The transverse momentum p T spectra of particles produced in heavy-ion collisions at high energies are important observables and can provide valuable information about the collision system. They are often used to discuss the particle-production properties of the collision system. A large amount of p T experimental data in pp collisions at different energies and nucleus–nucleus (AA) collisions for different centralities at different energies has been measured using the Relativistic Heavy Ion Collider (RHIC) [1, 2] and the Large Hadron Collider (LHC) [3, 4]. The initial distribution of the high-energy particles can be parameterized by the Tsallis distribution [5, 6], which extracted the Tsallis temperature T and a nonextensivity parameter q . The nonextensivity parameter was used to describe the degree of a non-equilibrium of the system. A thermodynamically consistent form of the Tsallis distribution was taken to fit the transverse momentum spectra. As a new matter, quark–gluon plasma (QGP) is a thermalized system composed of strongly coupled quarks and gluons in a finite area. The high-energy particles finally lose energy when they interact with the QGP medium, which is formed due to collision of heavy ions with each other. The distribution modification due to energy loss reveals the characteristics of the matter produced in collisions. The effects of the energy loss and the dynamics of hadronization can be studied using the nuclear modification factor R A A , which compares the transverse momentum differential production yields in nucleus–nucleus collisions ( d 2 N A A / d y d p T ) with the transverse momentum differential production yields in inelastic proton–proton collisions ( d 2 σ p p / d y d p T ).

Particle distribution is a significant value observed in the LHC experiment. Many phenomenological models were proposed to discuss the abundant experimental data. However, it is very difficult to uniformly describe the whole properties of particle distribution and to analyze the whole process of matter evolution in relativistic heavy-ion collisions by using only one method. In recent years, some different methods were combined with each other in order to figure out the multiparticle production in heavy-ion collisions at high energies. Recently, different statistic-based models were proposed to understand the transverse momentum p T distribution of final-state particles in high-energy collisions, such as the statistical thermal model [7, 8], the statistical hadronization model [9], Tsallis statistics [10], the wounded quark model [11], Boltzmann statistics [12], the multisource thermal model [13], Rayleigh distribution [14], and Erlang distribution [15]. In particular, Tsallis statistics has successfully described the experimental p T distribution, longitudinal momentum fraction distribution, and the rapidity distribution of hadrons produced in high-energy collisions [16, 17]. It was widely applied by STAR [18] and PHENIX [19] collaborations at RHIC and by ALICE [20–25], ATLAS [26], and CMS [27, 28] collaborations at LHC.

Heavy-flavor quarks are mostly produced in the initial stage of collisions in hard scattering processes between nucleus partons. Heavy-flavor quarks can undergo the whole evolution process of QGP created in ultra-relativistic heavy-ion collisions. Therefore, heavy-flavor mesons carry the substantial information of the hot-dense QCD medium and hadron production and are recognized as important probes of the QGP. It is crucial to study the interaction between heavy-flavor quarks and the strongly interacting medium by the differential production yield, the nuclear modification factor, and the anisotropic collective flow of heavy-flavor mesons. Thermodynamic properties were obtained via the comparison of theoretical models with transverse momentum spectra p _T of heavy-flavor mesons measured in collisions. The nuclear modification factor R _AA is a key observable, allowing us to discuss the mechanisms of the particle production in proton–proton collisions and heavy-ion collisions at high energies and understand the effects of energy loss and the dynamics of the heavy-quark hadronization. In the investigation of the transverse momentum p _T spectra of heavy-flavor mesons, some parameters required in the model calculation of the nuclear modification factor R _AA may be extracted synchronously.

The transverse momentum spectra of final-state particles can give the significant information of the produced matter in high-energy collisions. In our previous work [29], the temperature parameters of particle-emission sources were determined qualitatively in the geometrical manner of the multisource thermal model, and thermodynamic properties of these emission sources were determined from the central axis to the side-surface of the source cylinder. In this paper, we will investigate the transverse momentum distributions of prompt D 0 , D + , D * + , and D s + mesons for different centralities and prompt J / ψ in Pb–Pb collisions at s N N = 5.02  TeV . Furthermore, the transverse momentum p T distribution of charged particles for different centralities in Pb–Pb collisions at s N N = 2.76  TeV is compared with the transverse momentum up to 100 GeV / c . Based on the analysis, the nuclear modification factors will be reproduced. In most of our previous works [29, 30], the multisource thermal model was used mainly to discuss the transverse momentum spectra in different collisions at high energies. In this work, we will combine a new method with the multisource production to investigate the distribution of particles produced in Pb–Pb collisions at s N N = 5.02 TeV and s N N = 2.76 TeV. This work is a new attempt and will help us understand the properties of particle distribution in high-energy heavy-ion collisions from more different perspectives.

In the multisource thermal model [29], the projectile and target cylinders were supposed to be formed in nucleus–nucleus collisions at high energy. In the rapidity space, the projectile cylinder and the target cylinder lie in the rapidity range [-Y, Y]. The final-state particles are produced from different emission sources in the cylinders. Final-state particles emit anisotropically from these emission sources in different longitudinal locations. On the other hand, the projectile and target cylinder are thought to be composed of a series of emission sources with different rapidity shifts. The model is commonly known as a multisource thermal model. The simple model can only describe transverse momentum spectra of particles and can only identify the qualitative temperature parameters of emission sources by transverse momentum spectra. The limitation of the model is very difficult to avoid. In this work, the multisource production will be considered in Tsallis statistics, which is a thermodynamic formalism of describing the fractal structure of Yang–Mills fields [31, 32]. In addition, the relaxation time approximation of the collision term in the Boltzmann transport equation will be introduced into the model in order to describe the p T distribution and the nuclear modification factor R A A . Using the improved model, the particle distribution in experiments is explained in the new formalism. The different interpretations complement one another and allow us to understand the particle production from various perspectives. In the improved model, the thermodynamic properties of the multiparticle system are discussed further compared to our previous works.

In the calculation, we consider a thermodynamically consistent form of the Tsallis distribution, which was described in detail in Refs [5, 33]. From the Tsallis distribution, the correlative thermodynamic quantities can be extracted. According to Tsallis statistics, the particle number is given by N = g V ∫ d 3 p 2 π 3 1 + q − 1 E − μ T − q q − 1 , (1) where g , V , p , E , and μ are the degeneracy factor, volume, particle momentum, energy, and chemical potential, respectively. The parameter T is a Tsallis temperature, and q is a nonextensivity parameter. The corresponding momentum distribution is given by d 3 N d 3 p = g V 2 π 3 1 + q − 1 E − μ T − q q − 1 . (2)

When the nonextensivity parameter q tends to 1, the distribution function is the Boltzmann distribution, given by lim q → 1 d 3 N d 3 p = g V 2 π 3 exp − E − μ T . (3)

Considering the multisource emission [29, 30], the transverse momentum distribution of initial-state particles can be written as f i t = ∫ − Y Y g V 2 π 2 p T m T 1 + q − 1 m T T − q q − 1 d y . (4)

In the calculation, Eq. 4 is regarded as an initial distribution of the Boltzmann transport equation. By the relaxation time approximation of the collision term, the Boltzmann transport equation is solved in order to obtain a distribution of final-state particles.

The nuclear modification factor R A A is R A A = f f t f i t , (5) where f f t is a distribution function of final-state particles.

An evolution of the particle distribution f x → , p → , t is given by the Boltzmann transport equation: d f x → , p → , t d t = ∂ f ∂ t + υ → ⋅ ∇ → x f + F → ⋅ ∇ → p f , (6) where υ → is a velocity and F → is an external force. In order to account for the collision, the equation is written as d f x → , p → , t d t = ∂ f ∂ t c o l l . (7)

By considering the relaxation time approximation, the collision term ∂ f ∂ t c o l l is given by ∂ f ∂ t c o l l = − f − f e t τ , (8) where τ is a relaxation time. The function f e t is a distribution function of particles in the Boltzmann local equilibrium state: f e t = g V 2 π 2 p T m T e − m T T e t , (9) where T e t is the equilibrium temperature. Considering a homogeneous distribution and F → = 0 in Eq. 6, the distribution of final-state particles produced in collisions is f f t = f e t + f i t − f e t e − t f t τ . (10)

By using Eq. 5, the nuclear modification factor [6] is given by R A A = f f t f i t = f e t f i t + 1 − f e t f i t e − t f t τ . (11)

Figure 1 shows the transverse momentum spectra of prompt D 0 , D + , D * + , and D s + mesons for 0%–10%, 30%–50%, and 60%–80% centrality classes in Pb–Pb collisions at s N N = 5.02 TeV. Figures 1A–D show D 0 , D + , D * + , and D s + , respectively. The scattered symbols indicate the experimental data obtained from ALICE collaboration [34]. The lines represent the model results, which are in agreement with the experimental results. For low- p T data, the model results are inconsistent with experimental data in central collisions. However, the major trend of the data change is similar. The parameter values taken in the calculation are listed in Table 1. For the same meson, the temperature T and t f t / τ increase with the centrality. The reaction system is subjected to high excitation because the number of participating nucleons increases with centrality. It is shown that the reaction system in central collisions requires less time to reach the local equilibrium due to the initial distribution. Using the parameters and Eq. 11, the nuclear modification factor R A A of these mesons for different centralities is obtained. Figure 2 shows the nuclear modification factor R A A of prompt D 0 , D + , D * + , and D s + mesons in the 0%–10%, 30%–50%, and 60%–80% centrality classes in Pb–Pb collisions at s N N = 5.02 TeV. The scattered symbols represent the experimental data obtained from ALICE collaboration [34]. The lines represent the model results. Compared with Figure 1, the model results of Figure 2 are approximately consistent with the experimental data. The model values are not close to the low- p T experimental data due to other processes, such as regeneration and shadowing.

Figure 3 shows the transverse momentum spectra of the prompt J / ψ meson in Pb–Pb collisions at s N N = 5.02 TeV. The scattered symbols indicate the experimental data obtained from CMS collaboration [35]. The maximum of the transverse momentum is 50 GeV / c . The lines represent the model results, which are in agreement with the experimental results. The parameter values are listed in Table 1. The nuclear modification factor R A A of J / ψ is shown in Figure 4.

To further test the capacity of the model, we analyze other particles at a higher energy. Figure 5 shows the transverse momentum spectra of charged particles for 0%–5%, 5%–10%, 10%–30%, 30%–50%, 50%–70%, and 70%–90% centrality classes in Pb–Pb collisions at s N N = 2.76 TeV. The scattered symbols indicate the experimental data obtained from CMS collaboration [36]. The lines represent the model results, which are in agreement with the experimental results. The parameter values and χ²/NDF (number of degrees of freedom) are listed in Table 2. The temperature T and t f t / τ increase with the centrality. Overall, the values of parameters T, q, and t f t / τ are smaller than those at s N N = 5.02 TeV. In the calculation, charged particles π ± , K ± , p , and p ¯ are considered [37].

By analyzing the results, it is revealed that the improved model can explain the transverse momentum p T spectra of particles produced in collisions and reproduce R A A approximately. Furthermore, thermodynamic properties of the multiparticle system are discussed.

In our previous works [29, 30], the multisource production of final-state particles in high-energy nuclear collisions was proposed in several versions, which can be applied to study the transverse momentum distributions, elliptic flows, and so on. Final-state particles emit from different emission sources in the model, which can only identify the qualitative temperature parameters of emission sources. In recent years, Tsallis statistics is widely used in the investigation of particle distribution in high-energy collisions. In this paper, we combine Tsallis statistics with the multisource model. Moreover, the relaxation time approximation of the collision term in the Boltzmann transport equation is applied in the improved model. We study the transverse momentum spectra for different centrality classes in Pb–Pb collisions at s N N = 5.02 TeV and s N N = 2.76 TeV. The model results are in agreement with experimental data measured by ALICE and CMS collaborations. The values of parameters T , q , and t f t / τ are obtained. On this basis, the nuclear modification factor R A A is reproduced.

The temperature T increases with collision centrality and collision energy due to the excitation degree of the multiparticle system. For the same reason, t f t / τ increases with the collision centrality and the collision energy. The non-equilibrium degree q at s N N = 5.02  TeV is larger than that at s N N = 2.76  TeV . The multiparticle system at the larger collision energy deviates farther from the equilibrium state. These thermodynamic properties may shed light on some information carried by particle distribution and are helpful in the better understanding of the particle production in high-energy collisions.

In the multisource thermal model, final-state particles emit from different emission sources, which are expected to be formed in collisions. The model is still in development. The present work will further be improved in the framework of multisource production. The interaction of emission sources is related to the hot dense matter in the sources and also results in the azimuthally anisotropic expansion in the momentum space. The momentum asymmetry will be used to describe the elliptic flows of particles produced in ultra-relativistic heavy-ion collisions. Considering different rapidity shifts of anisotropic emission sources, the particle distribution in the rapidity space can be discussed. In the future, more properties of the multiparticle system will be found in the model and some thermodynamic quantities (such as the heat capacity, speed of sound, and conformal symmetry breaking measure) can be calculated.

Altogether, this work is a new attempt to study the properties of particle distribution using the improved method.