Event patterns extracted from top quark-related spectra in proton-proton collisions at 8 TeV

We analyze the transverse momentum ($p_T$) and rapidity ($y$) spectra of top quark pairs, hadronic top quarks, and top quarks produced in proton-proton ($pp$) collisions at center-of-mass energy $\sqrt{s}=8$ TeV. For $p_T$ spectra, we use the superposition of the inverse power-law suggested by the QCD (quantum chromodynamics) calculus and the Erlang distribution resulting from a multisource thermal model. For $y$ spectra, we use the two-component Gaussian function resulting from the revised Landau hydrodynamic model. The modelling results are in agreement with the experimental data measured at the detector level, in the fiducial phase-space, and in the full phase-space by the ATLAS Collaboration at the Large Hadron Collider (LHC). Based on the parameter values extracted from $p_T$ and $y$ spectra, the event patterns in three-dimensional velocity ($\beta_x$-$\beta_y$-$\beta_z$), momentum ($p_x$-$p_y$-$p_z$), and rapidity ($y_1$-$y_2$-$y$) spaces are obtained, and the probability distributions of these components are also obtained.


Introduction
The top quark is the heaviest particle in the standard model and very different from the other quarks. It is expected that the top quark may have some special characteristics and be related to the new physics beyond the standard model. Therefore, it is very significant to study its characteristics more. The top quark was first found by the Tevatron at the Fermi National Accelerator Laboratory [1][2][3][4], and the successful operation of the CERN Large Hadron Collider (LHC) makes the study of the top quark enter a more precise measurement period. High energy collisions are the only way to study the top quark in experiments. Generally, due to the complexity of the process in an extremely short time, theoretical physicists need to use models to analyze the properties of observables instead of to study the interacting systems directly.
Among the kinematic observables, the transverse momentum (p T ) and rapidity (y) are always the hot topics for the theoretical physicists. Some phenomenological models and formulas were proposed to fit the spectra of p T and y. For p T spectra, many formulas can be used, such as the standard (Fermi-Dirac, Bose-Einstein, or Boltzmann) distribution [5][6][7][8], the Tsallis statistics [8][9][10][11][12][13][14], the inverse power-law [15][16][17], the Er-lang distribution [18], the Schwinger mechanism [19][20][21][22], and the combination of them, while y spectra can be described by the one-, two-, or three-component Gaussian function. Except for these analytic expressions, many models based on the Monte Carlo method have reported their arithmetic solutions on the spectra of p T and y, and other interesting results which contain, but are not limited to, chemical and kinetic freeze-out temperatures, chemical potential, transverse flow velocity, particle ratio, and so forth.
In order to understand the sophisticated collision process and mechanism intuitively, we can use the method of event pattern (particle scatter plot) at kinetic freeze-out stage to obtain the information of interaction system. Using this method, we have analyzed the scatter plots of net-baryons produced in central gold-gold (Au-Au) collisions at BNL Relativistic Heavy Ion Collider (RHIC) energies in three-dimensional momentum (p x -p y -p z ) space, three-dimensional momentum-rapidity (p x -p y -y) space, and three-dimensional velocity (β x -β yβ z ) space [23]; charged particles produced in protonproton (pp) and lead-lead (Pb-Pb) collisions at 2.76 TeV (one of LHC energies) in three-dimensional p x -p yp z space and β x -β y -β z space [24]; as well as Z bosons and quarkonium states produced in pp and Pb-Pb collisions at LHC energies in two-dimensional transverse momentum-rapidity (p T -y) space and three-dimensional β x -β y -β z space [25]. Due to the variousness of produced particles, further analyses on the event patterns of other particles such as the top quarks are needed.
In this paper, we mainly use the superposition of the inverse power-law suggested by the QCD (quantum chromodynamics) calculus [15][16][17] and the Erlang distribution resulted from a multisource thermal model [18] and the two-component Gaussian function resulted from the revised Landau hydrodynamic model [26][27][28][29] to fit the p T and y spectra of the top quark-related products produced in pp collisions at the center-of-mass energy √ s = 8 TeV measured at the detector level, in the fiducial phase-space, and in the full phase-space by the AT-LAS Collaboration at the LHC [30]. The related parameters can be extracted from the fitting. Based on the parameters and using the Monte Carlo method, we can obtain the event patterns in three-dimensional β xβ y -β z , p x -p y -p z , and rapidity (y 1 -y 2 -y) spaces, and the probability distributions of these components can be obtained, too.
The rest part of this paper is structured as followings. A briefly description of the model and method is given in section 2. Then, the results and discussion are presented in section 3. Finally, we summarize our main observations and conclusions in section 4.

The model and method
As the heaviest particle in the standard model, the formation of top quark is expected to be through the hard scattering process among partons (quarks and gluons) with high energy. Viewing the top quark-related p T spectra, one can see a very wide range of distribution. This means that in some cases the spectra can be in fact divided into two parts, one is in the relative high p T region and mainly contributed by the real hard (the harder) scattering process, and the other one is in the relative low p T process and mainly contributed by the not too hard (the hard) scattering process.
For the harder and hard processes, we have to choose a superposition distribution which has two components to describe the p T spectra. In the two components, one is for the harder scattering process and the other one is for the hard scattering process. The relative contributions of the harder scattering process are expected to be different for different products such as the top quark pairs (the tt systems), the hadronic top (hadronic t) quarks (the hadronically decaying top quarks), and the top (t) quarks (the semileptonically and hadronically decaying top quarks).
For the harder scattering process, we can use the inverse power-law which is resulted from the QCD calculus [15][16][17] in high energy collisions to describe the p T spectra. For the hard scattering process, we can use the Erlang distribution which is resulted from a multisource thermal model [18] to describe the p T spectra. Although the Erlang distribution is not sure to be the best choice for the hard scattering process, it is a good one to fit many data. In fact, the Erlang distribution is also used for the soft excitation process. The inverse power-law plays a significant role in the region of relative high p T and the Erlang distribution contributes mainly in the region of relative low p T .
According to the QCD calculus [15][16][17], we have the inverse power-law in the form where A denotes the normalization constant which makes ∞ 0 f 1 (p T )dp T = 1, and p 0 and n are free parameters and influence the value of A.
According to the multisource thermal model [18], the spectra of p T for a given set of data selected in a special condition can be described by the Erlang distribution where p T i and m are free parameters. In particular, p T i denotes the average value of p T i , where i = 1 to m, and m denotes the number of contribution sources which are in fact the participant partons which contribute the same exponential function to p T . Generally, m = 2 or 3 due to only two or three partons taking part in the formation of each particle. The top quark-related p T spectra is a superposition of the inverse power-law and the Erlang distribution. Let k denote the relative contribution of the inverse power-law. Then, the relative contribution of the Erlang distribution is 1 − k. We have the normalized distribution In many cases, the spectra of p T in experiments are presented in terms of non-normalized distribution. To give a comparison with the experimental data, the normalized constant (N pT ) is needed. According to the Landau hydrodynamic model and its revisions [26][27][28][29], we have the y spectrum to be a Gaussian function [28,29] where y C denotes the mid-rapidity (peak position) and σ y denotes the distribution width. In the center-of-mass reference frame, y C = 0 corresponds to the symmetric collisions such as the pp collisions considered in the present work. In many cases, the Gaussian function cannot describe the y spectra very well, we need at least two Gaussian functions for the y spectrum. That is where k B (1 − k B ), y B (y F ), and σ yB (σ yF ) denote respectively the relative contribution ratio, peak position, and distribution width of the first (second) component which distributes in the backward (forward) rapidity region. Due to the symmetry of pp collisions, we have k B = 1 − k B = 0.5, y B = −y F , and σ yB = σ yF . When comparing with experimental data, the normalization constant (N y ) is needed.
In the present work, we use the Monte Carlo method to get the discrete values which are used in the event patterns in three-dimensional β x -β y -β z , p x -p y -p z , and y 1 -y 2 -y spaces. Let R, r i , and R 1−5 denote random numbers distributed evenly in [0, 1]. To get the discrete values, the variable p T in Eq. (1) or in the first component in Eq. (3) obeys the following formula The variable p T in Eq. (2) or in the second component in Eq.
(3) is obtained by As for the variable y in the first and second components in Eq. 5, we have and y = σ y −2 ln R 3 cos(2πR 4 ) + y F respectively. An isotropic emission in the transverse plane results in the azimuthal angle ϕ to be which distributes evenly in [0, 2π].
In this way, we can get the energy E to be where, m 0 is the peak mass in the invariant mass spectrum of the tt systems [30], or the rest mass of top quark in the case of considering the hadronic top quarks or top quarks. The momentum components are and The velocity components are and As for y 1 and y 2 , we use the definitions and for the rapidities in the directions of ox and oy axes respectively.
In the concrete calculation, we need firstly to fit the p T and y spectra to get the values of free parameters and normalization constants. Then, we use the values of free parameters obtained by the fitting in the first step to get the discrete values of different kinds of kinematic variables, and repeat for 1000 times to get the event patterns in three-dimensional β x -β y -β z , p x -p y -p z , and y 1 -y 2 -y spaces. By way of parenthesis, the probability distributions of the velocity, momentum, and rapidity components can be obtained from more discrete values.

Results and discussion
Before describing the comparisons with experimental data, we introduce firstly the meanings of "the detector level", "the fiducial phase-space, and "the full phase-space". According to ref. [30] which is quoted in the present work, "the event selection consists of a set of requirements based on the general event quality and on the reconstructed objects", defined by definite conditions, "that characterize the final-state event topology". Each requirement or condition for quantities of considered event has to be detected, identified, and selected by various types of detectors, which refers to the detector level. The fiducial phase space for the measurements presented in ref. [30] is defined "using a series of requirements applied to particle-level objects close to those used in the selection of the detector-level objects". The full phase space for the measurements presented in ref. [30] is defined "by the set of tt pairs in which one top quark decays semileptonically (including τ leptons) and the other decays hadronically".
On the short-cut process of the rest mass for the tt systems in the cases of measuring the invariant mass spectra by the three requirements which are (i) at the detector level, (ii) in the fiducial phase-space, and (iii) in the full phase-space, we take m 0 = 463.2, 382.5, and 372.5 GeV, respectively, due to the quoted ref. [30]. As for the rest mass of (hadronic) top quark, we take m 0 = 172.5 GeV due to the same literature. It is very important to use a correct rest mass in the extraction of event pattern. In the case of giving the spectra of p T and y, the most important issue is the rest mass. Figure 1 shows the event yields about (a)(c) p T and (b)(d) y of (a)(b) the tt systems and (c)(d) the hadronic top quarks produced in pp collisions at √ s = 8 TeV, where y spectra are presented in terms of absolute values. The symbols represent the experimental data of the ATLAS Collaboration [30] measured in the combined electron and muon selections at the detector level, and the error bars are the combined statistical and systematic uncertainties, where the integral luminosity corresponds to 20.3 fb −1 . The solid curves are our results calculated by using (a) the inverse power-law, (c) the superposition of the inverse power-law and the Erlang distribution, and (b)(d) the two-component Gaussian distribution, respectively. The dashed curve in (a) is the result calculated by using the Erlang distribution with m = 1, which is in fact the exponential distribution for the purpose of comparison. In the calculation, we use the method of least squares to determine the values of parameters. The values of free parameters [p 0 , n, k, p T i , y F (= −y B ), and σ yF (= σ yB )], normalization constants (N pT and N y ), and χ 2 per degree of freedom (χ 2 /dof) are listed in Tables 1-3 for different sets of products and parameters, where k = 1 for the tt systems and m = 3 for the hadronic top quarks.
One can see from Figure 1 and Tables 1-3 that the results calculated by the hybrid model are in agreement with the experimental p T and y spectra of both the tt systems and hadronic top quarks produced in pp collisions at √ s = 8 TeV measured at the detector level by the ATLAS Collaboration at the LHC. In particular, the tt systems show only the contribution of harder scattering process (with k = 1), which results in p T i and m for the tt systems are not available. The hadronic top quarks show mainly the contribution of hard scattering process (with a small k). These render that more collision energies are needed to create the tt system. As a part of the tt system, the hadronic top quark apportioned partly the energy of tt system, which results in a not too hard scattering process. Figure 2 is the same as that for Figure 1, but it shows the fiducial phase-space normalized differential cross-sections for (a)(b) the tt systems and (c)(d) the hadronic top quarks, where σ denotes the cross-section. Figure 3 is also the same as that for Figure 1, but it shows the full phase-space normalized differential crosssections for (a)(b) the tt systems and (c)(d) the top quarks. The corresponding parameters are also listed in Tables 1-3 for different sets of products and parameters, where k = 1 for the tt systems and m = 3 for the hadronic top quarks and top quarks, which are listed only in the captions of Tables 1 and 2, but not in the column.
One can also see from Figures 2 and 3 and Tables 1-3 that the modelling results are in agrement with the experimental data of the fiducial phase-space normalized differential cross-sections for the tt systems and the hadronic top quarks, and the full phase-space normalized differential cross-sections for the tt systems and the top quarks, produced in pp collisions at 8 TeV measured by the ATLAS Collaboration. In particular, for the tt systems in both the fiducial and full phase-spaces, we also have k = 1. For both the hadronic top quarks in the fiducial phase-space and the top quarks in the full phase-space, we also have a small k.
As for the tendencies of the free parameters, one can see from Tables 1-3 that, for both the tt systems and the (hadronic) top quarks, p 0 and n decrease when the experimental requirement changes from the detector level to the fiducial phase-space and then to the full phasespace. Only for the (hadronic) top quarks, k slightly increases, and p T i and m almost do not change when the requirement changes from the detector level to the full phase-space. At the same time, for both the tt systems and the (hadronic) top quarks, both y F and σ yF increase when the requirement changes from the detector level to the full phase-space. These tendencies may           Table 4. Values of the root-mean-squares β 2 x for βx, β 2 y for βy, and β 2 z for βz, as well as the maximum |βx|, |βy|, and |βz| (|βx|max, |βy|max, and |βz|max) corresponding to the scatter plots for different types of products, where the corresponding scatter plots are presented in Figure 4. Both the root-mean-squares and the maximum velocity components are in the units of c.

Figure
Type have no obvious meaning due to little relations among these requirements. However, because these tendencies, we can obtain abundant structures of event patterns. Tables 1-3, the Monte Carlo calculation can be performed and the values of a series of kinematical quantities can be obtained. Thus, we can get different kinds of diagrammatic sketches at the kinetic freeze-out of the interacting system formed in pp collisions. Figures 4-6 give the event patterns which are displayed by the particle scatter plots in the threedimensional β x -β y -β z , p x -p y -p z , and y 1 -y 2 -y spaces, respectively. In these figures, panels (a)-(f) correspond to the results for the tt systems at the detector level, the hadronic top quarks at the detector level, the tt systems in the fiducial phase-space, the hadronic top quarks in the fiducial phase-space, the tt systems in the full phase-space, and the top quarks in the full phasespace, respectively. The total number of particles for each panel is 1000. The blue and red globules represent the contributions of inverse power-law and Erlang distribution respectively. The values of root-mean-squares ( β 2

Based on the parameter values obtained from Figures 1-3 and listed in
x for β x , β 2 y for β y , and β 2 z for β z ,) and the maximum |β x |, |β y |, and |β z | (|β x | max , |β y | max , and |β z | max ) are listed in Table 4. The values of root-mean-squares ( p 2 x for p x , p 2 y for p y , and p 2 z for p z ), and the maximum |p x |, |p y |, and |p z | (|p x | max , |p y | max , and |p z | max ) are listed in Table 5. The values of root-mean-squares ( y 2 1 for y 1 and y 2 2 for y 2 ) and the maximum |y 1 | and |y 2 | (|y 1 | max and |y 2 | max ) are listed in Table 6. The values of these kinematical quantities listed in Tables 4-6 are obtained by higher statistics.
From Figures 4-6 and Tables 4-6, one can see that the event patterns in the three-dimensional β x -β y -β z space for the tt systems in the three requirements are rough cylinders with β 2 x ≈ β 2 y ≪ β 2 z and |β x | max ≈ |β y | max < |β z | max , though little differences among the three requirements are observed. The event patterns for the (hadronic) top quarks in the three re-quirements are rough ellipsoids with the similar relations among these quantities and little differences among the three requirements. An obvious difference between the event patterns for the tt systems and the (hadronic) top quarks are observed due to their different production processes. Meanwhile, both the quantities on the rootmean-squares and the maximums for the tt systems are less than those for the (hadronic) top quarks, and the differences in relative sizes between transverse and longitudinal quantities for the tt systems are larger than those for the (hadronic) top quarks.
The event patterns in the three-dimensional p x -p yp z space for the tt systems in the three requirements are relative thin and very rough ellipsoids with p 2 x ≈ p 2 y ≪ p 2 z and |p x | max ≈ |p y | max ≪ |p z | max , though little differences among the three requirements are observed. The event patterns for the (hadronic) top quarks in the three requirements are relative fat and very rough ellipsoids with the similar relations among these quantities and little differences among the three requirements. An obvious difference between the event patterns for the tt systems and the (hadronic) top quarks are observed. Meanwhile, the transverse quantities for the tt systems are less than those for the (hadronic) top quarks, and the situations of longitudinal quantities are opposite. The differences in relative sizes between transverse and longitudinal quantities for the tt systems are larger than those for the (hadronic) top quarks. The maximum quantities do not show an obvious tendency for the tt systems and the (hadronic) top quarks.
The event patterns in the three-dimensional y 1 -y 2 -y space for the tt systems in the three requirements are very rough ellipsoids with y 2 1 ≈ y 2 2 ≪ y 2 and |y 1 | max ≈ |y 2 | max ≪ |y| max , though little differences among the three requirements are observed. The event patterns for the (hadronic) top quarks in the three requirements are very rough rhombogen with the similar relations among these quantities and little differences among the three requirements. An obvious difference     Table 5. Values of the root-mean-squares p 2 x for px, p 2 y for py, and p 2 z for pz , as well as the maximum |px|, |py|, and |pz| (|px|max, |py|max, and |pz|max) corresponding to the scatter plots for different types of products, where the corresponding scatter plots are presented in Figure 5. Both the root-mean-squares and maximum momentum components are in the units of GeV/c.

Figure
Type  Table 6. Values of the root-mean-squares y 2 1 for y1, y 2 2 for y2, and y 2 for y, as well as the maximum |y1|, |y2|, and |y| (|y1|max, |y2|max, and |y|max) corresponding to the scatter plots for different types of products, where the corresponding scatter plots are presented in Figure 6.

Figure
Type between the event patterns for the tt systems and the (hadronic) top quarks are observed. Meanwhile, both the quantities on the root-mean-squares and the maximums for the tt systems are obviously less than those for the (hadronic) top quarks. The differences in relative sizes between transverse and longitudinal quantities for the tt systems are larger than those for the (hadronic) top quarks.
According to these scatter plots (Figures 4-6), we can obtain the probability distributions of the considered quantities. Using higher statistics, Figures 7-9 present the probability distributions of β i (i = x, y, and z), p i (i = x, y, and z), and y i (i = 1 and 2), respectively, where N denotes the number of particles. Different panels correspond to different requirements and different curves correspond to different quantities shown in the panels. One can see that the distributions of x and y components are almost the same, if not equal to each other at the pixel level, due to the assumption of isotropic emission in the transverse plane. All distributions of x, y and z components are symmetric at zero.
On the velocity components, β x and β y distribute only in a small region near zero. The tt systems have a more narrow region and a higher peak than the (hadronic) top quarks. For the requirements from the detector level to the fiducial phase-space then to the full phase-space, the peak value increases obviously. β z distributes almost uniformly in a wide region for different particles and requirements. In particular, the distributions of |β z | of the (hadronic) top quarks increase slightly with the increase of |β z |, while the tt systems show an opposite or placid tendency. Moreover, the tendency in the full phase-space is more obvious than those in the fiducial phase-space and at the detector level. The main differences appear near |β z | max and are caused by different m 0 .
On the momentum components, all the distributions of p x , p y , and p z for different particles and requirements have a peak at zero, though the distribution of p z has a wider range and a lower peak than those of p x and p y . The tt systems have a higher and higher peak in p x and p y distributions in proper order at the detector level, in the fiducial phase-space, and in the full phase-space, while other three types of distributions have severally similar shapes and do not show obviously the tendency of higher and higher peak.
As for the distributions of y 1 and y 2 , one can see a higher and higher peak for the tt systems for the requirements from the detector level to the fiducial phasespace then to the full phase-space. For the hadronic top quarks, the distributions of y 1 and y 2 at the detector level and in the fiducial phase-space are almost the same, and they are different in shape and slope around the peak region from the distributions for the top quarks.
For comparisons with our recent works [23][24][25], as an example, we can see the similarity and difference in the three-dimensional β x -β y -β z space. The scatter plots of tt systems and (hadronic) top quarks are similar to those of Z bosons and quarkonium states [25] due to they being heavy particles. In fact, the scatter plots of heavy particles show that the root-mean square velocities form a rough cylinder or ellipsoid surface and the maximum velocities form a fat cylinder or ellipsoid surface, due to their productions being at the initial stage of collisions. The scatter plots of charged particles show that the root-mean-square velocities form an ellipsoid surface and the maximum velocities form a spherical surface [23,24], due to their productions being mostly at the intermediate stage of collisions and suffering particularly the processes of thermalization and expansion of the interacting system.

Conclusions
We summarize here our main observations and conclusions.
(a) We have used the hybrid model to describe the top quark-related spectra of p T and y, which include the spectra of tt systems, hadronic top quarks, and top quarks produced in pp collisions at √ s = 8 TeV measured by the ATLAS Collaboration at the LHC. The hybrid model uses the superposition of the inverse powerlaw and the Erlang distribution for the description of p T spectra and the two-component Gaussian function for the description of y spectra. The inverse power-law, the Erlang distribution, and the two-component Gaussian function are derived from the QCD calculus, the multisource thermal model, and the Landau hydrodynamic model, respectively. We have used the inverse power-law and the Erlang distribution to describe the harder and hard scattering processes respectively.
(b) The modelling results are in agreement with the experimental data of the tt systems and the hadronic top quarks measured at the detector level, the fiducial phase-space normalized differential cross-sections for the tt systems and the hadronic top quarks, and the full phase-space normalized differential cross-sections for the tt systems and the top quarks. The tt systems show only the contribution of harder scattering process (with k = 1). The (hadronic) top quarks show mainly the contribution of hard scattering process (with a small k). These render that more collision energies are needed to create the tt system. As a part of the tt system, the (hadronic) top quark apportioned partly the energy of tt system, which results in a not too hard scattering process.
(c) When the experimental requirement changes from the detector level to the fiducial phase-space and then to the full phase-space, for both the tt systems and the (hadronic) top quarks, p 0 and n decrease, and y F and σ yF increase. Only for the (hadronic) top quarks, k slightly increases, and p T i and m almost do not change. Although these tendencies of the parameters may have no obvious meaning due to little relations among these experimental requirements, these parameters can be used in the extraction of discrete values of some kinematic quantities. In fact, based on these parameters, we have obtained some discrete values of the velocity, momentum, and rapidity components. Based on these discrete values, the event patterns in some three-dimensional spaces are obtained.
(d) The event patterns in the three-dimensional β xβ y -β z space for the tt systems in the three requirements are rough cylinders with β 2 x ≈ β 2 y ≪ β 2 z and |β x | max ≈ |β y | max < |β z | max . The event patterns for the (hadronic) top quarks in the three requirements are rough ellipsoids with the similar relations among these quantities. Both the quantities on the root-meansquares and the maximums for the tt systems are less than those for the (hadronic) top quarks, and the differences in relative sizes between transverse and longitudinal quantities for the tt systems are larger than those for the (hadronic) top quarks. (e) The event patterns in the three-dimensional p xp y -p z space for the tt systems in the three requirements are relative thin and very rough ellipsoids with p 2 x ≈ p 2 y ≪ p 2 z and |p x | max ≈ |p y | max ≪ |p z | max . The event patterns for the (hadronic) top quarks in the three requirements are relative fat and very rough ellipsoids with the similar relations among these quantities. The transverse quantities for the tt systems are less than those for the (hadronic) top quarks, and the situations of longitudinal quantities are opposite. The differences in relative sizes between transverse and longitudinal quantities for the tt systems are larger than those for the (hadronic) top quarks. The maximum quantities do not show an obvious tendency for the tt systems and the (hadronic) top quarks.
(f) The event patterns in the three-dimensional y 1y 2 -y space for the tt systems in the three requirements are very rough ellipsoids with y 2 1 ≈ y 2 2 ≪ y 2 and |y 1 | max ≈ |y 2 | max ≪ |y| max . The event patterns for the (hadronic) top quarks in the three requirements are very rough rhombogen with the similar relations among these quantities. Both the quantities on the root-meansquares and the maximums for the tt systems are obviously less than those for the (hadronic) top quarks. The differences in relative sizes between transverse and longitudinal quantities for the tt systems are larger than those for the (hadronic) top quarks.
(g) According to these scatter plots, we have obtained the probability distributions of the considered quantities such as β i , p i , and y i . The distributions of x and y components are almost the same, if not equal to each other at the pixel level, due to the assumption of isotropic emission in the transverse plane. β x and β y distribute only in a small region near zero. The tt systems have a more narrow region and a higher peak than the (hadronic) top quarks. β z distributes almost uniformly in a wide region for different particles and requirements. In particular, the distributions of |β z | of the (hadronic) top quarks increase slightly with the increase of |β z |, while the tt systems show an opposite or placid tendency.
(h) All the distributions of p x , p y , and p z for different particles and requirements have a peak at zero, though the distribution of p z has a wider range and a lower peak than those of p x and p y . The tt systems have a higher peak in p x and p y distributions, while other three types of distributions have severally similar shapes and do not show obviously such a high peak. In the distributions of y 1 and y 2 , a higher peak for the tt systems than for the (hadronic) top quarks is observed. For the hadronic top quarks, the distributions of y 1 and y 2 at the detector level and in the fiducial phase-space are almost the same, and they are different in shape and slope around the peak region from the distributions for the top quarks.