# Event Patterns Extracted from Transverse Momentum and Rapidity Spectra ofZ Bosons and Quarkonium States Produced in pp and Pb-Pb Collisions at LHC

Authors: Ya-Hui Chen; Fu-Hu Liu; Roy A. Lacey
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

Transverse momentum (pT) and rapidity (y) spectra of Z bosons and quarkonium states (some charmonium cc¯ mesons such as J/ψ and ψ(2S) and some bottomonium bb¯ mesons such as Υ(1S), Υ(2S), and Υ(3S)) produced in proton-proton (pp) and lead-lead (Pb-Pb) collisions at the large hadron collider (LHC) are uniformly described by a hybrid model of two-component Erlang distribution for pT spectrum and two-component Gaussian distribution for y spectrum. The former distribution results from a multisource thermal model, and the latter one results from the revised Landau hydrodynamic model. The modelling results are in agreement with the experimental data measured in pp collisions at center-of-mass energies s=2.76 and 7 TeV and in Pb-Pb collisions at center-of-mass energy per nucleon pair sNN=2.76 TeV. Based on the parameter values extracted from pT and y spectra, the event patterns (particle scatter plots) in two-dimensional pT-y space and in three-dimensional velocity space are obtained.


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## Body

## 1. Introduction

High energy nucleus-nucleus collisions at the relativistic heavy ion collider (RHIC) [1–4] and large hadron collider (LHC) [5–8] provide excellent environment and condition of high temperature and density [9], where a new state of matter, namely, the quark-gluon plasma (QGP) [10–12], is expected to form and to live for a very short time. It is regretful that the QGP cannot be directly measured in experiments due to its very short lifetime. Instead, to understand the formation and properties of QGP, the distribution laws of final-state particles are studied. Because of the complexities and difficulties in experiments, the observables are limited. To obtain more information from limited observables, we need more modelling and theoretical analyses.Generally, an interacting system of nucleus-nucleus collisions undergoes a few stages which include, but are not limited to, the scattering, color glass condensate, thermalization, hadronization, chemical equilibrium and freeze-out, and kinetic equilibrium and freeze-out. The distribution laws of final-state particle transverse momentum and rapidity spectra reflect the situation of interacting system at the stage of kinetic freeze-out, while the feed-down corrected yields and the ratios of those yields of different particles reflect the situation at the stage of chemical freeze-out. Based on the descriptions of transverse momentum (pT) and rapidity (y) spectra, one can extract some information of transverse excitation and longitudinal expansion of interacting system. Thus, other pieces of information such as the event pattern (particle scatter plot) in two-dimensional pT-y space and three-dimensional velocity space can be extracted from parameters fitted to pT and y spectra [13, 14].In peripheral nucleus-nucleus collisions, less nucleons take part in the interactions. The most peripheral nucleus-nucleus collisions contain only two nucleons which are from the two collision nuclei, respectively. Proton-proton (pp) collisions are similar to the most peripheral nucleus-nucleus collisions in the case of neglecting the spectator (cold nuclear) effect. As an input quantity and a basic collision process, pp collisions can be used to give comparisons with nucleus-nucleus collisions. We are interested in both pp collisions and lead-lead (Pb-Pb) collisions at the LHC.To understand the stage of kinetic freeze-out in high energy collisions, we can analyze thepT and y spectra to obtain the probability density functions fpT(pT) for pT and fy(y) for y. Nevertheless, these probability density functions cannot directly give us a whole and perceptual picture of the interacting system at the stage of kinetic freeze-out. In fact, a whole and perceptual picture can help us understand the interacting mechanisms in detail. Fortunately, we can use the Monte Carlo method to extract some discrete values of pT and y based on fpT(pT) and fy(y). Other quantities such as energy, momentum components, velocity, and velocity components can be obtained according to some definitions and assumptions. Because fpT(pT) for pT and fy(y) for y are based on descriptions of experimental spectra, the extracted discrete values are independent of models.In this paper, based on a hybrid model of two-component Erlang distribution forpT spectrum (which results from a multisource thermal model [15–17]) and two-component Gaussian distribution for y spectrum (which results from the Landau hydrodynamic model and its revisions [18–26]), we analyze together pT and y spectra of Z bosons and quarkonium states (some charmonium cc¯ mesons such as J/ψ and ψ(2S) and some bottomonium bb¯ mesons such as Υ(1S), Υ(2S), and Υ(3S)) produced in pp collisions at center-of-mass energies s=2.76 and 7 TeV and in Pb-Pb collisions at center-of-mass energy per nucleon pair sNN=2.76 TeV. The modelling results are in agreement with the experimental data measured at the LHC. Based on the parameters extracted from pT and y spectra, the event patterns (particle scatter plots) at kinetic freeze-out in two-dimensional pT-y space and in three-dimensional velocity space are obtained.The structure of the present work is as follows. The model and method are shortly described in Section2. Results and discussion are given in Section 3. In Section 4, we summarize our main observations and conclusions.


## 2. The Model and Method

Firstly, we need modelling descriptions of pT and y spectra. In the framework of multisource thermal model [15–17], we can obtain an Erlang distribution or a two-, three-, or multi-component Erlang distribution to fit pT spectrum. According to the model, many (m) emission sources which stay at the same excitation state are assumed to form in high energy collisions. Each (the ith) source is assumed to contribute to transverse momentum pTi by an exponential function:(1)fpTipTi=1pTiexp⁡-pTipTi,where 〈pTi〉 denotes the average value of pTi, which results in (1) being a probability distribution and ∫0∞‍fpTi(pTi)dpTi=1. The m sources which contribute to pT result in an Erlang distribution [17]:(2)fpTpT=pTm-1m-1!pTimexp⁡-pTpTiwhich is the folding of m exponential distributions and has the average transverse momentum 〈pT〉=m〈pTi〉.In the case of considering the two-component Erlang distribution, we have(3)fpTpT=k1pTm1-1m1-1!pTi1m1exp⁡-pTpTi1+1-k1pTm2-1m2-1!pTi2m2exp⁡-pTpTi2which has the average transverse momentum 〈pT〉=k1m1〈pTi〉1+(1-k1)m2〈pTi〉2, where k1 and 1-k1 denote the relative contributions of the first and second components which contribute to the low- and high-pT regions, respectively, and the subscripts 1 and 2 denote the quantities related to the first and second components, respectively. Equations (2) and (3) are probability distributions which are normalized to 1. When we compare them with experimental data, normalization constant (NpT) which is used to fit the data is needed.Ony spectrum, we choose the Landau hydrodynamic model and its revisions [18–26] which are called the revised Landau hydrodynamic model in the present work. In the model, the interacting system is described by the hydrodynamics. The y spectrum can be described by a Gaussian function [25, 26]:(4)fyy=12πσyexp⁡-y-yC22σy2,where σy denotes the rapidity distribution width and yC denotes the midrapidity (peak position). In symmetric collisions, yC=0 is in the center-of-mass reference frame.In the case of considering the two-component Gaussian function fory spectrum, we have(5)fyy=kB2πσyBexp⁡-y-yB22σyB2+1-kB2πσyFexp⁡-y-yF22σyF2,where kB (1-kB), yB (yF), and σyB (σyF) denote, respectively, the relative contribution, peak position, and distribution width of the first (second) component which distributes in the backward (forward) rapidity region. In symmetric collisions such as pp and Pb-Pb collisions which are considered in the present work, we have kB=1-kB=0.5, yB=-yF, and σyB=σyF. As probability distributions, (4) and (5) are normalized to 1. When we compare them with experimental data, a normalization constant (Ny) which is used to fit the data is needed.Secondly, we need discrete values of pT and y. The related calculation is performed by a Monte Carlo method. Let ri, r1i, r2i, and R1,2,…,7 denote random numbers in [0,1]. Equations (2)–(5) result in(6)pT=-pTi∑i=1mln⁡ri=-pTiln⁡∏i=1mri,(7)pT=-pTi1∑i=1m1ln⁡r1i=-pTi1ln⁡∏i=1m1r1ifor the first component in the low-pT region, or(8)pT=-pTi2∑i=1m2ln⁡r2i=-pTi2ln⁡∏i=1m2r2ifor the second component in the high-pT region,(9)y=σy-2ln⁡R1cos⁡2πR2+yC,(10)y=σyB-2ln⁡R3cos⁡2πR4+yBfor the first component in the backward rapidity region, or(11)y=σyF-2ln⁡R5cos⁡2πR6+yFfor the second component in the forward rapidity region, respectively.It should be clarified that the random numbers used above are independent in[0,1]. Through the conversion equations (7), (8), (10), and (11), we can obtain a series of new values which are no longer independent in [0,1] and obey statistically (3) and (5), respectively. In the Monte Carlo method, (6)–(8) are accustomed expressions which result from the Erlang distribution, and (9)–(11) are accustomed expressions which result from the Gaussian distribution. If we use y=σy-2ln⁡R1 instead of (9), we obtain an accustomed expression which result from the Rayleigh distribution f(y)=y/σy2exp⁡(-y2/2σy2) which is different from the Gaussian function.The energyE is given by(12)E=pT2+m02cosh⁡y,where m0 denotes the rest mass of the considered particle. The x-, y-, and z-components of momentum p are given by(13)px=pTcos⁡φ,py=pTsin⁡φ,pz=pT2+m02sinh⁡y,respectively, where φ=2πR7 is the azimuthal angle to distribute evenly in [0,2π]. Combining with (12) and (13), we have the velocity(14)β=pE=pT2+pz2E=px2+py2+pz2Eand its components(15)βx=pxE,βy=pyE,βz=pzE,respectively. All velocity and its components are in the units of c which is the speed of light in vacuum and equal to 1 in natural units.Thirdly, we describe the fitting and structuring method step by step. (i) We fit the pT and y spectra by using (3) and (5), respectively. In the fit, the method of least square method is used to determine the values of parameters. The minimum χ2 per degree freedom (χ2/dof) corresponds to the best values of parameters. Appropriate increases or decreases in parameters determine the uncertainties on parameters, where an appropriate large χ2/dof is used as a limitation. Because there are correlations among parameters accounted for, we have to adjust the parameters and their uncertainties again and again. The best way is to use a multicirculation in the calculation by the computer. (ii) Using the best values of parameters, the discrete values of pT and y are obtained by (7) and (8), as well as (10) and (11), respectively. The discrete values of momentum and velocity components are obtained by (13) and (15), respectively. (iii) Repeating step (ii) many times, we can obtain a series of discrete values. Then, particle scatter plots, that is, event patterns, can be structured by graphic software.


## 3. Results and Discussion

Figure1 presents (a)-(b) transverse momentum spectra, d2σ/(dydpT), and (c)-(d) rapidity spectra, dσ/dy, of Z bosons produced in pp collisions at s=2.76 TeV for the (a)-(c) dimuon (μμ) and (b)-(d) dielectron (ee) decay channels in |y|<2.0 and |y|<1.44, respectively, where σ on the vertical axis denotes the cross section, and the integral luminosity Lint=5.4 pb-1. The closed squares represent the experimental data of the CMS Collaboration [27], and the error bars are only the statistical uncertainties. The curves are our results calculated by using the two-component Erlang distribution for pT spectrum and the two-component Gaussian distribution for y spectrum, which are the results of the multisource thermal model [15–17] and the revised Landau hydrodynamic model [18–26], respectively. The values of free parameters [m1, 〈pTi〉1, k1, m2, 〈pTi〉2, yF (=-yB), and σyF (=σyB)], normalization constants (NpT and Ny), and χ2/dof are listed in Tables 1 and 2, where the normalization constant NpT (or Ny) is used to give comparison between the normalized curves with experimental pT (or y) spectrum. One can see that the results calculated by using the hybrid model are in agreement with the experimental data of Z bosons produced in pp collisions at s=2.76 TeV measured by the CMS Collaboration. In some cases, the values of χ2 are very large due to very small experimental errors.Table 1
Values of free parameters (m1, 〈pTi〉1, k1, m2, and 〈pTi〉2), normalization constant (NpT), and χ2/dof corresponding to the two-component Erlang distribution in Figures 1(a), 1(b), 2(a), 2(b), 5(a), 5(b), 8(a), 11(a), and 11(c), where both the uncertainties on m1 and m2 are 0 which are not listed in the columns. In some cases, dof is less than 1.
Figure Type m 1 〈 p T i 〉 1 (GeV/c) k 1 m 2 〈 p T i 〉 2 (GeV/c) N p T χ 2/dof Figure1(a) p p, Z→μμ 2 4.85 ± 0.20 0.78 ± 0.02 2 15.80 ± 0.79 55.00 ± 2.75 22.945/1 Figure1(b) p p, Z→ee 2 4.10 ± 0.20 0.60 ± 0.02 2 11.00 ± 0.55 63.00 ± 3.15 7.495/1  Figure2(a) Pb-Pb,Z→μμ 2 3.35 ± 0.17 0.45 ± 0.02 2 10.60 ± 0.53 ( 3.57 ± 0.18 ) × 1 0 - 7 10.850/1 Figure2(b) Pb-Pb,Z→ee 2 3.05 ± 0.15 0.54 ± 0.02 2 10.70 ± 0.53 ( 3.85 ± 0.19 ) × 1 0 - 7 2.606/1  Figure5(a) p p, J/ψ 2 1.15 ± 0.06 0.95 ± 0.02 2 2.34 ± 0.12 5.00 ± 0.25 41.272/28 Figure5(b) p p, ψ(2S) 2 1.30 ± 0.07 0.90 ± 0.02 2 2.50 ± 0.13 0.75 ± 0.04 2.587/15  Figure8(a) p p, Υ(1S) 2 2.30 ± 0.12 0.65 ± 0.02 3 1.75 ± 0.09 2.38 ± 0.12 1.997/9 p p, Υ(2S) 3 1.78 ± 0.09 0.68 ± 0.02 2 2.80 ± 0.14 0.56 ± 0.03 0.506/6 p p, Υ(3S) 3 2.00 ± 0.10 0.40 ± 0.02 2 3.20 ± 0.16 0.29 ± 0.01 5.800/5  Figure11(a) Pb-Pb,J/ψ, 0–20% 2 0.89 ± 0.04 0.50 ± 0.02 3 0.69 ± 0.03 ( 2.50 ± 0.13 ) × 1 0 - 2 12.872/7 Pb-Pb,J/ψ, 20–40% 2 0.90 ± 0.05 0.40 ± 0.02 3 0.78 ± 0.04 ( 9.00 ± 0.45 ) × 1 0 - 3 18.291/7 Pb-Pb,J/ψ, 40–90% 2 0.93 ± 0.05 0.40 ± 0.02 3 0.78 ± 0.04 ( 1.60 ± 0.08 ) × 1 0 - 3 29.495/7 Figure11(c) Pb-Pb,Υ(1S) 3 1.50 ± 0.08 0.35 ± 0.02 2 3.00 ± 0.15 0.30 ± 0.02 11.024/-1 Pb-Pb,Υ(2S) 3 1.50 ± 0.08 0.25 ± 0.02 2 4.00 ± 0.20 ( 1.50 ± 0.08 ) × 1 0 - 2 0.287/-3Table 2
Values of free parameter [yF (=-yB) and σyF (=σyB)], normalization constant (Ny), and χ2/dof corresponding to the two-component Gaussian distribution in Figures 1(c), 1(d), 2(c), 2(d), 5(c), 5(d), 8(b), 8(c), 8(d), 11(b), and 11(d). In some cases, dof is less than 1.
Figure Type y F (=-yB) σ y F (=σyB) N y χ 2/dof Figure1(c) p p, Z→μμ 1.10 ± 0.03 1.00 ± 0.05 300 ± 15 8.622/5 Figure1(d) p p, Z→ee 1.10 ± 0.03 1.05 ± 0.05 330 ± 17 2.074/2  Figure2(c) Pb-Pb,Z→μμ 1.12 ± 0.03 1.00 ± 0.05 1.80 ± 0.09 2.860/5 Figure2(d) Pb-Pb,Z→ee 1.12 ± 0.03 1.00 ± 0.05 1.95 ± 0.10 5.626/2  Figure5(c) p p, J/ψ 1.90 ± 0.03 1.82 ± 0.09 54 ± 3 8.141/14 Figure5(d) p p, ψ(2S) 1.90 ± 0.03 1.85 ± 0.10 8.8 ± 0.4 1.338/3  Figure8(b) p p, Υ(1S) 1.58 ± 0.03 1.65 ± 0.08 13.00 ± 0.65 0.492/2 p p, Υ(2S) 1.55 ± 0.03 1.48 ± 0.07 3.20 ± 0.16 0.657/2 p p, Υ(3S) 1.55 ± 0.03 1.60 ± 0.08 1.50 ± 0.08 0.395/2 Figure8(c) p p, Υ(1S) 1.58 ± 0.03 1.65 ± 0.08 510 ± 26 22.156/17 Figure8(d) p p, Υ(2S) 1.55 ± 0.03 1.48 ± 0.07 160 ± 8 5.760/9  Figure11(b) Pb-Pb,J/ψ, inclusive 1.40 ± 0.03 1.70 ± 0.09 17.00 ± 0.85 0.504/0 Pb-Pb,J/ψ, prompt 1.51 ± 0.03 1.70 ± 0.09 13.70 ± 0.69 0.899/0 Figure11(d) Pb-Pb,Υ(1S) 1.35 ± 0.03 1.25 ± 0.06 1.95 ± 0.10 1.070/3 Pb-Pb,Υ(2S) 1.38 ± 0.03 1.22 ± 0.06 0.16 ± 0.008 0.085/-1Figure 1
(a)-(b) Transverse momentum spectra and (c)-(d) rapidity spectra ofZ bosons produced in pp collisions at s=2.76 TeV, for the (a)-(c) dimuon (μμ) and (b)-(d) dielectron (ee) decay channels in |y|<2.0 and |y|<1.44, respectively, where σ on the vertical axis denotes the cross section, and the integral luminosity Lint=5.4 pb-1. The closed squares represent the experimental data of the CMS Collaboration [27], and the error bars are only the statistical uncertainties. The curves are our results calculated by using the (a)-(b) two-component Erlang distribution and (c)-(d) two-component Gaussian distribution, respectively.
(a) (b) (c) (d)Figure2 is similar to Figure 1, but it shows the results in Pb-Pb collisions with 0–100% centrality at sNN=2.76 TeV, where the per-event yield N on the vertical axis is used instead of the cross section σ, and Lint=150 μb−1. The values of free parameters, normalization constants, and χ2/dof are listed in Tables 1 and 2. One can see that the results calculated by using the hybrid model are in agreement with the experimental data of Z bosons produced in Pb-Pb collisions at sNN=2.76 TeV measured by the CMS Collaboration. It should be noted that the ATLAS Collaboration has also published similar experimental results [28, 29]. We can do similar fits for the spectra of pT and y, which are not presented in the present work by design due to the similarity.Figure 2
Same as Figure1, but showing the results in Pb-Pb collisions with 0–100% centrality at sNN=2.76 TeV, where the per-event yield N on the vertical axis is used instead of the cross section σ, and Lint=150 μb−1.
(a) (b) (c) (d)Based on the parameter values obtained from Figures1 and 2, we can perform a Monte Carlo calculation and obtain a series of kinematic quantities. For example, event patterns in two-dimensional pT-y space are presented in Figure 3 for (a)-(c) Z→μμ and (b)-(d) Z→ee channels in (a)-(b) pp collisions at s=2.76 TeV and (c)-(d) Pb-Pb collisions at sNN=2.76 TeV. The total number of particles for each panel is 1000. The squares and circles represent the contributions of the first and second components in the two-component Erlang distribution for pT spectrum, respectively. Most particles appear in the region of low-pT and low-|y|. The values of root-mean-squares (pT2¯ for pT and y2¯ for y) are listed in Table 3. In the calculation, the contributions of the first and second components in the two-component Gaussian distribution for y spectrum are not distinguished by design.Table 3
Values of the root-mean-squarespT2¯ for pT and y2¯ for y corresponding to the particle scatter plots in Figures 3, 6, 9, and 12.
Figure Type p T 2 ¯ (GeV/c) y 2 ¯ Figure3(a) p p, Z→μμ 18.91 ± 0.72 1.50 ± 0.03 Figure3(b) p p, Z→ee 19.05 ± 0.69 1.49 ± 0.03 Figure3(c) Pb-Pb,Z→μμ 20.38 ± 0.66 1.50 ± 0.03 Figure3(d) Pb-Pb,Z→ee 18.82 ± 0.64 1.47 ± 0.03  Figure6(a) p p, J/ψ 3.11 ± 0.09 2.68 ± 0.05 Figure6(b) p p, ψ(2S) 3.40 ± 0.09 2.67 ± 0.05  Figure9(a) p p, Υ(1S) 5.53 ± 0.11 2.26 ± 0.05 Figure9(b) p p, Υ(2S) 6.03 ± 0.11 2.13 ± 0.04 Figure9(c) p p, Υ(3S) 6.49 ± 0.11 2.23 ± 0.04  Figure12(a) Pb-Pb,Υ(1S) 6.26 ± 0.14 1.88 ± 0.04 Figure12(b) Pb-Pb,Υ(2S) 7.80 ± 0.15 1.83 ± 0.03Figure 3
Event patterns (particle scatter plots) in two-dimensionalpT-y space in (a)-(b) pp collisions at s=2.76 TeV and (c)-(d) Pb-Pb collisions at sNN=2.76 TeV, for the (a)-(c) dimuon and (b)-(d) dielectron decay channels. The number of Z bosons for each panel is 1000. The squares and circles represent the results of the first and second components in the two-component Erlang distribution for pT, respectively. The contributions of the first and second components in the two-component Gaussian distribution for y are not distinguished by design.
(a) (b) (c) (d)Figure4 is another example of event pattern in three-dimensional velocity (βx-βy-βz) space for (a)-(c) Z→μμ and (b)-(d) Z→ee channels in (a)-(b) pp collisions at s=2.76 TeV and (c)-(d) Pb-Pb collisions at sNN=2.76 TeV. The total number of particles for each panel is 1000. The red and blue globules represent the contributions of the first and second components in the two-component Erlang distribution for pT spectrum, respectively. Most particles appear in the region of low-|βx| and low-|βy|. The values of root-mean-squares (βx2¯ for βx, βy2¯ for βy, and βz2¯ for βz) and the maximum |βx|, |βy|, and |βz| (|βx|max, |βy|max, and |βz|max) are listed in Table 4. One can see that βx2¯≈βy2¯≪βz2¯, and |βx|max≈|βy|max<|βz|max. The event pattern in velocity (or coordinate) space looks like a rough cylinder along z-axis (the beam direction) when it emits Z bosons, and the maximum velocity surface (the surface consisted of the maximum velocities in different directions) is a fat cylinder which has the length being 1.6–2.2 times the diameter. As for Figure 3, the contributions of the first and second components in the two-component Gaussian distribution for y spectrum are not distinguished by design.Table 4
Values of the root-mean-squaresβx2¯ for βx, βy2¯ for βy, and βz2¯ for βz, as well as the maximum |βx|, |βy|, and |βz| (|βx|max, |βy|max, and |βz|max) corresponding to the particle scatter plots in Figures 4, 7, 10, and 13.
Figure Type β x 2 ¯ (c) β y 2 ¯ (c) β z 2 ¯ (c) | β x | m a x (c) | β y | m a x (c) | β z | m a x (c) Figure4(a) p p, Z→μμ 0.08 ± 0.01 0.09 ± 0.01 0.76 ± 0.01 0.56 0.47 1.00 Figure4(b) p p, Z→ee 0.09 ± 0.01 0.08 ± 0.01 0.76 ± 0.01 0.56 0.60 1.00 Figure4(c) Pb-Pb,Z→μμ 0.09 ± 0.01 0.10 ± 0.01 0.75 ± 0.01 0.65 0.54 1.00 Figure4(d) Pb-Pb,Z→ee 0.08 ± 0.01 0.08 ± 0.01 0.76 ± 0.01 0.54 0.45 1.00  Figure7(a) p p, J/ψ 0.16 ± 0.01 0.17 ± 0.01 0.90 ± 0.01 0.83 0.76 1.00 Figure7(b) p p, ψ(2S) 0.20 ± 0.01 0.21 ± 0.01 0.88 ± 0.01 0.84 0.77 1.00  Figure10(a) p p, Υ(1S) 0.14 ± 0.01 0.14 ± 0.01 0.85 ± 0.01 0.79 0.61 1.00 Figure10(b) p p, Υ(2S) 0.13 ± 0.01 0.13 ± 0.01 0.83 ± 0.01 0.67 0.67 1.00 Figure10(c) p p, Υ(3S) 0.19 ± 0.01 0.19 ± 0.01 0.84 ± 0.01 0.74 0.76 1.00  Figure13(a) Pb-Pb,Υ(1S) 0.19 ± 0.01 0.17 ± 0.01 0.81 ± 0.01 0.81 0.78 1.00 Figure13(b) Pb-Pb,Υ(2S) 0.21 ± 0.01 0.22 ± 0.01 0.81 ± 0.01 0.79 0.83 1.00Figure 4
Event patterns in three-dimensional velocity (βx-βy-βz) space in (a)-(b) pp collisions at s=2.76 TeV and (c)-(d) Pb-Pb collisions at sNN=2.76 TeV, for the (a)-(c) dimuon and (b)-(d) dielectron decay channels. The number of Z bosons for each panel is 1000. The red and blue globules represent the results of the first and second components in the two-component Erlang distribution for pT, respectively. The contributions of the first and second components in the two-component Gaussian distribution for y are not distinguished by design.
(a) (b) (c) (d)In Figure5, the spectra of inclusive (a) J/ψpT, (b) ψ(2S)pT, (c) J/ψy, and (d) ψ(2S)y mesons in pp collisions at s=7 TeV are displayed. Different solid symbols represent different data measured by the ALICE [30–32] and LHCb Collaborations [33, 34] in different conditions shown in the panels, and the error bars are only the statistical uncertainties. The curves for pT and y spectra are our results calculated by using the two-component Erlang distribution for pT spectrum and the two-component Gaussian distribution for y spectrum, respectively. The values of free parameters, normalization constants, and χ2/dof are listed in Tables 1 and 2. One can see that the results calculated by using the hybrid model are in agreement with the experimental data of quarkonium states (charmonium cc¯ mesons J/ψ and ψ(2S)) produced in pp collisions at s=7 TeV measured by the ALICE and LHCb Collaborations.Figure 5
(a)-(b) Transverse momentum spectra and (c)-(d) rapidity spectra of (a)-(c) inclusiveJ/ψ mesons and (b)-(d) inclusive ψ(2S) mesons produced in pp collisions at s=7 TeV. The closed symbols represent the experimental data of the ALICE [30–32] and LHCb Collaboration [33, 34] in different conditions shown in the panels, and the error bars are only the statistical uncertainties. The curves are our results calculated by using the hybrid model.
(a) (b) (c) (d)Based on the parameter values obtained from Figure5, we can obtain some event patterns for J/ψ and ψ(2S). In Figures 6 and 7, the event patterns in pT-y and βx-βy-βz spaces for (a) inclusive J/ψ mesons and (b) inclusive ψ(2S) mesons are given. The red squares and globules represent the contribution of the first component, and the blue circles and globules represent the contribution of the second component, in the two-component Erlang distribution for pT spectrum. The corresponding values of root-mean-squares and the maximum velocity components are listed in Tables 3 and 4. One can see that pT2¯ (y2¯) for charmonium cc¯ mesons is less (greater) than that for Z bosons. Once again, βx2¯≈βy2¯≪βz2¯, and |βx|max≈|βy|max<|βz|max. At the same time, βx2¯ and |βx|max (βy2¯ and |βy|max, βz2¯) for charmonium cc¯ mesons are larger than those for Z bosons, and the two types of particles have nearly the same |βz|max. The event pattern in velocity (or coordinate) space looks like a rough cylinder along the beam direction when it emits charmonium cc¯ mesons, and the maximum velocity surface is a fat cylinder which has the length being 1.2–1.4 times the diameter.Figure 6
Same as Figure3, but showing the results for (a) inclusive J/ψ mesons and (b) inclusive ψ(2S) mesons produced in pp collisions at s=7 TeV.
(a) (b)Figure 7
Same as Figure4, but showing the results for (a) inclusive J/ψ mesons and (b) inclusive ψ(2S) mesons produced in pp collisions at s=7 TeV.
(a) (b)In Figures8(a) and 8(b), the spectra of (a) pT for Υ(1S), Υ(2S), and Υ(3S) and (b) y in backward (forward) region for Υ(1S), Υ(2S), and Υ(3S) produced in pp collisions at s=7 TeV are presented, where σiS and BiS on the vertical axis denote the cross section and branching fractions of Υ(iS) (i=1, 2, and 3), respectively. The closed circles, squares, and triangles represent the experimental data of Υ(1S), Υ(2S), and Υ(3S), respectively, measured by the LHCb Collaboration [35], and the error bars are the total uncertainties which combine the systematic and statistical uncertainties to be equal to their root-sum-square. The curves for pT and y spectra are our results calculated by using the two-component Erlang distribution and the two-component Gaussian distribution, respectively. The values of free parameters, normalization constants, and χ2/dof are listed in Tables 1 and 2. One can see that the results calculated by using the hybrid model are in agreement with the experimental data of quarkonium states (bottomonium bb¯ mesons Υ(1S), Υ(2S), and Υ(3S)) measured by the LHCb Collaboration.Figure 8
(a) Transverse momentum spectra and (b) backward (forward) region rapidity spectra of bottomoniumbb- mesons Υ(1S), Υ(2S), and Υ(3S), as well as whole space rapidity spectra of (c) Υ(1S) and (d) Υ(2S) mesons produced in pp collisions at s=7 TeV, where σiS and BiS on the vertical axis in (a)-(b) denote the cross section and branching fractions of Υ(iS) (i = 1, 2, and 3), respectively. The closed symbols represent the experimental data of the LHCb [35], ALICE [30], and CMS Collaborations [36, 37]. The error bars in (a)-(b) are the total uncertainties which combine the systematic and statistical uncertainties to be equal to their root-sum-square, and the error bars in (c)-(d) are only the systematic uncertainties. The curves are our results calculated by using the hybrid model.
(a) (b) (c) (d)To study further they spectra of bottomonium bb¯ mesons, Figures 8(c) and 8(d) give the results for Υ(1S) and Υ(2S) in pp collisions at 7 TeV, respectively. Different closed symbols represent different data measured by the ALICE [30], LHCb [35], and CMS Collaborations [36, 37] in different conditions shown in the panels, and the error bars are only the systematic uncertainties. The curves are our results calculated by using the two-component Gaussian distribution. The values of free parameters, normalization constants, and χ2/dof are listed in Table 2. One can see that the results calculated by using the revised Landau hydrodynamic model are indeed in agreement with the experimental data of bottomonium bb¯ mesons Υ(1S) and Υ(2S).Based on the parameter values obtained from Figures8(a) and 8(b), we can obtain some event patterns for Υ(1S), Υ(2S), and Υ(3S). In Figures 9 and 10, the event patterns in pT-y and βx-βy-βz spaces for (a) Υ(1S), (b) Υ(2S), and (c) Υ(3S) are given. The red squares and globules represent the contribution of the first component, and the blue circles and globules represent the contribution of the second component in the two-component Erlang distribution. The corresponding values of root-mean-squares and the maximum velocity components are listed in Tables 3 and 4. One can see that pT2¯ (y2¯) for bottomonium bb¯ mesons is close to that for charmonium cc¯ mesons and less (greater) than that for Z bosons. Once more, βx2¯≈βy2¯≪βz2¯, and |βx|max≈|βy|max<|βz|max. At the same time, βx2¯ and |βx|max (βy2¯ and |βy|max, βz2¯) for bottomonium bb¯ mesons are close to those for charmonium cc¯ mesons and larger than those for Z bosons. The three types of particles have nearly the same |βz|max. The event pattern in velocity (or coordinate) space looks like a rough cylinder along the beam direction when it emits bottomonium bb¯ mesons, and the maximum velocity surface is a fat cylinder which has the length being 1.3–1.6 times the diameter.Figure 9
Same as Figure3, but showing the results for (a) Υ(1S), (b) Υ(2S), and (c) Υ(3S) mesons produced in pp collisions at s=7 TeV.
(a) (b) (c)Figure 10
Same as Figure4, but showing the results for (a) Υ(1S), (b) Υ(2S), and (c) Υ(3S) mesons produced in pp collisions at s=7 TeV.
(a) (b) (c)The distributions of inclusiveJ/ψpT, inclusive and prompt J/ψy, Υ(1S) and Υ(2S)pT, and Υ(1S) and Υ(2S)y in Pb-Pb collisions at sNN=2.76 TeV are presented in Figures 11(a)–11(d), respectively, where Y on the vertical axis denotes the yield and TAA denotes the nuclear overlap function which depends on the collision centrality. The closed symbols represent the experimental data of the ALICE [38] and CMS Collaborations [39, 40] measured in different conditions shown in the panels, and the error bars are only the systematic uncertainties. The curves are our results calculated by using the hybrid model in which the pT and y spectra are described by the two-component Erlang distribution and the two-component Gaussian distribution, respectively. The values of free parameters, normalization constants, and χ2/dof are listed in Tables 1 and 2. One can see that the results calculated by using the hybrid model are in agreement with the experimental data of quarkonium states (J/ψ, Υ(1S), and Υ(2S)) measured by the ALICE and CMS Collaborations.Figure 11
(a) Transverse momentum spectra and (b) rapidity spectra ofJ/ψ mesons, as well as (c) transverse momentum spectra and (d) rapidity spectra of Υ(1S) and Υ(2S) mesons, produced in Pb-Pb collisions at sNN=2.76 TeV, where Y on the vertical axis denotes the yield and TAA denotes the nuclear overlap function which depends on the collision centrality. The closed symbols represent the (preliminary) experimental data of the ALICE [38] and CMS Collaborations [39, 40] in different conditions shown in the panels, and the error bars are only the systematic uncertainties. The curves are our results calculated by using the hybrid model.
(a) (b) (c) (d)Because Figures11(a) and 11(b) are the data set from different experimental conditions, we do not extract the event patterns from them. Based on the parameter values obtained from Figures 11(c) and 11(d), we obtain some event patterns for (a) Υ(1S) and (b) Υ(2S) in Figures 12 and 13, where the event patterns in pT-y and βx-βy-βz spaces are given, respectively. The red squares and globules represent the contribution of the first component, and the blue circles and globules represent the contribution of the second component, in the two-component Erlang distribution. The corresponding values of root-mean-squares and the maximum velocity components are listed in Tables 3 and 4. One can see the same or similar conclusion as that from Figures 9 and 10. The maximum velocity surface is a fat cylinder which has the length being 1.2–1.3 times the diameter.Figure 12
Same as Figure3, but showing the results for (a) Υ(1S) and (b) Υ(2S) mesons produced in Pb-Pb collisions at sNN=2.76 TeV.
(a) (b)Figure 13
Same as Figure4, but showing the results for (a) Υ(1S) and (b) Υ(2S) mesons produced in Pb-Pb collisions at sNN=2.76 TeV.
(a) (b)We can give qualitatively a prediction on the prospects to actually getJ/ψ rapidity distributions in Figure 11(b) for the given centrality classes listed in Figure 11(a). From Table 1 we see that 〈pTi〉1 and 〈pTi〉2 for Figure 11(a) increase slightly with decrease of the centrality. As a nonleading particle, the rapidity distribution of J/ψ does not depend obviously on the centrality. These two characteristics render that in the derived event pattern; the particles will scatter slightly in higher pT region in peripheral collisions. From central to peripheral collisions, the increase in pT is slight due to the slight increases in 〈pTi〉1 and 〈pTi〉2.The most important set of parameters in Table1 are the mean contributions, 〈pTi〉1 and 〈pTi〉2, of each source for pT spectrum in the first and second components. Then, the mean contribution of each source in the first + second components is 〈pTi〉=〈pT〉/[k1m1+(1-k1)m2]. The mean transverse momenta contributed by the first and second components are m1〈pTi〉1 and m2〈pTi〉2, respectively. The most important set of parameters in Table 2 are the peak position yF (yB) and distribution width σyF (σyB) for the forward (backward) region in the two-component Gaussian function for y spectrum. To see the dependencies of these parameters on m0, Figure 14 shows the relations between (a) 〈pTi〉 (〈pTi〉1, 〈pTi〉2) and m0, (b) 〈pT〉 (m1〈pTi〉1, m2〈pTi〉2) and m0, (c) yF and m0, and (d) σyF and m0, in pp and Pb-Pb collisions. Different symbols represent different parameters for different collisions shown in the panels. One can see an obvious increase in 〈pTi〉, 〈pTi〉1, 〈pTi〉2, 〈pT〉, m1〈pTi〉1, and m2〈pTi〉2 and a decrease in yF and σyF, when m0 changes from ~3 GeV to ~90 GeV. This mass scale imposes a difference on the observed data from the presence of a quantum chromodynamics (QCD) hard scale.Figure 14
Relations between (a)〈pTi〉 (〈pTi〉1, 〈pTi〉2) and m0, (b) 〈pT〉 (m1〈pTi〉1, m2〈pTi〉2) and m0, (c) yF and m0, and (d) σyF and m0, in pp and Pb-Pb collisions. Different symbols represent different parameters for different collisions shown in the panels, where 〈pTi〉, 〈pTi〉1, and 〈pTi〉2 denote the source’s mean contributions in the first + second, first, and second components, respectively; and 〈pT〉, m1〈pTi〉1, and m2〈pTi〉2 denote the mean transverse momenta contributed by the first + second, first, and second components, respectively.
(a) (b) (c) (d)From Table1 one can see that in some cases k1 is less than 0.5, and 〈pTi〉1 is greater than 〈pTi〉2. These results do not mean a reversal of the first and second components, or a reversal of the low- and high-pT regions. In fact, the first component that contributes to the low-pT region is determined by both m1 and 〈pTi〉1, and the second component that contributes to the high-pT region is determined by both m2 and 〈pTi〉2. The mean transverse momenta, m1〈pTi〉1 and m2〈pTi〉2, contributed by the first and second components, respectively, play a determinate action. Generally, we restrict m1〈pTi〉1<m2〈pTi〉2, which ensures the first component contribution to the low-pT region and the second component contribution to the high-pT region. The size of k1 is not related to the relative sizes of m1〈pTi〉1 and m2〈pTi〉2.We would like to point out thatZ bosons and quarkonium states are chosen by us in the present work due to their productions being through hard scattering and at initial state, which are different from those of light flavor particles which are mostly produced through soft excitation and at intermediate state. Although we describe uniformly the presence of a variable mass scale from 3 to 90 GeV, we cannot provide further information in theory. In fact, the method used in the present work is independent of models, which is only based on the description of probability distribution of experimental data. If we do not use formulas, but discrete probabilities based on data, similar event patters can be obtained. In addition, our choices on data sets are not sole. In fact, other data sets which contained both pT and y spectra can be used to structure the event patterns.It should be noted that in the calculation ofχ2 in Figures 1, 2, 5, 8, and 11, we have to determine the corresponding relation between a theoretical point and an experimental data in a given bin of a variable. According to [41], there are more than one method to determine the corresponding relation, which include considerations in variable and frequency, respectively. In the present work, we have simply used the method of histograms, that is, presenting the theoretical point and experimental data in the form of histograms when we calculate χ2. In most cases, if the bin is narrow enough, to use the closest theoretical point and experimental data is an approximate alternative method.It should also be noted that Figures3, 4, 6, 7, 9, 10, 12, and 13 are supposed to only give pictorial representations of the particle scatter plots at the stage of kinetic freeze-out of the interacting system. These pictorial representations are based on the experimental data themselves. They are independent of models due to (3) and (5) being only parameterizations of the experimental data and (7), (8), (10), and (11) being only stochastic extractions based on (3) and (5). Any other event generators that describe simultaneously the spectra of pT and y can give similar pictorial representations. As we know, concrete and ready-made pictorial representations of the particle scatter plots at the stage of kinetic freeze-out given by other event generators are not available at present.It seems that possible effects from kinematic cuts are not considered for the extracted discrete values and event patterns. As we know, all the experimental data are subject to trigger and event selection restrictions, which have subsequent effects on the kinematic region in which the cross sections are defined. In fact, because our analyses are based on the experimental data themselves, the extracted parameters imply these restrictions. This renders that there are considerations on the impacts of these restrictions on the subsequent event shapes. Because the differences in experimental restrictions forpp and Pb-Pb collisions as well as for spectra of Z bosons and quarkonium states are small, the impacts on the values of parameters, discrete values, and event patterns are small in some details. Generally, these small differences in restrictions do not affect largely the global shapes and contour profiles of events.


## 4. Conclusions

We summarize here our main observations and conclusions.(a)
The transverse momentum and rapidity spectra ofZ bosons produced in pp and Pb-Pb collisions at 2.76 TeV, the same spectra of charmonium cc¯ mesons (J/ψ and ψ(2S)) and bottomonium bb¯ mesons (Υ(1S), Υ(2S), and Υ(3S)) produced in pp collisions at 7 TeV, and the same spectra of J/ψ, Υ(1S), and Υ(2S) produced in Pb-Pb collisions at 2.76 TeV at the LHC are uniformly described by the hybrid model of two-component Erlang distribution for pT spectrum and the two-component Gaussian distribution for y spectrum. The former distribution results from the multisource thermal model, and the latter one results from the revised Landau hydrodynamic model. The modelling results are in agreement with the experimental data measured at the LHC.(b)
In the two-component Erlang distribution (3), the first component that contributes to the low-pT region is determined by both m1 and 〈pTi〉1, and the second component that contributes to the high-pT region is determined by both m2 and 〈pTi〉2. The mean transverse momenta contributed by the first and second components are m1〈pTi〉1 and m2〈pTi〉2, respectively, and the mean transverse momentum contributed by the two components is 〈pT〉=k1m1〈pTi〉1+(1-k1)m2〈pTi〉2. Generally, we restrict m1〈pTi〉1<m2〈pTi〉2, which ensures the first component contribution to the low-pT region and the second component contribution to the high-pT region. The two-component Gaussian function (5) is a revision of the Landau hydrodynamic model.(c)
The most important set of parameters are the mean contributions,〈pTi〉1 and 〈pTi〉2, of each source in the first and second components of the two-component Erlang distribution for pT spectrum and the peak position yF (yB) and distribution width σyF (σyB) for the forward (backward) region in the two-component Gaussian function for y spectrum. The mean contribution of each source in the two-component Erlang distribution is 〈pTi〉=〈pT〉/[k1m1+(1-k1)m2]. From the dependences of related parameters on m0, one can see an obvious increase in 〈pTi〉, 〈pTi〉1, 〈pTi〉2, 〈pT〉, m1〈pTi〉1, and m2〈pTi〉2, and a decrease in yF and σyF, when m0 changes from ~3 GeV to ~90 GeV. This mass scale imposes a difference on the observed data from the presence of a QCD hard scale. Because of some nonviolent collision nucleons existing in the overlapping region, Pb-Pb collisions show more or less lower parameter values than pp collisions in some cases.(d)
Based on the parameter values extracted frompT and y spectra, the event patterns (or particle scatter plots) in two-dimensional pT-y space and in three-dimensional velocity (or coordinate) space are obtained. As a result of hard scattering process at the initial state, both the Z bosons and quarkonium states present some similarities and differences in event patterns. Generally, βx2¯≈βy2¯≪βz2¯, as well as |βx|max≈|βy|max<|βz|max, which renders the fact that the event pattern in velocity (or coordinate) space looks like a rough cylinder along the beam direction when it emits Z bosons and quarkonium states, and the maximum velocity surface is a fat cylinder which has the length being 1.2–2.2 times the diameter. The situations for pp and Pb-Pb collisions do not show an obvious difference.(e)
The cylinder diameters (2βx2¯ or 2|βx|max) or lengths (2βz2¯ or 2|βz|max) corresponding to emissions of charmonium cc¯ mesons and bottomonium bb¯ mesons are nearly the same. Because of large mass, Z bosons correspond to a thinner and shorter cylinder than quarkonium states in terms of 2βx2¯ and 2βz2¯ in velocity space, though the values of |βz|max for both the types of particles are almost the same. Meanwhile, the values of y2¯ for Z bosons are less than those for quarkonium states due to the shorter cylinder corresponding to Z bosons. The values of pT2¯ for Z bosons are greater than those for quarkonium states due to the excitation degree of emission sources for Z bosons being higher than that for quarkonium states. The situations for pp and Pb-Pb collisions do not show an obvious difference.(f)
The present work provides a reference in methodology which displays event patterns (particle scatter plots) for different particles in three-dimensional velocity space or other available spaces based on the transverse momentum and rapidity spectra of considered particles. Because the present analyses are based on the experimental data themselves, the extracted parameters imply experimental restrictions. This renders that the discrete values and event patterns extracted by us are accurate to the best of our ability. It is expected that different particles correspond to different event patterns in different shapes with different sizes. To give a better comparison, the same experimental restrictions for different particles are needed in the future.



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