Can Tsallis Distribution Fit All the Particle Spectra Produced at RHIC and LHC?

Abstract

The Tsallis distribution has been tested to fit all the particle spectra at mid-rapidity from central events produced in d + Au, Cu + Cu, and Au + Au collisions at RHIC and p + Pb, Pb + Pb collisions at LHC. Even though there are strong medium effects in Cu + Cu and Au + Au collisions, the results show that the Tsallis distribution can be used to fit most of particle spectra in the collisions studied except in Au + Au collisions where some deviations are seen for proton and Λ at low [subscript] p T [/subscript] . In addition, as the Tsallis distribution can only fit part of the particle spectra produced in Pb + Pb collisions where [subscript] p T [/subscript] is up to 20 GeV/c, a new formula with one more fitting degree of freedom is proposed in order to reproduce the entire [subscript] p T [/subscript] region.

H. Zheng 1 and Lilin Zhu 2

Academic Editor:Xiaochun He

1, INFN, Laboratori Nazionali del Sud, Via Santa Sofia 62, 95123 Catania, Italy
2, Department of Physics, Sichuan University, Chengdu 610064, China

Received 24 April 2015; Revised 15 June 2015; Accepted 9 July 2015; 29 July 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³ .

1. Introduction

The heavy-ion collision experiments at RHIC and LHC give us the opportunity to study the phase transition from nuclear matter to quark gluon plasma (QGP), the collective motion, the nuclear medium effects, and so on. The particle spectrum is one of the basic quantities measured in experiments to address the questions raised in such studies. Recently, the Tsallis distribution has attracted the attention of many theorists and experimentalists in high energy heavy-ion collisions [1-25]. It has been applied to particle spectra produced in different reaction systems, from pp, pA to AA, to understand the particle production mechanism and extract physical quantities, for example, temperature [2-4, 7, 12-14, 17-21, 24] and chemical potential [26]. In pp collisions, the excellent ability to fit the spectra of identified hadrons and charged particles in a large range of [figure omitted; refer to PDF] up to 200 GeV/c is quite impressive [2, 23-25]. A systematic investigation of particle spectra in p + p collisions at RHIC and LHC has been conducted in [2]. The results show that the Tsallis distribution can fit all the particle spectra at different energies in p + p collisions. A possible cascade particle production mechanism is proposed. Recently, a Tsallis distribution scaling function was found for charged hadron spectra in p + p and [figure omitted; refer to PDF] collisions [1]. Comparing to nucleus-nucleus collisions, the pp collision is very simple. It has been used as a baseline for nucleus-nucleus collisions. A nuclear modification factor [figure omitted; refer to PDF] or [figure omitted; refer to PDF] was proposed to show the nuclear medium effects in pA or AA collisions referring to pp collisions [5, 10-12, 27-34]. The nuclear modification factor different from unity is a manifestation of medium effects. Many authors have successfully applied Tsallis distribution to fit particle spectra in pA and AA collisions even though the spectra were affected by nuclear medium modification [3-8, 10, 11, 18, 21]. We also notice that many works only show the small [figure omitted; refer to PDF] part of the particle spectra, while the exponential distribution also can fit the low [figure omitted; refer to PDF] region [11, 35]. It should cover all [figure omitted; refer to PDF] regions of particle spectra, available in experiment, in order to show the advantage and/or the fitting power of the Tsallis distribution. In recent years, the experimental groups at RHIC and LHC have published the wide [figure omitted; refer to PDF] range of particle spectra for different particles in different reaction systems. Such data allow us to conduct the systematic study of particle spectra in heavy-ion collisions at RHIC and LHC using the Tsallis distribution, as we have done for p + p collisions [2].

In this work, we would like to test whether the Tsallis distribution can fit all the particle spectra produced at RHIC and LHC, which can help us to understand the particle production mechanism. Before we start to conduct our investigation, we can get some clues to estimate whether it can fit the particle spectrum or not from the nuclear modification factor. If [figure omitted; refer to PDF] or [figure omitted; refer to PDF] is flat for the whole [figure omitted; refer to PDF] region, according to its definition, this means that the particle spectrum is similar in shape and only differs in magnitude to the one in p + p collisions. Based on the previous studies [1, 2], we are sure that the Tsallis distribution can fit the particle spectrum since it can fit all the particle spectra produced in p + p collisions, especially up to extremely high [figure omitted; refer to PDF] [2, 23-25]. In the pA reactions, [figure omitted; refer to PDF] are flat and very close to 1 for most of the produced particles at different centralities [5, 10, 11, 27, 29, 34], while in the AA collisions the nuclear medium effects play an important role. [figure omitted; refer to PDF] increases from the most central collisions to peripheral collisions [11, 12, 28, 30-33]. Since the nuclear modification factors of different particles are almost 1 in peripheral heavy-ion collisions [5, 10-12, 27-34], as we discussed, Tsallis distribution should be able to fit the particle spectra. Therefore we will only focus on the particle spectra at the most central collisions where the Tsallis distribution may not fit all of them. We have collected data of particle spectra from d + Au, Cu + Cu, and Au + Au collisions at RHIC and p + Pb, Pb + Pb collisions at LHC and select the data for most of the particles with the highest [figure omitted; refer to PDF] GeV to conduct this study.

The paper is organized as follows. In Section 2, we show different versions of the Tsallis distribution used in the literature. More details can be found in [2]. We also give the form of Tsallis distribution used in our analysis. In Section 3, we show our results of particle spectra from d + Au, p + Pb, Cu + Cu, Au + Au, and Pb + Pb. Another distribution is proposed to fit the particle spectra in Pb + Pb collisions since Tsallis distribution can only fit part of the particle spectra in the case. A brief conclusion is given in Section 4.

2. Tsallis Distributions

In the literature, several versions of Tsallis distribution with different arguments can be found [2-25]. The asymptotic behaviors of these distributions at low and high [figure omitted; refer to PDF] limits can be found in [2]. We only briefly show them here.

The STAR [9] and PHENIX [5, 12] Collaborations at RHIC along with ALICE [13, 14] and CMS [15] Collaborations at LHC adopted this form of Tsallis distribution: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the transverse mass and [figure omitted; refer to PDF] is the mass of the particle. [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are fitting parameters.

In [8, 19-22, 36], the following Tsallis form is used: [figure omitted; refer to PDF] based on thermodynamic consistency arguments. Where [figure omitted; refer to PDF] is the degeneracy of the particle, [figure omitted; refer to PDF] is the volume, [figure omitted; refer to PDF] is the rapidity, [figure omitted; refer to PDF] is the chemical potential, [figure omitted; refer to PDF] is the temperature, and [figure omitted; refer to PDF] is a parameter. In (2), there are four parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . [figure omitted; refer to PDF] was assumed to be 0 in [19-21, 36] which is a reasonable assumption because the energy is high enough and the chemical potential is small compared to temperature. In the mid-rapidity [figure omitted; refer to PDF] region, (2) is reduced to [figure omitted; refer to PDF] In [4, 21], (2) has been rewritten as [figure omitted; refer to PDF] to take into account the width of the corresponding rapidity distribution of the particles.

In [17], Sena and Deppman applied the nonextensive formalism to obtain the probability of particle with momentum [figure omitted; refer to PDF] as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the normalization constant, [figure omitted; refer to PDF] is a parameter, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is the mass of particle. With the approximation [figure omitted; refer to PDF] very large compared to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] [37], (5) can be rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the beta-function.

In [24], Wong and Wilk proposed a new form of the Tsallis distribution function to take into account the rapidity cut: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] with [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is assumed to be a constant.

In [2], we have obtained [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the fitting parameters. This is equivalent to (1) but in a simpler form. We adopt (10) to do the analysis here. We notice that (10) has been used by the CMS Collaboration [16, 38] and by Wong et al. in their recent paper [25]. The STAR Collaboration has also applied a formula which is very close to (10) [29].

3. Results

We have selected the data of particle spectra from the most central collisions with the highest [figure omitted; refer to PDF] GeV for most of the particles in d + Au, p + Pb, Cu + Cu, A u + Au, and Pb + Pb at RHIC and LHC. We fit the center values of the experimental points. The fit metric used is defined by [figure omitted; refer to PDF]

As we discussed before, the Tsallis distribution should be able to fit the particle spectra from d + Au and p + Pb. One good example has been shown in [4, 5] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in d + Au at [figure omitted; refer to PDF] GeV where their spectra can be obtained by multiplying the particle spectra in p + p collisions with [figure omitted; refer to PDF] . In Figures 1 and 2, our results for d + Au at [figure omitted; refer to PDF] GeV and p + Pb at [figure omitted; refer to PDF] TeV using (10) have been shown. In order to see the agreement between the data and the Tsallis distribution in linear scale, a ratio data/fit is defined. As shown in Figures 1 and 2, the fits for all particles are good. For the left collision systems, we also do the same comparisons. We would like to emphasize that [figure omitted; refer to PDF] of charged particle spectrum is up to 45 GeV/c.

Figure 1: The data are from [10, 11, 27, 39-42] for d + Au at [figure omitted; refer to PDF] GeV. The curves are the analytical results with Tsallis distribution Equation (10). The corresponding fitting parameters and [figure omitted; refer to PDF] /ndf are given in Table 1. For a better visualization both the data and the analytical curves have been scaled by a constant as indicated. The ratios of data/fit are shown at the bottom.

[figure omitted; refer to PDF]

Table 1: The fitting parameters and the corresponding [figure omitted; refer to PDF] /ndf for various particles in different collision systems with Tsallis distribution Equation (10).

System	Particle	Centrality	[figure omitted; refer to PDF]	[figure omitted; refer to PDF] (GeV)	[figure omitted; refer to PDF]	[figure omitted; refer to PDF] /ndf
[figure omitted; refer to PDF] + Au [figure omitted; refer to PDF] GeV	[figure omitted; refer to PDF]	Minimum bias	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-20%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-20%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-20%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-20%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-20%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-20%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-20%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-20%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF] + Pb [figure omitted; refer to PDF] TeV	Charged	Minimum bias	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-5%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-5%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-5%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Au + Au [figure omitted; refer to PDF] GeV	[figure omitted; refer to PDF]	0-10%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-5%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-10%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-5%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-5%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-20%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Cu + Cu [figure omitted; refer to PDF] GeV	[figure omitted; refer to PDF]	0-10%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-10%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-10%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-20%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-10%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-10%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-10%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-10%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-20%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Au + Au [figure omitted; refer to PDF] GeV	[figure omitted; refer to PDF] low [figure omitted; refer to PDF]	0-20%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF] high [figure omitted; refer to PDF]	0-5%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-10%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-10%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-5%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-20%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-12%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-10%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-5%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	0-20%	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Figure 2: The data are from [34, 43] for p + Pb at [figure omitted; refer to PDF] TeV. The curves are the analytical results with Tsallis distribution Equation (10). The corresponding fitting parameters and [figure omitted; refer to PDF] /ndf are given in Table 1. For a better visualization both the data and the analytical curves have been scaled by a constant as indicated. The ratios of data/fit are shown at the bottom.

[figure omitted; refer to PDF]

Now let us turn to the AA collisions. First we use Tsallis distribution Equation (10) to fit the particle spectra in Au + Au collisions at [figure omitted; refer to PDF] GeV in Figure 3. All the particle spectra are well fitted except the proton spectrum at [figure omitted; refer to PDF] GeV/c. This makes a little difference of AA collisions from p + p collisions. We want to check whether this deviation will become larger at higher colliding energy in AA collisions. We considered the particle spectra from Cu + Cu collisions at [figure omitted; refer to PDF] GeV. The results are shown in Figure 4. The fitting with (10) for different particle spectra is very well. But we do not know whether there is deviation or not for proton at low [figure omitted; refer to PDF] since the data for [figure omitted; refer to PDF] GeV/c are not available. Fortunately, the data for different particle spectra at low [figure omitted; refer to PDF] in Au + Au collisions at [figure omitted; refer to PDF] GeV are given. In Figure 5, one can see the deviations of particle spectra of proton and [figure omitted; refer to PDF] at low [figure omitted; refer to PDF] from the Tsallis distribution Equation (10). While a deviation is observed for proton at [figure omitted; refer to PDF] GeV/c which becomes a little larger than the one in Au + Au at [figure omitted; refer to PDF] GeV, all other particle spectra are well fitted. This makes us curious to fit the particle spectra in Pb + Pb collisions at [figure omitted; refer to PDF] TeV. With the successful running at LHC, the identified hadron particle spectra data in Pb + Pb collisions at [figure omitted; refer to PDF] TeV are available up to 20 GeV/c. The data satisfy two criteria. One is that there are strong nuclear medium effects in Pb + Pb collisions which can be seen from [figure omitted; refer to PDF] and the other is that the transverse momenta of the particles reach high values. This gives us an opportunity to test the fitting power of the Tsallis distribution. When we use Tsallis distribution Equation (10) to fit pion spectrum, we find that (10) can only fit part of it. If we choose to fit the low [figure omitted; refer to PDF] region, (10) can fit the particle spectrum up to 10 GeV/c, as shown in Figure 6 with red dashed line. The blue dotted-dashed line shows the fit for high [figure omitted; refer to PDF] region which starts from 4 GeV/c.

Figure 3: The data are from [30, 44] for Au + Au at [figure omitted; refer to PDF] GeV. The curves are the analytical results with Tsallis distribution Equation (10). The corresponding fitting parameters and [figure omitted; refer to PDF] /ndf are given in Table 1. For a better visualization both the data and the analytical curves have been scaled by a constant as indicated. The ratios of data/fit are shown at the bottom (Color online).

[figure omitted; refer to PDF]

Figure 4: The data are from [11, 31, 40, 45-47] for Cu + Cu at [figure omitted; refer to PDF] GeV. The curves are the analytical results with Tsallis distribution Equation (10). The corresponding fitting parameters and [figure omitted; refer to PDF] /ndf are given in Table 1. For a better visualization both the data and the analytical curves have been scaled by a constant as indicated. The ratios of data/fit are shown at the bottom.

[figure omitted; refer to PDF]

Figure 5: The data are from [11, 32, 46, 48-52] for Au + Au at [figure omitted; refer to PDF] GeV. The curves are the analytical results with Tsallis distribution Equation (10). The corresponding fitting parameters and [figure omitted; refer to PDF] /ndf are given in Table 1. For a better visualization both the data and the analytical curves have been scaled by a constant as indicated. The ratios of data/fit are shown at the bottom.

[figure omitted; refer to PDF]

Figure 6: The data are from [33, 53-55] for Pb + Pb at [figure omitted; refer to PDF] TeV. The red dashed line is the results fitting the low [figure omitted; refer to PDF] region and the blue dotted-dashed line is the results fitting the high [figure omitted; refer to PDF] region using (10). The solid curves are the analytical results with (12). The corresponding fitting parameters and [figure omitted; refer to PDF] /ndf are given in Table 2. For a better visualization both the data and the analytical curves have been scaled by a constant as indicated. The ratios of data/fit are shown at the bottom.

[figure omitted; refer to PDF]

Table 2: The fitting parameters and the corresponding [figure omitted; refer to PDF] /ndf for different particles in Pb + Pb at [figure omitted; refer to PDF] TeV with (12).

System	Particle	Centrality	[figure omitted; refer to PDF]	[figure omitted; refer to PDF] (GeV)	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF] /ndf
Pb + Pb [figure omitted; refer to PDF] TeV	Charged [figure omitted; refer to PDF]	0-5%	2049.91	0.252	2.195	0.886	174.54/59
Charged [figure omitted; refer to PDF]	0-5%	112.68	0.346	1.776	1.150	35.37/54
Charged [figure omitted; refer to PDF]	0-5%	10.55	0.710	1.845	1.605	42.29/45
[figure omitted; refer to PDF]	0-5%	2.11	0.749	1.281	1.080	1.84/4
[figure omitted; refer to PDF]	0-5%	3.525	0.761	1.907	1.679	23.15/27
[figure omitted; refer to PDF]	0-10%	0.376	0.774	2.003	1.665	32.95/23
[figure omitted; refer to PDF]	0-10%	0.0615	0.658	2.098	1.324	2.41/9

The exponential form equation was used to fit the particle spectra at RHIC when only the low [figure omitted; refer to PDF] data are available [11, 35]. With the upgrade of detectors, we have a better ability to measure the particle spectra. When the intermediate [figure omitted; refer to PDF] data are available, the Tsallis distribution is used to understand the particle spectra and extract physical information. The two-Boltzmann distribution was also used in [7]. But in both cases the number of free fitting parameters increases from 2 to 3. One fitting degree of freedom is increased in this transition. In [6], a double Tsallis formula was proposed to fit particle spectra obtained from central events in Pb + Pb collisions. In this case, three fitting degrees of freedom are increased. Here we will follow the same logic of the transition from exponential distribution to Tsallis distribution to propose a new form equation to fit the particle spectra in Pb + Pb collisions at [figure omitted; refer to PDF] TeV by only increasing one fitting degree of freedom. We increase the number of free fitting parameters from 3 to 4 and the proposed formula is [figure omitted; refer to PDF] There are four parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . We are inspired by the solution of Fokker-Planck equation [56]. We change the power from 2 in [56] to 4 in (12) in order to fit well all the particle spectra with one equation. Figure 6 shows that the fits with (12) are excellent. We would like to mention that when [figure omitted; refer to PDF] , (12) becomes [figure omitted; refer to PDF] and when [figure omitted; refer to PDF] , [figure omitted; refer to PDF] Equation (12) has the same asymptotic behaviors as (10).

4. Conclusions

In this paper, we have tested the fitting ability of Tsallis function by fitting different particle spectra produced at the most central collisions in d + Au, p + Pb, Cu + Cu, Au + Au, and Pb + Pb at RHIC and LHC. The Tsallis distribution is able to fit all the particle spectra in d + Au and p + Pb collisions where the medium effects are very weak. This information can be obtained by the nuclear modification factor. In the AA collisions, the Tsallis distribution can fit all the particle spectra very well at RHIC energies except the little deviation observed for proton and [figure omitted; refer to PDF] at low [figure omitted; refer to PDF] . However the Tsallis distribution can only fit part of the particle spectra in Pb + Pb at [figure omitted; refer to PDF] TeV, either in the low or in the high [figure omitted; refer to PDF] region. We have proposed a new formula in order to fit all the particle spectra in Pb + Pb by increasing one fitting degree of freedom from Tsallis distribution. This follows the same idea of the transition from the exponential distribution to Tsallis distribution when intermediate [figure omitted; refer to PDF] data are available in experiments.

According to the results in this paper and [2], we conclude that we can do the systematic analysis of particle spectra with Tsallis distribution in p + p, pA at RHIC and LHC. In the AA collisions at [figure omitted; refer to PDF] GeV/c, we can do the same analysis as in p + p and pA at RHIC and LHC. But when we consider Pb + Pb collisions, the Tsallis distribution fails.

Acknowledgments

The authors thank Dr. J. Mabiala for reading carefully their paper. This work was supported, in part, by the NSFC of China under Grant no. 11205106.

Conflict of Interests

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

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