# Initial- and Final-State Temperatures of Emission Source from Differential Cross-Section in Squared Momentum Transfer in High-Energy Collisions

Authors: Qi Wang; Fu-Hu Liu; Khusniddin K. Olimov
Journal: Advances in High Energy Physics
 (2021)

---

## Abstract

The differential cross-section in squared momentum transfer ofρ, ρ0, ω, ϕ, f0980, f11285, f01370, f11420, f01500, and J/ψ produced in high-energy virtual photon-proton (γ∗p), photon-proton (γp), and proton-proton (pp) collisions measured by the H1, ZEUS, and WA102 Collaborations is analyzed by the Monte Carlo calculations. In the calculations, the Erlang distribution, Tsallis distribution, and Hagedorn function are separately used to describe the transverse momentum spectra of the emitted particles. Our results show that the initial- and final-state temperatures increase from lower squared photon virtuality to a higher one and decrease with the increase of center-of-mass energy.


---

## Body

## 1. Introduction

In high-energy collisions, it is interesting for us to describe the excitation and equilibrium degrees of an interacting system because of the two degrees related to the reaction mechanism and evolution process of the collision system [1–10]. In the progress of describing the excitation degree and structure character of the system, temperature is an important quantity in physics in view of intuitiveness and representation. In high-energy collisions, different types of temperature are used [11–18], which usually refer to the initial temperature Ti, quark-hadron transition temperature Ttr, chemical freeze-out temperature Tch, kinetic freeze-out or final-state temperature (“confinement” temperature) Tkin or T0, and effective temperature Teff or T, etc. In this work, we emphatically discuss the initial- and final-state temperatures, though other types of temperatures are also important.The initial temperatureTi is the temperature of emission source or interacting system when a projectile particle or nucleus and a target particle or nucleus undergo the initial stage of a collision. It represents the excitation degree of the emission source or that of an interacting system in the initial state of collisions, and it is usually meant as describing the interacting system after thermalization. The initial temperature Ti can be extracted by fitting the transverse momentum pT spectra of particles by using some distributions such as the Erlang distribution [19–21], Tsallis distribution [22, 23], Hagedorn function [24], and Lévy–Tsallis function [25]. Here, both the names of distribution and the function are used according to the various accepted terminologies in the literature, though they represent the similar probability density function in fitting the particle spectra. Meanwhile, the average transverse momentum pT can be obtained from the same function.The final-state temperatureT0 is usually known as the kinetic freeze-out temperature, which refers to the temperature of emission source when the inelastic collisions ceased and there are only elastic collisions among particles. In the last stage of collisions, the momentum distribution of particles is fixed and the transverse momentum spectra can be measured in experiments. The excitation degree of the system in the last stage can be described by the final-state temperature T0 in which the influence of flow effect is excluded. The temperature or related main parameters used in the Erlang distribution [19–21], Tsallis distribution [22, 23], Hagedorn function [24], and Lévy–Tsallis function [25] are not T0, but the effective temperature T in which the influence of flow effect is not excluded.The Mandelstam variables [26] consist of the four-momentum of particles in a two-body reaction. Both the squared momentum transfer and the transverse momentum can represent the kinetic character of particles. Let us use the squared momentum transfer to replace the transverse momentum in fitting the particle spectra. Then, we can fit the squared momentum transfer spectra with the related distributions to obtain the initial temperature Ti, average transverse momentum pT, and other quantities. Of course, in fitting the squared momentum transfer spectra, the above-mentioned distributions cannot be used directly. In fact, we have to use the Monte Carlo method to obtain the concrete value of a transverse momentum for a given particle from the mentioned distributions. Then, the concrete value of squared momentum transfer can be obtained from the definition.Except for the temperature parameter, other parameters also describe partly the characters of the interacting system. For instance, the entropy indexq which describes the degree of equilibrium can be extracted from the Tsallis distribution [22, 23] considering the particle mass. Meanwhile, q can be extracted from the Hagedorn function [24] which is the same as the Lévy–Tsallis function [25] for a particle neglecting its mass. If there is relation between the Tsallis distribution and the Hagedorn function, we may say that the former one covers the latter one in which the mass is neglected. Because the universality, similarity, or common characteristics exist in high-energy collisions [27–36], some distributions used in the large collision system can be also used in a small collision system.In this paper, the differential cross-section in the squared momentum transfer ofρ, ρ0, ϕ, and J/ψ produced in virtual photon-proton (γ∗p) collisions and ω and J/ψ produced in photon-proton (γp) collisions, as well as f0980, f11285, f01370, f11420, and f01500 produced in proton-proton (pp) collisions measured by the H1 [37, 38], ZEUS [39–42], and WA102 Collaborations [43, 44] is fitted with the results from the Monte Carlo calculations. Firstly, the transverse momenta satisfied with the Erlang distribution, Tsallis distribution, and Hagedorn function are generated. Secondly, these special transverse momenta are transformed to the squared momentum transfers. Thirdly and lastly, the distribution of squared momentum transfers is obtained and fitted to the experimental data by the least squares method.


## 2. Formalism and Method

### 2.1. The Erlang Distribution

The Erlang distribution is the convolution of multiple exponential distributions. In the framework of a multisource thermal model [19–21], we may think that more than one parton (or parton-like) contribute to the transverse momentum of the considered particle. The j-th parton (or parton-like) is assumed to contribute to the transverse momentum to be ptj which obeys an exponential distribution with the average pt which is j-ordinal number independent. We have the probability density function obeyed by ptj to be(1)fptj=1ptexp−ptjpt.The averagept reflects the excitation degree of contributor parton and can be regarded as the effective temperature T.The contribution of allns partons to pT is the sum of various ptj. The distribution of pT is then the convolution of ns exponential functions [19–21]. We have the pT distribution (the probability density function of pT) of final-state particles to be the Erlang distribution(2)f1pT=1NdNdpT=pTns−1ns−1!ptnsexp−pTpt,where N denotes the number of all considered particles and pT has an average of pT=∫0∞pTf1pTdpT=nspt. Equation (2) is naturally normalized to be 1. In Equation (2), there are two free parameters, ns and pt.


### 2.2. The Tsallis Distribution

The Tsallis distribution [22, 23] has more than one form, which are widely used in the field of high-energy collisions. Conveniently, we use the following form(3)f2pT=1NdNdpT=CpT1+mT−m0nT−n,where C is the normalization constant, mT=pT2+m02 is the transverse mass, m0 is the rest mass, n=1/q−1, and q is the entropy index [22, 23]. Equation (3) is valid only at midrapidity (y≈0) which results in coshy≈1 and the particle energy E=mTcoshy≈mT.In Equation (3), a large n corresponds to a q that is close to 1, and the source or system approaches to equilibrium. The larger the parameter n is, the closer to 1 the entropy index q is, with the source or system being at a higher degree of equilibrium. There is no exact minimum n or maximum q [22–25] which is a limit for approximate equilibrium. Empirically, in the case of n≥4 or q≤1.25 which is 25% more than 1 (even n≥5 or q≤1.2 which is 20% more than 1), the source or system can be regarded as being in a state of approximate (local) equilibrium. Usually, in high-energy collisions, the source or system is approximately in equilibrium due to n being large enough.


### 2.3. The Hagedorn Function

The Hagedorn function [24] is an inverse power law which has the probability density function of pT to be(4)f3pT=1NdNdpT=ApT1+pTp0−n0,where A is the normalization constant, n0 is a free parameter which is similar to n in the Tsallis distribution [22, 23], and p0 is a free parameter which is similar to the product of nT in the Tsallis distribution. Note here that it appears as if p0=nT is a perfect liquid-like relation; however, p0 is a transverse momentum and n is a dimensionless number. This is not meant in a perfect liquid sense, but the letters are just randomly coinciding.It should be noted that the Hagedorn function is a special case of the Tsallis distribution in whichm0 can be neglected. Generally, at high pT, we may neglect m0, observing the two distributions being very similar to each other. At low pT, the two distributions have obvious differences due to nonignorable m0. To build a connection with the entropy index q, we have n0≈1/q−1. To build a connection with the effective temperature T, we have p0≈n0T≈T/q−1.


### 2.4. The Squared Momentum Transfer

In the center-of-mass reference frame, in a two-body reaction1+2⟶3+4 or in a two-body-like reaction, it is supposed that particle 1 is incident along the z direction and particle 2 is incident along the opposite direction. In addition, particle 3 is emitted with angle θ relative to the z direction and particle 4 is emitted along the opposite direction. According to Ref. [26], three Mandelstam variables are defined as(5)s=−P1+P22=−P3+P42,(6)t=−P1−P32=−−P2+P42,(7)u=−P1−P42=−−P2+P32,where P1, P2, P3, and P4 are four-momenta of particles 1, 2, 3, and 4, respectively.In the Mandelstam variables, slightly varying the form,s is the center-of-mass energy, −t is the squared momentum transfer between particles 1 and 3, and −u is the squared momentum transfer between particles 1 and 4. Conveniently, let ∣t∣ be the squared momentum transfer between particles 1 and 3. We have(8)∣t∣=∣E1−E32−p⟶1−p⟶32∣=m12+m32−2E1p3Tsinθ2+m32+2E12−m12p3Ttanθ.Here,E1 and E3, p⟶1 and p⟶3, and m1 and m3 are the energy, momentum, and rest mass of particles 1 and 3, respectively. In particular, p3T is the transverse momentum of particle 3, which is referred to be perpendicular to the z direction.As the energy of incoming photon in the center-of-mass reference frame of the reaction,E1 in Equation (8) should be a fixed value. However, E1 has a slight shift from the peak value due to different experiments and selections. To obtain a good fit, we treat E1 as a parameter which is the same or has small difference in the same/similar reactions. p3T obeys one of Equations (2)–(4) and θ obeys an isotropic assumption in the center-of-mass reference frame, which will be discussed later in this section. To obtain ∣t∣, we may perform the Monte Carlo calculations. Note that we may calculate ∣t∣ from two particles, i.e., particles 1 and 3, but not from one particle. Instead, for one calculation, ∣t∣ means the squared momentum transfer in an event. For many calculations, ∣t∣ distribution can be obtained from the statistics. For convenience in the description, the transverse momentum and rest mass of particle 3 are also denoted by pT and m0, respectively.Based on the experiments cited from literature [37–44], we have used two main selection factors for the data. (1) The squared photon virtuality Q2=−Pγ2, where Pγ denotes the four-momentum of the photon. (2) The center-of-mass energy s or W, i.e., W=s=−P1+P22. Let x denote the Bjorken scaling variable; one has W2≃Q2/x.


### 2.5. The Initial- and Final-State Temperatures

According to Refs. [45–47], in a color string percolation approach, the initial temperature Ti can be estimated as(9)Ti=pT22,where pT2 is the root mean square of pT and pT2=∫0maxpT2f1,2,3pTdpT. In the expression of the initial temperature, we have used a single string in the cluster for a given particle production [48], though more than two partons or partons-like take possible part in the formation of the string. That is, we have used the color suppression factor Fξ to be 1 in the color string percolation model [48]. Other strings, even if they exist, do not affect noticeably the production of a given particle. If other strings are considered, i.e., if we take the minimum Fξ to be 0.6 [48], a higher Ti can be obtained by multiplying a revised factor, 1/Fξ=1.291, in Equation (9).The extraction of final-state temperatureT0 is more complex than that of the initial temperature Ti. Generally, one may introduce the transverse flow velocity βT in the considered function and obtain T0 and βT simultaneously [49–57], in which the effective temperature T no longer appears. Alternatively, the intercept in T versus m0 is assumed to be T0 [50, 58–63], and the slope in pT versus m¯ is assumed to be βT [62–66], where m¯ denotes the average energy. However, the alternative method using intercept and slope is not suitable for us due to the fact that the spectra of more than two types of particles (e.g., pions, kaons, and protons) are needed in the extraction which is not our case.In theγ∗p, γp, and pp collisions discussed in the present work, the flow effect is not considered by us due to the collective effect being small in the two-body process. This means that T0≈T in the considered processes. Here, T appears as that in Equation (3). Meanwhile, T can be also approximated by pt in Equation (1) and p0/n0 in Equation (4). Generally, we may regard different distributions or functions as different “thermometers.” Just like the Celsius thermometer and the Fahrenheit thermometer, different thermometers measure different temperatures, though they can be transformed from one to another according to conversion rules. Although we may approximately regard T in Equation (3) as T0, a smaller T0 can be obtained if the flow effect is considered.As mentioned above,T=pt=pT/ns in Equation (1) and the Erlang distribution, and Ti=pT2/2. We have T2ns2=2Ti2−σpT2, so this would mean that T is basically encoded in σpT2, the squared variance of pT in the distribution. This also means that Ti and T are related through pT. It is understandable, because they reflect the violent degrees of collisions at different stages. Generally, Ti>T; this is natural.Note that although we may use the final-state temperature, it is not a freeze-out temperature for the small system discussed in this paper. In particular, forγ∗p and γp reactions, these are just a process describable in terms of perturbative quantum chromodynamics (pQCD) and factorization [67], but not a process in which deconfinement or freeze-out is involved. The meaning of the final-state temperature for the large system such as heavy-ion collisions or the small system such as pp collisions with high multiplicity is somehow different from here. At least, for the large system, we may consider the deconfinement- or freeze-out-involved picture. Meanwhile, the flow effect in the large system cannot be neglected.


### 2.6. The Process of Monte Carlo Calculations

In an analytical calculation, the function Equations (2)–(4) on pT distribution are difficult to use in Equation (8) to obtain the ∣t∣ distribution. Instead, we may perform the Monte Carlo calculations. Let R1,2 and r1,2,3,⋯,ns be random numbers distributed evenly in [0,1]. To use Equation (8), we have to know the changeable p3T (i.e., pT) and θ. Other quantities such as E1, m1, and m3 in the equation are fixed, though E1 is treated by us as a parameter with slight variety.To obtain a concrete value ofpT, we need one of Equations (2)–(4). Solving the equation(10)∫0pTfip′Tdp′T<R1<∫0pT+δpTfip′Tdp′T,where i=1, 2, and 3 and δpT is a small shift relative to pT; we may obtain concrete pT. It seems that Equation (10) directly means that the integral of f1pT, f2pT, and f3pT is the same for the 0,pT interval, which essentially means that the three functions are equal (except for a null measure set). In fact, the three functions are different in forms because of Equations (2)–(4), and we need to distinguish them.In particular, forf1pT, we have a simpler expression. Let us solve the equation(11)∫0ptjfp′tjdp′tj=rjj=1,2,3,⋯,ns.We have(12)ptj=−ptlnrjj=1,2,3,⋯,ns,due to Equation (1) being used, where rj in Equation (12) replaced 1−rj because both of them are random numbers in [0,1]. The simpler expression is(13)pT=−pt∏j=1nslnrj,due to pT being the sum of ns random ptj.To obtain a concrete value ofθ, we need the function(14)fθθ=12sinθ,which is obeyed by θ under the assumption of isotropic emission in the center-of-mass reference frame. Solving the equation(15)∫0θfθθ′dθ′=R2,we have(16)θ=2arcsinR2,which is needed by us.According to the concrete values ofpT and θ, and using other quantities, the value of ∣t∣ can be obtained from Equation (8). After repeating the calculations many times, the distribution of ∣t∣ is obtained statistically. Based on the method of least squares, the related parameters are obtained naturally. Meanwhile, Ti can be obtained from Equation (9). pT and pT2 can be obtained from one of Equations (2)–(4) or from the statistics. The errors of parameters are obtained by the general method of statistical simulation.


## 2.1. The Erlang Distribution

The Erlang distribution is the convolution of multiple exponential distributions. In the framework of a multisource thermal model [19–21], we may think that more than one parton (or parton-like) contribute to the transverse momentum of the considered particle. The j-th parton (or parton-like) is assumed to contribute to the transverse momentum to be ptj which obeys an exponential distribution with the average pt which is j-ordinal number independent. We have the probability density function obeyed by ptj to be(1)fptj=1ptexp−ptjpt.The averagept reflects the excitation degree of contributor parton and can be regarded as the effective temperature T.The contribution of allns partons to pT is the sum of various ptj. The distribution of pT is then the convolution of ns exponential functions [19–21]. We have the pT distribution (the probability density function of pT) of final-state particles to be the Erlang distribution(2)f1pT=1NdNdpT=pTns−1ns−1!ptnsexp−pTpt,where N denotes the number of all considered particles and pT has an average of pT=∫0∞pTf1pTdpT=nspt. Equation (2) is naturally normalized to be 1. In Equation (2), there are two free parameters, ns and pt.


## 2.2. The Tsallis Distribution

The Tsallis distribution [22, 23] has more than one form, which are widely used in the field of high-energy collisions. Conveniently, we use the following form(3)f2pT=1NdNdpT=CpT1+mT−m0nT−n,where C is the normalization constant, mT=pT2+m02 is the transverse mass, m0 is the rest mass, n=1/q−1, and q is the entropy index [22, 23]. Equation (3) is valid only at midrapidity (y≈0) which results in coshy≈1 and the particle energy E=mTcoshy≈mT.In Equation (3), a large n corresponds to a q that is close to 1, and the source or system approaches to equilibrium. The larger the parameter n is, the closer to 1 the entropy index q is, with the source or system being at a higher degree of equilibrium. There is no exact minimum n or maximum q [22–25] which is a limit for approximate equilibrium. Empirically, in the case of n≥4 or q≤1.25 which is 25% more than 1 (even n≥5 or q≤1.2 which is 20% more than 1), the source or system can be regarded as being in a state of approximate (local) equilibrium. Usually, in high-energy collisions, the source or system is approximately in equilibrium due to n being large enough.


## 2.3. The Hagedorn Function

The Hagedorn function [24] is an inverse power law which has the probability density function of pT to be(4)f3pT=1NdNdpT=ApT1+pTp0−n0,where A is the normalization constant, n0 is a free parameter which is similar to n in the Tsallis distribution [22, 23], and p0 is a free parameter which is similar to the product of nT in the Tsallis distribution. Note here that it appears as if p0=nT is a perfect liquid-like relation; however, p0 is a transverse momentum and n is a dimensionless number. This is not meant in a perfect liquid sense, but the letters are just randomly coinciding.It should be noted that the Hagedorn function is a special case of the Tsallis distribution in whichm0 can be neglected. Generally, at high pT, we may neglect m0, observing the two distributions being very similar to each other. At low pT, the two distributions have obvious differences due to nonignorable m0. To build a connection with the entropy index q, we have n0≈1/q−1. To build a connection with the effective temperature T, we have p0≈n0T≈T/q−1.


## 2.4. The Squared Momentum Transfer

In the center-of-mass reference frame, in a two-body reaction1+2⟶3+4 or in a two-body-like reaction, it is supposed that particle 1 is incident along the z direction and particle 2 is incident along the opposite direction. In addition, particle 3 is emitted with angle θ relative to the z direction and particle 4 is emitted along the opposite direction. According to Ref. [26], three Mandelstam variables are defined as(5)s=−P1+P22=−P3+P42,(6)t=−P1−P32=−−P2+P42,(7)u=−P1−P42=−−P2+P32,where P1, P2, P3, and P4 are four-momenta of particles 1, 2, 3, and 4, respectively.In the Mandelstam variables, slightly varying the form,s is the center-of-mass energy, −t is the squared momentum transfer between particles 1 and 3, and −u is the squared momentum transfer between particles 1 and 4. Conveniently, let ∣t∣ be the squared momentum transfer between particles 1 and 3. We have(8)∣t∣=∣E1−E32−p⟶1−p⟶32∣=m12+m32−2E1p3Tsinθ2+m32+2E12−m12p3Ttanθ.Here,E1 and E3, p⟶1 and p⟶3, and m1 and m3 are the energy, momentum, and rest mass of particles 1 and 3, respectively. In particular, p3T is the transverse momentum of particle 3, which is referred to be perpendicular to the z direction.As the energy of incoming photon in the center-of-mass reference frame of the reaction,E1 in Equation (8) should be a fixed value. However, E1 has a slight shift from the peak value due to different experiments and selections. To obtain a good fit, we treat E1 as a parameter which is the same or has small difference in the same/similar reactions. p3T obeys one of Equations (2)–(4) and θ obeys an isotropic assumption in the center-of-mass reference frame, which will be discussed later in this section. To obtain ∣t∣, we may perform the Monte Carlo calculations. Note that we may calculate ∣t∣ from two particles, i.e., particles 1 and 3, but not from one particle. Instead, for one calculation, ∣t∣ means the squared momentum transfer in an event. For many calculations, ∣t∣ distribution can be obtained from the statistics. For convenience in the description, the transverse momentum and rest mass of particle 3 are also denoted by pT and m0, respectively.Based on the experiments cited from literature [37–44], we have used two main selection factors for the data. (1) The squared photon virtuality Q2=−Pγ2, where Pγ denotes the four-momentum of the photon. (2) The center-of-mass energy s or W, i.e., W=s=−P1+P22. Let x denote the Bjorken scaling variable; one has W2≃Q2/x.


## 2.5. The Initial- and Final-State Temperatures

According to Refs. [45–47], in a color string percolation approach, the initial temperature Ti can be estimated as(9)Ti=pT22,where pT2 is the root mean square of pT and pT2=∫0maxpT2f1,2,3pTdpT. In the expression of the initial temperature, we have used a single string in the cluster for a given particle production [48], though more than two partons or partons-like take possible part in the formation of the string. That is, we have used the color suppression factor Fξ to be 1 in the color string percolation model [48]. Other strings, even if they exist, do not affect noticeably the production of a given particle. If other strings are considered, i.e., if we take the minimum Fξ to be 0.6 [48], a higher Ti can be obtained by multiplying a revised factor, 1/Fξ=1.291, in Equation (9).The extraction of final-state temperatureT0 is more complex than that of the initial temperature Ti. Generally, one may introduce the transverse flow velocity βT in the considered function and obtain T0 and βT simultaneously [49–57], in which the effective temperature T no longer appears. Alternatively, the intercept in T versus m0 is assumed to be T0 [50, 58–63], and the slope in pT versus m¯ is assumed to be βT [62–66], where m¯ denotes the average energy. However, the alternative method using intercept and slope is not suitable for us due to the fact that the spectra of more than two types of particles (e.g., pions, kaons, and protons) are needed in the extraction which is not our case.In theγ∗p, γp, and pp collisions discussed in the present work, the flow effect is not considered by us due to the collective effect being small in the two-body process. This means that T0≈T in the considered processes. Here, T appears as that in Equation (3). Meanwhile, T can be also approximated by pt in Equation (1) and p0/n0 in Equation (4). Generally, we may regard different distributions or functions as different “thermometers.” Just like the Celsius thermometer and the Fahrenheit thermometer, different thermometers measure different temperatures, though they can be transformed from one to another according to conversion rules. Although we may approximately regard T in Equation (3) as T0, a smaller T0 can be obtained if the flow effect is considered.As mentioned above,T=pt=pT/ns in Equation (1) and the Erlang distribution, and Ti=pT2/2. We have T2ns2=2Ti2−σpT2, so this would mean that T is basically encoded in σpT2, the squared variance of pT in the distribution. This also means that Ti and T are related through pT. It is understandable, because they reflect the violent degrees of collisions at different stages. Generally, Ti>T; this is natural.Note that although we may use the final-state temperature, it is not a freeze-out temperature for the small system discussed in this paper. In particular, forγ∗p and γp reactions, these are just a process describable in terms of perturbative quantum chromodynamics (pQCD) and factorization [67], but not a process in which deconfinement or freeze-out is involved. The meaning of the final-state temperature for the large system such as heavy-ion collisions or the small system such as pp collisions with high multiplicity is somehow different from here. At least, for the large system, we may consider the deconfinement- or freeze-out-involved picture. Meanwhile, the flow effect in the large system cannot be neglected.


## 2.6. The Process of Monte Carlo Calculations

In an analytical calculation, the function Equations (2)–(4) on pT distribution are difficult to use in Equation (8) to obtain the ∣t∣ distribution. Instead, we may perform the Monte Carlo calculations. Let R1,2 and r1,2,3,⋯,ns be random numbers distributed evenly in [0,1]. To use Equation (8), we have to know the changeable p3T (i.e., pT) and θ. Other quantities such as E1, m1, and m3 in the equation are fixed, though E1 is treated by us as a parameter with slight variety.To obtain a concrete value ofpT, we need one of Equations (2)–(4). Solving the equation(10)∫0pTfip′Tdp′T<R1<∫0pT+δpTfip′Tdp′T,where i=1, 2, and 3 and δpT is a small shift relative to pT; we may obtain concrete pT. It seems that Equation (10) directly means that the integral of f1pT, f2pT, and f3pT is the same for the 0,pT interval, which essentially means that the three functions are equal (except for a null measure set). In fact, the three functions are different in forms because of Equations (2)–(4), and we need to distinguish them.In particular, forf1pT, we have a simpler expression. Let us solve the equation(11)∫0ptjfp′tjdp′tj=rjj=1,2,3,⋯,ns.We have(12)ptj=−ptlnrjj=1,2,3,⋯,ns,due to Equation (1) being used, where rj in Equation (12) replaced 1−rj because both of them are random numbers in [0,1]. The simpler expression is(13)pT=−pt∏j=1nslnrj,due to pT being the sum of ns random ptj.To obtain a concrete value ofθ, we need the function(14)fθθ=12sinθ,which is obeyed by θ under the assumption of isotropic emission in the center-of-mass reference frame. Solving the equation(15)∫0θfθθ′dθ′=R2,we have(16)θ=2arcsinR2,which is needed by us.According to the concrete values ofpT and θ, and using other quantities, the value of ∣t∣ can be obtained from Equation (8). After repeating the calculations many times, the distribution of ∣t∣ is obtained statistically. Based on the method of least squares, the related parameters are obtained naturally. Meanwhile, Ti can be obtained from Equation (9). pT and pT2 can be obtained from one of Equations (2)–(4) or from the statistics. The errors of parameters are obtained by the general method of statistical simulation.


## 3. Results and Discussion

### 3.1. Comparison with Data

Figure1 shows the differential cross-section in squared momentum transfer, dσ/d∣t∣, of (a) γ∗p⟶ρp, (b) γ∗p⟶ρY, and (c) γ∗p⟶ρ0p produced in electron-proton (ep) collisions at photon-proton center-of-mass energy (a, b) W=75 GeV and (c) W=90 GeV, where σ denotes the cross-section and Y in Figure 1(b) denotes an “elastic” scattering proton or a diffractively excited “proton dissociation” [37]. The experimental data points from (a, b) nonexclusive and (c) exclusive productions are measured by the H1 [37] and ZEUS Collaborations [39], respectively, with different average squared photon virtuality (a) Q2=3.3, 6.6, 11.5, 17.4, and 33.0 GeV2; (b) Q2=3.3, 6.6, and 15.8 GeV2; and (c) Q2=2.7, 5.0, 7.8, 11.9, 19.7, and 41.0 GeV2. The data points are fitted by the Monte Carlo calculations with the Erlang distribution Equation (2) (the solid curves), the Tsallis distribution Equation (3) (the dashed curves), and the Hagedorn function Equation (4) (the dotted curves) for p3T in Equation (8). Some data are scaled by different quantities marked in the panels for clear visibility. In the calculations, the method of least squares is used to obtain the parameter values. The values of E1, pt, ns, Ti, T, n, p0, and n0 are listed in Tables 1–3 with χ2 and number of degree of freedom (ndof). One can see that in most cases, the calculations based on Equation (8) with Equations (2)–(4) for p3T can fit approximately the experimental data measured by the H1 and ZEUS Collaborations.Figure 1
The differential cross-section in squared momentum transfer of (a)γ∗p⟶ρp, (b) γ∗p⟶ρY, and (c) γ∗p⟶ρ0p produced in ep collisions at (a, b) W=75 GeV and (c) W=90 GeV. The experimental data points from (a, b) nonexclusive and (c) exclusive productions are measured by the H1 [37] and ZEUS Collaborations [39], respectively, with different Q2 marked in the panels. The data points are fitted by the Monte Carlo calculations with the Erlang distribution Equation (2) (the solid curves), the Tsallis distribution Equation (3) (the dashed curves), and the Hagedorn function Equation (4) (the dotted curves) for p3T in Equation (8).
(a)
(b)
(c)Table 1
Values ofE1, pt, ns, Ti, and χ2/ndof corresponding to the solid curves in Figures 1–3, where ns=5 that is not listed in the table to avoid trivialness. In some cases, ndof is less than 1, which is denoted by “−” in the last column, and the corresponding curve is only to guide the eyes. For Figure 3(a), the first and second Q2=6.8 GeV2 are averaged from the ranges of Q2=2–100 and 5–10 GeV2, respectively.
Figure
Reaction
Main selection
E
1 (GeV)
p
t (GeV/c)
T
i (GeV)
χ
2/ndof
Figure1(a)
γ
∗
p
⟶
ρ
p
Q
2
=
3.3 GeV2
0.960
±
0.003
0.045
±
0.003
0.174
±
0.011
17
/
3
6.6
0.960
±
0.003
0.048
±
0.001
0.186
±
0.004
8
/
3
11.5
0.960
±
0.001
0.052
±
0.001
0.201
±
0.003
3
/
3
17.4
0.960
±
0.003
0.057
±
0.002
0.221
±
0.008
9
/
3
33.0
0.960
±
0.001
0.061
±
0.002
0.236
±
0.007
5
/
3
Figure1(b)
γ
∗
p
⟶
ρ
Y
Q
2
=
3.3 GeV2
0.960
±
0.020
0.085
±
0.010
0.329
±
0.039
47
/
7
6.6
0.960
±
0.020
0.100
±
0.012
0.387
±
0.046
95
/
7
15.8
0.960
±
0.020
0.115
±
0.020
0.445
±
0.078
86
/
7
Figure1(c)
γ
∗
p
⟶
ρ
0
p
Q
2
=
2.7 GeV2
0.961
±
0.004
0.047
±
0.001
0.182
±
0.004
22
/
1
5.0
0.961
±
0.001
0.052
±
0.001
0.201
±
0.003
5
/
1
7.8
0.961
±
0.001
0.052
±
0.002
0.201
±
0.007
11
/
1
11.9
0.961
±
0.001
0.055
±
0.001
0.213
±
0.004
2
/
1
19.7
0.961
±
0.001
0.056
±
0.002
0.217
±
0.008
1
/
1
41.0
0.961
±
0.002
0.058
±
0.002
0.225
±
0.008
3
/
1
Figure2(a)
γ
p
⟶
ω
p
70
GeV
<
W
<
90 GeV
0.960
±
0.010
0.036
±
0.005
0.139
±
0.019
5
/
2
Figure2(b)
γ
∗
p
⟶
ϕ
p
Q
2
=
3.3 GeV2
0.970
±
0.010
0.045
±
0.005
0.174
±
0.020
22
/
3
6.6
0.970
±
0.010
0.048
±
0.003
0.186
±
0.012
11
/
3
15.8
0.970
±
0.010
0.050
±
0.002
0.194
±
0.008
9
/
3
γ
∗
p
⟶
ϕ
Y
W
=
75 GeV
0.970
±
0.030
0.095
±
0.007
0.368
±
0.027
4
/
−
Figure2(c)
γ
∗
p
⟶
ϕ
p
Q
2
=
2.4 GeV2
0.960
±
0.010
0.058
±
0.002
0.225
±
0.008
4
/
−
3.6
0.960
±
0.010
0.060
±
0.005
0.232
±
0.020
7
/
−
5.2
0.960
±
0.010
0.063
±
0.002
0.244
±
0.008
4
/
−
6.9
0.960
±
0.011
0.065
±
0.003
0.252
±
0.012
6
/
−
9.2
0.960
±
0.010
0.067
±
0.002
0.259
±
0.008
7
/
−
12.6
0.960
±
0.012
0.070
±
0.002
0.271
±
0.008
5
/
−
19.7
0.960
±
0.020
0.072
±
0.005
0.279
±
0.019
5
/
−
Figure2(d)
p
p
⟶
p
p
f
0
980
s
N
N
=
29.1 GeV
0.960
±
0.001
0.062
±
0.002
0.240
±
0.008
15
/
2
p
p
⟶
p
p
f
1
1285
1.018
±
0.001
0.045
±
0.002
0.174
±
0.007
4
/
2
p
p
⟶
p
p
f
0
1370
1.033
±
0.002
0.035
±
0.001
0.136
±
0.004
86
/
2
p
p
⟶
p
p
f
1
1420
1.050
±
0.001
0.058
±
0.002
0.225
±
0.008
21
/
2
p
p
⟶
p
p
f
0
1500
1.056
±
0.001
0.063
±
0.002
0.244
±
0.008
16
/
2
Figure3(a)
γ
∗
p
⟶
J
/
ψ
p
Q
2
=
3.1 GeV2
1.690
±
0.020
0.020
±
0.001
0.077
±
0.004
3
/
−
6.8
1.690
±
0.010
0.022
±
0.001
0.085
±
0.004
5
/
−
6.8
1.690
±
0.010
0.022
±
0.001
0.085
±
0.004
1
/
−
16.0
1.690
±
0.010
0.023
±
0.001
0.089
±
0.004
8
/
−
Figure3(b)
γ
p
⟶
J
/
ψ
p
Q
2
=
0.05 GeV2
1.700
±
0.005
0.018
±
0.001
0.070
±
0.004
35
/
−
3.2
1.700
±
0.001
0.020
±
0.001
0.077
±
0.004
2
/
−
7.0
1.700
±
0.004
0.022
±
0.002
0.085
±
0.008
10
/
−
22.4
1.700
±
0.001
0.024
±
0.002
0.093
±
0.008
3
/
4
Figure3(c)
γ
p
⟶
J
/
ψ
p
W
=
45 GeV
1.700
±
0.012
0.026
±
0.004
0.101
±
0.016
12
/
1
55
1.700
±
0.018
0.024
±
0.004
0.093
±
0.016
15
/
1
65
1.700
±
0.006
0.022
±
0.002
0.085
±
0.008
14
/
1
75
1.700
±
0.015
0.021
±
0.002
0.081
±
0.008
16
/
1
85
1.700
±
0.010
0.020
±
0.001
0.077
±
0.004
6
/
1
95
1.700
±
0.010
0.018
±
0.002
0.070
±
0.008
5
/
1
Figure3(d)
γ
p
⟶
J
/
ψ
p
W
=
105 GeV
1.700
±
0.006
0.017
±
0.001
0.066
±
0.004
25
/
1
119
1.700
±
0.001
0.016
±
0.001
0.062
±
0.004
18
/
1
144
1.700
±
0.002
0.015
±
0.001
0.058
±
0.004
12
/
1
181
1.700
±
0.002
0.014
±
0.001
0.054
±
0.004
46
/
1
251
1.700
±
0.001
0.013
±
0.001
0.050
±
0.004
15
/
1Table 2
Values ofE1, T, n, and χ2/ndof corresponding to the dashed curves in Figures 1–3, where “−” in the last column denotes the case of ndof<1 and the corresponding curve is only to guide the eyes.
Figure
Reaction
Main selection
E
1 (GeV)
T (GeV)
n
χ
2/ndof
Figure1(a)
γ
∗
p
⟶
ρ
p
Q
2
=
3.3 GeV2
0.950
±
0.003
0.032
±
0.002
18.0
±
1.5
27
/
3
6.6
0.950
±
0.002
0.037
±
0.002
17.5
±
2.0
5
/
3
11.5
0.950
±
0.002
0.039
±
0.001
17.0
±
1.0
6
/
3
17.4
0.950
±
0.005
0.049
±
0.002
16.0
±
2.0
6
/
3
33.0
0.950
±
0.005
0.054
±
0.002
15.0
±
2.0
5
/
3
Figure1(b)
γ
∗
p
⟶
ρ
Y
Q
2
=
3.3 GeV2
0.950
±
0.010
0.065
±
0.015
5.0
±
1.0
55
/
7
6.6
0.950
±
0.010
0.075
±
0.011
4.5
±
1.1
102
/
7
15.8
0.950
±
0.010
0.085
±
0.020
4.2
±
2.0
68
/
7
Figure1(c)
γ
∗
p
⟶
ρ
0
p
Q
2
=
2.7 GeV2
0.950
±
0.002
0.035
±
0.001
17.0
±
2.0
35
/
1
5.0
0.950
±
0.001
0.039
±
0.002
16.0
±
1.0
8
/
1
7.8
0.950
±
0.003
0.041
±
0.002
15.0
±
0.5
8
/
1
11.9
0.950
±
0.002
0.043
±
0.002
14.0
±
1.0
9
/
1
19.7
0.950
±
0.002
0.045
±
0.001
13.0
±
1.0
2
/
1
41.0
0.950
±
0.001
0.047
±
0.001
12.0
±
1.0
2
/
1
Figure2(a)
γ
p
⟶
ω
p
70
GeV
<
W
<
90
GeV
0.960
±
0.010
0.021
±
0.003
20.0
±
2.0
5
/
2
Figure2(b)
γ
∗
p
⟶
ϕ
p
Q
2
=
3.3 GeV2
0.970
±
0.010
0.025
±
0.002
10.0
±
1.5
9
/
3
6.6
0.970
±
0.010
0.027
±
0.003
9.0
±
1.0
11
/
3
15.8
0.970
±
0.010
0.028
±
0.003
8.0
±
1.0
23
/
3
γ
∗
p
⟶
ϕ
Y
W
=
75 GeV
0.970
±
0.010
0.060
±
0.008
5.0
±
1.0
5
/
−
Figure2(c)
γ
∗
p
⟶
ϕ
p
Q
2
=
2.4 GeV2
0.961
±
0.001
0.040
±
0.001
9.0
±
0.1
2
/
−
3.6
0.961
±
0.001
0.041
±
0.001
8.3
±
0.1
1
/
−
5.2
0.961
±
0.001
0.043
±
0.001
7.4
±
0.3
3
/
−
6.9
0.961
±
0.001
0.043
±
0.001
7.2
±
0.2
3
/
−
9.2
0.961
±
0.001
0.045
±
0.001
7.0
±
0.2
3
/
−
12.6
0.961
±
0.001
0.046
±
0.002
6.7
±
0.3
3
/
−
19.7
0.961
±
0.001
0.048
±
0.003
5.4
±
0.3
1
/
−
Figure2(d)
p
p
⟶
p
p
f
0
980
s
N
N
=
29.1 GeV
0.965
±
0.001
0.047
±
0.003
8.0
±
1.0
9
/
2
p
p
⟶
p
p
f
1
1285
1.017
±
0.001
0.021
±
0.001
12.6
±
0.2
6
/
2
p
p
⟶
p
p
f
0
1370
1.033
±
0.001
0.013
±
0.001
12.0
±
0.3
102
/
2
p
p
⟶
p
p
f
1
1420
1.050
±
0.004
0.030
±
0.004
9.5
±
0.7
17
/
2
p
p
⟶
p
p
f
0
1500
1.062
±
0.001
0.034
±
0.003
7.0
±
0.5
15
/
2
Figure3(a)
γ
∗
p
⟶
J
/
ψ
p
Q
2
=
3.1 GeV2
1.680
±
0.010
0.0022
±
0.0001
22.0
±
1.0
6
/
−
6.8
1.680
±
0.004
0.0024
±
0.0001
20.0
±
0.3
8
/
−
6.8
1.680
±
0.004
0.0024
±
0.0001
20.0
±
0.3
2
/
−
16.0
1.680
±
0.003
0.0026
±
0.0002
16.0
±
1.0
9
/
−
Figure3(b)
γ
p
⟶
J
/
ψ
p
Q
2
=
0.05 GeV2
1.685
±
0.001
0.0019
±
0.0001
25.0
±
3.0
33
/
−
3.2
1.685
±
0.005
0.0020
±
0.0001
22.0
±
2.0
8
/
−
7.0
1.685
±
0.002
0.0022
±
0.0002
20.0
±
2.0
6
/
−
22.4
1.685
±
0.003
0.0026
±
0.0001
16.0
±
2.0
3
/
4
Figure3(c)
γ
p
⟶
J
/
ψ
p
W
=
45 GeV
1.690
±
0.010
0.0042
±
0.0001
18.0
±
2.0
27
/
1
55
1.690
±
0.010
0.0038
±
0.0006
19.0
±
4.0
35
/
1
65
1.690
±
0.010
0.0033
±
0.0005
20.0
±
3.0
32
/
1
75
1.690
±
0.030
0.0030
±
0.0007
21.0
±
5.0
28
/
1
85
1.690
±
0.003
0.0026
±
0.0003
22.0
±
3.0
18
/
1
95
1.690
±
0.007
0.0020
±
0.0003
23.0
±
3.0
12
/
1
Figure3(d)
γ
p
⟶
J
/
ψ
p
W
=
105 GeV
1.690
±
0.003
0.0013
±
0.0003
24.0
±
3.0
34
/
1
119
1.690
±
0.001
0.0012
±
0.0004
25.0
±
2.0
23
/
1
144
1.690
±
0.005
0.0011
±
0.0004
26.0
±
3.0
20
/
1
181
1.690
±
0.001
0.0010
±
0.0003
27.0
±
3.0
40
/
1
251
1.690
±
0.001
0.0009
±
0.0003
28.0
±
2.0
17
/
1Table 3
Values ofE1, p0, n0, and χ2/ndof corresponding to the dotted curves in Figures 1–3, where “−” in the last column denotes the case of ndof<1 and the corresponding curve is only to guide the eyes.
Figure
Reaction
Main selection
E
1 (GeV)
p
0 (GeV/c)
n
0
χ
2/ndof
Figure1(a)
γ
∗
p
⟶
ρ
p
Q
2
=
3.3 GeV2
0.960
±
0.010
1.70
±
0.11
19.0
±
1.5
66
/
3
6.6
0.960
±
0.003
1.85
±
0.02
18.0
±
0.8
19
/
3
11.5
0.960
±
0.007
1.90
±
0.10
17.5
±
2.0
16
/
3
17.4
0.960
±
0.007
2.02
±
0.05
15.8
±
0.8
16
/
3
33.0
0.960
±
0.005
2.12
±
0.10
15.0
±
1.0
9
/
3
Figure1(b)
γ
∗
p
⟶
ρ
Y
Q
2
=
3.3 GeV2
0.960
±
0.020
2.10
±
0.37
15.0
±
2.7
78
/
7
6.6
0.960
±
0.020
2.30
±
0.17
14.0
±
0.9
118
/
7
15.8
0.960
±
0.020
2.50
±
0.27
13.5
±
1.0
74
/
7
Figure1(c)
γ
∗
p
⟶
ρ
0
p
Q
2
=
2.7 GeV2
0.950
±
0.003
1.59
±
0.10
18.5
±
0.5
55
/
1
5.0
0.950
±
0.001
1.61
±
0.04
18.0
±
0.5
34
/
1
7.8
0.950
±
0.002
1.63
±
0.04
17.0
±
1.0
33
/
1
11.9
0.950
±
0.003
1.63
±
0.10
16.5
±
1.0
29
/
1
19.7
0.950
±
0.002
1.63
±
0.05
16.0
±
1.3
16
/
1
41.0
0.950
±
0.002
1.65
±
0.05
15.8
±
0.4
2
/
1
Figure2(a)
γ
p
⟶
ω
p
70 GeV<W<90 GeV
0.960
±
0.010
1.30
±
0.05
21.0
±
2.0
7
/
2
Figure2(b)
γ
∗
p
⟶
ϕ
p
Q
2
=
3.3 GeV2
0.970
±
0.010
1.75
±
0.05
20.0
±
2.0
12
/
3
6.6
0.970
±
0.010
1.80
±
0.04
19.0
±
1.0
22
/
3
15.8
0.970
±
0.010
1.85
±
0.03
18.0
±
1.0
37
/
3
γ
∗
p
⟶
ϕ
Y
W
=
75 GeV
0.970
±
0.020
1.50
±
0.20
11.0
±
1.0
6
/
−
Figure2(c)
γ
∗
p
⟶
ϕ
p
Q
2
=
2.4 GeV2
0.962
±
0.002
2.20
±
0.03
16.2
±
0.4
10
/
−
3.6
0.962
±
0.002
2.22
±
0.02
16.0
±
0.5
5
/
−
5.2
0.962
±
0.002
2.24
±
0.04
15.7
±
0.8
4
/
−
6.9
0.962
±
0.002
2.26
±
0.05
15.5
±
0.5
5
/
−
9.2
0.962
±
0.001
2.29
±
0.03
15.0
±
0.3
7
/
−
12.6
0.962
±
0.001
2.31
±
0.03
14.7
±
0.2
5
/
−
19.7
0.962
±
0.001
2.34
±
0.02
14.4
±
0.2
1
/
−
Figure2(d)
p
p
⟶
p
p
f
0
980
s
N
N
=
29.1 GeV
0.971
±
0.001
1.53
±
0.10
13.5
±
0.7
56
/
2
p
p
⟶
p
p
f
1
1285
1.023
±
0.001
2.00
±
0.10
22.0
±
0.5
12
/
2
p
p
⟶
p
p
f
0
1370
1.036
±
0.002
1.30
±
0.10
21.0
±
2.0
132
/
2
p
p
⟶
p
p
f
1
1420
1.049
±
0.003
2.10
±
0.20
16.0
±
1.5
21
/
2
p
p
⟶
p
p
f
0
1500
1.065
±
0.001
2.30
±
0.20
16.0
±
1.0
20
/
2
Figure3(a)
γ
∗
p
⟶
J
/
ψ
p
Q
2
=
3.1 GeV2
1.678
±
0.001
1.25
±
0.05
28.0
±
1.0
5
/
−
6.8
1.678
±
0.001
1.29
±
0.02
26.0
±
3.0
12
/
−
6.8
1.678
±
0.001
1.29
±
0.02
26.0
±
3.0
4
/
−
16.0
1.678
±
0.001
1.31
±
0.02
24.0
±
1.0
12
/
−
Figure3(b)
γ
p
⟶
J
/
ψ
p
Q
2
=
0.05 GeV2
1.700
±
0.001
1.20
±
0.02
29.0
±
0.5
64
/
−
3.2
1.700
±
0.001
1.30
±
0.02
27.0
±
0.7
10
/
−
7.0
1.700
±
0.001
1.40
±
0.03
25.0
±
2.0
5
/
−
22.4
1.700
±
0.001
1.50
±
0.01
23.0
±
0.1
1
/
4
Figure3(c)
γ
p
⟶
J
/
ψ
p
W
=
45 GeV
1.680
±
0.003
1.38
±
0.10
22.0
±
3.0
19
/
1
55
1.680
±
0.010
1.33
±
0.20
23.0
±
4.0
26
/
1
65
1.680
±
0.010
1.29
±
0.17
24.0
±
4.0
21
/
1
75
1.680
±
0.008
1.27
±
0.10
24.0
±
3.0
21
/
1
85
1.680
±
0.002
1.25
±
0.15
25.0
±
3.0
21
/
1
95
1.680
±
0.002
1.21
±
0.10
26.0
±
3.0
13
/
1
Figure3(d)
γ
p
⟶
J
/
ψ
p
W
=
105 GeV
1.680
±
0.001
1.18
±
0.06
27.0
±
1.0
33
/
1
119
1.680
±
0.001
1.15
±
0.12
28.0
±
1.0
25
/
1
144
1.680
±
0.001
1.12
±
0.11
29.0
±
2.0
22
/
1
181
1.680
±
0.001
1.11
±
0.07
30.0
±
1.0
41
/
1
251
1.680
±
0.001
1.07
±
0.07
31.0
±
3.0
13
/
1Figure2 presents the differential cross-section in squared momentum transfer, dσ/d∣t∣, of (a) γp⟶ωp, (b) γ∗p⟶ϕp and γ∗p⟶ϕY, (c) γ∗p⟶ϕp, and (d) pp⟶ppV (V=f0980, f11285, f01370, f11420, and f01500) produced in (a–c) ep and (d) pp collisions in (a) 70GeV<W<90GeV, at (b, c) W=75 GeV, and at (d) proton-proton center-of-mass energy per nucleon pair sNN=29.1 GeV. The experimental data points from (a, c) exclusive, (b) nonexclusive, and (d) exclusive productions are measured by the ZEUS [40, 41], H1 [37], and WA102 Collaborations [43, 44], respectively, with different Q2 for only Figure 2(b) (Q2=3.3, 5, 6.6, and 15.8 GeV2) and Figure 2(c) (Q2=2.4, 3.6, 5.2, 6.9, 9.2, 12.6, and 19.7 GeV2). Similar to Figure 1, the data points are fitted by the Monte Carlo calculations based on Equation (8). The values of parameters are listed in Tables 1–3 with χ2/ndof. One can see that in most cases, the calculations based on Equation (8) with Equations (2)–(4) for p3T can fit approximately the experimental data measured by the H1 and ZEUS Collaborations.Figure 2
The differential cross-section in squared momentum transfer of (a)γp⟶ωp, (b) γ∗p⟶ϕp and γ∗p⟶ϕY, (c) γ∗p⟶ϕp, and (d) pp⟶ppV (V=f0980, f11285, f01370, f11420, and f01500) produced in (a–c) ep and (d) pp collisions in (a) 70GeV<W<90GeV, at (b, c) W=75 GeV, and at (d) sNN=29.1 GeV. The experimental data points from (a, c) exclusive, (b) nonexclusive, and (d) exclusive productions are measured by the ZEUS [40, 41], H1 [37], and WA102 Collaborations [43, 44], respectively, with different Q2 for only (b) and (c). Similar to Figure 1, the data points are fitted by the Monte Carlo calculations based on Equation (8).
(a)
(b)
(c)
(d)Figure3 displays the differential cross-section in squared momentum transfer, dσ/d∣t∣, of (a) γ∗p⟶J/ψp and (b–d) γp⟶J/ψp produced in ep collisions at (a) W=90 GeV, in (b) 40GeV<W<160GeV, and at (c, d) Q2=0.05 GeV2. The experimental data points from (a) exclusive and (b–d) nonexclusive productions are measured by the ZEUS [42] and H1 Collaborations [38], respectively, with (a) Q2=3.1, 6.8 averaged in 2–100, 6.8 averaged in 5–10, and 16 GeV2 and (b) Q2=0.05, 3.2, 7.0, and 22.4 GeV2, as well as with (c) W=45, 55, 65, 75, 85, and 95 GeV and (d) W=105, 119, 144, 181, and 251 GeV. Similar to Figures 1 and 2, the data points are fitted by the Monte Carlo calculations based on Equation (8). The values of parameters are listed in Tables 1–3 with χ2/ndof. One can see that in most cases, the calculations based on Equation (8) with Equations (2)–(4) for p3T can fit approximately the experimental data measured by the H1 and ZEUS Collaborations.Figure 3
The differential cross-section in squared momentum transfer of (a)γ∗p⟶J/ψp and (b–d) γp⟶J/ψp produced in ep collisions at (a) W=90 GeV, in (b) 40GeV<W<160GeV, and at (c, d) Q2=0.05 GeV2. The experimental data points from (a) exclusive and (b–d) nonexclusive productions are measured by the ZEUS [42] and H1 Collaborations [38], respectively, with different Q2 marked in panels (a) and (b), as well as with different W marked in (c) and (d), where in (a), the first and second Q2=6.8 GeV2 are averaged from the ranges of Q2=2–100 and 5–10 GeV2, respectively. Similar to Figures 1 and 2, the data points are fitted by the Monte Carlo calculations based on Equation (8).
(a)
(b)
(c)
(d)From the above comparisons, we see that some fits have largeχ2 compared to ndof, corresponding to low confidence levels. The parameters obtained from these fits are not representing the data well. We would like to say here that these values are used only for the qualitative description of the data tendencies, but not the quantitative interpretation of the data size. In some cases, ndof<1, which means that there were at least as many parameters as data points. This means that a perfect fit should have been found. However, this was not the case here. The reason is that we have used given functions, but not any function such as a polynomial.


### 3.2. Tendency of Parameters

The dependencies of energyE1 of particle 1 on rest mass m0 of particle 3 for different two-body reactions are given in Figure 4, where Figures 4(a)–4(c) correspond to the results from the Erlang distribution, Tsallis distribution, and Hagedorn function, respectively. The types of reactions are marked in the panels. Different symbols represent the results from different reactions or collaborations. One can see that the production of particle 3 with larger m0 needs the participation of particle 1 with larger E1.Figure 4
The dependencies ofE1 on m0 for different two-body reactions which are marked in the panels. (a–c) correspond to the results from the Erlang distribution, Tsallis distribution, and Hagedorn function, respectively.
(a)
(b)
(c)The tendency ofE1 versus m0 presented in Figure 4 is natural due to the conservation of energy. The results from the three distributions or functions are almost the same, if not equal to each other, due to the same experimental data considered. In fact, E1 should be a fixed value for a given reaction in the present work. However, because different selections such as different Q2 and W are used in experiments, E1 has a slight shift from the peak value. Thus, we may regard E1 as a parameter and obtain it from the fits.The dependencies of (a)pT, (b) Ti, (c) T0, (d) n, (e) p0, and (f) n0 on average squared photon virtuality Q2 for different two-body reactions are shown in Figure 5. The types of reactions are marked in the panels. Different symbols for different reactions represent the parameter values extracted from Figures 1–3 and listed in Tables 1–3, where the Erlang distribution, Tsallis distribution, and Hagedorn function in the ranges of available data are used. In particular, pT=nspt from Table 1 and T0=T from Table 2. One can see that pT, Ti, T0, and p0 increase generally with increases in Q2, and n and n0 decrease significantly with an increase in Q2.Figure 5
The dependencies of (a)pT, (b) Ti, (c) T0, (d) n, (e) p0, and (f) n0 on Q2 for different two-body reactions. The symbols represent the parameter values extracted from Figures 1–3 and listed in Tables 1–3. Here, pT=nspt from Table 1 and T0=T from Table 2.
(a)
(b)
(c)
(d)
(e)
(f)Because ofQ2 being a reflection of a hard scale of reaction, this is natural that a harder scale results in a higher excitation degree and then a larger pT, Ti, and T0. In most cases, one can see a large enough n or n0. This means that q is close to 1 and the reaction systems stay in an approximate equilibrium state. At a harder scale, the degree of equilibrium decreases due to more disturbance to the equilibrated residual partons in the target particle. Then, one has a larger q and smaller n or n0 when compared with those at the softer scale.Figure6 shows the excitation functions of related parameters, i.e., the dependencies of (a) pT, (b) Ti, (c) T0, (d) n, (e) p0, and (f) n0 on the photon-proton center-of-mass energy W for γp⟶J/ψp reactions. The symbols represent the parameter values extracted from Figure 3 and listed in Tables 1–3. Again, pT=nspt from Table 1 and T0=T from Table 2. One can see that pT, Ti, T0, and p0 decrease with an increase in W, and n and n0 increase with an increase in W.Figure 6
The dependencies of (a)pT, (b) Ti, (c) T0, (d) n, (e) p0, and (f) n0 on W for γp⟶J/ψp reactions. The symbols represent the parameter values extracted from Figure 3 and listed in Tables 1–3. Here, pT=nspt from Table 1 and T0=T from Table 2.
(a)
(b)
(c)
(d)
(e)
(f)Inγp⟶J/ψp reactions, at a higher center-of-mass energy, the incident photon has a higher energy. Although the emitted J/ψ also has a higher energy, it is more inclined to have a smaller angle. As a comprehensive result, the transverse momentum of J/ψ is smaller, and then, Ti and T0, which are obtained from the transverse momentum, are also smaller. In addition, larger n and n0 at a higher collision energy means more equilibrium due to the shorter collision time and then less disturbance to the equilibrated residual partons in the target particle. This situation is different from nucleus-nucleus collisions in which a cold or spectator nuclear effect has to be considered.In fact, in nucleus-nucleus collisions, secondary cascade collisions may happen among produced particles and spectator nucleons. The secondary collisions may cause the emission angle to increase and then the transverse momentum to increase. The effect of secondary collisions is more obvious or nearly saturated at a higher energy. In nucleus-nucleus collisions at a lower energy, the system approaches equilibrium more easily due to a longer interaction time. Conversely, at a higher energy, the system does not approach equilibrium more easily due to the shorter interaction time for secondary collisions.


### 3.3. Further Discussion

Before the summary and conclusions, we would like to point out that the concept of temperature used in the present work is valid. Generally, the concept of temperature is used in a large system with multiple particles, which stays in an equilibrium state or approximate (local) equilibrium state. From the macroscopic point of view, the systems ofγ∗p, γp, and pp reactions are indeed small. However, we know that there are lots of events under the same condition in the experiments. These events obey the law of grand canonical ensemble in which the concept of temperature is applicable.Because the same experimental condition is used in statistics, lots of events are in equilibrium if they consist of a large statistical system which can be described by the grand canonical ensemble. Particles in the large statistical system obey the same distribution law such as the same transverse momentum distribution. From the statistical point of view, particle productions in high-energy collisions are a statistical behavior, and the temperature reflects the width of distribution. The higher the temperature is, the wider the distribution is.The temperature is also a reflection of the average kinetic energy based on a large statistical system or a single particle. For a single particle, if the distribution law of kinetic energies or transverse momenta is known, the temperature of emission source or interacting system is known, where the source or system means the large thermal source from the ensemble. Generally, we say the temperature of source or system, not saying the temperature of a given particle, from the point of view of statistical significance of temperature. Based on the temperature, we may compare the experimental spectra of different particles in different experiments.However, different methods have used different distributions or functions, i.e., different “thermometers.” To unify these “thermometers” or to find transformations among them, one has to perform quite extensive analysis. Although one may use as far as possible the standard distribution such as the Boltzmann, Fermi-Dirac, or Bose-Einstein distribution to fit the experimental spectra, it is regretful that a single standard distribution cannot fit the experimental spectra very well in general. Naturally, one may use a two-, three-, or even multicomponent standard distribution to fit the experimental spectra, though more parameters are introduced.In fact, the two-, three-, or multicomponent standard distribution can be fitted satisfactorily by the Tsallis distribution withq>1, because the standard distribution is narrower than the Tsallis distribution [68]. In particular, the standard distribution is equivalent to the Tsallis distribution with q=1. It is natural to use the Tsallis distribution to replace the standard distribution. That is, one may use the Tsallis distribution with q>1 to fit the experimental spectra and obtain the temperature, though the Tsallis temperature is less than the standard one.As mentioned in the first section and discussed above, some distributions applied in a large collision system can be also applied in a small collision system due to the universality, similarity, or common characteristics existing in high-energy collisions [27–36]. Based on the same reason, some statistical or hydrodynamic models applied in the large system should be also applied in the small system. Of course, lots of events are needed in experiments and high statistics is needed in calculation if performing a Monte Carlo code.


## 3.1. Comparison with Data

Figure1 shows the differential cross-section in squared momentum transfer, dσ/d∣t∣, of (a) γ∗p⟶ρp, (b) γ∗p⟶ρY, and (c) γ∗p⟶ρ0p produced in electron-proton (ep) collisions at photon-proton center-of-mass energy (a, b) W=75 GeV and (c) W=90 GeV, where σ denotes the cross-section and Y in Figure 1(b) denotes an “elastic” scattering proton or a diffractively excited “proton dissociation” [37]. The experimental data points from (a, b) nonexclusive and (c) exclusive productions are measured by the H1 [37] and ZEUS Collaborations [39], respectively, with different average squared photon virtuality (a) Q2=3.3, 6.6, 11.5, 17.4, and 33.0 GeV2; (b) Q2=3.3, 6.6, and 15.8 GeV2; and (c) Q2=2.7, 5.0, 7.8, 11.9, 19.7, and 41.0 GeV2. The data points are fitted by the Monte Carlo calculations with the Erlang distribution Equation (2) (the solid curves), the Tsallis distribution Equation (3) (the dashed curves), and the Hagedorn function Equation (4) (the dotted curves) for p3T in Equation (8). Some data are scaled by different quantities marked in the panels for clear visibility. In the calculations, the method of least squares is used to obtain the parameter values. The values of E1, pt, ns, Ti, T, n, p0, and n0 are listed in Tables 1–3 with χ2 and number of degree of freedom (ndof). One can see that in most cases, the calculations based on Equation (8) with Equations (2)–(4) for p3T can fit approximately the experimental data measured by the H1 and ZEUS Collaborations.Figure 1
The differential cross-section in squared momentum transfer of (a)γ∗p⟶ρp, (b) γ∗p⟶ρY, and (c) γ∗p⟶ρ0p produced in ep collisions at (a, b) W=75 GeV and (c) W=90 GeV. The experimental data points from (a, b) nonexclusive and (c) exclusive productions are measured by the H1 [37] and ZEUS Collaborations [39], respectively, with different Q2 marked in the panels. The data points are fitted by the Monte Carlo calculations with the Erlang distribution Equation (2) (the solid curves), the Tsallis distribution Equation (3) (the dashed curves), and the Hagedorn function Equation (4) (the dotted curves) for p3T in Equation (8).
(a)
(b)
(c)Table 1
Values ofE1, pt, ns, Ti, and χ2/ndof corresponding to the solid curves in Figures 1–3, where ns=5 that is not listed in the table to avoid trivialness. In some cases, ndof is less than 1, which is denoted by “−” in the last column, and the corresponding curve is only to guide the eyes. For Figure 3(a), the first and second Q2=6.8 GeV2 are averaged from the ranges of Q2=2–100 and 5–10 GeV2, respectively.
Figure
Reaction
Main selection
E
1 (GeV)
p
t (GeV/c)
T
i (GeV)
χ
2/ndof
Figure1(a)
γ
∗
p
⟶
ρ
p
Q
2
=
3.3 GeV2
0.960
±
0.003
0.045
±
0.003
0.174
±
0.011
17
/
3
6.6
0.960
±
0.003
0.048
±
0.001
0.186
±
0.004
8
/
3
11.5
0.960
±
0.001
0.052
±
0.001
0.201
±
0.003
3
/
3
17.4
0.960
±
0.003
0.057
±
0.002
0.221
±
0.008
9
/
3
33.0
0.960
±
0.001
0.061
±
0.002
0.236
±
0.007
5
/
3
Figure1(b)
γ
∗
p
⟶
ρ
Y
Q
2
=
3.3 GeV2
0.960
±
0.020
0.085
±
0.010
0.329
±
0.039
47
/
7
6.6
0.960
±
0.020
0.100
±
0.012
0.387
±
0.046
95
/
7
15.8
0.960
±
0.020
0.115
±
0.020
0.445
±
0.078
86
/
7
Figure1(c)
γ
∗
p
⟶
ρ
0
p
Q
2
=
2.7 GeV2
0.961
±
0.004
0.047
±
0.001
0.182
±
0.004
22
/
1
5.0
0.961
±
0.001
0.052
±
0.001
0.201
±
0.003
5
/
1
7.8
0.961
±
0.001
0.052
±
0.002
0.201
±
0.007
11
/
1
11.9
0.961
±
0.001
0.055
±
0.001
0.213
±
0.004
2
/
1
19.7
0.961
±
0.001
0.056
±
0.002
0.217
±
0.008
1
/
1
41.0
0.961
±
0.002
0.058
±
0.002
0.225
±
0.008
3
/
1
Figure2(a)
γ
p
⟶
ω
p
70
GeV
<
W
<
90 GeV
0.960
±
0.010
0.036
±
0.005
0.139
±
0.019
5
/
2
Figure2(b)
γ
∗
p
⟶
ϕ
p
Q
2
=
3.3 GeV2
0.970
±
0.010
0.045
±
0.005
0.174
±
0.020
22
/
3
6.6
0.970
±
0.010
0.048
±
0.003
0.186
±
0.012
11
/
3
15.8
0.970
±
0.010
0.050
±
0.002
0.194
±
0.008
9
/
3
γ
∗
p
⟶
ϕ
Y
W
=
75 GeV
0.970
±
0.030
0.095
±
0.007
0.368
±
0.027
4
/
−
Figure2(c)
γ
∗
p
⟶
ϕ
p
Q
2
=
2.4 GeV2
0.960
±
0.010
0.058
±
0.002
0.225
±
0.008
4
/
−
3.6
0.960
±
0.010
0.060
±
0.005
0.232
±
0.020
7
/
−
5.2
0.960
±
0.010
0.063
±
0.002
0.244
±
0.008
4
/
−
6.9
0.960
±
0.011
0.065
±
0.003
0.252
±
0.012
6
/
−
9.2
0.960
±
0.010
0.067
±
0.002
0.259
±
0.008
7
/
−
12.6
0.960
±
0.012
0.070
±
0.002
0.271
±
0.008
5
/
−
19.7
0.960
±
0.020
0.072
±
0.005
0.279
±
0.019
5
/
−
Figure2(d)
p
p
⟶
p
p
f
0
980
s
N
N
=
29.1 GeV
0.960
±
0.001
0.062
±
0.002
0.240
±
0.008
15
/
2
p
p
⟶
p
p
f
1
1285
1.018
±
0.001
0.045
±
0.002
0.174
±
0.007
4
/
2
p
p
⟶
p
p
f
0
1370
1.033
±
0.002
0.035
±
0.001
0.136
±
0.004
86
/
2
p
p
⟶
p
p
f
1
1420
1.050
±
0.001
0.058
±
0.002
0.225
±
0.008
21
/
2
p
p
⟶
p
p
f
0
1500
1.056
±
0.001
0.063
±
0.002
0.244
±
0.008
16
/
2
Figure3(a)
γ
∗
p
⟶
J
/
ψ
p
Q
2
=
3.1 GeV2
1.690
±
0.020
0.020
±
0.001
0.077
±
0.004
3
/
−
6.8
1.690
±
0.010
0.022
±
0.001
0.085
±
0.004
5
/
−
6.8
1.690
±
0.010
0.022
±
0.001
0.085
±
0.004
1
/
−
16.0
1.690
±
0.010
0.023
±
0.001
0.089
±
0.004
8
/
−
Figure3(b)
γ
p
⟶
J
/
ψ
p
Q
2
=
0.05 GeV2
1.700
±
0.005
0.018
±
0.001
0.070
±
0.004
35
/
−
3.2
1.700
±
0.001
0.020
±
0.001
0.077
±
0.004
2
/
−
7.0
1.700
±
0.004
0.022
±
0.002
0.085
±
0.008
10
/
−
22.4
1.700
±
0.001
0.024
±
0.002
0.093
±
0.008
3
/
4
Figure3(c)
γ
p
⟶
J
/
ψ
p
W
=
45 GeV
1.700
±
0.012
0.026
±
0.004
0.101
±
0.016
12
/
1
55
1.700
±
0.018
0.024
±
0.004
0.093
±
0.016
15
/
1
65
1.700
±
0.006
0.022
±
0.002
0.085
±
0.008
14
/
1
75
1.700
±
0.015
0.021
±
0.002
0.081
±
0.008
16
/
1
85
1.700
±
0.010
0.020
±
0.001
0.077
±
0.004
6
/
1
95
1.700
±
0.010
0.018
±
0.002
0.070
±
0.008
5
/
1
Figure3(d)
γ
p
⟶
J
/
ψ
p
W
=
105 GeV
1.700
±
0.006
0.017
±
0.001
0.066
±
0.004
25
/
1
119
1.700
±
0.001
0.016
±
0.001
0.062
±
0.004
18
/
1
144
1.700
±
0.002
0.015
±
0.001
0.058
±
0.004
12
/
1
181
1.700
±
0.002
0.014
±
0.001
0.054
±
0.004
46
/
1
251
1.700
±
0.001
0.013
±
0.001
0.050
±
0.004
15
/
1Table 2
Values ofE1, T, n, and χ2/ndof corresponding to the dashed curves in Figures 1–3, where “−” in the last column denotes the case of ndof<1 and the corresponding curve is only to guide the eyes.
Figure
Reaction
Main selection
E
1 (GeV)
T (GeV)
n
χ
2/ndof
Figure1(a)
γ
∗
p
⟶
ρ
p
Q
2
=
3.3 GeV2
0.950
±
0.003
0.032
±
0.002
18.0
±
1.5
27
/
3
6.6
0.950
±
0.002
0.037
±
0.002
17.5
±
2.0
5
/
3
11.5
0.950
±
0.002
0.039
±
0.001
17.0
±
1.0
6
/
3
17.4
0.950
±
0.005
0.049
±
0.002
16.0
±
2.0
6
/
3
33.0
0.950
±
0.005
0.054
±
0.002
15.0
±
2.0
5
/
3
Figure1(b)
γ
∗
p
⟶
ρ
Y
Q
2
=
3.3 GeV2
0.950
±
0.010
0.065
±
0.015
5.0
±
1.0
55
/
7
6.6
0.950
±
0.010
0.075
±
0.011
4.5
±
1.1
102
/
7
15.8
0.950
±
0.010
0.085
±
0.020
4.2
±
2.0
68
/
7
Figure1(c)
γ
∗
p
⟶
ρ
0
p
Q
2
=
2.7 GeV2
0.950
±
0.002
0.035
±
0.001
17.0
±
2.0
35
/
1
5.0
0.950
±
0.001
0.039
±
0.002
16.0
±
1.0
8
/
1
7.8
0.950
±
0.003
0.041
±
0.002
15.0
±
0.5
8
/
1
11.9
0.950
±
0.002
0.043
±
0.002
14.0
±
1.0
9
/
1
19.7
0.950
±
0.002
0.045
±
0.001
13.0
±
1.0
2
/
1
41.0
0.950
±
0.001
0.047
±
0.001
12.0
±
1.0
2
/
1
Figure2(a)
γ
p
⟶
ω
p
70
GeV
<
W
<
90
GeV
0.960
±
0.010
0.021
±
0.003
20.0
±
2.0
5
/
2
Figure2(b)
γ
∗
p
⟶
ϕ
p
Q
2
=
3.3 GeV2
0.970
±
0.010
0.025
±
0.002
10.0
±
1.5
9
/
3
6.6
0.970
±
0.010
0.027
±
0.003
9.0
±
1.0
11
/
3
15.8
0.970
±
0.010
0.028
±
0.003
8.0
±
1.0
23
/
3
γ
∗
p
⟶
ϕ
Y
W
=
75 GeV
0.970
±
0.010
0.060
±
0.008
5.0
±
1.0
5
/
−
Figure2(c)
γ
∗
p
⟶
ϕ
p
Q
2
=
2.4 GeV2
0.961
±
0.001
0.040
±
0.001
9.0
±
0.1
2
/
−
3.6
0.961
±
0.001
0.041
±
0.001
8.3
±
0.1
1
/
−
5.2
0.961
±
0.001
0.043
±
0.001
7.4
±
0.3
3
/
−
6.9
0.961
±
0.001
0.043
±
0.001
7.2
±
0.2
3
/
−
9.2
0.961
±
0.001
0.045
±
0.001
7.0
±
0.2
3
/
−
12.6
0.961
±
0.001
0.046
±
0.002
6.7
±
0.3
3
/
−
19.7
0.961
±
0.001
0.048
±
0.003
5.4
±
0.3
1
/
−
Figure2(d)
p
p
⟶
p
p
f
0
980
s
N
N
=
29.1 GeV
0.965
±
0.001
0.047
±
0.003
8.0
±
1.0
9
/
2
p
p
⟶
p
p
f
1
1285
1.017
±
0.001
0.021
±
0.001
12.6
±
0.2
6
/
2
p
p
⟶
p
p
f
0
1370
1.033
±
0.001
0.013
±
0.001
12.0
±
0.3
102
/
2
p
p
⟶
p
p
f
1
1420
1.050
±
0.004
0.030
±
0.004
9.5
±
0.7
17
/
2
p
p
⟶
p
p
f
0
1500
1.062
±
0.001
0.034
±
0.003
7.0
±
0.5
15
/
2
Figure3(a)
γ
∗
p
⟶
J
/
ψ
p
Q
2
=
3.1 GeV2
1.680
±
0.010
0.0022
±
0.0001
22.0
±
1.0
6
/
−
6.8
1.680
±
0.004
0.0024
±
0.0001
20.0
±
0.3
8
/
−
6.8
1.680
±
0.004
0.0024
±
0.0001
20.0
±
0.3
2
/
−
16.0
1.680
±
0.003
0.0026
±
0.0002
16.0
±
1.0
9
/
−
Figure3(b)
γ
p
⟶
J
/
ψ
p
Q
2
=
0.05 GeV2
1.685
±
0.001
0.0019
±
0.0001
25.0
±
3.0
33
/
−
3.2
1.685
±
0.005
0.0020
±
0.0001
22.0
±
2.0
8
/
−
7.0
1.685
±
0.002
0.0022
±
0.0002
20.0
±
2.0
6
/
−
22.4
1.685
±
0.003
0.0026
±
0.0001
16.0
±
2.0
3
/
4
Figure3(c)
γ
p
⟶
J
/
ψ
p
W
=
45 GeV
1.690
±
0.010
0.0042
±
0.0001
18.0
±
2.0
27
/
1
55
1.690
±
0.010
0.0038
±
0.0006
19.0
±
4.0
35
/
1
65
1.690
±
0.010
0.0033
±
0.0005
20.0
±
3.0
32
/
1
75
1.690
±
0.030
0.0030
±
0.0007
21.0
±
5.0
28
/
1
85
1.690
±
0.003
0.0026
±
0.0003
22.0
±
3.0
18
/
1
95
1.690
±
0.007
0.0020
±
0.0003
23.0
±
3.0
12
/
1
Figure3(d)
γ
p
⟶
J
/
ψ
p
W
=
105 GeV
1.690
±
0.003
0.0013
±
0.0003
24.0
±
3.0
34
/
1
119
1.690
±
0.001
0.0012
±
0.0004
25.0
±
2.0
23
/
1
144
1.690
±
0.005
0.0011
±
0.0004
26.0
±
3.0
20
/
1
181
1.690
±
0.001
0.0010
±
0.0003
27.0
±
3.0
40
/
1
251
1.690
±
0.001
0.0009
±
0.0003
28.0
±
2.0
17
/
1Table 3
Values ofE1, p0, n0, and χ2/ndof corresponding to the dotted curves in Figures 1–3, where “−” in the last column denotes the case of ndof<1 and the corresponding curve is only to guide the eyes.
Figure
Reaction
Main selection
E
1 (GeV)
p
0 (GeV/c)
n
0
χ
2/ndof
Figure1(a)
γ
∗
p
⟶
ρ
p
Q
2
=
3.3 GeV2
0.960
±
0.010
1.70
±
0.11
19.0
±
1.5
66
/
3
6.6
0.960
±
0.003
1.85
±
0.02
18.0
±
0.8
19
/
3
11.5
0.960
±
0.007
1.90
±
0.10
17.5
±
2.0
16
/
3
17.4
0.960
±
0.007
2.02
±
0.05
15.8
±
0.8
16
/
3
33.0
0.960
±
0.005
2.12
±
0.10
15.0
±
1.0
9
/
3
Figure1(b)
γ
∗
p
⟶
ρ
Y
Q
2
=
3.3 GeV2
0.960
±
0.020
2.10
±
0.37
15.0
±
2.7
78
/
7
6.6
0.960
±
0.020
2.30
±
0.17
14.0
±
0.9
118
/
7
15.8
0.960
±
0.020
2.50
±
0.27
13.5
±
1.0
74
/
7
Figure1(c)
γ
∗
p
⟶
ρ
0
p
Q
2
=
2.7 GeV2
0.950
±
0.003
1.59
±
0.10
18.5
±
0.5
55
/
1
5.0
0.950
±
0.001
1.61
±
0.04
18.0
±
0.5
34
/
1
7.8
0.950
±
0.002
1.63
±
0.04
17.0
±
1.0
33
/
1
11.9
0.950
±
0.003
1.63
±
0.10
16.5
±
1.0
29
/
1
19.7
0.950
±
0.002
1.63
±
0.05
16.0
±
1.3
16
/
1
41.0
0.950
±
0.002
1.65
±
0.05
15.8
±
0.4
2
/
1
Figure2(a)
γ
p
⟶
ω
p
70 GeV<W<90 GeV
0.960
±
0.010
1.30
±
0.05
21.0
±
2.0
7
/
2
Figure2(b)
γ
∗
p
⟶
ϕ
p
Q
2
=
3.3 GeV2
0.970
±
0.010
1.75
±
0.05
20.0
±
2.0
12
/
3
6.6
0.970
±
0.010
1.80
±
0.04
19.0
±
1.0
22
/
3
15.8
0.970
±
0.010
1.85
±
0.03
18.0
±
1.0
37
/
3
γ
∗
p
⟶
ϕ
Y
W
=
75 GeV
0.970
±
0.020
1.50
±
0.20
11.0
±
1.0
6
/
−
Figure2(c)
γ
∗
p
⟶
ϕ
p
Q
2
=
2.4 GeV2
0.962
±
0.002
2.20
±
0.03
16.2
±
0.4
10
/
−
3.6
0.962
±
0.002
2.22
±
0.02
16.0
±
0.5
5
/
−
5.2
0.962
±
0.002
2.24
±
0.04
15.7
±
0.8
4
/
−
6.9
0.962
±
0.002
2.26
±
0.05
15.5
±
0.5
5
/
−
9.2
0.962
±
0.001
2.29
±
0.03
15.0
±
0.3
7
/
−
12.6
0.962
±
0.001
2.31
±
0.03
14.7
±
0.2
5
/
−
19.7
0.962
±
0.001
2.34
±
0.02
14.4
±
0.2
1
/
−
Figure2(d)
p
p
⟶
p
p
f
0
980
s
N
N
=
29.1 GeV
0.971
±
0.001
1.53
±
0.10
13.5
±
0.7
56
/
2
p
p
⟶
p
p
f
1
1285
1.023
±
0.001
2.00
±
0.10
22.0
±
0.5
12
/
2
p
p
⟶
p
p
f
0
1370
1.036
±
0.002
1.30
±
0.10
21.0
±
2.0
132
/
2
p
p
⟶
p
p
f
1
1420
1.049
±
0.003
2.10
±
0.20
16.0
±
1.5
21
/
2
p
p
⟶
p
p
f
0
1500
1.065
±
0.001
2.30
±
0.20
16.0
±
1.0
20
/
2
Figure3(a)
γ
∗
p
⟶
J
/
ψ
p
Q
2
=
3.1 GeV2
1.678
±
0.001
1.25
±
0.05
28.0
±
1.0
5
/
−
6.8
1.678
±
0.001
1.29
±
0.02
26.0
±
3.0
12
/
−
6.8
1.678
±
0.001
1.29
±
0.02
26.0
±
3.0
4
/
−
16.0
1.678
±
0.001
1.31
±
0.02
24.0
±
1.0
12
/
−
Figure3(b)
γ
p
⟶
J
/
ψ
p
Q
2
=
0.05 GeV2
1.700
±
0.001
1.20
±
0.02
29.0
±
0.5
64
/
−
3.2
1.700
±
0.001
1.30
±
0.02
27.0
±
0.7
10
/
−
7.0
1.700
±
0.001
1.40
±
0.03
25.0
±
2.0
5
/
−
22.4
1.700
±
0.001
1.50
±
0.01
23.0
±
0.1
1
/
4
Figure3(c)
γ
p
⟶
J
/
ψ
p
W
=
45 GeV
1.680
±
0.003
1.38
±
0.10
22.0
±
3.0
19
/
1
55
1.680
±
0.010
1.33
±
0.20
23.0
±
4.0
26
/
1
65
1.680
±
0.010
1.29
±
0.17
24.0
±
4.0
21
/
1
75
1.680
±
0.008
1.27
±
0.10
24.0
±
3.0
21
/
1
85
1.680
±
0.002
1.25
±
0.15
25.0
±
3.0
21
/
1
95
1.680
±
0.002
1.21
±
0.10
26.0
±
3.0
13
/
1
Figure3(d)
γ
p
⟶
J
/
ψ
p
W
=
105 GeV
1.680
±
0.001
1.18
±
0.06
27.0
±
1.0
33
/
1
119
1.680
±
0.001
1.15
±
0.12
28.0
±
1.0
25
/
1
144
1.680
±
0.001
1.12
±
0.11
29.0
±
2.0
22
/
1
181
1.680
±
0.001
1.11
±
0.07
30.0
±
1.0
41
/
1
251
1.680
±
0.001
1.07
±
0.07
31.0
±
3.0
13
/
1Figure2 presents the differential cross-section in squared momentum transfer, dσ/d∣t∣, of (a) γp⟶ωp, (b) γ∗p⟶ϕp and γ∗p⟶ϕY, (c) γ∗p⟶ϕp, and (d) pp⟶ppV (V=f0980, f11285, f01370, f11420, and f01500) produced in (a–c) ep and (d) pp collisions in (a) 70GeV<W<90GeV, at (b, c) W=75 GeV, and at (d) proton-proton center-of-mass energy per nucleon pair sNN=29.1 GeV. The experimental data points from (a, c) exclusive, (b) nonexclusive, and (d) exclusive productions are measured by the ZEUS [40, 41], H1 [37], and WA102 Collaborations [43, 44], respectively, with different Q2 for only Figure 2(b) (Q2=3.3, 5, 6.6, and 15.8 GeV2) and Figure 2(c) (Q2=2.4, 3.6, 5.2, 6.9, 9.2, 12.6, and 19.7 GeV2). Similar to Figure 1, the data points are fitted by the Monte Carlo calculations based on Equation (8). The values of parameters are listed in Tables 1–3 with χ2/ndof. One can see that in most cases, the calculations based on Equation (8) with Equations (2)–(4) for p3T can fit approximately the experimental data measured by the H1 and ZEUS Collaborations.Figure 2
The differential cross-section in squared momentum transfer of (a)γp⟶ωp, (b) γ∗p⟶ϕp and γ∗p⟶ϕY, (c) γ∗p⟶ϕp, and (d) pp⟶ppV (V=f0980, f11285, f01370, f11420, and f01500) produced in (a–c) ep and (d) pp collisions in (a) 70GeV<W<90GeV, at (b, c) W=75 GeV, and at (d) sNN=29.1 GeV. The experimental data points from (a, c) exclusive, (b) nonexclusive, and (d) exclusive productions are measured by the ZEUS [40, 41], H1 [37], and WA102 Collaborations [43, 44], respectively, with different Q2 for only (b) and (c). Similar to Figure 1, the data points are fitted by the Monte Carlo calculations based on Equation (8).
(a)
(b)
(c)
(d)Figure3 displays the differential cross-section in squared momentum transfer, dσ/d∣t∣, of (a) γ∗p⟶J/ψp and (b–d) γp⟶J/ψp produced in ep collisions at (a) W=90 GeV, in (b) 40GeV<W<160GeV, and at (c, d) Q2=0.05 GeV2. The experimental data points from (a) exclusive and (b–d) nonexclusive productions are measured by the ZEUS [42] and H1 Collaborations [38], respectively, with (a) Q2=3.1, 6.8 averaged in 2–100, 6.8 averaged in 5–10, and 16 GeV2 and (b) Q2=0.05, 3.2, 7.0, and 22.4 GeV2, as well as with (c) W=45, 55, 65, 75, 85, and 95 GeV and (d) W=105, 119, 144, 181, and 251 GeV. Similar to Figures 1 and 2, the data points are fitted by the Monte Carlo calculations based on Equation (8). The values of parameters are listed in Tables 1–3 with χ2/ndof. One can see that in most cases, the calculations based on Equation (8) with Equations (2)–(4) for p3T can fit approximately the experimental data measured by the H1 and ZEUS Collaborations.Figure 3
The differential cross-section in squared momentum transfer of (a)γ∗p⟶J/ψp and (b–d) γp⟶J/ψp produced in ep collisions at (a) W=90 GeV, in (b) 40GeV<W<160GeV, and at (c, d) Q2=0.05 GeV2. The experimental data points from (a) exclusive and (b–d) nonexclusive productions are measured by the ZEUS [42] and H1 Collaborations [38], respectively, with different Q2 marked in panels (a) and (b), as well as with different W marked in (c) and (d), where in (a), the first and second Q2=6.8 GeV2 are averaged from the ranges of Q2=2–100 and 5–10 GeV2, respectively. Similar to Figures 1 and 2, the data points are fitted by the Monte Carlo calculations based on Equation (8).
(a)
(b)
(c)
(d)From the above comparisons, we see that some fits have largeχ2 compared to ndof, corresponding to low confidence levels. The parameters obtained from these fits are not representing the data well. We would like to say here that these values are used only for the qualitative description of the data tendencies, but not the quantitative interpretation of the data size. In some cases, ndof<1, which means that there were at least as many parameters as data points. This means that a perfect fit should have been found. However, this was not the case here. The reason is that we have used given functions, but not any function such as a polynomial.


## 3.2. Tendency of Parameters

The dependencies of energyE1 of particle 1 on rest mass m0 of particle 3 for different two-body reactions are given in Figure 4, where Figures 4(a)–4(c) correspond to the results from the Erlang distribution, Tsallis distribution, and Hagedorn function, respectively. The types of reactions are marked in the panels. Different symbols represent the results from different reactions or collaborations. One can see that the production of particle 3 with larger m0 needs the participation of particle 1 with larger E1.Figure 4
The dependencies ofE1 on m0 for different two-body reactions which are marked in the panels. (a–c) correspond to the results from the Erlang distribution, Tsallis distribution, and Hagedorn function, respectively.
(a)
(b)
(c)The tendency ofE1 versus m0 presented in Figure 4 is natural due to the conservation of energy. The results from the three distributions or functions are almost the same, if not equal to each other, due to the same experimental data considered. In fact, E1 should be a fixed value for a given reaction in the present work. However, because different selections such as different Q2 and W are used in experiments, E1 has a slight shift from the peak value. Thus, we may regard E1 as a parameter and obtain it from the fits.The dependencies of (a)pT, (b) Ti, (c) T0, (d) n, (e) p0, and (f) n0 on average squared photon virtuality Q2 for different two-body reactions are shown in Figure 5. The types of reactions are marked in the panels. Different symbols for different reactions represent the parameter values extracted from Figures 1–3 and listed in Tables 1–3, where the Erlang distribution, Tsallis distribution, and Hagedorn function in the ranges of available data are used. In particular, pT=nspt from Table 1 and T0=T from Table 2. One can see that pT, Ti, T0, and p0 increase generally with increases in Q2, and n and n0 decrease significantly with an increase in Q2.Figure 5
The dependencies of (a)pT, (b) Ti, (c) T0, (d) n, (e) p0, and (f) n0 on Q2 for different two-body reactions. The symbols represent the parameter values extracted from Figures 1–3 and listed in Tables 1–3. Here, pT=nspt from Table 1 and T0=T from Table 2.
(a)
(b)
(c)
(d)
(e)
(f)Because ofQ2 being a reflection of a hard scale of reaction, this is natural that a harder scale results in a higher excitation degree and then a larger pT, Ti, and T0. In most cases, one can see a large enough n or n0. This means that q is close to 1 and the reaction systems stay in an approximate equilibrium state. At a harder scale, the degree of equilibrium decreases due to more disturbance to the equilibrated residual partons in the target particle. Then, one has a larger q and smaller n or n0 when compared with those at the softer scale.Figure6 shows the excitation functions of related parameters, i.e., the dependencies of (a) pT, (b) Ti, (c) T0, (d) n, (e) p0, and (f) n0 on the photon-proton center-of-mass energy W for γp⟶J/ψp reactions. The symbols represent the parameter values extracted from Figure 3 and listed in Tables 1–3. Again, pT=nspt from Table 1 and T0=T from Table 2. One can see that pT, Ti, T0, and p0 decrease with an increase in W, and n and n0 increase with an increase in W.Figure 6
The dependencies of (a)pT, (b) Ti, (c) T0, (d) n, (e) p0, and (f) n0 on W for γp⟶J/ψp reactions. The symbols represent the parameter values extracted from Figure 3 and listed in Tables 1–3. Here, pT=nspt from Table 1 and T0=T from Table 2.
(a)
(b)
(c)
(d)
(e)
(f)Inγp⟶J/ψp reactions, at a higher center-of-mass energy, the incident photon has a higher energy. Although the emitted J/ψ also has a higher energy, it is more inclined to have a smaller angle. As a comprehensive result, the transverse momentum of J/ψ is smaller, and then, Ti and T0, which are obtained from the transverse momentum, are also smaller. In addition, larger n and n0 at a higher collision energy means more equilibrium due to the shorter collision time and then less disturbance to the equilibrated residual partons in the target particle. This situation is different from nucleus-nucleus collisions in which a cold or spectator nuclear effect has to be considered.In fact, in nucleus-nucleus collisions, secondary cascade collisions may happen among produced particles and spectator nucleons. The secondary collisions may cause the emission angle to increase and then the transverse momentum to increase. The effect of secondary collisions is more obvious or nearly saturated at a higher energy. In nucleus-nucleus collisions at a lower energy, the system approaches equilibrium more easily due to a longer interaction time. Conversely, at a higher energy, the system does not approach equilibrium more easily due to the shorter interaction time for secondary collisions.


## 3.3. Further Discussion

Before the summary and conclusions, we would like to point out that the concept of temperature used in the present work is valid. Generally, the concept of temperature is used in a large system with multiple particles, which stays in an equilibrium state or approximate (local) equilibrium state. From the macroscopic point of view, the systems ofγ∗p, γp, and pp reactions are indeed small. However, we know that there are lots of events under the same condition in the experiments. These events obey the law of grand canonical ensemble in which the concept of temperature is applicable.Because the same experimental condition is used in statistics, lots of events are in equilibrium if they consist of a large statistical system which can be described by the grand canonical ensemble. Particles in the large statistical system obey the same distribution law such as the same transverse momentum distribution. From the statistical point of view, particle productions in high-energy collisions are a statistical behavior, and the temperature reflects the width of distribution. The higher the temperature is, the wider the distribution is.The temperature is also a reflection of the average kinetic energy based on a large statistical system or a single particle. For a single particle, if the distribution law of kinetic energies or transverse momenta is known, the temperature of emission source or interacting system is known, where the source or system means the large thermal source from the ensemble. Generally, we say the temperature of source or system, not saying the temperature of a given particle, from the point of view of statistical significance of temperature. Based on the temperature, we may compare the experimental spectra of different particles in different experiments.However, different methods have used different distributions or functions, i.e., different “thermometers.” To unify these “thermometers” or to find transformations among them, one has to perform quite extensive analysis. Although one may use as far as possible the standard distribution such as the Boltzmann, Fermi-Dirac, or Bose-Einstein distribution to fit the experimental spectra, it is regretful that a single standard distribution cannot fit the experimental spectra very well in general. Naturally, one may use a two-, three-, or even multicomponent standard distribution to fit the experimental spectra, though more parameters are introduced.In fact, the two-, three-, or multicomponent standard distribution can be fitted satisfactorily by the Tsallis distribution withq>1, because the standard distribution is narrower than the Tsallis distribution [68]. In particular, the standard distribution is equivalent to the Tsallis distribution with q=1. It is natural to use the Tsallis distribution to replace the standard distribution. That is, one may use the Tsallis distribution with q>1 to fit the experimental spectra and obtain the temperature, though the Tsallis temperature is less than the standard one.As mentioned in the first section and discussed above, some distributions applied in a large collision system can be also applied in a small collision system due to the universality, similarity, or common characteristics existing in high-energy collisions [27–36]. Based on the same reason, some statistical or hydrodynamic models applied in the large system should be also applied in the small system. Of course, lots of events are needed in experiments and high statistics is needed in calculation if performing a Monte Carlo code.


## 4. Summary and Conclusions

In summary, the differential cross-section in the squared momentum transfer ofρ, ρ0ω, ϕ, f0980, f11285, f01370, f11420, f01500, and J/ψ produced in γ∗p, γp, and pp collisions has been analyzed by the Monte Carlo calculations in which the Erlang distribution, Tsallis distribution, and Hagedorn function (inverse power law) are separately used to describe the transverse momentum spectra of the emitted particles. In most cases, the model results are approximately in agreement with the experimental data measured by the H1, ZEUS, and WA102 Collaborations. In some cases, the fits show qualitatively the data tendencies. The values of the initial- and final-state temperatures and other related parameters are extracted from the fitting process. The squared photon virtuality Q2 and center-of-mass energy W-dependent parameters are obtained.With an increase inQ2, the quantities pT, Ti, T0, and p0 increase generally, and the quantities n and n0 decrease significantly. Q2 is a reflection of a hard scale of reaction. A harder scale results in a higher excitation degree and then a larger pT, Ti, and T0. In most cases, the reaction system can be regarded as an equilibrium state. At a harder scale (larger Q2), the degree of equilibrium decreases due to more disturbance to the equilibrated residual partons in the target particle, though the degree of excitation is high.With the increase ofW, the quantities pT, Ti, T0, and p0 decrease, and the quantities n and n0 increase. In γp⟶J/ψp reactions at a high energy, the emitted J/ψ is more inclined to have a small angle and hence small pT, Ti, and T0. In addition, the system stays in a state with a higher degree of equilibrium at high energy due to less disturbance to the equilibrated residual partons in the target particle. This situation is different from nucleus-nucleus collisions in which the influence of a cold or spectator nuclear effect is existent.



---

*Source: 6677885-2021-06-17.xml*