Advances in High Energy Physics

Volume 2018, Article ID 7895967, 12 pages

https://doi.org/10.1155/2018/7895967

## Comparing Standard Distribution and Its Tsallis Form of Transverse Momenta in High Energy Collisions

Correspondence should be addressed to Hui-Ling Li; moc.anis@xyxgniliuhil and Fu-Hu Liu; moc.361@uiluhuf

Received 26 October 2017; Accepted 2 January 2018; Published 19 March 2018

Academic Editor: Lorenzo Bianchini

Copyright © 2018 Rui-Fang Si et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The experimental (simulated) transverse momentum spectra of negatively charged pions produced at midrapidity in central nucleus-nucleus collisions at the Heavy-Ion Synchrotron (SIS), Relativistic Heavy-Ion Collider (RHIC), and Large Hadron Collider (LHC) energies obtained by different collaborations are selected by us to investigate, where a few simulated data are taken from the results of FOPI Collaboration which uses the IQMD transport code based on Quantum Molecular Dynamics. A two-component standard distribution and the Tsallis form of standard distribution are used to fit these data in the framework of a multisource thermal model. The excitation functions of main parameters in the two distributions are analyzed. In particular, the effective temperatures extracted from the two-component standard distribution and the Tsallis form of standard distribution are obtained, and the relation between the two types of effective temperatures is studied.

#### 1. Introduction

High energy heavy-ion (nucleus-nucleus) collisions are an important method to simulate and study the big bang in the early universe, properties of new matter created in extreme conditions, accompanying phenomena in the creation, and physics mechanisms of the creation. Some models based on the quantum chromodynamics (QCD) and/or thermal and statistical methods can be used to analyze the equation of state (EoS) at finite temperature and density, properties of chemical and kinetic freeze-outs in collision process, distribution laws of different particles in final state, and universality of hadroproduction in different systems [1–5]. The properties of nuclear matter and its phase transition to quark-gluon plasma (QGP) at high temperature and density can be obtained. With the developments in the methodologies of experimental techniques and theoretical studies, the collision energy per nucleon pair in the center-of-mass system increases from high energy range which has a few to several hundred GeV to ultrahigh energy range which has presently a few to over ten TeV.

The temperature and density described the EoS showing that the new matter created in high and ultrahigh energy ranges is not similar to the ideal gas-like state of quarks and gluons expected by early theoretical models. Instead, the effects of strong dynamical coupling, long-range interactions, local memory, and others appear in the interior of interacting system. The rapid evolution of interacting system and the indirect measurements of some observable quantities result in that one can use the statistical method to study the distribution properties of some observable quantities such as (pseudo) rapidity, (transverse) momentum, (transverse) energy, azimuthal angle, elliptic flow, multiplicity, and others of final-state fragments and particles [1–5]. Thus, some quantitative or qualitative results related to the properties of interacting system and particle production can be observed.

As the quantities which can be early measured in experiments, that is, the so-called “the first day” measureable quantities, the rapidity and transverse momentum distributions attract wide attentions due to their carryovers on the information of longitudinal extension and transverse expansion of the emission source in interacting system. With the increasing collision energy, the rapidity distribution range extends from a few rapidity units to over ten rapidity units, and the transverse momentum distribution range increases from 0 until a few GeV/c to 0 until over hundred GeV/c. Different functions and methods are used by different researchers to describe rapidity and transverse momentum distributions as well as other distributions which can be measured in experiments [1–5]. Based on a multisource thermal model [6–9], the rapidity and transverse momentum distributions obtained in experiments at different collision energies are studied by us in terms of two-cylinder, Rayleigh, Boltzmann, Tsallis, and other distributions. In particular, comparing with rapidity distribution, transverse momentum distribution contains more abundant information and attracts wider attentions. Although one has Monte Carlo and other indirect methods to describe transverse momentum distributions, analytical functions are more expected to use.

Because of the same transverse momentum distribution being described by different functions to obtain values of different parameters, possible relations existing among different parameters can be studied. In this paper, based on the multisource thermal model [6–9], the standard distribution (Boltzmann, Fermi-Dirac, and Bose-Einstein distributions) and its Tsallis form are used to describe the transverse momentum distribution of final-state particles produced in high energy nucleus-nucleus collisions. The excitation functions of effective temperatures obtained by the two distributions are extracted and the relation between the two effective temperatures is studied.

The rest part of this paper is structured as follows. A brief description of the model and method is presented in Section 2. Results on comparisons with experimental (simulated) data and discussion are given in Section 3. Finally, we summarize our main observations and conclusions in Section 4.

#### 2. The Model and Method

According to the multisource model [6–9], a few emission sources of produced particles are assumed to form in interacting system due to different reaction mechanisms and/or data examples. For each emission source, the thermal model or other similar models and distributions can be used to perform calculation on the production of particles. The potential models include [10], but are not limited to, ideal gas-like model, ideal hydrodynamic model, and viscous hydrodynamic model. In these models, the relativistic effect has to be particularly considered, and the quantum effect can be usually neglected. If we study in detail the interacting system and final-state particles, both the relativistic and quantum effects have to be considered.

In the middle stage of collision process, the interacting system and emission sources in it can be regarded as to stay at the hydrodynamic state. After the stage of chemical freeze-out, in particular after the stage of kinetic freeze-out, the interacting system and emission sources in it should stay at the gas-like state. Otherwise, it is difficult to understand the kinetic information of singular particle measured in experiments. What had happened during the phase transition from the liquid-like state at the middle stage to the gas-like state at the final stage and why is beyond the focus of the present work. We shall not discuss this issue here.

According to the ideal gas model with the relativistic and quantum effects, the particle spectra can be described by the standard distribution. The number of particles is [11]where is the degeneracy factor, is the volume, is the momentum, is the energy, is the rest mass, is the chemical potential, and is the effective temperature; , , and correspond to the Boltzmann, Fermi-Dirac, and Bose-Einstein statistics, respectively; corresponds to plus +, and corresponds to minus −. The invariant momentum distribution of particles isThe normalized probability density distribution of particle momenta can be written aswhere is the normalized constant in the standard probability density distribution of momenta. It is related to the selection of parameters.

The normalized joint probability density distribution of particle rapidities and transverse momenta iswhere corresponds to plus + and corresponds to minus −. The normalized probability density distribution of particle rapidities is then written to bewhere denotes the maximum transverse momentum. This rapidity distribution is only for an emission source. In the case of considering multiple sources, we have to consider sources distribution in the rapidity space [2–4, 12–16]. This issue is beyond the focus of the present work, and we shall not discuss it anymore. The normalized probability density distribution of particle transverse momenta is written to bewhere and denote the maximum and minimum rapidities, respectively.

It should be noted that, in the above formulas, although the same symbol is used to represent the normalized constants in different formulas, these constants may be different from each other. In the case of considering multisource emission, we have to use the multicomponent distribution to describe the transverse momentum distribution of final-state particles. If emission sources are considered, we havewhere denotes the normalized constant for the th component in components, denotes the contribution fraction of the th component in final-state distribution, and denotes the effective temperature corresponding to the th component. There are temperature fluctuations among different components. In the case of considering multisource emission, we have the effective temperature of interacting system as . Generally, two or three emission sources are enough to describe the experimental data obtained in soft excitation process. That is, 2 or 3 in most cases.

If we consider the Tsallis form of standard distribution, the number of particles is [11, 17]where is an entropy index which characterizes the departing degree of the interacting system from the equilibrium state. Generally, we have ; if , the system stays in the equilibrium state. is the effective temperature. Other symbols have the same meanings as (1). The invariant momentum distribution of particles isThe normalized probability density distribution of particle momenta is

The normalized joint probability density distribution of particle rapidities and transverse momenta isThen, the normalized probability density distribution of particle rapidities isThe normalized probability density distribution of particle transverse momenta isIn the above formulas, although the same symbol is used to represent the normalized constants in different formulas, these constants may be different from each other. As discussed in [17], the Tsallis form has at least four types of function representations, though we choose only one that contains after and the index . We do not need to consider a multisource for the Tsallis form due to it covering a two- or three-component standard distribution, and the two- or three-component standard distribution describes well the transverse momentum spectrum of particles produced in soft excitation process.

It should be noted again that the above multicomponent (two- or three-component) standard distribution and the Tsallis form of standard distribution can describe only the transverse momentum spectrum of particles produced in soft excitation process. The transverse momentum spectrum produced in soft excitation process covers a narrow range. For the transverse momentum spectrum covering a wide range, we have to consider the contribution of hard scattering process. According to the QCD calculus [18–20], we have an inverse power-lawto describe the transverse momentum spectrum produced in hard scattering process, where and are free parameters and is the normalized constant which is related to the free parameters. It is obvious that a two-component function is needed for a wide transverse momentum spectrum. The first component is the multicomponent (two- or three-component) standard distribution or Tsallis form which describes the soft process, and the second component is the inverse power-law which describes the hard process. The application of the inverse power-law is beyond the focus of the present work. We shall not discuss it anymore.

In the above discussions, to obtain chemical potential of a given particle, the chemical freeze-out temperature of the emission source is needed to know first of all. In the case of assuming the same chemical freeze-out moment, the emission source has the sole . According to [21, 22], there is a relation among , the yield and mass of the first particle, the yield and mass of the second particle, and the ratio . We havewhere denote fermion and boson, respectively. If the fermion and boson are not needed to distinguish each other, we have . This results in a simple expression for (15); that is, .

In the framework of a statistical thermal model of noninteracting gas particles with the assumption of standard Maxwell-Boltzmann statistics, there is an empirical expression for the chemical freeze-out temperature [23–26],where denotes the energy per nucleon pair in the center-of-mass system. Both the units of and are in GeV. The limiting value of is GeV.

In the framework of a thermal model with standard distribution, the chemical potentials of some particles can be obtained from the ratios of negatively to positively charged particles. According to [27], we havewhere the symbol of a given particle is used for its yield for the purpose of simplicity. Further, the chemical potentials of the mentioned particles areEmpirically, the chemical potential for baryon is [23–26]which is also obtained in the framework of a statistical thermal model of noninteracting gas particles with the assumption of standard Maxwell-Boltzmann statistics, where both the units of and are in GeV.

We would like to point out that (16) and (23) should be modified in the framework of generalized nonextensive statistics when we use the Tsallis form of standard distribution. At the same time, (17)–(22) should be generalized within an analysis with the Tsallis form. To modify (16)–(23) is beyond our focus and ability. We shall not discuss these modifications here. Instead, as an approximate treatment, we use and obtained within an analysis with the standard distribution as those within the Tsallis form. In fact, the absolute value of is very small, and its effect on the transverse momentum spectra can be neglected. Therefore, this approximate treatment is acceptable.

It should be noted once more that, as mentioned in the above discussions, what we extract from the multicomponent standard distribution or the Tsallis form of standard distribution is the effective temperature, but not the real temperature of emission source. Generally, the transverse momentum spectrum contains both the contributions of thermal motion and flow effect. The real temperature is only a reflection of purely thermal motion, and the flow effect should not be included in it. As for the methods to obtain the real temperature by disengaging the contributions of thermal motion and flow effect, we can use the blast-wave model based on the Boltzmann distribution [28–30], the blast-wave model based on the Tsallis distribution [31], the improved Tsallis distribution [32, 33], some alternative methods [21, 29, 34–36], and others [37–40]. These methods themselves are beyond the focus of the present work. We shall not discuss them anymore.

#### 3. Results and Discussion

The transverse momentum spectra of negatively charged pions produced in midrapidity range in 2.24 and 2.52 GeV central gold-gold (Au-Au) collisions [41] measured (simulated) by the FOPI Collaboration at the Heavy-Ion Synchrotron (SIS), 11.5 [42], 62.4, 130, and 200 GeV central Au-Au collisions [29] measured by the STAR Collaboration at the Relativistic Heavy-Ion Collider (RHIC), 22.5 GeV central copper-copper (Cu-Cu) [43] and 200 GeV central Au-Au collisions [27] measured by the PHENIX Collaboration at the RHIC, and 2.76 TeV central lead-lead (Pb-Pb) collisions [44] measured by the ALICE Collaboration at the Large Hadron Collider (LHC) are selected to investigate. Among them, the results of FOPI Collaboration are given in Figure 1 with the simulated data (the last eight circles) of the IQMD transport code [45] which is based on Quantum Molecular Dynamics [46]. To avoid confusion, most results of the STAR Collaboration are given in Figure 2, and the results corresponding to 11.5 GeV are given in Figure 3. The results of PHENIX and ALICE Collaborations are given in Figures 3 and 4, respectively. In each figure, the symbols represent the experimental (simulated) data scaled by different amounts in some cases. The collision energy and type, centrality and midrapidity ranges, and scaled amount if not 1 are marked in the panel. The dashed and solid curves denote the results fitted by the two-component standard distribution and the Tsallis form of standard distribution. The values of parameters, , and degree of freedom (dof) are listed in Table 1 ordered by the energy from low to high. In particular, is the average weighted by the fractions of different components, is obtained by (16) and (22), and the values of in (22) at different energies are obtained from [47]. As a preliminary result, the values of for the first and second standard distributions and the Tsallis form are assumed to be the same. In the fitting, the method of least square is used to obtain the best parameter values. One can see that the two-component standard distribution and the Tsallis form of standard distribution describe approximately the transverse momentum spectra of negatively charged pions produced in central nucleus-nucleus collisions in the energy range from SIS to LHC.