Excitation function of initial temperature of heavy flavor quarkonium emission source in high energy collisions

The transverse momentum spectra of $J/\psi$, $\psi(2S)$, and $\Upsilon(nS, n=1,2,3)$ produced in proton-proton ($p$+$p$), proton-antiproton ($p$+$\overline{p}$), proton-lead ($p$+Pb), gold-gold (Au+Au), and lead-lead (Pb+Pb) collisions over a wide energy range are analyzed by the (two-component) Erlang distribution, the Hagedorn function (the inverse power-law), and the Tsallis-Levy function. The initial temperature is obtained from the color string percolation model due to the fit by the (two-component) Erlang distribution in the framework of multisource thermal model. The excitation functions of some parameters such as the mean transverse momentum and initial temperature increase from dozens of GeV to above 10 TeV. The mean transverse momentum and initial temperature decrease (increase slightly or do not change obviously) with the increase of rapidity (centrality). Meanwhile, the mean transverse momentum of $\Upsilon(nS, n=1,2,3)$ is larger than that of $J/\psi$ and $\psi(2S)$, and the initial temperature for $\Upsilon(nS, n=1,2,3)$ emission is higher than that for $J/\psi$ and $\psi(2S)$ emission, which shows a mass-dependent behavior.


Introduction
The excitation functions of some physical quantities are significative to help us to understand the nuclear reaction mechanism, and how the interacting system evolution from the stage of initial state to that of kinetic freeze-out. For instance, the mean transverse momentum (mean p T , i.e. p T ) represents the kinetic state of particles. The initial temperature (T i ) [1,2,3,4,5] shows the violent degree of collisions or the excitation degree of emission source. By the analysis of the excitation functions of p T and T i , we can learn more about the process of high energy collisions. In high energy collisions, the excitation functions of some parameters such as p T and T i can be obtained from the p T spectra of produced particles.
In a data-driven reanalysis, to obtain p T and T i , at the first place, we need the p T spectra of particles in experiments. At the second place, we should choose appropriate functions such as the Erlang distribution [6,7,8], the Hagedorn function or the inverse power-law [9,10], the Tsallis-Levy function [11,12], and others. At the last place, we use the chosen functions to fit the experi-ential data on particles. By describing the p T spectra, the parameters from the selected functions can be extracted. By comparing the parameters obtained from the experiential data at different energies, centralities, and rapidities, we can find out the dependences of parameters on these quantities. These dependences are related to excitation and expansion degrees of emission source, which is beneficial to understand the mechanisms of nuclear reactions and system evolutions.
Except for the two derived parameters p T and T i , we can obtain other related parameters by using the method which is similar to abstract p T and T i . For example, using the Hagedorn function or the inverse power-law [9,10] and the Tsallis-Levy function [11,12] to fit p T spectra, some free parameters such as p 0 , n 0 , T , and n in the mentioned functions which will be discussed in section 2 can be abstracted. These free parameters are also useful to understand particle productions and system evolution. Not only the excitation functions of derived parameters p T and T i but also the trends of free parameters p 0 , n 0 , T , and n can be studied from the fit to p T spectra.

Formalism and method
i) The (two-component) Erlang distribution According to the multisource thermal model [6,7,8], a given particle is produced in the collision process where a few partons have taken part in. Each (the ith) parton is assumed to contribute to an exponential function [f i (p t )] of transverse momentum (p t ) distribution. Let p t denotes the mean transverse momentum contributed by the i-th parton, we have the probability density function of p t to be which is normalized to 1. The contribution p T of all N partons which have taken part in the collision process is the folding of N exponential functions [6,7,8]. We have the p T distribution f E (p T ) (the probability density function of p T ) of final state particles to be the Erlang distribution which is naturally normalized to 1. The mean p T is p T = N p t . In the two-component Erlang distribution, we have where k E denotes the contribution fraction of the first component, N 1 (N 2 ) denotes the number of partons in the first (second) component, and p t 1 ( p t 2 ) denotes the mean transverse momentum contributed by each parton in the first (second) component. The mean p T is p T = k E N 1 p t 1 + (1 − k E )N 2 p t 2 , where N 1 = 2-5 and N 2 = 2 in most cases.
ii) The (two-component) Hagedorn function The Hagedorn function is an inverse power-law which is suitable to describe wide p T spectra of particles produced in hard scattering process. In refs. [9,10], the Hagedorn function or the inverse power-law shows the probability density function of p T to be where p 0 and n 0 are the free parameters and A is the normalization constant which is related to p 0 and n 0 and results in ∞ 0 f H (p T )dp T = 1. Eq. (6) is an empirical formula inspired by quantum chromodynamics (QCD). We call Eq. (4) the Hagedorn function or the inverse power-law [9,10].
In the case of using two-component Hagedorn function, we have where k H denotes the contribution fraction of the first component, A 1 (A 2 ) is the normalization constant which results in the first (second) component to be normalized to 1, and p 01 (p 02 ) and n 01 (n 02 ) are free parameters related to the first (second) component. To combine the free parameters of the two components, we have p 0 = k H p 01 +(1−k H )p 02 and n 0 = k H n 01 +(1−k H )n 02 . Generally, Eq. (4) is possible to describe the spectra in both the low-and high-p T regions. In fact, the spectra in the low-and high-p T regions represent similar trend in some cases. This is caused due to the similarity [35,36,37,38,39,40,41,42,43,44,45] which is widely existent in high energy collisions. In addition, one can revise Eq. (4) if needed in different ways [46,47,48,49,50,51,52] which result in low probability in low-or high-p T region according to experimental spectra. To discuss various revisions of the Hagedorn function or the inverse power-law [9,10] is beyond the focus of this paper. We shall not discuss anymore on this issue. For a very wide p T spectrum, Eq. (5) is possibly needed.
iii) The (two-component) Tsallis-Levy function The Tsallis statistics [11] has wide applications in high energy collisions. There are various forms of the Tsallis distribution or function. In this work, we use the Tsallis-Levy function [12] f L (p T ) = Cp T 1 + where T and n are free parameters, p 2 T + m 2 0 ≡ m T is the transverse mass, m 0 is the rest mass of the considered particle, and C is the normalized constant which is related to T , n, and m 0 and results in ∞ 0 f L (p T )dp T = 1.
We notice that f L (p T ) is related to particle mass m 0 , which is not the case of f E (p T ) and f H (p T ) presented in Eqs. (2) and (4) respectively. Although f L (p T ) is related to m 0 , this relation is not strong due to m 0 appearing only in p 2 T + m 2 0 − m 0 . In the case of using two-component Tsallis-Levy function, we have where k L denotes the contribution fraction of the first component, C 1 (C 2 ) is the normalization constant which results in the first (second) component to be normalized to 1, and T 1 (T 2 ) and n 1 (n 2 ) are free parameters. To combine the free parameters of the two components, we have T = k L T 1 + (1 − k L )T 2 and n = k L n 1 + (1 − k L )n 2 .
iv) The initial temperature According to the color string percolation model [53,54,55], the initial temperature of the emission source is determined by where is the square of the root-mean-square of p T due to ∞ 0 f 1,2,3 (p T )dp T = 1. If the x-component (p x ) and y-component (p y ) of momentum p are considered before or after the transformation of reference frame, we have Obviously, T i is invariant in the transformation of reference frame. In the source's rest frame and under the assumption of isotropic emission, if the z-component of momentum is p ′ z , we also have
The p T spectra, d 2 σ/dp T dy, of (a)(c)(e)(g) prompt      tion [19,20,21,22] in the rapidity intervals of (a)(b) 1.5 < y < 2.0, 2.0 < y < 2.5, 2.5 < y < 3.0, 3.0 < y < 3.5, and 3.5 < y < 4.0 and (c)-(h) 2.0 < y < 2.5, 2.5 < y < 3.0, 3.0 < y < 3.5, 3.5 < y < 4.0, and 4.0 < y < 4.5, and scaled by different amounts marked in the panels. The solid, dashed, and dotted curves rep- σ/dp T dy (nb/(GeV/c))      Levy function (Eq. (6)), respectively. The method of least square is used to obtain the best parameter values which are listed in Tables A1 and A2 with χ 2 and ndof. One can see that the experimental p T spectra of J/ψ via different production modes in different rapidity intervals in p+p and p+Pb collisions at high energies are approximately fitted by the Erlang distribution, the Hagedorn function, and the Tsallis-Levy function. Figure 3 shows the p T spectra, (a)(c)-(g) d 2 σ/dp T dy and (b) dσ/dp T , of ψ(2S) via different production modes. The different symbols represent the experimen-   σ/dp T dy (pb/(GeV/c)) CDF p+p 1.8 TeV |y| < 0.4 dσ/dp T (nb/(GeV/c))  σ/dp T dy (pb/(GeV/c))   and (d) 0.0 < |y| < 0.3, 0.3 < |y| < 0.6, 0.6 < |y| < 0.9, and 0.9 < |y| < 1.2, where some data points are scaled by different amounts marked in the panels. The data points are fitted by the Erlang distribution (Eq. (3), the solid curve), the Hagedorn function (Eq. (4), the dashed curves), and the Tsallis-Levy function (Eq. (6), the dotted curves). The values of free parameters are listed in Tables A1 and A2 with χ 2 and ndof. For Fig. 4(d), the two-component Eqs. (5) and (7) are used, where the free parameters for the first (second) component are listed in the first (second) row. One can see that the experimental p T spectra of ψ(2S) via different production modes in different rapidity intervals in p+p collisions at high energies are also approximately fitted by the Erlang distribution, the (two-component) Hagedorn function, and the (two-component) Tsallis-Levy function.
In Fig. 5, the p T spectra, (a)(c)-(f) d 2 σ/dp T dy and (b) dσ/dp T , of (a)-(c) Υ(nS, n = 1, 2, σ/dp T dy (pb/(GeV/c)) LHCb p+p 8 TeV σ/dp T dy (pb/(GeV/c)) σ/dp T dy (pb/(GeV/c)) σ/dp T dy (pb/(GeV/c)) In Fig. 6, the p T spectra, d 2 σdp T dy, of (a)(d)(e) (a)-(c) and (e)-(f) are 2.0 < y < 2.5, 2.5 < y < 3.0, 3.0 < y < 3.5, 3.5 < y < 4.0, and 4.0 < y < 4.5. The rapidity intervals for panel (d) are 1.5 < y < 2.0 2.0 < y < 2.5, 2.5 < y < 3.0, 3.0 < y < 3.5, and 3.5 < y < 4.0. Different sets of data points are scaled by different amounts shown in the panels. The data points  Table A1 and A2 with χ 2 and ndof. Once again, one can see that the experimental p T spectra of Υ(nS, n = 1, 2, 3) → µ + µ − in different rapidity intervals in p+p and p+Pb collisions at high energies are approximately fitted by the Erlang distribution, the Hagedorn function, and the Tsallis-Levy function.    with the increases of collision energy and particle mass.   = 1, 2, 3). One can see that p 0 (n 0 ) increases (decreases) slightly with the increase of event centrality from peripheral to central collisions, and decreases (increase) (slightly) with the increase of rapidity from mid-rapidity to forward rapidity. Meanwhile, p 0 (n 0 ) increases with the increases of collision energy and particle mass.   = 1, 2, 3). One can see that T (n) increases (decreases) slightly with the increase of event centrality from peripheral to central collisions, and decreases (increase) (slightly) with the increase of rapidity from mid-rapidity to forward rapid-ity. Meanwhile, T (n) increases with the increases of collision energy and particle mass. We may explain the tendency of derived p T and T i which have similar tendency with p T . With the increase of collision energy, the violent degree of collisions increases obviously due to large energy transfer, which results in the obvious increase of p T . With the increase of centrality, the degree of multiple-scattering increases due to more participant nucleons and produced particles taking part in the scattering process, which results in slight increase of emission angle and then slight increase of p T . With the increase of rapidity, the energy transfer decreases due to larger penetrability between participant nucleons. Meanwhile, the degree of multiplescattering also decreases due to less produced particles taking part in the scattering process. These two factors result in the decrease of p T . It is natural that p T increases with the increase of m 0 .
The tendency of p 0 (T ) and n 0 (n) with collision energy are also explained by more violent collision at higher energy. Both p 0 (T ) and n 0 (n) increase with the increase of collision energy. This means that p 0 (T ) and n 0 (n) are positively correlative at different energies. Meanwhile, for a given p T spectrum or in given collisions, an increase in p 0 (T ) is concomitant with a decrease in n 0 (n). This means that p 0 (T ) and n 0 (n) are negatively correlative in given collisions (at given energy). There are entanglements when determine p 0 (T ) and n 0 (n).
The correlation between p 0 (T ) and n 0 (n) is similar to that between kinetic freeze-out temperature and transverse flow velocity [56,57] which also show positive correlation at different energies and negative correlation in given spectrum. If p 0 (T ) corresponds to kinetic freeze-out temperature, n 0 (n) should correspond to transverse flow velocity. Meanwhile, the results obtained in this work are in agreement with our recent work [58], which shows mass-dependent parameters. In particular, with the increase of particle mass, p T , T i , p 0 , and n 0 increase.

Summary and conclusions
To summary, the transverse momentum spectra of J/ψ, ψ(2S), and Υ(nS, n = 1, 2, 3) produced in p+p, p+p, p+Pb, Au+Au, and Pb+Pb collisions over an energy range from dozens of GeV to above 10 TeV have been analyzed by the (two-component) Erlang distribution, the Hagedorn function, and the Tsallis-Levy function. The function results are approximately in agreement with the experimental data measured by several international collaborations. The values of related parameters are extracted from the fit process and the excitation functions of these parameters are obtained.
The excitation functions of parameters p T and T i increase from dozens of GeV to above 10 TeV. Meanwhile, p T and T i increase (slightly) with event centrality and particle mass and decrease from mid-rapidity to forward rapidity. These tendencies render that these parameters describe the excitation and expansion degrees of the system. At higher energy, larger energy transfer had happened, which results in higher excitation and expansion degrees of the system. In central collisions and at mid-rapidity, larger energy transfer and further multiple-scattering had happened, which also results in higher excitation and expansion degrees of the system.
The parameters p 0 (T ) and n 0 (n) increase with the collision energy, which reflects the degree of energy deposition and transfer. In given collisions, p 0 (T ) and n 0 (n) are negatively correlative. At different energies, p 0 (T ) and n 0 (n) are positively correlative. There are entanglements when determine p 0 (T ) and n 0 (n). The correlation between p 0 (T ) and n 0 (n) is similar to that between kinetic freeze-out temperature and transverse flow velocity. If p 0 (T ) corresponds to kinetic freeze-out temperature, n 0 (n) should correspond to transverse flow velocity. Conflict of Interest: The authors declare that there are no conflicts of interest regarding the publication of this paper. The funding agencies have no role in the design of the study; in the collection, analysis, or interpretation of the data; in the writing of the manuscript, or in the decision to publish the results.
Data Availability: The data used to support the findings of this study are included within the article and are cited at relevant places within the text as references.
Compliance with Ethical Standards: The authors declare that they are in compliance with ethical standards regarding the content of this paper.