#### Abstract

We study the transverse momentum and pseudorapidity spectrum of the top quark and their decay products, the system, and the total number of jets in proton-proton (pp) collisions at 13 TeV by using the Tsallis-Pareto-type function and the three-source Landau hydrodynamic model, respectively. The related parameters, such as the effective temperature of the interacting system (), the nonextensivity of the process (), and the width () of pseudorapidity distribution, are extracted.

#### 1. Introduction

The top quark is an interesting particle. It was first found by the Tevatron detector of Fermi National Accelerator Laboratory in 1995 [1–3]. As the heaviest particle in the Standard Model, the top quark is very different from the other quark. Its biggest mass is one of the hotspots for physicists. And it is the only one quark which will decay before it becomes hadron. These unique properties make top quark occupy a very important position in particle physics.

The Large Hadron Collider (LHC) has created a lot of top quark events since 2010. Physicists could study the top quark in detail with the help of the LHC. By studying the final state particles produced in high-energy collisions, physicists could obtain some information about the evolution of the collision system. The spectra of transverse momenta () and (pseudo)rapidity (/*y*) are important quantities measured in experiments. Many models and functions are used to describe the transverse momentum and pseudorapidity spectra.

In this paper, we use the Tsallis-Pareto-type function [4, 5] and the three-source Landau hydrodynamic model [6] to describe the transverse momentum and pseudorapidity spectra of the top quark, lepton, and bottom quark produced in proton-proton (pp) collisions at the center of mass energy TeV measured at the parton level and particle level, in the full phase space and in the fiducial phase by CMS Collaboration at the LHC [7]. The values of related parameters are extracted and analyzed.

The present paper is organized as follows. We briefly introduce the Tsallis-Pareto-type function and the three-source Landau hydrodynamic model in Section 2. The result of comparisons with experimental data is given in Section 3. Moreover, we outlined the conclusions in Section 4.

#### 2. The Model and Formalism

##### 2.1. The Tsallis-Pareto-Type Function

There are some models and functions that could be used to analyze the transverse momentum distribution, such as the Erlang distribution [8–10], the inverse power law [11–14], the Schwinger mechanism [9, 10, 15, 16], Lévy distribution [16], and the Tsallis statistics [17–22]. In a lot of models and formulas, the Tsallis-Pareto-type function is a good choice for transverse momentum spectra. Whether in the relative high region or the relative low region, people could use the Tsallis-Pareto-type function to fit the experimental data well [5]. So, in this work, we chose the Tsallis-Pareto-type function to describe the transverse momentum spectra. The Tsallis-Pareto-type function can be written as [5] where is the rest mass of each particle; and are given by

and are free parameters. means the mean effective temperature of the interacting system, and it is connected with the average particle energy. reveals the nonextensivity of the process, and it denotes the departure of the spectra from the Boltzmann distribution [5]. We could obtain values of the two free parameters ( and ) by using the Tsallis-Pareto-type function to fit the spectra.

##### 2.2. The Three-Source Landau Hydrodynamic Model

The three-source Landau hydrodynamic model has become a mature theoretical model in research for the pseudorapidity distributions at nuclear collisions. In our previous work, this model has described the experimental data successfully [6]. In the present work, we will use it to fit the pseudorapidity distributions of particles (top quark, lepton, and bottom quark) produced in pp collisions at TeV again. The following is a brief description of the three-source Landau hydrodynamic model.

The source means particle emission source. We think the rapidity distribution of particles produced in high-energy collisions is contributed by the three emission sources. The three emission sources are a central source (), a target source (), and a projectile (). The central source is located at the central of rapidity distribution and covers the whole rapidity range. The target source and projectile source are located at the left and right side of the central source, respectively. And the target source and projectile source are revisions for the central source. For the rapidity distribution of particles produced in each emission source, we can use a Gaussian form of the Landau solution to describe it [23–25]: where is the normalization constant, is rapidity, denote the width of rapidity distribution, and represents the type of emission source.

Actually, the three-source Landau hydrodynamic model can be written as a form of a superposition of three-Gaussian forms of the Landau solution:

A given parameter is the contribution of each emission source, . For the symmetric collision, we think . Because of at very high energy, we could describe the pseudorapidity distributions of lepton by the following formula:

Significantly, as a revision of the central source, the contributions of the target source and projectile source are small.

#### 3. Comparisons with Experimental Data

From the experimental point of view, the particle-level definition is expected to result in decreased uncertainties with respect to a definition at the parton level [26]. So, in this work, we use the same approach to analyze the experimental data.

In this section, we study the transverse momentum and pseudorapidity spectrum of the top quark, lepton, and jet in proton-proton collisions at LHC. Based on the Tsallis-Pareto-type function, we describe the transverse momentum spectra of the related product. And we use the three-source Landau hydrodynamic model to analyze the pseudorapidity and rapidity distribution.

Figures 1 and 2 show the transverse momentum and rapidity cross sections of (a) the top quark (), (b) the top antiquark (), (c) the top quark or top antiquark with the largest ( (leading)), (d) the top quark or top antiquark with the second-largest ( (trailing)), and (e) the top quark in the rest frame of the system ( RF) at the parton level in the full phase space produced in pp collisions at TeV. The same as Figures 1 and 2, but Figures 3 and 4 show the results at the particle level in the fiducial phase space. In these four figures, the solid circles represent the experimental data recorded by the CMS experiment at the LHC and correspond to an integrated luminosity of 35.9 fb^{-1} in 2016 [7]. In Figures 1 and 3, the curves are our results calculated by the Tsallis-Pareto-type function. In Figures 2 and 4, the curves show the calculate results of the three-Gaussian functions. Obviously, our results fit well with the experimental data. One can see that the Tsallis-Pareto-type function and the three-Gaussian functions are very helpful approaches to describe the transverse momentum and rapidity cross sections of particles, respectively. The related parameter values (, , , , ) extracted from the transverse momentum and rapidity cross sections and are given in Tables 1 and 2. We could find that the values ofandof(leading) are the biggest in the five-type particles at the parton and particle level in the full and fiducial phase space. The values of are the same in the scope of consideration. The values of , , and are basically the same within the margin of error in the , , (leading), and (trailing), but these three parameter values decreased significantly in the rest frame of the system.

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Figures 5 and 6 show the transverse momentum and rapidity cross sections of (a) the lepton (), (b) the antilepton (), (c) the lepton or antilepton with the largest ( (leading)), and (d) the lepton or antilepton with the second-largest ( (trailing)), and Figure 5(e) shows the transverse momentum cross sections of the dilepton system () at the particle level in the fiducial phase space produced in pp collisions at TeV. The solid circles and curves represent the experimental data and our calculated results, respectively. One can see that the results calculated by using the theoretical functions are in agreement with the experimental data. The related parameters are extracted and listed in Tables 1 (5(a)–(e)) and 2 (6(a)–(d)). The values of and of the system are bigger than the other particles. The values of , , , and of the considered particles are not different obviously.

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The results of transverse momentum and rapidity cross sections of (a) the jet with the largest ( (leading)) and (b) the jet with the second-largest ( (trailing)) are showed in Figures 7 and 8, and Figure 7(c) shows the transverse momentum cross sections of the system at the particle level in the fiducial phase space produced in pp collisions at TeV. We have extracted the related parameters by describing the transverse momentum and rapidity cross section distributions and listed them in Tables 1 (7(a)–(c)) and 2 (8(a)–(b)). The values of parameters are not in an obvious relationship.

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Generally, the initial temperature () and mean transverse momentum () are important physical quantities to understand the excitation degree of the interacting system. The two physical quantities are not dependent on the selected models and functions; they are only dependent on the experimental data. According to the literatures [27–29], can be described by the ratio of root-mean-square to () approximately. So, we calculated the values of and by the curves in Figures 1, 3, 5, and 7 and listed them in Table 3. Figures 9 and 10 show the and of each particle. From Figures 9 and 10, we could observe that theandof each particle are big. This phenomenon means that the excitation degree of the interacting system is violent.

In order to find the relationship between the free parameter and collision energy, we have extracted the related parameters from the transverse momentum and rapidity cross sections at 7 TeV and 8 TeV (Figures 11–16), together. In Figures 11 and 12, the solid circles represent the experimental data recorded by the CMS experiment at 7 TeV and correspond to an integrated luminosity of 5.0 fb^{-1} in 2011 [30]. In Figures 13–16, the solid circles represent the experimental data recorded by the CMS experiment at 8 TeV and correspond to an integrated luminosity of 19.7 fb^{-1} in 2012 [31]. From the transverse momentum and rapidity cross sections of final state particles, we extracted the values of free parameters and listed them in Tables 1 and 2. And we plot the , , and values listed in Tables 1 and 2 in Figure 17. In the panel, the symbols represent the values of , , and , and the lines are linear fitting functions. The intercepts, slopes, and corresponding to the lines are listed in Table 4. From Figure 17 and Table 4, we found that the values of show a slight increase with the collision energy increased; this phenomenon may be affected by QGP. It can be explained by the influence of jet quenching effect (quickly energy loss) in the case of high-energy quark and gluon jets penetrating through the dense deconfined matter [32]. shows to be increased with the collision energy increased. The values of about and do not show obvious change, and the values of about decreased when the collision energy increased. Besides, we found that there is no linear relationship and , so the values of for this relationship are large which are listed in Table 4.

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#### 4. Conclusion

We summarize here our main observations and conclusions. (a)We use the Tsallis-Pareto-type function to describe the transverse momentum cross-section spectrum of different cross sections in pp collisions at 7, 8, and 13 TeV. The values of the effective temperature of the interacting system () and the nonextensivity of the process () parameters are extracted and listed in Table 1(b)The rapidity cross sections of the spectrum of different cross sections in pp collisions at 7, 8, and 13 TeV are analyzed by the three-source Landau hydrodynamic model. We have obtained the values of the contribution of central emission source () and the width of rapidity distribution (, ) and calculated the values of . The values of related parameters and are given in Table 2. In fact, because of the relationship , , and (for the symmetric collision), we can calculate the contributions of the target and projectile source and the width of rapidity distribution of the target source(c)We have plotted the relationship of free parameters and collision energy in Figure 17. For the particles of , , and , shows an increasing trend with the collision energy increased. And shows a slight increase with the collision energy increased; this phenomenon may be affected by QGP(d)As has been mentioned in Section 3, the initial temperature () can be described by approximately. In order to understand the excitation degree of the interacting system, we calculated the values of and by the curves in Figures 1, 3, 5, and 7. We found that the values of and are large; this could mean the high excitation degree of the interacting system

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Ethical Approval

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

#### Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work is supported by Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi under Grant No. 2019L0804, the National Natural Science Foundation of China under Grant No. 11847003, and the university student innovation and entrepreneurship training program of Taiyuan Normal University No. CXCY 2084. This work has been published at http://arxiv.org/abs/2007.12445.