Advances in High Energy Physics

Volume 2019, Article ID 6739315, 17 pages

https://doi.org/10.1155/2019/6739315

## Rapidity Dependent Transverse Momentum Spectra of Heavy Quarkonia Produced in Small Collision Systems at the LHC

^{1}Department of Physics, Taiyuan Normal University, Jinzhong, Shanxi 030619, China^{2}Institute of Theoretical Physics and State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan, Shanxi 030006, China

Correspondence should be addressed to Fu-Hu Liu; moc.361@uiluhuf

Received 14 March 2019; Accepted 5 May 2019; Published 22 May 2019

Academic Editor: Carlos Pajares

Copyright © 2019 Li-Na Gao 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

The rapidity dependent transverse momentum spectra of heavy quarkonia ( and mesons) produced in small collision systems such as proton-proton (*pp*) and proton-lead (*p*-Pb) collisions at center-of-mass energy (per nucleon pair) () = 5-13 TeV are described by a two-component statistical model which is based on the Tsallis statistics and inverse power-law. The experimental data measured by the LHCb Collaboration at the Large Hadron Collider (LHC) are well fitted by the model results. The related parameters are obtained and the dependence of parameters on rapidity is analyzed.

#### 1. Introduction

The study of high energy proton-proton, proton-nucleus, and nucleus-nucleus collisions [1–5] can provide a unique opportunity for ones to understand the strong interaction theory and nuclear reaction mechanism [6–10] and analyze the evolution processes of interacting system and quark-gluon plasma (QGP). At the same time, by this study, one can examine the standard model and other phenomenological models or statistical methods [11–14] and search for new physics beyond the standard model. This study also provides new information for people to understand the origin of the universe. As the basic element in nuclear collisions, proton-proton collisions are worth studying. Meanwhile, as a transition from proton-proton collisions to nucleus-nucleus collisions, proton-nucleus collisions are also worth studying.

With the development of modern experimental and detecting technology, the collision energy has been continuously improved. Meanwhile, more and more information about collision process can be accurately measured in experiments [15–19]. Because the collision time of interacting system is very short, one can only analyze the characteristics of final particles produced in the collisions to obtain the mechanisms of nuclear reactions and the properties of formed matter such as QGP.

Generally, the information of nuclear reactions in experiments can be obtained by measuring the transverse momentum spectrum and correlation, pseudorapidity or rapidity spectrum and correlation, anisotropic flow distribution and correlation, multiplicity distribution and correlation, nuclear modified factor, and so forth [15–19]. The transverse momentum spectrum is one of the most general objects in the study. It is measured by experiments and provides information about temperature and excitation degree of interacting system at the stage of kinetic freeze-out. Therefore, the study of transverse momentum spectrum of final particles is greatly significative in analyzing the mechanisms of nuclear reactions and the properties of QGP.

Many theoretical models and formulas have been applied for the descriptions of transverse momentum spectra. These models and formulas include, but are not limited to, the Boltzmann-Gibbs statistics [1–3], Lévy distribution [4, 5], Erlang distribution [6], Tsallis statistics [7–14], and so on. In this paper, we use a two-component statistical model to describe the experimental transverse momentum spectra of heavy quarkonia and mesons produced in small collision systems such as proton-proton (*pp*) and proton-lead (*p*-Pb) collisions. The data quoted by us are measured by the LHCb Collaboration [15–18] at the Large Hadron Collider (LHC), though other data are available [19]. The two-component statistical model is based on the Tsallis statistics and inverse power-law.

In the following sections, we describe the formulism of the two-component statistical model in Section 2. The results and discussion are given in Section 3. Finally, the conclusions of the present work are given in Section 4.

#### 2. The Formulism

Within the framework of the multisource thermal model [20–22], the emission sources of final particles produced in high energy collisions can be divided into several groups due to different interacting mechanisms, impact parameter ranges (centrality classes), or event samples. A typical classification is soft excitation and hard scattering processes [23–26], and even including very-soft excitation and very-hard scattering processes. Generally, one can use different models and formulas to describe different processes. In some cases, one can use the same model and formula to describe different processes. In other cases, one can use different models and formulas to describe the same process.

The Tsallis statistics has been widely applied for high energy collisions [27–31]. It describes different particle spectra in different processes, but not the heavy quarkonium spectra in very-hard process in some cases. For the soft and very-soft processes, the Boltzmann-Gibbs statistics [1–3] also play a main role in the description. For the hard and very-hard processes, an inverse power-law [32–35] play the main role in the description. For the transverse momentum () spectra of heavy quarkonia ( and mesons) produced in collisions at the LHC, we need a superposition of the Tsallis statistics and the inverse power-law, which is a two-component statistical model.

In the Tsallis statistics [27–30], the invariant momentum (*p*) distribution iswhere* E* is the energy,* N* is the particle number, is the degeneracy factor,* V* is the volume, is the transverse mass, is the rest mass,* T* is the temperature parameter,* q* is the entropy index, and is the chemical potential. The normalized distribution can be given byIn the mid-rapidity (*y* = 0) region, the formulism of Tsallis statistics can be given by [31]where is the normalization constant related to the free parameters. When the collision energy is high enough, the chemical potential is especially small. In the energy range of LHC, the value of approximately is zero [27–29].

In some cases, the experimental data are presented in a given rapidity range, which is generally not in the mid-rapidity region. We have to shift simply the given rapidity range to the mid-rapidity region by subtracting the mid-value of the given rapidity range and use (3) directly. If we consider the differences of rapidity in the given rapidity range or in the mid-rapidity region, a more accurate equation (2) which includes the integral for the rapidity can be used. If we consider the given rapidity range in the more accurate equation (2), the kinetic energy of directional movement will be included in the temperature, which causes a larger temperature and is not correct. In fact, in the mid-rapidity region, the difference between the minimum (maximum) rapidity and 0 is neglected. The more accurate equation (2) is not needed.

It should be noted that when we use the multisource thermal model and the Tsallis statistics, each group or process is assumed to stay in a local equilibrium state. The excitation degree of each group or process is described by the temperature parameter* T*, and the equilibrium degree is described by the entropy index* q*. A large* T* corresponds to a high excitation degree, and a large* q* () corresponds to a far away from the equilibrium state. The closer to 1 the* q* is, the closer to equilibrium the group or process becomes. In an equilibrium state, one has . Generally,* q* is not too large. This means that each group or process stays approximately in a local equilibrium state.

The inverse power-law can describe the hard and very-hard processes. In [32–34], the inverse power-law is described by the Hagedorn function [35]; its parameterized form is expressed aswhere and* n* are free parameters and* A* is the normalization constant related to the free parameters.

In the Hagedorn function, scattering between nucleons may be thought of in terms of valence quarks. To measure the scattering strength, the parameters and* n* can be used. A large and a small* n* describe a wide range which means a violent scattering. Impact between quarks may also be described via pQCD (perturbative quantum chromodynamics), which gives an inverse power-law spectrum [32–34] which is the same as the Hagedorn function [35]. The pQCD also gives rapidity dependent spectra which results in rapidity dependent and* n*.

According to (3) and (4), we can structure a superposition of the Tsallis statistics and the inverse power-law, which results in a two-component statistical model aswhere* k* is the contribution ratio of the first component. Naturally, (5) is normalized to 1 due to the fact that (3) and (4) are normalized to 1. Although the Tsallis statistics has more than one forms and the inverse power-law has different modified forms, we shall not discuss them further. In fact, (5) structured through (3) and (4) is enough to use in the present work.

It should be noted that there are two types of superposition for two components. Except for (5), another superposition is the step function or the Hagedorn model [35]where and are constants which ensure the contributions of two components are the same at , and if , and if . Although there are entanglements in determining parameters by (5), the curve at is not smooth due to (6). Our very recent work [36] shows that (5) and (6) result in similar values of parameters, especially for the trends. To obtain a smooth curve, (5) is used in the present work.

For a real fit process, we may select firstly a set of free parameters. Then, we may use the selected set of parameters in (3) and (4), and let the two equations be normalized to 1, respectively. The normalization constants and* A* can be determined and used back in (3) and (4) so that the two equations can be used in (5). In the determination for the parameters, the method of least squares can be used. The errors of the parameters can be determined to let the confidence levels of fittings be 95% in most cases and 90% in a few cases if existent.

#### 3. Results and Discussion

Figure 1 shows the transverse momentum spectra, , of mesons produced in* pp* collisions at center-of-mass energy TeV, where denotes the cross section. Figures 1(a)–1(d) present the results of the prompt with no polarisation, from* b* with no polarisation, prompt with full transverse polarisation, and prompt with full longitudinal polarisation, respectively. The symbols represent the experimental data measured by the LHCb Collaboration [15] at the LHC. In order to see clearly, different symbols are used to distinguish the different rapidity ranges in the panels. The curves are our results fitted by (5). The values of free parameters (*k*,* T*,* q*, , and* n*) and (degree of freedom) corresponding to each curve in Figure 1 are listed in Table 1, where the normalization constants which reflect the areas under the curves are not listed to avoid trivialness. For the same reason, the concrete confidence levels are not listed in the table one by one. One can see that the experimental data measured by the LHCb Collaboration are well fitted by the two-component statistical model. The behaviors of parameters will be discussed later.