Advances in High Energy Physics

Volume 2016, Article ID 1505823, 15 pages

http://dx.doi.org/10.1155/2016/1505823

## Comparing Erlang Distribution and Schwinger Mechanism on Transverse Momentum Spectra in High Energy Collisions

Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China

Received 13 October 2015; Accepted 9 December 2015

Academic Editor: Ming Liu

Copyright © 2016 Li-Na Gao and Fu-Hu Liu. 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

We study the transverse momentum spectra of and mesons by using two methods: the two-component Erlang distribution and the two-component Schwinger mechanism. The results obtained by the two methods are compared and found to be in agreement with the experimental data of proton-proton (), proton-lead (-Pb), and lead-lead (Pb-Pb) collisions measured by the LHCb and ALICE Collaborations at the large hadron collider (LHC). The related parameters such as the mean transverse momentum contributed by each parton in the first (second) component in the two-component Erlang distribution and the string tension between two partons in the first (second) component in the two-component Schwinger mechanism are extracted.

#### 1. Introduction

In the last century, scientists predicted that a new state of matter could be produced in relativistic heavy-ion (nucleus-nucleus) collisions or could exist in quark stars owing to high temperature and high density [1–3]. The new matter is named the quark-gluon plasma (QGP) or quark matter. This prediction makes the research of high energy collisions develop rapidly. A lot of physics researchers devoted themselves to researching the mechanisms of particle productions and the properties of QGP formation. Because of the fact that the reaction time of the impacting system is very short, people could not make a direct measurement for the collision process. So, only by researching the final state particles, people can presume the evolutionary process of collision system. For this reason, people proposed many models to simulate the process of high energy collisions [4].

The transverse momentum (mass) spectra of particles in final state are an important observation. They play one of the major roles in high energy collisions. Other quantities which also play major roles include, but are not limited to, pseudorapidity (or rapidity) distribution, azimuthal distribution (anisotropic flow), particle ratio, and various correlations [5]. Presently, many formulas such as the standard (Fermi-Dirac, Bose-Einstein, or Boltzmann) distribution [6], the Tsallis statistics [7–11], the Tsallis form of standard distribution [11], the Erlang distribution [12], and the Schwinger mechanism [13–16] are used in describing the transverse momentum spectra. It is expected that the excitation degree (effective temperature and kinetic freeze-out temperature), radial flow velocity, and other information can be obtained by analyzing the particle transverse momentum spectra.

In this paper, we use two methods, the two-component Erlang distribution and the two-component Schwinger mechanism, to describe the transverse momentum spectra of and mesons produced in proton-proton (), proton-lead (-Pb), and lead-lead (Pb-Pb) collisions measured by the LHCb and ALICE Collaborations at the large hadron collider (LHC) [17–20]. The related parameters such as the mean transverse momentum contributed by each parton in the first (second) component in the two-component Erlang distribution and the string tension between two partons in the first (second) component in the two-component Schwinger mechanism are extracted.

#### 2. Formulism

We assume that the basic impacting process in high energy collisions is binary parton-parton collision. We have two considerations on the description of violent degree of the collision. A consideration is the mean transverse momentum contributed by each parton. The other one is the string tension between two partons. The former consideration can be studied in the framework of Erlang distribution. The latter one results in the Schwinger mechanism. Considering the wide transverse momentum spectra in experiments, we use the two-component Erlang distribution and the two-component Schwinger mechanism. Generally, the first component describes the region of low transverse momentum, and the second one describes the high transverse momentum region.

*Firstly*, we consider the two-component Erlang distribution. Let and denote the mean transverse momentums contributed by each parton in the first component and second component, respectively. Each parton is assumed to contribute an exponential transverse momentum () spectrum. For the th ( and 2) parton in the th component, we have the distributionThe transverse momentum () in final state is . The transverse momentum distribution in final state is the folding of and . Considering the contribution ratio of the first component, we have the (simplest) two-component Erlang distribution to beIn Monte Carlo method, let denote random numbers in . For the th component, we haveWe would like to point out that the folding of multiple exponential distributions with the same parameter results in the ordinary Erlang distribution which is not used in the present work. Monte Carlo method performs a simpler calculation for the folding.

*Secondly*, we consider the two-component Schwinger mechanism. Let and denote the string tensions between the two partons in the first component and second component, respectively, , , and denotes the rest of the mass of a parton. According to [13–16], for the th ( and 2) parton in the given string in the th component, we have the distributionwhereis the normalization constant. Considering the contribution ratio of the first component, we have the two-component Schwinger mechanismIn Monte Carlo method, let denote random numbers in . For the th component, we have

#### 3. Results

The transverse momentum spectra, , of mesons produced in different data samples in collision at center-of-mass energy TeV are shown in Figure 1, where and denote the cross section and rapidity, respectively. The symbols represent the experimental data measured by the LHCb Collaboration [17] in different rapidity ranges and scaled by different amounts marked in the panels. The dashed and solid curves are our results calculated by using the two-component Erlang distribution and the two-component Schwinger mechanism, respectively. From Figures 1(a)–1(d), the data samples are prompt with no polarisation, from with no polarisation, prompt with full transverse polarisation, and prompt with full longitudinal polarisation, respectively. The values of free parameters and per degree of freedom () are listed in Table 1. One can see that the two methods describe the experimental data of the LHCb Collaboration.