions result possibly in three main stages [16,17,18]: i) The initial stage: at this stage the collisions are in the beginning. The temperature at this stage is called the initial temperature which is one of the main factors to affect the particle spectra, which is less studied in the community comparatively. After the initial state, the "fireball" leads to a decrease in the temperature and finally to the hadronization.
ii) The chemical freeze-out stage: at this stage the inner collisions among various particles are elastic and the yield ratios of differential types of particles remain invariant. The chemical freeze-out temperature T ch can be obtained from the particle ratios, which is much studied in the community comparatively.
iii) The kinetic freeze-out stage: at this stage the scattering processes stop and the hadrons decouple from the rest of the system and the hadron's energy/momentum spectra freeze in time. The tem-1 perature at this stage is known as the kinetic freezeout temperature T 0 which can be obtained from the p T spectra.
When one studies T 0 from the p T spectra, the effect of β T should be eliminated. If the effect of β T is not eliminated in the temperature, this temperature is called the effective temperature (T ef f or T ). At the stage of kinetic freeze-out, T 0 and β T are two important parameters which describe the thermal motion of the produced particles and the collective expansion of the emission source respectively. The spectra in low-p T region (p T = 2-3 GeV/c) which is mainly contributed by the soft excitation process essentially separate the contribution of the thermal motion and the collective expansion, if one only extracts T 0 and β T . The spectra in high-p T region are contributed by the hard scattering process which is not needed in extracting T 0 and β T .
We are very interested in the extraction of T 0 and β T in collisions at the RHIC-BES energies which are very suitable to study the spectra in low-p T region, where the spectra in high-p T region are not produced due to not too high energies. In this work, the double differential p T spectra of charged particles dependences on collision energy and event centrality in goldgold (Au-Au) collisions are analyzed by the blast-wave model with Boltzmann-Gibbs statistics by means of data-driven analysis. The model results are compared with the data measured by the STAR Collaboration at the RHIC-BES energies [19,20].
The remainder of this work consists of the method and formalism, results and discussion as well as conclusions. We shall describe the remanent parts orderly.

The method and formalism
Various methods can be used for the extraction of T 0 and β T , e.g. the blast-wave model with Boltzmann-Gibbs statistics [21,22,23], the blast-wave model with Tsallis statistics [24,25,26], an alternative method by using the Boltzmann-Gibbs statistics [22,27,28,29,30,31,32,33] and the alternative method by using Tsallis distribution [33,34,35,36,37,38,39]. In this work, we choose the blast-wave model with Boltzmann-Gibbs statistics due to its similarity with the ideal gas model in thermodynamics and few parameters. However, these methods only describe the spectra in low-p T region. For the spectra in high-p T region if available, the Hagedorn function which is know as the inverse power-law [40,41] can be used. We shall discuss these issues in detail as follows.
In general, there are two main processes responsible in the contribution of p T spectra. They are i) the soft excitation process which contributes the soft component in low-p T region and ii) the hard scattering process which contributes the hard component in high-p T region.
For the soft component, according to refs. [21,22,23], the probability density function of the p T spectra in the blast-wave model with Boltzmann-Gibbs statisitcs results in where N is the number of particles, C is the normalization constant, m T = p 2 T + m 2 0 is the transverse mass, m 0 is the rest mass of the considered particle, r and R are the radial position and the maximum radial position respectively, I 0 and K 1 are the modified Bessel functions of the first and second kinds respectively, ρ = tanh −1 [β(r)] is the boost angle, β(r) = β S (r/R) n0 is a self-similar flow profile, β S is the flow velocity on the surface, and n 0 = 2 is used in original form [21]. Particularly, β T = (2/R 2 ) R 0 rβ(r)dr = 2β S /(n 0 + 2)= 0.5β S . The parameter n 0 is used different in different works, e.g. n 0 = 1 or non-integer in refs. [24,42], which corresponds to the centrality from center to periphery. Equation (1) and similar or related functions are not enough to describe the whole p T spectra. In particular, the maximum p T reaches up to 100 GeV/c in collisions at the LHC [43]. Then, one needs other functions such as the Tsallis-Lévy [44,45] or Tsallis-Pareto-type function [44,46] and the Hagedorn function [40,41] or inverse power law [47,48,49] to the spectra in high and very high-p T regions. In this work, the hard component is simply represented by the inverse power law. That is where p 0 and n are free parameters and A is the normalization constant which is related to the free parameters. However, the structure of p T spectra is very complex. In fact, several regions have been observed and analyzed in ref. [50]. These regions include the first one with p T < 4-6 GeV/c, the second one with 4-6 GeV/c < p T < 17-20 GeV/c and the the third one with p T > 17-20 GeV/c. Different regions maybe correspond to different mechanisms. The first p T region in our discussion is regarded as the region of soft excitation process, while the second and third p T regions are regarded as the regions of hard and very hard excitation process respectively. In particular, a special region with p T < 0.2-0.3 GeV/c is considered due to the resonant production in some cases, and it is regarded as the region of very soft excitation process.
Generally, all the p T regions discussed above can be unifiedly superposed by two methods: i) the general superposition in which the contribution regions of different components overlap each other and ii) the Hagedorn model (the usual step function) [40] in which there is no overlapping of different regions of different components.
Considering f 1 (p T ), f 2 (p T ), f V S (p T ) and f V H (p T ) which denote the probability density functions by the soft, hard, very soft and very hard components respectively, where f V S (p T ) and f V H (p T ) are assumed to be in the form of f 1 (p T ) and f 2 (p T ) respectively, the unified superposition according to the first method is where k V S is the contribution fraction of very soft component, while k and k V H denote the contributions of soft and very hard components respectively. The step function can be used to structure the superposition according to Hagedorn model [40], i.e.
where A V S , A 1 , A 2 and A V H are the constants which make the interfacing components link to each other perfectly. Particularly, if the contributions of very soft and very hard components can be neglected, Eqs. (3) and (4) are simplified to be and respectively. Further, if the contribution of hard component at the RHIC-BES energies can be neglected, Eqs. (5) and (6) are simplified to be the same form This work deals with Au-Au collisions at the RHIC-BES energies, for which Eq. (7) i.e. Eq. (1) is suitable. In the following section, we shall use Eq. (1) to fit the experimental data measured by the STAR Collaboration at the RHIC-BES energies [19,20]. Figure 1 presents the event centrality dependent double differential p T spectra, (1/2πp T )d 2 N/dp T dy, of π + , K + and p produced in the mid-rapidity interval |y| < 0.1 in Au-Au collisions at the center-of-mass energy per nucleon pair √ s N N = 7.7 GeV at the RHIC-BES, where y denotes the rapidity. The symbols represent the experimental data measured by the STAR Collaboration [19] and the curves are our fitting results by using the blast-wave model with Boltzmann-Gibbs statistics, Eq. (1) [21,22,23]. The spectra in centrality class 0-5%, 5-10%, 10-20%, 20-30%, 30-40%, 40-50%, 50-60%, 60-70% and 70-80% are scaled by 1, 1/2, 1/4, 1/6, 1/8, 1/10, 1/12, 1/14 and 1/16 respectively. The related parameters along with χ 2 and degree of freedom (dof) are listed in Table 1, where the centrality classes are listed together. One can see that Eq. (1) fits well the data in Au-Au collisions at 7.7 GeV at the RHIC. Figure 2 is the same as Fig. 1, but it shows the p T spectra at √ s N N = 11.5 GeV. One can see that Eq. (1) fits well the data in Au-Au collisions at 11.5 GeV at the RHIC-BES. Figure 3 is also the same as Fig. 1, but it shows the p T spectra at √ s N N = 14.5 GeV, where the data are cited from ref. [20]. Once again, Eq. (1) fits well the data in Au-Au collisions at 14.5 GeV at the RHIC-BES. It is noteworthy to point out that Eq. (1) for the blast-wave model in the system is assumed to be in local thermodynamic equilibrium and therefore, a single T 0 and β T should be obtained by the weight average of different particles species. To see clearly the trends of weight average parameters, Figs. 7(a) and 7(b) show the dependences of weight averages T 0 and β T on √ s N N for   Fig. 1. Transverse momentum spectra of (a)-(c) π + , K + and p produced in different centrality bins in Au-Au collisions at √ s N N = 7.7 GeV. The symbols represent the experimental data measured by the STAR Collaboration in the mid-rapidity interval |y| < 0.1 [19]. The curves are our fitted results by Eq. (1). The ratios of Data/Fit corresponding to the panels (a)-(c) are presented by panels (a*)-(c*) respectively. different event centralities. The symbols represent the parameter values averaged by weighting the yields of different particles which are listed in Table 1. One can see that T 0 and β T increase with the increase of √ s N N from 7.7 to 39 GeV. Meanwhile, T 0 and β T increase with the increase of event centrality from periphery to center.

Results and discussion
In addition, the variation of weight averages T 0 on β T for different collision energies and event centralities are displayed in Fig. 7(c), where the symbols represent the parameter values averaged by weighting the yields of different particles. One can see that T 0 increases with the increase of β T . At higher energy and in central collisions, one see larger T 0 and β T . There is a positive correlation between T 0 and β T .
The dependences of mean transverse momentum ( p T ) and initial temperature (T i = p 2 T /2 [51, 52, 53]) on √ s N N for different event centralities obtained by weighting the yields of different particles are shown in Figs. 8(a) and 8(b) respectively. One can see that p T and T i increase with the increase of √ s N N from 7.7 to 39 GeV. Meanwhile, p T and T i increase with the increase of event centrality from periphery to center.
The reason for increasing of T 0 and β T with the increase of collision energy is due to the fact that more energies are deposited in collisions at higher energy in the considered RHIC-BES energy range. Meanwhile, the system size at higher energy decreases due to relativistic constriction effect, which results in a smaller volume, then a larger energy density and larger T 0 . Meanwhile, at higher energy, the squeeze is more violent, which re- sults in a rapider expansion and larger β T .
The reason for increasing of T 0 and β T with the increase of event centrality is due to the fact that the central collisions contain more nucleons than the peripheral collisions, then more energies are deposited in central collisions. Meanwhile a rapider expansion appears due to more violent squeeze in central collisions, comparatively to peripheral collisions. As a result, T 0 and β T in central collisions are larger than those in peripheral collisions.
Because of p T and T i being positive correlation with T 0 and β T , the increasing of T 0 and β T with the increases of collision energy and event centrality result naturally in the increasing of p T and T i with the increases of collision energy and event centrality. This work shows that the two free parameters T 0 and β T and the two derived parameters p T and T i ap-pear similar law on the dependences of collision energy and event centrality. In particular, p T and T i are model-independent, though we obtain them from modeldependent free parameters T 0 and β T in this work. In fact, p T and T i can be obtained by the p T data themselves if the data are across the possible p T range.
It should be noted that there is entanglement in the extraction of T 0 and β T . In fact, if one uses a smaller T 0 and a larger β T for central collisions, a decreasing trend for T 0 from peripheral to central collisions can be obtained. Meanwhile, a negative correlation between T 0 and β T can also be obtained. Thus, this situation is in agreement with some current references [19,54,55]. If one even uses an almost invariant or slightly larger T 0 and a properly larger β T for central collisions, an almost invariant or slightly increase trend for T 0 from peripheral to central collisions can be obtained [56]. To show the flexibility in the extraction of T 0 and β T , this work has reported an increasing trend for T 0 from peripheral to central collisions, and a positive correlation between T 0 and β T . This whole phenomenal analysis results in degree of thermal motion and collective expansion, that are reflected by T 0 and β T . With the increasing collision energy, the system may undergo different evolution processes. In the considered RHIC-BES energy range, the violent degree of collisions increase with increasing the collision energy. The trends of T 0 and β T show approximately monotonous increase in which large fluctuation does not appear. The evolution processes at the considered six energies show similar behaviors to each other.

Conclusions
The main observations and conclusions are summa-rized here.
(a) Based on the data-driven analysis, the blast-wave model with Boltzmann-Gibbs statistics is used to analyze the collision energy dependent and event centrality dependent double-differential transverse momentum spectra of charged particles (π + , K + and p) produced in the mid-rapidity interval in Au-Au collisions at the RHIC-BES energies. The contribution of soft excitation is considered in this work, but the contribution of hard process is not excluded if available.
(b) As the free parameters, the kinetic freeze-out temperature T 0 and transverse flow velocity β T are extracted by the blast-wave model. Both T 0 and β T increase with the increase of collision energy due to more violent collisions at higher energy. The two parameters also increase with the increase of centrality, as the central collisions contain more nucleons which means more energy deposited and more violent collisions and squeeze, comparing with peripheral collisions. (c) As the derived parameters, the mean transverse momentum p T and initial temperature T i appear similar law to the free parameters T 0 and β T when we study the dependences of parameters on collision energy and event centrality. Although T 0 and β T are model-dependent, p T and T i are generally modelindependent. There is no large fluctuation in the excitation function of the considered parameters at the RHIC-BES, which means similar collision mechanism. funders had no role in the design of the study; in the collection, analyses, or interpretation of the data; in the writing of the manuscript, or in the decision to publish the results.   Table 1.  Table 1.