1 Introduction

Kinetic freeze-out temperature and flow velocity extracted from transverse momentum spectra of final-state light flavor particles produced in collisions at RHIC and LHC

Hua-Rong Wei, Fu-Hu Liu111E-mail: fuhuliu@163.com; fuhuliu@sxu.edu.cn, and Roy A. Lacey222E-mail: Roy.Lacey@Stonybrook.edu


Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China

Departments of Chemistry & Physics, Stony Brook University, Stony Brook, NY 11794, USA


Abstract: The transverse momentum spectra of final-state light flavor particles produced in proton-proton (-), copper-copper (Cu-Cu), gold-gold (Au-Au), lead-lead (Pb-Pb), and proton-lead (-Pb) collisions for different centralities at relativistic heavy ion collider (RHIC) and large hadron collider (LHC) energies are studied in the framework of a multisource thermal model. The experimental data measured by the STAR, CMS, and ALICE Collaborations are consistent with the results calculated by the multi-component Erlang distribution and Tsallis Statistics. The effective temperature and real temperature (kinetic freeze-out temperature) of interacting system at the stage of kinetic freeze-out, the mean transverse flow velocity and mean flow velocity of particles, and the relationships between them are extracted. The dependences of effective temperature and mean (transverse) momentum on rest mass, moving mass, centrality, and center-of-mass energy, and the dependences of kinetic freeze-out temperature and mean (transverse) flow velocity on centrality, center-of-mass energy, and system size are obtained.

Keywords: Kinetic freeze-out temperature, Flow and transverse flow velocities, Transverse momentum spectrum, Erlang distribution, Tsallis statistics

PACS: 12.38.Mh, 25.75.Dw, 24.10.Pa


1 Introduction

Since the relativistic heavy ion collider (RHIC) and large hadron collider (LHC) successfully run, the evolution process of collision system and properties of quark-gluon plasma (QGP) [1] formed in a high-temperature and high-density extreme condition, attract more interests and studies. Many theoretical and experimental methods are used to study the high energy nucleus-nucleus collisions which result in multi-particle productions. The model analysis is a simple but effective method in the study of high energy nucleus-nucleus collisions. It can extract the information of interacting system and QGP by analyzing various spectra of final-state products with distribution laws of different models. For example, by using the blast-wave model [2], thermal and statistical model [3], Landau hydrodynamic model [4–7], multisource thermal model [8–10], and so forth, one can describe transverse momentum (mass) spectrum, azimuthal distribution, and rapidity distribution of final-state products to extract temperature, non-equilibrium degree, longitudinal extension, and speed of sound of interacting system, and flow velocity and chemical potential of particles [11–14], especially at the state of kinetic freeze-out, which renders that model analyzes have made great contributions to study the properties of reaction system and QGP, as well as new physics.

In high energy collisions, the interacting system at the kinetic freeze-out (the last stage of collisions) stays at a thermodynamic equilibrium state or local equilibrium state, when the particle emission process is influenced by not only the thermal motion but also the flow effect. In other words, it is interesting to study the temperature of interacting system and flow velocity of particles at the stage of kinetic freeze-out. The effective temperature extracted from the transverse momentum spectrum [15–25] includes thermal motion and flow effect of particles, where the thermal motion is actually the reflection of the real temperature of emission source. From dissecting the effective temperature, it is possible to obtain the real temperature of interacting system (kinetic freeze-out temperature) and mean (transverse) flow velocity of particles. The relationships between effective temperature, real temperature, flow velocity are not totally clear. Although the theories of studying kinetic freeze-out temperature are many, their results are different from each other in some cases. This indicates that studying more their relations is important and needful.

In the present work, we use the Erlang distribution with one-, two-, or three-component and Tsallis statistics in the multisource thermal model [8–10] to describe the transverse momentum spectra of final-state particles produced in proton-proton (-), copper-copper (Cu-Cu), gold-gold (Au-Au), lead-lead (Pb-Pb), and proton-lead (-Pb) collisions with different centrality intervals over a (center-of-mass energy per nucleon pair) or (only for - collision in some cases) range from 0.2 to 7 TeV. The Monte Carlo method is used to calculate the results and to see the statistical fluctuations in the process of calculation. The calculated results are compared with the experimental data of the STAR [15–19], CMS [20], and ALICE [21–25] Collaborations. From comparison and analysis, the kinetic freeze-out temperature of interacting system, mean transverse flow velocity and mean flow velocity of particles, and their relations are then extracted.

The structure of the present work is as followings. The model and formulism are shortly described in section 2. Results and discussion are given in section 3. In section 4, we summarize our main observations and conclusions.

2 The model and formulism

The present work is based on the multisource thermal model [8–10], which assumes that many emission sources are formed in high energy collisions. Due to the existent of different interacting mechanisms in collisions and measurement of different event samples in experiments, these sources are classified into a few groups. Generally, we assume that sources in the same group stay at a local equilibrium state, which means they have the same excitation degree and a common temperature. The emission process of all the sources in different groups results in the final-state spectrum, which can be described by a multi-component distribution law, because of a local equilibrium state corresponding to a singular distribution. Using different distributions to describe transverse momentum () spectra of final-state particles can obtain different information. For example, a multi-component Erlang distribution can directly give the mean transverse momentum of each group, while the Tsallis statistics can show the effective temperature of the whole interacting system which may have group-by-group fluctuations in different local thermal equilibriums.

According to the model [10], particles generated from one emission source is assumed to obey an exponential function of transverse momentum distribution

(1)

where is the transverse momentum contributed by the th source in the th group, and is the mean value of . All the sources in the th group results in the folding of exponential functions

(2)

where is the source number in the th group, and denotes the transverse momentum contributed by the sources. This is the Erlang distribution. The contribution of all groups of sources can be expressed as

(3)

where denotes the relative weight contributed by the th group and meets the normalization . This is the multi-component Erlang distribution. Then, using the inverse slope parameter , we can obtain the mean transverse momentum of final-state particles to be

(4)

The Tsallis statistics is in fact the sum of contributions of two or three standard distributions. Although the Tsallis statistics does not give each local temperature like multi-component distribution which reveals fluctuations from a local equilibrium state to another one, it use an average temperature of the whole interacting system to describe the effect of local temperature fluctuations. So, the Tsallis statistics is widely used in high energy collisions [26–35].

According to the Tsallis statistics [26–31], we use the formalism of unit-density function of and rapidity ()

(5)

where is the number of particles, is the normalization constant, and are degeneracy factor and volume respectively, is the rest mass of considered particle, is the mean effective temperature over fluctuations in different groups, and () is the factor (entropy index) to characterize the degree of non-equilibrium among different groups. With an integral for in Eq. (5), the normalized Tsallis distribution is obtained and can be written as

(6)

where denotes the normalization constant which results in , is the minimum rapidity, and is the maximum rapidity.

Based on the above two distributions, we can use the Monte Carlo method to obtain a series of . Under the assumption of isotropic emission in the source rest frame, the space angle and azimuthal angel of particles satisfy the distributions of and respectively. By the Monte Carlo method, a series of and are obtained. Correspondingly, the -component, -component, and (longitudinal) -component of momentum are , , and , respectively. Then, the momentum or , the energy , the Lorentz factor , and the moving mass , as well as their averages , , , and are acquired. Particularly, the values of mean transverse momentums according to analytical function and Monte Carlo method are almost the same.

3 Results and discussion

Figure 1 presents the transverse momentum spectra of various identified hadrons produced in - collision at center-of-mass energy (a) , (b) (c) , (d) , and (e) (f) TeV, where on the axis denotes the number of inelastic collisions events. The symbols represent the experimental data of (a) , , and measured by the STAR Collaboration at midrapidity [15], (b) (d) (e) , , and measured by the CMS Collaboration in the range [20], (c) , , and measured by the ALICE Collaboration in the range [21], as well as (f) and measured by the ALICE Collaboration in the range [22]. The errors include both the statistical and systematic errors. The solid and dashed curves are our results calculated by using the one- or two-component Erlang distribution and Tsallis statistics, respectively. The values of free parameters (, , , , and ), normalization constant (), and per degree of freedom (/dof) corresponding to the one- or two-component Erlang distribution are listed in Table 1, and the values of free parameters ( and ), normalization constant (), and /dof corresponding to the Tsallis statistics are given in Table 2, where the normalization constants ( and ) are used for comparisons between the normalized curves and data points. In particular, the value of /dof for in Figure 1(c) is in fact the value of due to the number of data points being less than that of parameters. One can see that the one- or two-component Erlang distribution and Tsallis statistics describe the experimental data of the considered particles in - collision at different energies. From Tables 1 and 2, one can see that the numbers of sources in different groups are 2, 3, or 4. The effective temperature increases and the non-equilibrium degree parameter decreases with increase of rest mass, which reflects non-simultaneous productions of different types of particles, while and increase with increase of center-of-mass energy. The normalization constants and decrease with increase of rest mass, and increase with increase of center-of-mass energy. It is interesting to note that the product of and increases with increase of rest mass and center-of-mass energy.

Figure 2 shows the transverse momentum spectra of (a) , (b) , (c) , and (d) produced in Cu-Cu collisions at TeV. The symbols represent the experimental data of the STAR Collaboration in and different centrality () intervals of 0–10%, 20–30%, and 40–60% [16]. The error bars are combined statistical and systematic errors. The solid and dashed curves are our results calculated by using the one- or two-component Erlang distribution and Tsallis statistics, respectively. For clarity, the results for different intervals are scaled by different amounts shown in the panels. The values of free parameters, normalization constants, and /dof are displayed in Tables 1 and 2. Obviously, the one- or two-component Erlang distribution and Tsallis statistics describe well the experimental data of the considered particles in 0.2 TeV Cu-Cu collisions with different centrality intervals. The numbers of sources in different groups are 1, 2, 3, or 4. The parameter increases and the parameter decreases with increases of rest mass and centrality. The product of and increases with increases of rest mass and centrality. The parameters and decrease with increases of rest mass and decrease of centrality.

The spectra of (a) for , (b) for , (c) for , (d) for , (e) for , and (f) for produced in Au-Au collisions at TeV as a function of centrality are given in Figure 3. The experimental data were recorded by the STAR Collaboration [17–19], and scale factors for different centralities are applied to the spectra in the panels for clarity. The uncertainties on the data points for , , , and are statistical and systematic combined. While the uncertainties for and are only statistical uncertainties and systematic uncertainties respectively. The results calculated by using the one-, two-, or three-component Erlang distribution and Tsallis statistics are shown in the solid and dashed curves, respectively. The values of free parameters (, , , , and ), normalization constant, and /dof corresponding to the one- or two-component Erlang distribution in Figures 3(b)–3(f) are listed in Table 1. The values of free parameters (, , , , , , , and ), normalization constant, and /dof corresponding to the three-component Erlang distribution in Figure 3(a) are listed in Table 3. The values of free parameters, normalization constant, and /dof corresponding to the Tsallis statistics are given in Table 2. Once more, the two types of distributions are in good agreement with the experimental data of the considered particles in 0.2 TeV Au-Au collisions with different centrality intervals. The numbers of sources in different groups are 1, 2, 3, or 4. The parameter increases and the parameter decreases with increases of rest mass and centrality. The product of and increases with increases of rest mass and centrality. The normalization constants and decrease with increase of rest mass and decrease of centrality.

The spectra of (a) , (b) , (c) , and (d) produced in central (0–5%), semi-central (50–60%), and peripheral (80–90%) Pb-Pb collisions at TeV are displayed in Figure 4. The symbols represent the experimental data measured by the ALICE Collaboration at midrapidity [23, 24]. The uncertainties on the data points are combined statistical and systematic ones. The fitted results with the one- or two-component Erlang distribution and Tsallis statistics are plotted by the solid and dashed curves, respectively. The values of free parameters, normalization constants, and /dof are exhibited in Tables 1 and 2. As can be seen, the two types of distribution laws are consistent with the experimental data of the considered particles in 2.76 TeV Pb-Pb collisions with different centrality classes. The numbers of sources in different groups are 1, 2, or 3. The effective temperature increases with increases of rest mass and centrality. The parameter decreases with increases of rest mass and centrality. The product of and increases with increases of rest mass and centrality. The parameters and decrease with increase of rest mass and decrease of centrality.

Figure 5 exhibits the spectra of (a) , (b) , (c) , and (d) produced in central (0–5%), semi-central (40–60%), and peripheral (80–100%) -Pb collisions at TeV. The ALICE experimental data in are represented by different symbols [25]. The error bars are combined statistical and systematic errors. Our results analyzed by the two-component Erlang distribution and Tsallis statistics are given by the solid and dashed curves, respectively. The values of free parameters, normalization constants, and /dof are summarized in Tables 1 and 2. Once again, the experimental data can be well described by the two types of fit functions for spectra in all centrality bins. The numbers of sources in different groups are 2, 3, or 4. The effective temperature increases with increases of rest mass and centrality. With increases of rest mass and centrality, the parameters , , and decrease, and the product of and increases.

In the above descriptions, it is easy to notice that the numbers of sources in the th group are in the range . The is so small that we think these sources corresponding to a few partons which include sea and valent quarks and gluons in high-energy collisions. Generally, the transverse momentum spectrum is contributed by the sum of soft and hard parts. The soft excitation process is a strong interacting process where a few sea quarks and gluons taken part in, and the hard scattering process is a more violent collision among a few valent quarks in incident nucleons. In spectrum, the soft excitation and hard scattering processes correspond to a narrow low- and a wide high- regions respectively [36–38]. And in the describing by two-component Erlang distribution, the first and second components correspond to the soft and hard processes respectively. Although the low- region contributed by soft process is narrow, the contribution of soft excitation is main, which can be seen from the relative weight . Particularly, when the region of spectrum is narrow enough, the two-component Erlang distribution actually is the (one-component) Erlang distribution for the contribution of the second component being neglected. On the contrary, when the high- region is very wide, the hard process can not be described by one component distribution, which means the two-component distribution would expand to the three-component Erlang distribution.

The temperature parameter extracted from the Tsallis distribution is actually the effective temperature of emission sources, which can not reflect the transverse excitation of interacting system at the stage of kinetic freeze-out. In order to obtain the real temperature (kinetic freeze-out temperature) of emission sources and (transverse) flow velocity of final-state particles, we study the linear dependences of mean transverse momentum , mean momentum , and effective temperatures on particle rest mass and mean moving mass . Figures 6–8 show the center-of-mass energy and centrality dependences of , , and on respectively, and Figures 9–11 show that on respectively. One can see that the three quantities , , and increase with increase of and . From - collision, the three quantities increase with increase of center-of-mass energy, and from Cu-Cu, Au-Au, Pb-Pb, and -Pb collisions, the three quantities increase with increase of centrality.

To see clearly the dependences of , , and on and , we fit the linear relationships which can be written as

(7)
(8)
(9)
(10)
(11)

and

(12)

respectively, where the units of temperature, momentum, velocity, and mass are GeV, GeV/, , and GeV/, respectively, where is in natural units; The intercepts (, , , , £¬and ) have the same units as corresponding dependent variables; The slopes , , , and are in the units of , while and are in the units of . The values of intercepts, slopes, and are given in Table 4.

From Figures 6–11 and Table 4, we can see the intercepts and slopes in different linear correlations. In all cases, the intercepts with different center-of-mass energies (- collision) or centrality bins (Cu-Cu, Au-Au, Pb-Pb, or -Pb collisions) have the tendency of converging to one point, which means the intercepts are nearly equal to each other or do not change obviously with center-of-mass energy or centrality, while the slopes have the tendencies of increasing with center-of-mass energy and centrality. The intercepts and slopes obtained from , , and correlations are larger than those from , , and correlations, but the changes about intercepts are much larger than those about slopes. Besides, the increments of slopes with center-of-mass energy or centrality in , , and correlations are larger than those in , , and correlations, which renders slow changes in the latter cases.

In function , the quantity extracted directly from the distribution mentioned above, is the effective temperature which includes thermal motion and flow effect of particles. As the temperature corresponding to massless () particle, is regarded as the source real temperature at the kinetic freeze-out (or the kinetic freeze-out temperature of interacting system) [39–43]. The flow effect of particles is shown in quantity of , where the slope has the dimension of the square of velocity. At the same time, in function , although the intercept has the same dimension as temperature, it’s values shown in Figure 11 and Table 4 approximately equal to zero, which have no physics meaning. In correlations , , , and , the slopes (, , , and ) have the dimension of velocity and are considered to be related to mean transverse flow velocity and mean flow velocity. It is interesting to find the relationships between the intercepts and slopes, especially relationships between the several flow velocities and (or ).

Figure 12 exhibits the correlations between intercepts (a) , (b) , and (c) in different collision systems with different centrality classes at different energies, and corresponding fitting are executed. One can see that and , and have explicit linear relations and with almost zero , which is due to the assumption of isotropic emission in the source rest frame. In correlation, the increases with increase of , and they are basically compatible with the linear relation with /dof = 0.237. Other relations among intercepts do not show an obvious law and are not shown in the panels due to trivialness.

At the same time, the correlations between different slopes (a) , (b) , (c) , (d) , and (e) in different collision systems with different centrality classes at different energies, as well as corresponding fitting are shown in Figure 13. Once again, based on the assumption of isotropic emission in the source rest frame, the and from and , and from and also have the same explicit linear relations and with almost zero . From Figures 13(b) and 13(d), one can see that (or ) increases with increase of (or ), and the parameters are in good agreement with the fitted power function relations with /dof = 1.546 and with /dof = 1.472, respectively. The relationship between and respectively from and correlations is fitted by the line with /dof = 0.043. Other relations among slopes do not show an obvious law and are not shown in the panels due to trivialness. In addition, we would like to point out that in the Monte Carlo calculation, some conservation laws (such as energy conservation and momentum conservation) and physics limitation (such as flow velocity ) are used so that we can obtain reasonable values and relations.

It is noticed that most of values (0.339–0.522) extracted from are slightly less than 0.5 and (0.382–0.944) from , while most of values (0.532–0.820) extracted from is more close to 0.5 and less than (0.546–1.424) from . Considering overlarge and , we regard and as the mean transverse flow velocity and mean flow velocity of particles respectively, and they obey the relation of . In addition, and approximatively obey .

The mean of interacting system obtained from does dot change in error range or approximates independent of energy, centrality, particle type, and system size. The mean corresponding to the Tsallis distribution is () GeV, which is less than that [()] GeV from the Boltzmann distribution in previous work [43], and is also less than that (0.177 GeV) from an exponential shape of transverse mass spectrum [41]. Our result is close to that (0.098 GeV) from the blast-wave model [44]. Comparing with chemical freeze-out temperature, the kinetic freeze-out temperature in present work is obviously less than that (0.170 GeV) of theoretical critical point of the QCD (quantum chromodynamics) prediction [45–47], and that (0.156 GeV) from particle ratios in a thermal and statistical model [3].

In the above discussions, we present in fact a method to extract the kinetic freeze-out temperature and (transverse) flow velocity from transverse momentum spectra in the multisource thermal model [8–10], in which the sources are described by different distribution laws. The method to extract the kinetic freeze-out temperature is also used in other literature [39–43], and the method to extract the (transverse) flow velocity is seldom discussed elsewhere. Although the method used in the present work is different from the blast-wave model [2] which discusses radial flow, it offers anyhow another way to study the kinetic freeze-out temperature and (transverse) flow velocity. In our recent work [48], the multisource thermal model is revised with the blast-wave picture, where we assume that the fragments and particles produced by thermal reason in the sources are pushed away by a blast-wave. The blast-wave causes the final-state products to be effected by isotropic and anisotropic flows.

4 Conclusions

From the above discussions, we obtain following conclusions.

(a) The transverse momentum distributions of final-state particles produced in -, Cu-Cu, Au-Au, Pb-Pb, and -Pb collisions with different centrality bins over an energy range from 0.2 to 7 TeV, can be described by one-, two-, or three-component Erlang distribution and Tsallis statistics in the framework of multisource thermal model. The calculated results are consistent with the experimental data of , , , , , , , , , and , etc. measured by the STAR, CMS, and ALICE Collaborations.

(b) In most cases, the data of spectra are fitted by the two-component Erlang distribution, where the first component corresponding to a narrow low- region is contributed by the soft excitation process in which a few sea quarks and gluons taken part in and accounts for a larger proportion, and the second component for a wide high- region indicates hard scattering process which is a more violent collision among a few valent quarks in incident nucleons. The mean transverse momentum extracted from multi-component Erlang distribution increases with increases of center-of-mass energy, particle mass, event centrality, and system size.

(c) The Tsallis distribution uses two free parameters and to describe the effective temperature and the non-equilibrium degree of the interacting system respectively. The present work shows that decreases with increases of particle mass and event centrality, and increases with increase of center-of-mass energy. In physics, corresponds to an equilibrium state and a large corresponds to a state departing far from equilibrium. Our study indicates that a high center-of-mass energy results in interacting system deviating from the equilibrium state, and the interacting system of central collisions and heavier particles are closer to the equilibrium state. The extracted increases with increases of center-of-mass energy, particle mass, event centrality, and system size.

(d) The intercept in correlation is regarded as the mean kinetic freeze-out temperature of interacting system, which do not depend on center-of-mass energy, event centrality, and system size in error range, which renders that different interacting systems at the stage of kinetic freeze-out stay at the same phase state. The average of corresponding to the Tsallis distribution is GeV, which is less than the effective temperature, chemical freeze-out temperature, and expected critical temperature (130–165 MeV) of the QGP formation [49].

(e) The mean transverse flow velocity and mean flow velocity of particles are obtained from and correlations respectively. The present work shows that (0.339-0.522) and (0.532-0.820) have narrower range and are more close to 0.5 than those from and , respectively. On the assumption of isotropic emission in the source rest frame, there is . Besides, and from corresponding correlation approximatively meet , which means and can be obtained from correlation based on and relationships. In addition, the mean transverse flow velocity and mean flow velocity have a slightly increase tendency with center-of-mass energy, event centrality, and system size.

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. 11575103 and the US DOE under contract DE-FG02-87ER40331.A008.


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Table 1. Values of free parameters, normalization constant, and /dof corresponding to one- or two-component Erlang distribution in Figures 1–5 except Figure 3(a). The value of /dof for in Figure 1(c) is in fact the value of due to less data points.

Figure Type (GeV/) (GeV/) /dof
Figure 1(a) 2 1 - - 0.032
2 1 - - 0.009
2 1 - - 0.027
Figure 1(b) 3 2 0.018
3 4 0.074
4 4 0.016
Figure 1(c) 2 1 - - 0.310
2 1 - - 0.622
2 1 - - (0.218)
Figure 1(d) 3 2 0.025
3 4 0.015
3 4 0.057
Figure 1(e) 3 2 0.099
3 4 0.015
3 4 0.017
Figure 1(f) 3 2 0.056
2 2 0.064
Figure 2(a) 0-10% 2 1 0.301
20-30% 2 1 0.214
40-60% 2 1 0.368
Figure 2(b) 0-10% 4 3 0.296
20-30% 4 3 0.088
40-60% 4 3 0.233
Figure 2(c) 0-10% 3 1 - - 0.655
20-30% 3 1 - - 1.353
40-60% 3 1 - - 0.708
Figure 2(d) 0-10% 4 1 - - 0.419
20-30% 3 1 - - 0.139
40-60% 3 1 - - 0.121
Figure 3(b) 0-5% 2 1 0.034
10-20% 2 1 0.028
20-40% 2 1 0.015
40-60% 2 1 0.056
60-80% 2 1 0.042
Figure 3(c) 0-12% 4 1 0.254
10-20% 4 1 0.136
20-40% 4 1 0.188
40-60% 4 1 0.195
60-80% 4 1 0.117
Figure 3(d) 0-5% 3 1 - - 0.019
10-20% 3 1 - - 0.031
30-40% 3 1 - - 0.026
50-60% 2 1 - - 0.071
70-80% 2 1 - - 0.110
Figure 3(e) 0-5% 4 3 0.035
10-20% 4 3 0.049
20-40% 4 3 0.069
40-60% 4 3 0.009
60-80% 4 3 0.059
Figure 3(f) 0-5% 4 1 - - 0.008
10-20% 4 1 - - 0.044
20-40% 4 1 - - 0.052
40-60% 4 1 - - 0.136
60-80% 4 1 - - 0.402
Figure 4(a) 0-5% 2 1 0.058
50-60% 2 1 0.038
80-90% 2 1 0.037
Figure 4(b) 0-5% 3 2 0.025
50-60% 3 2 0.022
80-90% 3 2 0.054
Figure 4(c) 0-5% 3 1 - - 0.275
50-60% 3 3 0.041
80-90% 3 3 0.053
Figure 4(d) 0-5% 3 1 - - 0.289
50-60% 2 1 - - 0.272
80-90% 2 1 - - 0.344
Figure 5(a) 0-5% 2 2 0.176
40-60% 2 2 0.083
80-100% 2 2 0.094
Figure 5(b) 0-5% 3 2 0.020
40-60% 3 2 0.031
80-100% 3 2 0.011
Figure 5(c) 0-5% 3 2 0.036
40-60% 3 3 0.012
80-100% 3 3 0.022
Figure 5(d) 0-5% 4 2 0.026
40-60% 4 2 0.082
80-100% 3 2 0.074

Table 2. Values of free parameters, normalization constants, and /dof corresponding to Tsallis distribution in Figures 1–5. The value of /dof for in Figure 1(c) is the value of due to less data points.

Figure Type (GeV) /dof
Figure 1(a) 0.018
0.037
0.085
Figure 1(b) 0.077
0.017
0.084
Figure 1(c) 1.192
0.804
(0.348)
Figure 1(d) 0.131
0.014
0.081
Figure 1(e) 0.143
0.022
0.042
Figure 1(f) 0.583
0.037
Figure 2(a) 0-10% 0.242
20-30% 0.290
40-60% 0.195
Figure 2(b) 0-10% 0.789
20-30% 0.275
40-60% 0.331
Figure 2(c) 0-10% 0.167
20-30% 0.379
40-60% 0.343
Figure 2(d) 0-10% 0.858
20-30% 0.089
40-60% 0.080
Figure 3(a) 0-12% 0.115
10-20% 0.124
20-40% 0.058
40-60% 0.028
60-80% 0.084
Figure 3(b) 0-5% 0.142
10-20% 0.056
20-40% 0.032
40-60% 0.049
60-80% 0.046
Figure 3(c) 0-12% 1.785
10-20% 0.442
20-40% 0.562
40-60% 0.296
60-80% 0.283
Figure 3(d) 0-5% 0.041
10-20% 0.066
30-40% 0.079
50-60% 0.061
70-80% 0.105
Figure 3(e) 0-5% 0.378
10-20% 0.161
20-40% 0.370
40-60% 0.161
60-80% 0.198
Figure 3(f) 0-5% 0.061
10-20% 0.059
20-40% 0.034
40-60% 0.140
60-80% 0.248
Figure 4(a) 0-5% 0.211
50-60% 0.161
80-90% 0.114
Figure 4(b) 0-5% 0.009
50-60% 0.019
80-90% 0.048
Figure 4(c) 0-5% 0.362
50-60% 0.041
80-90% 0.104
Figure 4(d) 0-5% 0.223
50-60% 0.302
80-90% 0.182
Figure 5(a) 0-5% 0.371
40-60% 0.304
80-100% 0.014
Figure 5(b) 0-5% 0.007
40-60% 0.010
80-100% 0.031
Figure 5(c) 0-5% 0.102
40-60% 0.032
80-100% 0.018
Figure 5(d) 0-5% 0.025
40-60% 0.018
80-100% 0.026

Table 3. Values of free parameters, normalization constant, and /dof corresponding to the three-component Erlang distribution in Figure 3(a), where which are not listed in the column.

Figure Type (GeV/) (GeV/) (GeV/) /dof
Figure 3(a) 0-12% 0.043
10-20% 0.119
20-40% 0.086
40-60% 0.080
60-80% 0.183

Table 4. Values of intercepts, slopes, and corresponding to the lines in Figures 6–11.

Figure Correlation Type Intercept Slope
Figure 6(a) - 0.2 TeV 0.175
- 0.9 TeV 1.001
- 2.76 TeV 0.050
- 7 TeV 0.187
Figure 6(b) 0-10% 0.154
20-30% 0.184
40-60% 0.077
Figure 6(c) 0-12% 0.160
10-20%