Advances in High Energy Physics

Volume 2016, Article ID 9876253, 19 pages

http://dx.doi.org/10.1155/2016/9876253

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

^{1}Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China^{2}Departments of Chemistry & Physics, Stony Brook University, Stony Brook, NY 11794, USA

Received 23 April 2016; Revised 26 August 2016; Accepted 21 September 2016

Academic Editor: Burak Bilki

Copyright © 2016 Ya-Hui Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Transverse momentum () and rapidity () spectra of bosons and quarkonium states (some charmonium mesons such as and and some bottomonium mesons such as , , and ) 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 spectrum and two-component Gaussian distribution for 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 and 7 TeV and in Pb-Pb collisions at center-of-mass energy per nucleon pair TeV. Based on the parameter values extracted from and spectra, the event patterns (particle scatter plots) in two-dimensional - space and in three-dimensional velocity space are obtained.

#### 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 () and rapidity () 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 - space and three-dimensional velocity space can be extracted from parameters fitted to and 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 the and spectra to obtain the probability density functions for and for . 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 and based on and . Other quantities such as energy, momentum components, velocity, and velocity components can be obtained according to some definitions and assumptions. Because for and for 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 for spectrum (which results from a multisource thermal model [15–17]) and two-component Gaussian distribution for spectrum (which results from the Landau hydrodynamic model and its revisions [18–26]), we analyze together and spectra of bosons and quarkonium states (some charmonium mesons such as and and some bottomonium mesons such as , , and ) produced in pp collisions at center-of-mass energies and 7 TeV and in Pb-Pb collisions at center-of-mass energy per nucleon pair TeV. The modelling results are in agreement with the experimental data measured at the LHC. Based on the parameters extracted from and spectra, the event patterns (particle scatter plots) at kinetic freeze-out in two-dimensional - 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 Section 2. 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 and 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 spectrum. According to the model, many () emission sources which stay at the same excitation state are assumed to form in high energy collisions. Each (the th) source is assumed to contribute to transverse momentum by an exponential function:where denotes the average value of , which results in (1) being a probability distribution and . The sources which contribute to result in an Erlang distribution [17]:which is the folding of exponential distributions and has the average transverse momentum .

In the case of considering the two-component Erlang distribution, we havewhich has the average transverse momentum , where and denote the relative contributions of the first and second components which contribute to the low- and high- regions, respectively, and the subscripts and 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 () which is used to fit the data is needed.

On 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 spectrum can be described by a Gaussian function [25, 26]:where denotes the rapidity distribution width and denotes the midrapidity (peak position). In symmetric collisions, is in the center-of-mass reference frame.

In the case of considering the two-component Gaussian function for spectrum, we havewhere (), (), and () 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 , , and . As probability distributions, (4) and (5) are normalized to 1. When we compare them with experimental data, a normalization constant () which is used to fit the data is needed.

*Secondly*, we need discrete values of and . The related calculation is performed by a Monte Carlo method. Let , , , and denote random numbers in . Equations (2)–(5) result infor the first component in the low- region, orfor the second component in the high- region,for the first component in the backward rapidity region, orfor the second component in the forward rapidity region, respectively.

It should be clarified that the random numbers used above are independent in . Through the conversion equations (7), (8), (10), and (11), we can obtain a series of new values which are no longer independent in 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 instead of (9), we obtain an accustomed expression which result from the Rayleigh distribution which is different from the Gaussian function.

The energy is given bywhere denotes the rest mass of the considered particle. The -, -, and -components of momentum are given byrespectively, where is the azimuthal angle to distribute evenly in . Combining with (12) and (13), we have the velocityand its componentsrespectively. All velocity and its components are in the units of 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 and 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 per degree freedom (/dof) corresponds to the best values of parameters. Appropriate increases or decreases in parameters determine the uncertainties on parameters, where an appropriate large /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 and 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

Figure 1 presents (a)-(b) transverse momentum spectra, , and (c)-(d) rapidity spectra, , of bosons produced in collisions at TeV for the (a)-(c) dimuon () and (b)-(d) dielectron () decay channels in and , respectively, where on the vertical axis denotes the cross section, and the integral luminosity . 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 spectrum and the two-component Gaussian distribution for 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 [, , , , , (), and ()], normalization constants ( and ), and /dof are listed in Tables 1 and 2, where the normalization constant (or ) is used to give comparison between the normalized curves with experimental (or ) spectrum. One can see that the results calculated by using the hybrid model are in agreement with the experimental data of bosons produced in collisions at TeV measured by the CMS Collaboration. In some cases, the values of are very large due to very small experimental errors.