#### Abstract

The differential cross-section in squared momentum transfer of , , , , , , , , , and produced in high-energy virtual photon-proton (), photon-proton (), and proton-proton () 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.

#### 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 , quark-hadron transition temperature , chemical freeze-out temperature , kinetic freeze-out or final-state temperature (“confinement” temperature) or , and effective temperature or , etc. In this work, we emphatically discuss the initial- and final-state temperatures, though other types of temperatures are also important.

The initial temperature 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 can be extracted by fitting the transverse momentum 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 can be obtained from the same function.

The final-state temperature 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 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 , but the effective temperature 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 , average transverse momentum , 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 index which describes the degree of equilibrium can be extracted from the Tsallis distribution [22, 23] considering the particle mass. Meanwhile, 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 of , , , and produced in virtual photon-proton () collisions and and produced in photon-proton () collisions, as well as , , , , and produced in proton-proton () 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 -th parton (or parton-like) is assumed to contribute to the transverse momentum to be which obeys an exponential distribution with the average which is -ordinal number independent. We have the probability density function obeyed by to be

The average reflects the excitation degree of contributor parton and can be regarded as the effective temperature .

The contribution of all partons to is the sum of various . The distribution of is then the convolution of exponential functions [19–21]. We have the distribution (the probability density function of ) of final-state particles to be the Erlang distributionwhere denotes the number of all considered particles and has an average of . Equation (2) is naturally normalized to be 1. In Equation (2), there are two free parameters, and .

##### 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 formwhere is the normalization constant, is the transverse mass, is the rest mass, , and is the entropy index [22, 23]. Equation (3) is valid only at midrapidity () which results in and the particle energy .

In Equation (3), a large corresponds to a that is close to 1, and the source or system approaches to equilibrium. The larger the parameter is, the closer to 1 the entropy index is, with the source or system being at a higher degree of equilibrium. There is no exact minimum or maximum [22–25] which is a limit for approximate equilibrium. Empirically, in the case of or which is 25% more than 1 (even or 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 being large enough.

##### 2.3. The Hagedorn Function

The Hagedorn function [24] is an inverse power law which has the probability density function of to bewhere is the normalization constant, is a free parameter which is similar to in the Tsallis distribution [22, 23], and is a free parameter which is similar to the product of in the Tsallis distribution. Note here that it appears as if is a perfect liquid-like relation; however, is a transverse momentum and 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 which can be neglected. Generally, at high , we may neglect , observing the two distributions being very similar to each other. At low , the two distributions have obvious differences due to nonignorable . To build a connection with the entropy index , we have . To build a connection with the effective temperature , we have .

##### 2.4. The Squared Momentum Transfer

In the center-of-mass reference frame, in a two-body reaction or in a two-body-like reaction, it is supposed that particle 1 is incident along the direction and particle 2 is incident along the opposite direction. In addition, particle 3 is emitted with angle relative to the direction and particle 4 is emitted along the opposite direction. According to Ref. [26], three Mandelstam variables are defined aswhere , , , and are four-momenta of particles 1, 2, 3, and 4, respectively.

In the Mandelstam variables, slightly varying the form, is the center-of-mass energy, is the squared momentum transfer between particles 1 and 3, and is the squared momentum transfer between particles 1 and 4. Conveniently, let be the squared momentum transfer between particles 1 and 3. We have

Here, and , and , and and are the energy, momentum, and rest mass of particles 1 and 3, respectively. In particular, is the transverse momentum of particle 3, which is referred to be perpendicular to the direction.

As the energy of incoming photon in the center-of-mass reference frame of the reaction, in Equation (8) should be a fixed value. However, has a slight shift from the peak value due to different experiments and selections. To obtain a good fit, we treat as a parameter which is the same or has small difference in the same/similar reactions. obeys one of Equations (2)–(4) and obeys an isotropic assumption in the center-of-mass reference frame, which will be discussed later in this section. To obtain , we may perform the Monte Carlo calculations. Note that we may calculate from two particles, i.e., particles 1 and 3, but not from one particle. Instead, for one calculation, means the squared momentum transfer in an event. For many calculations, 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 and , 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 , where denotes the four-momentum of the photon. (2) The center-of-mass energy or , i.e., . Let denote the Bjorken scaling variable; one has .

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

According to Refs. [45–47], in a color string percolation approach, the initial temperature can be estimated aswhere is the root mean square of and . 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 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 to be 0.6 [48], a higher can be obtained by multiplying a revised factor, , in Equation (9).

The extraction of final-state temperature is more complex than that of the initial temperature . Generally, one may introduce the transverse flow velocity in the considered function and obtain and simultaneously [49–57], in which the effective temperature no longer appears. Alternatively, the intercept in versus is assumed to be [50, 58–63], and the slope in versus is assumed to be [62–66], where 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 the , , and 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 in the considered processes. Here, appears as that in Equation (3). Meanwhile, can be also approximated by in Equation (1) and 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 in Equation (3) as , a smaller can be obtained if the flow effect is considered.

As mentioned above, in Equation (1) and the Erlang distribution, and . We have , so this would mean that is basically encoded in , the squared variance of in the distribution. This also means that and are related through . It is understandable, because they reflect the violent degrees of collisions at different stages. Generally, ; 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, for and 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 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 distribution are difficult to use in Equation (8) to obtain the distribution. Instead, we may perform the Monte Carlo calculations. Let and be random numbers distributed evenly in [0,1]. To use Equation (8), we have to know the changeable (i.e., ) and . Other quantities such as , , and in the equation are fixed, though is treated by us as a parameter with slight variety.

To obtain a concrete value of , we need one of Equations (2)–(4). Solving the equationwhere , 2, and 3 and is a small shift relative to ; we may obtain concrete . It seems that Equation (10) directly means that the integral of , , and is the same for the 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, for , we have a simpler expression. Let us solve the equation

We havedue to Equation (1) being used, where in Equation (12) replaced because both of them are random numbers in [0,1]. The simpler expression isdue to being the sum of random .

To obtain a concrete value of , we need the functionwhich is obeyed by under the assumption of isotropic emission in the center-of-mass reference frame. Solving the equationwe havewhich is needed by us.

According to the concrete values of and , and using other quantities, the value of can be obtained from Equation (8). After repeating the calculations many times, the distribution of is obtained statistically. Based on the method of least squares, the related parameters are obtained naturally. Meanwhile, can be obtained from Equation (9). and 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

Figure 1 shows the differential cross-section in squared momentum transfer, , of (a) , (b) , and (c) produced in electron-proton () collisions at photon-proton center-of-mass energy (a, b) GeV and (c) GeV, where denotes the cross-section and 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) , 6.6, 11.5, 17.4, and 33.0 GeV^{2}; (b) , 6.6, and 15.8 GeV^{2}; and (c) , 5.0, 7.8, 11.9, 19.7, and 41.0 GeV^{2}. 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 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 , , , , , , , and are listed in Tables 1–3 with 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 can fit approximately the experimental data measured by the H1 and ZEUS Collaborations.

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Figure 2 presents the differential cross-section in squared momentum transfer, , of (a) , (b) and , (c) , and (d) (, , , , and ) produced in (a–c) and (d) collisions in (a) , at (b, c) GeV, and at (d) proton-proton center-of-mass energy per nucleon pair 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 for only Figure 2(b) (, 5, 6.6, and 15.8 GeV^{2}) and Figure 2(c) (, 3.6, 5.2, 6.9, 9.2, 12.6, and 19.7 GeV^{2}). 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 /ndof. One can see that in most cases, the calculations based on Equation (8) with Equations (2)–(4) for can fit approximately the experimental data measured by the H1 and ZEUS Collaborations.

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Figure 3 displays the differential cross-section in squared momentum transfer, , of (a) and (b–d) produced in collisions at (a) GeV, in (b) , and at (c, d) GeV^{2}. 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) , 6.8 averaged in 2–100, 6.8 averaged in 5–10, and 16 GeV and (b) , 3.2, 7.0, and 22.4 GeV^{2}, as well as with (c) , 55, 65, 75, 85, and 95 GeV and (d) , 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 /ndof. One can see that in most cases, the calculations based on Equation (8) with Equations (2)–(4) for can fit approximately the experimental data measured by the H1 and ZEUS Collaborations.

**(a)**

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From the above comparisons, we see that some fits have large 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, , 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 energy of particle 1 on rest mass 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 needs the participation of particle 1 with larger .

**(a)**

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The tendency of versus 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, should be a fixed value for a given reaction in the present work. However, because different selections such as different and are used in experiments, has a slight shift from the peak value. Thus, we may regard as a parameter and obtain it from the fits.

The dependencies of (a) , (b) , (c) , (d) , (e) , and (f) on average squared photon virtuality 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, from Table 1 and from Table 2. One can see that , , , and increase generally with increases in , and and decrease significantly with an increase in .

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Because of 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 , , and . In most cases, one can see a large enough or . This means that 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 and smaller or when compared with those at the softer scale.

Figure 6 shows the excitation functions of related parameters, i.e., the dependencies of (a) , (b) , (c) , (d) , (e) , and (f) on the photon-proton center-of-mass energy for reactions. The symbols represent the parameter values extracted from Figure 3 and listed in Tables 1–3. Again, from Table 1 and from Table 2. One can see that , , , and decrease with an increase in , and and increase with an increase in .

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In reactions, at a higher center-of-mass energy, the incident photon has a higher energy. Although the emitted also has a higher energy, it is more inclined to have a smaller angle. As a comprehensive result, the transverse momentum of is smaller, and then, and , which are obtained from the transverse momentum, are also smaller. In addition, larger and 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 of , , and 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 with , because the standard distribution is narrower than the Tsallis distribution [68]. In particular, the standard distribution is equivalent to the Tsallis distribution with . It is natural to use the Tsallis distribution to replace the standard distribution. That is, one may use the Tsallis distribution with 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 of , , , , , , , , and produced in , , and 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 and center-of-mass energy -dependent parameters are obtained.

With an increase in , the quantities , , , and increase generally, and the quantities and decrease significantly. is a reflection of a hard scale of reaction. A harder scale results in a higher excitation degree and then a larger , , and . In most cases, the reaction system can be regarded as an equilibrium state. At a harder scale (larger ), 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 of , the quantities , , , and decrease, and the quantities and increase. In reactions at a high energy, the emitted is more inclined to have a small angle and hence small , , and . 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.

#### Data Availability

This manuscript has no associated data or the data will not be deposited. (Authors’ comment: the data used to support the findings of this study are included within the article and are cited at relevant places within the text as references.)

#### Ethical Approval

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

#### Disclosure

The funding agencies have no role in the design of the study; in the collection, analysis, or interpretation of the data; in the writing of the manuscript; or in the decision to publish the results.

#### Conflicts of Interest

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

#### Acknowledgments

The work of Q.W. and F.H.L. was supported by the National Natural Science Foundation of China under Grant Nos. 12047571, 11575103, and 11947418; the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (STIP) under Grant No. 201802017; the Shanxi Provincial Natural Science Foundation under Grant No. 201901D111043; and the Fund for Shanxi “1331 Project” Key Subjects Construction. The work of K.K.O. was supported by the Ministry of Innovative Development of the Republic of Uzbekistan within the fundamental project on analysis of open data on heavy-ion collisions at RHIC and LHC.