# Random Statistical Analysis of Transverse Momentum Spectra of Strange Particles and Dependence of Related Parameters on Centrality in High Energy Collisions at the LHC

Authors: Xu-Hong Zhang; Fu-Hu Liu; Khusniddin K. Olimov; Airton Deppman
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

We have studied the transverse momentum (pT) spectra of the final-state strange particles, including K±, ϕ, Ξ, and Ω, produced in high energy lead–lead (Pb–Pb), proton–lead (p–Pb), xenon–xenon (Xe–Xe) collisions at the Large Hadron Collider (LHC). Taking into account the contribution of multiquark composition, whose probability density distribution is described by the modified Tsallis–Pareto-type function; we simulate the pT spectra of the final-state strange particles by a Monte Carlo method, which is shown to be in good agreement with the experimental data in most the cases. The kinetic freeze-out parameters are obtained. The present method provides a new tool for studying the spectra of various particles produced in high energy collisions, reflecting more realistically the collision process, which is of great significance to study the formation and properties of the produced particles.


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## Body

## 1. Introduction

Plenty of final-state particles are produced in relativistic heavy-ion collisions [1, 2] in various collision processes, resulting in different configurations of final-state particles. In high energy collisions, novel physical phenomena can appear with the most representative being formation of quark-gluon plasma (QGP) [3, 4]. In addition, multiparticle production provides abundant information on thermal and statistical properties of a system. Although quantum chromodynamics (QCD) can provide an important theoretical basis to study strong interactions among quarks and features of collision system [5, 6], thermal and statistical method is still a powerful tool for such analysis.Investigation of the transverse momentum (pT) (or transverse mass, mT) spectra of the final-state particles is an effective and fast method [1, 2] to study the processes of multiparticle production and system evolution. This method can help to obtain the thermodynamic parameters of the final-state particles and collision system. The pT spectra of different final-state particles have been studied extensively. In addition to baryons, leptons, and other elementary particles have also attracted much attention.In recent years, the production of single and multistrange particles [7–10] has attracted an increased interest, being extensively researched. Based on the exotic properties [11], the pT spectra of strange particles measured in high energy collisions by various collaborations have been analyzed and predicted utilizing Tsallis function [12–17], nonequilibrium chemical or kinetic freeze-out model [8], etc.Relative to pions, enhanced production of multistrange hadrons [10–21] in high-multiplicity proton–proton collisions has been observed experimentally for the first time, and it is found that the integrated yields of strange particles increases significantly with the event charged-particle multiplicity. Besides, the azimuthal angular correlation and the mass dependence of pT due to the existence of QGP in high-energy collisions have been also reported. Further researches on strange particles can better explore and reveal the properties of QGP.We are interested in the exploration of the properties of different strange particles. In contrast to our previous work [22–24], in this work we adopt a new algorithm to simulate and analyze the pT spectra of strange particles. We aim to restore the collision process more realistically and extract more accurate characteristic parameters. The multiquark composition of baryons is considered. A Monte Carlo method [25–27] is used to simulate different transverse momenta carried by various quarks. The modified Tsallis–Pareto-type function [22–24] is used to define the contribution of multiquarks, from which the kinetic freeze-out temperature (T0) [28–31], average transverse flow velocity (βt) [32, 33], and other related parameters can be directly extracted.This paper is structured as follows. The formalism and Monte Carlo method are briefly introduced in Section2. The simulated or fitted results, their comparison with the data, and discussion are given in Section 3. Finally, in Section 4 we summarize our main observations and conclusions.


## 2. Formalism and Method

To deal withpT of strange particles produced in high energy collisions, we briefly introduce the analysis, based on the Monte Carlo method. Generally, the modified Tsallis–Pareto-type function [22–24].
(1)fpT=1NdNdpT=CpTa01+mT−m0nT−n,is suitable to characterize the pT spectra of final-state particles in low- and intermediate-pT regions. Here, N is the number of particles; C is the normalization constant; T is the effective temperature of a collision system; n is an entropy-related index, which is used to describe the degree of nonequilibrium of the system; mT=pT2+m02 is the transverse mass of a particle; m0 is the rest mass of the particle, and a0 is the correction index, which makes the function to fit better the spectra in low-pT region.It should be noted that when we seta0=1. Equation (1) is naturally converged to the Tsallis–Pareto-type function. That is to say, the introduction of a0≠1 in Equation (1) is a modification of Tsallis–Pareto-type function [17]. To extract characteristic parameters such as the thermal or kinetic freeze-out temperature T0 of a collision system [28–31] and the average transverse flow velocity βt of the produced particles [32, 33] at the quark level, following References [34–37], we perform the Lorentz-like transformation on mT and pT in Equation (1) by using mT⟶γtmT−pTβt and pT⟶γtpT−mTβt, where γt=1/1−βt2 is the Lorentz factor.Thus, the transverse momentumpTi of the i-th quark contributed to the particle pT is assumed to obey the new modified probability density function
(2)fipTi=Ciγta0+1mTimTi−pTiβtpTi−mTiβta0×1+γtmTi−pTiβt−m0inT0−n.The temperature parameter at this time has been naturally converted fromT in Equation (1) to the kinetic freeze-out temperature T0. In Equation (2), m0i is the constituent mass of the i-th quark. As shown in the literature [38], we have m01=0.31GeV/c2 for up and down quarks and m02=0.5 GeV/c2 for a strange quark. Thus, we have constructed the probability density function, which satisfies multiquark states.Within the defined interval0,∞ of pTi of the i-th quark, we have the normalization condition
(3)∫0∞fipTidpTi=1,where the upper limit is usually quite a large value, but not infinite. In a Monte Carlo calculation, Ri denotes the random number in 0,1, and we have the relation satisfied by pTi to be
(4)∫0pTifipTi′dp′Ti<Ri<∫0pTi+δpTifipTi′dp′Ti,where δpTi is a small shift from pTi. According to Equation (4), one can obtain a series of discrete values of pTi which satisfy Equation (2). In the calculation, we take δpTi=0.01 GeV/c. Considering the contributions of multiquarks, we have i=1 to 2 for mesons, and i=1 to 3 for baryons, because meson consists of two quarks, and baryon has three quarks.In general condition, the particle transverse momentumpT is the vector superposition of pTi of two or three quarks. In the right-handed Cartesian coordinate system O-xyz, let the Oz axis be the beam direction, xOz plane be the reaction plane, and xOy plane be the transverse one. In the source rest frame, pTi is assumed to be isotropic. The movement of the source along the Oz axis constitutes the longitudinal flow, and the interactions among different sources causes the transverse flow. The longitudinal flow is not the focus of the present work, though it can be described by the rapidity distribution. The transverse flow is described by the average transverse flow velocity βt that is mentioned in the above discussion.For thei-th quark, we have the x- and y-components of pTi to be
(5)pTix=pTicosφi=pTicos2πri,pTiy=pTisinφi=pTisin2πri,where the isotropic azimuthal angle φi is distributed uniformly in 0,2π. Using a random number ri distributed uniformly in 0,1, one has φi=2πri in the Monte Carlo calculation. According to the principle of vector superposition, we obtain the expressions of pT’s components and pT of the final-state particle to be
(6)pTx=∑i=12,3pTicos2πri,pTy=∑i=12,3pTisin2πri,pT=pTx2+pTy2,where the sums are considered due to the contributions of 2 or 3 quarks; the upper limit “2” for the sums corresponds to the meson, and the limit “3” for the sums corresponds to the baryon in the first two relations in Equation (6).According to the above analysis, we may obtain a lot of discrete values ofpT by the iterative calculations. Finally, we may perform a statistical analysis on the discrete values of pT and obtain suitable distributions such as 1/2πpTd2N/dydpT and d2N/dydpT, which can be compared with the experimental data, where y denotes the rapidity, and dy is the width of rapidity bin at midrapidity. In calculations, the statistical interval of pT is taken to be 0.1 GeV/c.Although we may simply use the modified Tsallis–Pareto-type function to fit the experimental data and adopt theχ2-minimisation scheme to obtain the fit parameters, the results obtained by the simple method are at the particle level, which seem not to be deeper insight compared to the results at the quark and gluon level. To obtain the results at the partonic-level, we may consider the contribution of multiple partons to particle’s transverse momentum. However, the analytical expression obtained at the partonic-level is not available due to the complex calculations for the superposition of multiple transverse momenta with randomized azimuthal angles. Instead, we may use the Monte Carlo method to obtain numerical results. This is suitable for the case discussed in this paper.It should be noted that several restrictions are used in the Monte Carlo calculations.First, the constituent masses of the considered quarks are determinate. These masses are taken from the literature [38]. Second, pTi of a quark is restricted to obey a given function. This function is modified from the current Tsallis-like function. Third, an isotropic emission in the rest frame of emission source is assumed. The momentum in the rest frame of emission source is then obtained. Fourth, the movement of emission source is restricted by its rapidity which is assumed to be evenly distributed in the projectile or target thermalized region in the rapidity space. The rapidity of emission source is also related to longitudinal flow, which restricts the rapidity range. Fifth, the conservation of energy and momentum is obeyed. The combination of pTi of 2- or 3-quarks with randomized azimuthal angles into pT of hadron can be conducted, in which any energy beyond hadron’s rest mass is converted into hadron’s kinetic energy. Sixth, the distribution of pT is restricted by the experimental data.


## 3. Results and Discussion

### 3.1. Comparison with Data

In the study, we analyze thepT spectra of strange particles generated in high energy lead–lead (Pb–Pb), proton–lead (p–Pb), and xenon–xenon (Xe–Xe) collisions at the Large Hadron Collider (LHC). Figure 1 shows the pT spectra of (a) K++K−, (b) K∗0+K¯∗0/2, (c) KS0, (d) ϕ, (e) Λ, (f) Ξ−, (g) Ξ¯+, (h) Ω−, and (i) Ω¯+ produced in Pb–Pb collisions with different centrality intervals marked in the panels at the collision energy sNN=2.76 TeV per nucleon pair. The symbols represent the data measured by the ALICE Collaboration [39–41]. The mid-pseudorapidity range of panel (a) is η<0.2, and those of other panels are y<0.5. For better distinguishing and comparing, the data with different centralities in different panels are multiplied by 10 to the n-th power. In addition, the gray rectangles represent the uncertainty of the experimental data, whose length is the uncertainty of the abscissa, whereas the width is the uncertainty of the ordinate, which denote the quadratic sum of the statistical and systematical uncertainties. The solid curves represent the Monte Carlo results calculated from the probability density function in Equation (2) and the pT expression in Equation (6).Figure 1
ThepT spectra of (a) K++K−, (b) K∗0+K¯∗0/(b), (c) KS0, (d) ϕ, (e) Λ, (f) Ξ−, (g) Ξ¯+, (h) Ω−, and (i) Ω¯+ in Pb–Pb collisions with different centrality and mid-pseudorapidity intervals at sNN=2.76 TeV. The symbols represent the data measured by the ALICE Collaboration [39–41]. The solid curves represent the Monte Carlo results, based on the probability density function Equation (2) and the pT expression in Equation (6).
(a)(b)(c)(d)(e)(f)(g)(h)(i)We useχ2=ΣjDataj−Fitj2/Uncertaintyj2 to quantify the deviation of the Monte Carlo results from the experimental data, where j denotes the order number of the data. The smaller the mean of χ2 is, the closer the Monte Carlo results to the experimental data are. The free parameters, the entropy-related index n, the kinetic freeze-out temperature T0, the correction index a0, and the average transverse flow velocity βt are extracted by the method of least squares. The values of free parameters, χ2 and the number of degree of freedom (ndof) are listed in Table 1. One can see that in most cases the Monte Carlo results agree well or approximately with the experimental pT spectra of strange particles. In few cases, the Monte Carlo results are in qualitative agreement with the experimental data, for which the values of χ2/ndof are quite large.Table 1
Values ofn, T0, a0, βt, χ2, and ndof corresponding to the solid curves in Figure 1.
FigureParticlesNN (TeV)SelectionnT0 (GeV)a0βt (c)χ2/ndofFigure1(a)K++K−2.760–5%7.1±0.20.221±0.003−0.022±0.0040.247±0.0051199/53η<0.25–10%7.0±0.20.219±0.003−0.024±0.0040.246±0.0051021/53Pb–Pb10–20%6.9±0.20.216±0.003−0.026±0.0040.244±0.005887/5320–40%6.7±0.20.212±0.003−0.029±0.0040.241±0.005793/5340–60%6.2±0.20.209±0.003−0.032±0.0040.238±0.005323/5360–80%6.1±0.20.208±0.003−0.035±0.0040.234±0.005115/53Figure1(b)K∗0+K¯∗0/22.760–20%7.7±0.20.174±0.0031.042±0.0100.309±0.00621/3y<0.520–40%7.4±0.20.170±0.0031.037±0.0100.306±0.00622/3Pb–Pb40–60%7.2±0.20.164±0.0031.032±0.0100.301±0.0063/360–80%6.9±0.20.152±0.0031.027±0.0100.294±0.0063/3Figure1(c)KS02.760–5%8.1±0.20.179±0.0030.227±0.0050.310±0.006357/28y<0.55–10%8.0±0.20.177±0.0030.226±0.0050.307±0.006337/28Pb–Pb10–20%7.8±0.20.175±0.0030.224±0.0050.305±0.006348/2820–40%7.5±0.20.174±0.0030.220±0.0050.303±0.006318/2840–60%7.1±0.20.171±0.0030.218±0.0050.301±0.006221/2860–80%6.7±0.20.169±0.0030.214±0.0050.298±0.006151/2880–90%6.4±0.20.168±0.0030.210±0.0050.295±0.00646/28Figure1(d)ϕ2.760–5%8.6±0.20.183±0.0031.013±0.0090.311±0.00614/3y<0.55–10%8.5±0.20.181±0.0031.012±0.0090.309±0.00631/3Pb–Pb10–20%8.3±0.20.179±0.0031.011±0.0090.308±0.00620/320–30%8.2±0.20.176±0.0031.009±0.0090.305±0.0067/330–40%7.9±0.20.174±0.0031.006±0.0090.302±0.0068/340–50%7.7±0.20.171±0.0030.999±0.0090.299±0.0065/350–60%7.4±0.20.166±0.0030.996±0.0090.296±0.0063/360–70%7.1±0.20.162±0.0030.991±0.0090.287±0.0061/370–80%6.9±0.20.158±0.0030.987±0.0090.285±0.0064/380–90%6.6±0.20.154±0.0030.982±0.0090.282±0.0062/3Figure1(e)Λ2.760–5%10.0±0.20.137±0.0021.429±0.0120.218±0.004272/26y<0.55–10%9.8±0.20.135±0.0021.426±0.0120.216±0.004211/26Pb–Pb10–20%9.7±0.20.134±0.0021.420±0.0120.213±0.004197/2620–40%9.5±0.20.132±0.0021.419±0.0120.212±0.004127/2640–60%9.2±0.20.128±0.0021.416±0.0120.210±0.00432/2660–80%9.0±0.20.126±0.0021.413±0.0120.207±0.00438/2680–90%8.8±0.20.122±0.0021.410±0.0120.204±0.00460/24Figure1(f)Ξ−2.760–10%11.9±0.30.137±0.0021.925±0.0140.238±0.005126/22y<0.510–20%11.8±0.30.135±0.0021.922±0.0140.236±0.00563/22Pb–Pb20–40%11.4±0.20.132±0.0021.918±0.0140.233±0.00529/2240–60%10.8±0.20.126±0.0021.912±0.0140.230±0.00521/2060–80%10.5±0.20.120±0.0021.904±0.0140.227±0.00537/15Figure1(g)Ξ¯+2.760–10%12.1±0.30.138±0.0021.924±0.0140.235±0.005122/22y<0.510–20%12.0±0.30.136±0.0021.922±0.0140.233±0.00544/22Pb–Pb20–40%11.8±0.30.134±0.0021.918±0.0140.231±0.00533/2240–60%11.4±0.30.130±0.0021.911±0.0140.227±0.00529/2060–80%11.1±0.20.126±0.0021.910±0.0140.225±0.00572/15Figure1(h)Ω−2.760–10%10.7±0.20.133±0.0022.026±0.0150.248±0.0055/8y<0.510–20%10.4±0.20.131±0.0022.024±0.0150.247±0.00513/8Pb–Pb20–40%10.3±0.20.129±0.0022.022±0.0150.245±0.00513/840–60%10.1±0.20.126±0.0022.020±0.0150.242±0.00511/760–80%9.8±0.20.124±0.0022.018±0.0150.237±0.0057/5Figure1(i)Ω¯+2.760–10%10.9±0.20.135±0.0022.022±0.0150.247±0.0057/8y<0.510–20%10.8±0.20.134±0.0022.020±0.0150.245±0.00513/8Pb–Pb20–40%10.6±0.20.132±0.0022.017±0.0150.241±0.0056/840–60%10.3±0.20.130±0.0022.014±0.0150.239±0.0057/760–80%9.8±0.20.122±0.0022.004±0.0150.235±0.0057/5ThepT spectra of (a) K++K−, (b) K∗0+K¯∗0/2, and (c) ϕ are displayed in Figure 2 for Pb–Pb collisions at sNN=5.02 TeV, quoted from the ALICE Collaboration [42, 43]. The mid-pseudorapidity is η<0.2 in panel (a) and y<0.5 in panels (b) and (c). Similarly to Figure 1, the data with different centralities are represented by different symbols, and the gray rectangles represent the uncertainty of the data. The solid curves show the Monte Carlo results. For clarity, we use a power index amplification from 101 to 109. The free parameters n, T0, a0, and βt, as well as χ2/nd of are listed in Table 2. One can see that the Monte Carlo results show an approximate agreement with the data in some cases and a qualitative agreement with the data in the other cases.Figure 2
ThepT spectra of (a) K++K−, (b) K∗0+K¯∗0/2, and (c) ϕ in Pb–Pb collisions at sNN=5.02 TeV. The symbols represent the data measured by the ALICE Collaboration [42, 43]. The solid curves represent the Monte Carlo results, based on the probability density function Equation (2) and the pT expression in Equation (6).
(a)(b)(c)Table 2
Values ofn, T0, a0, βt, χ2, and ndof corresponding to the solid curves in Figure 2.
FigureParticlesNN (TeV)SelectionnT0 (GeV)a0βt (c)χ2/ndofFigure2(a)K++K−5.020–5%6.8±0.20.228±0.0030.050±0.0050.251±0.0052716/48η<0.85–10%6.7±0.20.227±0.0030.048±0.0050.250±0.0052292/48Pb–Pb10–20%6.5±0.20.225±0.0030.046±0.0050.248±0.0052239/4820–30%6.3±0.20.221±0.0030.042±0.0050.246±0.0051814/4830–40%6.2±0.20.218±0.0030.039±0.0050.244±0.0051468/4840–50%5.9±0.20.215±0.0030.035±0.0050.238±0.005996/4850–60%5.7±0.20.211±0.0030.032±0.0050.231±0.005621/4860–70%5.5±0.20.209±0.0030.029±0.0050.227±0.005267/4870–80%5.3±0.20.207±0.0030.027±0.0050.224±0.005219/4880–90%5.1±0.20.204±0.0030.023±0.0050.221±0.005487/48Figure2(b)K∗0+K¯∗0/25.020–10%7.0±0.20.180±0.0031.028±0.0100.313±0.006103/10y<0.510–20%6.8±0.20.176±0.0031.034±0.0100.309±0.00681/10Pb–Pb20–30%6.6±0.20.173±0.0031.018±0.0100.305±0.00695/1030–40%6.5±0.20.171±0.0031.015±0.0100.302±0.00672/1040–50%6.4±0.20.171±0.0031.011±0.0100.300±0.00688/1050–60%6.3±0.20.168±0.0031.008±0.0100.298±0.00666/1060–70%6.2±0.20.166±0.0031.005±0.0100.297±0.00674/970–80%6.2±0.20.164±0.0031.000±0.0100.295±0.00632/8Figure2(c)ϕ5.020–10%8.1±0.20.184±0.0031.017±0.0100.302±0.006323/19y<0.510–20%8.0±0.20.182±0.0031.014±0.0100.301±0.006137/19Pb–Pb20–30%7.9±0.20.180±0.0031.009±0.0100.299±0.006144/1930–40%7.7±0.20.177±0.0031.006±0.0100.295±0.00687/1940–50%7.6±0.20.174±0.0031.002±0.0090.293±0.00677/1850–60%7.4±0.20.169±0.0030.996±0.0090.291±0.00612/1660–70%7.2±0.20.165±0.0030.991±0.0090.288±0.00616/1670–80%6.9±0.20.158±0.0030.987±0.0090.285±0.00616/15In addition to Pb–Pb collisions, the experimentalpT spectra in other collisions are also studied in order to better explore the properties of strange particles. Figure 3 shows the pT spectra of (a) K++K−, (b) K∗0+K¯∗0/2, (c) KS0, (d) ϕ, (e) Λ+Λ¯, (f) Ξ−+Ξ¯+/2, and (g) Ω−+Ω¯+/2 in p–Pb collisions at sNN=5.02 TeV. The experimental data in the panels are all quoted from the ALICE Collaboration when the midrapidity is 0<y<0.5 (a, c, e) and −0.5<y<0 (b, d, f, g) [44, 45]. Different symbols indicate the different centrality intervals. The Monte Carlo results are given by the solid curves. The coefficients in parentheses are magnification factors for better distinguishing the spectra in different centrality intervals. The relevant parameters n, T0, a0, and βt, as well as χ2/nd of are listed in Table 3. One can see that the Monte Carlo results are in good agreement with the data in most cases and approximately in agreement with the data in few cases.Figure 3
ThepT spectra of (a) K++K−, (b) K∗0+K¯∗0/2, (c) KS0, (d) ϕ, (e) Λ+Λ¯, (f) Ξ−+Ξ¯+/2, and (g) Ω−+Ω¯+/2 in p–Pb collisions at sNN=5.02 TeV. The symbols represent the data measured by the ALICE Collaboration [44, 45]. The solid curves represent the Monte Carlo results, based on the probability density function Equation (2) and the pT expression in Equation (6).
(a)(b)(c)(d)(e)(f)(g)Table 3
Values ofn, T0, a0, βt, χ2, and ndof corresponding to the solid curves in Figure 3.
FigureParticlesNN (TeV)SelectionnT0 (GeV)a0βt (c)χ2/ndofFigure3(a)K++K−5.020–5%5.5±0.20.223±0.0030.229±0.0050.257±0.0057/260<y<0.55–10%5.3±0.20.215±0.0030.226±0.0050.252±0.0056/26p–Pb10–20%5.1±0.20.210±0.0030.221±0.0050.248±0.0054/2620–40%4.9±0.20.199±0.0030.214±0.0050.241±0.0053/2640–60%4.7±0.20.188±0.0030.205±0.0050.230±0.0052/2660–80%4.5±0.20.167±0.0030.197±0.0050.221±0.0052/2680–100%4.3±0.20.139±0.0030.183±0.0050.212±0.0055/26Figure3(b)K∗0+K¯∗0/25.020–20%6.8±0.20.238±0.0030.550±0.0070.267±0.00572/17−0.5<y<020–40%6.7±0.20.231±0.0030.541±0.0070.258±0.00556/17p–Pb40–60%6.4±0.20.225±0.0030.536±0.0070.254±0.00516/1760–80%6.2±0.20.211±0.0030.528±0.0070.249±0.00511/1780–100%5.7±0.20.183±0.0030.511±0.0070.233±0.00519/12Figure3(c)KS05.020–5%6.6±0.20.219±0.0030.202±0.0050.328±0.00678/290<y<0.55–10%6.5±0.20.216±0.0030.200±0.0050.325±0.00639/29p–Pb10–20%6.4±0.20.214±0.0030.199±0.0050.323±0.00655/2920–40%6.2±0.20.209±0.0030.197±0.0050.321±0.00645/2940–60%6.0±0.20.204±0.0030.192±0.0050.317±0.00661/2960–80%5.7±0.20.189±0.0030.186±0.0050.312±0.00683/2980–100%5.8±0.20.174±0.0030.184±0.0050.304±0.006145/29Figure3(d)ϕ5.020–5%6.7±0.20.199±0.0030.812±0.0090.332±0.00618/15−0.5<y<05–10%6.6±0.20.197±0.0030.810±0.0090.330±0.00626/15p–Pb10–20%6.4±0.20.194±0.0030.806±0.0090.324±0.00629/1520–40%6.3±0.20.192±0.0030.803±0.0090.322±0.00625/1540–60%6.1±0.20.189±0.0030.799±0.0080.319±0.00620/1560–80%5.5±0.20.165±0.0030.783±0.0080.312±0.00638/1480–100%5.1±0.20.147±0.0030.775±0.0080.305±0.00663/14Figure3(e)Λ+Λ¯5.020–5%7.7±0.20.132±0.0021.426±0.0120.226±0.0046/150<y<0.55–10%7.6±0.20.130±0.0021.424±0.0120.224±0.0044/15p–Pb10–20%7.5±0.20.127±0.0021.421±0.0120.221±0.0042/1520–40%7.3±0.20.123±0.0021.417±0.0120.218±0.0044/1540–60%7.0±0.20.112±0.0021.410±0.0120.211±0.0044/1560–80%6.7±0.20.104±0.0021.407±0.0120.206±0.00422/1580–100%6.4±0.20.089±0.0021.403±0.0120.202±0.00423/15Figure3(f)Ξ−+Ξ¯+/25.020–5%9.0±0.20.179±0.0030.703±0.0080.225±0.00415/12−0.5<y<05–10%8.9±0.20.177±0.0030.701±0.0080.223±0.00414/12p–Pb10–20%8.4±0.20.171±0.0030.692±0.0080.218±0.0049/1220–40%8.0±0.20.165±0.0030.688±0.0080.215±0.00418/1240–60%7.2±0.20.150±0.0030.678±0.0080.209±0.00440/1260–80%7.0±0.20.130±0.0020.669±0.0080.203±0.00419/1180–100%6.4±0.20.119±0.0020.662±0.0080.194±0.00425/10Figure3(g)Ω−+Ω¯+/25.020–5%7.6±0.20.174±0.0030.788±0.0080.240±0.0055/3−0.5<y<05–10%7.3±0.20.173±0.0030.786±0.0080.238±0.0054/3p–Pb10–20%7.2±0.20.170±0.0030.783±0.0080.235±0.0054/320–40%7.0±0.20.167±0.0030.779±0.0080.232±0.0041/340–60%6.7±0.20.150±0.0030.769±0.0080.223±0.0049/360–80%6.4±0.20.141±0.0020.761±0.0080.214±0.0045/380–100%5.9±0.20.132±0.0020.755±0.0080.206±0.0042/−Besides, we have also studied thepT spectra of (a) K++K− and (b) ϕ produced in Xe–Xe collisions at sNN=5.44 TeV in Figure 4. The symbols represent the data measured by the ALICE Collaboration, and the mid-pseudorapidity is η<0.8 [46]. Similarly, the data with different centralities are expressed as different symbols with gray rectangles, and the solid curves show the Monte Carlo results. Another, 10n in legends are magnification factors for better distinguishing the spectra. The relevant parameters n, T0, a0, and βt, as well as χ2/nd of are listed in Table 4. Once again, one can see that the Monte Carlo results are in good agreement with the experimental data.Figure 4
ThepT spectra of (a) K++K− and (b) ϕ produced in Xe–Xe collisions at sNN=5.44 TeV. The symbols represent the data measured by the ALICE Collaboration [46]. The solid curves represent the Monte Carlo results, based on the probability density function Equation (2) and the pT expression in Equation (6).
(a)(b)Table 4
Values ofn, T0, a0, βt, χ2, and ndof corresponding to the solid curves in Figure 4.
FigureParticlesNN (TeV)SelectionnT0 (GeV)a0βt (c)χ2/ndofFigure4(a)K++K−5.440–5%8.3±0.20.199±0.0030.399±0.0060.257±0.00573/33η<0.85–10%8.1±0.20.198±0.0030.396±0.0060.255±0.00558/33Xe–Xe10–20%7.9±0.20.194±0.0030.395±0.0060.254±0.00555/3320–30%7.7±0.20.192±0.0030.392±0.0060.252±0.00537/3330–40%7.5±0.20.187±0.0030.390±0.0060.250±0.00520/3340–50%7.3±0.20.183±0.0030.386±0.0060.247±0.00516/3350–60%7.2±0.20.179±0.0030.384±0.0060.244±0.00514/3360–70%7.1±0.20.174±0.0030.381±0.0060.238±0.00516/3270–90%6.9±0.20.162±0.0030.368±0.0060.230±0.00527/32Figure4(b)ϕ5.440–10%9.0±0.20.219±0.0030.810±0.0090.338±0.00613/6η<0.810–30%8.9±0.20.216±0.0030.804±0.0090.334±0.00610/6Xe–Xe30–50%8.7±0.20.213±0.0030.800±0.0080.332±0.0067/650–70%8.5±0.20.209±0.0030.797±0.0080.327±0.00611/670–90%8.3±0.20.203±0.0030.792±0.0080.317±0.00610/4


### 3.2. Tendencies of Parameters

To further understand the regularities shown by thepT spectra of strange particles, we analyze the tendencies of various parameters with changing centrality observed in the work. Figure 5 shows the dependence of entropy-related index (a, b) n on centrality and C in Pb–Pb, (c) p–Pb, and (d) Xe–Xe collisions at the LHC energies. The symbols are quoted from Tables 1–4, where the results from different particles are represented by different symbols. As an entropy-related index, n=1/q−1 or n=q′/q′−1 describes the degree of nonequilibrium of the system, where q (q′) is the entropy index. Generally, a larger n or a q (q′) closer to 1 corresponds to a higher degree of equilibrium. The present work shows that n is large enough or q (q′) is close to 1, and the system stays in an approximate equilibrium. Meanwhile, the system in central collisions stays in higher degree of equilibrium. In addition, as the rest mass of strange particle increases, the value of n increases. The multistrange particles correspond to larger n than the single-strange particles. These results imply that the system stays in larger degree of equilibrium when it forms multistrange particles.Figure 5
Dependence of entropy-related index (a, b)n on centrality and C in Pb–Pb, (c) p–Pb, and (d) Xe–Xe collisions.
(a)(b)(c)(d)Similar to Figure5, Figure 6 shows the dependence of kinetic freeze-out temperature T0 on centrality C in the mentioned collisions at the LHC. One can see that T0 decreases slightly with the decrease of centrality from central to peripheral collisions in most cases. In few cases, the decrease is significant. In addition, T0 extracted from the single-strange particle spectra is larger than that from the multistrange particle spectra. This finding suggests that the single-strange particles are formed earlier than the multistrange particles, though the latter may leave the system earlier than the former in the hydrodynamic evolution due to different masses.Figure 6
Dependence of kinetic freeze-out temperature (a, b)T0 on centrality and C in Pb–Pb, (c) p–Pb, and (d) Xe–Xe collisions.
(a)(b)(c)(d)As shown in Figure7, we present the changing law of the correction index a0 on centrality C in the mentioned collisions at the LHC. One can see that with the decrease of centrality from central to peripheral collisions, a0 is almost invariant, though in few cases a0 shows very slight decrease. In most cases, the values of a0 are far from 1, which means that the introduction of a0 is necessary. We have also compared the present fits with a0≠1 with those by a0=1. An obvious difference appears in the low-pT region. This implies that the fits by a0=1 are not suitable. In fact, in the fits of the present work, we use a0=1 at the first, then we change a0 to fit the pT spectra if a0=1 is not satisfactory.Figure 7
Dependence of correction index (a, b)a0 on centrality and C in Pb–Pb, (c) p–Pb, and (d) Xe–Xe collisions.
(a)(b)(c)(d)The dependence of the average transverse flow velocityβt on centrality C in the mentioned collisions at the LHC is displayed in Figure 8. Similar conclusions to the dependences of n on C and T0 on C can be obtained. That is, βt decreases slightly with the decrease of centrality from central to peripheral collisions in most cases. In few cases, the decrease is significant. In addition, βt extracted from the single-strange particle spectra is larger than that from the multistrange particle spectra. This finding also suggests that the single-strange particles are formed earlier than the multistrange particles.Figure 8
Dependence of average transverse flow velocity (a, b)βt on centrality and C in Pb–Pb, (c) p–Pb, and (d) Xe–Xe collisions.
(a)(b)(c)(d)In order to further study the final states of the strange particles and the disorder degree of the system, we give the results of pseudoentropyShadron′=−ΣfpT/ΣfpTlnfpT/ΣfpT [28], which is based on the probability density function fpT of pT, inspired by the entropy Shadron=−ΣPNlnPN [47–49], based on the probability density function PN of multiplicity N, where pT is in the units of GeV/c. As is done in our recent work [24], the width of pT bin is taken to be 0.1 GeV/c, though the width is changeable. The unit of fpT is neglected due to the fact that PN is dimensionless.In Figure9, we demonstrate the relation of the pseudoentropy Shadron′ versus the centrality C. It is seen from the results that Shadron′ decreases slightly with the decrease of centrality from central to peripheral collisions in most cases. Only in few cases, Shadron′ decreases significantly with the decrease of C. The dependence of Shadron′ on C is similar to those of n, T0, and βt on C. Generally, the multistrange particles show larger Shadron′ than the single-strange particles, though both Shadron′ for multi and single-strange particles are negative. This also reflects the mass-dependence of Shadron′, which is similar to those of the other parameters.Figure 9
Dependence of pseudoentropy (a, b)Shadron′ on centrality C in Pb–Pb, (c) p–Pb, and (d) Xe–Xe collisions.
(a)(b)(c)(d)Analyzing Figures5–9, one can see that the correlations between n and T0, n and βt, n and Shadron′, T0 and βt, T0 and Shadron′, and βt and Shadron′ are positive; while the correlations between a0 and n, a0 and T0, a0 and βt, and a0 and Shadron′ are very small or negligible. These results are understandable due to the fact that all n, T0, and βt determine mainly, and Shadron′ is mostly defined by, the shape of the spectra in intermediate- and high-pT regions; while a0 determines mainly the spectra in low- or even very low-pT region. Of course, because of the requirement of normalization, all parameters affect the spectra in the whole pT region, though the level of influence in different pT regions are different.


### 3.3. Further Discussion

Although the extraction ofT0 and βt in present work is performed at the partonic-level, most extractions in the literature have been done at the particle level, showing inconsistent trends. For example, in terms of dependence of parameters on centrality from central to peripheral collisions, some works show an increase in T0 and a decrease in βt, and βt≈0 in small system or peripheral collisions [50–52]. Other work shows a decrease in both T0 and βt, and βt is considerable in small system or peripheral collisions [53]. Generally, the relative difference between two T0 in central and peripheral collisions is small (~10%).We observe a slightly higherT0 in central collisions at both the partonic- and particle level. If so, the centrality dependence of T0 is similar to those of the chemical freeze-out temperature and effective temperature. However, it is still an open question which T0 is larger. In any case, the explanation is understandable. A higher T0 in central collisions means a higher excitation, and a lower T0 in central collisions means a longer lifetime, of the hot and dense matter. Although we may extract the thermal parameters at the particle level, we aim to perform it at the partonic-level. The reason is that the similarity, commonality, and universality available in high energy collisions [54–61] should be related to partons, which are a deeper level in the structure of matter compared to the particle level.The present work shows that central collisions correspond to a higher excitation degree and a larger blast due to the more energy being deposited. Meanwhile, central collisions correspond to higher degree of equilibrium due to many particles being produced. Compared with the parameters at the particle level, the parameters at the partonic-level are extracted at earlier time moment and correspond to largerT0 and βt. However, the system at the particle level is at higher degree of equilibrium due to the fact that longer time is taken to approach the chemical and kinetic freeze-out. However, because the time span from parton phase to particle phase is very small, the difference between two sets of parameters at the partonic- and particle level is very small.In some cases, the fitting quality is not so good due to largeχ2 values. This is because only experimental uncertainty is considered. In fact, we have used the Monte Carlo method, which causes additional uncertainty. If we take the uncertainty induced by the Monte Carlo method to be approximately the same as the experimental uncertainty, χ2 values will be reduced by 1/2≈70.7%. In the case of large χ2 values, we may regard the fits as the qualitative ones. In most cases, the fits are good or approximate, and in few cases the fits are qualitative.Before summarizing and concluding, we would like to conduct some further discussions on the Monte Carlo results. As we know, most of baryons consist of up and down quarks, and the number of strange quarks is very small. Only when there is a formation of QGP, the system is likely to produce a lot of strange quarks and strange antiquarks, and then, strange quarks and strange antiquarks are able to combine with other neighbouring partons to form strange particles. Thus, the abundant yield of the final-state strange particles is an important signal of the existence of QGP matter [62–67]. Because of the formation of QGP with abundance of strange particles, the implementation of this work has become possible due to sufficient statistics.In the above discussions, we have used different sets of the parameters to fit thepT spectra of different strange particles. This means that we have used the multiscenario of kinetic freeze-out. In some cases, the two-scenario of kinetic freeze-out is also applicable, if we consider the single-strange and multistrange particles, respectively. In our opinion, the single-scenario is a rough description, the two-scenario is a slight refined description, and the multiple-scenario is a more refined description, of the process of kinetic freeze-out. This situation is analogous to the atomic spectra and their fine structures.


## 3.1. Comparison with Data

In the study, we analyze thepT spectra of strange particles generated in high energy lead–lead (Pb–Pb), proton–lead (p–Pb), and xenon–xenon (Xe–Xe) collisions at the Large Hadron Collider (LHC). Figure 1 shows the pT spectra of (a) K++K−, (b) K∗0+K¯∗0/2, (c) KS0, (d) ϕ, (e) Λ, (f) Ξ−, (g) Ξ¯+, (h) Ω−, and (i) Ω¯+ produced in Pb–Pb collisions with different centrality intervals marked in the panels at the collision energy sNN=2.76 TeV per nucleon pair. The symbols represent the data measured by the ALICE Collaboration [39–41]. The mid-pseudorapidity range of panel (a) is η<0.2, and those of other panels are y<0.5. For better distinguishing and comparing, the data with different centralities in different panels are multiplied by 10 to the n-th power. In addition, the gray rectangles represent the uncertainty of the experimental data, whose length is the uncertainty of the abscissa, whereas the width is the uncertainty of the ordinate, which denote the quadratic sum of the statistical and systematical uncertainties. The solid curves represent the Monte Carlo results calculated from the probability density function in Equation (2) and the pT expression in Equation (6).Figure 1
ThepT spectra of (a) K++K−, (b) K∗0+K¯∗0/(b), (c) KS0, (d) ϕ, (e) Λ, (f) Ξ−, (g) Ξ¯+, (h) Ω−, and (i) Ω¯+ in Pb–Pb collisions with different centrality and mid-pseudorapidity intervals at sNN=2.76 TeV. The symbols represent the data measured by the ALICE Collaboration [39–41]. The solid curves represent the Monte Carlo results, based on the probability density function Equation (2) and the pT expression in Equation (6).
(a)(b)(c)(d)(e)(f)(g)(h)(i)We useχ2=ΣjDataj−Fitj2/Uncertaintyj2 to quantify the deviation of the Monte Carlo results from the experimental data, where j denotes the order number of the data. The smaller the mean of χ2 is, the closer the Monte Carlo results to the experimental data are. The free parameters, the entropy-related index n, the kinetic freeze-out temperature T0, the correction index a0, and the average transverse flow velocity βt are extracted by the method of least squares. The values of free parameters, χ2 and the number of degree of freedom (ndof) are listed in Table 1. One can see that in most cases the Monte Carlo results agree well or approximately with the experimental pT spectra of strange particles. In few cases, the Monte Carlo results are in qualitative agreement with the experimental data, for which the values of χ2/ndof are quite large.Table 1
Values ofn, T0, a0, βt, χ2, and ndof corresponding to the solid curves in Figure 1.
FigureParticlesNN (TeV)SelectionnT0 (GeV)a0βt (c)χ2/ndofFigure1(a)K++K−2.760–5%7.1±0.20.221±0.003−0.022±0.0040.247±0.0051199/53η<0.25–10%7.0±0.20.219±0.003−0.024±0.0040.246±0.0051021/53Pb–Pb10–20%6.9±0.20.216±0.003−0.026±0.0040.244±0.005887/5320–40%6.7±0.20.212±0.003−0.029±0.0040.241±0.005793/5340–60%6.2±0.20.209±0.003−0.032±0.0040.238±0.005323/5360–80%6.1±0.20.208±0.003−0.035±0.0040.234±0.005115/53Figure1(b)K∗0+K¯∗0/22.760–20%7.7±0.20.174±0.0031.042±0.0100.309±0.00621/3y<0.520–40%7.4±0.20.170±0.0031.037±0.0100.306±0.00622/3Pb–Pb40–60%7.2±0.20.164±0.0031.032±0.0100.301±0.0063/360–80%6.9±0.20.152±0.0031.027±0.0100.294±0.0063/3Figure1(c)KS02.760–5%8.1±0.20.179±0.0030.227±0.0050.310±0.006357/28y<0.55–10%8.0±0.20.177±0.0030.226±0.0050.307±0.006337/28Pb–Pb10–20%7.8±0.20.175±0.0030.224±0.0050.305±0.006348/2820–40%7.5±0.20.174±0.0030.220±0.0050.303±0.006318/2840–60%7.1±0.20.171±0.0030.218±0.0050.301±0.006221/2860–80%6.7±0.20.169±0.0030.214±0.0050.298±0.006151/2880–90%6.4±0.20.168±0.0030.210±0.0050.295±0.00646/28Figure1(d)ϕ2.760–5%8.6±0.20.183±0.0031.013±0.0090.311±0.00614/3y<0.55–10%8.5±0.20.181±0.0031.012±0.0090.309±0.00631/3Pb–Pb10–20%8.3±0.20.179±0.0031.011±0.0090.308±0.00620/320–30%8.2±0.20.176±0.0031.009±0.0090.305±0.0067/330–40%7.9±0.20.174±0.0031.006±0.0090.302±0.0068/340–50%7.7±0.20.171±0.0030.999±0.0090.299±0.0065/350–60%7.4±0.20.166±0.0030.996±0.0090.296±0.0063/360–70%7.1±0.20.162±0.0030.991±0.0090.287±0.0061/370–80%6.9±0.20.158±0.0030.987±0.0090.285±0.0064/380–90%6.6±0.20.154±0.0030.982±0.0090.282±0.0062/3Figure1(e)Λ2.760–5%10.0±0.20.137±0.0021.429±0.0120.218±0.004272/26y<0.55–10%9.8±0.20.135±0.0021.426±0.0120.216±0.004211/26Pb–Pb10–20%9.7±0.20.134±0.0021.420±0.0120.213±0.004197/2620–40%9.5±0.20.132±0.0021.419±0.0120.212±0.004127/2640–60%9.2±0.20.128±0.0021.416±0.0120.210±0.00432/2660–80%9.0±0.20.126±0.0021.413±0.0120.207±0.00438/2680–90%8.8±0.20.122±0.0021.410±0.0120.204±0.00460/24Figure1(f)Ξ−2.760–10%11.9±0.30.137±0.0021.925±0.0140.238±0.005126/22y<0.510–20%11.8±0.30.135±0.0021.922±0.0140.236±0.00563/22Pb–Pb20–40%11.4±0.20.132±0.0021.918±0.0140.233±0.00529/2240–60%10.8±0.20.126±0.0021.912±0.0140.230±0.00521/2060–80%10.5±0.20.120±0.0021.904±0.0140.227±0.00537/15Figure1(g)Ξ¯+2.760–10%12.1±0.30.138±0.0021.924±0.0140.235±0.005122/22y<0.510–20%12.0±0.30.136±0.0021.922±0.0140.233±0.00544/22Pb–Pb20–40%11.8±0.30.134±0.0021.918±0.0140.231±0.00533/2240–60%11.4±0.30.130±0.0021.911±0.0140.227±0.00529/2060–80%11.1±0.20.126±0.0021.910±0.0140.225±0.00572/15Figure1(h)Ω−2.760–10%10.7±0.20.133±0.0022.026±0.0150.248±0.0055/8y<0.510–20%10.4±0.20.131±0.0022.024±0.0150.247±0.00513/8Pb–Pb20–40%10.3±0.20.129±0.0022.022±0.0150.245±0.00513/840–60%10.1±0.20.126±0.0022.020±0.0150.242±0.00511/760–80%9.8±0.20.124±0.0022.018±0.0150.237±0.0057/5Figure1(i)Ω¯+2.760–10%10.9±0.20.135±0.0022.022±0.0150.247±0.0057/8y<0.510–20%10.8±0.20.134±0.0022.020±0.0150.245±0.00513/8Pb–Pb20–40%10.6±0.20.132±0.0022.017±0.0150.241±0.0056/840–60%10.3±0.20.130±0.0022.014±0.0150.239±0.0057/760–80%9.8±0.20.122±0.0022.004±0.0150.235±0.0057/5ThepT spectra of (a) K++K−, (b) K∗0+K¯∗0/2, and (c) ϕ are displayed in Figure 2 for Pb–Pb collisions at sNN=5.02 TeV, quoted from the ALICE Collaboration [42, 43]. The mid-pseudorapidity is η<0.2 in panel (a) and y<0.5 in panels (b) and (c). Similarly to Figure 1, the data with different centralities are represented by different symbols, and the gray rectangles represent the uncertainty of the data. The solid curves show the Monte Carlo results. For clarity, we use a power index amplification from 101 to 109. The free parameters n, T0, a0, and βt, as well as χ2/nd of are listed in Table 2. One can see that the Monte Carlo results show an approximate agreement with the data in some cases and a qualitative agreement with the data in the other cases.Figure 2
ThepT spectra of (a) K++K−, (b) K∗0+K¯∗0/2, and (c) ϕ in Pb–Pb collisions at sNN=5.02 TeV. The symbols represent the data measured by the ALICE Collaboration [42, 43]. The solid curves represent the Monte Carlo results, based on the probability density function Equation (2) and the pT expression in Equation (6).
(a)(b)(c)Table 2
Values ofn, T0, a0, βt, χ2, and ndof corresponding to the solid curves in Figure 2.
FigureParticlesNN (TeV)SelectionnT0 (GeV)a0βt (c)χ2/ndofFigure2(a)K++K−5.020–5%6.8±0.20.228±0.0030.050±0.0050.251±0.0052716/48η<0.85–10%6.7±0.20.227±0.0030.048±0.0050.250±0.0052292/48Pb–Pb10–20%6.5±0.20.225±0.0030.046±0.0050.248±0.0052239/4820–30%6.3±0.20.221±0.0030.042±0.0050.246±0.0051814/4830–40%6.2±0.20.218±0.0030.039±0.0050.244±0.0051468/4840–50%5.9±0.20.215±0.0030.035±0.0050.238±0.005996/4850–60%5.7±0.20.211±0.0030.032±0.0050.231±0.005621/4860–70%5.5±0.20.209±0.0030.029±0.0050.227±0.005267/4870–80%5.3±0.20.207±0.0030.027±0.0050.224±0.005219/4880–90%5.1±0.20.204±0.0030.023±0.0050.221±0.005487/48Figure2(b)K∗0+K¯∗0/25.020–10%7.0±0.20.180±0.0031.028±0.0100.313±0.006103/10y<0.510–20%6.8±0.20.176±0.0031.034±0.0100.309±0.00681/10Pb–Pb20–30%6.6±0.20.173±0.0031.018±0.0100.305±0.00695/1030–40%6.5±0.20.171±0.0031.015±0.0100.302±0.00672/1040–50%6.4±0.20.171±0.0031.011±0.0100.300±0.00688/1050–60%6.3±0.20.168±0.0031.008±0.0100.298±0.00666/1060–70%6.2±0.20.166±0.0031.005±0.0100.297±0.00674/970–80%6.2±0.20.164±0.0031.000±0.0100.295±0.00632/8Figure2(c)ϕ5.020–10%8.1±0.20.184±0.0031.017±0.0100.302±0.006323/19y<0.510–20%8.0±0.20.182±0.0031.014±0.0100.301±0.006137/19Pb–Pb20–30%7.9±0.20.180±0.0031.009±0.0100.299±0.006144/1930–40%7.7±0.20.177±0.0031.006±0.0100.295±0.00687/1940–50%7.6±0.20.174±0.0031.002±0.0090.293±0.00677/1850–60%7.4±0.20.169±0.0030.996±0.0090.291±0.00612/1660–70%7.2±0.20.165±0.0030.991±0.0090.288±0.00616/1670–80%6.9±0.20.158±0.0030.987±0.0090.285±0.00616/15In addition to Pb–Pb collisions, the experimentalpT spectra in other collisions are also studied in order to better explore the properties of strange particles. Figure 3 shows the pT spectra of (a) K++K−, (b) K∗0+K¯∗0/2, (c) KS0, (d) ϕ, (e) Λ+Λ¯, (f) Ξ−+Ξ¯+/2, and (g) Ω−+Ω¯+/2 in p–Pb collisions at sNN=5.02 TeV. The experimental data in the panels are all quoted from the ALICE Collaboration when the midrapidity is 0<y<0.5 (a, c, e) and −0.5<y<0 (b, d, f, g) [44, 45]. Different symbols indicate the different centrality intervals. The Monte Carlo results are given by the solid curves. The coefficients in parentheses are magnification factors for better distinguishing the spectra in different centrality intervals. The relevant parameters n, T0, a0, and βt, as well as χ2/nd of are listed in Table 3. One can see that the Monte Carlo results are in good agreement with the data in most cases and approximately in agreement with the data in few cases.Figure 3
ThepT spectra of (a) K++K−, (b) K∗0+K¯∗0/2, (c) KS0, (d) ϕ, (e) Λ+Λ¯, (f) Ξ−+Ξ¯+/2, and (g) Ω−+Ω¯+/2 in p–Pb collisions at sNN=5.02 TeV. The symbols represent the data measured by the ALICE Collaboration [44, 45]. The solid curves represent the Monte Carlo results, based on the probability density function Equation (2) and the pT expression in Equation (6).
(a)(b)(c)(d)(e)(f)(g)Table 3
Values ofn, T0, a0, βt, χ2, and ndof corresponding to the solid curves in Figure 3.
FigureParticlesNN (TeV)SelectionnT0 (GeV)a0βt (c)χ2/ndofFigure3(a)K++K−5.020–5%5.5±0.20.223±0.0030.229±0.0050.257±0.0057/260<y<0.55–10%5.3±0.20.215±0.0030.226±0.0050.252±0.0056/26p–Pb10–20%5.1±0.20.210±0.0030.221±0.0050.248±0.0054/2620–40%4.9±0.20.199±0.0030.214±0.0050.241±0.0053/2640–60%4.7±0.20.188±0.0030.205±0.0050.230±0.0052/2660–80%4.5±0.20.167±0.0030.197±0.0050.221±0.0052/2680–100%4.3±0.20.139±0.0030.183±0.0050.212±0.0055/26Figure3(b)K∗0+K¯∗0/25.020–20%6.8±0.20.238±0.0030.550±0.0070.267±0.00572/17−0.5<y<020–40%6.7±0.20.231±0.0030.541±0.0070.258±0.00556/17p–Pb40–60%6.4±0.20.225±0.0030.536±0.0070.254±0.00516/1760–80%6.2±0.20.211±0.0030.528±0.0070.249±0.00511/1780–100%5.7±0.20.183±0.0030.511±0.0070.233±0.00519/12Figure3(c)KS05.020–5%6.6±0.20.219±0.0030.202±0.0050.328±0.00678/290<y<0.55–10%6.5±0.20.216±0.0030.200±0.0050.325±0.00639/29p–Pb10–20%6.4±0.20.214±0.0030.199±0.0050.323±0.00655/2920–40%6.2±0.20.209±0.0030.197±0.0050.321±0.00645/2940–60%6.0±0.20.204±0.0030.192±0.0050.317±0.00661/2960–80%5.7±0.20.189±0.0030.186±0.0050.312±0.00683/2980–100%5.8±0.20.174±0.0030.184±0.0050.304±0.006145/29Figure3(d)ϕ5.020–5%6.7±0.20.199±0.0030.812±0.0090.332±0.00618/15−0.5<y<05–10%6.6±0.20.197±0.0030.810±0.0090.330±0.00626/15p–Pb10–20%6.4±0.20.194±0.0030.806±0.0090.324±0.00629/1520–40%6.3±0.20.192±0.0030.803±0.0090.322±0.00625/1540–60%6.1±0.20.189±0.0030.799±0.0080.319±0.00620/1560–80%5.5±0.20.165±0.0030.783±0.0080.312±0.00638/1480–100%5.1±0.20.147±0.0030.775±0.0080.305±0.00663/14Figure3(e)Λ+Λ¯5.020–5%7.7±0.20.132±0.0021.426±0.0120.226±0.0046/150<y<0.55–10%7.6±0.20.130±0.0021.424±0.0120.224±0.0044/15p–Pb10–20%7.5±0.20.127±0.0021.421±0.0120.221±0.0042/1520–40%7.3±0.20.123±0.0021.417±0.0120.218±0.0044/1540–60%7.0±0.20.112±0.0021.410±0.0120.211±0.0044/1560–80%6.7±0.20.104±0.0021.407±0.0120.206±0.00422/1580–100%6.4±0.20.089±0.0021.403±0.0120.202±0.00423/15Figure3(f)Ξ−+Ξ¯+/25.020–5%9.0±0.20.179±0.0030.703±0.0080.225±0.00415/12−0.5<y<05–10%8.9±0.20.177±0.0030.701±0.0080.223±0.00414/12p–Pb10–20%8.4±0.20.171±0.0030.692±0.0080.218±0.0049/1220–40%8.0±0.20.165±0.0030.688±0.0080.215±0.00418/1240–60%7.2±0.20.150±0.0030.678±0.0080.209±0.00440/1260–80%7.0±0.20.130±0.0020.669±0.0080.203±0.00419/1180–100%6.4±0.20.119±0.0020.662±0.0080.194±0.00425/10Figure3(g)Ω−+Ω¯+/25.020–5%7.6±0.20.174±0.0030.788±0.0080.240±0.0055/3−0.5<y<05–10%7.3±0.20.173±0.0030.786±0.0080.238±0.0054/3p–Pb10–20%7.2±0.20.170±0.0030.783±0.0080.235±0.0054/320–40%7.0±0.20.167±0.0030.779±0.0080.232±0.0041/340–60%6.7±0.20.150±0.0030.769±0.0080.223±0.0049/360–80%6.4±0.20.141±0.0020.761±0.0080.214±0.0045/380–100%5.9±0.20.132±0.0020.755±0.0080.206±0.0042/−Besides, we have also studied thepT spectra of (a) K++K− and (b) ϕ produced in Xe–Xe collisions at sNN=5.44 TeV in Figure 4. The symbols represent the data measured by the ALICE Collaboration, and the mid-pseudorapidity is η<0.8 [46]. Similarly, the data with different centralities are expressed as different symbols with gray rectangles, and the solid curves show the Monte Carlo results. Another, 10n in legends are magnification factors for better distinguishing the spectra. The relevant parameters n, T0, a0, and βt, as well as χ2/nd of are listed in Table 4. Once again, one can see that the Monte Carlo results are in good agreement with the experimental data.Figure 4
ThepT spectra of (a) K++K− and (b) ϕ produced in Xe–Xe collisions at sNN=5.44 TeV. The symbols represent the data measured by the ALICE Collaboration [46]. The solid curves represent the Monte Carlo results, based on the probability density function Equation (2) and the pT expression in Equation (6).
(a)(b)Table 4
Values ofn, T0, a0, βt, χ2, and ndof corresponding to the solid curves in Figure 4.
FigureParticlesNN (TeV)SelectionnT0 (GeV)a0βt (c)χ2/ndofFigure4(a)K++K−5.440–5%8.3±0.20.199±0.0030.399±0.0060.257±0.00573/33η<0.85–10%8.1±0.20.198±0.0030.396±0.0060.255±0.00558/33Xe–Xe10–20%7.9±0.20.194±0.0030.395±0.0060.254±0.00555/3320–30%7.7±0.20.192±0.0030.392±0.0060.252±0.00537/3330–40%7.5±0.20.187±0.0030.390±0.0060.250±0.00520/3340–50%7.3±0.20.183±0.0030.386±0.0060.247±0.00516/3350–60%7.2±0.20.179±0.0030.384±0.0060.244±0.00514/3360–70%7.1±0.20.174±0.0030.381±0.0060.238±0.00516/3270–90%6.9±0.20.162±0.0030.368±0.0060.230±0.00527/32Figure4(b)ϕ5.440–10%9.0±0.20.219±0.0030.810±0.0090.338±0.00613/6η<0.810–30%8.9±0.20.216±0.0030.804±0.0090.334±0.00610/6Xe–Xe30–50%8.7±0.20.213±0.0030.800±0.0080.332±0.0067/650–70%8.5±0.20.209±0.0030.797±0.0080.327±0.00611/670–90%8.3±0.20.203±0.0030.792±0.0080.317±0.00610/4


## 3.2. Tendencies of Parameters

To further understand the regularities shown by thepT spectra of strange particles, we analyze the tendencies of various parameters with changing centrality observed in the work. Figure 5 shows the dependence of entropy-related index (a, b) n on centrality and C in Pb–Pb, (c) p–Pb, and (d) Xe–Xe collisions at the LHC energies. The symbols are quoted from Tables 1–4, where the results from different particles are represented by different symbols. As an entropy-related index, n=1/q−1 or n=q′/q′−1 describes the degree of nonequilibrium of the system, where q (q′) is the entropy index. Generally, a larger n or a q (q′) closer to 1 corresponds to a higher degree of equilibrium. The present work shows that n is large enough or q (q′) is close to 1, and the system stays in an approximate equilibrium. Meanwhile, the system in central collisions stays in higher degree of equilibrium. In addition, as the rest mass of strange particle increases, the value of n increases. The multistrange particles correspond to larger n than the single-strange particles. These results imply that the system stays in larger degree of equilibrium when it forms multistrange particles.Figure 5
Dependence of entropy-related index (a, b)n on centrality and C in Pb–Pb, (c) p–Pb, and (d) Xe–Xe collisions.
(a)(b)(c)(d)Similar to Figure5, Figure 6 shows the dependence of kinetic freeze-out temperature T0 on centrality C in the mentioned collisions at the LHC. One can see that T0 decreases slightly with the decrease of centrality from central to peripheral collisions in most cases. In few cases, the decrease is significant. In addition, T0 extracted from the single-strange particle spectra is larger than that from the multistrange particle spectra. This finding suggests that the single-strange particles are formed earlier than the multistrange particles, though the latter may leave the system earlier than the former in the hydrodynamic evolution due to different masses.Figure 6
Dependence of kinetic freeze-out temperature (a, b)T0 on centrality and C in Pb–Pb, (c) p–Pb, and (d) Xe–Xe collisions.
(a)(b)(c)(d)As shown in Figure7, we present the changing law of the correction index a0 on centrality C in the mentioned collisions at the LHC. One can see that with the decrease of centrality from central to peripheral collisions, a0 is almost invariant, though in few cases a0 shows very slight decrease. In most cases, the values of a0 are far from 1, which means that the introduction of a0 is necessary. We have also compared the present fits with a0≠1 with those by a0=1. An obvious difference appears in the low-pT region. This implies that the fits by a0=1 are not suitable. In fact, in the fits of the present work, we use a0=1 at the first, then we change a0 to fit the pT spectra if a0=1 is not satisfactory.Figure 7
Dependence of correction index (a, b)a0 on centrality and C in Pb–Pb, (c) p–Pb, and (d) Xe–Xe collisions.
(a)(b)(c)(d)The dependence of the average transverse flow velocityβt on centrality C in the mentioned collisions at the LHC is displayed in Figure 8. Similar conclusions to the dependences of n on C and T0 on C can be obtained. That is, βt decreases slightly with the decrease of centrality from central to peripheral collisions in most cases. In few cases, the decrease is significant. In addition, βt extracted from the single-strange particle spectra is larger than that from the multistrange particle spectra. This finding also suggests that the single-strange particles are formed earlier than the multistrange particles.Figure 8
Dependence of average transverse flow velocity (a, b)βt on centrality and C in Pb–Pb, (c) p–Pb, and (d) Xe–Xe collisions.
(a)(b)(c)(d)In order to further study the final states of the strange particles and the disorder degree of the system, we give the results of pseudoentropyShadron′=−ΣfpT/ΣfpTlnfpT/ΣfpT [28], which is based on the probability density function fpT of pT, inspired by the entropy Shadron=−ΣPNlnPN [47–49], based on the probability density function PN of multiplicity N, where pT is in the units of GeV/c. As is done in our recent work [24], the width of pT bin is taken to be 0.1 GeV/c, though the width is changeable. The unit of fpT is neglected due to the fact that PN is dimensionless.In Figure9, we demonstrate the relation of the pseudoentropy Shadron′ versus the centrality C. It is seen from the results that Shadron′ decreases slightly with the decrease of centrality from central to peripheral collisions in most cases. Only in few cases, Shadron′ decreases significantly with the decrease of C. The dependence of Shadron′ on C is similar to those of n, T0, and βt on C. Generally, the multistrange particles show larger Shadron′ than the single-strange particles, though both Shadron′ for multi and single-strange particles are negative. This also reflects the mass-dependence of Shadron′, which is similar to those of the other parameters.Figure 9
Dependence of pseudoentropy (a, b)Shadron′ on centrality C in Pb–Pb, (c) p–Pb, and (d) Xe–Xe collisions.
(a)(b)(c)(d)Analyzing Figures5–9, one can see that the correlations between n and T0, n and βt, n and Shadron′, T0 and βt, T0 and Shadron′, and βt and Shadron′ are positive; while the correlations between a0 and n, a0 and T0, a0 and βt, and a0 and Shadron′ are very small or negligible. These results are understandable due to the fact that all n, T0, and βt determine mainly, and Shadron′ is mostly defined by, the shape of the spectra in intermediate- and high-pT regions; while a0 determines mainly the spectra in low- or even very low-pT region. Of course, because of the requirement of normalization, all parameters affect the spectra in the whole pT region, though the level of influence in different pT regions are different.


## 3.3. Further Discussion

Although the extraction ofT0 and βt in present work is performed at the partonic-level, most extractions in the literature have been done at the particle level, showing inconsistent trends. For example, in terms of dependence of parameters on centrality from central to peripheral collisions, some works show an increase in T0 and a decrease in βt, and βt≈0 in small system or peripheral collisions [50–52]. Other work shows a decrease in both T0 and βt, and βt is considerable in small system or peripheral collisions [53]. Generally, the relative difference between two T0 in central and peripheral collisions is small (~10%).We observe a slightly higherT0 in central collisions at both the partonic- and particle level. If so, the centrality dependence of T0 is similar to those of the chemical freeze-out temperature and effective temperature. However, it is still an open question which T0 is larger. In any case, the explanation is understandable. A higher T0 in central collisions means a higher excitation, and a lower T0 in central collisions means a longer lifetime, of the hot and dense matter. Although we may extract the thermal parameters at the particle level, we aim to perform it at the partonic-level. The reason is that the similarity, commonality, and universality available in high energy collisions [54–61] should be related to partons, which are a deeper level in the structure of matter compared to the particle level.The present work shows that central collisions correspond to a higher excitation degree and a larger blast due to the more energy being deposited. Meanwhile, central collisions correspond to higher degree of equilibrium due to many particles being produced. Compared with the parameters at the particle level, the parameters at the partonic-level are extracted at earlier time moment and correspond to largerT0 and βt. However, the system at the particle level is at higher degree of equilibrium due to the fact that longer time is taken to approach the chemical and kinetic freeze-out. However, because the time span from parton phase to particle phase is very small, the difference between two sets of parameters at the partonic- and particle level is very small.In some cases, the fitting quality is not so good due to largeχ2 values. This is because only experimental uncertainty is considered. In fact, we have used the Monte Carlo method, which causes additional uncertainty. If we take the uncertainty induced by the Monte Carlo method to be approximately the same as the experimental uncertainty, χ2 values will be reduced by 1/2≈70.7%. In the case of large χ2 values, we may regard the fits as the qualitative ones. In most cases, the fits are good or approximate, and in few cases the fits are qualitative.Before summarizing and concluding, we would like to conduct some further discussions on the Monte Carlo results. As we know, most of baryons consist of up and down quarks, and the number of strange quarks is very small. Only when there is a formation of QGP, the system is likely to produce a lot of strange quarks and strange antiquarks, and then, strange quarks and strange antiquarks are able to combine with other neighbouring partons to form strange particles. Thus, the abundant yield of the final-state strange particles is an important signal of the existence of QGP matter [62–67]. Because of the formation of QGP with abundance of strange particles, the implementation of this work has become possible due to sufficient statistics.In the above discussions, we have used different sets of the parameters to fit thepT spectra of different strange particles. This means that we have used the multiscenario of kinetic freeze-out. In some cases, the two-scenario of kinetic freeze-out is also applicable, if we consider the single-strange and multistrange particles, respectively. In our opinion, the single-scenario is a rough description, the two-scenario is a slight refined description, and the multiple-scenario is a more refined description, of the process of kinetic freeze-out. This situation is analogous to the atomic spectra and their fine structures.


## 4. Summary and Conclusions

The transverse momentum spectra of strange particles, includingK±, ϕ, Ω, Ξ, and so on, in high energy collisions are analyzed by considering the contributions of constituent quarks. Each constituent quark contributes to the transverse momentum obeying the modified Tsallis–Pareto-type function with random azimuthal angle. The transverse momentum of strange particle is the vector superposition of the transverse momenta of the two or three constituent quarks. The results calculated by the Monte Carlo method are in good agreement with the experimental data in most cases in Pb–Pb, p–Pb, and Xe–Xe collisions at a few TeV, measured by the ALICE Collaboration at the LHC. In few cases, the agreement is qualitative due to quite large χ2 values.With the decrease of centrality from central to peripheral collisions, the free parametersn, T0, and βt, as well as the derived parameter Shadron′ decrease slightly in most cases and decrease significantly in few cases. Meanwhile, the free parameter a0 is almost invariant in most cases and decreases slightly in few cases. These results imply that central collisions stay in the state with larger degree of equilibrium, higher excitation, and larger blast than peripheral collisions, though both the central and peripheral collisions stay in approximate equilibrium when the system produces a lot of strange particles.The correlations betweenn and T0, n and βt, n and Shadron′, T0 and βt, T0 and Shadron′, and βt and Shadron′ are positive, because all n, T0, and βt determine mainly, and Shadron′ is mostly defined by the shape of the spectra in intermediate- and high-pT regions. The correlations between a0 and n, a0 and T0, a0 and βt, and a0 and Shadron′ are very small or negligible, because a0 determines mainly the spectra in (very) low-pT region.We have used the multiscenario of kinetic freeze-out, though in some cases the two-scenario of kinetic freeze-out may also be applicable if we consider a part ofpT region. Meanwhile, if we consider the single-strange and multistrange particles, respectively, the single-strange particles are shown to form earlier than the multistrange ones. We may regard the relations of various scenarios as follows: the single-scenario is a rough description, the two-scenario is a slightly refined description, and the multiple-scenario is a more refined description of the process of kinetic freeze-out.



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*Source: 5949610-2022-11-02.xml*