Abstract
Transverse momentum distributions of deuterons and antideuterons in Au + Au collisions at , 62.4, and 200 GeV with different centrality are studied in the framework of the multisource thermal model. Transverse momentum spectra are conformably and approximately described by the Tsallis distribution. The dependence of parameters (average transverse momenta, effective temperature, and entropy index) on event centrality is obtained. It is found that the parameters increase and decrease with increase of the average number of particles involved in collisions, which reveals the transverse excitation degree increases with collision centrality.
1. Introduction
The study of strongly interacting matter at extreme temperatures and densities is provided a chance by heavy ion collisions at ultrarelativistic energies [1–5]. The production mechanism of nuclei in ultrarelativistic heavy ion collisions deserves more investigation since it may give important message on the quantum chromodynamics (QCD) phase transition from quark-gluon plasma (QGP) to hadron gas (HG) [6, 7]. The RHIC is scheduled to run at the energies which are around the critical energy of phase transition from hadronic matter to QGP [8]. The theoretical study of nuclei and antinuclei has been undertaken for many years, for example, the thermal model and coalescence model [9–13]. In particular, the study of transport phenomena is major important to the understanding of many fundamental properties [14]. The spectra of transverse momentum of particles produced in high energy collisions are of high interest as soon as they provide us with an important information of the kinetic freeze-out state of the interacting system [15]. At the stage of kinetic freeze-out, the effective temperature is not a real temperature, and it describes the sum of excitation degree of the interacting system and the effect of transverse flow [16].
In this paper, using the Tsallis distribution [17–19] in the multisource thermal model to simulate the transverse momentum distributions of (anti-)deuterons in Au + Au collisions at RHIC, we compare them with experiment data taken from the STAR Collaboration [20]. The main purpose of this work is to extract the information on effective temperature, because it allows us to extract the kinetic freeze-out temperature.
2. The Model and Method
The model used in the present work is the multisource thermal model [21–23]. In this model, many emission sources are formed in high-energy nucleus-nucleus collisions. The different distributions can describe the emission sources and particle spectra, such as the Tsallis distribution, the standard (Boltzmann, Fermi-Dirac, and Bose-Einstein) distributions, the Tsallis+standard distributions [24–29], and the Erlang distribution [21]. The Tsallis distribution can be described by two or three standard distributions.
The experimental data of the transverse momentum spectrum of the particles are fitted by using the Tsallis distribution which can describe the temperature fluctuation in a few sources to give an average value. The Tsallis distribution has many function forms [17–19, 24–31]. In the rest frame of a considered source, we choose a simplified form of the joint probability density function of transverse momentum () and rapidity () [8],
Here, is the particle number, is the degeneracy factor, is the volume of emission sources, is the rest mass of the studied particle, is the temperature which describes averagely a few sources (local equilibrium states), is the entropy index which describes the degree of nonequilibrium among different states, and is the chemical potential which is related to [32]. In the RHIC energy region, the values of are shown in Table 1 [33]. We can extract the values of , , and from reproducing the particle spectra, where and are fitted independently for the studied particle and is related to other parameters.
The Monte Carlo distribution generating method is used to obtain . Let denote the random numbers distributed uniformly in . A series of values of can be obtained by
Here, is the transverse momentum probability density function which is an alternative representation of the Tsallis distribution as follows: where and are the maximum and minimum rapidity, respectively.
Under the assumption of isotropic emission in the source rest frame, we use the Monte Carlo method to acquire the polar angle:
Here, denotes the random numbers distributed uniformly in . Thus, we can obtain a series of values of momentum and energy due to the momentum and the energy . Therefore, the corresponding values of rapidity can be obtained according to the definition of rapidity.
3. Results and Discussion
3.1. Transverse Momentum Spectra
Figure 1 demonstrates midrapidity () transverse momentum spectra for deuterons in Au + Au collisions at GeV for 0-10%, 10-20%, 20-40%, 40-60%, and 60-80% centralities. The symbols represent the experimental data of STAR Collaboration [20]. The solid lines are our calculated results fitted by using the Tsallis distribution based on eq. (1) in the region of midrapidity. The values of the related parameters and are given in Table 2 along with the ( and number of degree of freedom). It is found that the calculations of the Tsallis distribution are in keeping with the experimental data well.
In Figures 2 and 3, the curves and symbols are similar to Figure 1. Figure 2 demonstrates midrapidity () transverse momentum spectra for deuterons in Au + Au collisions at GeV for 0-10%, 10-20%, 20-40%, 40-60%, and 60-80% centralities. The values of the related parameters and are given in Tables 3 and 4 along with the . It is found that the calculations of the Tsallis distribution are in keeping with the experimental data well.
In Figures 4–6 demonstrates midrapidity () transverse momentum spectra for antideuterons in Au + Au collisions at , 62.4, and 200 GeV for 0-10%, 10-20%, 20-40%, 40-60%, and 60-80% centralities. The curves and symbols are similar to Figure 1. One can see that the calculations also can describe approximately the experimental data of antideuterons with different centrality intervals of event. The values of the related parameters and are given in Tables 2–4.
3.2. Average Transverse Momenta
Figure 7 presents the centrality dependence of deuterons and antideuterons average transverse momenta () at the midrapidity () for , 62.4, and 200 GeV. The hollow symbols are the experiment data taken from the Figures 1–6, and the solid symbols are the calculations of the Tsallis distribution. The calculations can be obtained by Here, is the value of transverse momentum corresponding to the experimental data, and is the value of that corresponds to the .In this figure, one sees that the calculations can describe the experimental data well in the range of the errors permitted. For deuterons, the values of average transverse momenta in the different incident energy get closer with decrease of centrality percentage. It has indicated that the transverse excitation degree increases with collision centrality.
3.3. Dependence of Parameters on Number of Participating Nucleons
Figures 8 and 9 give the change trends of parameters ( and ) with the average number of participants for deuterons and antideuterons produced in Au + Au collision at the midrapidity () for , 62.4, and 200 GeV. The symbols represent the parameter values extracted from Figures 1–6 and listed in Tables 2–4.
From Figures 8 and 9, we can see that the values of parameters increase with decrease of centrality percentage, and the values of parameters increase with increase of centrality percentage. Entropy is a physical quantity that represents the degree of chaos in the system. When a central collision occurs, the motion law of the final state particles is complex, and the whole system is in a higher state of order, so the entropy value is small. In the central region where the collision occurs, with the increase of the intensity of the collision, the corresponding effective temperature increases. The dependence of effective temperature on collision energy increases with the increase of collision energy. Under the same collision parameters, the entropy increases with the increase of collision energy, indicating that the higher the collision energy is, the more different microscopic states the particle may have, and the more disordered the system becomes. The kinetic freeze-out temperature can be extracted from the effective temperature; the correlation between Kinetic freeze-out temperature and centrality will be focused in the future work.
4. Summary and Outlook
In summary, we have presented the transverse momentum distributions of (anti-)deuterons in Au + Au collisions at , 62.4, and 200 GeV for 0-10%, 10-20%, 20-40%, 40-60%, and 60-80% centralities. The Tsallis distribution in the multisource thermal model has been used in all calculations. Based on this model, we have investigated transverse momentum distributions of (anti-)deuterons and the law about effective temperature and entropy with the centrality of collision. In conclusion, it can give the agreement between calculation results and the experimental data. The effective temperature extracted from and increases with decrease of centrality percentage at the same incident energy, and the entropy index decreases with decrease of centrality percentage at the same incident energy. And at the same collision centrality, they increase with increase of incident energy. But the Kinetic freeze-out temperature and the evolution of time during the collision have yet to be studied in depth.
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.
Conflicts of Interest
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Introduction of Doctoral Starting Funds of Scientific Research of Guangxi University of Chinese Medicine under Grant No. 2018BS024, the Natural Science Foundation of Guangxi Zhuangzu Autonomous Region of China under Grant no. 2012GXNSFBA053011, the Research support project of Guangxi institutions of higher learning (No. 200103YB071), and the Open Project of Guangxi Key Laboratory of Nuclear Physics and Nuclear Technology (No. NLK2020-03).