Description of Midrapidity p_T Spectra of Primary Charged Particles with Changing Centrality and Energy of Pb+Pb and Xe+Xe Collisions at the LHC with a Two-Component Model

1. Introduction

Quark-gluon plasma (QGP) is a plasma state of matter consisting of nearly free quarks and gluons, which is believed to be a matter of a newly born Universe a few microseconds after the so called Bing-Bang − starting point of Universe creation. QGP was generated [1,2,3,4,5,6,7,8,9] in high-energy heavy-ion collisions at the Large Hadron Collider (LHC, Switzerland) and the Relativistic Heavy-ion Collider (RHIC, USA). With a very short lifetime of the order of ~ 10⁻²³ seconds, this extremely dense and hot QGP matter has demonstrated almost perfect fluid behavior, characterized by very low viscosity. During its short existence, QGP undergoes a process called hadronization, converting colored quarks and gluons into many color-neutral particles, most of which are charged and neutral pions. Following this, the hot and dense cluster (“fireball”) of newly produced hadrons expands (flows) longitudinally and radially, cooling down rapidly and passing through the chemical and kinetic freeze-out stages. Chemical freeze-out occurs when the hadrons cease inelastic interactions, which determines the final numbers of produced various particle species. The chemical potential and the temperature of the system at this moment are estimated by analyzing the yield ratios of different particle species with the help of statistical hadronization or thermal models [10,11,12,13,14,15]. Finally, kinetic freeze-out stage is reached when elastic interactions among particles in the expanding fireball stop, which determines the particles’ final momenta and energies as measured by detectors surrounding the interaction zone. Analysis of the experimental transverse momentum distributions of produced hadrons in the low (p_T ≤ 2−3 GeV/c) and intermediate (2−3 GeV/c ≤ p_T ≤ 5−6 GeV/c) p_T range is needed for evaluating the thermodynamic and hydrodynamic properties of a collision system at the moment of kinetic freeze-out. Processes of quark recombination in production of hadrons, which result in enhancement of proton-to-pion yield ratio in high-energy nuclear collisions compared to proton-proton collisions, contribute to intermediate p_T range of transverse momentum spectra of hadrons. QCD (Quantum-Chromo-Dynamics) hard scatterings, quark and gluon fragmentation, production of quark-gluon jets and their energy loss in the QGP medium, are responsible for and determine mostly the high-p_T part (p_T > 5−6 GeV/c) of hadron spectra [16]. While parton (quark or gluon) fragmentation dominates high-p_T hadron production, the dense QGP matter produced in heavy-ion collisions allows for partons to recombine, which favors baryon (proton) production over meson (pion) production mostly in intermediate p_T range. Decays of higher mass resonances also contribute to the measured hadron spectra [16]. Hence, analysis of transverse momentum spectra of hadrons and their dependencies on collision centrality, or average number of participant nucleons (or charged-particle multiplicity density), and their nuclear modifications allow one to extract valuable information on collective properties of the medium and various mechanisms of hadron production in proton-proton, proton-nucleus and nucleus-nucleus collisions at high energies [10,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41].

Transverse momentum spectra of hadrons, produced in high-energy collisions, demonstrate exponential behavior in the low-p_T part and power-like distribution, typical for pQCD (perturbative QCD), in the high-p_T part. There are no exact p_T borders between various processes and mechanisms, which contribute to the measured p_T spectra of hadrons overlapping significantly in different p_T regions. Therefore, these various phenomena, contributing differently to the final p_T spectra of hadrons, still lack unambiguous theoretical explanation and can be reproduced by various phenomenological models only. Different phenomenological models have been proposed to describe the transverse momentum spectra of hadrons in high-energy proton-proton, proton-nucleus and nucleus-nucleus collisions. The most widely used model, which combines both exponential and power-like shapes, is the non-extensive Tsallis distribution function and its various modifications [42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63]. The QCD inspired inverse power-law Hagedorn function [64,65] could reproduce quite well the high p_T part of p_T spectra of particles generated in high-energy nucleon-nucleon collisions. Various power-law or Tsallis type models can describe well the whole measured p_T spectra in minimum bias or low multiplicity proton-proton collisions, or in low multiplicity peripheral proton-nucleus and nucleus-nucleus collisions. However, these models fail to describe accurately the p_T spectra in high-multiplicity proton-proton and central heavy-ion collisions in the whole measured p_T range, suggesting more complex models [20,21,22,24,26,27,30,31,37,38,63,66,67,68,69], including the hydrodynamical flow effects in the blast-wave model (BWM) [24,26,27]. The BWM model itself is used to fit the p_T spectra in the limited p_T range up to 3 GeV/c [24,26,27]. The Hagedorn and Tsallis functions with embedded transverse flow [20,21,22,30,31,37,38,63,66], Blast-Wave model with Boltzmann Gibbs statistics [24,26,27], the Improved Tsallis function with an effect of transverse flow [67,68,69], and other functions incorporating transverse flow have been used to estimate the average transverse flow velocity and temperature of a collision system at the moment of kinetic freeze-out. However, these phenomenological model functions, which incorporate transverse flow, could describe well just the low and intermediate p_T range up to 5 GeV/c only.

In Ref. [17] transverse momentum spectra of charged hadrons in proton-proton collisions at the LHC have been analyzed in terms of a two-component model [17], parameterizing the charged-particle spectrum by the sum of an exponential (Boltzmann-Gibbs like distribution) and a power-law term. The exponential term of this model has been associated with thermalized production of hadrons by valence quarks and cloud of quarks and gluons coupled to them [17]. The power-law term has been related to the mini-jet fragmentation of the virtual partons (pomerons in pQCD) exchanged between two colliding systems of partons [17]. Variations of the model parameters as a function of pseudorapidity interval and center-of-mass collision energy in proton-proton collisions have been investigated. However, this two-component model function could fit well the charged-hadron spectra only in minimum bias proton-proton collisions at high energies [17]. In Ref. [18] a three-component model function, which is a sum of the hydrodynamical blast-wave term, Tsallis function term and $p_T^2$-dependent power-law term, has been proposed for an improved fitting of the hadron spectra measured at midrapidity for arbitrary transverse momenta in proton-proton and heavy-ion collisions at the LHC. It has been concluded that this model function is effectively two-component “soft+hard”

model, because of mainly auxiliary character of the Tsallis term, which vanishes for heavy particles and presumably describes mainly the contribution from decays of resonances [18]. In Ref. [70] a unified non-extensive statistical approach, using the Pearson distribution, has been proposed to describe p_T spectra of hadrons produced in high-energy collisions. Though this unified model function reproduces accurately hadron p_T spectra up to the largest p_T values in minimum-bias high-energy proton-proton collisions, it could fit well p_T distributions of particles in various centralities of high-energy heavy-ion collisions in the limited p_T range up to 5 GeV/c only [70].

In present work, we propose a new two-component model function, which is a sum of thermodynamically consistent [45] Tsallis function with embedded transverse flow, introduced for the first time in Ref. [21] and shown [21,36,37,66] to reproduce quite well the midrapidity p_T spectra of identified hadrons in various centralities of high-energy heavy-ion collisions in p_T range up to 5 GeV/c, and inverse power-law function, which is asymptotic of Hagedorn function at very high p_T, aimed to reproduce hadron p_T spectra in p_T > 5 GeV/c range up to the highest measured p_T values. We call this two-component model function as Tsallis-Hagedorn model with transverse flow (THMTF) in present analysis. In present work, we analyze, using proposed THMTF function, the midrapidity experimental p_T spectra of primary charged hadrons up to the highest p_T values, as measured by ALICE collaboration, in various centrality intervals of high-energy Xe+Xe (at $\sqrt{s_{nn}}$=5.44 TeV) [71] and Pb+Pb (at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV) [72] collisions at the LHC. We study also the midrapidity experimental p_T spectra of primary charged hadrons produced in minimum bias proton-proton (at $\sqrt{s}$=2.76 and 5.02 TeV) and p+Pb (at $\sqrt{s_{nn}}$= 5.02 TeV) collisions at the LHC, measured [72] by ALICE collaboration, as the reference data for a comparison.

2. Experimental Data and Two-Component (THMTF) Model

In present work, we analyze the transverse momentum spectra of primary charged particles from Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV [72] and Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV [71] measured by the ALICE Collaboration in nine centrality intervals in midpseudorapidity region |η| < 0.8 and in the transverse momentum ranges 0.15 < p_T < 15 GeV/c (for Pb+Pb collisions) and 0.15 < p_T < 50 GeV/c (for Xe+Xe collisions). We also use experimental transverse momentum spectra of primary charged particles measured [72] by the ALICE Collaboration in minimum bias proton-proton (at $\sqrt{s}$=2.76 and 5.02 TeV) and p+Pb (at $\sqrt{s_{nn}}$= 5.02 TeV) collisions as the reference data for comparison with those in Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV [72] and Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV [71]. The p_T spectra of primary charged particles in minimum bias proton-proton collisions at $\sqrt{s}$=2.76 and 5.02 TeV have been measured [72] by ALICE collaboration in p_T range 0.15 < p_T < 50 GeV/c, and in p_T range 0.15 < p_T < 15 GeV/c in minimum bias p+Pb collisions at $\sqrt{s_{nn}}$= 5.02 TeV. A primary charged particle is [72] taken to be a charged particle with a mean proper lifetime largerthan 1 cm/c which is either produced directly in the interaction, or from decays of particles with lifetimesmaller than 1 cm/c, excluding particles produced in interactions with the detector material (here c is the speed of light in vacuum). The criteria for selection of tracks have been identical for all data sets and optimized for best track quality and minimal contamination from secondary particles. The average values of the number of participant nucleons ($〈N_{part}〉$) and charged-particle (pseudo-rapidity) multiplicity density ($\langle d{N}_{ch}/d\eta \rangle$) obtained [71,72,73] by the ALICE Collaboration, using the Glauber - Monte Carlo model calculations, in the analyzed centrality classes of Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV and Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV are presented in Table 1.

The following version of Tsallis function [45,46,47,50], calculated for a zero chemical potential (μ ≈ 0 at the TeV collision energies) at midrapidity (<y>=0), agrees with the thermodynamic relations for pressure, temperature, particle number, energy, and entropy densities [45]:

$${\displaystyle \frac{d^{2}N}{2\pi N_{ev}{p}_{\mathrm{T}}d{p}_{\mathrm{T}}dy}}=C_q{m_{\mathrm{T}}\left(1+(q-1)·{\displaystyle \frac{m_{\mathrm{T}}}{T}}\right)}^{-q/(q-1)},$$

(1)

where C_q represents the fitting constant, N_ev indicates the total number of inelastic collision events,$m_{\mathrm{T}}=\sqrt{p_{\mathrm{T}}^2+m_0^2}$ – denotes the transverse energy (mass), and m₀ is the hadron's rest mass. Furthermore, T shows the effective temperature, while q refers to the non-extensivity parameter. This parameter, q, which is fundamental for Tsallis distribution, is assumed to represent the fluctuations in system’s temperature via the relationship $q-1={\displaystyle \frac{Var\left(T\right)}{{<T>}^2}}$. The deviation, q−1, from unity of this parameter is believed to quantify the extent of non-thermalization and non-equilibrium within the system. As q approaches one (1), the Tsallis distribution converges to the exponential Boltzmann-Gibbs distribution, applicable to systems in equilibrium. The closer the q parameter is to one (1), the higher the level of thermalization (equilibrium) of the system. Since the majority (~ 85%) of primary charged particles generated in high-energy p+p, p+Pb, Xe+Xe, and Pb+Pb collisions at LHC energies are charged pions [19,25,74], we use charged-pion mass, m₀ = 139.57 MeV/c², for all primary charged particles spectra across all the analyzed collision systems in present study. The function represented in Eq. (1) is termed the thermodynamically consistent Tsallis function. The proof of thermodynamic consistency of the function in Eq. (1) was illustrated in Ref. [45]. The normalization constant is assumed to have a linear relationship with the system’s volume (V) [45,46] through expression C_q=gV/(2π)³, where g is the degeneracy factor of particle species.

In Eq. (1), the effective temperature T includes contributions from both chaotic thermal motion and transverse radial flow. To distinguish these two contributions, the velocity of transverse flow must be plugged into the Tsallis distribution function. The transverse flow can be incorporated into the Tsallis distribution with thermodynamic consistency in Eq. (1) by substituting $m_{\mathrm{T}}\to \langle {\gamma }_{\mathrm{T}}\rangle (m_{\mathrm{T}}-p_{\mathrm{T}}\langle {\beta }_{\mathrm{T}}\rangle )$, as was first done in Ref. [21], to derive the function:

$${\displaystyle \frac{d^{2}N}{{2\pi N}_{ev}{p}_{\mathrm{T}}d{p}_{\mathrm{T}}dy}}=C_q·{〈{\gamma }_{\mathrm{T}}〉·\left(m_{\mathrm{T}}-p_{\mathrm{T}}·〈{\beta }_{\mathrm{T}}〉\right)\left(1+〈{\gamma }_{\mathrm{T}}〉·{\displaystyle \frac{\left(q-1\right)·\left(m_{\mathrm{T}}-p_{\mathrm{T}}·〈{\beta }_{\mathrm{T}}〉\right)}{T_0}}\right)}^{-q/(q-1)},$$

(2)

This substitution, $m_T\to \langle {\gamma }_T\rangle (m_T-p_T\langle {\beta }_T\rangle )$, means [21] simply the Lorentz transformation to a system co-moving with an average transverse flow velocity, $\langle {\beta }_T\rangle$, of the system of particles in the transverse plane assuming that such a flow of particles exists in this system. The thermodynamically consistent Tsallis function with embedded transverse flow in Eq. (2) has reproduced [21,36,37,66] quite well the midrapidity p_T spectra of identified charged hadrons in the low and intermediate p_T ranges up to 5 GeV/c in high-energy p+Pb, Xe+Xe, and Pb+Pb collisions at the LHC, allowing for the physically meaningful interpretation of the extracted collective parameters and their dependencies on system size, collision centrality and average charged-particle multiplicity density.

The high p_T part of transverse momentum spectra of hadrons in high-energy nucleon-nucleon collisions could be reproduced well by the inverse power law − QCD inspired Hagedorn function [62,64,65]:

$${\displaystyle \frac{d^{2}N}{{2\pi N}_{ev}{p}_{\mathrm{T}}d{p}_{\mathrm{T}}dy}}=C_n{·\left(1+{\displaystyle \frac{p_T}{P_0}}\right)}^{-n}$$

(3)

where $C_n$ is the fitting constant, proportional to the system’s volume at freeze-out, n and P₀ are free parameters, which have distinct physical meanings related to underlying production mechanisms of hadrons. Parameter $P_0$, being characteristic momentum scale, can represent the typical momentum scale where the production mechanism transitions from soft (thermal/exponential) to hard (power-law/perturbative) processes. In some modified versions of Hagedorn function, $P_0$is related to the temperature of the system (T) and the power-index n, expressed as $P_0=nT$. It also characterizes the average transverse momentum contributed by partons (quarks or gluons) before they fragment into hadrons. The power-index parameter n describes the "hardness" of the interaction. A small value of n represents a "hard" spectrum dominated by perturbative QCD (pQCD) scattering and jet production. The QCD cross sections at high energies scale as ${\left(p_T\right)}^{-n}$ with the power-index n modelled as $n=2n_{\alpha }-4$, where $n_{\alpha }$ represents the number of active flavors or the number of partons participating in a scattering [62,75]. In the simple scattering (leading twist), the dominant process is a simple point-like scattering between two quarks, $q+q\to q+q$, with the number of participant quarks ($n_{\alpha }$) equal four. This results in the power-index parameter $n=2n_{\alpha }-4=4$. In the multiple scattering (higher twist), multiple quark-quark scatterings or quark-hadron scatterings occur, and the value of n increases. This reflects the involvement of more constituent participants in the processes of particle production. Obviously, the value of parameter $n$ changes significantly depending on whether the produced particle is a meson (two constituent quarks) or a baryon (three constituent quarks). These differences are due to the differing quark structure and distinct production mechanisms of mesons and baryons − specifically, how many constituent quarks (partons) must recombine or "coalesce" to form the final hadron. In a statistical framework, Hagedorn parameter $n$ is inversely connected to the Tsallis non-extensivity parameter q through the relation $n=1/(q-1)$.     

For very large values of transverse momentum ($p_{\mathrm{T}}$) the Hagedorn function in Eq. (3) reduces to the following asymptotic, called (simple) inverse power-law function:

$${\displaystyle \frac{d^{2}N}{{2\pi N}_{ev}{p}_{\mathrm{T}}d{p}_{\mathrm{T}}dy}}=C_n{\left({\displaystyle \frac{p_T}{P_0}}\right)}^{-n}.$$

(4)

To reproduce well the measured long $p_{\mathrm{T}}$ ranges of midrapidity transverse momentum distributions of charged particles at various centralities of high-energy heavy-ion collisions, we propose the following two-component model function, which is the sum of the thermodynamically consistent Tsallis function with embedded transverse flow, given in Eq. (2), and (simple) inverse power-law function, shown in Eq. (4):

$$\begin{gathered} {\displaystyle \frac{d^{2}N}{{2\pi N}_{ev}{p}_{\mathrm{T}}d{p}_{\mathrm{T}}dy}}= \\ =\theta \left(x_{thr}-p_T\right)C_q{〈{\gamma }_{\mathrm{T}}〉\left(m_{\mathrm{T}}-p_{\mathrm{T}}〈{\beta }_{\mathrm{T}}〉\right)\left(1+〈{\gamma }_{\mathrm{T}}〉{\displaystyle \frac{\left(q-1\right)\left(m_{\mathrm{T}}-p_{\mathrm{T}}〈{\beta }_{\mathrm{T}}〉\right)}{T_0}}\right)}^{-q/(q-1)}+{\theta \left(p_T-x_{thr}\right)C}_n{\left({\displaystyle \frac{p_T}{P_0}}\right)}^{-n}, \end{gathered}$$

(5)

where $\theta \left(x\right)$ is the Heaviside function with the following properties: $\theta \left(x\right)=0$ if $x<0,$and $\theta \left(x\right)=1$ if $x>0$. This new two-component model function in Eq. (5) is called Tsallis-Hagedorn model with transverse flow (THMTF) in present work. Parameter $x_{thr}$ represents the optimal value of $p_{\mathrm{T}}$border between the corresponding $p_{\mathrm{T}}$ fit ranges of the first and second term in Eq. (5). In our work, we use fixed $x_{thr}=5$ GeV/c, because thermodynamically consistent Tsallis function with embedded transverse flow, given in Eq. (2), could reproduce quite well the midrapidity transverse momentum distributions of identified charged particles at various centralities of high-energy heavy-ion collisions in the low and intermediate $p_{\mathrm{T}}$ range up to $p_{\mathrm{T}}$ = 5 GeV/c [21,36,37,66]. Using $x_{thr}=5$ GeV/c in Eq. (5) means that the hadron $p_{\mathrm{T}}$ spectrum is described by thermodynamically consistent Tsallis function with transverse flow (Eq. (2)) in $p_{\mathrm{T}}$ ≤ 5 GeV/c range and by inverse power-law function (Eq. (4)) in $p_{\mathrm{T}}$ > 5 GeV/c region. For the measured midrapidity $p_{\mathrm{T}}$ spectra, ${\displaystyle \frac{d^{2}N}{N_{ev}d{p}_{\mathrm{T}}dy}}$ , of particles the THMTF function in Eq. (5) becomes:

$$\begin{gathered} {\displaystyle \frac{d^{2}N}{N_{ev}d{p}_{\mathrm{T}}dy}=} \\ =\theta \left(x_{thr}-p_T\right)2\pi {p_{\mathrm{T}}C}_q{〈{\gamma }_{\mathrm{T}}〉\left(m_{\mathrm{T}}-p_{\mathrm{T}}〈{\beta }_{\mathrm{T}}〉\right)\left(1+〈{\gamma }_{\mathrm{T}}〉{\displaystyle \frac{\left(q-1\right)\left(m_{\mathrm{T}}-p_{\mathrm{T}}〈{\beta }_{\mathrm{T}}〉\right)}{T_0}}\right)}^{-q/(q-1)}+{\theta \left(p_T-x_{thr}\right)2\pi p_{\mathrm{T}}C}_n{\left({\displaystyle \frac{p_T}{P_0}}\right)}^{-n} \end{gathered}$$

(6)

In present work, we have fitted, using the newly introduced THMTF function (Eq. (6)), the midrapidity experimental p_T spectra of primary charged hadrons up to the highest p_T values, measured by ALICE collaboration, in nine centrality intervals (see Table 1) of high-energy Xe+Xe (at $\sqrt{s_{nn}}$=5.44 TeV) and Pb+Pb (at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV) collisions at the LHC. We have also fitted using THMTF function the midrapidity experimental p_T spectra of primary charged hadrons produced in minimum bias proton-proton (at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV) and minimum bias p+Pb (at $\sqrt{s_{nn}}$= 5.02 TeV) collisions at the LHC, measured by ALICE collaboration, as a baseline data for comparison. The minimum χ² fits of the experimental p_T spectra with two-component model (THMTF) function in Eq. (6) have been done using Nonlinear Curve Fitting of the Origin 9.1 Graphing and Data Analysis Software. The error bars for the experimental data points in the figures of present work show the added (combined) statistical and systematic errors. These combined errors are mostly determined by the systematic uncertainties, which are much larger than the statistical errors. Minimum χ² fit procedures have been conducted taking into account the weights (1/(combined error)²) of the experimental data points. The region p_T < 0.5 GeV/c in spectra of primary charged particles has been excluded from fitting procedures, as also done previously in Refs. [19,21,25,36,37,66], because of the significant contribution to production of primary charged particles (mostly pions) from decays of baryon resonances in this low p_T range. Because the midrapidity experimental p_T spectra of primary charged hadrons produced in various centrality intervals of Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV in Ref. [72], measured by ALICE collaboration, are available up to 15 GeV/c, in present work we analyze and compare the parameters of the THMTF function extracted from fitting the experimental p_T spectra in the same p_T range from 0.5 to 15 GeV/c across all the analyzed Xe+Xe, Pb+Pb, proton-proton and p+Pb collision systems.

3. Analysis and Results

The parameters obtained from minimum ${\chi }^2$ fits with Tsallis-Hagedorn model with transverse flow (THMTF) (Eq. (6)) of experimental midrapidity $p_T$ spectra of primary charged particles in minimum bias p+p collisions at $\sqrt{s}$=2.76 and 5.02 TeV and minimum bias p+Pb collisions at $\sqrt{s_{nn}}$= 5.02 TeV for the fitted p_T range from 0.5 to 15 GeV/c are presented in Table 2. The corresponding results for the fitted p_T range from 0.5 to 50 GeV/c in minimum bias p+p collisions at $\sqrt{s}$=2.76 and 5.02 TeV are also shown in this table. The corresponding minimum ${\chi }^2$ fit curves of experimental midrapidity $p_T$ spectra of primary charged particles in minimum bias p+p collisions at $\sqrt{s}$=2.76 and 5.02 TeV and minimum bias p+Pb collisions at $\sqrt{s_{nn}}$= 5.02 TeV are demonstrated in Figure 1. As observed from Figure 1, THMTF function in Eq. (6) describes quite well the experimental $p_T$ spectra of primary charged particles in minimum bias p+p collisions at $\sqrt{s}$=2.76 and 5.02 TeV and minimum bias p+Pb collisions at $\sqrt{s_{nn}}$= 5.02 TeV. As seen from Table 2, the parameters of THMTF function remain practically the same, not changing within fit errors in minimum bias p+p collisions at $\sqrt{s}$=2.76 and 5.02 TeV when the fitted p_T range changes from [0.5−15] GeV/c to [0.5−50] GeV/c.

Table 2 demonstrates that the power-index $n$ decreases and momentum scale parameter, $P_0$, slightly increases in minimum bias proton-proton collisions with an increase in center-of-mass collision energy, $\sqrt{s}$, from 2.76 to 5.02 TeV. Decrease of parameter $n$ is obviously caused by the hardening of p_T spectrum because of more frequent high-momentum partonic scatterings at higher proton-proton collision energy ($\sqrt{s}$=5.02 TeV) with respect to lower collision energy ($\sqrt{s}$=2.76 TeV). The increase in $P_0$parameter with an increase in proton-proton collision energy reflects higher energy transfer (higher energy density) and multiple parton scatterings during the initial collision, leading to a higher average transverse momentum at $\sqrt{s}$=5.02 TeV as compared to that at $\sqrt{s}$=2.76 TeV. As seen from Table 2, non-extensivity parameter q increases significantly as proton-proton c.m. collision energy, $\sqrt{s}$, increases from 2.76 to 5.02 TeV. As mentioned earlier, non-extensivity parameter q measures the degree of non-extensivity or deviation of the system from standard Boltzmann-Gibbs thermal equilibrium. As the proton-proton collision energy increases, parameter q typically increases [31,63] because the system moves further away from equilibrium due to several physical reasons. At higher collision energies, particle production is increasingly governed by perturbative QCD (pQCD) processes rather than soft thermal-like interactions. These hard scattering events produce power-law tails in the transverse momentum spectra, which are mathematically represented by a higher q value. The parameter q can also be interpreted as a measure of temperature fluctuations within the system. Higher collision energies lead to more violent and diverse initial conditions, resulting in larger fluctuations of the local "temperature," which pushes parameter q further away from unity. In addition, systems at higher collision energies undergo a more rapid expansion and evolution process. This could mean that the system has less time to thermalize before "freeze-out" occurs, leaving it in a more non-equilibrium state characterized by a larger q value. Because the nuclear modification factor (R_pPb) in p+Pb collisions is close to unity at high p_T values (p_T > 8 GeV/c), the value of parameter $n$ in minimum bias p+Pb collisions at $\sqrt{s_{nn}}$= 5.02 TeV does not deviate significantly from $n$ values in minimum bias p+p collisions at $\sqrt{s}$=2.76 and 5.02 TeV in Table 2. This is obviously due to the nuclear modification factor (R_pPb) in p+Pb collisions being close to unity at high p_T values (p_T > 8 GeV/c), and, hence, similar power-law tails (governed by similar values of parameter $n$) in p+p and p+Pb collisions at comparable c.m. collision energies. The significant increase in $P_0$parameter in minimum bias p+Pb collisions at $\sqrt{s_{nn}}$= 5.02 TeV as compared to that in minimum bias p+p collisions at $\sqrt{s}$=5.02 and 2.76 TeV in Table 2 is obviously due to the stronger radial flow and collective expansion in the larger (p+Pb) nuclear system than that in smaller p+p system. As seen from Table 2, the temperature parameter, T₀, has proved to be significantly larger in minimum bias p+Pb collisions at $\sqrt{s_{nn}}$= 5.02 TeV as compared to that in minimum bias p+p collisions at $\sqrt{s}$=5.02 and 2.76 TeV. This can be explained as follows. In p+Pb collisions, the incoming proton undergoes multiple scatterings with nucleons inside the lead nucleus. This "random walk" of impinging proton increases the transverse momentum of the partons (the Cronin effect), which results in a "harder" spectrum and a higher extracted temperature parameter in the final state as compared to p+p collisions at the same collision energy. The non-extensivity parameter q in minimum bias p+Pb collisions at $\sqrt{s_{nn}}$= 5.02 TeV has proved to be significantly smaller than parameter q in minimum bias p+p collisions at $\sqrt{s}$=5.02 TeV. This is because in p+Pb collisions the larger system size and higher particle density lead to more frequent secondary collisions among the produced particles, which pushes the system closer to thermal equilibrium, resulting in a q value closer to 1. On the contrary, the p+p collision system is smaller and more dilute, meaning that the particles are more likely to escape without sufficient re-scattering to reach equilibrium, leading to a higher q value further away from 1.

The parameters extracted from minimum ${\chi }^2$ fits with two-component model (THMTF) function (Eq. (6)) of experimental midrapidity $p_T$ spectra of primary charged particles in different centrality classes of Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV and Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV for the fitted p_T range from 0.5 to 15 GeV/c are presented in Table 3. It is necessary to mention that the corresponding parameter <β_T> values in Table 3 have been fixed to those extracted earlier [21,36,37] from simultaneous minimum ${\chi }^2$ fits with thermodynamically consistent Tsallis function with transverse flow (Eq. (2)) of experimental midrapidity transverse momentum distributions of the charged pions and kaons, protons and antiprotons in corresponding centrality classes of Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV [21] and Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 TeV [37], and Pb+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV [36] in the fitted p_T range up to 5 GeV/c. The parameter <β_T> values have been fixed during minimum ${\chi }^2$ fit procedures because of the strong negative correlation between the parameters <β_T> and T₀ [21,36,37] of thermodynamically consistent Tsallis function with embedded transverse flow. The resulting fit curves with the THMTF function of experimental midrapidity $p_T$ spectra of primary charged particles in various centralities of Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV for the fitted p_T range from 0.5 to 15 GeV/c are shown in Figure 2. The corresponding results for the fitted p_T range from 0.5 to 50 GeV/c in various centralities of Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV are also demonstrated in Figure 2. The fit curves with the THMTF function of experimental midrapidity $p_T$ spectra of primary charged particles in various centralities of Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV for the fitted p_T range from 0.5 to 15 GeV/c are given in Figure 3. As seen from Figure 2 (a) and Figure 2 (b), the THMTF function in Eq. (6) reproduces quite accurately the experimental midrapidity $p_T$ spectra of primary charged particles in various centralities of Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV in both fitted p_T ranges: p_T = 0.5−15 GeV/c and p_T = 0.5−50 GeV/c. It is necessary to mention that the parameters of THMTF function remain practically the same, not changing within fit errors in all the analyzed centrality classes of Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV when the fitted p_T range changes from [0.5−15] GeV/c to [0.5−50] GeV/c. As observed from Figure 3 (a) and Figure 3 (b), the THMTF function describes quite well the experimental midrapidity $p_T$ spectra of primary charged particles in various centralities of Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV in the fitted p_T range from 0.5 to 15 GeV/c.

To consider the correlation between two sets of parameters x and y, in present work we calculate the Pearson linear correlation coefficient as follows:

$$R(x,y)={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n}(x_i-<x>)·(y_i-<y>)}{\sqrt{{\displaystyle {\sum }_{i=1}^n}{(x_i-<x>)}^2·{\displaystyle {\sum }_{i=1}^n}{(y_i-<y>)}^2}}},$$

(7)

where $<x>={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n}{x}_i}{n}}$ and $<y>={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n}{y}_i}{n}}$ are the average values of the parameters x and y. The Pearson correlation coefficient, $R(x,y)$, being a statistical measure of a linear correlation between two sets of data, changes from −1 to +1. The values $R(x,y)$ = ±1 mean that the dependence between x and y is perfectly described by a linear equation, and all data points (x_i, y_i) are lying on a line in XY plane. The value $R(x,y)$ = 0 implies an absence of a linear correlation between x and y. The positive and negative values of $R(x,y)$ denote the positive and negative (linear) correlation, respectively, between x and y.

To evaluate the uncertainty in the $R(x,y)$ values, we calculate the standard error of Pearson correlation coefficient as

$$s_r=\sqrt{{\displaystyle \frac{1-R^2(x,y)}{n-2}}}$$

(8)

The formula in Eq. (8) has been obtained from an assumption that the data are normally distributed and with the null hypothesis that there is a zero correlation between x and y. Pearson correlation coefficient, $R(x,y)$, and its corresponding standard error between various parameters of the THMTF function (Eq. (6)) extracted (see Table 3) in Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV and Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV in present work are given in Table 4.

The dependencies of the obtained parameters q, T₀, P₀, and n of two-component model (THMTF) function, given in Table 3, on the average number of participant nucleons ($\langle N_{part}\rangle$) in Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV and Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV are illustrated in Figure 4. The corresponding dependencies of the same parameters, given in Table 3, on the average value of charged-particle multiplicity density ($\langle d{N}_{ch}/d\eta \rangle$) in the analyzed collisions are shown in Figure 5. As seen from Figure 4 (a) and Figure 5 (a), non-extensivity parameter q, decreases with an increase in $〈N_{part}〉$ and $\langle d{N}_{ch}/d\eta \rangle$, that is with increasing collision centrality, in both Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV. Parameter q for Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV also shows on the whole a decreasing trend with an increase in $〈N_{part}〉$ and $\langle d{N}_{ch}/d\eta \rangle$ values. This is further supported by the strong negative correlation observed between parameter q and <N_part> as well as between parameter q and <dN_ch/dη> in the analyzed collisions with Pearson correlation coefficient $R(x,y)$ ~ −0.9, as seen in Table 4. This confirms quantitatively that non-extensivity parameter q decreases with increasing $〈N_{part}〉$ and $\langle d{N}_{ch}/d\eta \rangle$ (with an increase in collision centrality) in both Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV. This demonstrates clear increase in degree of system thermalization with increasing centrality of Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV and Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV at the LHC, which agrees with the similar result of previous works [21,36,37].

It is interesting to mention that the parameter q values in Figure 4 (a) coincide within uncertainties in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV in region $〈N_{part}〉$ > 109. As observed from Figure 4 (b), the kinetic freeze-out temperature parameter, T₀, decreases drastically with an increase in $〈N_{part}〉$ up to $〈N_{part}〉$ = 109±8 ($〈N_{part}〉$ = 71±7) values in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV (Pb+Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV), then T₀ parameter does not change, remaining constant within uncertainties, in region $〈N_{part}〉$ > 109 ($〈N_{part}〉$ > 71). Parameter T₀ decreases also drastically with an increase in $〈N_{part}〉$ up to $〈N_{part}〉$ = 68±5 values in Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV. It is interesting to mention that in the analyzed collisions the power-index parameter n shows the similar dependence on collision centrality, $〈N_{part}〉$, in Figure 4 (d) as parameter T₀ in Figure 4 (b). This is further supported by the strong positive linear correlation observed between parameters n and T₀ (R(n, T₀) ~ +0.95) in Table 4. Decreasing significantly with an increase in $〈N_{part}〉$ at low $〈N_{part}〉$ values, parameter n does not change, reaching a plateau within uncertainties starting from $〈N_{part}〉$ ≈ 109±8, $〈N_{part}〉$ ≈ 71±7, and $〈N_{part}〉$ ≈ 68±5 in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV, and in Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV, respectively. Parameter n reflects “hardness” of the spectrum and corresponding production mechanisms of high $p_T$ hadrons, being directly connected to the number of quarks (gluons) involved in initial hard scattering and in-medium energy loss (jet quenching) of produced high $p_T$ partons. Therefore, the observed $〈N_{part}〉$ dependencies of the parameter n in Figure 4 (d) coupled with similar T₀ versus $〈N_{part}〉$ dependencies in Figure 4 (b) could indicate that $〈N_{part}〉$ ≈ 109±8, $〈N_{part}〉$ ≈ 71±7, and $〈N_{part}〉$ ≈ 68±5 are estimated border (threshold) values of collision centrality for crossover phase transition from the dense hadronic state to that of QGP in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV, and in Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV, respectively. These probable border (threshold) values of collision centrality for crossover phase transition estimated in present work have proved to be compatible with those estimated in Refs. [21,36,37] from combined (simultaneous) model analysis of the evolution of the experimental midrapidity transverse momentum distributions of the charged pions and kaons, protons and antiprotons with changing collision centrality in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV, and in Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV. The results for Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV also agree well with those of recent work [41].

As observed from Figure 4 (c) and Figure 5 (c), momentum scale parameter, P₀, of the inverse power-law component of THMTF function in Eq. (6) increases consistently with an increase in both $〈N_{part}〉$ and $\langle d{N}_{ch}/d\eta \rangle$ (with an increase in collision centrality) in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV, and in Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV. This is further supported by the strong positive correlation observed between parameter P₀ and <N_part> as well as between parameter P₀ and <dN_ch/dη> in the analyzed collisions with Pearson correlation coefficient $R(x,y)$ > 0.9, as seen in Table 4. As seen from Figure 4 (c), parameter P₀ demonstrates quite high rate of increase with $〈N_{part}〉$ in region up to $〈N_{part}〉$ ≈ 100 and then its rate of increase slows down significantly in region $〈N_{part}〉$ > 100. As can be seen from Table 4, parameter P₀ and average transverse (radial) flow velocity, <β_T>, demonstrate very strong positive linear correlation in all analyzed Xe+Xe and Pb+Pb collisions at the LHC with Pearson correlation coefficient $R(x,y)$ ≈ +1. Parameter P₀ values in Xe+Xe and Pb+Pb collisions at the LHC, given in Table 3, have proved to be significantly larger than P₀ values in similar c.m. collision energies (per nucleon pair) in minimum bias proton-proton and minimum bias p+Pb collisions at the LHC, given in Table 2. This can serve as a signature of the strong radial flow in Xe+Xe and Pb+Pb collisions at the LHC, which pushes (boosts) particles to higher p_T values and therefore shifts the transition point from “soft” to “hard” part of the spectrum towards larger values of p_T as compared to proton-proton collisions. With an increase in centrality of Xe+Xe and Pb+Pb collisions at the LHC, the pressure gradient in the hot and dense overlap zone of the colliding nuclei increases significantly, resulting in the stronger collective flow in more central collisions. This could explain consistent increase of parameter P₀ (similarly with <β_T>) with an increase in both $〈N_{part}〉$ and $\langle d{N}_{ch}/d\eta \rangle$ (with an increase in collision centrality) observed in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV, and in Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV in Figure 4 (c) and Figure 5 (c). The values of P₀ in Figure 4 (c) and Figure 5 (c) have proved to be very close to each other in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV and Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV at the same values of $〈N_{part}〉$ as well as $\langle d{N}_{ch}/d\eta \rangle$. This is due to closeness of c.m. collision energies (per nucleon pair) and hence closeness of pressure gradients and corresponding energy densities of Xe+Xe and Pb+Pb collisions at the same $〈N_{part}〉$ as well as $\langle d{N}_{ch}/d\eta \rangle$ values. On the other hand, as seen from Figure 4 (c) and Figure 5 (c), values of P₀ in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV have proved to be significantly smaller than P₀ in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV and Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV at the same values of $〈N_{part}〉$ as well as $\langle d{N}_{ch}/d\eta \rangle$. This is obviously due to the significantly smaller c.m. collision energy (per nucleon pair) of Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV as compared to Pb+Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV and Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV.

The power-index n, which describes “hardness” of p_T spectrum and reflects high p_T hadron production, also shows clear dependence on c.m. collision energy (per nucleon pair) in Figure 4 (d) and Figure 5 (d). The parameter n values have proved to be systematically and significantly larger in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV as compared to n values in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV and Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV at the same values of $〈N_{part}〉$ as well as $\langle d{N}_{ch}/d\eta \rangle$. As collisions move from $\sqrt{s_{nn}}$ = 2.76 TeV to $\sqrt{s_{nn}}$ = 5.02 TeV or $\sqrt{s_{nn}}$=5.44 TeV, the parameter n generally decreases, signifying a power-law exponent that produces harder spectra, consistent with higher effective temperatures, due to the harder initial parton scatterings, which produce larger number of high p_T hadrons making high p_T spectrum flatter at higher ($\sqrt{s_{nn}}$ = 5.02 TeV or $\sqrt{s_{nn}}$=5.44 TeV) collision energy as compared to the lower ($\sqrt{s_{nn}}$ = 2.76 TeV) one. The values of parameter n in Figure 4 (d) and Figure 5 (d) have proved to be very close to each other in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV and Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV at the same values of $〈N_{part}〉$ as well as $\langle d{N}_{ch}/d\eta \rangle$, which is due to closeness of c.m. collision energies (per nucleon pair) of Pb+Pb and Xe+Xe collisions. It is interesting to mention that the values of the parameter n extracted in minimum bias p+p collisions at $\sqrt{s}$=2.76 and 5.02 TeV (see Table 2) have coincided within uncertainties with the corresponding n values in most peripheral ((70-80)% centrality) Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV (see Table 3), respectively. This is because only few participant nucleons at the edges of the nuclei actually collide in a peripheral collision. This makes the peripheral collision event with too small overlap volume as a collection of independent binary nucleon-nucleon collisions, where produced particles, including high p_T partons, escape without significant loss of energy or flow as contrasted to central heavy-ion collisions with quite large and dense overlap zone resulting in significant medium effects (jet quenching and parton energy loss) and strong radial flow. Using model expression [62,75] for power-index n in high-energy collisions, $n=2n_{\alpha }-4$, where $n_{\alpha }$ is the number of partons participating in a scattering, we can estimate $n_{\alpha }$ in the analyzed collisions. As can be seen from Table 2 and Table 3, the average value of $n$ is approximately 6 in the analyzed collision types. The value $n=6$ corresponds to the number of participant partons, $n_{\alpha }$=5. Considering that most of primary charged hadrons in the analyzed collisions at the LHC [19,25,74] are charged pions (~ 85%) and kaons (~ 10%), $n_{\alpha }$=5 corresponds to $g+q\to meson+q$ and $g+g\to meson+g$ (here g is the gluon and q − quark) gluon fragmentation reactions as the dominant mechanism for high p_T hadron production in the analyzed p+p, p+Pb, Xe+Xe, and Pb+Pb collisions at the LHC. This result agrees with earlier result [76] stating that gluon fragmentation is the dominant mechanism for producing high p_T mesons, particularly in the central rapidity region, in high-energy collisions at the LHC. This can be explained by that the gluons carry a higher color charge (3) as compared to that of quarks (4/3), making them more likely to radiate and fragment into a larger shower of particles. Besides it, the cross-sections for gluon-gluon ($g+g$) and quark-gluon ($q+g$) scatterings are significantly higher than that for quark-quark ($q+q$) scattering at a few TeV collision energy ($\sqrt{s_{nn}}$) range at the LHC.

4. Summary and Conclusions

To reproduce accurately the measured long $p_{\mathrm{T}}$ ranges of the midrapidity transverse momentum distributions of charged particles at various centralities of high-energy heavy-ion collisions at the LHC, the two-component model function, which is the sum of thermodynamically consistent Tsallis function with embedded transverse flow and inverse power-law function (which is asymptotic of Hagedorn function at high p_T values), has been proposed. The first and second terms of this two-component model function are assumed to describe the “soft” and “hard” components, respectively, of midrapidity transverse momentum distributions of hadrons in high-energy collisions. This two-component model function, called Tsallis-Hagedorn model with transverse flow (THMTF), has described quite well the experimental midrapidity transverse momentum spectra of primary charged particles in the whole $p_{\mathrm{T}}$ range (up to 50 GeV/c), measured by ALICE Collaboration in minimum bias p+p collisions at $\sqrt{s}$=2.76 and 5.02 TeV [72], minimum bias p+Pb collisions at $\sqrt{s_{nn}}$= 5.02 TeV [72], and nine collision centrality intervals of Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV [71] and Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV [72]. We have analyzed the dependencies of the extracted parameters of THMTF function on Xe+Xe and Pb+Pb collision centrality at the LHC, expressed by $〈N_{part}〉$ as well as $\langle d{N}_{ch}/d\eta \rangle$, and c.m. collision energy (per nucleon pair). The midrapidity experimental p_T spectra of primary charged hadrons produced in minimum bias proton-proton (at $\sqrt{s}$=2.76 and 5.02 TeV) and p+Pb (at $\sqrt{s_{nn}}$= 5.02 TeV) collisions at the LHC, measured by ALICE collaboration, have been used as the reference data for a comparison.

The power-index $n$ has decreased and parameter $P_0$ slightly increased in minimum bias proton-proton collisions with an increase in center-of-mass collision energy, $\sqrt{s}$, from 2.76 to 5.02 TeV. Decrease of parameter $n$ could be explained by the hardening of p_T spectrum due to more frequent high-momentum partonic scatterings at higher proton-proton collision energy ($\sqrt{s}$=5.02 TeV) with respect to lower collision energy ($\sqrt{s}$=2.76 TeV). The observed increase in momentum scale parameter, $P_0$, with an increase in $\sqrt{s}$ could reflect higher energy transfer (higher energy density) and multiple parton scatterings during the initial collision, leading to a higher average transverse momentum at $\sqrt{s}$=5.02 TeV as compared to that at $\sqrt{s}$=2.76 TeV.

The non-extensivity parameter q in minimum bias p+Pb collisions at $\sqrt{s_{nn}}$= 5.02 TeV has proved to be significantly smaller than parameter q in minimum bias p+p collisions at $\sqrt{s}$=5.02 TeV, because the larger system size and higher particle density in p+Pb collisions lead to more frequent secondary collisions among the produced particles, which pushes the system closer to thermal equilibrium, resulting in a q value closer to 1.

Parameter q has shown a decreasing trend with an increase in $〈N_{part}〉$ and $\langle d{N}_{ch}/d\eta \rangle$ values, demonstrating clear increase in degree of system thermalization with increasing centrality of Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV and Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV at the LHC, in agreement with the similar result of previous works. This has been further supported by the strong negative correlation observed between parameter q and <N_part> as well as between parameter q and <dN_ch/dη> in the analyzed collisions with corresponding Pearson correlation coefficient $R(x,y)$ ~ −0.9.

Decreasing significantly with an increase in $〈N_{part}〉$ at low $〈N_{part}〉$ values, parameters T₀ and n of the THMTF function have not changed, reaching a plateau within uncertainties starting from $〈N_{part}〉$ ≈ 109±8, $〈N_{part}〉$ ≈ 71±7, and $〈N_{part}〉$ ≈ 68±5 in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV, and in Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV, respectively. The analogous $〈N_{part}〉$ dependencies of the power-index n and kinetic freeze-out temperature parameter, T₀ , have been further supported by the strong positive linear correlation observed between parameters n and T₀ (R(n, T₀) ~ +0.95).

Parameter n reflects “hardness” of the spectrum (and encompasses the corresponding production mechanisms of high $p_T$ hadrons), being directly related to the number of quarks (gluons) involved in initial hard scattering and in-medium energy loss (jet quenching) of produced high $p_T$ partons. Therefore, the observed $〈N_{part}〉$ dependencies of the parameter n coupled with analogous T₀ versus $〈N_{part}〉$ dependencies could indicate that $〈N_{part}〉$ ≈ 109±8, $〈N_{part}〉$ ≈ 71±7, and $〈N_{part}〉$ ≈ 68±5 are estimated border values of collision centrality for probable crossover phase transition from the dense hadronic state to that of QGP in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV, and in Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV, respectively. These probable border (threshold) values of collision centrality for crossover phase transition estimated in present work have proved to be compatible with those evaluated recently in Refs. [21,36,37] from combined (simultaneous) model analysis of evolution of the experimental midrapidity transverse momentum distributions of the charged pions and kaons, protons and antiprotons with changing collision centrality in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV, and in Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV. The results for Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV also agree well with those of recent work [41].

Momentum scale parameter, P₀, of the inverse power-law component of THMTF function has increased consistently with an increase in both $〈N_{part}〉$ and $\langle d{N}_{ch}/d\eta \rangle$ in Pb+Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV, and in Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV, which could likely be due to the higher pressure gradient in overlap zone and, hence, stronger radial flow developed in more central collisions. Parameter P₀ has demonstrated quite high rate of increase with $〈N_{part}〉$ in region up to $〈N_{part}〉$ ≈ 100 and then its rate of increase has slowed down significantly in region $〈N_{part}〉$ > 100. Parameter P₀ values in Xe+Xe and Pb+Pb collisions at the LHC have proved to be significantly larger than P₀ values in similar c.m. collision energies (per nucleon pair) in minimum bias proton-proton and minimum bias p+Pb collisions at the LHC, which could serve as a signature of significantly stronger radial flow developed in Xe+Xe and Pb+Pb collisions as compared to p+Pb collisions at the LHC, which pushes (boosts) particles to higher p_T values and therefore shifts the transition point from “soft” to “hard” part of the spectrum towards larger values of p_T. Parameter P₀ and average transverse (radial) flow velocity, <β_T>, have demonstrated very strong positive linear correlation in analyzed Xe+Xe and Pb+Pb collisions at the LHC with Pearson correlation coefficient being R(<β_T>, P₀) ≈ +1.

Both P₀ and n parameters of the inverse power-law component of THMTF function have shown clear $\sqrt{s_{nn}}$ dependencies in Xe+Xe and Pb+Pb collisions at the LHC. The P₀ parameter has increased and parameter n decreased with an increase in $\sqrt{s_{nn}}$. As collisions move from $\sqrt{s_{nn}}$ = 2.76 TeV to $\sqrt{s_{nn}}$ = 5.02 TeV or $\sqrt{s_{nn}}$=5.44 TeV, the parameter n generally decreases because of the hardening of $p_T$ spectra, consistent with higher effective temperatures, due to the harder initial parton scatterings, which produce larger number of high p_T hadrons at higher $\sqrt{s_{nn}}$.

Using the average value of power-index $n$ ≈ 6 extracted in present work, we have estimated the average number of partons participating in a scattering, $n_{\alpha }$=5, in the analyzed collisions. Taking into account that most of primary charged hadrons in the analyzed collisions at the LHC are charged pions (~ 85%) and kaons (~ 10%), $n_{\alpha }$=5 matches well with $g+q\to meson+q$ and $g+g\to meson+g$ gluon fragmentation reactions as the dominant mechanism for high p_T hadron production in the analyzed p+p, p+Pb, Xe+Xe, and Pb+Pb collisions at the LHC. This result agrees with earlier result that gluon fragmentation is the dominant mechanism for producing high p_T mesons, particularly in the central rapidity region, in high-energy collisions at the LHC.