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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

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## Abstract

**:**

_{T}⟩ in regions $\u27e8{N}_{part}\u27e9$ < 71 and $\u27e8{N}_{part}\u27e9$ > 71 with the temperature T

_{0}becoming constant in region $\u27e8{N}_{part}\u27e9$ > 71 has been observed. This could indicate that $\u27e8{N}_{part}\u27e9$ = 71 ± 5 (corresponding to $\u27e8d{N}_{ch}/d\eta \u27e9$ = 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 T

_{0}and $\u27e8{\beta}_{T}\u27e9$ parameters in the corresponding $\u27e8{\beta}_{T}\u27e9$ < 0.44 and $\u27e8{\beta}_{T}\u27e9$ > 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 T

_{0}. Analysis of q versus $\u27e8{N}_{part}\u27e9$ 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 $\u27e8{N}_{part}\u27e9$ > 160.

## 1. Introduction

^{−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.

_{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.

_{ch}/dη>) in proton–proton collisions at $\sqrt{s}$ = 7 and 13 TeV, respectively. The combined minimum χ

^{2}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, T

_{0}, 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].

^{2}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, T

_{0}, 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.

^{2}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

_{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.

_{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.

_{0}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.

_{0}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 χ

^{2}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 χ

^{2}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

_{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.

_{0}, 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.

_{0}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 T

_{0}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 T

_{0}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,

_{1}and A

_{2}—fitting (normalization) constants, and α

_{1}and α

_{2}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. 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.

_{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 T

_{0}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 T

_{0}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.

^{208}Pb and

^{132}Xe 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.

_{0}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 T

_{0}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 T

_{0}and 〈β

_{T}〉 are highly anticorrelated. As observed from Figure 4a,b, the T

_{0}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 T

_{0}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 T

_{0}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 T

_{0}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 T

_{0}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.

_{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 T

_{0}, q((anti)protons) versus T

_{0}, and q(kaons) versus T

_{0}dependencies are displayed in Figure 8a–c, respectively.

_{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 T

_{0}in all three cases.

## 4. Summary and Conclusions

^{2}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 T

_{0}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}〉).

_{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.

_{T}〉 versus $\langle {N}_{part}\rangle $ as well as T

_{0}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.

_{0}and 〈β

_{T}〉 parameters are strongly anticorrelated in region $\langle {\beta}_{T}\rangle $ < 0.44, and the linear correlation between T

_{0}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 T

_{0}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.

_{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, T

_{0}.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The resulting fit curves by Tsallis function with thermodynamical consistence with included transverse flow (Equation (5)) of the experimental midrapidity transverse momentum spectra of the charged pions (

**●**), kaons (Δ), and protons and antiprotons (

**▪**) in Pb + Pb collisions at $\sqrt{{s}_{nn}}$ = 2.76 TeV at various centralities: 10–20% (

**a**), 40–50% (

**b**), 50–60% (

**c**), and 80–90% (

**d**).

**Figure 2.**The resulting fit curves by non-consistent Tsallis function with included transverse flow (Equation (4)) of the experimental midrapidity transverse momentum spectra of the charged pions (

**●**), kaons (Δ), and protons and antiprotons (

**▪**) in Pb + Pb collisions at $\sqrt{{s}_{nn}}$ = 2.76 TeV at various centralities: 10–20% (

**a**), 40–50% (

**b**), 50–60% (

**c**), and 80–90% (

**d**).

**Figure 3.**The $\langle {N}_{part}\rangle $ dependencies of the T

_{0}(

**a**) and $\langle {\beta}_{T}\rangle $ (

**b**) parameters (●) in Pb + Pb collisions at $\sqrt{{s}_{nn}}$ = 2.76 TeV extracted using Equation (5) and given in Table 2; (

**c**)—the $\langle {N}_{part}\rangle $ dependence for the extracted q values given in Table 2 for the charged pions (●), kaons (▲), and protons and antiprotons (▪). The results (Table 3) obtained for the respective particles in Pb + Pb collisions at $\sqrt{{s}_{nn}}$ = 2.76 TeV using Equation (4) are shown by the corresponding open symbols. The dashed and solid curves in panel (

**b**) are minimum $\chi 2$ fits by the simple single-power (Equation (6)) and two-power (Equation (7)) functions, respectively, of the $\langle {\beta}_{T}\rangle $ versus $\langle {N}_{part}\rangle $ dependence, respectively, extracted using Equation (5). The data (open symbols), obtained using Equation (4), are slightly shifted along the positive direction of $\langle {N}_{part}\rangle $ axis for better visibility.

**Figure 4.**(

**a**)—Dependence (●) of T

_{0}versus 〈β

_{T}〉 parameters extracted using Equation (5) and presented in Table 2. (

**b**)—Dependence (●) of T

_{0}versus 〈β

_{T}〉 parameters extracted using Equation (4) and presented in Table 3. The 1-sigma confidence ellipse (which corresponds to a 68% confidence interval) of the covariance of parameters T

_{0}and 〈β

_{T}〉 and the calculated Pearson correlation coefficient, r

_{xy}, between T

_{0}and 〈β

_{T}〉 are also given in the figures.

**Figure 5.**(

**a**)—Dependence (●) of T

_{0}versus 〈β

_{T}〉 parameters, extracted using Equation (5) and presented in Table 2, in region 〈β

_{T}〉 < 0.44. (

**b**)—Dependence (●) of T

_{0}versus 〈β

_{T}〉 parameters, extracted using Equation (5) and presented in Table 2, in region 〈β

_{T}〉 > 0.44. The corresponding 1-sigma confidence ellipses and Pearson correlation coefficients are also shown in the figures.

**Figure 6.**The q(pions) versus q(kaons) (

**a**), q(pions) versus q((anti)protons) (

**b**), and q(kaons) versus q((anti)protons) (

**c**) dependencies, obtained using Equation (5) and presented in Table 2. The corresponding 1-sigma confidence ellipses and Pearson correlation coefficients are also shown in the figures.

**Figure 7.**The q(pions) versus 〈β

_{T}〉 (

**a**), q((anti)protons) versus 〈β

_{T}〉 (

**b**), and q(kaons) versus 〈β

_{T}〉 (

**c**) dependencies, obtained using Equation (5) and presented in Table 2. The corresponding 1-sigma confidence ellipses and Pearson correlation coefficients are also shown in the figures.

**Figure 8.**The q(pions) versus T

_{0}(

**a**), q((anti)protons) versus T

_{0}(

**b**), and q(kaons) versus T

_{0}(

**c**) dependencies, obtained using Equation (5) and presented in Table 2. The corresponding 1-sigma confidence ellipses and Pearson correlation coefficients are also shown in the figures.

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 |

**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 | $q({\pi}^{+}+{\pi}^{-})$ | $q({K}^{+}+{K}^{-})$ | $q(p+\overline{p})$ | T_{0} (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) |

**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}^{-})$ | $q(p+\overline{p})$ | T_{0} (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) |

**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) | A_{1} = 0.073 ± 0.012α _{1} = 0.432 ± 0.044A _{2} = 0.244 ± 0.024α _{2} = 0.152 ± 0.018 | 1.50 (6) |

**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 |

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**MDPI and ACS Style**

Olimov, K.K.; Lebedev, I.A.; Fedosimova, A.I.; Liu, F.-H.; Kanokova, S.Z.; Shodmonov, M.Z.; Tukhtaev, B.J.
Simultaneous Analysis of Midrapidity *Universe* **2022**, *8*, 655.
https://doi.org/10.3390/universe8120655

**AMA Style**

Olimov KK, Lebedev IA, Fedosimova AI, Liu F-H, Kanokova SZ, Shodmonov MZ, Tukhtaev BJ.
Simultaneous Analysis of Midrapidity *Universe*. 2022; 8(12):655.
https://doi.org/10.3390/universe8120655

**Chicago/Turabian Style**

Olimov, Khusniddin K., Igor A. Lebedev, Anastasiya I. Fedosimova, Fu-Hu Liu, Shakhnoza Z. Kanokova, Maratbek Z. Shodmonov, and Boburbek J. Tukhtaev.
2022. "Simultaneous Analysis of Midrapidity *Universe* 8, no. 12: 655.
https://doi.org/10.3390/universe8120655