cpcChinese Physics CChin. Phys. C1674-1137Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltdcpc_42_3_03310110.1088/1674-1137/42/3/03310142/3/033101PaperGeneralEvent patterns extracted from top quark-related spectra in proton-proton collisions at 8 TeV*
Supported by National Natural Science Foundation of China (11575103, 11747319), the Shanxi Provincial Natural Science Foundation (201701D121005), the Fund for Shanxi “1331 Project” Key Subjects Construction and the US DOE (DE-FG02-87ER40331.A008)
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 s=8 TeV. For pT 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 (β_{x}-β_{y}-β_{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.
top quark-related spectraevent patternthree-dimensional space14.65.Ha25.75.Ag25.75.Dw
Article funded by SCOAP^{3}
arxivppt1611.10150Introduction
The top quark is the heaviest particle in the standard model, and is very different from the other quarks. It is expected that the top quark may have some special characteristics and be related to new physics beyond the standard model. Therefore, it is very important to study its characteristics more. The top quark was first found by the Tevatron at the Fermi National Accelerator Laboratory [1–4], and the successful operation of the CERN Large Hadron Collider (LHC) has brought the study of the top quark into 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 studying the interacting systems directly.
Among the kinematic observables, the transverse momentum (p_{T}) and rapidity (y) are always hot topics for theoretical physicists. Some phenomenological models and formulas have been proposed to fit the p_{T} and y spectra. For p_{T} spectra, many formulas can be used, such as the standard (Fermi-Dirac, Bose-Einstein, or Boltzmann) distribution [5–8], Tsallis statistics [8–14], the inverse power-law [15–17], the Erlang distribution [18], the Schwinger mechanism [19–22], and combinations of these, 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 been used to find arithmetic solutions of 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.
High energy collisions are complex processes in which the production mechanisms of different types of particles are different. In particular, at the same stage of a collision process, different types of particles may undergo different ways of propagation. Although some spectra of different types of particles can be described or fitted by particular theoretical models or functions, we are interested in the differences in their production. For example, before leaving the interacting region, most light flavor particles undergo the stage of kinetic freeze-out and local thermal equilibrium. Heavy flavor particles do not undergo this stage. It is hard to learn more information from the limited spectra available in experiments. We hope to use a simple method to extract some intuitive pictures from the limited spectra, so that some differences in the production of different types of particles can be observed. These differences are useful in understanding the production mechanisms of different types of particles.
In order to understand the sophisticated collision process and mechanism intuitively, we can use the method of event pattern (particle scatter plot) at the last stage of particle production to obtain information about the interacting 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_{y}-β_{z}) space [23]; charged particles produced in proton-proton (pp) and lead-lead (Pb-Pb) collisions at 2.76 TeV (one of the LHC energies) in three-dimensional p_{x}-p_{y}-p_{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 variety of produced particles, further analyses of 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 [15–17] and the Erlang distribution resulting from a multisource thermal model [18], and the two-component Gaussian function resulting from the revised Landau hydrodynamic model [26–29] 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 ATLAS 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_{x}-β_{y}-β_{z}, p_{x}-p_{y}-p_{z}, and rapidity (y_{1}-y_{2}-y) spaces. The probability distributions of these components can also be obtained.
The remainder of this paper is structured as follows. A brief 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.
Model and method
As the heaviest particle in the standard model, the formation of the top quark is expected to be through the hard scattering process among partons (quarks and gluons) with high energy. The top quark-related p_{T} spectra have a very wide range of distributions. This means that in some cases the spectra can in fact be divided into two parts. One part is in the relatively high p_{T} region and mainly contributed by the real hard (the harder) scattering process, and the other part is in the relatively 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. Of the two components, one is for the harder scattering process and the other for the hard scattering process. The relative contributions of the harder scattering process are expected to be different for different products such as top quark pairs (tt¯ systems), hadronic top (hadronic t) quarks (hadronically decaying top quarks), and top (t) quarks (semileptonically and hadronically decaying top quarks).
For the harder scattering process, we can use the inverse power-law which results from the QCD calculus [15–17] in high energy collisions to describe the p_{T} spectra. For the hard scattering process, we can use the Erlang distribution which results from a multisource thermal model [18]. 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 relatively high p_{T}, and the Erlang distribution contributes mainly in the region of relatively low p_{T}.
According to the QCD calculus [15–17], we have the inverse power-law in the formf1(pT)=ApT(1+pTp0)−n,where A denotes the normalization constant which makes ∫0∞f1(pT)dpT=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 distributionf2(pT)=pTm−1(m−1)!〈pTi〉mexp(−pT〈pTi〉),where 〈p_{Ti}〉 and m are free parameters. In particular, 〈p_{Ti}〉 denotes the average value of p_{Ti}, 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} spectrum 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 distributionf0(pT)=kf1(pT)+(1−k)f2(pT).In many cases, the spectra of p_{T} in experiments are presented in terms of non-normalized distributions. To give a comparison with the experimental data, the normalized constant (N_{pT}) is needed.
According to the Landau hydrodynamic model and its revisions [26–29], the y spectrum is a Gaussian function [28, 29]fy(y)=12πσyexp[−(y−yC)22σy2],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 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, and we need at least two Gaussian functions for the y spectrum. That is,fy(y)=kB2πσyBexp[−(y−yB)22σyB2]+1−kB2πσyFexp[−(y−yF)22σyF2],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, distributed 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 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∫0pTf1(pT)dpT<R<∫0pT+dpTf1(pT)dpT.The variable p_{T} in Eq. (2) or in the second component in Eq. (3) is obtained bypT=−〈pTi〉∑i=1mlnri=−〈pTi〉ln∏i=1mri.As for the variable y in the first and second components in Eq. (5), we havey=σy−2lnR1cos(2πR2)+yBandy=σy−2lnR3cos(2πR4)+yFrespectively.
An isotropic emission in the transverse plane results in the azimuthal angle φ beingφ=2πR5which is distributed evenly in [0,2π]. The momentum components arepx=pTcosφ,py=pTsinφ,andpz=pT2+m02sinhy,where m_{0} is the peak mass in the invariant mass spectrum of the tt¯ systems [30], or the rest mass of the top quark in the case of hadronic top quarks or top quarks.
The energy E isE=pT2+m02coshy.The velocity components areβx=pxE,βy=pyE,andβz=pzE.
For y_{1} and y_{2}, we use the definitionsy1=12ln(E+pxE−px)andy2=12ln(E+pyE−py)for the rapidities in the directions of the ox and oy axes respectively. We get y using Eq. (8) or (9) directly.
In the concrete calculation, we need a few steps to generate the event patterns, i.e. the three-dimensional distributions of particle scatters.
1) We use Eq. (3) to fit the p_{T} spectra and Eq. (5) to fit the y spectra so that the values of free parameters and normalization constants can be obtained;
2) According to the values of the free parameters obtained by the fitting in the first step, we may use Eq. (6) or (7) on the basis of k to get a discrete value of p_{T}, and Eq. (8) or (9) with an equal probability to get a discrete value of y;
3) We use Eq. (10) to get a discrete value of φ, and Eqs. (11)–(13) to get a set of discrete values of (p_{x},p_{y},p_{z});
4) We use Eq. (14) to get a discrete value of E, and Eqs. (15)–(17) to get a set of discrete values of (β_{x},β_{y},β_{z});
5) We use Eq. (18) to get a discrete value of y_{1}, Eq. (19) to get a discrete value of y_{2}, and the discrete value of y is obtained directly from Eq. (8) or (9). Then, a set of discrete values of (y_{1},y_{2},y) is obtained;
6) Repeating steps 2) to 5) 1000 times, we can obtain 1000 sets of (p_{x},p_{y},p_{z}), 1000 sets of (β_{x},β_{y},β_{z}), and 1000 sets of (y_{1},y_{2},y);
7) Finally, the three-dimensional distributions of particle scatters are plotted in three three-dimensional spaces.
The probability distributions of the various components are obtained by statistics.
Results and discussion
Before describing the comparisons with experimental data, we first introduce the meanings of “the detector level”, “the fiducial phase-space”, and “the full phase-space”. According to Ref. [30], “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 the considered event has to be detected, identified, and selected by various types of detectors, which is referred to as 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”.
Although we intend to “describe” separately the spectra of top quark pairs, hadronic top quarks, and top quarks at the “detector level”, in the “fiducial” phase-space, and in the “full” phase-space, these concepts are not independent. In fact, the differences between them arise due to the decay of the top quark and also due to the finite coverage of the detector. Predicting the relationship between them is one of the great successes of perturbative QCD: the experiments only measure particle data at the detector-level, which is then extrapolated with theoretical tools to the level of top quarks and, if desired, can also be corrected to the fiducial or full phase-space.
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 (1) at the detector level, (2) in the fiducial phase-space, and (3) in the full phase-space, we take m_{0} = 463.2, 382.5, and 372.5 GeV, respectively, as in Ref. [30]. For the rest mass of (hadronic) top quark, we take m_{0} = 172.5 GeV, also from Ref. [30]. It is very important to use the correct rest mass in the extraction of event patterns. 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 for (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_{Ti}〉, 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.
(color online) (a)(c) Transverse momentum and (b)(d) rapidity spectra of (a)(b) the tt¯ systems and (c)(d) the hadronic top quarks produced in pp collisions at s=8 TeV, where the rapidity 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. The solid curves are our results calculated by using the (a) inverse power-law, (c) superposition of inverse power-law and Erlang distribution, and (b)(d) 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.
Values of free parameters (p_{0} and n), normalization constant (N_{pT}), and χ^{2}/dof corresponding to the curves in Figs. 1(a), 2(a), and 3(a), where k = 1, which means that 〈p_{Ti}〉 and m are not available.
Figure
type
p_{0}/(GeV/c)
n
N_{pT}
χ^{2}/dof
Figure 1(a)
tt¯
162.86±4.20
7.70±0.40
(1.93±0.05)×10^{5}
2.88
Figure 2(a)
tt¯
52.73±2.60
4.34±0.20
1.00
4.26
Figure 3(a)
tt¯
45.83±2.50
4.30±0.20
1.00
1.43
Values of free parameter [y_{F} (= − y_{B}) and σ_{yF} (=σ_{yB})], normalization constant (N_{η}), and χ^{2}/dof corresponding to the curves in Figs. 1(b), 1(d), 2(b), 2(d), 3(b), and 3(d).
Figure
type
y_{F} (= − y_{B})
σ_{yF} (=σ_{yB})
N_{y}
χ^{2}/dof
Figure 1(b)
tt¯
0.36±0.02
0.53±0.02
(196.00±2.00)×10^{3}
2.27
Figure 1(d)
hadronic t
0.53±0.02
0.68±0.02
(199.00±2.00)×10^{3}
2.21
Figure 2(b)
tt¯
0.39±0.02
0.55±0.02
1.00
7.88
Figure 2(d)
hadronic t
0.57±0.02
0.69±0.02
1.01±0.01
13.46
Figure 3(b)
tt¯
0.52±0.02
0.79±0.03
1.04±0.01
4.64
Figure 3(d)
t
0.58±0.02
0.98±0.03
1.04±0.01
6.55
One can see from Fig. 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 the harder scattering process (with k = 1), which means that 〈p_{Ti}〉 and m for the tt¯ systems are not available. The hadronic top quarks show mainly the contribution of the hard scattering process (with a small k). This implies that more collision energy is needed to create the tt¯ system. As part of the tt¯ system, the hadronic top quark takes up part of the energy of the tt¯ system, which results in a not too hard scattering process.
Figure 2 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 shows the full phase-space normalized differential cross-sections 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 as a separate column in the tables.
(color online) (a)(c) Transverse momentum and (b)(d) rapidity normalized differential cross-sections of (a)(b) the tt¯ systems and (c)(d) the hadronic top quarks produced in pp collisions at s=8 TeV. The symbols represent the experimental data of the ATLAS Collaboration [30] measured in the combined electron and muon selections at the fiducial phase-space level. The solid curves are our results calculated by using the (a) inverse power-law, (c) superposition of inverse power-law and Erlang distribution, and (b)(d) 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.
(color online) (a)(c) Transverse momentum and (b)(d) rapidity normalized differential cross-sections of (a)(b) the tt¯ systems and (c)(d) the top quarks produced in pp collisions at s=8 TeV. The symbols represent the experimental data of the ATLAS Collaboration [30] measured in the combined electron and muon selections at the full phase-space level. The solid curves are our results calculated by using the (a) inverse power-law, (c) superposition of inverse power-law and Erlang distribution, and (b)(d) 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.
Values of free parameters (p_{0}, n, k, 〈p_{Ti}〉, and m), normalization constant (N_{pT}), and χ^{2}/dof corresponding to the curves in Figs. 1(c), 2(c), and 3(c), where the values of m in the Erlang distribution are invariably taken to be 3 and are not listed in a separate column.
Figure
type
p_{0}/(GeV/c)
n
k
〈p_{Ti}〉 (GeV/c)
N_{pT}
χ^{2}/dof
Figure 1(c)
hadronic t
225.00±22.50
6.50±1.00
0.10±0.02
39.70±0.50
(1.99±0.05)×10^{5}
11.34
Figure 2(c)
hadronic t
204.00±20.40
6.05±1.00
0.13±0.02
40.50±0.05
1.00
14.33
Figure 3(c)
t
165.00±16.50
5.60±1.00
0.14±0.02
39.20±0.05
1.00
3.76
One can also see from Figs. 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 phase-space. Only for the (hadronic) top quarks, k slightly increases, and there is almost no change in 〈p_{Ti}〉 and m 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 have no obvious meaning due to there being little relation among these requirements. However, because of these tendencies, we can obtain abundant structures of event patterns.
Based on the parameter values obtained from Figs. 1–3 and listed in Tables 1–3, 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 last stage of particle production in the interacting system formed in pp collisions. Figures 4–6 give the event patterns which are displayed by the particle scatter plots in the three-dimensional 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 phase-space, respectively. The total number of particles for each panel is 1000. The blue and red globules represent the contributions of the inverse power-law and Erlang distribution respectively. The values of root-mean-squares βx2¯ for β_{x}, βy2¯ for β_{y}, and βz2¯ for β_{z}, as well as the maximum |β_{x}|, |β_{y}|, and |β_{z}| (i.e. |β_{x}|_{max}, |β_{y}|_{max}, and |β_{z}|_{max}) are listed in Table 4. The values of root-mean-squares px2¯ for p_{x}, py2¯ for p_{y}, and pz2¯ for p_{z}, as well as the maximum |p_{x}|, |p_{y}|, and |p_{z}| (i.e. |p_{x}|_{max}, |p_{y}|_{max}, and |p_{z}|_{max}) are listed in Table 5. The values of root-mean-squares y12¯ for y_{1}, y22¯ for y_{2}, and y2¯ for y, as well as the maximum |y_{1}|, |y_{2}|, and |y| (i.e. |y_{1}|_{max}, |y_{2}|_{max}, and |y|_{max}) are listed in Table 6.
(color online) Event patterns (particle scatter plots) in three-dimensional β_{x}-β_{y}-β_{z} space in pp collisions at s=8 TeV (a)(b) at the detector level, (c)(d) in the fiducial phase-space, and (e)(f) in the full phase-space, for (a)(c)(e) the tt¯ systems, (b)(d) the hadronic top quarks, and (f) the top quarks. The number of particles for each panel is 1000. The blue and red globules represent the results corresponding to the inverse power-law function and the Erlang distribution for p_{T}, respectively.
(color online) Event patterns (particle scatter plots) in three-dimensional p_{x}-p_{y}-p_{z} space in pp collisions at s=8 TeV (a)(b) at the detector level, (c)(d) in the fiducial phase-space, and (e)(f) in the full phase-space, for (a)(c)(e) the tt¯ systems, (b)(d) the hadronic top quarks, and (f) the top quarks. The number of particles for each panel is 1000. The blue and red globules represent the results corresponding to the inverse power-law function and the Erlang distribution for p_{T}, respectively.
(color online) Event patterns (particle scatter plots) in three-dimensional y_{1}-y_{2}-y space in pp collisions at s=8 TeV (a)(b) at the detector level, (c)(d) in the fiducial phase-space, and (e)(f) in the full phase-space, for (a)(c)(e) the tt¯ systems, (b)(d) the hadronic top quarks, and (f) the top quarks. The number of particles for each panel is 1000. The blue and red globules represent the results corresponding to the inverse power-law function and the Erlang distribution for p_{T}, respectively.
Values of the root-mean-squares βx2¯ for β_{x}, βy2¯ for β_{y}, and βz2¯ for β_{z}, as well as the maximum |β_{x}|, |β_{y}|, and |β_{z}| (|β_{x}|_{max}, |β_{y}|_{max}, and |β_{z}|_{max}) corresponding to the scatter plots for different types of products, where the corresponding scatter plots are presented in Fig. 4. Both the root-mean-squares and the maximum velocity components are in units of c.
Figure
type
βx2¯
βy2¯
βz2¯
|β_{x}|_{max}
|β_{y}|_{max}
|β_{z}|_{max}
Figure 4(a)
tt¯
0.11±0.01
0.11±0.01
0.50±0.01
0.58
0.56
0.96
Figure 4(b)
hadronic t
0.31±0.01
0.32±0.01
0.59±0.01
0.79
0.82
0.99
Figure 4(c)
tt¯
0.13±0.01
0.14±0.01
0.52±0.01
0.76
0.67
0.97
Figure 4(d)
hadronic t
0.31±0.01
0.32±0.01
0.61±0.01
0.81
0.87
0.99
Figure 4(e)
tt¯
0.10±0.01
0.10±0.01
0.62±0.01
0.52
0.46
0.99
Figure 4(f)
t
0.28±0.01
0.29±0.01
0.67±0.01
0.77
0.84
1.00
Values of the root-mean-squares px2¯ for p_{x}, py2¯ for p_{y}, and pz2¯ for p_{z}, as well as the maximum |p_{x}|, |p_{y}|, and |p_{z}| (|p_{x}|_{max}, |p_{y}|_{max}, and |p_{z}|_{max}) corresponding to the scatter plots for different types of products, where the corresponding scatter plots are presented in Fig. 5. Both the root-mean-squares and maximum momentum components are in units of GeV/c.
Figure
type
px2¯
py2¯
pz2¯
|p_{x}|_{max}
|p_{y}|_{max}
|p_{z}|_{max}
Figure 5(a)
tt¯
63.2±2.8
63.9±2.8
373.4±11.0
427.4
469.0
1545.0
Figure 5(b)
hadronic t
97.0±2.7
98.8±2.5
279.1±10.8
433.6
384.8
1347.2
Figure 5(c)
tt¯
75.1±4.7
74.9±4.4
338.2±10.3
589.3
632.2
1399.1
Figure 5(d)
hadronic t
100.9±3.1
102.8±3.0
304.9±13.4
498.6
560.5
1999.9
Figure 5(e)
tt¯
52.4±2.1
53.2±2.0
574.6±23.7
273.5
255.6
3189.3
Figure 5(f)
t
95.0±2.6
96.8±2.5
513.9±28.8
393.8
423.9
3079.7
Values of the root-mean-squares y12¯ for y_{1}, y22¯ for y_{2}, and y2¯ for y, as well as the maximum |y_{1}|, |y_{2}|, and |y| (|y_{1}|_{max}, |y_{2}|_{max}, and |y|_{max}) corresponding to the scatter plots for different types of products, where the corresponding scatter plots are presented in Fig. 6.
Figure
type
y12¯
y22¯
y2¯
|y_{1}|_{max}
|y_{2}|_{max}
|y|_{max}
Figure 6(a)
tt¯
0.11±0.01
0.11±0.01
0.65±0.01
0.67
0.63
1.92
Figure 6(b)
hadronic t
0.34±0.01
0.36±0.01
0.87±0.02
1.08
1.16
2.53
Figure 6(c)
tt¯
0.14±0.01
0.15±0.01
0.68±0.01
1.01
0.81
2.01
Figure 6(d)
hadronic t
0.34±0.01
0.36±0.01
0.90±0.02
1.12
1.34
2.60
Figure 6(e)
tt¯
0.10±0.01
0.11±0.01
0.95±0.02
0.58
0.49
2.84
Figure 6(f)
t
0.30±0.01
0.33±0.01
1.15±0.03
1.02
1.22
3.46
From Figs. 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 βx2¯≈βy2¯≪βz2¯ and |β_{x}|_{max} ≈ |β_{y}|_{max} < |β_{z}|_{max}, though few differences among the three requirements are observed. The event patterns for the (hadronic) top quarks in the three requirements are rough ellipsoids, with similar relations among these quantities and few differences between the three requirements. An obvious difference between the event patterns for the tt¯ systems and the (hadronic) top quarks is observed due to their different production processes. Meanwhile, both the root-mean-squares and the maxima 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_{y}-p_{z} space for the tt¯ systems in the three requirements are relatively thin and very rough ellipsoids with px2¯≈py2¯≪pz2¯ and |p_{x}|_{max} ≈ |p_{y}|_{max} ≪ |p_{z}|_{max}, though few differences among the three requirements are observed. The event patterns for the (hadronic) top quarks in the three requirements are relatively fat and very rough ellipsoids with similar relations among these quantities and few differences between the three requirements. An obvious difference between the event patterns for the tt¯ systems and the (hadronic) top quarks is observed. Meanwhile, the transverse quantities for the tt¯ systems are less than those for the (hadronic) top quarks, while the opposite is true for the longitudinal quantities. 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 y12¯≈y22¯≪y2¯ and |y_{1}|_{max} ≈ |y_{2}|_{max} ≪ |y|_{max}, though few differences among the three requirements are observed. The event patterns for the (hadronic) top quarks in the three requirements are very rough rhomboids with similar relations among these quantities and few differences between the three requirements. An obvious difference between the event patterns for the tt¯ systems and the (hadronic) top quarks is observed. Meanwhile, both the root-mean-squares and the maxima 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 (Figs. 4–6), we can obtain the probability distributions of the considered quantities. Using higher statistics, Figs. 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.
(color online) Distributions of velocity components β_{x}, β_{y}, and β_{z} in pp collisions at s=8 TeV (a)(b) at the detector level, (c)(d) in the fiducial phase-space, and (e)(f) in the full phase-space, for (a)(c)(e) the tt¯ systems, (b)(d) the hadronic top quarks, and (f) the top quarks. The dotted, dashed, and solid curves correspond to the distributions of β_{x}, β_{y}, and β_{z}, respectively, where the distributions of β_{x} and β_{y} are nearly the same.
(color online) Distributions of momentum components p_{x}, p_{y}, and p_{z} in pp collisions at s=8 TeV (a)(b) at the detector level, (c)(d) in the fiducial phase-space, and (e)(f) in the full phase-space, for (a)(c)(e) the tt¯ systems, (b)(d) the hadronic top quarks, and (f) the top quarks. The dotted, dashed, and solid curves correspond to the distributions of p_{x}, p_{y}, and p_{z}, respectively, where the distributions of p_{x} and p_{y} are nearly the same.
(color online) Distributions of y_{1} and y_{2} in pp collisions at s=8 TeV (a)(b) at the detector level, (c)(d) in the fiducial phase-space, and (e)(f) in the full phase-space, for (a)(c)(e) the tt¯ systems, (b)(d) the hadronic top quarks, and (f) the top quarks. The dotted, dashed, and solid curves correspond to the distributions of y_{1} and y_{2}, respectively, where the two distributions are nearly the same.
On the velocity components, β_{x} and β_{y} are distributed only in a small region near zero. The tt¯ systems have a narrower 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} is distributed 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 static tendency. Moreover, the tendency in the full phase-space is more obvious than 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¯ system has an increasingly high peak in the p_{x} and p_{y} distributions at the detector level, in the fiducial phase-space, and in the full phase-space, while the other three types of distributions have similar shapes and do not show an obvious tendency to increasing peak height.
For the distributions of y_{1} and y_{2}, one can see an increasingly high peak for the tt¯ system going from the detector level to the fiducial phase-space 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.
It should be noted that, in the above discussions, we have produced fits to the ATLAS 8 TeV data on p_{T} and y of the top quarks and the tt¯ pairs. The fits for these one-dimensional kinematic distributions are then used in such a way that fully differential distributions for the same final states are “predicted”. It seems that, in general, it is impossible to achieve this because one cannot reconstruct a generic multidimensional distribution from its one-dimensional (marginal) projections due to the correlations between variables not being known. In fact, our procedure of reconstruction works well due to the correlations between variables being considered through Eqs. (11) and (14).
For the correlations between x- and y-components, we have used an isotropic assumption in the transverse plane. This results in Eqs. (10)–(12) for the azimuthal angle, p_{x}, and p_{y}, respectively. We would like to point out that the effect of elliptic flow for the top quarks and the tt¯ pairs are neglected due to this effect appearing mainly in the soft process. In the case of considering the elliptic flow for the soft process, we are expected to study the fine-structure of event patterns [31], which is beyond the focus of the present work, though this effect is small and can be neglected. In any case, conservation of energy and momentum is satisfied in the calculation.
For comparisons with our recent works [23–25], as an example, we can see the similarities and differences 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 them 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 production 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 production being mostly at the intermediate stage of collisions and suffering particularly the processes of thermalization and expansion of the interacting system.
As for comparisons with other modelling or theoretical works, although thousands of papers on top quark-related subjects have been published since (at least) the 1980s, and the number of top quark-related publications from the LHC is also in the hundreds (for example, see Refs. [32–36]), few of them are directly related to the event patterns or particle scatter plots. In fact, we cannot give a direct comparison with other works due to the available results not being obtained. In addition, although one might just use some event generators such as Pythia, JETSET, and HERWIG instead [37–41], they are not just fitted to transverse momentum and rapidity spectra and require information about the underlying event, pileup, and so on. The present work provides a simple and alternative method to structure event patterns displayed by the scatter plots of different particles. Using this alternative method, one can obtain some direct and ikonic pictures for production of different particles.
Although we have used the hybrid model to fit the experimental p_{T} and y spectra to extract the parameter values and to restructure the event patterns, the event patterns we have obtained are model-independent. In particular, similar or related experimental spectra are also described or predicted by other perturbative QCD calculations such as the Next-to-Leading Order (NLO) in QCD [42–44], Next-to-Next-to-Leading Order (NNLO) in QCD [45–50], Next-to-Next-to-Leading Logarithms (NNLL) [51–53], etc. The event patterns are independent of these theories. In fact, the event patterns are only dependent on the discrete values of experimental probability density distributions of p_{T} and y. What we fitted in the above by using the hybrid model is only parameterizations for the p_{T} and y spectra. These parameterizations smooth only the experimental probability density distributions and help us to restructure the event patterns.
Before giving conclusions, we would like to emphasize briefly the significance of the present work. In our opinion, the present work supports the methodology which restructures the event patterns or particle scatter plots from both p_{T} and y spectra. This method can be used in the studies of other particles [23–25, 31] as discussed above, which allows us to give comparisons of the production of different types of particles. Indeed, from the three-dimensional distribution, we have obviously observed some differences for different type of particles. These differences are useful in the understanding of particle production and event reconstruction.
Conclusions
We summarize here our main observations and conclusions.
(a) We have used the hybrid model to fit 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 power-law 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 fit 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 the harder scattering process (with k = 1). The (hadronic) top quarks show mainly the contribution of the hard scattering process (with a small k). This implies that more collision energy is needed to create the tt¯ system. As a part of the tt¯ system, the (hadronic) top quark takes up part of the energy of the 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, while there is almost no change in 〈p_{Ti}〉 and m. Although these tendencies of the parameters may have no obvious meaning, due to there being little relation 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_{x}-β_{y}-β_{z} space for the tt¯ systems in the three requirements are rough cylinders with βx2¯≈βy2¯≪βz2¯ 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 similar relations among these quantities. Both the root-mean-squares and the maxima 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_{x}-p_{y}-p_{z} space for the tt¯ systems in the three requirements are relatively thin and very rough ellipsoids with px2¯≈py2¯≪pz2¯ and |p_{x}|_{max} ≈ |p_{y}|_{max} ≪ |p_{z}|_{max}. The event patterns for the (hadronic) top quarks in the three requirements are relatively 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_{1}-y_{2}-y space for the tt¯ systems in the three requirements are very rough ellipsoids with y12¯≈y22¯≪y2¯ and |y_{1}|_{max} ≈ |y_{2}|_{max} ≪ |y|_{max}. The event patterns for the (hadronic) top quarks in the three requirements are very rough rhomboids with similar relations among these quantities. Both the root-mean-squares and the maxima 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} are distributed only in a small region near zero. The tt¯ systems have a narrower region and a higher peak than the (hadronic) top quarks. β_{z} is distributed 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 static 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 the p_{x} and p_{y} distributions, while the other three types of distributions have similar shapes and do not obviously show 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.