Excitation functions of related parameters from transverse momentum (mass) spectra in high energy collisions

Transverse momentum (mass) spectra of positively and negatively charged pions ($\pi^+$ and $\pi^-$), positively and negatively charged kaons ($K^+$ and $K^-$), protons and antiprotons ($p$ and $\bar p$) produced at mid-(pseudo)rapidity in various collisions at high energies are analyzed in this work. The experimental data measured in central gold-gold (Au-Au) collisions at the Alternating Gradient Synchrotron (AGS) by the E866, E895, and E802 Collaborations and at the Relativistic Heavy Ion Collider (RHIC) by the STAR and PHENIX Collaborations, in central lead-lead (Pb-Pb) collisions at the Super Proton Synchrotron (SPS) by the NA49 Collaboration and at the Large Hadron Collider (LHC) by the ALICE Collaboration, as well as in inelastic (INEL) proton-proton ($pp$) collisions at the SPS by the NA61/SHINE Collaboration, at the RHIC by the PHENIX Collaboration, and at the LHC by the CMS Collaboration are studied. The (two-component) standard distribution is used to fit the data and the excitation function of effective temperature is obtained. Then, the excitation functions of kinetic freeze-out temperature, transverse flow velocity, and initial temperature are extracted. In the considered collisions, the four parameters increase with the increase of collision energy in general, and the kinetic freeze-out temperature appears the trend of saturation at the top RHIC and LHC.


Introduction
It is believed that the environment of high temperature and high density is formed in the system evolution process of central nucleus-nucleus (AA) collisions at high energy [1,2,3], in which quark-gluon plasma (QGP) is created and many particles are produced [4,5,6]. At present, it is impossible to detect directly the system evolution process of collisions due to very short time interval. Instead, the particle spectra at the stage of kinetic freeze-out can be measured in experiments and the mechanisms of system evolutions and particle productions can be studied indirectly [7,8,9], though the particle ratios reflect the property at the stage of chemical freeze-out. As for peripheral AA collisions and small collision system, the situation is similar if the multi-plicity is high enough due to small system also appears collective behavior [10,11].
Although there are different stages in the system evolution [1,2,3], the initial state is the most important due to its determining effect to the system evolution. In addition, chemical and kinetic freeze-outs are two important stages in the system evolution. At the stage of chemical freeze-out, the system had happened the phase transition from QGP to hadronic matter, and the constituents and ratios of various particles do not change anymore. At the stage of kinetic freeze-out, the collisions among various particles are elastic, and the transverse momentum spectra of various particles are fixed [2,7]. In small system with low multiplicity, QGP is not expected to create in it due to a very small vol-ume of the violent collision region. From the similar multiplicity at the energy up to 200 GeV, small system is more similar to peripheral AA collisions, but not to central AA collisions [12]. At the energy down to 10 or several GeV, the situation is different due to the fact that baryon-dominated effect plays more important role in AA collisions [13].
The temperatures at the stages of kinetic freezeout, chemical freeze-out, and initial state are called the kinetic freeze-out temperature (T 0 or T kin ), chemical freeze-out temperature (T ch ), and initial temperature (T i ), respectively. Besides, one also has the effective temperature (T ) in which the contribution of transverse flow is not excluded. It is expected that various temperatures can be extracted from particle spectra, which are usually model dependent. Generally, T is unavoidably model dependent, and T ch extracted from particle ratios in the statistical thermal model [14,15,16,17] is also model dependent. We hope to use a less model dependent method to extract T 0 , β T , and T i . The quantities used in the method are expected to relate to experimental data as much as possible, though they can be calculated from models in some cases.
To perform a less model dependent method, we would like to use the standard distribution or its twocomponent form to obtain T by fitting the experimental transverse momentum (p T ) or transverse mass (m T ) spectra of various particles. The standard distribution includes the Bose-Einstein, Fermi-Dirac, and Boltzmann distributions, in which the effective temperature parameter T are the closest to that in the ideal gas model when comparing T with those in other distributions. After the fitting, we hope to extract T 0 and β T from the relation to average p T ( p T ) due to the Erlang distribution in multisource thermal model [18,19,20] and T i from the relation to root-mean-square p T ( p 2 T ) due to the color string percolation model [21,22,23]. Obviously, p T and p 2 T depend on the data themselves, though they can be calculated from models.
In this work, the p T (m T ) spectra of positively and negatively charged pions (π + and π − ), positively and negatively charged kaons (K + and K − ), protons and antiprotons (p andp) produced at mid-(pseudo)rapidity (mid-y or mid-η) measured in central gold-gold (Au-Au) collisions at the Alternating Gradient Synchrotron (AGS) by the E866 [24], E895 [25,26], and E802 [27,28] Collaborations and at the Relativistic Heavy Ion Collider (RHIC) by the STAR [29,30,31] and PHENIX [32,33] Collaborations, in central leadlead (Pb-Pb) collisions at the Super Proton Synchrotron (SPS) by the NA49 Collaboration [34,35,36] and at the Large Hadron Collider (LHC) by the ALICE Collaboration [37], as well as in inelastic (INEL) proton-proton (pp) collisions at the SPS by the NA61/SHINE Collaboration [38,39], at the RHIC by the PHENIX Collaboration [40], and at the LHC by the CMS Collaboration [41,42] are studied. The (two-component) standard distribution is used to fit the data and to extract T , T i , T 0 , and β T , as well as the excitation functions of parameters (the energy dependent parameters).
The remainder of this paper is structured as follows. The formalism and method are shortly described in Section 2. Results and discussion are given in Section 3. In Section 4, we summarize our main observations and conclusions.

Formalism and method
In high energy collisions, the soft excitation and hard scattering processes are two main processes of particle productions. Most light flavor particles are produced in the soft excitation process and distribute in a narrow p T range which is less than 2 ∼ 3 GeV/c or a little more. Some light flavor particles are produced in the hard scattering process and distribute in a wide p T range. In collisions at not too high energies, the contribution of hard scattering process can be neglected and the main contributor to produced particles is the soft excitation process. In collisions at high energy, the contribution of hard scattering process cannot be neglected, though the main contributor to produced particles is also the soft excitation process. It is expected that the contribution fraction of hard scattering process increases with the increase of collisions energy.
The contributions of soft excitation and hard scattering processes can be described by similar or different probability density functions. Generally, the hard scattering process does not contribute mainly to the temperature and flow velocity due to its small fraction in a narrow p T range. We can neglect the contribution of hard scattering process if we study the spectra in a not too wide p T range. On the contribution of soft excitation process, we have more than one functions to describe the p T spectra. These functions include, but are not limited to, the standard distribution [43], the Tsallis statistics [43,44,45,46], the Erlang distribution [18,19,20], the Schwinger mechanism [47,48,49,50], the blast-wave model with Boltzmann statistics [51,52], the blast-wave model with Tsallis statistics [53,54,55], the Hagedorn thermal distribution [56], and their superposition with two-or three-component. These functions also describe partly the p T spectra of hard scattering process in most cases.
In our opinion, in the case of fitting the data with acceptable representations, various distributions show similar behaviors which result in similar p T ( p 2 T ) with different parameters. To be the closest to the temperature concept in the ideal gas model, we choose the standard distribution in which the chemical potential µ and spin property S are included. That is, one has the probability density function in terms of p T to be [43] m 0 is the rest mass, N denotes the particle number, y min (y max ) is the minimum (maximum) value in the rapidity interval, S = −1 (+1) is for bosons (fermions), and C is the normalization constant. Similarly, the probability density function in terms of m T is In some cases, the independent variable m T in Eq. (3) is replaced by m T − m 0 which starts at 0. Both m T and m T − m 0 show the same distribution shape. The chemical potential µ in Eqs. (1) and (3) is particle dependent. For the particle type i (i = π, K, and p in this work), its chemical potential µ i is expressed by [33,57,58] where k i denotes the ratio of negative to positive particle numbers, is empirically the chemical freeze-out temperature in the statistical thermal model [14,15,16,17], T lim = 0.158 GeV is the limiting or saturation temperature [3], and √ s N N is the center-of-mass energy per nucleon pair in the units of GeV.
Generally, one needs one or two standard distributions to fit the p T (m T ) spectra in a narrow range. In particular, if the resonance decays contribute a large fraction, a two-component distribution is indeed needed. Or, if the hard scattering process contributes a sizable fraction in the considered p T (m T ) range, a twocomponent distribution is also needed. In the case of using two-component standard distribution, one has the p T and m T distributions to be and respectively, where k denotes the contribution fraction of the first component, and f pT (p T , T 1 ) [f pT (p T , T 2 )] and f mT (m T , T 1 ) [f mT (m T , T 2 )] are given in Eqs. (1) and (3) respectively. Correspondingly, is averaged by weighting the two fractions. The temperature T defined in Eq. (8) reflects the common effective temperature of the two components in the case of the two components are assumed to stay in equilibrium. According to Hagedorn model [56], one may also use the usual step function θ(x) to superpose the two stan- and where A 1 and A 2 are constants which result in the two components to be equal to each other at p T = p 1 and m T = m 1 . The contribution fractions of the first component in Eqs. (9) and (10) are and respectively, where p T max and m T max denote the maximum p T and m T respectively. Eq. (8) is also suitable for the superposition in terms of Hagedorn model [56]. The two superpositions show respective advantages and disadvantages. The first superposition can fit the data by a smooth curve. However, there are entanglements in determining T 1 and T 2 . The second superposition can determine T 1 and T 2 without entanglements. However, the curves are possibly not smooth at p 1 or m 1 . In the case of obtaining p T and p 2 T , it does not matter which superposition is used, though the two T are slightly different. In this work, we use the first superposition to obtain smooth curves. One has and if pT max 0 f pT (p T )dp T = 1.
Based on m T spectrum, we may use the same parameters to obtain p T and p 2 T from the related formula of p T distribution.
It should be noted that, since we aim to extract the parameters in a less model dependent way, we shall obtain p T and p 2 T from the combination of data points and fit function in this work. In fact, we may divide p T (m T ) spectrum into two or three regions according to the measured and un-measured p T (m T ) ranges. To obtain p T and p 2 T , we may use the data points in the measured p T (m T ) range and only use the fit function to extrapolate to the un-measured p T (m T ) range.
In each nucleon-nucleon collision in AA and pp collisions, the projectile and target participant sources contribute equally to p T . In the framework of multisource thermal model [18,19,20], each projectile and target source contribute a fraction of 1/2 to p T , i.e. p T /2 which is contributed together by the thermal motion and transverse flow. Let k 0 (1 − k 0 ) denote the contribution fraction of thermal motion (transverse flow), we define empirically and where γ is the mean Lorentz factor of the considered particles and is an empirical representation in this work. In Eq. (18), √ s N N is in the units of GeV as that in Eq. (5).
To continue this work, we need some assumptions and a coordinate system. In the source rest frame, the particles are assumed to emit isotropically. Meanwhile, the interactions among various sources are neglected, which affects slightly the p T (m T ) spectra, though which affects largely anisotropic flows [59]. A right-handed coordinate system O-xyz is established in the source rest frame, where Oz axis is along the beam direction, xOy plane is the transverse plane, and xOz plane is the reaction plane.
We can obtain γ by a Monte Carlo method. Let R 1,2,3 denote random number distributed evenly in where δp T denotes a small shift relative to p T . Each concrete emission angle θ satisfies due to the fact that θ obeys the probability density function f θ (θ) = (1/2) sin θ in [0, π] in the case of isotropic assumption in the source rest frame. The solution of the equation (20). We give up to use rapidity due to the fact that it is unnecessary here. Each concrete momentum p, energy E, and Lorentz factor γ can be obtained by and respectively. After multiple repeating calculations due to the Monte Carlo method, we have where E denotes the mean E for a given type of particle.
In addition, each concrete azimuthal angle φ satisfies due to the fact that φ obeys the probability density func- in the case of isotropic assumption in the source rest frame. The solution of the equation . Each concrete momentum components p x , p y , and p z can be obtained by and respectively. By using the components and E, p, and θ, we can obtain other quantities such as (pseudo)rapidity and event structure [59] which are beyond the focus of this work and will not be studied anymore. According to the color string percolation model [21,22,23], one has Meanwhile, we have the relation between the three components p x , p y , and p z of the momentum p to be in which the root-mean-square components p 2 x , p 2 y , and p 2 z are used. Naturally, T i can be given by one of the root-mean-square components.

Results and discussion
where the particle types, y or η intervals, centrality classes, and collision energies are marked in the panels. The closed and open symbols represent respectively the experimental data of positively and negatively charged particles measured by the E866 [24], E895 [25,26], E802 [27,28], STAR [29,30,31], and PHENIX [32,33] Collaborations marked in the panels, where in Figs. 1(a)-1(d) the data for π ± and K + are taken from the E866 Collaboration [24] and the data for p are taken from the E895 Collaboration [25,26]. The solid and dashed curves are our results fitted by Eqs. (6) or (7) for positively and negatively charged particles respectively. The values of free parameters (T 1 , T 2 if available, k), derived parameter (T ), normalization constant (N 0 ), χ 2 , and degreeof-freedom (dof) are listed in Table 1. The dot-dashed curves are our results fitted by using the single component function with the weighted average parameter T which will be discussed later. In the fitting process for the solid and dashed curves, the least squares method is used to determine the best parameter values. The experimental global uncertainties used in the calculation of χ 2 are taken to be the root sum square of statistical uncertainties and point-by-point systematic uncertainties. The best parameters are determined due to the limitation of the minimum χ 2 . The global uncertainties of parameters are obtained using the method of statistical simulation [60]. We note that χ 2 per dof (χ 2 /dof) in a few cases is larger than 10, which is caused by very small experimental uncertainties. One can see that the (two-component) standard distribution fits approximately the p T (m T − m 0 ) spectra of π ± , K ± , p, andp measured at mid-y or mid-η in central Au-Au collisions over an energy range from 2.7 to 200 GeV. (1/2πm (1/2πm at mid-y or mid-η in central Au-Au collisions at high √ sNN . The closed and open symbols represent respectively the experimental data of positively and negatively charged particles measured by the E866 [24], E895 [25,26], E802 [27,28], STAR [29,30,31], and PHENIX [32,33] Collaborations marked in the panels, where in Figs. 1(a)-1(d) the data for π ± and K + are taken from the E866 Collaboration [24] and the data for p are taken from the E895 Collaboration [25,26]. The solid and dashed curves are our results fitted by Eqs. (6) or (7) for positively and negatively charged particles respectively.
The dot-dashed curves are our results fitted by using the weighted average parameter T . Figure 2 is similar to Fig. 1    Collab.  ing the spectra of various particles produced at mid-y or mid-η in INEL pp collisions at high center-of-mass energies √ s, where the factor (1/2πp T ) on the vertical axis is removed and the invariant cross-section Ed 3 σ/dp 3 is used in some cases. The symbols represent the experimental data measured by the NA61/SHINE [38,39], PHENIX [40], and CMS [41,42] Collaborations. The values of various parameters, χ 2 , and dof for fitting the solid and dashed curves are listed in Table 3, where σ 0 is used as the normalization constant if the spectrum is Ed 3 σ/dp 3 . One can see that the (two-component) standard distribution fits approximately the p T (m T − m 0 ) spectra of π ± , K ± , p, andp measured at mid-y or mid-η in central Pb-Pb and INEL pp collisions over an energy range from 6.3 to 13000 GeV.
To study the dependence of main parameter T on collision energy The results obtained from the spectra of π + , π − , K + , K − , p, andp are displayed by different symbols marked in the panels. The asterisks represent T which is the average T by weighting different masses and yields of the six particles. In the case of one of the six particles is absent, T is not available. One can see that T and T increase with the increase of ln( √ s N N ) [ln( √ s)]. Meanwhile, T increases with the increase of particle mass.
To better determine the kinetic freeze-out information, we now fit simultaneously the spectra of π + , π − , K + , K − , p, andp in different p T ranges using the same set of parameters. In Figs. 1-3, the dot-dashed curves are the results using the weighted average T which is energy dependent, though in fact we may use other T to obtain a little better result in some cases. In the refit, the normalization constant N 0 for different spectra is adjustable to fit suitable p T range. One can see that the same set of T can fit only a part of p T range in some cases. The fit results using the same set of parameters are not ideal. These results do not support the single scenario for kinetic freeze-out. We are not inclined to fit simultaneously the spectra of different particles. Conversely, we use different T for different spectra in this which can be removed. The symbols represent the experimental data measured by the NA49 [34,35,36] and ALICE [37] Collaborations.
work. in general, and T 0 for pion emission and T 0 appears the trend of saturation at the RHIC and LHC. Meanwhile, T 0 , T 0 , T i , and T i increase, and β T and β T decrease, with the increase of particle mass.
It is regretful that some particles are absent in experimental measurements at the energies blow 10 GeV. This renders that T , T 0 , β T , and T i are not available in the energy range of several GeV. From the trends Ed 3 σ/dp 3 is used in some cases. The symbols represent the experimental data measured by the NA61/SHINE [38,39], PHENIX [40], and CMS [41,42] Collaborations.
of available T , T 0 , β T , and T i , we may estimate that T , T 0 , β T , and T i increase (quickly) with the increase of ln( √ s N N ) [ln( √ s)] in the energy range of several GeV.
In particular, some excitation functions show little peak around 10 GeV, which should be studied further in future. Meanwhile, we hope to obtain more data in the energy range of several GeV in future.
The trends of excitation functions render that the collision system undergoes different evolution processes.
From several GeV to about 10 GeV, the violent degree of collisions increases with increasing the energy and matter density of the collision system. The hadronic matter in the collision system stays at a state with ever higher density and temperature. At about 10 GeV, the energy and matter density of the collision system reaches just right to a high enough value. The density and temper- ature of the collision system are so high that the phase transition from hadronic matter to QGP had happened just right. At above 10 GeV, the energy and matter density of the collision system reaches to a higher value. The density and temperature of the collision system are also higher. However, because the phase transition from hadronic matter to QGP had happened, the density and temperature are limited, which results in the levels of T 0 for pion emission and T 0 had stabilized.
Before summary and conclusions, we would like to point out that we have used a new method to extract T 0 , β T , and T i . After fitting the p T (m T ) spectra by using the two-component standard distribution Eqs. (6) or (7) in which the free parameters are the effective temperatures T 1 and T 2 and the contribution fraction k of the first component, the derived parameter, the effective temperature T , can be obtained from the weighted average formula Eq. (8). Then, the derived parameters, the kinetic freeze-out temperature T 0 and transverse flow velocity β T , can be obtained respectively from Eqs. (16) and (17) which are related to the mean transverse momentum p T . The derived parameter, the initial tem- perature T i , can be obtained from Eq. (29) which is related the root-mean-square According to the analysis of the spectra of six hadron species listed in Tables 1 and 2, one can see that pions, kaons, and (anti)protons correspond to different temperatures of emission source. This shows a mass-dependent multiple scenario for kinetic freeze-out. Moreover, in AA collisions at energies below the LHC, charged pions can be redistributed between two sources, one is hot and another is cold, while kaons and (anti)protons are located in single (hot) sources (though with different temperatures) in most cases. This is understandable. That charged pions come from resonance decays contribute a relative large fraction in low-p T region, which cannot be concealed in single source and appears in cold source together with low-p T pions from non-resonance decay. As an ensemble, Eq. (1) describes the cold source with low T for all low-p T pions. However, the resonance decays for kaons and (anti)protons are relatively small comparing to those for pions in low-p T region, which are concealed in single source.
Naturally, if we regard T , T 0 , β T , and T i as common quantities corresponding to emissions of various hadron species, we may use the mass-independent single scenario for kinetic freeze-out and other system evolution stages such as chemical freeze-out and initial state. It is contentious that the mass-independent single scenario or mass-dependent multiple scenario is right due to different physics thinkings. In our opinion, the mass-independent single scenario is a very ideal situation which is similar to the equilibrium state of mixture gas. And the mass-dependent multiple scenario describes a fine emission process which "shows massive particles coming out of the system earlier in time with smaller radial flow velocities, which is hydrodynamic behavior" [61]. The temperatures discussed in this work reflect mainly the kinetic energies of various hadron species, but do not have certainly the statistical sense.
We note that, in the mass-dependent multiple scenario for kinetic freeze-out (Figs. 4 and 5), the obtained temperature for proton emission is much larger than that for pion emission. This reflects that protons coming out of the system is much earlier than pions due to much larger mass of proton comparing to pion. This phenomenon is a hydrodynamic behavior [61], in which massive particles are early leaved behind in the evolution process of collision system. In other words, massive  particles are not emitted from the system on their owns initiative due to high T 0 , but are leaved behind under compulsion due to low β T and large m 0 . In fact, some protons existed in projectile and target nuclei appear in rapidity space as leading protons outside the fireball. This issue also results in protons coming out of the system to be earlier than pions.
Because T 0 and β T are model dependent, this work is different from Figs. 37 and 39 in ref. [31], though this work is less model dependent and ref. [31] is much model dependent. In ref. [31], a flow velocity profile parameter n is used in the extraction of T 0 and β T . The parameter n can be largely changed from 0 to 2 in AA collisions and can be above 4 in pp collisions, which is mutable and debatable. The pion spectra in low-p T region (< 0.5 GeV/c) are excluded from the fit due to resonance decay, which overrates T 0 and β T . Our work shows that T 0 (β T ) in pp collisions is slightly smaller than (almost equal to) that in AA collisions, which is in agreement with our recent work [12] in which the intercept in the linear relation of T versus m 0 is regarded as T 0 and the slope in the linear relation of p T versus m 0 γ is regarded as β T . This result is understandable due to similar collective behavior as in AA collisions appearing in pp collisions [10].
We would like to emphasize that this work is a datadriven reanalysis based on some physics considerations, but not a simple fit to the data. From the data-driven reanalysis, the excitation functions of some quantities such as the effective temperature T and its weighted average T , the kinetic freeze-out temperature T 0 and its weighted average T 0 , the transverse flow velocity β T and its weighted average β T , as well as the initial temperature T i and its weighted average T i have been obtained. These excitation functions have appeared some obvious laws with the increase of collision energy.

Summary and conclusion
We summarize here our main observations and conclusions.
(a) The transverse momentum or mass spectra of π + , π − , K + , K − , p, andp at mid-y or mid-η produced in central Au-Au (Pb-Pb) collisions over an energy range from 2.7 to 200 (6.3 to 2760) GeV have been analyzed in this work. Meanwhile, the spectra in INEL pp collisions over an energy range from 6.3 to 13000 GeV have also been analyzed. The experimental data measured by the E866, E895, E802, NA49, NA61/SHINE, STAR, PHENIX, ALICE, and CMS Collaborations are approximately fitted by the (two-component) standard distribution in which the temperature concept is the closest to the ideal gas model.
(b) The effective temperature and its excitation function are obtained from the transverse momentum spectra of identified particles produced in collisions at high energies. The kinetic freeze-out temperature and transverse flow velocity and their excitation functions are extracted from the formulas related to the average transverse momentum, which is based on the multisource thermal model. The initial temperature and its excitation function are extracted from the formula related to the root-mean-square transverse momentum, which is based on the color string percolation model.
(c) With the increase of collision energy, the four derived parameters and each average increase (quickly) from a few GeV to about 10 GeV, then increases slowly after 10 GeV. In particular, the kinetic freeze-out temperature for pion emission and its average finally appear the trend of saturation at the RHIC and LHC. Mean-while, the three derived temperatures increases and the derived transverse flow velocity decreases with the increase of particle mass, which result in a mass-dependent multiple scenario for kinetic freeze-out and other system evolution stages such as chemical freeze-out and initial state.

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.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper. The funders had 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.