Abstract

In Tsallis statistics, we investigate charged pion and proton production for pCu and pPb interactions at 3, 8, and 15 GeV/c. Two versions of Tsallis distribution are implemented in a multisource thermal model. A comparison with experimental data of the HARP-CDP group shows that they both can reproduce the transverse momentum spectra, but the improved form gives a better description. It is also found that the difference between q and q′ is small when the temperature T = T′ for the same incident momentum and angular interval, and the value of q is greater than q′ in most cases.

1. Introduction

Heavy ion collisions at Large Hadron Collider (LHC) are essential for the investigation of strongly interacting matter at high-energy density [1]. Proton-nucleus () program is used as a baseline measurement for nucleus-nucleus () collisions and is also crucial to discuss the various domains of quantum chromodynamics (QCD). So, the collision has been considered as an important component of the heavy ion collisions [2]. Measurements of the transverse momentum spectra of identified particles, to some extent, can provide an insight into the dynamics of the colliding systems.

In recent years, many models have been proposed in the interpretation of the spectra in high-energy collisions, such as multisource thermal model [3–5], diffusion model [6], and Tsallis statistics [7–11]. In particular, Tsallis distribution has successfully reproduced the spectra and has aroused interest of scientists recently. The statistics can extract two parameters, Tsallis temperature and , which is used to characterize a degree of nonequilibrium in the system. For example, a Tsallis-like distribution has given excellent descriptions to the experimental data, which have been measured by the STAR [7] and PHENIX [8] collaborations at the RHIC and by the ALICE [9], ATLAS [10], and CMS [11] collaborations at the LHC. Generally, the Tsallis parameter goes to 1. A thermodynamically consistent form of Tsallis statistics has also been proposed to fit the transverse momentum spectra [12, 13]. In our previous work [14], we have consistently embedded the improved form of the Tsallis distribution into a multisource thermal model to describe systematically pseudorapidity distributions in (), AuAu, CuCu, and PbPb collisions at RHIC and LHC energies. The result shows that a rapidity shift of longitudinal sources needs to be considered. In this paper, we will use the Tsallis distributions with the rapidity shift to analyze proton and charged pion distributions in and interactions at 3, 8, and 15 GeV/c in the hadron production (HARP) experiment at CERN [15]. The results obtained from the two forms of Tsallis distribution are compared in detail.

The paper is organized as follows: in Section 2, the improved Tsallis distribution is introduced and the results are compared with the experimental data; at the end, we give discussions and conclusions in Section 3.

2. Tsallis Statistics Description of the Transverse Momentum Spectra

According to Tsallis statistics, the particle number iswhere , , , , and are the degeneracy factor, the volume, the momentum, the energy, and the chemical potential, respectively. The parameters and are temperature and nonequilibrium factor, respectively. The distribution of the corresponding momentum is given byIn terms of the transverse momentum and the rapidity , the distribution function is

For , at midrapidity , the transverse momentum distribution isThe Tsallis distribution is a quantum form, which can meet the thermodynamic consistency [10, 11]. Approximately, the can equal 1 and the distribution is given by The two distribution functions both represent a single spectrum of one source at . Therefore, we need to consider the distribution width of the rapidity of final-state particles [14]. Equations (4) and (5) will be used in the following analysis.

Figures 1, 2, and 3 show , , and double-differential cross-sections as a function of the transverse momentum in collisions, respectively. From left to right, the incident proton momenta are 3, 8, and 15 GeV/c, respectively. And from top to bottom, the angular intervals are 30°–40°, 60°–75°, and 105°–125°, respectively. The symbols denote the experimental data measured in the hadron production (HARP) experiment at CERN [15]. The solid lines are results fitted by (4) and the dashed lines are results fitted by (5). The Tsallis parameters , , , and are given in Tables 1 and 2. From the figures, one can see that the results of (4) and (5) are in agreement with the experimental data in the whole observed region, but (4) can give a better fit. The values of and show slight difference when for the fixed incident momentum and angular interval. And, in most cases, . The minimum difference is 0.002 and the maximum difference is 0.02. For proton production, the scaling properties behave well at . The case is similar for production at 60°–75° and 105°–125°.

Table 1 
Values of , , , and taken in Figures 1 and 2.
Table 2 
Values of , , , and taken in Figures 3 and 4.
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Figure 1 
Transverse momentum spectra of protons produced in collisions at 3, 8, and 15 GeV/c at different angular intervals. The symbols represent the HARP-CDP experimental data [15]. The solid curves and dashed lines are the results calculated by (4) and (5), respectively.
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Figure 2 
Same as Figure 1, but for production.
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Figure 3 
Same as Figure 1, but for production.

The transverse momentum spectra of , , and produced in interactions are displayed in Figures 4, 5, and 6, respectively. The symbols denote the HARP-CDP experimental data [15]. The solid lines and dashed lines are fitting results from (4) and (5), respectively. Both the solid lines and dashed lines pass through the experimental data points, but the results of (4) are in better agreement with the data. The values of , , , and taken in the calculations are listed in Tables 2 and 3. The parameters and correspond to (4) and and correspond to (5). Similar to Figures 1–3, the difference between and is small when for the same incident momentum and angular interval, and in most cases. The maximum value of the difference is 0.02.

Table 3 
Values of , , , and taken in Figures 5 and 6.
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Figure 4 
Same as Figure 1, but for collisions.
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Figure 5 
Same as Figure 4, but for production.
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Figure 6 
Same as Figure 4, but for production.

3. Discussions and Conclusions

We combine the picture of the multisource thermal model and Tsallis statistics to investigate the transverse momentum spectra of protons and charged pions produced in the collisions of 3, 8, and 15 GeV/c protons on Cu and Pb at fixed angles of 30°–40°, 60°–75°, and 105°–125°. In practice, we choose (5), where tends to 1. Equation (4) has been improved from (5) to satisfy the thermodynamic consistency [12, 13]. By comparing their results to the HARP-CDP data, it is found that they both agree with the experimental data and (4) can better reproduce the transverse momentum spectra. In addition, we notice that the values of and show slight difference when for the same incident momentum and angular interval, and the value of is greater than in most cases.

In the present work, we focus on the two versions of Tsallis distribution in the picture of the multisource production for the description of the transverse momentum spectra of produced particles. According to the multisource thermal model [16, 17], many emission sources of produced particles and nuclear fragments are formed in and interactions. Every source was regarded as a thermal equilibrium system, which is comprised approximately of ideal gases. Understandably, the Maxwell distribution was selected in the discussion of the particle production of high-energy collisions [18, 19]. So, it is only an approximate classical method. If the relativity effect and quantum effect are considered, the improved Tsallis distribution [20–22] is a better choice. The observed particles are emitted isotropically in the rest frame of emission sources with the different excitation degree. The Tsallis distributions are embedded consistently into the model. The rapidity location in the framework of the Tsallis description is tightly linked to the rapidity (pseudorapidity) shift of the emission sources. And, the rapidity width is taken into account in the analysis of final-state particles. By the multisource-production discussion, the Tsallis statistics can not only describe the transverse momentum spectra but also obtain the underlying physical picture of the particle production in high-energy collisions.

Conflict of Interests

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

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grants no. 11247250, 11005071, and 10975095; the National Fundamental Fund of Personnel Training under Grant no. J1103210; the Shanxi Provincial Natural Science Foundation under Grants no. 2013021006 and 2011011001.