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

Authors: Li-Li Li; Fu-Hu Liu; Muhammad Waqas; Rasha Al-Yusufi; Altaf Mujear
Journal: Advances in High Energy Physics
 (2020)

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## 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.


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## Body

## 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 (T0 or Tkin), chemical freeze-out temperature (Tch), and initial temperature (Ti), respectively. Besides, one also has the effective temperature (T) 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, T is unavoidably model dependent, and Tch 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 T0, βT, and Ti. 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 obtainT by fitting the experimental transverse momentum (pT) or transverse mass (mT) spectra of various particles. The standard distribution includes the Bose-Einstein, Fermi-Dirac, and Boltzmann distributions, in which the effective temperature parameter T is the closest to that in the ideal gas model when comparing T with those in other distributions. After the fitting, we hope to extract T0 and βT from the relation to average pT (pT) due to the Erlang distribution in the multisource thermal model [19–21] and Ti from the relation to root-mean-square pT (pT2) due to the color string percolation model [22–24]. Obviously, pT and pT2 depend on the data themselves, though they can be calculated from the models.In this work, thepT (mT) spectra of positively and negatively charged pions (π+ and π−) and positively and negatively charged kaons (K+ and K−), protons, and antiprotons (p and p¯) 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 [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 (pp) 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 T, Ti, T0, and βT, 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 Section2. 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 narrowpT 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 distributed in a wide pT 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 narrowpT range. We can neglect the contribution of the hard scattering process if we study the spectra in a not too wide pT range. On the contribution of the soft excitation process, we have more than one functions to describe the pT 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 pT 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 similarpT (pT2) 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 pT to be [44]
(1)fpTpT,T=1NdNdpT=CpTmT∫yminymaxcoshy×expmTcoshy−μT+S−1dy,where
(2)mT=pT2+m02,m0 is the rest mass, N denotes the particle number, ymin (ymax) 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 mT is
(3)fmTmT,T=1NdNdmT=CmT2∫yminymaxcoshy×expmTcoshy−μT+S−1dy.In some cases, the independent variable mT in Equation (3) is replaced by mT−m0 which starts at 0. Both mT and mT−m0 show the same distribution shape. As probability density functions, the integrals of Equations (1) and (3) are naturally normalized to 1, respectively.The chemical potentialμ in Equations (1) and (3) is particle dependent. For the particle type i (i=π, K, and p in this work), its chemical potential μi is expressed by [34, 58, 59]
(4)μi=−12Tchlnki,where ki denotes the ratio of negative to positive particle numbers,
(5)Tch=Tlim1+exp2.60−lnsNN/0.45is empirically the chemical freeze-out temperature in the statistical thermal model [15–18], Tlim=0.158 GeV is the limiting or saturation temperature [3], and sNN is the center-of-mass energy per nucleon pair in the units of GeV.Generally, one needs one or two standard distributions to fit thepT (mT) 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 pT (mT) 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-pT region (<0.2~0.3 GeV/c), one has the probability density functions of pT and mT to be
(6)fpTpT=kfpTpT,T1+1−kfpTpT,T2,(7)fmTmT=kfmTmT,T1+1−kfmTmT,T2,respectively, where k denotes the contribution fraction of the first component, and fpTpT,T1 [fpTpT,T2] and fmTmT,T1 [fmTmT,T2] are given in Equations (1) and (3), respectively. The integrals of Equations (6) and (7) are also normalized to 1, respectively. Correspondingly,
(8)T=kT1+1−kT2is averaged by weighting the two fractions. The temperature T 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 θx to superpose the two standard distributions, where θx=0 if x<0 and θx=1 if x≥0. Thus, we have new probability density functions of pT and mT to be
(9)fpTpT=A1θp1−pTfpTpT,T1+A2θpT−p1fpTpT,T2,(10)fmTmT=A1θm1−mTfmTmT,T1+A2θmT−m1fmTmT,T2,respectively, where A1 and A2 are constants which result in the two components to be equal to each other at pT=p1 and mT=m1. 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
(11)k=∫0p1A1fpTpT,T1dpT=1−∫p1pTmaxA2fpTpT,T2dpT,(12)k=∫m0m1A1fmTmT,T1dmT=1−∫m1mTmaxA2fmTmT,T2dmT,respectively, where pTmax and mTmax denote the maximum pT and mT, 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 determiningT1 and T2. The second superposition can determine T1 and T2 without correlations. However, the curves are possibly not smooth at p1 or m1. In the case of obtaining pT and pT2, 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
(13)pT=∫0pTmaxpTfpTpTdpT,(14)pT2=∫0pTmaxpT2fpTpTdpT,due to
(15)∫0pTmaxfpTpTdpT=1.Based on themT spectrum, we may use the same parameters to obtain pT and pT2 from the related formula of pT distribution.It should be noted that, since we aim to extract the parameters in a less model-dependent way, we shall obtainpT and pT2 from the combination of data points and fit function in this paper. In fact, we may divide pT (mT) spectrum into two or three regions according to the measured and unmeasured pT (mT) ranges. To obtain pT and pT2, we may use the data points in the measured pT (mT) range and only use the fit function to extrapolate to the unmeasured pT (mT) range.In each nucleon-nucleon collision in AA and pp collisions, the projectile and target participant sources contribute equally topT. In the framework of the multisource thermal model [19–21], each projectile and target source contribute a fraction of 1/2 to pT, i.e., pT/2 which is contributed together by the thermal motion and flow effect. Let k0 (1−k0) denote the contribution fraction of thermal motion (flow effect); we define empirically
(16)T0≡k0pT2,(17)βT≡1−k0pT2m0γ¯,where γ¯ is the mean Lorentz factor of the considered particles and
(18)k0=0.30−0.01lnsNNis 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), sNN 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 pT 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 T0∝pT (Equation (16)) used in this paper. Considering each projectile and target source contributing pT/2 [19–21], we have concretely T0∝pT/2. The remainder in pT/2 is naturally contributed by the transverse flow. This confirms Equations (16) and (17) to be justified, though k0 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 thepT(mT) spectra, through which affects largely anisotropic flows [61]. A right-handed coordinate system O–xyz is established in the source rest frame, where the Oz axis is along the beam direction, the xOy plane is the transverse plane, and the xOz plane is the reaction plane.We can obtainγ¯ by a Monte Carlo (MC) method. Let R1,2,3 denote a random number distributed evenly in 0,1; each concrete pT satisfies
(19)∫0pTfpTp′T,Tdp′T<R1<∫0pT+δpTfpTp′T,Tdp′T,where δpT denotes a small shift relative to pT. Each concrete emission angle θ satisfies
(20)θ=2arcsinR2,due to the fact that θ obeys the probability density function fθθ=1/2sinθ in 0,π in the case of isotropic assumption in the source rest frame. The solution of the equation ∫0θfθ′θ′dθ′=R2 is 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
(21)p=pTcscθ,(22)E=p2+m02,(23)γ=Em0,respectively. After multiple repeating calculations due to the MC method, we have
(24)γ¯=E¯m0,where E¯ denotes the mean E for a given type of particle.In addition, each concrete azimuthal angleϕ satisfies
(25)ϕ=2πR3,due to the fact that ϕ obeys the probability density function fϕϕ=1/2π in 0,2π in the case of isotropic assumption in the source rest frame. The solution of the equation ∫0ϕfϕ′ϕ′dϕ′=R3 is Equation (25). Each concrete momentum components px, py, and pz can be obtained by
(26)px=pTcosϕ,(27)py=pTsinϕ,(28)pz=pTcotθ=pcosθ,respectively. By using the components and E, p, and θ, 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
(29)Ti≡pT22.Meanwhile, we have the relation between the three components px, py, and pz of the momentum p to be
(30)px2=py2=pz2=pT22,in which the root-mean-square components px2, py2, and pz2 are used. Naturally, Ti 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 pT and mT 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

Figures1(a)–1(q) show the pT (mT−m0) spectra, 1/2πpT·d2N/dydpT [1/2πmT·d2N/dydmT], of π+, π−, K+, K−, p, and p¯ produced at mid-y or mid-η in central Au-Au collisions at different sNN, 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 [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 π± and K+ are taken from the E866 Collaboration [25] and the data for p 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 (T1, T2 if available, and k), derived parameter (T), normalization constant (N0), χ2, 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 T which will be discussed later. The dotted curves and asterisks in Figures 1(a)–1(f) represent the MC results for K+ with high (106 particles) and low (104 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 χ2 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 χ2. The global uncertainties of parameters are obtained using the method of statistical simulation [62]. We note that χ2 per dof (χ2/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 pT (mT−m0) spectra of π±, K±, p, and p¯ measured at mid-y or mid-η in central Au-Au collisions over an energy range from 2.7 to 200 GeV in most cases.Figure 1
ThepT (mT−m0) spectra, 1/2πpT·d2N/dydpT [1/2πmT·d2N/dydmT], of π+, π−, K+, K−, p, and p¯ produced 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 (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 π± and K+ are taken from the E866 Collaboration [25] and the data for p 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 T. The dotted curves and asterisks in (a)–(f) represent the MC results for K+ with high (106 particles) and low (104 particles) statistics, respectively.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)(q)Table 1
Values of free parameters (T1, T2, if available, and k), derived parameter [T=kT1+1−kT2], normalization constant (N0), χ2, and dof corresponding to the solid and dashed curves in Figure 1 in which different data in central Au-Au collisions are measured in different mid-(pseudo)rapidity intervals at different energies by different collaborations. The values for positive and negative particles are listed in terms of value1/value2, if the two values appear together.
Collab.sNN (GeV)ParticleT1 (MeV)T2 (MeV)kT (MeV)N0χ2dofE866/E8952.7π±80±4/62±4190±5/165±50.88±0.02/0.88±0.0293±4/74±41.19±0.03/2.08±0.0415/2219/26K+128±4−1128±40.16±0.0148p192±6−1192±31.22±0.02278383.32π±85±4/71±4209±12/163±80.78±0.03/0.79±0.02112±4/90±42.84±0.02/3.96±0.048/3224/36K+131±5−1131±51.35±0.02210p213±4−1213±46.99±0.05401383.84π±88±3/75±3230±5/170±50.74±0.03/0.73±0.02125±3/101±34.00±0.03/5.16±0.046/5419/36K+167±5−1167±53.08±0.0119p210±4−1210±46.32±0.05592384.3π±90±4/77±4229±4/171±30.70±0.02/0.71±0.01132±3/104±34.88±0.03/6.30±0.0410/5016/36K+172±5−1172±55.59±0.0617p224±5−1224±56.09±0.07204384.85π+96±3214±50.73±0.03128±3319.96±1.12116K+170±5−1170±54.66±0.1029E8025.03π±103±3/93±4210±4/194±50.73±0.03/0.72±0.02132±2/121±322.00±3.00/27.12±3.0017/4530/29K±172±5/166±5−/−1/1172±5/166±52.28±0.05/0.45±0.013/49/9p230±4−1230±412.59±1.0313627STAR7.7π±115±10/115±10199±3/198±30.68±0.02/0.70±0.02142±10/140±918.21±3.00/18.83±4.0014/1622/22K±185±5/170±5−/−1/1185±5/170±54.19±0.50/1.49±0.3011/1221/21p/p¯226±7/251±6−/−1/1226±7/251±611.01±0.70/0.084±0.00211/327/1311.5π±124±10/123±10211±3/209±30.72±0.02/0.73±0.02148±9/146±922.82±0.50/23.58±0.708/1022/22K±190±5/178±5−/−1/1190±5/178±54.93±0.20/2.43±0.107/423/21p/p¯223±7/223±7−/−1/1223±7/223±79.41±0.60/0.30±0.0113/3226/2114.5π±124±10/123±10213±3/212±30.72±0.02/0.70±0.02149±9/150±927.58±1.30/27.38±1.201/124/24K±183±9/173±8−/−1/1183±9/173±85.49±0.03/3.07±0.021/116/16p/p¯230±9/229±7−/−1/1230±9/229±78.39±0.06/0.52±0.011/123/2319.6π±124±10/123±10216±3/216±30.70±0.03/0.70±0.02152±9/151±929.98±5.10/30.00±5.406/822/21K±193±6/189±5−/−1/1193±6/189±55.99±0.05/3.77±0.0235/3624/24p/p¯237±8/243±9−/−1/1237±8/243±97.17±0.07/0.85±0.016/1527/2027π±125±10/124±10220±3/217±30.68±0.02/0.69±0.02155±10/153±931.86±4.10/32.86±4.505/1021/22K±199±6/190±6−/−1/1199±6/190±66.07±0.05/4.57±0.0319/3224/22p/p¯242±10/251±12−/−1/1242±10/251±126.31±0.07/1.20±0.016/1521/2039π±125±10/125±10225±3/224±30.66±0.02/0.68±0.01159±9/157±933.80±5.10/34.12±5.305/622/22K±203±7/203±6−/−1/1203±7/203±66.23±0.07/4.91±0.0316/2824/24p/p¯257±11/259±10−/−1/1257±11/259±104.91±0.05/1.72±0.015/1920/2162.4π±123±10/123±10220±3/220±30.71±0.01/0.72±0.01151±9/150±942.30±5.12/43.12±5.2022/256/6K±210±8/210±8−/−1/1210±8/210±87.61±0.30/6.50±0.271/108/8p/p¯320±11/346±11−/−1/1320±11/346±116.25±0.20/3.10±0.05170/10213/14130π±120±6/123±5232±9/234±100.68±0.01/0.69±0.01156±6/157±551.86±5.22/51.92±5.6639/746/6K±216±8/214±9−/−1/1216±8/214±99.32±0.28/8.48±0.152/310/11p/p¯353±9/359±8−/−1/1353±9/359±86.64±0.15/4.68±0.1157/13410/11200π±126±9/127±10236±12/235±120.67±0.01/0.70±0.01162±9/159±1059.00±3.33/59.12±3.7018/237/7K±221±13/230±13−/−1/1221±13/230±139.48±0.23/9.50±0.253/28/8p/p¯376±16/372±11−/−1/1376±16/372±117.56±0.18/5.68±0.144/3114/15PHENIX130π±123±10/121±10220±3/229±30.70±0.02/0.73±0.01152±9/150±10167.72±10.50/167.72±11.00146/18210/10K±201±10/213±13−/−1/1210±10/213±1327.31±2.10/21.36±1.5317/1511/11p/p¯276±10/267±11−/−1/1276±10/267±1118.77±1.25/13.80±1.1615/1315/15200π±136±4/136±4259±3/259±30.74±0.02/0.73±0.02168±4/169±4199.02±16.01/196.43±15.10286/20524/24K±197±12/215±13339±11/346±120.73±0.01/0.81±0.01235±12/240±1332.56±2.10/30.04±2.0028/2212/12p/p¯311±7/314±8−/−1/1311±7/314±811.49±0.72/8.62±0.0669/9020/20Figure2 is similar to Figure 1, but it is showing the spectra of various particles produced at mid-y in central Pb-Pb collisions at high sNN, where the factor 1/2π on the vertical axis is removed in some cases and Nevt 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, χ2, 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-y or mid-η in INEL pp collisions at high center-of-mass energies s, where the factor 1/2πpT on the vertical axis is removed and the invariant cross-section Ed3σ/dp3 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, χ2, and dof for fitting the solid and dashed curves are listed in Table 3, where the cross-section σ0 is used as the normalization constant if the spectrum is Ed3σ/dp3. One can see that the (two-component) standard distribution fits approximately the pT (mT−m0) spectra of π±, K±, p, and p¯ 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 in most cases.Figure 2
Same as in Figure1 but showing the spectra of various particles produced at mid-y in central Pb-Pb collisions at high sNN, where the factor 1/2π on the vertical axis is removed in some cases and Nevt if available denotes the particle number which can be removed. The symbols represent the experimental data measured by (a–e) the NA49 [35–37] and (f) the ALICE [38] Collaborations.
(a)(b)(c)(d)(e)(f)Table 2
The same as Table1 but showing the results for central Pb-Pb collisions which are shown in Figure 2.
Collab.sNN (GeV)ParticleT1 (MeV)T2 (MeV)kT (MeV)N0χ2dofNA496.3π±97±5/86±4185±5/181±40.64±0.02/0.65±0.01129±5/119±414.50±1.10/16.98±1.2369/4312/12K±182±5/172±8−/−1/1182±5/172±83.33±0.03/1.14±0.0122/258/8p/p¯227±10/259±12−/−1/1227±10/259±122.05±0.01/0.0023±0.000114/212/57.6π±97±6/86±4195±4/187±30.61±0.01/0.58±0.01135±6/128±417.44±1.15/19.50±2.2024/2612/12K±193±5/181±6−/−1/1193±5/181±64.11±0.33/1.63±0.0530/608/18p/p¯242±6/284±11−/−1/1242±6/284±113.74±0.08/0.014±0.00110/112/58.7π−84±5188±40.56±0.02130±521.40±0.714410K±185±6/187±5−/−1/1185±6/187±54.30±0.05/1.58±0.0137/338/8p/p¯244±7/280±10−/−1/1244±7/280±103.56±0.03/0.030±0.00114/1012/812.3π−84±6189±50.56±0.01130±627.80±2.116010K±185±7/183±5−/−1/1185±7/183±54.92±0.08/2.24±0.0210/5412/16p/p¯244±8/260±14−/−1/1244±8/260±145.48±0.09/0.13±0.015/112/817.3π−84±6200±50.57±0.01134±635.80±1.804710K±193±7/185±6−/−1/1193±7/185±65.52±0.30/3.26±0.0927/1710/10p/p¯280±13/280±14−/−1/1280±13/280±148.86±0.23/0.56±0.039/310/8ALICE2760π±127±4/127±4309±3/309±30.65±0.01/0.65±0.01191±4/191±4762.61±33.11/762.61±33.1174/7337/37K±290±6/275±4409±10/409±30.82±0.02/0.75±0.02311±6/309±4108.82±5.3/108.49±5.149/432/32p/p¯428±9/428±6−/−1/1428±9/428±632.81±2.10/32.91±1.9163/4940/40Figure 3
Same as Figure1 but showing 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 1/2πpT on the vertical axis is removed and the invariant cross-section Ed3σ/dp3 is used in some cases. The symbols represent the experimental data measured by (a–e) the NA61/SHINE [39, 40], (f, g) the PHENIX [41], and (h–k) the CMS [42, 43] Collaborations, where panels (g–k) appear.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)Table 3
The same as Table1 but showing the results for INEL pp collisions which are shown in Figure 3, where the normalization constant is σ0 for the cases of 62.4 and 200 GeV presented in Figures 3(f) and 3(g), respectively.
Collab.s (GeV)ParticleT1 (MeV)T2 (MeV)kT (MeV)N0 (σ0)χ2dofNA61/SHINE6.3π−110±7169±90.82±0.01121±70.086±0.0061214K±113±6/123±6−/−1/1113±6/123±60.0096±0.0003/0.0030±0.00012/14/8p125±5−1125±50.027±0.0016957.7π±103±10/113±10177±9/178±90.78±0.01/0.81±0.01119±10/125±100.16±0.01/0.098±0.0031/161/16K±131±6/127±6−/−1/1131±6/127±60.0013±0.0001/0.0050±0.00024/17/5p/p¯136±10/104±12−/−1/1136±10/104±120.027±0.002/0.00066±0.000014/59/78.8π±105±10/113±10182±9/178±90.78±0.02/0.81±0.02122±10/125±100.16±0.01/0.11±0.011/332/18K±126±6/123±6−/−1/1126±6/123±60.013±0.001/0.0056±0.000224/69/7p/p¯136±9/102±12−/−1/1136±9/102±120.027±0.001/0.00078±0.000041/39/312.3π±108±10/113±10190±9/184±90.77±0.02/0.79±0.01127±10/128±100.16±0.01/0.12±0.011/232/16K±135±6/127±6−/−1/1135±6/127±60.015±0.001/0.0074±0.000310/98/7p/p¯150±10/106±12−/−1/1150±10/106±120.017±0.001/0.0018±0.00016/911/917.3π±111±8/113±10180±9/190±80.76±0.01/0.79±0.01128±8/129±100.16±0.01/0.13±0.011/81/19K±130±6/138±6−/−1/1130±6/138±60.015±0.001/0.010±0.0012/28/8p/p¯150±10/128±12−/−1/150±10/128±120.015±0.001/0.0034±0.00012/48/8PHENIX62.4π±118±5/113±4240±4/229±40.82±0.01/0.81±0.01140±5/135±417.21±3.00/18.40±5.0014/2522/22K±147±6/149±5284±6/278±50.76±0.01/0.78±0.01180±6/177±51.84±0.50/1.58±0.405/212/12p/p¯161±10/163±12283±4/278±50.80±0.00/0.81±0.01182±10/185±111.05±0.40/0.57±0.0312/1423/23200π±117±5/115±5278±5/273±50.85±0.00/0.82±0.01141±5/145±525.26±5.00/22.61±3.00101/8323/23K±151±4/149±5303±8/301±50.72±0.01/0.72±0.01193±4/192±52.78±0.40/2.76±0.7010/712/12p/p¯162±5/160±6316±7/312±50.73±0.00/0.76±0.00204±5/197±61.23±0.20/1.09±0.10108/9130/30CMS900π±115±4/117±5264±5/270±60.68±0.01/0.68±0.01163±4/165±53.76±0.50/3.70±0.5013/1218/18K±156±6/154±6343±10/340±100.59±0.01/0.58±0.01233±6/232±60.47±0.03/0.46±0.034/613/13p/p¯177±6/177±5331±9/328±80.55±0.01/0.56±0.01246±6/243±50.21±0.02/0.20±0.0119/1723/232760π±116±3/116±4275±5/281±50.64±0.01/0.65±0.01173±3/174±44.68±0.90/4.60±0.8019/1418/18K±165±7/164±8351±9/350±90.56±0.01/0.56±0.01247±7/246±80.60±0.03/0.59±0.035/1013/13p/p¯182±6/183±7406±8/392±70.54±0.01/0.55±0.01285±6/277±70.27±0.01/0.26±0.0128/3223/237000π±116±4/115±4292±5/288±40.63±0.02/0.62±0.01181±4/181±45.82±0.80/5.76±0.8034/2218/18K±178±6/175±6420±10/393±100.59±0.02/0.56±0.02277±6/271±60.76±0.03/0.75±0.033/813/13p/p¯217±7/202±7429±9/430±90.57±0.02/0.53±0.01308±7/309±70.34±0.02/0.33±0.0228/2223/2313000π±114±4/114±6300±5/291±60.61±0.01/0.60±0.01187±4/185±65.46±0.70/5.38±0.609/1018/18K±180±10/180±10360±12/383±120.59±0.01/0.60±0.01254±10/261±100.65±0.03/0.65±0.033/213/13p/p¯229±10/231±11425±3/425±90.63±0.01/0.62±0.01302±10/305±100.30±0.02/0.29±0.0225/1922/22To study the dependence of the main parameterT on collision energy sNN or s, the excitation functions of T for central Au-Au (Pb-Pb) collisions and INEL pp collisions are shown in Figures 4(a) and 4(b), respectively. The results obtained from the spectra of π+, π−, K+, K−, p, and p¯ 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 where one of the six particles is absent, T is not available. One can see that T and T increase with the increase of lnsNN [lns]. Meanwhile, T increases with the increase of particle mass.Figure 4
Excitation functions ofT and T in (a) central Au-Au (Pb-Pb) collisions and (b) INEL pp collisions. The symbols corresponding to identified particles are extracted from experimental spectra, where the symbols with small size are used for Pb-Pb collisions.
(a)(b)To better determine the kinetic freeze-out information, we now fit simultaneously the spectra ofπ+, π−, K+, K−, p, and p¯ in different pT ranges using the same set of parameters. In Figures 1–3, the dot-dashed curves are the results using the weighted average T which is energy dependent, though in fact we may use the other T to obtain a little better result in some cases. In the refit, the normalization constant N0 for different spectra is adjustable to fit the suitable pT range. In despite of mass-dependent (two-)temperature in the nonsimultaneous fit, the simultaneous fit is done using the same and only set of T 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 T can fit only a part of the pT 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 paper.Figures5(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 T0 (βT) and T0 (βT). 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 Ti and Ti. One can see that T0, T0, βT, βT, Ti, and Ti increase with the increase of lnsNN [lns] in general, and T0 for pion emission and T0 appears in the trend of saturation at the RHIC and LHC. Meanwhile, T0, T0, Ti, and Ti increase, and βT and βT decrease, with the increase of the particle mass.Figure 5
Same as Figure4 but showing the excitation functions of (a) and (b) T0 and T0, as well as (c) and (d) βT and βT.
(a)(b)(c)(d)Figure 6
Same as Figure4 but showing the excitation functions of Ti and Ti.
(a)(b)It is regretful that some particles are absent in experimental measurements at the energies below 10 GeV. This renders thatT, T0, βT, and Ti are not available in the energy range of several GeV. From the trends of available T, T0, βT, and Ti, we may estimate that T, T0, βT, and Ti increase (quickly) with the increase of lnsNN [lns] 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 T0 and βT, it seems that one can obtain arbitrary results by choosing another parametrization of k0. In particular, it seems possibly to choose a parametrization of k0 so that π+ (π−), K+ (K−), and p (p¯) could have similar T0 and βT even though the combined temperature T (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 pT 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 ofT0 for pion emission and T0 had stabilized.Before summary and conclusions, we would like to point out that we have used a new method to extractT0, βT, and Ti. After fitting the pT (mT) spectra by using the two-component standard distribution Equations (6) or (7) in which the free parameters are the effective temperatures T1 and T2 and the contribution fraction k of the first component, the derived parameter, the effective temperature T, can be obtained from the weighted average formula Equation (8). Then, the derived parameters, the kinetic freeze-out temperature T0, and transverse flow velocity βT, can be obtained, respectively, from Equations (16) and (17) which are related to the mean transverse momentum pT. The derived parameter, the initial temperature Ti, can be obtained from Equation (29) which is related to the root-mean-square pT2.According to the analysis of the spectra of six hadron species listed in Tables1 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-pT region, which can be described by the first component in Equation (6) or (7). Some low-pT pions from nonresonance decay can be also described by the first component. As an ensemble, Equation (1) describes the cold source with low T for all low-pT pions. However, the resonance decays for kaons and (anti)protons are relatively small comparing to those for pions in the low-pT region, which are concealed in a single source.Naturally, if we regardT, T0, βT, and Ti 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 (Figures4 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 T0 but are left behind under compulsion due to low βT and large m0. 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.BecauseT0 and βT 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 n is used in the extraction of T0 and βT. The parameter n 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-pT region (<0.5 GeV/c) are excluded from the fit [32] due to resonance decay, which overrates T0 and βT. Our work shows that T0 (βT) in pp 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 T versus m0 is regarded as T0 and the slope in the linear relation of pT versus m0γ¯ is regarded as βT. 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 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 temperatureT and its weighted average T, the kinetic freeze-out temperature T0 and its weighted average T0, the transverse flow velocity βT and its weighted average βT, and the initial temperature Ti and its weighted average Ti have been obtained. These excitation functions have shown some obvious laws with the increase of collision energy.In the above discussions, to obtainγ¯ then βT, 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 pT (mT−m0) 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 K+ 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 ofπ+, π−, K+, K−, p, and p¯ 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. 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.



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*Source: 5356705-2020-06-10.xml*