Comparing Erlang distribution and Schwinger mechanism on transverse momentum spectra in high energy collisions

We study the transverse momentum spectra of J/psi and Upsilon 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 (pp), proton-lead (p-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.


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][2][3]. The new matter is named the quark-gluon plasma (QGP) or quark matter. This prediction makes the research of high energy collisions developing rapidly. A lot of physics researchers devote in research the mechanisms of particle productions and the properties of QGP formation. Because of the reaction time of the impacting system being very short, people could not make a direct measurement for the collision process. So, only by researching the final state particles, people can presumed 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 major roles in high energy collisions. Other quantities which also play major roles include, but not limited to, pseudorapidity (or rapidity) distribution, azimuthal distribution (anisotropic flow), particle ratio, various correlations, etc. [5]. Presently, many formulas such as the standard (Fermi-Dirac, Bose-Einstein, or Boltzmann) distribution [6], the Tsallis statistics [7][8][9][10][11], the Tsallis form of standard distribution [11], the Erlang distribution [12], the Schwinger mechanism [13][14][15][16][17], and others 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 J  and  mesons produced in proton-proton (pp), proton-lead (p-Pb), and lead-lead (Pb-Pb) collisions measured by the LHCb and ALICE Collaborations at the large hadron collider (LHC) [18][19][20][21]. 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.

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 denote the mean transverse momentums contributed by each parton in the first and second component respectively. Each parton is assumed to contribute an exponential transverse momentum ( t p ) spectrum. For the jth (j=1 and 2) parton in the ith component, we have the distribution The transverse momentum ( T p ) in final state is The transverse momentum distribution in final state is the folding of 1 1 ( ) In the Monte Carlo method, let 1,2 R denote random numbers in [0,1]. For the ith component, we have We 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. The Monte Carlo method performs a simpler calculation for the folding. Secondly, we consider the two-component Schwinger mechanism. Let 1  and 2  denote the string tensions between the two partons in the first and second component respectively, 1 , and 0 m denotes the rest mass of a parton. According to [13][14][15][16][17], for the jth (j=1 and 2) parton in the given string in the ith component, we have the distribution In the Monte Carlo method, let 3  ln cos 2 ln cos 2

Results
The transverse momentum spectra,    Table 1. One can see that the two methods describe the experimental data of the LHCb Collaboration.  present the transverse momentum spectra, mesons produced in the same collision respectively, where n=1, 2, and 3 correspond to the three mesons respectively, and B denotes the branching ratio. The symbols represent the experimental data of the LHCb Collaboration [19] and the curves are our results.
mesons produced in pp collision at s  8 TeV. The symbols represent the experimental data of the LHCb Collaboration [19]. The dashed and solid curves are our results calculated by using the two-component Erlang distribution and the two-component Schwinger mechanism respectively. 6     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][8][9][10][11] in Figure 5 by the dotted curves. A simplified form of the Tsallis transverse momentum distribution is used, where T is the temperature parameter, q is the entropy index,  is the chemical potential which is regarded as 0 at the LHC, and T C is the normalization constant. In Eq. (6), the effect of longitudinal motion is subtracted by using 0 y  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 0.285±0.008, 0.273±0.006, and 0.264±0.009 GeV, the entropy index is taken to be 1.067±0.003, 1.080±0.004, and 1.085±0.007, with 2 dof  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 2 dof  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, as we as parameters and others, we plot the values listed in Table 1 in Figures 6-11, which correspond to the relationships related to parameters

Discussions
To discuss on the Schwinger mechanism, if the charmonium ( cc ) or bottomonium ( bb ) can be produced in the collision, the minimum distance between the two partons, for the ith component, is [17] 2 2 min, 0 where 0 m is the mass of produced charmed or bottom quark, but not that of the participant parton. The mean minimum distance 2 2 min, 0 The minimal minimum distance   min, 0 min 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    min, 1 i N dN dL , where N denotes the number of charmoniums. 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 is 2 2 max, min, 0 The mean maximum potential energy 2 2 max, min, 0 The maximal maximum potential energy  , and 2  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 2  , 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. Partons in projectile nucleon have large probability to close exceedingly to partons in target nucleons. At the same time, both the contribution ratios of the two second components in the two calculation methods are large and cannot be neglected.

Conclusions
From the above discussions, we obtain the following conclusions: (a) The transverse momentum spectra of J  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