Excitation Functions of Related Parameters from Transverse Momentum (Mass) Spectra in High-Energy Collisions

Li, Li-Li; Liu, Fu-Hu; Waqas, Muhammad; Al-Yusufi, Rasha; Mujear, Altaf

doi:https://doi.org/10.1155/2020/5356705

Advances in High Energy Physics

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Research Article | Open Access

Volume 2020 | Article ID 5356705 | https://doi.org/10.1155/2020/5356705

Excitation Functions of Related Parameters from Transverse Momentum (Mass) Spectra in High-Energy Collisions

Li-Li Li,¹Fu-Hu Liu,¹Muhammad Waqas,¹Rasha Al-Yusufi,^1,2and Altaf Mujear^1,2

Academic Editor: Torsten Asselmeyer-Maluga

Received29 Jan 2020

Accepted06 May 2020

Published10 Jun 2020

Abstract

Transverse momentum (mass) spectra of positively and negatively charged pions and of positively and negatively charged kaons, protons, and antiprotons produced at mid-(pseudo)rapidity in various collisions at high energies are analyzed in this work. The experimental data measured in central gold-gold, central lead-lead, and inelastic proton-proton collisions by several international collaborations are studied. The (two-component) standard distribution is used to fit the data and extract the excitation function of effective temperature. Then, the excitation functions of kinetic freeze-out temperature, transverse flow velocity, and initial temperature are obtained. In the considered collisions, the four parameters increase with the increase of collision energy in general, and the kinetic freeze-out temperature appears at the trend of saturation at the top Relativistic Heavy Ion Collider and the Large Hadron Collider.

1. 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 [1–3], in which quark-gluon plasma (QGP) is possibly created and many particles are produced [4–6]. 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 [7–9], 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 multiplicity is high enough due to the small system which also appears collective behavior [10, 11].

Although there are different stages in the system evolution [1–3], 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 a 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 the small system with low multiplicity, QGP is not expected to create in it due to a very small volume of the violent collision region. From the similar multiplicity at the energy up to 200 GeV, the small system is more similar to peripheral AA collisions, but not to central AA collisions [12, 13]. At the energy down to 10 or several GeV, the situation is different due to the fact that the baryon-dominated effect plays more important role in AA collisions [14].

The temperatures at the stages of kinetic freeze-out, chemical freeze-out, and initial state are called the kinetic freeze-out temperature ( or ), chemical freeze-out temperature (), and initial temperature (), respectively. Besides, one also has the effective temperature () in which both the contributions of thermal motion and flow effect are included. It is expected that various temperatures can be extracted from particle spectra, which are usually model dependent. Generally, is unavoidably model dependent, and extracted from particle ratios in the statistical thermal model [15–18] is also model dependent. We hope to use a less model-dependent method to extract , , and . 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 two-component form to obtain by fitting the experimental transverse momentum () or transverse mass () spectra of various particles. The standard distribution includes the Bose-Einstein, Fermi-Dirac, and Boltzmann distributions, in which the effective temperature parameter is the closest to that in the ideal gas model when comparing with those in other distributions. After the fitting, we hope to extract and from the relation to average () due to the Erlang distribution in the multisource thermal model [19–21] and from the relation to root-mean-square () due to the color string percolation model [22–24]. Obviously, and depend on the data themselves, though they can be calculated from the models.

In this work, the () spectra of positively and negatively charged pions ( and ) and positively and negatively charged kaons ( and ), protons, and antiprotons ( and ) produced at mid-(pseudo)rapidity (mid- or mid-) measured in central gold-gold (Au-Au) collisions at the Alternating Gradient Synchrotron (AGS) by the E866 [25], E895 [26, 27], and E802 [28, 29] Collaborations and at the Relativistic Heavy Ion Collider (RHIC) by the STAR [30–32] and PHENIX [33, 34] Collaborations, in central lead-lead (Pb-Pb) collisions at the Super Proton Synchrotron (SPS) by the NA49 Collaboration [35–37] and at the Large Hadron Collider (LHC) by the ALICE Collaboration [38], as well as in inelastic (INEL) proton-proton () collisions at the SPS by the NA61/SHINE Collaboration [39, 40], at the RHIC by the PHENIX Collaboration [41], and at the LHC by the CMS Collaboration [42, 43], are studied. The (two-component) standard distribution is used to fit the data and to extract , , , and , as well as the excitation functions (energy dependences) of 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.

2. 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 distributed in a narrow range which is less than GeV/ or a little more. Some light flavor particles are produced in the hard scattering process and distributed in a wide range. In collisions at not too high energies, the contribution of the hard scattering process can be neglected and the main contributor that produced particles is the soft excitation process. In collisions at a high energy, the contribution of the hard scattering process cannot be neglected, though the main contributor that produced particles is also the soft excitation process. It is expected that the contribution fraction of the hard scattering process increases with the increase of collision 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 range. We can neglect the contribution of the hard scattering process if we study the spectra in a not too wide range. On the contribution of the soft excitation process, we have more than one functions to describe the spectra. These functions include, but are not limited to, the standard distribution [44], the Tsallis statistics [44–47], the Erlang distribution [19–21], the Schwinger mechanism [48–51], the blast-wave model with Boltzmann statistics [52, 53], the blast-wave model with Tsallis statistics [54–56], the Hagedorn thermal distribution [57], and their superposition with two- or three-component. These functions also describe partly the spectra of the 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 () 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 and spin property are included. That is, one has the probability density function in terms of to be [44] where is the rest mass, denotes the particle number, () is the minimum (maximum) value in the rapidity interval, (+1) is for bosons (fermions), and is the normalization constant. Similarly, the probability density function in terms of is In some cases, the independent variable in Equation (3) is replaced by which starts at 0. Both and show the same distribution shape. As probability density functions, the integrals of Equations (1) and (3) are naturally normalized to 1, respectively.

The chemical potential in Equations (1) and (3) is particle dependent. For the particle type (, , and in this work), its chemical potential is expressed by [34, 58, 59] where denotes the ratio of negative to positive particle numbers, is empirically the chemical freeze-out temperature in the statistical thermal model [15–18], GeV is the limiting or saturation temperature [3], and 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 () 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 () range, a two-component distribution is also needed. In the case of using the two-component standard distribution in which the contributions from resonance decay are naturally included in the first component which covers the spectra in the low- region ( GeV/), one has the probability density functions of and to be respectively, where denotes the contribution fraction of the first component, and [] and [] are given in Equations (1) and (3), respectively. The integrals of Equations (6) and (7) are also normalized to 1, respectively. Correspondingly, is averaged by weighting the two fractions. The temperature defined in Equation (8) reflects the common effective temperature of the two components in the case that the two components are assumed to stay in equilibrium.

According to the Hagedorn model [57], one may also use the usual step function to superpose the two standard distributions, where if and if . Thus, we have new probability density functions of and to be respectively, where and are constants which result in the two components to be equal to each other at and . The integrals of Equations (9) and (10) should be normalized to 1, respectively, due to the fact that they are probability density functions. The contribution fractions of the first component in Equations (9) and (10) are respectively, where and denote the maximum and , respectively. Equation (8) is also suitable for the superposition in terms of the Hagedorn model [57].

The two superpositions show respective advantages and disadvantages. The first superposition can fit the data by a smooth curve. However, there are correlations in determining and . The second superposition can determine and without correlations. However, the curves are possibly not smooth at or . In the case of obtaining and , it does not matter which superposition is used, though the two are slightly different. In this work, we use the first superposition to obtain smooth curves. One has due to

Based on the spectrum, we may use the same parameters to obtain and from the related formula of distribution.

It should be noted that, since we aim to extract the parameters in a less model-dependent way, we shall obtain and from the combination of data points and fit function in this paper. In fact, we may divide () spectrum into two or three regions according to the measured and unmeasured () ranges. To obtain and , we may use the data points in the measured () range and only use the fit function to extrapolate to the unmeasured () range.

In each nucleon-nucleon collision in AA and pp collisions, the projectile and target participant sources contribute equally to . In the framework of the multisource thermal model [19–21], each projectile and target source contribute a fraction of 1/2 to , i.e., which is contributed together by the thermal motion and flow effect. Let () denote the contribution fraction of thermal motion (flow effect); we define empirically where is the mean Lorentz factor of the considered particles and is a parameterized representation in this paper due to our comparison with the results [12, 13] from the blast-wave model [52–56]. In Equation (18), is in the units of GeV as that in Equation (5).

In a recent work [60], it is shown that the effective temperature is proportional to and the kinetic freeze-out temperature is proportional to the effective temperature, though the effective temperature used in Ref. [60] is different from this paper. This confirms the relation of (Equation (16)) used in this paper. Considering each projectile and target source contributing [19–21], we have concretely . The remainder in is naturally contributed by the transverse flow. This confirms Equations (16) and (17) to be justified, though is an empirical representation.

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 () spectra, through which affects largely anisotropic flows [61]. A right-handed coordinate system – is established in the source rest frame, where the axis is along the beam direction, the plane is the transverse plane, and the plane is the reaction plane.

We can obtain by a Monte Carlo (MC) method. Let denote a random number distributed evenly in ; each concrete satisfies where denotes a small shift relative to . Each concrete emission angle satisfies due to the fact that obeys the probability density function in in the case of isotropic assumption in the source rest frame. The solution of the equation is Equation (20). We give up to use rapidity due to the fact that it is unnecessary here. Each concrete momentum , energy , and Lorentz factor can be obtained by respectively. After multiple repeating calculations due to the MC method, we have where denotes the mean for a given type of particle.

In addition, each concrete azimuthal angle satisfies due to the fact that obeys the probability density function in in the case of isotropic assumption in the source rest frame. The solution of the equation is Equation (25). Each concrete momentum components , , and can be obtained by respectively. By using the components and , , and , we can obtain other quantities such as (pseudo)rapidity and event structure [61] which are beyond the focus of this work and will not be studied anymore.

According to the color string percolation model [22–24], one has Meanwhile, we have the relation between the three components , , and of the momentum to be in which the root-mean-square components , , and are used. Naturally, can be given by one of the root-mean-square components.

We would like to point out that the above isotropic assumption is only performed in the source rest frame. It is expected that many sources are formed in high-energy collisions according to the multisource model [19–21]. These sources distribute at different rapidities in the rapidity space, which appear at the effect of longitudinal flow. The two-component and spectra render that these sources stay in two different excitation states or have two different decay mechanisms. The interactions among these sources also affect anisotropic flows in transverse plane [61].

3. Results and Discussion

Figures 1(a)–1(q) show the () spectra, [], of , , , , , and produced at mid- or mid- in central Au-Au collisions at different , where the particle types, or 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 [25], E895 [26, 27], E802 [28, 29], STAR [30–32], and PHENIX [33, 34] Collaborations marked in the panels, where in Figures 1(a)–1(d), the data for and are taken from the E866 Collaboration [25] and the data for are taken from the E895 Collaboration [26, 27]. The solid and dashed curves are our results fitted by Equation (6) or (7) for positively and negatively charged particles, respectively. The values of free parameters (, if available, and ), derived parameter (), normalization constant (), , and degree-of-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 which will be discussed later. The dotted curves and asterisks in Figures 1(a)–1(f) represent the MC results for with high ( particles) and low ( particles) statistics, respectively, which will be discussed at the end of this section. 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 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 . The global uncertainties of parameters are obtained using the method of statistical simulation [62]. We note that per dof (/dof) in a few cases is larger than 10, which renders that the fit is not too good. One can see that the (two-component) standard distribution fits approximately the () spectra of , , , and measured at mid- or mid- in central Au-Au collisions over an energy range from 2.7 to 200 GeV in most cases.

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Figure 1

The () spectra, [], of , , , , , and produced at mid- or mid- in central Au-Au collisions at high . The closed and open symbols represent, respectively, the experimental data of positively and negatively charged particles measured by (a–e) the E866 [25] and E895 [26, 27], (f) the E802 [28, 29], (g–o) the STAR [30–32], and (p–q) the PHENIX [33, 34]. Collaborations marked in the panels which appear mostly (g–q), where in (a)–(d), the data for and are taken from the E866 Collaboration [25] and the data for are taken from the E895 Collaboration [26, 27]. The solid and dashed curves are our results fitted by Equation (6) or (7) for positively and negatively charged particles, respectively. The dot-dashed curves are our results fitted by using the weighted average parameter . The dotted curves and asterisks in (a)–(f) represent the MC results for with high ( particles) and low ( particles) statistics, respectively.

Figure 2 is similar to Figure 1, but it is showing the spectra of various particles produced at mid- in central Pb-Pb collisions at high , where the factor on the vertical axis is removed in some cases and if available denotes the particle number which can be removed. The symbols represent the experimental data measured by the NA49 [35–37] and ALICE [38] Collaborations. The values of various parameters, , and dof for fitting the solid and dashed curves are listed in Table 2. Figure 3 is similar to Figures 1 and 2, but it showing the spectra of various particles produced at mid- or mid- in INEL collisions at high center-of-mass energies , where the factor on the vertical axis is removed and the invariant cross-section is used in some cases. The symbols represent the experimental data measured by the NA61/SHINE [39, 40], PHENIX [41], and CMS [42, 43] Collaborations. The values of various parameters, , and dof for fitting the solid and dashed curves are listed in Table 3, where the cross-section is used as the normalization constant if the spectrum is . One can see that the (two-component) standard distribution fits approximately the () spectra of , , , and measured at mid- or mid- in central Pb-Pb and INEL collisions over an energy range from 6.3 to 13000 GeV in most cases.

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To study the dependence of the main parameter on collision energy or , the excitation functions of for central Au-Au (Pb-Pb) collisions and INEL collisions are shown in Figures 4(a) and 4(b), respectively. The results obtained from the spectra of , , , , , and are displayed by different symbols marked in the panels. The asterisks represent which is the average by weighting different masses and yields of the six particles. In the case where one of the six particles is absent, is not available. One can see that and increase with the increase of []. Meanwhile, increases with the increase of particle mass.

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To better determine the kinetic freeze-out information, we now fit simultaneously the spectra of , , , , , and in different ranges using the same set of parameters. In Figures 1–3, the dot-dashed curves are the results using the weighted average which is energy dependent, though in fact we may use the other to obtain a little better result in some cases. In the refit, the normalization constant for different spectra is adjustable to fit the suitable range. In despite of mass-dependent (two-)temperature in the nonsimultaneous fit, the simultaneous fit is done using the same and only set of for the six species of particles at each energy. That is to say that we use one component function to fit the spectra of the six species of particles at each energy. And the temperature is the same for each particle, and the normalization factor is adjusted. It seems that our treatment is not a fair comparison of the nonsimultaneous fit and simultaneous fit, where the nonsimultaneous fit uses the two-component in some cases. However, the simultaneous fit seeks for the same temperature which does not allow the two-component for some particles due to other particles corresponding to one component. In our opinion, our comparison is fair for the nonsimultaneous fit and the simultaneous fit. One can see that the same set of can fit only a part of the 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 for different spectra in this paper.

Figures 5(a) and 5(b) (Figures 5(c) and 5(d)) are similar to Figures 4(a) and 4(b), respectively, but they are showing the excitation functions of () and (). Figures 6(a) and 6(b) are also similar to Figures 4(a) and 4(b), respectively, but they are showing the excitation functions of and . One can see that , , , , , and increase with the increase of [] in general, and for pion emission and appears in the trend of saturation at the RHIC and LHC. Meanwhile, , , , and increase, and and decrease, with the increase of the particle mass.

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It is regretful that some particles are absent in experimental measurements at the energies below 10 GeV. This renders that , , , and are not available in the energy range of several GeV. From the trends of available , , , and , we may estimate that , , , and increase (quickly) with the increase of [] 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.

It should be noted that since empirical Equation (18) is essential for obtaining the energy dependence of and , it seems that one can obtain arbitrary results by choosing another parametrization of . In particular, it seems possibly to choose a parametrization of so that (), (), and () could have similar and even though the combined temperature (in Figure 4) for these particles are quite different. This case corresponds to the a single scenario for kinetic freeze-out, which is not consistent with two or multiple scenarios observed in other studies [63–66]. We are inclined to use multiple scenarios [67] due to a more accurate fit to the wider spectra. To coordinate the single and multiple scenarios, we may regard the multiple scenarios as a refined situation of the single scenario, which will be discussed later in detail.

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 the increasing of the energy and matter density of the collision system. The hadronic matter in the collision system stays at a state with an ever higher density and temperature. At about 10 GeV, the energy and matter density of the collision system reaches to a high value. The temperature is also high, which is needed to come into notice. At above 10 GeV, the energy and matter density of the collision system reaches to a higher value. The temperature is also higher. However, because the phase transition from hadronic matter to QGP had happened possibly, the temperature is limited, which results in the levels of for pion emission and had stabilized.

Before summary and conclusions, we would like to point out that we have used a new method to extract , , and . After fitting the () spectra by using the two-component standard distribution Equations (6) or (7) in which the free parameters are the effective temperatures and and the contribution fraction of the first component, the derived parameter, the effective temperature , can be obtained from the weighted average formula Equation (8). Then, the derived parameters, the kinetic freeze-out temperature and transverse flow velocity , can be obtained, respectively, from Equations (16) and (17) which are related to the mean transverse momentum . The derived parameter, the initial temperature , can be obtained from Equation (29) which is related to 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 the low- region, which can be described by the first component in Equation (6) or (7). Some low- pions from nonresonance decay can be also described by the first component. As an ensemble, Equation (1) describes the cold source with low for all low- pions. However, the resonance decays for kaons and (anti)protons are relatively small comparing to those for pions in the low- region, which are concealed in a single source.

Naturally, if we regard , , , and 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 refined emission process which “shows massive particles coming out of the system earlier in time with smaller radial flow velocities, which is hydrodynamic behavior” [68]. The temperatures discussed in this paper 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 (Figures 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 [68], in which massive particles are early left behind in the evolution process of the collision system. In other words, massive particles are not emitted from the system on their own initiative due to high but are left behind under compulsion due to low and large . In fact, some protons which 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 and are model dependent, this paper is different from Figures 37 and 39 in Ref. [32], though this paper is less model dependent and Ref. [32] is much model dependent. In Ref. [32], a flow velocity profile parameter is used in the extraction of and . The parameter can be largely changed from 0 to 2 in AA collisions and above 4 in pp collisions, which is mutable and debatable. The pion spectra in the low- region ( GeV/) are excluded from the fit [32] due to resonance decay, which overrates and . Our work shows that () in collisions is slightly smaller than (almost equal to) those in AA collisions, which is in agreement with our recent work [12] in which the intercept in the linear relation of versus is regarded as and the slope in the linear relation of versus is regarded as . This result is understandable due to similar collective behavior as in AA collisions appearing in collisions [10].

We would like to emphasize that this paper is a data-driven 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 and its weighted average , the kinetic freeze-out temperature and its weighted average , the transverse flow velocity and its weighted average , and the initial temperature and its weighted average have been obtained. These excitation functions have shown some obvious laws with the increase of collision energy.

In the above discussions, to obtain then , we have used the MC method. Figures 5(c) and 5(d) are a direct result by the MC method. As a statistical model is implemented for Figures 1–3, we can also obtain the curves by Equation (19), which is in terms of the MC method. In fact, in the calculation by the MC method, after the shuffled treatment due to the randomicity by the Matlab code, we can obtain a lot of “simulated data”. Then, we may count them in different () bins and obtain similar or the same results to the curves in Figures 1–3. In case in the event that the numbers are being not too large, we shall observe fluctuations around the curves. As an example, for spectra in Figures 1(a)–1(f), the dotted curves and crosses represent the MC results with high and low statistics, respectively. The two results from the analytical function and MC method confirmed each other.

4. Summary and Conclusion

We summarize here our main observations and conclusions. (a)The transverse momentum or mass spectra of , , , , , and at mid- or mid- 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 collisions over an energy range from 6.3 to 13000 GeV have also been analyzed. In most cases, 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 or mass 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 at the trend of saturation at the RHIC and LHC. Meanwhile, the three derived temperatures increase and the derived transverse flow velocity decrease 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.

Ethical Approval

The authors declare that they are in compliance with ethical standards regarding the content of this paper.

Disclosure

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.

Conflicts of Interest

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

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11575103 and 11947418, the Chinese Government Scholarship (China Scholarship Council), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (STIP) under Grant No. 201802017, the Shanxi Provincial Natural Science Foundation under Grant No. 201901D111043, and the Fund for Shanxi “1331 Project” Key Subjects Construction.

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Copyright © 2020 Li-Li Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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