Abstract

The transverse momentum distributions of final-state particles are very important for high energy collision physics. In this work, we investigate and meson distributions in the framework of a particle-production source, where Tsallis statistics are consistently incorporated. The results are in good agreement with the experimental data in and -Pb collisions at LHC energies. The temperature of the emission source and the nonequilibrium degree of the collision system are extracted.

1. Introduction

The investigation of nuclear matter at high energy densities is the main purpose of Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) [1–7]. As a new matter state, quark-gluon plasma (QGP) is a thermalized system, which consists of strongly coupled quarks and gluons in a limited region. The suppression of meson with respect to proton-proton () collisions is regarded as a distinctive signature of the QGP formation and brings valuable insight into properties of the nuclear matter. In proton-nucleus collisions, the prompt meson suppression has also been observed at large rapidity [8]. The low-viscosity QGP may be created in the small system, 3He + Au collisions [9]. The heavy quarkonium production can be suppressed by the suppression cold-nuclear-matter (CNM) effects, such as nuclear absorption, nuclear shadowing (antishadowing), and parton energy loss.

The transverse momentum spectra of identified particles produced in the collisions are a vital research for physicists. Now, different models have been suggested to describe the distributions of the final-state particles in high energy collisions [10–13], such as Boltzmann distribution, Rayleigh distribution, Erlang distribution, the multisource thermal model, and Tsallis statistics. Different phenomenological models of initial coherent multiple interactions and particle transport have been proposed to discuss the particle production in high energy collisions. Tsallis statistics can deal with nonequilibrated complex systems in condensed-matter research [14]. It is developed to describe the particle production in recent years [15–24].

In our previous work [12], the temperature information was understood indirectly by an excitation degree. We have obtained the emission source location dependence of the exciting degree specifically. In this paper, the temperature of the emission source is given directly by combining a picture of the particle-production source with Tsallis statistics. We discuss the transverse momentum distributions of in collisions at  TeV,  TeV, and  TeV and -Pb collisions at  TeV. And the distributions in collisions at  TeV are also taken into account for comparison.

2. Tsallis Statistics in an Emission Source

According to the multisource thermal model [12] and the nuclear geometry picture, at the initial stage of the collision, two cylinder-shaped groups are formed along the beam direction. In the laboratory reference frame, it is assumed that the projectile cylinder is at the positive space and the target cylinder is at the negative space. The cylinders are not a real shape and are understood to be a rapidity range of the emission source. The projectile and target cylinders can be regarded as an emission source with a rapidity width. The observed particles are emitted from the emission source.

With Tsallis statistics’ success in dealing with nonequilibrated complex systems in condensed-matter research [14], it has been used to understand the particle production in high energy physics. In order to describe the transverse momentum spectra in high energy collisions, several versions of Tsallis distribution are proposed [15–24]. Recently, an improved form of the Tsallis distribution was proposed [18–20] and can meet the thermodynamic consistency. The meson number [16, 17] is given bywhere , , , , and are the degeneracy factor, the volume, the momentum, the energy, and the chemical potential, respectively. The parameter is the temperature of the emission source and the parameter is the nonequilibrium degree. Generally, is greater than 1 and is close to 1. The corresponding momentum distribution is given byThen, the transverse momentum distribution isWhen the chemical potential is neglected, at midrapidity , the distribution isIt is worth noting that the distribution function of is only the distribution of mesons emitted from an emission point at in the emission source, not the final-state distribution due to the nonzero rapidity width of the emission source. By summing the contributions of all emission points, the transverse momentum distribution is rewritten as where is a normalize constant and and are the maximum and minimum values of the observed rapidity.

3. Transverse Momentum Spectra and Discussions

Figure 1 shows the double-differential cross section of mesons in collisions at  TeV. Figures 1(a), 1(b), 1(c), and 1(d) present prompt with no polarisation, from with no polarisation, prompt with full transverse polarisation, and prompt with full longitudinal polarisation, respectively. The experimental data in bins are from the LHCb Collaboration [25]. The solid lines indicate our model results, which are in good agreement with the experimental data in all rapidity ranges. The parameters and taken for the calculation are listed in Table 1. In different rapidity ranges, the values of the temperature are different and decrease with increasing the rapidity bins in all four figures of Figure 1. The closer the emission source is to the center , the larger the excitation degree is. The values of do not change regularly with the rapidity bins.

Table 1 
Values of and used in Figures 1-2.
(a)
(a)
(b)
(b)
(c)
(c)
(d)
(d)
Figure 1 
Double-differential cross section as a function of in bins of for (a) prompt with no polarisation, (b) from with no polarisation, (c) prompt with full transverse polarisation, and (d) prompt with full longitudinal polarisation in collisions at  TeV. The symbols indicate the experimental data of the LHCb Collaboration [15]. The solid lines indicate our theoretical calculations.

Figure 2 presents the double-differential cross section of mesons in collisions at  TeV. Figures 2(a) and 2(b) are prompt and from , respectively. Figure 3 presents the double-differential cross section of mesons in the same collision, where 1, 2, and 3 correspond to , , and mesons, respectively, and is the dimuon branching fraction. The experimental data are from [26]. The symbols and lines represent the same meanings as those in Figure 1. The model results are also in agreement with the experiment data. The parameters and for the calculation are listed in Tables 1 and 2. The freeze-out temperature also decreases with increasing the rapidity bins for both and . In the same ranges, the values are smaller for than that for . For , the values are generally larger than that at  TeV. for and for are 1.003–1.038 and 1.170–1.127, respectively. At  GeV, the agreement is not very good in Figures 1-2 and the values are in the range 0.926–1.385.

Table 2 
Values of and used in Figures 3–5.
(a)
(a)
(b)
(b)
Figure 2 
Double-differential cross section as a function of in bins of for (a) prompt and (b) from in collisions at 8 TeV. The symbols indicate the experimental data of the LHCb Collaboration [25]. The solid lines indicate our theoretical calculations.
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(a)
(b)
(b)
(c)
(c)
Figure 3 
Double-differential cross section as a function of in bins of for (a) , (b) , and (c) mesons in collision at 8 TeV. The symbols indicate the experimental data of the LHCb Collaboration [25]. The solid lines indicate our theoretical calculations.

Figure 4 presents the double-differential cross section of mesons produced in collisions at  TeV. The experimental data are from [27]. The symbols and lines represent the same meanings as those in Figure 1. The results are also in agreement with the experiment data. The and values taken for the calculation are listed in Table 2. The temperature decreases with increasing the rapidity bins. As the emission source draws closer to the center, the excitation degree gets larger. The values are larger than that at  TeV in the same range. The behavior is similar to that in Figures 1 and 2.

(a)
(a)
(b)
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Figure 4 
Same as Figure 2, but for  TeV. The experimental data are taken from the LHCb Collaboration [26].

For comparison, Figure 5 presents the double-differential cross section of mesons in -Pb collisions at  TeV. The experimental data are from [8]. The symbols and lines represent the same meanings as those in Figure 1. The results are also in agreement with the experiment data. The parameters and taken for the calculation are listed in Table 2. The temperature of the emission source decreases with increasing the rapidity bins. The values are between 1.031 and 1.065. In the calculation, the least-square-fitting testing provides a statistical indication of the most probable values of the two parameters. The values of corresponding to the curves in Figures 1–5 are given in Tables 1 and 2. By the comparison between the model results and experimental data, we obtain the temperature of the emission source. For particles in the same bins, the temperature increases with increasing the collision energy. In collision at  TeV, the temperature is smaller for than that for in the same ranges.

(a)
(a)
(b)
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Figure 5 
Same as Figure 2, but for -Pb collisions at  TeV. The experimental data are taken from the LHCb Collaboration [27].

4. Conclusions

Final-state particle production in high energy collisions has attracted much attention since the attempt has been made to understand the properties of strongly coupled QGP. Thermal-statistical models have been successful in describing particle yields in various systems at different energies. The emission source temperature is very important for understanding the matter evolution in collisions at high energy. In the rapidity space, different final-state particles are emitted from different positions due to stronger longitudinal flow. In our previous work [12], we have studied the transverse momentum spectra of strange particles produced in collisions at and 200 GeV in the improved fireball model. The temperature of the emission source was characterized indirectly by the excitation degree, which varies with location in the cylinder. In the present work, we can directly extract the specific temperature by Tsallis statistics.

In this work, we have embedded consistently the Tsallis statistics into the picture of the multisource thermal model for descriptions of the transverse momentum spectra of and mesons produced in and -Pb collisions over an energy range from 5 to 13 TeV, with different rapidity intervals. In most cases, the model results are in good agreement with the experimental data of the LHCb Collaboration. The agreement is not very good at  GeV in Figures 1-2. In other collisions, the maximum value is 0.531 and the minimum value is 0.124. By comparing with the experimental data, the temperature of the emission source and the nonequilibrium degree of the collision system are extracted. In and -Pb collisions, the temperature decreases with increasing the rapidity bins. The closer the emission source is to the center, the larger the excitation degree is. For particles in the same bins, the temperature increases with increasing the collision energy. So, the excitation degree of the emission source also increases with increasing the collision energy. In collision at  TeV, the temperature is smaller for than that for in the same ranges. The parameter does not change significantly, which means the collision system is not very unstable.

In summary, the transverse momentum spectra of and mesons produced in and -Pb collisions at high energies have been studied by combining a picture of the particle-production source with Tsallis statistics. The model results are compared with the experimental data of the LHCb Collaboration. Our investigations show the improved model is successful in the description of the transverse momentum distributions. At the same time, it can offer the temperature of the emission source and the information about the nonequilibrium degree in the collisions.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant nos. 11575103, 11247250, and 10975095, the National Fundamental Fund of Personnel Training under Grant no. J1103210, and the Shanxi Provincial Natural Science Foundation under Grant no. 2013021006.