#### Abstract

Transverse momentum distributions of final-state particles produced in soft process in proton-proton (*pp*) and nucleus-nucleus (*AA*) collisions at Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies are studied by using a multisource thermal model. Each source in the model is treated as a relativistic and quantum ideal gas. Because the quantum effect can be neglected in investigation on the transverse momentum distribution in high energy collisions, we consider only the relativistic effect. The concerned distribution is finally described by the Boltzmann or two-component Boltzmann distribution. Our modeling results are in agreement with available experimental data.

#### 1. Introduction

High energy collisions are an important research topic in particle and nuclear physics. The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) did firstly collider experiments on heavy ions [1], and the center-of-mass energy per nucleon pair at the RHIC reached highly 200 GeV [2]. The Large Hadron Collider (LHC) at European Laboratory for Particle Physics (CERN) renovated value of to TeV region [3]. It seems that a new state of matter, namely, Quark-Gluon Plasma (QGP), is possibly formed in heavy ion collisions at RHIC and LHC energies due to high temperature and density [4, 5]. At initial stage of high energy collisions, another possible new state of matter, namely, color glass condensate (CGC), is caused by strong color fields in the low-gluon realm [6, 7], wheredenotes the ratio of quark or gluon momentum to hadron one. A CGC is in fact a region of the nuclear wave function at low-andand exists already before the collisions, wheredenotes the square momentum of virtual photon. On the other hand, the CGC may not be a new state, but more like a model or calculation for initial state hadron behavior. One cannot measure the QGP and CGC directly. However, one can measure final-state particle spectra at freeze-out to extract thermal and other characteristics of interacting system and give a judgment on formation and property of the new matters.

The final-state particle spectra include rapidity (or pseudorapidity ) [8, 9], transverse momentum (or transverse mass) [10, 11], transverse energy [12, 13], and other distributions [14]. It is known that and distributions reflect, respectively, the degrees of longitudinal extension and transverse excitation of interacting system. Especially, for transverse excitation, soft excitation and hard scattering processes can affect, respectively, distributions in low- and high- ranges. The soft and hard processes correspond to different physics mechanisms and distribution laws [15]. In low energy collisions, the soft process is main process, and the hard process can be neglected due to almost zero contribution. In high energy collisions, although the hard process cannot be neglected, the soft process is still main process.

To understand the transverse excitation, we need firstly to study the soft excitation process. A lot of models have been introduced to describe the soft process, although some of them can be used to describe the hard process too [16, 17]. Among the models, the multisource thermal model proposed by us is a very simple one and can be used to describe spectra in both the soft and hard processes if source’s contribution is given by an Erlang distribution [18]. Finally, the considered distribution is described by a multicomponent Erlang distribution [19, 20]. Different from some simulation codes, our model gives directly a few statistical laws by analytical expressions in describing some quantities. In the case of being incapable of analytical expressions, we could use a Monte Carlo method to give a numerical result. Our model is easy to be used by experimental experts.

Due to significances of the considered model and topic, in this paper, based on Boltzmann distribution for a single source, we describe spectra of final-state particles produced in soft process in proton-proton () and nucleus-nucleus () collisions at RHIC and LHC energies. Some interesting results are obtained.

#### 2. The Model

According to the multisource thermal model, many emission sub-sources of final-state particles are assumed to form in high energy collisions [19, 20]. These multiple sub-sources can be different regions in the overlap region or different mechanisms, and these particles can be created/emitted at different times in the collisions. In fact, these sub-sources can be divided into different groups (sources) due to different interacting mechanisms or event samples. Obviously, soft process corresponds to sources with low degree of excitation or to particles with low transverse momentum, and hard process corresponds to sources with high degree of excitation or to particles with high transverse momentum, where the excitation means to create particles through string breaking, direct scattering, recombination, and their hybrid.

In the rest frame of a source, we consider the source as a thermodynamic system of relativistic and quantum ideal gas. The momentum () distribution of final-state particles in the natural unit system is given by [21] where is the number of particles, is the normalization constant, is the rest mass of a considered particle, is the chemical potential, is the temperature parameter, denotes fermions, and denotes bosons, respectively. Our calculations show that at RHIC and LHC energies the quantum effect and chemical potential can be neglected compared to the relativistic effect [22]. Then, we have a simple expression for momentum distribution to be [23, 24] where and is the modified Bessel function of order 2.

The distribution can be written as a Boltzmann distribution [25]: where is the normalization constant. Because of interactions among different sources, the considered source has a deformation and/or movement in the transverse plane. Letdenote a relative deformation and letdenote an absolute movement of the source; that is, we use instead of in (3). The revised distribution can be given by In the case of considering multiple sources, we have or where , , and denote the contribution ratio, normalization constant, and temperature of theth source, respectively. Because the effects of deformation and movement of the source can be neglected in the calculation of transverse momentum, we take the default values of and in the revised distribution, which results from the Boltzmann distribution or a multicomponent Boltzmann distribution.

We should have a few sources to describe the soft and hard processes. For the soft process, the number of sources is generally 1 or 2. For the hard process, the number of sources is also 1 or 2. The total number of sources will be from 2 to 4 for a wide distribution. In this paper, we pay our attention on the soft process which has a narrow distribution. It is hard to say that what the distribution range is for the soft process. What we can say is that for low energy collisions the distribution range is narrower. In the present work, we regard the distribution range as 0–10 GeV/c. The difference between the single and multisource models is obvious. The former one describes a narrower distribution which corresponds to an equilibrium state with a lower degree of excitation. The latter one describes a wider distribution which corresponds to a few local equilibrium sates with different excitations.

#### 3. Comparisons with Experimental Data

Figure 1 presents the transverse momentum distributions of (a) , and , (b) , and , (c) , (d) , and (e) produced in collision at center-of-mass energy GeV with different ranges and magnifications shown in the figure. The symbols represent the experimental data of the STAR [26] (Figures 1(a) and 1(b)) and BRAHMS Collaborations [27] (Figures 1(c)–1(e)), and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution. In the calculation, we have used a fitting method to obtain parameter values which are shown in Table 1 with values of per degree of freedom (). To give a short presentation, the values corresponding to “negative/positive” charged particles are given in terms of “the first value/the second value” or “value” in the case of the first value and the second value being the same. We would like to point out that the presenting styles of rapidity ranges for Figures 1(a) and 1(b) as well as for Figures 1(c)–1(e) are different due to different presentations in [26, 27]. One can see that the modeling results with 1 or 2 sources are in agreement with the experimental data. For emissions of , and , the temperature parameter increases with increase of particle mass (Figures 1(a) and 1(b)), which indicates the impact of radial flow and/or the early emission of heavy hadrons. The temperature does depend nonobviously on rapidity range (Figures 1(c)–1(e)).

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Figure 2 shows the distributions of produced in (a) and (b) collision at GeV, (c) *d*-Au collisions at GeV, and (d) *p*-Pb collisions at beam energy being 400 GeV with different *y* ranges, global scale uncertainty (GSU) (or invariant mass ()) ranges, and magnifications shown in the figure, where anddenote the cross section and dilepton branching ratio, respectively. The symbols represent the experimental data of the PHENIX [28] (Figures 2(a) and 2(c)) [29, 30] (Figure 2(b)) and NA50 Collaborations [31] (Figure 2(d)), and the curves are our results calculated by the Boltzmann or two-component Boltzmann distributions. The values of parameters and are given in Table 2. We see again that the model with 1 or 2 sources describes the experimental data. For emission of the temperature parameter does depend nonobviously on rapidity range (Figures 2(a) and 2(c)).

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The distributions of identified charged particles produced in-Au collisions at GeV, Cu-Cu collisions at GeV, Au-Au collisions at , 130, and 200 GeV with different centrality classes are displayed in Figures 3–7, respectively. The symbols represent the experimental data of the STAR [26] (Figures 3, 5, 6, and 7) and PHENIX Collaboration [32] (Figure 4), and the curves are our results calculated by the Boltzmann or two-component Boltzmann distributions. Correspondingly, the values of parameters and are given in Tables 3–7, respectively. Once more, the model with 1 or 2 sources describes the experimental data. From Tables 5, 6, and 7 we see clearly that for emissions of , , and (Figures 5(a), 6(a), and 7(a)), as well as , , and (Figures 5(b), 6(b), and 7(b)), the temperature parameter increases with increases of particle mass, impact centrality, and , where we would like to point out that a large centrality (a small percentage) corresponds to a small impact parameter. The similar conclusions can be obtained from Tables 3 and 4.

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Figure 8 gives the distributions of (a)–(c) identified particles and (d) charged particles in range of in nonsingle diffraction (NSD) produced in collision at GeV, where and denote numbers of events and charged particles, respectively. The symbols represent the experimental data of the ALICE Collaboration [33, 34] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution. The values of parameters and are given in Table 8. We see that the model with 1 or 2 sources describes the experimental data. For emissions of charged hadrons, the temperature parameter increases with increase of particle mass.

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The distributions of identified particles produced in (a) central Pb-Pb collisions at TeV, (b) Pb-Pb collisions with different centralities at TeV and inelastic collision at TeV, (c) central rapidity region in Pb-Pb collisions with different centralities at TeV, and (d) central rapidity region in collision at TeV are presented in Figure 9. The symbols represent the experimental data of the ALICE Collaboration [35–38] and the curves are our results calculated by the Boltzmann or two-component Boltzmann distribution. The values of parameters and are given in Table 9. We see that in most cases the model with 1 or 2 sources describes the experimental data. Especially, for emissions of , , and (Figure 9(a)), the temperature parameter increases with increase of particle mass; for emission of (Figure 9(b)), the temperature parameter does not depend on impact centrality; and for emission of (Figure 9(c)), the temperature parameter increases with increase of impact centrality (or with decrease of impact parameter).

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From Tables 1–9 we see that some values of are too low pointing to overestimated errors of the experimental points. In fact, in the case of errors being not available in related references, we have used a half size of the experimental points to give the errors. This treatment may cause larger errors in some cases.

#### 4. Discussions and Conclusions

From the above discussions we see that the model used in the present work is just a simple phenomenology which does not contain other processes such as parton-hadron string dynamics, hydrodynamic flows, and resonances. The successful description renders that the mentioned processes should contribute a higher transverse momentum at multi-GeV energy or a refined structure in distribution curve. In the concerned transverse momentum region and for the concerned distribution curves, we just need to consider the Maxwell-Boltzmann thermal law.

The present work is justified to compare fits in a low transverse momentum region (<10 GeV/c) for different particles by the same thermal law. Although the difference for charged and neutral particles is unlikely due to Coulomb effects which are important for very soft charged particles only, both the charged and neutral particles obey the same thermal law. The transverse momentum can extend to more than 100 GeV/c at multi-GeV energy. The distribution in the low transverse momentum region is mainly contributed by the soft processes. The hard processes which contribute high transverse momentums can be partly described by the thermal law.

To conclude, we have used the multisource thermal model to describe the transverse momentum distributions of particles produced in the soft process in and collisions at RHIC and LHC energies. For single source, the relativistic ideal gas model is applied in description of particle behavior. The concerned distribution is finally described by single source or two sources which result from a Boltzmann or two-component Boltzmann distributions. The modeling results are in agreement with available experimental data, which renders that an equilibrium or two local equilibriums are reached in high energy collisions. Because of the evolvement time of interesting system in collisions being very short, the particles should reach rapidly to the state of equilibrium.

The present work can be used to extract nuclear temperature for soft process. For emissions of charged hadrons, the temperature parameter increases with increases of particle mass, impact centrality, and center-of-mass energy and does depend nonobviously on rapidity range. That the temperature increases with particle mass indicates the impact of radial flow and/or the early emission of heavy hadrons. The temperature parameter for emission of does depend nonobviously on rapidity range too, which is consistent with charged hadrons. However, for emission of the temperature parameter does not depend on impact centrality, which is inconsistent with charged hadrons. Different behaviors for and render different production mechanisms. Especially, there are Coulomb corrections for emissions of charged particles, which affects the extraction of temperature [39].

The values of temperature parameter for emissions of are about 160–190 MeV which reaches the temperature (166–172 MeV) of creating QGP at zero baryon-chemical potential, where 172 MeV is the equilibrium phase transition temperature and 166 MeV is due to finite hadron size [40]. In most cases the temperature for emission of heavy hadrons is greater than that for pions, which renders the impact of radial flow and/or the early emission of heavy hadrons in collisions.

#### Conflict of Interests

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

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 10975095 and no. 11247250, the China National Fundamental Fund of Personnel Training under Grant no. J1103210, the Open Research Subject of the Chinese Academy of Sciences Large-Scale Scientific Facility under Grant no. 2060205, and the Shanxi Scholarship Council of China.