Simultaneous Analysis of Midrapidity $p_T$ Spectra of Identified Particle Species in Pb + Pb Collisions at $\sqrt{s_{nn}}$ = 2.76 TeV Using Tsallis Distribution with Transverse Flow

Abstract

The midrapidity transverse momentum distributions of the charged pions, kaons, protons, and antiprotons in ten groups of centrality of Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV, measured by the ALICE Collaboration, have been analyzed successfully using both thermodynamically consistent and non-consistent Tsallis distribution functions with transverse flow. The collision centrality dependencies of the extracted parameters of two kinds of Tsallis functions with transverse flow have been investigated. The significantly different behavior (growth rates) of ⟨β_T⟩ in regions $⟨N_{part}⟩$ < 71 and $⟨N_{part}⟩$ > 71 with the temperature T₀ becoming constant in region $⟨N_{part}⟩$ > 71 has been observed. This could indicate that $⟨N_{part}⟩$ = 71 ± 5 (corresponding to $⟨d{N}_{ch}/d\eta ⟩$ = 205 ± 15) is a threshold border value of collision centrality for crossover phase transition from the dense hadronic state to the QGP state (or a mixed state of QGP and hadrons) in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV. This conjecture is supported further by the observed, significantly different correlations between T₀ and $⟨{\beta }_T⟩$ parameters in the corresponding $⟨{\beta }_T⟩$ < 0.44 and $⟨{\beta }_T⟩$ > 0.44 ranges. The strong positive linear correlation between non-extensivity parameter q for pions and kaons, between q for pions and (anti)protons, and between q for kaons and (anti)protons has been obtained. The parameter q for all studied particle species has proven to be strongly anticorrelated with the average transverse flow velocity, ⟨β_T⟩. Quite a large positive linear correlation has been obtained between the q of the studied particle species and temperature parameter T₀. Analysis of q versus $⟨N_{part}⟩$ dependencies for the studied particle species suggests that the highly thermalized and equilibrated QGP is produced in central Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV with $⟨N_{part}⟩$ > 160.

1. Introduction

There is great importance and interest in investigating the properties of the short-lived and deconfined state of almost-free quarks and gluons, called quark–gluon plasma (QGP), which can be produced in high-energy nucleus–nucleus collisions. It is believed that the universe was in a state of QGP a few microseconds after the so-called Big Bang, considered to be the starting point in the birth of our universe. The striking evidence about the creation of QGP in high-energy heavy-ion collisions was reported in early works seen in Refs. [1,2,3,4,5,6,7,8,9]. It was found that the produced QGP was characterized by a fluid-like behavior with low viscosity. The modern experiments on high-energy heavy ions as well as proton–proton collisions at the RHIC (Relativistic heavy-ion collider, Brookhaven National Laboratory (BNL), Upton, NY, USA) and LHC (Large hadron collider, CERN, Switzerland) produce and measure tens of millions of collisions for the large HEP (high energy physics) community of scientists from around the world to analyze, and reveal various collective properties of the hot and dense matter, which has a very short lifetime of the order of 10⁻²³ s. Naturally, the properties of the created QGP and swiftly expanding matter found afterwards are deduced from the analysis of the kinematical characteristics of the final particles reaching the detectors with the help of the theoretical and phenomenological approaches, including those based on thermal and hydrodynamic models, and various models of extensive and non-extensive statistics. The hot and dense matter swiftly expands right after QGP’s creation, and it passes through the subsequent chemical and kinetic freeze-out stages, which fix the content of particle species and the final momenta of these particles, respectively. The large statistics, expressed by the huge number of particles produced in high-energy heavy-ion, and proton–proton collisions in the modern collider experiments, justifies the use of various statistical models and approaches for the HEP data analysis.

The chemical freeze-out parameters, such as the temperature T_ch and the baryochemical potential μ_b at the moment of chemical freeze-out are extracted by performing fits of the particle abundancies and their ratios with the thermal, statistical hadronization models [10,11,12,13,14,15]. The kinetic freeze-out parameters, such as the transverse flow velocity and the corresponding temperature at the moment of kinetic freeze-out, are obtained [10,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33] from fitting the transverse momentum ($p_T$) distributions of particles with the help of various theoretical and phenomenological model functions, including the hydro-inspired blast-wave models.

The non-extensive statistical distributions, particularly various forms of non-extensive Tsallis distribution functions, have been widely used to describe the $p_T$ distributions of particles in high-energy collisions [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]. Different kinds of Tsallis functions were extremely successful in describing the $p_T$ spectra of particles in high-energy proton–proton collisions up to the highest $p_T$ values measured at the RHIC and LHC experiments. The advantage of using the Tsallis distribution function is in its connection, via entropy, to thermodynamics [43], which is not the case with many other power law distributions. The important parameters of the Tsallis distribution function are the effective temperature (T) and the non-extensivity parameter q, which measures the degree of deviation of $p_T$ distribution from the exponential Boltzmann–Gibbs distribution. The q parameter is also said to characterize the degree of non-thermalization or non-equilibrium of the system [47]. Furthermore, the parameters q and T of the Tsallis function can be employed to identify the system size scaling and initial conditions [49]. Due to the effective temperature containing the contribution of both the chaotic thermal motion and transverse radial flow, the transverse flow should be embedded into the Tsallis distribution function in order to disentangle these two effects. For the description of the $p_T$ spectra of particles in high-energy heavy-ion collisions, different transverse expansion models are embedded into Tsallis statistics. To extract the kinetic freeze-out temperature and transverse flow velocity, different model functions have been used [10,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]: the Blast-Wave model with Boltzmann–Gibbs statistics (the BGBW model), the Blast-Wave model with Tsallis statistics (the TBW model), the improved Tsallis distribution with flow effect, Hagedorn formula with the embedded transverse flow, and recently [18,33], the Tsallis distribution function with included transverse flow.

The majority of works devoted to the analysis of $p_T$ spectra of particles produced in high-energy collisions with the help of various theoretical model functions, including Tsallis function and its modified forms, have separately fitted the $p_T$ distribution of each particle type in a given collision. As mentioned in Refs. [16,19,22,28,29], it is not possible to extract the collective properties of a system, such as kinetic freeze-out temperature or transverse flow velocity, from separate fits to the $p_T$ spectrum of each particle type. Only the theoretical model fits applied simultaneously to $p_T$ spectra of different particle species by assigning, for example, the transverse flow velocity and temperature as the common (global) fitted parameters for all particle types can yield the meaningful physical parameters of the studied collision system [16,19,22,28,29]. The collective parameters of the systems produced in high-energy collisions have been extracted and analyzed successfully by applying simultaneous (combined) model fits with just few parameters in Refs. [10,16,17,18,19,22,27,28,29,48].

This study continues from our recent papers [17,18,19,33] devoted to analysis, applying simultaneous theoretical model fits, of the centrality dependencies of transverse momentum spectra of the charged pions, kaons, protons, and antiprotons produced in high-energy proton–proton, Xe + Xe, and Pb + Pb collisions at the LHC, measured by ALICE Collaboration. Since the pions, kaons, and (anti)protons represent the largest predominant part of the final particles coming from high-energy proton–proton and heavy-ion collisions, we believe that the simultaneous analysis of these particle species can reveal the collective properties of the system produced in these collisions. In Refs. [17,19] we investigated the dependencies of midrapidity $p_T$ spectra of the charged pions, kaons, protons, and antiprotons on charged-particle multiplicity density (<dN_ch/dη>) in proton–proton collisions at $\sqrt{s}$ = 7 and 13 TeV, respectively. The combined minimum χ² fits with the thermodynamically consistent Tsallis function and Hagedorn function with transverse flow could quite well reproduce the $p_T$ distributions of the particle species in the analyzed groups of <dN_ch/dη> in proton–proton collisions at $\sqrt{s}$ = 7 and 13 TeV [17,19]. From the analysis of the <dN_ch/dη> dependencies of the extracted kinetic freeze-out temperature, T₀, and average transverse flow velocity, $\langle {\beta }_T\rangle$, it was estimated that the probable onset of the deconfinement phase transition in proton–proton collisions at $\sqrt{s}$ = 7 and 13 TeV occurs at <dN_ch/dη> ≈ 6.1 ± 0.3 [17] and <dN_ch/dη> ≈ 7.1 ± 0.2 [19], respectively. These estimates proved to be consistent with the calculations of Campanini in Refs. [50,51,52], made using a completely different method for high-energy proton–proton collisions, and theoretical predictions [52] in case of crossover transition from hadronic gas to the QGP phase starting at (dN_ch/dη) ≈ 6. The <dN_ch/dη> dependence of the effective temperature T of the Tsallis function with thermodynamical consistence in proton–proton collisions at both $\sqrt{s}$ = 7 and 13 TeV was described quite well by the simple power function, $T~<\frac{d{N}_{ch}}{d\eta }{>}^{1/3}$, with the same (≈1/3) exponent parameter [17,19].

In Refs. [18,33] we studied the dependencies of midrapidity $p_T$ spectra of the charged pions, kaons, protons, and antiprotons, measured by ALICE Collaboration, 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 $\sqrt{s_{nn}}$ = 5.02 TeV, respectively. The combined minimum χ² fits with thermodynamically consistent and non-consistent Tsallis function with embedded transverse flow could describe quite well the $p_T$ distributions of the studied particle species in various collision centralities, defined by $\langle N_{part}\rangle$, in Xe + Xe and Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.44 TeV and 5.02 TeV, respectively. From the analysis of the $\langle N_{part}\rangle$ dependencies of the extracted kinetic freeze-out temperature, T₀, and average transverse flow velocity, $\langle {\beta }_T\rangle$, it was deduced that $\langle N_{part}\rangle$ ≈ 44 ± 5 [18] and $\langle N_{part}\rangle$ ≈ 71 ± 7 [33] could be the threshold border values of collision centrality for a crossover transition from a dense hadronic state to the QGP phase (or mixed phase of QGP and hadrons) in Xe + Xe and Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.44 TeV and 5.02 TeV, respectively.

The present work is devoted to the analysis of midrapidity transverse momentum spectra of the charged pions, kaons, protons, and antiprotons at various centralities of Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV, measured by ALICE Collaboration [16]. We will implement the simultaneous minimum χ² model fits of the experimental midrapidity p_t distributions of the analyzed particle species in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV, using thermodynamically consistent as well as non-consistent Tsallis distribution functions with embedded transverse flow, as it was performed in our recent works [18,33] to analyze the midrapidity $p_T$ spectra of the same particle species in Xe + Xe and Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.44 TeV and 5.02 TeV, respectively. This will allow us to directly compare these three collision systems in order to find new regularities in the collision centrality and energy dependencies of the extracted system parameters.

2. The Data and Models

In this paper, we analyze the transverse momentum distributions of the charged pions, kaons, protons, and antiprotons produced at midrapidity (|y| < 0.5) at 10 centrality intervals in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV, measured by ALICE Collaboration and presented in Ref. [16]. The $p_T$ ranges measured by ALICE Collaboration have been as follows: [0.1–3.0] GeV/c for ${\pi }^{+}$ + ${\pi }^{-}$, [0.2–3.0] GeV/c for $K^{+}$ + $K^{-}$, and [0.3–4.6] GeV/c for p + $\overline{p}$. The averages of the number of participant nucleons ($\langle N_{part}\rangle$) and charged-particle (pseudo-rapidity) multiplicity density ($\langle d{N}_{ch}/d\eta \rangle$) calculated [16,53] using the Glauber–Monte Carlo model in the analyzed ten centrality groups of Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV are shown in

Table 1. The average number of participant nucleons and mean charged-particle multiplicity densities [16,53] in the analyzed groups of centralities of Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV.

Centr.	$\langle N_{part}\rangle$	$\langle d{N}_{ch}/d\eta \rangle$
0–5%	382 ± 14	1601 ± 60
5–10%	328 ± 13	1294 ± 49
10–20%	260 ± 10	966 ± 37
20–30%	187 ± 7	649 ± 23
30–40%	130 ± 5	426 ± 15
40–50%	87 ± 3	261 ± 9
50–60%	54 ± 2	149 ± 6
60–70%	31 ± 2	76 ± 4
70–80%	16 ± 2	35 ± 2
80–90%	7 ± 1	13 ± 2

.

There are several modifications [27,38,39,43,44,45,46,54,55] of the Tsallis distribution function, which provide comparably equal quality of fits of $p_T$ distributions of particles produced in high-energy collisions. One of the simple variants [45] of Tsallis function is represented at midrapidity (mid-y) (<y> = 0) as:

$$\frac{d^{2}N}{2\pi N_{ev}{p}_{T}d{p}_{T}dy}=C{\left(1+\left(q-1\right)\frac{m_T}{T}\right)}^{-\frac{q}{q-1}},$$

(1)

where C is the normalization constant, N_ev—is the total number of the inelastic events, $m_T=\sqrt{p_T^2+m_0^2}$ is the transverse mass (energy), and $m_0$—the rest mass of a hadron. The parameter T is the effective temperature, and q is the non-extensivity parameter. We call this function in Equation (1) the thermodynamically non-consistent Tsallis distribution or simple (non-consistent) Tsallis function in the present work. The q is an important parameter, which accounts for the deviation of the $p_T$ distribution from the exponential Boltzmann–Gibbs distribution. At the limit q → 1, the Tsallis function reduces to the exponential, or equilibrated, Boltzmann–Gibbs distribution. The closer the parameter q is to one (1), the more equilibrated and thermalized the system is.

The following version [38,43,44,45] of the Tsallis distribution derived at zero chemical potential (μ is approximately zero at the highest (TeV range) collision energies at the LHC) at mid-y satisfies the thermodynamical expressions for the particle number, temperature, pressure, entropy, and energy densities:

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

(2)

which we call a Tsallis function with thermodynamical consistence or (thermodynamically) consistent Tsallis function (distribution) in the present work. In Ref. [38] the proof of the thermodynamical consistence of the function in Equation (2) was presented. The fitting constant, $C_q$, is thought to be related linearly to the volume (V) of the system [38,43] via the relation C_q = gV/(2π)³, with g denoting the degeneracy factor. The degeneracy factor g is equal to two, three, or four for protons, pions, and kaons, respectively.

The effective temperature parameter T in Equations (1) and (2) contains the contribution of both the chaotic thermal motion and transverse flow. As already mentioned, the transverse flow velocity is incorporated into the Tsallis function in order to disentangle these two effects.

In Refs. [17,19,27,28,29,48] the transverse expansion velocity has been embedded into a QCD-inspired Hagedorn function, $\frac{d^{2}N}{2\pi N_{ev}{p}_{T}d{p}_{T}dy}=C_n{\left(1+\frac{m_T}{p_0}\right)}^{-n}$, by substituting $p_0=n{T}_0$ and implementing the Lorentz transformation $m_T\to \langle {\gamma }_T\rangle \left(m_T-p_T\langle {\beta }_T\rangle \right)$ to arrive, finally, at the Hagedorn function with (embedded) transverse flow:

$$\frac{d^{2}N}{2\pi N_{ev}{p}_{T}d{p}_{T}dy}=C_n{\left(1+\langle {\gamma }_T\rangle \frac{\left(m_T-p_T\langle {\beta }_T\rangle \right)}{n{T}_0}\right)}^{-n}.$$

(3)

Here, $C_n$ is the normalization constant, $〈{\gamma }_T〉=1 /\sqrt{1-{〈{\beta }_T^{}〉}^2}$, and $\langle {\beta }_T\rangle$ is the average transverse flow velocity; T₀ estimates the kinetic freeze-out temperature, and $n$ is the free parameter. The substitution $m_T\to \langle {\gamma }_T\rangle \left(m_T-p_T\langle {\beta }_T\rangle \right)$, applied to derive Equation (3), denotes simply the Lorentz transformation to a system co-moving with the average flow velocity, $\langle {\beta }_T\rangle$, of particles in the transverse plane assuming the existence of such a flow of particles in this co-moving frame [27]. The $p_T$ distributions of particles in proton+proton and heavy-ion collisions at the RHIC and LHC were reproduced quite well [17,19,27,28,29,48] by the combined fits with the Hagedorn function with transverse flow (Equation (3)), and the extracted parameters could be interpreted physically.

To describe midrapidity $p_T$ distributions of identified charged particles in various centralities of Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV in the present work, we incorporate the transverse flow velocity into the Tsallis function in Equation (1) by implementing, as was performed for the first time in Ref. [18], the Lorentz transformation $m_T\to \langle {\gamma }_T\rangle \left(m_T-p_T\langle {\beta }_T\rangle \right)$:

$$\frac{d^{2}N}{2\pi N_{ev}{p}_{T}d{p}_{T}dy}=C {\left(1+\langle {\gamma }_T\rangle \frac{\left(q-1\right)\left(m_T-p_T\langle {\beta }_T\rangle \right)}{T_0 }\right)}^{-q/\left(q-1\right)},$$

(4)

which we call the non-consistent or simple Tsallis function with transverse flow in present analysis.

Similarly, we embed the transverse flow into Tsallis distribution with thermodynamical consistence in Equation (2) by substituting $m_T\to \langle {\gamma }_T\rangle \left(m_T-p_T\langle {\beta }_T\rangle \right)$ to obtain [18] the function

$$\frac{d^{2}N}{2\pi N_{ev}{p}_{T}d{p}_{T}dy}=C_q\langle {\gamma }_T\rangle \left(m_T-p_T\langle {\beta }_T\rangle \right){\left(1+\langle {\gamma }_T\rangle \frac{\left(q-1\right)\left(m_T-p_T\langle {\beta }_T\rangle \right)}{T_0 }\right)}^{-q/\left(q-1\right)},$$

(5)

which we call a thermodynamically consistent or consistent Tsallis function with transverse flow in present analysis.

The combined (simultaneous) fits by the model functions in Equations (4) and (5) of midrapidity $p_T$ distributions of the analyzed particle species in each centrality group have been performed with the help of Nonlinear Curve Fitting of the Origin 9.1 Graphing and Data Analysis Software. During the combined fitting procedures, the parameters T₀ and $\langle {\beta }_T\rangle$ have been the common (shared) system parameters for the analyzed particle species. The error bars of the experimental data points, shown in the figures, represent the combined statistical and systematic errors (added in quadrature). These combined errors are mostly defined by the systematic ones with negligible statistical uncertainties. The details on the calculation of the systematic errors are given in Ref. [16]. The combined errors have been used to define the weights (1/(error)²) of the data points during the minimum χ² fitting procedures. The $p_T$ intervals measured by ALICE Collaboration in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV are [16]: [0.1–3.0] GeV/c for ${\pi }^{+}$ + ${\pi }^{-}$, [0.2–3.0] GeV/c for $K^{+}$ + $K^{-}$, and [0.3–4.6] GeV/c for p + $\overline{p}$. The region $p_T$ < 0.5 GeV/c in pion spectra is excluded from the fitting procedures, as done previously in Refs. [16,17,18,19,22,28,29,33,48], due to the significant contribution to pion production from the decays of baryon resonances in this range. Hence, in the present work we use the following $p_T$ intervals for combined minimum χ² fits with the model functions in Equations (4) and (5): [0.5–3.0] GeV/c for ${\pi }^{+}$ + ${\pi }^{-}$, [0.2–3.0] GeV/c for $K^{+}$ + $K^{-}$, and [0.3–4.6] GeV/c for p + $\overline{p}$.

3. Analysis and Results

The results obtained from combined minimum ${\chi }^2$ fitting procedures using thermodynamically consistent and non-consistent Tsallis functions with transverse flow (functions in Equations (5) and (4), respectively) are presented in

Table 2. The results obtained from combined minimum ${\chi }^2$ fits with thermodynamically consistent Tsallis function with transverse flow (Equation (5)) of midrapidity $p_T$ spectra of particles in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV. n.d.f. denotes the number of degrees of freedom.

Centrality	$q ({\pi }^{+}+{\pi }^{-})$	$q (K^{+}+K^{-})$	$q (p+\overline{p})$	T0 (MeV)	$\langle {\beta }_T\rangle$	${\chi }^2/n.d.f. (n.d.f.)$
0–5%	1.084 ± 0.004	1.083 ± 0.004	1.082 ± 0.002	78 ± 3	0.60 ± 0.01	1.21 (100)
5–10%	1.087 ± 0.004	1.086 ± 0.004	1.085 ± 0.002	78 ± 3	0.58 ± 0.01	1.18 (100)
10–20%	1.091 ± 0.004	1.089 ± 0.004	1.086 ± 0.002	78 ± 3	0.58 ± 0.01	1.10 (100)
20–30%	1.097 ± 0.004	1.094 ± 0.004	1.089 ± 0.002	78 ± 3	0.55 ± 0.01	0.99 (100)
30–40%	1.103 ± 0.004	1.099 ± 0.003	1.092 ± 0.002	78 ± 3	0.52 ± 0.01	0.89 (100)
40–50%	1.111 ± 0.003	1.108 ± 0.003	1.098 ± 0.002	78 ± 3	0.47 ± 0.01	0.65 (100)
50–60%	1.118 ± 0.003	1.115 ± 0.003	1.103 ± 0.002	80 ± 3	0.40 ± 0.01	0.54 (100)
60–70%	1.126 ± 0.003	1.124 ± 0.002	1.108 ± 0.001	80 ± 3	0.33 ± 0.01	0.37 (100)
70–80%	1.133 ± 0.002	1.133 ± 0.002	1.112 ± 0.001	82 ± 3	0.24 ± 0.01	0.25 (100)
80–90%	1.139 ± 0.002	1.143 ± 0.002	1.114 ± 0.001	84 ± 3	0.12 ± 0.03	0.24 (101)

and

Table 3. The results obtained from combined minimum ${\chi }^2$ fits with thermodynamically NON-consistent Tsallis function with transverse flow (Equation (4)) of midrapidity $p_T$ spectra of particles in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV.

Centrality	$q ({\pi }^{+}$$+ {\pi }^{-})$	$q (K^{+}$$+ K^{-})$	$q (p+\overline{p})$	T0 (MeV)	$\langle {\beta }_T\rangle$	${\chi }^2/n.d.f. (n.d.f.)$
0–5%	1.068 ± 0.006	1.068 ± 0.005	1.073 ± 0.003	120 ± 5	0.59 ± 0.01	1.02 (100)
5–10%	1.073 ± 0.006	1.072 ± 0.006	1.077 ± 0.003	121 ± 5	0.58 ± 0.01	1.01 (100)
10–20%	1.077 ± 0.006	1.076 ± 0.005	1.079 ± 0.003	121 ± 5	0.56 ± 0.01	0.93 (100)
20–30%	1.085 ± 0.006	1.082 ± 0.005	1.083 ± 0.003	122 ± 5	0.54 ± 0.01	0.99 (100)
30–40%	1.094 ± 0.006	1.091 ± 0.005	1.087 ± 0.003	123 ± 5	0.51 ± 0.01	0.74 (100)
40–50%	1.106 ± 0.005	1.102 ± 0.004	1.094 ± 0.002	124 ± 4	0.45 ± 0.01	0.53 (100)
50–60%	1.115 ± 0.005	1.113 ± 0.004	1.100 ± 0.002	129 ± 5	0.38 ± 0.01	0.43 (100)
60–70%	1.128 ± 0.004	1.127 ± 0.003	1.107 ± 0.002	130 ± 4	0.30 ± 0.01	0.29 (100)
70–80%	1.139 ± 0.003	1.141 ± 0.003	1.113 ± 0.002	134 ± 4	0.21 ± 0.01	0.20 (100)
80–90%	1.148 ± 0.003	1.156 ± 0.003	1.117 ± 0.002	137 ± 5	0.09 ± 0.02	0.21 (101)

, respectively. To show the quality of the combined fits with the functions in Equations (5) and (4), the experimental midrapidity transverse momentum distributions of the studied particle species in four centrality classes of Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV, along with the resulting fit curves, are shown in Figure 1 and Figure 2, respectively. As seen from Figure 1 and Figure 2 and $\frac{ {\chi }^2}{n.d.f.}$ values in Table 2 and Table 3, the combined fits by both thermodynamically consistent and non-consistent Tsallis functions with transverse flow reproduce quite well the experimental midrapidity p_T distributions of the charged pions, kaons, and (anti)protons in 10 centrality groups of Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV.

The $\frac{ {\chi }^2}{n.d.f.}$ values in Table 3 are slightly smaller than the corresponding values in Table 2, implying that, mathematically, the quality of the fits performed by non-consistent Tsallis function with transverse flow (Equation (4)) are slightly better than those performed by thermodynamically consistent Tsallis function with transverse flow (Equation (5)). However, from the physics perspective, it is preferable to use the thermodynamically consistent Tsallis function with transverse flow, because all the thermodynamic relations (those for the pressure, temperature, and particle and entropy densities) resulting from the first and second laws of thermodynamics are satisfied [38,43,44] by the thermodynamically consistent Tsallis function. Therefore, one can consider the temperature parameter T in the thermodynamically consistent Tsallis function [38] as being a true thermodynamic quantity.

To visualize and analyze the $\langle N_{part}\rangle$ dependencies of the parameters, the dependencies of q, T₀, and 〈β_T〉 on the average number of the participant nucleons, obtained from minimum ${\chi }^2$ fits by thermodynamically consistent as well as non-consistent Tsallis function with transverse flow and given in Table 2 and Table 3, are shown in Figure 3.

As seen from Figure 3a, the shapes of T₀ versus $\langle N_{part}\rangle$ dependencies are similar in case of fits by both thermodynamically consistent as well as non-consistent Tsallis function with transverse flow. At the same time, the absolute values of T₀ are consistently smaller in case of fits by the function in Equation (5), compared to those by the function in Equation (4), which is obviously due to the extra $\langle {\gamma }_T\rangle \left(m_T-p_T\langle {\beta }_T\rangle \right)$ factor in Equation (5) as compared to Equation (4). As observed from Figure 3a, the T₀ values decrease with increasing $\langle N_{part}\rangle$ in region $\langle N_{part}\rangle$ < 71 and remain constant in region $\langle N_{part}\rangle$ > 71. Additionally, as seen from Figure 3b, the parameter 〈β_T〉 shows significantly differing growth rates with increasing $\langle N_{part}\rangle$ in these two regions: a relatively higher increase rate in region $\langle N_{part}\rangle$ < 71 and a smaller one in region $\langle N_{part}\rangle$ > 71. The border $\langle N_{part}\rangle$ = 71 ± 5 between these two distinct regions was estimated as the middle value of $\langle N_{part}\rangle$ between the fourth and fifth points on Figure 3a, corresponding to the 50–60% and 40–50% centrality groups in Table 1. The corresponding border value $\langle d{N}_{ch}/d\eta \rangle$ = 205 ± 15 was calculated as the middle between the $\langle d{N}_{ch}/d\eta \rangle$ values for 50–60% and 40–50% centrality in Table 1. Analogously the corresponding border value of transverse flow velocity $\langle {\beta }_T\rangle$ = 0.44 ± 0.02 was estimated as the middle between the extracted $\langle {\beta }_T\rangle$ values for 50–60% and 40–50% centrality in Table 2. To quantify the different growth rates of 〈β_T〉 in regions $\langle N_{part}\rangle$ < 71 and $\langle N_{part}\rangle$ > 71, we have fitted the 〈β_T〉 versus $\langle N_{part}\rangle$ dependence in Figure 3b, obtained from fitting with thermodynamically consistent Tsallis functions with transverse flow,

$$\begin{gathered} \mathrm{using}\mathrm{single-power}\mathrm{function} \\ \langle {\beta }_T\rangle =A·{〈N_{part}〉}^{\alpha }·(A—\mathrm{fitting\; constant}, \alpha —\mathrm{exponent\; parameter}) \\ \mathrm{and}\mathrm{two-power}\mathrm{function} \end{gathered}$$

(6)

$$\langle {\beta }_T\rangle =A_1·u\left(71-〈N_{part}〉\right)·{〈N_{part}〉}^{{\alpha }_1}+A_2·u\left(〈N_{part}〉-71\right)·{〈N_{part}〉}^{{\alpha }_2},$$

(7)

where $u\left(t\right)$ is the Heaviside function ($u\left(t\right)=0$ if t < 0, and $u\left(t\right)=1$ if t > 0), A₁ and A₂—fitting (normalization) constants, and α₁ and α₂ are exponent parameters. The parameters of minimum ${\chi }^2$ fits with the functions in Equations (6) and (7) of $\langle {\beta }_T\rangle$ versus $\langle N_{part}\rangle$ dependence in Figure 3b, obtained using a consistent Tsallis function with transverse flow, are given in

Table 4. The parameters of minimum ${\chi }^2$ fits with the functions in Equations (6) and (7) of $\langle {\beta }_T\rangle$ versus $\langle N_{part}\rangle$ dependence in Figure 3b, obtained using consistent Tsallis function with transverse flow (Equation (5)).

Fitting Function	Parameter Values	${\chi }^2$/n.d.f. (n.d.f.)
Equation (6)	A = 0.143 ± 0.019 α = 0.249 ± 0.025	10.68 (8)
Equation (7)	A1 = 0.073 ± 0.012 α1 = 0.432 ± 0.044 A2 = 0.244 ± 0.024 α2 = 0.152 ± 0.018	1.50 (6)

. As seen in Table 4 and Figure 3b, the two-power function in Equation (7) fits the 〈β_T〉 versus $\langle N_{part}\rangle$ dependence quite well, while the single-power function in Equation (6) fails to fit this dependence.

The obtained significantly different behavior of 〈β_T〉 versus $\langle N_{part}\rangle$ in regions $\langle N_{part}\rangle$ < 71 and $\langle N_{part}\rangle$ > 71 with the temperature parameter becoming constant in region $\langle N_{part}\rangle$ > 71 could indicate that $\langle N_{part}\rangle$ = 71 ± 5 (corresponding to $\langle d{N}_{ch}/d\eta \rangle$ = 205 ± 15) is a border value of collision centrality for a crossover phase transition from the dense hadronic state to that of the QGP state (or mixed state of QGP and hadrons) in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV. It is important to note that the similar behaviors of 〈β_T〉 versus $\langle N_{part}\rangle$ as well as T₀ versus 〈β_T〉 dependencies with two distinct regions of $\langle N_{part}\rangle$ have been obtained recently in Refs. [18,33] from similar analyses, using the functions in Equations (4) and (5), of the experimental midrapidity transverse momentum distributions of identified charged particles in Xe + Xe collisions at $\sqrt{s_{nn}}$ = 5.44 TeV and Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV. It is interesting to note that the corresponding border between two distinct $\langle N_{part}\rangle$ regions with significantly different behaviors of 〈β_T〉 versus $\langle N_{part}\rangle$ as well as T₀ versus $\langle N_{part}\rangle$ dependencies was found to be between the 50–60% and 40–50% centrality classes in both Xe + Xe collisions at $\sqrt{s_{nn}}$ = 5.44 TeV [18] and Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV [33], coinciding with the corresponding border centrality intervals for Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV in the present work. A comparison of the estimated border values of $\langle N_{part}\rangle$, $\langle d{N}_{ch}/d\eta \rangle$, and $\langle {\beta }_T\rangle$ for probable crossover phase transition from the dense hadronic state to that of the QGP state (or mixed state of QGP and hadrons) in the present work on Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV with those estimated in Refs. [18,33] in Xe + Xe collisions at $\sqrt{s_{nn}}$ = 5.44 TeV and Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV, respectively, is presented in

Table 5. The comparison of the border values of $\langle N_{part}\rangle$, $\langle d{N}_{ch}/d\eta \rangle$, and $\langle {\beta }_t\rangle$, estimated in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV in the present work, with those extracted recently in Xe + Xe collisions at (s_nn)^1/2 = 5.44 TeV [18] and Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV [33].

Quantity	$\mathrm{Xe}+\mathrm{Xe} \mathrm{Collisions} \mathrm{at} \sqrt{s_{nn}}=5.44 \mathrm{TeV}$ [18]	$\mathrm{Pb}+\mathrm{Pb} \mathrm{Collisions} \mathrm{at} \sqrt{s_{nn}}=5.02 \mathrm{TeV}$ [33]	$\mathrm{Pb}+\mathrm{Pb} \mathrm{Collisions} \mathrm{at} \sqrt{s_{nn}}=2.76 \mathrm{TeV} (\mathrm{Present} \mathrm{Work})$
$\langle N_{part}\rangle$	44 ± 5	71 ± 7	71 ± 5
$\langle d{N}_{ch}/d\eta \rangle$	158 ± 20	251 ± 20	205 ± 15
$\langle {\beta }_T\rangle$	0.44 ± 0.02	0.46 ± 0.03	0.44 ± 0.02

.

As seen from Table 5, the estimated border values of $\langle N_{part}\rangle$ in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV proved to be significantly larger compared to those in Xe + Xe collisions at $\sqrt{s_{nn}}$ = 5.44 TeV. This can be explained by the significant difference in the mass numbers, A, of the Pb and Xe nuclei, and hence the larger number of the participant nucleons in the overlap zone of colliding nuclei in case of Pb + Pb compared to Xe + Xe collisions for the border value between the 50–60% and 40–50% centrality classes, coinciding for these three collision types. The coincidence of the border values of $\langle N_{part}\rangle$ in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV is obviously due to the same collision system and geometry for the coinciding estimated border between the 50–60% and 40–50% centrality groups in both cases. It should be noted that the ratio (1.61 ± 0.24) of the evaluated border values of $\langle N_{part}\rangle$ in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV and Xe + Xe collisions at $\sqrt{s_{nn}}$ = 5.44 TeV was found to coincide [33] with that (1.59 ± 0.24) of the respective border values of $\langle d{N}_{ch}/d\eta \rangle$ in these two collisions, and with the ratio of the mass numbers of the corresponding ²⁰⁸Pb and ¹³²Xe nuclei equal to $\frac{A\left({208}_{Pb}\right)}{A\left({132}_{Xe}\right)}\approx 1.58$. The approximate relation $\frac{\langle N_{part}\rangle {}_{Pb+Pb}}{\langle N_{part}\rangle {}_{Xe+Xe}}\approx \frac{{\langle d{N}_{ch}/d\eta \rangle }_{Pb+Pb}}{{\langle d{N}_{ch}/d\eta \rangle }_{Xe+Xe}}\approx \frac{A\left({208}_{Pb}\right)}{A\left({132}_{Xe}\right) }\approx 1.6$ was satisfied [33] for the border values estimated in Pb + Pb and Xe + Xe collisions $\sqrt{s_{nn}}$ = 5.02 and 5.44 TeV, respectively, with the close values of the center-of-mass collision energy per nucleon pair. The significantly larger estimated border value $\langle d{N}_{ch}/d\eta \rangle$ = 251 ± 20 in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV as compared to $\langle d{N}_{ch}/d\eta \rangle$ = 205 ± 15 in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV, as seen in Table 5, is likely due to the significantly higher energy density (hence, larger charged-particle multiplicity density) and more particles produced in cases of significantly larger value of center-of-mass collision energy, $\sqrt{s_{nn}}$ = 5.02 TeV, per nucleon pair.

As seen from Figure 3c, the non-extensivity parameter q decreases systematically for all studied particle species with increasing the centrality of Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV (with an increase in $\langle N_{part}\rangle$). Figure 3c also shows that the rate of decrease of q with increasing $\langle N_{part}\rangle$ in region $\langle N_{part}\rangle$ < 71 is noticeably greater than that in range $\langle N_{part}\rangle$ > 71. It is seen that q for pions coincides within uncertainties with q for kaons in the whole range of $\langle N_{part}\rangle$, and in region $\langle N_{part}\rangle$ < 100 the parameter q for (anti)protons is noticeably smaller than q for pions and kaons. It agrees well with the relation q(baryons) < q(mesons) obtained earlier in high-energy collisions in Refs. [17,18,19,28,33,45,46,48,49]. However, as observed from Figure 3c, on the whole the gap between parameter q for mesons and baryons decreases with increasing $\langle N_{part}\rangle$, and q for all studied particle species, i.e., for pions, kaons, and (anti)protons, becomes the same within uncertainties in region $\langle N_{part}\rangle$ > 160, in good agreement with the similar result obtained in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV in recent work [33]. The observed coincidence of the q parameter for the charged pions and kaons, protons and antiprotons at large $\langle N_{part}\rangle$ indicates likely that the highly thermalized and equilibrated QGP is produced in central Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV with $\langle N_{part}\rangle$ > 160 in agreement with the similar result found in central Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV in Ref. [33]. The similar decrease, with increasing $\langle N_{part}\rangle$, of the gap between q for (anti)protons, on the one hand, and q for pions and kaons, on the other hand, with parameter q coinciding for these mesons and baryons at large $\langle N_{part}\rangle$ was also observed in Xe + Xe collisions at $\sqrt{s_{nn}}$ = 5.44 TeV in Ref. [18].

Currently, it is important to study the correlations between the extracted parameters. The T₀ versus 〈β_T〉 dependencies, obtained from fits by non-consistent as well as thermodynamically consistent Tsallis function with transverse flow and presented in Table 2 and Table 3, are shown in Figure 4a,b, respectively. The obtained Pearson linear correlation coefficient, r_xy, between two parameters and the one-sigma confidence ellipse (covering a 68% confidence interval) of the covariance of the T₀ and $\langle {\beta }_T\rangle$ parameters are also shown in Figure 4. The corresponding 1-sigma confidence ellipse and calculated r_xy coefficient of a linear correlation between pairs of parameters are also presented in Figure 4. The Pearson correlation coefficient, r_xy, shows the magnitude of a linear correlation between two parameter sets, and it can vary between −1 and +1 values. The r_xy equal to −1 and +1 corresponds to the full negative linear correlation (anticorrelation) and full positive linear correlation, respectively, whereas r_xy = 0 denotes the absence of a linear correlation between two sets of data. As seen from the orientations and shapes of confidence ellipses and corresponding r_xy values in Figure 4a,b, the parameters T₀ and 〈β_T〉 are highly anticorrelated. As observed from Figure 4a,b, the T₀ versus $\langle {\beta }_T\rangle$ dependencies differ in regions $\langle {\beta }_T\rangle$ < 0.44 and $\langle {\beta }_T\rangle$ > 0.44, corresponding to Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV with $\langle N_{part}\rangle$ < 71 and $\langle N_{part}\rangle$ > 71, respectively. The T₀ versus 〈β_T〉 dependence (from Figure 4a) is shown separately for regions $\langle {\beta }_T\rangle$ < 0.44 and $\langle {\beta }_T\rangle$ > 0.44 in Figure 5a,b, respectively. As seen from Figure 5a, the T₀ and 〈β_T〉 parameters are strongly anticorrelated in region $\langle {\beta }_T\rangle$ < 0.44. On the other hand, as observed from Figure 5b, the linear correlation between T₀ and 〈β_T〉 is practically absent in region $\langle {\beta }_T\rangle$ > 0.44. The similar result has been obtained recently in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV [33] and Xe + Xe collisions at $\sqrt{s_{nn}}$ = 5.44 TeV [18]. Thus, in addition to significantly different growth rates of $\langle {\beta }_T\rangle$ in regions $\langle N_{part}\rangle$ < 71 ± 5 and $\langle N_{part}\rangle$ > 71 ± 5, we have observed the considerably different correlations between T₀ and $\langle {\beta }_T\rangle$ parameters in corresponding $\langle {\beta }_T\rangle$ < 0.44 and $\langle {\beta }_T\rangle$ > 0.44 ranges. This further substantiates our assumption that $\langle N_{part}\rangle$ ≈ 71 ± 5 ($\langle d{N}_{ch}/d\eta \rangle$≈ 205 ± 15) could be a border value of collision centrality for crossover transition from dense hadronic phase to the QGP state (or mixed state of QGP and hadrons) in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV.

It is important to also check the correlations between other pairs of the parameters in Table 2, extracted using a thermodynamically consistent Tsallis function with transverse flow. The q(pions) versus q(kaons), q(pions) versus q((anti)protons), and q(kaons) versus q((anti)protons) dependencies are shown in Figure 6a–c, respectively. The q(pions) versus 〈β_T〉, q((anti)protons) versus 〈β_T〉, and q(kaons) versus 〈β_T〉 dependencies are illustrated in Figure 7a–c, respectively. The q(pions) versus T₀, q((anti)protons) versus T₀, and q(kaons) versus T₀ dependencies are displayed in Figure 8a–c, respectively.

As seen from Figure 6a–c, the non-extensivity parameter q for pions and kaons, for pions and (anti) protons, as well as q for kaons and (anti) protons show the strong positive linear correlation with the Pearson correlation coefficient being very close to +1 for each pair of particle species. Figure 7a–c show that the non-extensivity parameter q is strongly negatively correlated (anticorrelated) with the average transverse flow velocity, 〈β_T〉, for all studied particle species, with the Pearson correlation coefficient being close to −1 in each case. On the other hand, as seen from Figure 8a–c, quite a large positive linear correlation (r_xy ≈ +0.9) exists between the q and temperature parameter T₀ in all three cases.

4. Summary and Conclusions

We have successfully analyzed the midrapidity transverse momentum distributions of the charged pions, kaons, protons, and antiprotons in 10 groups of centrality of Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV, measured by the ALICE Collaboration, using both thermodynamically consistent and non-consistent Tsallis distribution function with embedded transverse flow. We have performed the combined minimum χ² fits by the above model functions of the experimental midrapidity $p_T$ spectra of the analyzed particle species in each centrality class of Pb + Pb collisions to extract the global parameters T₀ and $\langle {\beta }_T\rangle$ of the system as well as non-extensivity parameter q for each particle type and study their dependencies on collision centrality (〈N_part〉).

We have observed significantly different behavior (growth rates) of 〈β_T〉 in regions $\langle N_{part}\rangle$ < 71 and $\langle N_{part}\rangle$ > 71 with the temperature parameter becoming constant in region $\langle N_{part}\rangle$ > 71. This could indicate that $\langle N_{part}\rangle$ = 71 ± 5 (corresponds to $\langle d{N}_{ch}/d\eta \rangle$ = 205 ± 15) is an estimated border value of collision centrality for crossover phase transition from the dense hadronic state to that of QGP state (or mixed state of QGP and hadrons) in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV.

The respective border between the two distinct $\langle N_{part}\rangle$ regions with different behaviors of 〈β_T〉 versus $\langle N_{part}\rangle$ as well as T₀ versus $\langle N_{part}\rangle$ dependencies was found to be between the 50–60% and 40–50% centrality classes in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV, coinciding with the corresponding border found to be also between 50–60% and 40–50% centrality in recent works [18,33] in both Xe + Xe collisions at $\sqrt{s_{nn}}$ = 5.44 TeV and Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV. The coincidence of the border values of $\langle N_{part}\rangle$ in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV has been observed, which is due to the same collision system and geometry. The estimated border values of $\langle N_{part}\rangle$ in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 and 5.02 TeV have been significantly larger compared to that in Xe + Xe collisions at $\sqrt{s_{nn}}$ = 5.44 TeV. The significantly larger estimated border value $\langle d{N}_{ch}/d\eta \rangle$ = 251 ± 20 in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV as compared to $\langle d{N}_{ch}/d\eta \rangle$ = 205 ± 15 in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV could be explained by the significantly greater energy density (hence, larger charged-particle multiplicity density) and more particles produced in case of significantly larger value of center-of-mass collision energy, $\sqrt{s_{nn}}$ = 5.02 TeV, per nucleon pair.

The non-extensivity parameter q has decreased systematically for all studied particle species with the increasing centrality of Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV in agreement with the similar tendency observed in recent works in Xe + Xe collisions at $\sqrt{s_{nn}}$ = 5.44 TeV as well as Pb + Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV. The gap between q(mesons) and q(baryons) has decreased with an increase in $\langle N_{part}\rangle$. We have observed coincidence of the q parameter for the analyzed mesons and baryons in more central collisions with large $\langle N_{part}\rangle$, suggesting that the highly thermalized and equilibrated QGP is produced in central Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV with $\langle N_{part}\rangle$ > 160.

The T₀ and 〈β_T〉 parameters are strongly anticorrelated in region $\langle {\beta }_T\rangle$ < 0.44, and the linear correlation between T₀ and 〈β_T〉 is practically absent in region $\langle {\beta }_T\rangle$ > 0.44. Hence, besides significantly different growth rates of $\langle {\beta }_T\rangle$ in regions $\langle N_{part}\rangle$ < 71 ± 5 and $\langle N_{part}\rangle$ > 71 ± 5, we have obtained considerably different correlations between T₀ and $\langle {\beta }_T\rangle$ parameters in the corresponding $\langle {\beta }_T\rangle$ < 0.44 and $\langle {\beta }_T\rangle$ > 0.44 ranges. This further supports our finding that $\langle N_{part}\rangle$ ≈ 71 ± 5 ($\langle d{N}_{ch}/d\eta \rangle$≈ 205 ± 15) could be a border value of collision centrality for crossover transition from dense hadronic phase to the QGP state (or mixed state of QGP and hadrons) in Pb + Pb collisions at $\sqrt{s_{nn}}$ = 2.76 TeV.

The strong positive linear correlation between non-extensivity parameter q for pions and kaons, between q for pions and (anti) protons, and between q for kaons and (anti)protons has been obtained with the corresponding Pearson correlation coefficient, r_xy, being very close to +1 for each pair of particle species. On the other hand, the parameter q for all studied particle species has proved to be strongly negatively correlated (anticorrelated) with the average transverse flow velocity, 〈β_T〉, with r_xy being close to −1 in each case. Quite a large positive linear correlation (r_xy ≈ +0.9) has been observed between the q for the studied particle species and temperature parameter, T₀.