#### 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.

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Figures 2(a) and 2(b) present the transverse momentum spectra, , of mesons produced in data samples, prompt and from , in collision at TeV, respectively. Figures 3(a), 3(b), and 3(c) present the transverse momentum spectra, , of , , and mesons produced in the same collision, respectively, where , 2, and 3 correspond to the three mesons, respectively, and denotes the branching ratio. The symbols represent the experimental data of the LHCb Collaboration [18] and the curves are our results. Figure 4 is similar to Figure 2, but it presents the results in -Pb collisions at center-of-mass energy per nucleon pair TeV, and the experiment data of the LHCb Collaboration are taken from [19]. The values of free parameters and corresponding to Figures 2–4 are listed in Table 1. Similar conclusion obtained from Figure 1 can be obtained from Figures 2–4, though some of fitting results are approximately in agreement with the data.

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The transverse momentum spectra, , of produced in three centrality classes and in Pb-Pb collisions at TeV are given in Figure 5, where denotes the yield. The symbols represent the experimental data of the ALICE Collaboration [20]. The dashed and solid curves are our results calculated by using the two-component Erlang distribution and the two-component Schwinger mechanism, respectively. The values of free parameters and are listed in Table 1. Similar conclusion obtained from Figure 1 can be obtained from Figure 5.

To give a comparison of fit quality with some other approach, as an example, we show the result of the Tsallis statistics [7–11] in Figure 5 by the dotted curves. A simplified form of the Tsallis transverse momentum distributionis used, where is the temperature parameter, is the entropy index, is the chemical potential which is regarded as 0 at the LHC, and is the normalization constant. In (8), the effect of longitudinal motion is subtracted by using to obtain the temperature parameter as accurately as possible. For the centrality from 0–20% to 40–90%, the temperature parameter is taken to be , , and GeV; the entropy index is taken to be , , and , with to be 0.969, 1.290, and 1.781, respectively. One can see the compatibility of the three approaches, which shows the multiformity of fit functions.

In the above fit to the experimental data of LHCb and ALICE Collaborations, the uncorrelated and correlated uncertainties in experimental data are together included in the calculation of by using the quadratic sums. No matter for the part of uncorrelated or correlated uncertainty, even for the part of correlated, especially multiplicative, common for all bins uncertainties, we just use the experimental values directly.

To see clearly the relationships between parameters and rapidity, parameters and centrality, and parameters and others, we plot the values listed in Table 1 in Figures 6–11, which correspond to the relationships related to parameters , , , , , and , respectively. In these figures, the symbols and lines are parameter values and fitting lines, respectively. The intercepts, slopes, and related to the lines in Figures 6–11 are listed in Table 2. In the error range, and do not show obvious change with rapidity, centrality, energy, and size. , , , and do not show obvious change with rapidity and centrality, or they decrease slightly with increases of rapidity and centrality.

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#### 4. Discussions

To discuss the Schwinger mechanism, if the charmonium () or bottomonium () can be produced in the collision, the minimum distance between the two partons, for the th component, is [16]where is the mass of produced charmed or bottom quark but not that of the participant parton. The mean minimum distance is as follows:The minimal minimum distance is as follows:We see that the string tension is an important parameter which is related to the minimum distance, the mean minimum distance, and the minimal minimum distance. Correspondingly, the distribution of the minimum distance can be obtained by , where denotes the number of charmonia.

Further, if the produced charmed or bottom quark stays at haphazard at the middle between the two participant partons, the maximum potential energy of the charmed quark staying in the colour field of the two partons isThe mean maximum potential energy is as follows:The maximal maximum potential energy is as follows:Correspondingly, the distribution of the maximum potential energy can be obtained by .

From Table 1, we see that the values of , , , and in the collisions at the LHC are large, which render that the interactions between partons are violent and the minimal minimum distance between the two interacting partons is small. According to , the minimal minimum distance is ~0.03–0.06 fm which is a few percent of nucleon size. We believe that, in the collisions at the LHC, nucleons penetrate through each other totally. Projectile partons are exceedingly close to target partons with large probability. At the same time, both the contribution ratios of the two second components in the two calculation methods are large and cannot be neglected.

#### 5. Conclusions

From the above discussions, we obtain the following conclusions:(a)The transverse momentum spectra of and mesons produced in high energy collisions are described by using both 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 , -Pb, and Pb-Pb collisions at the LHC.(b)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. At the same time, the contribution ratios and of the two first components in the two methods are obtained.(c)In the error range, and do not show obvious change with rapidity, centrality, energy, and size. , , , and do not show obvious change with rapidity and centrality, or they decrease slightly with increases of rapidity and centrality. Both the contribution ratios of the two second components cannot be neglected. The minimal minimum distance between interacting partons is ~0.03–0.06 fm which is a few percent of nucleon size.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant no. 11575103.