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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.

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 [

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 [

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 [

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 [

The rest part of this paper is structured as follows. A brief description of the model and method is presented in Section

According to the multisource model [

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 [

The normalized joint probability density distribution of particle rapidities and transverse momenta is

It should be noted that, in the above formulas, although the same symbol

If we consider the Tsallis form of standard distribution, the number of particles is [

The normalized joint probability density distribution of particle rapidities and transverse momenta is

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 [

In the above discussions, to obtain chemical potential of a given particle, the chemical freeze-out temperature

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 [

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 [

We would like to point out that (

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 [

The transverse momentum spectra of negatively charged pions produced in midrapidity range in

Values of free parameters

| | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|

2.24 GeV Au-Au | | | | | | | | | 24.043/32 |

2.52 GeV Au-Au | | | | | | | | | 16.856/43 |

11.5 GeV Au-Au | | | | | | | | | 2.850/19 |

22.5 GeV Cu-Cu | | | | | | | | | 1.360/20 |

62.4 GeV Au-Au | | | | | | | | | 1.561/7 |

130 GeV Au-Au | | | | | | | | | 42.448/7 |

200 GeV Au-Au | | | | | | | | | 10.736/8 |

200 GeV Au-Au | | | | | | | | | 7.650/25 |

2.76 TeV Pb-Pb | | | | | | | | | 9.766/38 |

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| | | | |

Transverse momentum spectra of

Same as Figure

Same as Figure

Same as Figure

To study the excitation functions of free parameters, that is, the dependence of free parameters on collision energy, the relations

Dependence of

Same as Figure

Dependence of

Dependence of

Our results show some interesting features. Actually, one could as well say that there is no difference in the particle production in central nucleus-nucleus collisions from a few GeV to a few TeV. This in some sense echoes recent studies of Sarkisyan et al. [

Our observation that

In the above analyses, for a not too wide transverse momentum spectrum, a standard distribution is usually not enough to describe the spectrum. Generally, we need a two-component standard distribution to describe the not too wide spectrum. It is expected that, in the case of studying a wider transverse momentum spectrum, we need a three-component standard distribution to describe the wider spectrum. If a set of experimental data is described by the two- or three-component standard distribution, it is also described by the Tsallis form of standard distribution [

Both the two- or three-component standard distribution and the Tsallis form of standard distribution describe only the results of soft excitation process. For the soft process, the particle spectrum appears with the characteristics of thermal emission phenomenon. Although the standard distribution describes the characteristics of thermal emission, some nonthermal emissions also obey the standard distribution. Even if the Tsallis form has less connection with thermal emission, they are relative due to the standard distribution. In the case of studying a very wide transverse momentum spectrum, for example, for a width of more than 5 GeV/c, to consider only the contribution of soft process is not enough in description of experimental data. To describe a wider transverse momentum spectrum, we have to consider simultaneously the contribution of hard scattering process. As mentioned in Section

In the above analyses, the temperature extracted by us is in fact the effective temperature

Generally speaking, the two-component standard distribution and the Tsallis form of standard distribution are the same in essentials while differing in minor points in the behaviors in the figures. The standard distribution corresponds to the classical statistical system which has short-range interactions in interior and non-multifractal structure in boundary. Some extensive thermodynamic quantities such as energy, momentum, internal energy, and entropy are linearly related to the system size and particle number. These quantities obey simply additive property. The statistical method and the microscopic description of system are adaptive. The entropy function is a power tool to study the microscopic dynamics of system under the macroscopic condition by describing the occupation number of phase spaces of the system. The Tsallis form breaks through the limitation of classical statistics by using the entropy index

In the above discussions, one can see that the two- or three-component standard distribution can be described by the Tsallis form of the standard distribution. It does not mean that the single standard distribution cannot be described by the Tsallis form. In fact, by using a lower temperature and an entropy index that is closer to 1, the Tsallis form describes well the single standard distribution. The standard distribution is successfully replaced by the Tsallis form due to

We summarize here our main observations and conclusions.

(a) The transverse momentum spectra of negatively charged pions produced in central nucleus-nucleus collisions measured (simulated) in midrapidity range by different collaborations at the SIS, RHIC, and LHC are studied by the two-component standard distribution and the Tsallis form of standard distribution which are fitted into the frame of multisource thermal model. The two distributions describe approximately the experimental (simulated) data.

(b) The excitation functions of related parameters are analyzed. The four effective temperatures

(c) There is no difference in the particle production in central nucleus-nucleus collisions from a few GeV to a few TeV. Combining with other works, one can say that the same or similar fits are good for proton-proton collisions. This suggests universality in particle production, as it is already obtained in mean multiplicity, pseudorapidity density, and multiplicity distribution, but now for transverse momentum distribution as well.

(d) The energy of

(e) To be closer to the classical situation, the two- or three-component standard distribution has an advantage over the Tsallis form of standard distribution due to similar statistics for the classical situation and standard distribution. However, the Tsallis form of standard distribution uses less parameter than the two- or three-component standard distribution. If the two- or three-component standard distribution describes a temperature fluctuation between two or among three sources, the Tsallis form of standard distribution describes a degree of nonequilibrium.

(f) In the considered energy range, different emission sources stay in an approximate equilibrium state or the whole interacting system stays in an approximate equilibrium state. There is no obvious boundary to distinguish extensive system and nonextensive system for a given interacting system. The interacting system stays in a transition gradation from extensive system to nonextensive system. To obtain only the kinetic freeze-out temperature, we would rather use the two- or three-component standard distribution due to it being closer to the classical situation.

The authors declare that they have no conflicts of interest.

Comments on the manuscript and relevant communications from Edward K. G. Sarkisyan and Ya-Hui Chen are highly acknowledged. This work was supported by the National Natural Science Foundation of China under Grants nos. 11575103 and 11747319, the Shanxi Provincial Natural Science Foundation under Grant no. 201701D121005, and the Fund for Shanxi “1331 Project” Key Subjects Construction.

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