Transverse Momentum and Pseudorapidity Spectrum of the Top Quark, Lepton, and b Jet in Proton-Proton Collisions at LHC

We study the transverse momentum and pseudorapidity spectrum of the top quark and their decay products, the t t 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 (T), the nonextensivity of the process (n), and the width (σ) of pseudorapidity distribution, are extracted.


Introduction
The top quark is an interesting particle. It was first found by the Tevatron detector of Fermi National Accelerator Laboratory in 1995 [1][2][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 (p T ) and (pseudo)rapidity (η/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 ffiffi s p = 13 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 threesource 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.

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][9][10], the inverse power law [11][12][13][14], the Schwinger mechanism [9,10,15,16], Lévy distribution [16], and the Tsallis statistics [17][18][19][20][21][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 p T region or the relative low p T region, people could use the Tsallis-Pareto-type function to fit the experimental data well  -(e) Figure 1: Transverse momentum distribution of (a) the top quark (p t T ), (b) the top antiquark (p t T ), (c) the top quark or top antiquark with the largest p T (p t T (leading)), (d) the top quark or top antiquark with the second-largest p T (p t T (trailing)), and (e) the top quark in the rest frame of the t t system (p t T (t tRF)) at the parton level in the full phase space produced in pp collisions at ffiffi s p = 13 TeV. The solid circles represent the experimental data of the CMS Collaboration in literature [7]; the curves are our results calculated by the Tsallis-Pareto-type function.  Figure 2: Rapidity distributions of (a) the top quark (y t ), (b) the top antiquark (y t ), (c) the top quark or top antiquark with the largest p T (y t (leading)), (d) the top quark or top antiquark with the second-largest p T (y t (trailing)), and (e) the t t system (y t t ) at the parton level in the full phase space produced in pp collisions at ffiffi s p = 13 TeV. The solid circles represent the experimental data of the CMS Collaboration in literature [7]; the curves are our results calculated by the three-Gaussian functions.     Figure 4: The same as Figure 2 but showing the results of (a) y t , (b) y t , (c) y t (leading), (d) y t (trailing), and (e) y t t at the particle level in the fiducial phase space. 5 Advances in High Energy Physics [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 m 0 is the rest mass of each particle; C and m T are given by T and n are free parameters. T means the mean effective temperature of the interacting system, and it is connected with the average particle energy. n 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 (T and n) by using the Tsallis-Pareto-type function to fit the p T spectra.

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 ffiffi s p = 13 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 highenergy collisions is contributed by the three emission sources. The three emission sources are a central source (C), a target source (T), and a projectile (P). 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][24][25]: where N 0 is the normalization constant, y is rapidity, σ denote the width of rapidity distribution, and X 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: Advances in High Energy Physics A given parameter k is the contribution of each emission source, k T + k C + k P = 1. For the symmetric collision, we think k T = k P . Because of y ≈ η 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.

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 b 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 (t), (b) the top antiquark ( t), (c) the top quark or top antiquark with

Figure
Type   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 (n, T, k, y P , σ) extracted from the transverse momentum and rapidity cross sections and χ 2 /dof are given in Tables 1 and 2. We could find that the values ofnandToft(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 y P are the same in the scope of consideration. The values of k, σðCÞ, and σðPÞ are basically the same within the margin of error in the t, t, t (leading), and t (trailing), but these three parameter values decreased significantly in the rest frame of the t t system. Figures 5 and 6 show the transverse momentum and rapidity cross sections of (a) the lepton (l), (b) the antilepton ( l), (c) the lepton or antilepton with the largest p T (l (leading)), and (d) the lepton or antilepton with the secondlargest p T (l (trailing)), and Figure 5  Generally, the initial temperature (T i ) and mean transverse momentum (hp T i) 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][28][29], T i can be described by the ratio of root-mean-square p T to ffiffi ffi 2 p ( ffiffiffiffiffiffiffiffiffiffiffiffi ffi hp 2 T i/2 p ) approximately. So, we calculated the values of h p T i and ffiffiffiffiffiffiffiffiffiffiffiffi ffi hp 2 T i/2 p by the curves in Figures 1, 3, 5, and 7 and listed them in Table 3. Figures 9 and 10 show the hp T i and ffiffiffiffiffiffiffiffiffiffiffiffi ffi hp 2 T i/2 p of each particle. From Figures 9 and 10, we could observe that thehp T iand ffiffiffiffiffiffiffiffiffiffiffiffi ffi hp 2 T i/2 p of 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 T, n, and σ values listed in Tables 1 and 2 in Figure 17. In the panel, the symbols represent the values of T, n, and σ, and the lines are linear fitting functions. The intercepts, slopes, and χ 2 /dof corresponding to the lines are listed in Table 4. From Figure 17 and Table 4, we found that the values of T 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]. σðCÞ shows to be increased with the collision