# A New Description of Transverse Momentum Spectra of Identified Particles Produced in Proton-Proton Collisions at High Energies

Authors: Pei-Pin Yang; Fu-Hu Liu; Raghunath Sahoo
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

The transverse momentum spectra of identified particles produced in high energy proton-protonp+p collisions are empirically described by a new method with the framework of the participant quark model or the multisource model at the quark level, in which the source itself is exactly the participant quark. Each participant (constituent) quark contributes to the transverse momentum spectrum, which is described by the TP-like function, a revised Tsallis–Pareto-type function. The transverse momentum spectrum of the hadron is the convolution of two or more TP-like functions. For a lepton, the transverse momentum spectrum is the convolution of two TP-like functions due to two participant quarks, e.g., projectile and target quarks, taking part in the collisions. A discussed theoretical approach seems to describe the p+p collisions data at center-of-mass energy s=200GeV, 2.76 TeV, and 13 TeV very well.


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## Body

## 1. Introduction

As one of the “first day” measurable quantities, the transverse momentum (pT) spectra of various particles produced in high energy proton-proton (p+p) (hadron-hadron), proton-nucleus (hadron-nucleus), and nucleus-nucleus collisions are of special importance because it reveals about the excitation degree and anisotropic collectivity in the produced systems. The distribution range of pT is generally very wide, from 0 to more than 100 GeV/c, which is collision energy dependent. In the very low-, low-, high-, and very high-pT regions [1], the shapes of pT spectrum for given particles are possibly different from each other. In some cases, the differences are very large, and the spectra show different empirical laws.Generally, the spectrum in (very) low-pT region contributed by (resonance decays or other) soft excitation process. The spectrum in (very) high-pT region is related to (very) hard scattering process (pQCD). There is no clear boundary in pT to separate soft and hard processes. At a given collision energy, for different collision species, looking into the spectral shape, a theoretical function that best fits to the pT-spectra is usually chosen to extract information like rapidity density, dN/dy, kinetic freeze-out temperature, Tkin or T0 and average radial flow velocity, βT orβT. The low-pT region up to ∼2–3 GeV/c is well described by a Boltzmann–Gibbs function, whereas the high-pT part is dominated by a power-law tail. It is interesting to note that there are many different functions, sometimes motivated by the experimental trend of the data or sometimes theoretically, to have a proper spectral description thereby leading to a physical picture. The widely used functions are:
(1)
An exponential function inpT or mT [2, 3](1)fpT=pT×A×e−pT/T×em0/TT2+Tm0,fpT=pT×A×e−mT/T×em0/TT2+Tm0.Here,A is the normalization constant, T is the effective temperature (thermal temperature and collective radial flow), and mT=pT2+m02 is the transverse mass, with m0 being the identified particle rest mass.
(2)
A Boltzmann distribution(2)fpT=pT×A×mT×e−mT/T×em0/T2T3+2T2m0+Tm02.(3)
Bose–Einstein/Fermi–Dirac distribution(3)fpT=pT×A×mT×1emT/T∓1×em0/T∓1.(4)
Power-law or Hagedorn function [4](4)fpT=pT×A×1+pTp0−n→exp−npTp0,forpT→0,p0pTn,forpT→∞,where p0 and n are fitting parameters. This becomes a purely exponential function for small pT and a purely power-law function for large pT values.
(5)
Tsallis–Levy [5, 6] or Tsallis–Pareto-type function [6–9](5)fpT=pT×An−1n−2nTnT+m0n−2×1+mT−m0nT−n.Note here that a multiplicative prefactor ofpT in the above functions are used assuming that the pT spectra do not have a pT factor in the denominator (see the expression for the invariant yield) and all the functions are normalized so that the integral of the functions provides the value of “A.” When the first three functions describe thepT-spectra up to a lowpTaround 2–3 GeV/c, the fourth function, i.e., the power-law, describes the highpTpart of the spectrum. The last two functions (power-law or Hagedorn function and Tsallis-L{\'e}vy or Tsallis–Pareto-type function), which are more empirical in nature, lack microscopic picture, however, describe a wide variety of identified particle spectra. The Tsallis distribution function, while describing the spectra in p+p collisions [10], has brought up the concept of nonextensive entropy, contrary to the low-pT domain pointing to an equilibrated system usually described by Boltzmann–Gibbs extensive entropy. In addition, the identified particle spectra are successfully explained in heavy ion collisions with the inclusion of radial flow in a Tsallis Blast Wave description [11].The two behaviors in (very) low- and (very) high-pT regions are difficult to fit simultaneously by a simple probability density function. Instead, one can use a two-component function [12], the first component f1pT is for the (very) low-pT region and the second component f2pT is for the (very) high-pT region, to superpose a new function fpT to fit the pT spectra. There are two forms of superpositions, fpT=kf1pT+1−kf2pT or fpT=A1θp1−pTf1pT+A2θpT−p1f2pT [4, 13, 14], where k denotes the contribution fraction of the first component, A1 and A2 are constants which make the two components equal to each other at pT=p1, and θx is the usual step function which satisfies θx=0 if x<0 and θx=1 if x≥0.It is known that there are correlations in determining parameters in the two components in the first superposition [13]. There is possibly a nonsmooth interlinkage at pT=p1 between the two components in the second superposition [14]. We do not expect these two issues. To avoid the correlations and nonsmooth interlinkage, we hope to use a new function to fit simultaneously the spectra in the whole pT region for various particles. After sounding many functions out, a Tsallis–Pareto-type function [6–9] which empirically describes both the low-pT exponential and the high-pT power-law [15–18] is the closest to our target, though the Tsallis–Pareto-type function is needed to revise its form in some cases.In this work, to describe the spectra in the wholepT range which includes (very) low and (very) high pT regions, the Tsallis–Pareto-type function is empirically revised by a simple method. To describe the spectra in the whole pT range as accurately as possible, the contribution of participant quark to the spectrum is also empirically taken to be the revised Tsallis–Pareto-type (TP-like) function with another set of parameters. Then, the pT distribution of given particles is a convolution of a few TP-like functions. To describe the spectra of identified particles in the whole pT range, both the TP-like function and the convolution of a few TP-like functions are used to fit the data measured in p+p collisions at the center-of-mass energy s=200GeV [19–23], 2.76 TeV [24–32], and 13 TeV [33–39] by different collaborations.The remainder of this paper is structured as follows. The formalism and method are described in Section1. The results and discussion are given in Section 2. In Section 3, we summarize our main observations and conclusions.


## 2. Formalism and Method

According to [6–9], the Tsallis–Pareto-type function which empirically describes both the low-pT exponential and the high-pT power-law can be simplified as presented in [15–18],
(6)fpT=C×pT×1+pT2+m02−m0nT−n,in terms of pT probability density function, where the parameter T describes the excitation degree of the considered source, the parameter n describes the degree of nonequilibrium of the considered source, and C is the normalization constant which depends on T, n, and m0. Equation (6) is in fact an improvement of Eq. (5).As an empirical formula, the Tsallis–Pareto-type function is successful in the description ofpT spectra in many cases. However, our exploratory analysis shows that Eq. (6) in some cases is not accurate in describing the spectra in the whole pT range. In particular, Eq. (6) is not flexible enough to describe the spectra in a very low-pT region, which is contributed by the resonance decays. We would like to revise empirically Eq. (6) by adding a power index a0 on pT. After the revision, we have
(7)fpT=C×pTa0×1+pT2+m02−m0nT−n,where C is the normalization constant which is different from that in Eq. (6). To be convenient, the two normalization constants in Eqs. (6) and (7) are denoted by the same symbol C. Equation (7) can be used to fit the spectra in the whole pT range. The revised Tsallis–Pareto-type function (Eq. (7)) is called the TP-like function by us.It should be noted that the indexa0 is a quantity with nondimension. Because of the introduction of a0, the dimension of pTa0 is GeV/ca0. The dimension of pTa0 does not affect the dimension (GeV/c)−1 of fpT. In fact, to fit the dimension of fpT, the dimension of the product CpTa0 is limited to be (GeV/c)−1. That is to say, the dimension of pTa0 is combined in the normalization constant so that we can obtain the consistent dimension for both sides of the equation. Due to the introduction of a0, for the spectra in the very low-pT region, not only the production of light particles via resonance decay but also the decay or absorbtion effect of heavy particles in the hot and dense medium in the participant region can be described.Our exploratory analysis shows that Eq. (7) is not accurate in describing the spectra in the whole pT range, too, though it is more accurate than Eq. (6). To obtain accurate results, the amount or portion (pti) contributed by the ith participant quark to pT is assumed to obey
(8)fipti=Ci×ptia0×1+pti2+m0i2−m0inT−n,where the subscript i is used for the quantities related to the participant quark i, and m0i is empirically the constituent mass of the considered quark i. The value of i can be 2 or 3 even 4 or 5 due to the number of participant (or constituent) quarks. Equation Eq. (8) is also the TP-like function with different mass from Eq. (7).It should be noted thatm0 in Eq. (7) is for a particle and m0i in Eq. (8) is for the quark i. For example, if we study the pT spectrum of protons, we have m0=0.938GeV/c2 and m01=m02=m03=0.31GeV/c2. In the case of studying the pT spectrum of photons, we have m0=0 and m01=m02=0.31GeV/c2 if we assume that the two lightest quarks take part in the collision with photon production.There are two participant quarks to constitute usually mesons, namely the quarks 1 and 2. ThepT spectra of mesons are the convolution of two TP-like functions.We have(9)fpT=∫0pTf1pt1f2pT−pt1dpt1=∫0pTf2pt2f1pT−pt2dpt2,in which f1pt1f2pt2 is the probability for the given pt1 and pt2. The total probability considered various pt1 and pt2 is given by Eq. (9) which is the convolution of distributions of two independent variables [40, 41]. The upper limit pT is not a cutoff, but the sum of pt1 and pt2, which is limited by physics. The lower limit 0 is also from the limitation related to the underlying physics. No matter how many leptons are produced in the process, two participant quarks are considered to contribute to the pT spectrum of each lepton.We would like to explain our treatment on Eq. (9) here. At least three relations between particle pT and quark pt1pt2 can be assumed. (i) If we regard pt1pt2 as the amount or portion contributed by the first (second) participant quark to pT, we have pT=pt1+pt2. (ii) If we regard the vector pt1pt2 as the component contributed by the first (second) participant quark to the vector pT, we have pT=pt12+pt22, where pt1 is perpendicular to pt2. (iii) In the second relation, it is not necessary that all the components are perpendicular, then we have pT=pt12+pt22+2pt1pt2cosϕ1−ϕ2, where ϕ1ϕ2 is the azimuthal angle of the first (second) participant quark. Different assumptions result in different relations. Of course, the three pt1pt2 in the three relations have different meanings, though the same symbol is used. In our opinion, at present, it is hard to say which relation is more correct. We need to test the three relations by more experimental data.In fact, all the three relations have still pending issues which needed further discussions. In the relation (i), althoughpT can be considered as the contribution of two energy sources: the first and second participant quarks that contribute the amounts or portions pt1 and pp2 to pT, respectively, the vector characteristic of transverse momentum is not used. In the relation (ii), as a vector, the transverse momentum is considered by two components: pt1 and pt2 which are contributed by the first and second participant quarks, respectively, though the origin of the third component of meson momentum is not clear. In addition, although the origin of three components of baryon momentum is clear, the physics picture is not consistent to the meson momentum. In the relation (iii), two more parameters ϕ1 and ϕ2 are introduced, which is not our expectation.This paper has used the relation (i) and Eq. (9) which is based on the probability theory [40–42]. However, in our recent work [43], we have used the relation (ii) and another functional form which is based on the vector and probability theory [41, 42]. We hope that we may use the relation (iii) in our future work by some limitations on ϕ1 and ϕ2. The relation (i) in terms of amount or portion is the same as or similar to the relation for multiplicity or transverse energy contributed by two sources [40]. This similarity reflects the law of universality existing in high energy collisions [44–49]. In fact, transverse momentum, multiplicity, and transverse energy reflect the amount of effective energy deposited in collisions [50, 51]. The effective energy through the participant quarks reflects the similarity or universality, which is not related to the production mechanisms for different particles. Then, different particles are described by the same type of model (formula).At the level of current knowledge, leptons have no further structures. However, to produce a lepton in a common process, two participant quarks, a projectile quark and a target quark, are assumed to take part in the interactions. ThepT spectra of leptons are in fact the convolution of two TP-like functions, that is Eq. (9) in which m01 and m02 are empirically the constituent mass of the lightest quark. To produce leptons in a special process such as in cc¯→μ+μ−, m01m02 is the constituent mass of the c quark.There are three participant quarks that constitute usually baryons, namely the quarks 1, 2, and 3. ThepT spectra of baryons are the convolution of three TP-like functions. We have the convolution of the first two TP-like functions to be
(10)f12pt12=∫0pt12f1pt1f2pt12−pt1dpt1=∫0pt12f2pt2f1pt12−pt2dpt2.The convolution of the first two TP-like functions and the third TP-like function is(11)fpT=∫0pTf12pt12f3pT−pt12dpt12=∫0pTf3pt3f12pT−pt3dpt3.Equation (7) can fit approximately the spectra in the whole pT range for various particles at the particle level, in which m0 is the rest mass of the considered particle. In principle, Eqs. (9) and (11) can fit the spectra in the whole pT range for various particles at the quark level, in which m0i is the constituent mass of the quark i. If Eq. (7) is more suitable than Eq. (6), Eqs. (9) and (11) are the results of the multisource model [52, 53] at the quark level. In the multisource model, one, two, or more sources are assumed to emit particles due to different production mechanisms, source temperatures, and event samples. In a given event sample, the particles with the same source temperature are assumed to emit from the same source by the same production mechanism. We can also call Eqs. (9) and (11) the results of the participant quark model due to the fact that they describe the contributions of participant quarks.It should be noted that, in principle, the three quarks should be symmetric in the formula for the production of baryons. Indeed, in Eqs. (10) and (11), the two momenta pt,1 and pt,2 are symmetric, and the third momentum pt,3 is also symmetric to the other two momenta. In fact, according to the rule of the convolution of three functions, we may also convolute firstly the last two functions, and then, we may convolute the result with the first function. Meanwhile, we may also convolute firstly the first and third functions, and then, we may convolute the result with the second function. We realize that the final result is not related to the order of convolution. The three functions contributed by the three quarks are indeed symmetric.We would like to explain the normalization constant in detail. As a probability density function,fpT=1/NdN/dpT cannot be used to compare directly with the experimental data presented in the literature in some cases, where N denotes the number of considered particles. Generally, the experimental data are presented in forms of (i) dN/dpT, (ii) d2N/dydpT, and (iii) 1/2πpTd2N/dydpT=Ed3N/dp3, where Ep denotes the energy (momentum) of the considered particle. One can use N0fpT, N0fpT/dy, and 1/2πpTN0fpT/dy to fit them accordingly, where N0 denotes the normalization constant.The data are usually in the form (i)dσ/dpT, (ii) d2σ/dydpT, and (iii) 1/2πpTd2σ/dydpT=Ed3σ/dp3, where σ denotes the cross-section. One can use σ0fpT, σ0fpT/dy, and 1/2πpTσ0fpT/dy to fit them accordingly, where σ0 denotes the normalization constant. The data presented in terms of mT can also be studied due to the conserved probability density and the relation between mT and pT. In particular, 1/2πpTd2σ/dydpT=1/2πmTd2σ/dydmT, where σ can be replaced by N.It should be noted that our treatment procedure means that the parameters are fitted for each energy and rapidity bin separately. This would limit the usefulness of the proposed parametrizations somewhat. However, after obtaining the relations between parameters and energy/rapidity, we can use the obtained fits to predictpT distributions at other energies/rapidities where the data are not available and the parameters are not fitted.


## 3. Results and Discussion

### 3.1. Comparison with Data

Figure1(a) shows the pT spectra (the invariant cross-sections), Ed3σ/dp3, of different hadrons with given combinations and decay channels including π++π−/2 plus π0→γγ, K++K−/2 plus KS0→π0π0, η→γγ plus η→π0π+π−; ω→e+e− plus ω→π0π+π− plus ω→π0γ, p+p¯/2, η′→ηπ+π−, ϕ→e+e− plus ϕ→K+K−, J/ψ→e+e−, and ψ′→e+e− produced in p+p collisions at 200 GeV. Different symbols represent different particles and their different decay channels measured by the PHENIX Collaboration [19] in the pseudorapidity range of η<0.35. The results corresponding to π, K, η, ω, p, and η′ are rescaled by multiplying by 106, 105, 104, 103, 102, and 10 factors, respectively. The results corresponding to ϕ, J/ψ, and ψ′ are not rescaled.Figure 1
(a) The invariant cross-sections of different hadrons with given combinations and decay channels produced inp+p collisions at 200 GeV. Different symbols represent different particles and their different decay channels in ∣η∣<0.35 measured by the PHENIX Collaboration [19]; some of them are scaled by different factors marked in the panel. The dotted and dashed curves are our fitted results by using Eqs. (7) and (9) or (11), respectively. (b) The ratio of data to fit obtained from Eq. (7). (c) The ratio of data to fit obtained from Eq. (9) or (11).In Figure1(a), the dotted and dashed curves are our fitted results by using Eqs. (7) (for mesons and baryons) and (9) (for mesons) or (11) (for baryons), respectively. The values of free parameters (T, n, and a0), normalization constant σ0, χ2, and the number of degree of freedom (ndof) obtained from Eq. (7) are listed in Table 1, while the values of parameters and χ2/ndof obtained from Eqs. (9) or (11) are listed in Table 2. In Eq. (7), m0 is taken to be the rest mass of π, K, η, ω, p, η′, ϕ, J/ψ, and ψ′ for the cases from π++π−/2 to ψ′→e+e− sequenced according to the order shown in Figure 1(a). In the fit process at the quark level, the quark structure of π0 results in its fpT to be the half of the sum of uu¯,sfpT and dd¯,sfpT. Because the constituent masses of u and d are the same [54], π0,sfpT is equal to uu¯,sfpT or dd¯,sfpT. The quark structure of η results in its fpT to be cos2ϕ×uu¯,sfpT+sin2ϕ×ss¯sfpT due to the quark structures of ηq and ηs, where ϕ=39.3∘±1.0∘ is the mixing angle [55]. The quark structure of η′ results in its fpT to be sin2ϕ×uu¯,sfpT+cos2ϕ×ss¯,sfpT.Table 1
Values ofT, n, a0, σ0, χ2, and ndof corresponding to the dotted curves in Figures 1(a), 3(a), and 5(a) which are fitted by the TP-like function (Eq. (7)). In the case of ndof being less than 1, it appears as “–” in the table.
FigureyηParticleT (GeV)na0σ0mbχ2/ndofFigure1(a)200 GeVη<0.35π++π−/2π00.129±0.0019.449±0.0200.890±0.00437.044±0.3485/39K++K−/2KS00.167±0.0029.529±0.0301.027±0.0043.122±0.0307/27η0.195±0.0029.889±0.0331.000±0.0031.755±0.0876/32ω0.193±0.0019.460±0.1000.900±0.0203.073±0.03023/34p+p¯/20.149±0.0029.100±0.0201.040±0.0031.291±0.00811/13η′0.210±0.00210.001±0.0230.980±0.0030.584±0.0044/8ϕ0.245±0.00210.559±0.0230.688±0.0030.334±0.00310/15J/ψ0.482±0.00216.778±0.0230.901±0.0045.320±0.132×10−44/22ψ′0.452±0.0028.349±0.0220.959±0.0039.234±0.008×10−51/−Figure3(a)2.76 TeVη<0.8π++π−0.130±0.0016.882±0.0210.937±0.0023.961±0.032×10246/59K++K−0.167±0.0016.985±0.0191.209±0.00347.404±0.64932/54p+p¯0.199±0.0017.870±0.0241.064±0.00323.645±0.12956/45y<0.5ρ07700.205±0.0016.987±0.0211.140±0.00314.167±0.0696/6ϕ0.245±0.0026.696±0.0211.010±0.0041.673±0.0167/17y<0.8η0.195±0.0016.910±0.0241.023±0.0033.340±0.064×10−107/72.5<y<4J/ψ0.482±0.0027.231±0.0251.225±0.0032.181±0.029×10−24/3Figure5(a)13 TeVy<1π++π−/20.129±0.0014.862±0.0210.806±0.0034.608±0.018×10257/18K++K−/20.167±0.0016.179±0.0181.261±0.0024.312±0.039×1013/13p+p¯/20.199±0.0024.768±0.0231.180±0.0042.211±0.013×1018/222<y<2.5J/ψprompt0.482±0.0016.729±0.0221.581±0.0031.097±0.001×10−2115/10J/ψfromb0.482±0.0015.529±0.0241.877±0.0021.850±0.020×10−335/10ψ2sprompt0.578±0.0017.603±0.0211.867±0.0031.509±0.019×10−333/13ψ2sfromb0.578±0.0015.989±0.0231.901±0.0044.907±0.104×10−433/13D0+D¯00.497±0.0016.624±0.0221.244±0.0035.573±0.063×10−15/14D++D−0.497±0.0026.446±0.0221.231±0.0052.805±0.059×10−111/13D∗++D∗−0.497±0.0016.566±0.0231.231±0.0042.622±0.054×10−116/11Ds++Ds−0.497±0.0019.259±0.0212.217±0.0047.876±0.149×0−25/12Table 2
Values ofT, n, a0, σ0, χ2, and ndof corresponding to the dashed curves in Figures 1(a), 3(a), and 5(a) which are fitted by the convolution (Eq. (9) or (11)) of two or three TP-like functions. The quark structures are listed together. In the case of ndof being less than 1, it appears as “–” in the table.
FigureyηParticleQuark structureT (GeV)na0σ0mbχ2/ndofFigure1(a)200 GeVη<0.35π++π−/2ud¯,du¯0.209±0.0027.774±0.025−0.540±0.02038.357±0.3506/39π0uu¯−dd¯/2K++K−/2us¯,su¯0.196±0.0017.816±0.030−0.091±0.0052.913±0.0264/27KS0ds¯η:ηq,ηsuu¯+dd¯/2,ss¯0.212±0.0018.109±0.0300.000±0.0041.838±0.0174/32ωuu¯+dd¯/20.222±0.0018.394±0.0130.000±0.0022.854±0.02620/34p+p¯/2uud,u¯u¯d¯0.162±0.0027.600±0.021−0.130±0.0031.282±0.0113/13η′:ηq,ηsuu¯+dd¯/2,ss¯0.233±0.0028.315±0.0250.000±0.0030.593±0.0043/8ϕss¯0.266±0.0019.022±0.022−0.107±0.0030.319±0.00310/15J/ψcc¯0.509±0.00214.545±0.0250.008±0.0035.275±0.004×10−44/22ψ′cc¯0.503±0.0027.025±0.0300.055±0.0049.232±0.008×10−51/−Figure3(a)2.76 TeVη<0.8π++π−ud¯,du¯0.209±0.0014.970±0.022−0.490±0.0034.474±0.021×10267/59K++K−us¯,su¯0.197±0.0025.223±0.017−0.058±0.00444.823±0.60530/54p+p¯uud,u¯u¯d¯0.194±0.0025.867±0.021−0.120±0.00324.401±0.11331/45y<0.5ρ0770uu¯+dd¯/20.252±0.0015.775±0.021−0.003±0.00314.514±0.0754/6ϕss¯0.266±0.0015.313±0.0190.018±0.0011.679±0.0456/17y<0.8η:ηq,ηsuu¯+dd¯/20.212±0.0025.341±0.022−0.021±0.0033.441±0.021×10−1010/72.5<y<4J/ψcc¯0.509±0.0026.181±0.0180.155±0.0042.179§0.007×1024/3Figure5(a)13 TeVy<1π++π−/2ud¯,du¯0.182±0.0014.401±0.024−0.390±0.0024.579±0.016×10254/18K++K−/2us¯,su¯0.196±0.0017.081±0.023−0.023±0.0024.384×0.016×1011/13p+p¯/2uud,u¯u¯d¯0.191±0.0013.832±0.021−0.090±0.0032.207±0.011×1018/222<y<2.5J/ψpromptcc¯0.509±0.0014.855±0.0220.291±0.0031.056±0.002×10−297/10J/ψfrombcc¯0.509±0.0013.904±0.0230.447±0.0021.812±0.016×10−328/10ψ2spromptcc¯0.675±0.0026.098±0.0220.435±0.0031.490±0.017×10−321/13ψ2sfrombcc¯0.675±0.0024.531±0.0190.431±0.0044.836±0.059×10−420/13D0+D¯0cu¯,c¯u0.545±0.0015.125±0.0170.104±0.0035.420±0.045×10−13/14D++D−cd¯,c¯d0.545±0.0024.929±0.0230.098±0.0042.718×0.043×10−18/13D∗++D∗−cd¯,c¯d0.545±0.0015.025±0.0190.101±0.0022.477±0.034×10−114/11Ds++Ds−cs¯,c¯s0.545±0.0016.515±0.0200.549±0.0037.406±0.099×10−25/12To show departures of the fit from the data, following Figure1(a), Figures 1(b) and 1(c) show the ratios of data to fit obtained from Eqs. (7) and (9) or (11), respectively. One can see that the fits are around the data in the whole pT range, except for a few sizeable departures. The experimental data for the mentioned hadrons measured in p+p collisions at 200 GeV by the PHENIX Collaboration [19] can be fitted by Eqs. (7) (for mesons and baryons) and (9) (for mesons) or (11) (for baryons). From the values of χ2 and the data over fit ratio, one can see that Eq. (9) or (11) can describe the data equally well as Eq. (7).It seems that Eq. (9) or (11) is not necessary due to Eq. (7) being good enough. In fact, the introduction of Eq. (9) or (11) does not contain more parameters compared with Eq. (7). Moreover, Eq. (9) or (11) can tell more about the underlying physics than Eq. (7). The effective temperature used in Eq. (9) or (11) is related to the excitation degree of quark matter, while the effective temperature in Eq. (7) is related to the excitation degree of hadronic matter. In our opinion, Eqs. (9) and (11) are necessary. We shall analyze sequentially the pT spectra of identified particles by using Eqs. (7) and (9) or (11) in the following text.Figure2(a) shows the invariant cross-sections of inclusive direct photons and different leptons with given combinations and production channels including e++e−/2, μ++μ−/2 (open heavy-flavor decays), Drell–Yan →μ+μ−, cc¯→μ+μ−, and bb¯→μ±μ± produced in p+p collisions at 200 GeV. Different symbols represent different particles and their production channels measured by the PHENIX Collaboration [20–23] in different η or y ranges. The dotted and dashed curves are our fitted results by using Eqs. (7) and (9), respectively, where two participant quarks are considered in the formation of the mentioned particles. The values of parameters and χ2/ndof obtained from Eqs. (7) and (9) are listed in Tables 3 and 4, respectively. In Eq. (7), m0 is taken to be the rest mass of γ, e, μ, 2μ, 2μ, and 4μ for the cases from inclusive direct γ to bb¯→μ±μ± sequenced according to the order shown in Figure 2(a), where 2μ is two times due to the continued two 2μ-related channels. In Eq. (9), m01+m02 are taken to be the constituent masses of u+u, u+u, u+c, u+u, c+c, and b+b sequenced according to the same order as particles.Figure 2
(a) The invariant cross-sections of photons and different leptons for a given combination of intermediate channel forp+p collisions at 200 GeV. Different symbols represent different particles, and their production channels in different η ranges measured by the PHENIX Collaboration [20–23]. The dotted and dashed curves are our fitted results by using Eqs. (7) and (9), respectively. (b) The ratio of data to fit obtained from Eq. (7). (c) The ratio of data to fit obtained from Eq. (9).Table 3
Values ofT, n, a0, σ0, χ2, and ndof corresponding to the dotted curves in Figures 2(a), 4(a), and 6(a) which are fitted by the TP-like function (Eq. (7)).
FigureyηParticleT (GeV)na0σ0mbχ2/ndofFigure2(a)200 GeVη<0.35Inclusivedirectγ0.258±0.0019.413±0.0201.750±0.0044.836±0.044×10−32/14e++e−/20.155±0.0028.460±0.0300.652±0.0031.105±0.009×10−28/241.4<y<2.0μ++μ−/2Openheavydecays0.125±0.0019.308±0.0220.799±0.0032.343±0.015×10−27/91.2<y<2.2Drell−Yan→μ+μ−0.349±0.0028.849±0.0232.200±0.0041.559±0.001×10−78/8cc¯→μ+μ−0.385±0.00213.983±0.0231.509±0.0034.227±0.004×10−910/11bb¯→μ±μ±0.445±0.00220.501±0.0502.260±0.0306.917±0.006×10−128/6Figure4(a)2.76 TeVy<0.9Inclusiveγ0.166±0.0016.791±0.0200.068±0.0023.565±0.035×10232/142.5<y<4μFromheavydecays0.345±0.0017.528±0.0210.000±0.0031.480±0.0027/12y<0.8eFrombeautydecays0.315±0.0016.094±0.0161.000±0.0047.686±0.0513/16eFromheavydecays0.165±0.0014.305±0.020−0.043±0.0043.701±0.063×1027/21e++e−/2Inclusive0.155±0.0025.554±0.019−0.05±0.0022.726±0.0577/15Figure6(a)13 TeVη<1.37H→diphoton0.150±0.00114.681±0.02212.257±0.0045.295±0.186×10−1116/9η<0.8Heavy dielectron0.125±0.0018.811±0.0192.281±0.0037.581±0.034×10−16/13Table 4
Values of T, n,a0, σ0, χ2 and ndof corresponding to the dashed curves in Figures 2(a), 4(a), and 6(a) which are fitted by the convolution (Eq. (9)) of two TP-like functions. The participant quarks are listed together.
FigureyηParticleQuarkT (GeV)na0σ0mbχ2/ndofFigure2(a)200 GeVη<0.35Inclusivedirectγuu¯0.383±0.0016.793±0.0240.060±0.0024.967±0.044×10−32/14e++e−/2uu¯0.236±0.0026.408±0.020−0.596±0.0031.192±0.008×10−25/241.4<y<2.0μ++μ−/2Openheavydecaysuc0.167±0.0016.035±0.025−0.802±0.0032.226±0.014×10−24/91.2<y<2.2Drell−Yan→μ+μ−uu¯0.418±0.0025.616±0.0230.398±0.0041.571±0.001×10−78/8cc¯→μ+μ−cc¯0.207±0.0024.072±0.0250.005±0.0044.206±0.004×10−98/11bb¯→μ±μ±bb¯0.207±0.0025.653±0.0240.049±0.0047.047±0.006×10−123/6Figure4(a)2.76 TeVy<0.9Inclusiveγuu¯0.233±0.0015.383±0.019−0.700±0.0033.345±0.020×10227/142.5<y<4μFromheavydecayscc¯0.309±0.0014.554±0.022−0.704±0.0031.364±0.0039/12y<0.8eFrombeautydecaysbb¯0.080±0.0011.993±0.018−0.150±0.0047.323±0.0424/16eFromheavydecayscc¯0.206±0.0022.441±0.017−0.894±0.0023.720±0.043×1025/21e++e−/2Inclusiveuu¯0.166±0.0013.724±0.018−0.700±0.0032.770±0.02112/15Figure6(a)13 TeVη<1.37H→diphotoncc¯0.702±0.0014.573±0.0192.550±0.0035.376±0.209×10−117/13η<0.8Heavy dielectroncc¯0.166±0.0013.128±0.021−0.520±0.0027.581±0.034×10−16/9Following Figure2(a), Figures 2(b) and 2(c) show the ratios of data to fit obtained from Eqs. (7) and (9), respectively. One can see that the fits of the data are rather good in the whole pT range, except for a few sizeable departures. The experimental data on the mentioned photons and leptons measured in p+p collisions at 200 GeV by the PHENIX Collaboration [20–23] can also be fitted by Eqs. (7) and (9). From the values of χ2 and the data over fit ratio, one can see that Eq. (9) can describe the data equally well as Eq. (7).Similar to Figure1(a), Figure 3(a) show the invariant cross-sections of various hadrons produced in p+p collisions at 2.76 TeV. Different symbols represent different particles measured by the ALICE Collaboration [24–28] in different η or y ranges. The values of parameters and χ2/ndof are listed in Table 1. The fit of ρ at the quark level is the same with π0. Other particles and corresponding quarks are discussed in Figure 1(a). Similarly, Figures 3(b) and 3(c) show the ratios of data to fit obtained from Eqs. (7) and (9) or (11), respectively. One can see that the fits of the data are rather good in the whole pT range, except for a few sizeable departures. The experimental data on the mentioned hadrons measured in p+p collisions at 2.76 TeV by the ALICE Collaboration [24–28] can be fitted by Eqs. (7) and (9) or (11). From the values of χ2 and the data over fit ratio, one can see that Eq. (9) or (11) can describe the data equally well as Eq. (7).Figure 3
(a) The invariant cross-sections of different hadrons produced inp+p collisions at 2.76 TeV. Different symbols represent different particles in different η or y ranges measured by the ALICE Collaboration [24–28] and scaled by different factors marked in the panel. The dotted and dashed curves are our fitted results by using Eqs. (7) and (9) or (11), respectively. (b) The ratio of data to fit obtained from Eq. (7). (c) The ratio of data to fit obtained from Eq. (9) or (11).Similar to Figure2(a), Figure 4(a) shows the invariant cross-sections of photons and different leptons with given combinations and production channels including inclusive γ, μ from heavy-flavor hadron decays, e from beauty hadron decays, efrom heavy-flavor hadron decays, and inclusive e++e−/2 produced in p+p collisions at 2.76 TeV. Different symbols represent different particles measured by the ALICE Collaboration [29–32] in different y ranges. The values of parameters and χ2/ndof are listed in Table 2. In Eq. (7), m0 is taken to be the rest mass of γ, μ, e, e, and e for the cases from inclusive γ to inclusive e++e−/2 sequenced according to the order shown in Figure 4(a), where e is three times due to the continued three e-related channels. In Eq. (9), m01+m02 are taken to be the constituent masses of u+u, c+c, b+b, c+c, and u+u sequenced according to the same order as particles. Following Figures 4(a)–4(c) show the ratios of data to fit obtained from Eqs. (7) and (9), respectively. One can see that the fits agree with the data in the whole pT range, except for a few departures. The experimental data on the mentioned photons and leptons measured in p+p collisions at 2.76 TeV by the ALICE Collaboration [29–32] can be fitted by Eqs. (8) and (10). From the values of χ2 and the data over fit ratio, one can see that Eq. (9) can describe the data equally well as Eq. (7).Figure 4
(a) The invariant cross-sections of photons and different leptons for a given combination of intermediate channel forp+p collisions at 2.76 TeV. Different symbols represent different particles and their production channels in different y ranges measured by the ALICE Collaboration [29–32] and scaled by different factors marked in the panel. The dotted and dashed curves are our fitted results by using Eqs. (7) and (9), respectively. (b) The ratio of data to fit obtained from Eq. (7). (c) The ratio of data to fit obtained from Eq. (9).Figure 5
(a) The invariant cross-sections of different hadrons produced inp+p collisions at 13 TeV. Different symbols represent different particles in different y ranges measured by the CMS [33] and LHCb [34–36] Collaborations and scaled by different factors marked in the panel. The dotted and dashed curves are our fitted results by using Eqs. (7) and (9) or (11), respectively. (b) The ratio of data to fit obtained from Eq. (7). (c) The ratio of data to fit obtained from Eq. (9) or (11).Similar to Figures1(a) and 3(a), Figure 5(a) shows the invariant cross-sections of different hadrons produced in p+p collisions at 13 TeV. Different symbols represent different particles measured by the CMS [33] or LHCb [34–36] Collaborations in different y ranges. The values of parameters and χ2/ndof are listed in Table 1. Except for the first five groups of particles and corresponding quarks which are discussed in Figure 1(a), m0 in Eq. (7) for other particles is taken to be the rest mass of ψ2s, ψ2s, D0, D+, D∗+, and Ds+ for the cases from ψ2s prompt to Ds++Ds− sequenced according to the order shown in Figure 5(a) from the left to right panels. Meanwhile, m01+m02 in Eq. (9) for other cases are taken to be the constituent masses of c+c, c+c, c+u, c+d, c+d, and c+s sequenced according to the same order as particles. Following Figures 5(a)–5(c) show the ratio of data to fit obtained from Eqs. (7) and (9) or (11), respectively. One can see that the fits are close to the data in the whole pT range, except for a few departures. The experimental data on the mentioned hadrons measured in p+p collisions at 13 TeV by the CMS [33] and LHCb [34–36] Collaborations can be fitted by Eqs. (7) and (9) or (11). From the values of χ2 and the data over fit ratio, one can see that Eq. (9) or (11) can describe the data equally well as Eq. (7).Similar to Figures2(a) and 4(a), Figure 6(a) shows the invariant cross-sections of H→diphotons and heavy flavor dielectrons produced in p+p collisions at 13 TeV. Different symbols represent different particles measured by the ATLAS [37] or ALICE [38] Collaborations in different η ranges. The values of parameters and χ2/ndof are listed in Table 2. In Eq. (7), m0 is taken to be the rest masses of 2γ=0 and 2e sequenced according to the order shown in Figure 6(a). In Eq. (9), both types of particles correspond to the same m01+m02, i.e., the constituent masses of c+c. Following Figures 6(a)–6(c) shows the ratios of data to fit obtained from Eqs. (7) and (9), respectively. One can see that the fits agree with the data in the whole pT range, except for a few departures. The experimental data of diphotons and dielectrons measured in p+p collisions at 13 TeV by the ATLAS [37] and ALICE [38] Collaborations can be fitted by Eqs. (8) and (10). From the values of χ2 and the data over fit ratio, one can see that Eq. (9) can describe the data equally well as Eq. (7).Figure 6
(a) The invariant cross-sections ofH→ diphotons and heavy-flavor dielectrons produced in p+p collisions at 13 TeV. Different symbols represent different particles in different η ranges measured by the ATLAS [37] and ALICE [38] Collaborations and scaled by different factors marked in the panel. The dotted and dashed curves are our fitted results by using Eqs. (7) and (9), respectively. (b) The ratio of data to fit obtained from Eq. (7). (c) The ratio of data to fit obtained from Eq. (9).


### 3.2. Discussion on Parameters

We now analyze the tendencies of the free parameters. The values of effective temperatureT for the emissions of different hadrons do not depend on collision energy. This situation is different for the emissions of photons and leptons, in which there is a clear dependence on energy. This reflects that the emission processes of photons and leptons are more complex than those of hadrons. In the central (pseudo) rapidity region, T shows an incremental tendency with the increase of particle or quark mass. This is understandable that more collision energies are deposited to produce massive hadrons or to drive massive quarks to take part in the process of photon and lepton production. In the forward/backward (pseudo) rapidity region, T is expected to be less than that in the central (pseudo) rapidity region due to less energy deposited.The values of power indexn are very large with small fluctuations in this study. In the Tsallis statistics [6–9, 15–18], n=1/q−1, where q is an entropy index that characterizes the degree of equilibrium or nonequilibrium. Generally, q=1 corresponds to an equilibrium state. A larger q than 1 corresponds to a nonequilibrium state. This study renders that the values of q are very close to 1, which means that the collision system considered by us is approximately in an equilibrium state. The functions based on statistical methods are applicable in this study. In particular, with the increasing collision energy, n decreases and then q increases slightly. This means that the collision system gets further away from the equilibrium state at higher energy.The values of revised indexa0, for the fits in Figures 1(a) and 3(a), listed in Table 1, show that maybe Eq. (7) is not useful because a0≈1. However, the values of a0 listed in Table 3 show that Eq. (7) is indeed necessary because a0≠1. The values of a0 for the fits in Figure 5(a) and listed in Table 1 are larger than 1 for nearly all heavy-flavor particles, while the values of a0 for others are around 1. The values of a0, for the fits in Figures 2(a), 4(a), and 6(a), listed in Tables 3 and 4, are not equal to 1 in most cases. In general, Eq. (7) is necessary in the data-driven analysis because a0≠1 in most cases. In fact, Tables 1–4 show specific a0 and corresponding collision energy, (pseudo) rapidity range, and particle type. Strictly, there are only two cases with a0=1, that is the meson η production in pp collisions with ∣η∣<0.35 at 200 GeV (Table 1) and electron e from beauty decays in pp collisions with ∣y∣<0.8 at 2.76 TeV (Table 3).To see the dependences of the spectra on free parameters, Figure7 presents various pion spectra with different parameters in Eqs. (7) and (9). From the upper panel (Figures 7(a)–7(c)) to the middle panel (Figures 7(d)–7(f)) and then to the lower panel (Figures 7(g)–7(i)), T changes from 0.1 GeV to 0.15 GeV and then to 0.2 GeV. From the left panel to the middle panel and then to the right panel, n changes from 5 to 10 and then to 15. In each panel, the solid, dotted, dashed, and dot-dashed curves without (with) open circles correspond to the spectra with a0=−0.1,0,1,and2, respectively, from Eq. (7) (Eq. (9)). One can see that the probability in the high pT region increases with increasing T, decreases with increasing n, and increases with increasing a0. From negative to positive, a0 determines the shape in the low-pT region.Figure 7
Various pion spectra with different parameters in Eqs. (7) and (9). From the upper panels ((a)–(c)) to the middle panels ((d)–(f)) and then to the lower panels ((g)–(i)), T changes from 0.1 GeV to 0.15 GeV and then to 0.2 GeV. From the left panels ((a), (d), (g)) to the middle panels ((b), (e), (h)) and then to the right panels ((c), (f), (i)), n changes from 5 to 10 and then to 15. In each panel, the solid, dotted, dashed, and dot-dashed curves without (with) open circles are obtained by a0=−0.1,0,1,and2, respectively, from Eq. (7) (Eq. (9)).From the shapes of curves in Figure7, one can see that the parameter a0 introduced in the TP-like function (Eq. (7)) by us determines mainly the trend of the curve in the low-pT region. If the production of light particles via resonance decay affects obviously the shape of the spectrum, one may use a more negative a0 in the fit. If the decay or absorbtion effect of heavy particles in the hot and dense medium in the participant region affects obviously the shape of the spectrum, one may use a more positive a0 in the fit. Due to the introduction of a0, the TP-like function is more flexible than the Tsallis–Pareto-type function. In fact, a0 is a sensitive quantity to describe the influence of the production of light particles via resonance decay and the decay or absorbtion effect of heavy particles in hot and dense medium. Indeed, the introduction of a0 is significant.Before the summary and conclusions, we would like to point out that [9] proposes an alternative form of parametrization for the Tsallis-like function which also well describes the spectra in the low-pT region, which we give as a major improvement of our fit. Indeed, although many theoretical or modelling works are proposed in high energy collisions, more works with different ideas are needed as the ways to systemize the experimental data in the field with fast progress.


## 3.1. Comparison with Data

Figure1(a) shows the pT spectra (the invariant cross-sections), Ed3σ/dp3, of different hadrons with given combinations and decay channels including π++π−/2 plus π0→γγ, K++K−/2 plus KS0→π0π0, η→γγ plus η→π0π+π−; ω→e+e− plus ω→π0π+π− plus ω→π0γ, p+p¯/2, η′→ηπ+π−, ϕ→e+e− plus ϕ→K+K−, J/ψ→e+e−, and ψ′→e+e− produced in p+p collisions at 200 GeV. Different symbols represent different particles and their different decay channels measured by the PHENIX Collaboration [19] in the pseudorapidity range of η<0.35. The results corresponding to π, K, η, ω, p, and η′ are rescaled by multiplying by 106, 105, 104, 103, 102, and 10 factors, respectively. The results corresponding to ϕ, J/ψ, and ψ′ are not rescaled.Figure 1
(a) The invariant cross-sections of different hadrons with given combinations and decay channels produced inp+p collisions at 200 GeV. Different symbols represent different particles and their different decay channels in ∣η∣<0.35 measured by the PHENIX Collaboration [19]; some of them are scaled by different factors marked in the panel. The dotted and dashed curves are our fitted results by using Eqs. (7) and (9) or (11), respectively. (b) The ratio of data to fit obtained from Eq. (7). (c) The ratio of data to fit obtained from Eq. (9) or (11).In Figure1(a), the dotted and dashed curves are our fitted results by using Eqs. (7) (for mesons and baryons) and (9) (for mesons) or (11) (for baryons), respectively. The values of free parameters (T, n, and a0), normalization constant σ0, χ2, and the number of degree of freedom (ndof) obtained from Eq. (7) are listed in Table 1, while the values of parameters and χ2/ndof obtained from Eqs. (9) or (11) are listed in Table 2. In Eq. (7), m0 is taken to be the rest mass of π, K, η, ω, p, η′, ϕ, J/ψ, and ψ′ for the cases from π++π−/2 to ψ′→e+e− sequenced according to the order shown in Figure 1(a). In the fit process at the quark level, the quark structure of π0 results in its fpT to be the half of the sum of uu¯,sfpT and dd¯,sfpT. Because the constituent masses of u and d are the same [54], π0,sfpT is equal to uu¯,sfpT or dd¯,sfpT. The quark structure of η results in its fpT to be cos2ϕ×uu¯,sfpT+sin2ϕ×ss¯sfpT due to the quark structures of ηq and ηs, where ϕ=39.3∘±1.0∘ is the mixing angle [55]. The quark structure of η′ results in its fpT to be sin2ϕ×uu¯,sfpT+cos2ϕ×ss¯,sfpT.Table 1
Values ofT, n, a0, σ0, χ2, and ndof corresponding to the dotted curves in Figures 1(a), 3(a), and 5(a) which are fitted by the TP-like function (Eq. (7)). In the case of ndof being less than 1, it appears as “–” in the table.
FigureyηParticleT (GeV)na0σ0mbχ2/ndofFigure1(a)200 GeVη<0.35π++π−/2π00.129±0.0019.449±0.0200.890±0.00437.044±0.3485/39K++K−/2KS00.167±0.0029.529±0.0301.027±0.0043.122±0.0307/27η0.195±0.0029.889±0.0331.000±0.0031.755±0.0876/32ω0.193±0.0019.460±0.1000.900±0.0203.073±0.03023/34p+p¯/20.149±0.0029.100±0.0201.040±0.0031.291±0.00811/13η′0.210±0.00210.001±0.0230.980±0.0030.584±0.0044/8ϕ0.245±0.00210.559±0.0230.688±0.0030.334±0.00310/15J/ψ0.482±0.00216.778±0.0230.901±0.0045.320±0.132×10−44/22ψ′0.452±0.0028.349±0.0220.959±0.0039.234±0.008×10−51/−Figure3(a)2.76 TeVη<0.8π++π−0.130±0.0016.882±0.0210.937±0.0023.961±0.032×10246/59K++K−0.167±0.0016.985±0.0191.209±0.00347.404±0.64932/54p+p¯0.199±0.0017.870±0.0241.064±0.00323.645±0.12956/45y<0.5ρ07700.205±0.0016.987±0.0211.140±0.00314.167±0.0696/6ϕ0.245±0.0026.696±0.0211.010±0.0041.673±0.0167/17y<0.8η0.195±0.0016.910±0.0241.023±0.0033.340±0.064×10−107/72.5<y<4J/ψ0.482±0.0027.231±0.0251.225±0.0032.181±0.029×10−24/3Figure5(a)13 TeVy<1π++π−/20.129±0.0014.862±0.0210.806±0.0034.608±0.018×10257/18K++K−/20.167±0.0016.179±0.0181.261±0.0024.312±0.039×1013/13p+p¯/20.199±0.0024.768±0.0231.180±0.0042.211±0.013×1018/222<y<2.5J/ψprompt0.482±0.0016.729±0.0221.581±0.0031.097±0.001×10−2115/10J/ψfromb0.482±0.0015.529±0.0241.877±0.0021.850±0.020×10−335/10ψ2sprompt0.578±0.0017.603±0.0211.867±0.0031.509±0.019×10−333/13ψ2sfromb0.578±0.0015.989±0.0231.901±0.0044.907±0.104×10−433/13D0+D¯00.497±0.0016.624±0.0221.244±0.0035.573±0.063×10−15/14D++D−0.497±0.0026.446±0.0221.231±0.0052.805±0.059×10−111/13D∗++D∗−0.497±0.0016.566±0.0231.231±0.0042.622±0.054×10−116/11Ds++Ds−0.497±0.0019.259±0.0212.217±0.0047.876±0.149×0−25/12Table 2
Values ofT, n, a0, σ0, χ2, and ndof corresponding to the dashed curves in Figures 1(a), 3(a), and 5(a) which are fitted by the convolution (Eq. (9) or (11)) of two or three TP-like functions. The quark structures are listed together. In the case of ndof being less than 1, it appears as “–” in the table.
FigureyηParticleQuark structureT (GeV)na0σ0mbχ2/ndofFigure1(a)200 GeVη<0.35π++π−/2ud¯,du¯0.209±0.0027.774±0.025−0.540±0.02038.357±0.3506/39π0uu¯−dd¯/2K++K−/2us¯,su¯0.196±0.0017.816±0.030−0.091±0.0052.913±0.0264/27KS0ds¯η:ηq,ηsuu¯+dd¯/2,ss¯0.212±0.0018.109±0.0300.000±0.0041.838±0.0174/32ωuu¯+dd¯/20.222±0.0018.394±0.0130.000±0.0022.854±0.02620/34p+p¯/2uud,u¯u¯d¯0.162±0.0027.600±0.021−0.130±0.0031.282±0.0113/13η′:ηq,ηsuu¯+dd¯/2,ss¯0.233±0.0028.315±0.0250.000±0.0030.593±0.0043/8ϕss¯0.266±0.0019.022±0.022−0.107±0.0030.319±0.00310/15J/ψcc¯0.509±0.00214.545±0.0250.008±0.0035.275±0.004×10−44/22ψ′cc¯0.503±0.0027.025±0.0300.055±0.0049.232±0.008×10−51/−Figure3(a)2.76 TeVη<0.8π++π−ud¯,du¯0.209±0.0014.970±0.022−0.490±0.0034.474±0.021×10267/59K++K−us¯,su¯0.197±0.0025.223±0.017−0.058±0.00444.823±0.60530/54p+p¯uud,u¯u¯d¯0.194±0.0025.867±0.021−0.120±0.00324.401±0.11331/45y<0.5ρ0770uu¯+dd¯/20.252±0.0015.775±0.021−0.003±0.00314.514±0.0754/6ϕss¯0.266±0.0015.313±0.0190.018±0.0011.679±0.0456/17y<0.8η:ηq,ηsuu¯+dd¯/20.212±0.0025.341±0.022−0.021±0.0033.441±0.021×10−1010/72.5<y<4J/ψcc¯0.509±0.0026.181±0.0180.155±0.0042.179§0.007×1024/3Figure5(a)13 TeVy<1π++π−/2ud¯,du¯0.182±0.0014.401±0.024−0.390±0.0024.579±0.016×10254/18K++K−/2us¯,su¯0.196±0.0017.081±0.023−0.023±0.0024.384×0.016×1011/13p+p¯/2uud,u¯u¯d¯0.191±0.0013.832±0.021−0.090±0.0032.207±0.011×1018/222<y<2.5J/ψpromptcc¯0.509±0.0014.855±0.0220.291±0.0031.056±0.002×10−297/10J/ψfrombcc¯0.509±0.0013.904±0.0230.447±0.0021.812±0.016×10−328/10ψ2spromptcc¯0.675±0.0026.098±0.0220.435±0.0031.490±0.017×10−321/13ψ2sfrombcc¯0.675±0.0024.531±0.0190.431±0.0044.836±0.059×10−420/13D0+D¯0cu¯,c¯u0.545±0.0015.125±0.0170.104±0.0035.420±0.045×10−13/14D++D−cd¯,c¯d0.545±0.0024.929±0.0230.098±0.0042.718×0.043×10−18/13D∗++D∗−cd¯,c¯d0.545±0.0015.025±0.0190.101±0.0022.477±0.034×10−114/11Ds++Ds−cs¯,c¯s0.545±0.0016.515±0.0200.549±0.0037.406±0.099×10−25/12To show departures of the fit from the data, following Figure1(a), Figures 1(b) and 1(c) show the ratios of data to fit obtained from Eqs. (7) and (9) or (11), respectively. One can see that the fits are around the data in the whole pT range, except for a few sizeable departures. The experimental data for the mentioned hadrons measured in p+p collisions at 200 GeV by the PHENIX Collaboration [19] can be fitted by Eqs. (7) (for mesons and baryons) and (9) (for mesons) or (11) (for baryons). From the values of χ2 and the data over fit ratio, one can see that Eq. (9) or (11) can describe the data equally well as Eq. (7).It seems that Eq. (9) or (11) is not necessary due to Eq. (7) being good enough. In fact, the introduction of Eq. (9) or (11) does not contain more parameters compared with Eq. (7). Moreover, Eq. (9) or (11) can tell more about the underlying physics than Eq. (7). The effective temperature used in Eq. (9) or (11) is related to the excitation degree of quark matter, while the effective temperature in Eq. (7) is related to the excitation degree of hadronic matter. In our opinion, Eqs. (9) and (11) are necessary. We shall analyze sequentially the pT spectra of identified particles by using Eqs. (7) and (9) or (11) in the following text.Figure2(a) shows the invariant cross-sections of inclusive direct photons and different leptons with given combinations and production channels including e++e−/2, μ++μ−/2 (open heavy-flavor decays), Drell–Yan →μ+μ−, cc¯→μ+μ−, and bb¯→μ±μ± produced in p+p collisions at 200 GeV. Different symbols represent different particles and their production channels measured by the PHENIX Collaboration [20–23] in different η or y ranges. The dotted and dashed curves are our fitted results by using Eqs. (7) and (9), respectively, where two participant quarks are considered in the formation of the mentioned particles. The values of parameters and χ2/ndof obtained from Eqs. (7) and (9) are listed in Tables 3 and 4, respectively. In Eq. (7), m0 is taken to be the rest mass of γ, e, μ, 2μ, 2μ, and 4μ for the cases from inclusive direct γ to bb¯→μ±μ± sequenced according to the order shown in Figure 2(a), where 2μ is two times due to the continued two 2μ-related channels. In Eq. (9), m01+m02 are taken to be the constituent masses of u+u, u+u, u+c, u+u, c+c, and b+b sequenced according to the same order as particles.Figure 2
(a) The invariant cross-sections of photons and different leptons for a given combination of intermediate channel forp+p collisions at 200 GeV. Different symbols represent different particles, and their production channels in different η ranges measured by the PHENIX Collaboration [20–23]. The dotted and dashed curves are our fitted results by using Eqs. (7) and (9), respectively. (b) The ratio of data to fit obtained from Eq. (7). (c) The ratio of data to fit obtained from Eq. (9).Table 3
Values ofT, n, a0, σ0, χ2, and ndof corresponding to the dotted curves in Figures 2(a), 4(a), and 6(a) which are fitted by the TP-like function (Eq. (7)).
FigureyηParticleT (GeV)na0σ0mbχ2/ndofFigure2(a)200 GeVη<0.35Inclusivedirectγ0.258±0.0019.413±0.0201.750±0.0044.836±0.044×10−32/14e++e−/20.155±0.0028.460±0.0300.652±0.0031.105±0.009×10−28/241.4<y<2.0μ++μ−/2Openheavydecays0.125±0.0019.308±0.0220.799±0.0032.343±0.015×10−27/91.2<y<2.2Drell−Yan→μ+μ−0.349±0.0028.849±0.0232.200±0.0041.559±0.001×10−78/8cc¯→μ+μ−0.385±0.00213.983±0.0231.509±0.0034.227±0.004×10−910/11bb¯→μ±μ±0.445±0.00220.501±0.0502.260±0.0306.917±0.006×10−128/6Figure4(a)2.76 TeVy<0.9Inclusiveγ0.166±0.0016.791±0.0200.068±0.0023.565±0.035×10232/142.5<y<4μFromheavydecays0.345±0.0017.528±0.0210.000±0.0031.480±0.0027/12y<0.8eFrombeautydecays0.315±0.0016.094±0.0161.000±0.0047.686±0.0513/16eFromheavydecays0.165±0.0014.305±0.020−0.043±0.0043.701±0.063×1027/21e++e−/2Inclusive0.155±0.0025.554±0.019−0.05±0.0022.726±0.0577/15Figure6(a)13 TeVη<1.37H→diphoton0.150±0.00114.681±0.02212.257±0.0045.295±0.186×10−1116/9η<0.8Heavy dielectron0.125±0.0018.811±0.0192.281±0.0037.581±0.034×10−16/13Table 4
Values of T, n,a0, σ0, χ2 and ndof corresponding to the dashed curves in Figures 2(a), 4(a), and 6(a) which are fitted by the convolution (Eq. (9)) of two TP-like functions. The participant quarks are listed together.
FigureyηParticleQuarkT (GeV)na0σ0mbχ2/ndofFigure2(a)200 GeVη<0.35Inclusivedirectγuu¯0.383±0.0016.793±0.0240.060±0.0024.967±0.044×10−32/14e++e−/2uu¯0.236±0.0026.408±0.020−0.596±0.0031.192±0.008×10−25/241.4<y<2.0μ++μ−/2Openheavydecaysuc0.167±0.0016.035±0.025−0.802±0.0032.226±0.014×10−24/91.2<y<2.2Drell−Yan→μ+μ−uu¯0.418±0.0025.616±0.0230.398±0.0041.571±0.001×10−78/8cc¯→μ+μ−cc¯0.207±0.0024.072±0.0250.005±0.0044.206±0.004×10−98/11bb¯→μ±μ±bb¯0.207±0.0025.653±0.0240.049±0.0047.047±0.006×10−123/6Figure4(a)2.76 TeVy<0.9Inclusiveγuu¯0.233±0.0015.383±0.019−0.700±0.0033.345±0.020×10227/142.5<y<4μFromheavydecayscc¯0.309±0.0014.554±0.022−0.704±0.0031.364±0.0039/12y<0.8eFrombeautydecaysbb¯0.080±0.0011.993±0.018−0.150±0.0047.323±0.0424/16eFromheavydecayscc¯0.206±0.0022.441±0.017−0.894±0.0023.720±0.043×1025/21e++e−/2Inclusiveuu¯0.166±0.0013.724±0.018−0.700±0.0032.770±0.02112/15Figure6(a)13 TeVη<1.37H→diphotoncc¯0.702±0.0014.573±0.0192.550±0.0035.376±0.209×10−117/13η<0.8Heavy dielectroncc¯0.166±0.0013.128±0.021−0.520±0.0027.581±0.034×10−16/9Following Figure2(a), Figures 2(b) and 2(c) show the ratios of data to fit obtained from Eqs. (7) and (9), respectively. One can see that the fits of the data are rather good in the whole pT range, except for a few sizeable departures. The experimental data on the mentioned photons and leptons measured in p+p collisions at 200 GeV by the PHENIX Collaboration [20–23] can also be fitted by Eqs. (7) and (9). From the values of χ2 and the data over fit ratio, one can see that Eq. (9) can describe the data equally well as Eq. (7).Similar to Figure1(a), Figure 3(a) show the invariant cross-sections of various hadrons produced in p+p collisions at 2.76 TeV. Different symbols represent different particles measured by the ALICE Collaboration [24–28] in different η or y ranges. The values of parameters and χ2/ndof are listed in Table 1. The fit of ρ at the quark level is the same with π0. Other particles and corresponding quarks are discussed in Figure 1(a). Similarly, Figures 3(b) and 3(c) show the ratios of data to fit obtained from Eqs. (7) and (9) or (11), respectively. One can see that the fits of the data are rather good in the whole pT range, except for a few sizeable departures. The experimental data on the mentioned hadrons measured in p+p collisions at 2.76 TeV by the ALICE Collaboration [24–28] can be fitted by Eqs. (7) and (9) or (11). From the values of χ2 and the data over fit ratio, one can see that Eq. (9) or (11) can describe the data equally well as Eq. (7).Figure 3
(a) The invariant cross-sections of different hadrons produced inp+p collisions at 2.76 TeV. Different symbols represent different particles in different η or y ranges measured by the ALICE Collaboration [24–28] and scaled by different factors marked in the panel. The dotted and dashed curves are our fitted results by using Eqs. (7) and (9) or (11), respectively. (b) The ratio of data to fit obtained from Eq. (7). (c) The ratio of data to fit obtained from Eq. (9) or (11).Similar to Figure2(a), Figure 4(a) shows the invariant cross-sections of photons and different leptons with given combinations and production channels including inclusive γ, μ from heavy-flavor hadron decays, e from beauty hadron decays, efrom heavy-flavor hadron decays, and inclusive e++e−/2 produced in p+p collisions at 2.76 TeV. Different symbols represent different particles measured by the ALICE Collaboration [29–32] in different y ranges. The values of parameters and χ2/ndof are listed in Table 2. In Eq. (7), m0 is taken to be the rest mass of γ, μ, e, e, and e for the cases from inclusive γ to inclusive e++e−/2 sequenced according to the order shown in Figure 4(a), where e is three times due to the continued three e-related channels. In Eq. (9), m01+m02 are taken to be the constituent masses of u+u, c+c, b+b, c+c, and u+u sequenced according to the same order as particles. Following Figures 4(a)–4(c) show the ratios of data to fit obtained from Eqs. (7) and (9), respectively. One can see that the fits agree with the data in the whole pT range, except for a few departures. The experimental data on the mentioned photons and leptons measured in p+p collisions at 2.76 TeV by the ALICE Collaboration [29–32] can be fitted by Eqs. (8) and (10). From the values of χ2 and the data over fit ratio, one can see that Eq. (9) can describe the data equally well as Eq. (7).Figure 4
(a) The invariant cross-sections of photons and different leptons for a given combination of intermediate channel forp+p collisions at 2.76 TeV. Different symbols represent different particles and their production channels in different y ranges measured by the ALICE Collaboration [29–32] and scaled by different factors marked in the panel. The dotted and dashed curves are our fitted results by using Eqs. (7) and (9), respectively. (b) The ratio of data to fit obtained from Eq. (7). (c) The ratio of data to fit obtained from Eq. (9).Figure 5
(a) The invariant cross-sections of different hadrons produced inp+p collisions at 13 TeV. Different symbols represent different particles in different y ranges measured by the CMS [33] and LHCb [34–36] Collaborations and scaled by different factors marked in the panel. The dotted and dashed curves are our fitted results by using Eqs. (7) and (9) or (11), respectively. (b) The ratio of data to fit obtained from Eq. (7). (c) The ratio of data to fit obtained from Eq. (9) or (11).Similar to Figures1(a) and 3(a), Figure 5(a) shows the invariant cross-sections of different hadrons produced in p+p collisions at 13 TeV. Different symbols represent different particles measured by the CMS [33] or LHCb [34–36] Collaborations in different y ranges. The values of parameters and χ2/ndof are listed in Table 1. Except for the first five groups of particles and corresponding quarks which are discussed in Figure 1(a), m0 in Eq. (7) for other particles is taken to be the rest mass of ψ2s, ψ2s, D0, D+, D∗+, and Ds+ for the cases from ψ2s prompt to Ds++Ds− sequenced according to the order shown in Figure 5(a) from the left to right panels. Meanwhile, m01+m02 in Eq. (9) for other cases are taken to be the constituent masses of c+c, c+c, c+u, c+d, c+d, and c+s sequenced according to the same order as particles. Following Figures 5(a)–5(c) show the ratio of data to fit obtained from Eqs. (7) and (9) or (11), respectively. One can see that the fits are close to the data in the whole pT range, except for a few departures. The experimental data on the mentioned hadrons measured in p+p collisions at 13 TeV by the CMS [33] and LHCb [34–36] Collaborations can be fitted by Eqs. (7) and (9) or (11). From the values of χ2 and the data over fit ratio, one can see that Eq. (9) or (11) can describe the data equally well as Eq. (7).Similar to Figures2(a) and 4(a), Figure 6(a) shows the invariant cross-sections of H→diphotons and heavy flavor dielectrons produced in p+p collisions at 13 TeV. Different symbols represent different particles measured by the ATLAS [37] or ALICE [38] Collaborations in different η ranges. The values of parameters and χ2/ndof are listed in Table 2. In Eq. (7), m0 is taken to be the rest masses of 2γ=0 and 2e sequenced according to the order shown in Figure 6(a). In Eq. (9), both types of particles correspond to the same m01+m02, i.e., the constituent masses of c+c. Following Figures 6(a)–6(c) shows the ratios of data to fit obtained from Eqs. (7) and (9), respectively. One can see that the fits agree with the data in the whole pT range, except for a few departures. The experimental data of diphotons and dielectrons measured in p+p collisions at 13 TeV by the ATLAS [37] and ALICE [38] Collaborations can be fitted by Eqs. (8) and (10). From the values of χ2 and the data over fit ratio, one can see that Eq. (9) can describe the data equally well as Eq. (7).Figure 6
(a) The invariant cross-sections ofH→ diphotons and heavy-flavor dielectrons produced in p+p collisions at 13 TeV. Different symbols represent different particles in different η ranges measured by the ATLAS [37] and ALICE [38] Collaborations and scaled by different factors marked in the panel. The dotted and dashed curves are our fitted results by using Eqs. (7) and (9), respectively. (b) The ratio of data to fit obtained from Eq. (7). (c) The ratio of data to fit obtained from Eq. (9).


## 3.2. Discussion on Parameters

We now analyze the tendencies of the free parameters. The values of effective temperatureT for the emissions of different hadrons do not depend on collision energy. This situation is different for the emissions of photons and leptons, in which there is a clear dependence on energy. This reflects that the emission processes of photons and leptons are more complex than those of hadrons. In the central (pseudo) rapidity region, T shows an incremental tendency with the increase of particle or quark mass. This is understandable that more collision energies are deposited to produce massive hadrons or to drive massive quarks to take part in the process of photon and lepton production. In the forward/backward (pseudo) rapidity region, T is expected to be less than that in the central (pseudo) rapidity region due to less energy deposited.The values of power indexn are very large with small fluctuations in this study. In the Tsallis statistics [6–9, 15–18], n=1/q−1, where q is an entropy index that characterizes the degree of equilibrium or nonequilibrium. Generally, q=1 corresponds to an equilibrium state. A larger q than 1 corresponds to a nonequilibrium state. This study renders that the values of q are very close to 1, which means that the collision system considered by us is approximately in an equilibrium state. The functions based on statistical methods are applicable in this study. In particular, with the increasing collision energy, n decreases and then q increases slightly. This means that the collision system gets further away from the equilibrium state at higher energy.The values of revised indexa0, for the fits in Figures 1(a) and 3(a), listed in Table 1, show that maybe Eq. (7) is not useful because a0≈1. However, the values of a0 listed in Table 3 show that Eq. (7) is indeed necessary because a0≠1. The values of a0 for the fits in Figure 5(a) and listed in Table 1 are larger than 1 for nearly all heavy-flavor particles, while the values of a0 for others are around 1. The values of a0, for the fits in Figures 2(a), 4(a), and 6(a), listed in Tables 3 and 4, are not equal to 1 in most cases. In general, Eq. (7) is necessary in the data-driven analysis because a0≠1 in most cases. In fact, Tables 1–4 show specific a0 and corresponding collision energy, (pseudo) rapidity range, and particle type. Strictly, there are only two cases with a0=1, that is the meson η production in pp collisions with ∣η∣<0.35 at 200 GeV (Table 1) and electron e from beauty decays in pp collisions with ∣y∣<0.8 at 2.76 TeV (Table 3).To see the dependences of the spectra on free parameters, Figure7 presents various pion spectra with different parameters in Eqs. (7) and (9). From the upper panel (Figures 7(a)–7(c)) to the middle panel (Figures 7(d)–7(f)) and then to the lower panel (Figures 7(g)–7(i)), T changes from 0.1 GeV to 0.15 GeV and then to 0.2 GeV. From the left panel to the middle panel and then to the right panel, n changes from 5 to 10 and then to 15. In each panel, the solid, dotted, dashed, and dot-dashed curves without (with) open circles correspond to the spectra with a0=−0.1,0,1,and2, respectively, from Eq. (7) (Eq. (9)). One can see that the probability in the high pT region increases with increasing T, decreases with increasing n, and increases with increasing a0. From negative to positive, a0 determines the shape in the low-pT region.Figure 7
Various pion spectra with different parameters in Eqs. (7) and (9). From the upper panels ((a)–(c)) to the middle panels ((d)–(f)) and then to the lower panels ((g)–(i)), T changes from 0.1 GeV to 0.15 GeV and then to 0.2 GeV. From the left panels ((a), (d), (g)) to the middle panels ((b), (e), (h)) and then to the right panels ((c), (f), (i)), n changes from 5 to 10 and then to 15. In each panel, the solid, dotted, dashed, and dot-dashed curves without (with) open circles are obtained by a0=−0.1,0,1,and2, respectively, from Eq. (7) (Eq. (9)).From the shapes of curves in Figure7, one can see that the parameter a0 introduced in the TP-like function (Eq. (7)) by us determines mainly the trend of the curve in the low-pT region. If the production of light particles via resonance decay affects obviously the shape of the spectrum, one may use a more negative a0 in the fit. If the decay or absorbtion effect of heavy particles in the hot and dense medium in the participant region affects obviously the shape of the spectrum, one may use a more positive a0 in the fit. Due to the introduction of a0, the TP-like function is more flexible than the Tsallis–Pareto-type function. In fact, a0 is a sensitive quantity to describe the influence of the production of light particles via resonance decay and the decay or absorbtion effect of heavy particles in hot and dense medium. Indeed, the introduction of a0 is significant.Before the summary and conclusions, we would like to point out that [9] proposes an alternative form of parametrization for the Tsallis-like function which also well describes the spectra in the low-pT region, which we give as a major improvement of our fit. Indeed, although many theoretical or modelling works are proposed in high energy collisions, more works with different ideas are needed as the ways to systemize the experimental data in the field with fast progress.


## 4. Summary and Conclusions

We summarize here our main observations and conclusions.(1)
The transverse momentum spectra in terms of the (invariant) cross-section of various particles (different hadrons with given combinations and decay channels, photons, and different leptons with given combinations and production channels) produced in high energy proton-proton collisions have been studied by a TP-like function (a revised Tsallis–Pareto-type function). Meanwhile, the transverse momentum spectra have also been studied by a new description in the framework of the participant quark model or the multisource model at the quark level. In the model, the source itself is exactly the participant quark. Each participant quark contributes to the transverse momentum spectrum to be the TP-like function(2)
For a hadron, the participant quarks are in fact constituent quarks. The transverse momentum spectrum of the hadron is the convolution of two or more TP-like functions. For a photon or lepton, the transverse momentum spectrum is the convolution of two TP-like functions due to two participant quarks, e.g., projectile and target quarks, taking part in the collisions. The TP-like function and the convolution of a few TP-like functions can fit the experimental data of various particles produced in proton-proton collisions at 200 GeV, 2.76 TeV, and 13 TeV measured by the PHENIX, ALICE, CMS, LHCb, and ATLAS Collaborations(3)
The values of effective temperature for the emissions of different hadrons do not depend on the collision energy, while for the emissions of photons and leptons, there is an obvious dependence on collision energy. This reflects the fact that the emission processes of photons and leptons are more complex than those of hadrons. In the central (pseudo) rapidity region, the effective temperature shows an increasing tendency with the increase of particle or quark mass. This reflects the fact that more collision energy is deposited to produce massive hadrons or to drive massive quarks to take part in the process of photon and lepton production(4)
The values of the power index are very large, which means that the values of the entropy index are very close to 1. The collision system considered in this study is approximately in an equilibrium state. The functions based on statistical methods are applicable in this study. In particular, with the increase of collision energy, the power index decreases and then the entropy index increases slightly. This means that the collision system gets further away from the equilibrium state at higher energy, though the entropy index is still close to 1 at the LHC(5)
The values of the revised index show that the TP-like function is indeed necessary due to the fact that this index is not equal to 1. In the TP-like function and its convolution, the effective temperature, power index, and revised index are sensitive to the spectra. In various pion spectra from the TP-like function and its convolution of two, the probability in the high transverse momentum region increases with the increase of effective temperature, decreases with the increase of the power index, and increases with the increase of the revised index. From negative to positive, the revised index determines the shape in the low transverse momentum region, which is sensitive to the contribution of resonance decays



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