Comparing Multicomponent Erlang Distribution and Lévy Distribution of Particle Transverse Momentums

The transverse momentum spectrums of final-state products produced in nucleus-nucleus and proton-proton collisions at different center-of-mass energies are analyzed by using a multicomponent Erlang distribution and the Lévy distribution. The results calculated by the two models are found in most cases to be in agreement with experimental data from the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC).The multicomponent Erlang distribution that resulted from a multisource thermalmodel seems to give a better description as comparedwith the Lévy distribution.The temperature parameters of interacting system corresponding to different types of final-state products are obtained. Light particles correspond to a low temperature emission, and heavy particles correspond to a high temperature emission. Extracted temperature from central collisions is higher than that from peripheral collisions.


Introduction
The Relativistic Heavy Ion Collider (RHIC) in USA and the Large Hadron Collider (LHC) in Switzerland have been built to study properties of matters formed in high-energy collisions.These collisions are helpful in understanding particles' statistical behavior, production process, interaction mechanism, and related phenomenon in high-density and high-temperature states.Such high-energy collisions offer us opportunities to carry out investigations not only on the Higgs and dark matter [1][2][3], but also on particle statistical behavior at ultrahigh energy.
Transverse momentum spectrums of final-state products are very important in high-energy collisions.Many models have been introduced to describe the transverse momentum spectrums of different final-state products [4].From the spectrums, one can extract temperature parameter of interacting system.It is expected that temperature parameters extracted from different particle spectrums are different due to different emission stages and regions in collisions.Although we can compare nuclear temperature with classical temperature, they have different physical meanings.
Temperature parameter in high-energy collisions is very important.Generally speaking, temperatures of interacting system at initial, intermediate, and final states are different [5].Since these temperatures cannot be measured directly, it may, therefore, be interesting to find out an indirect method for obtaining the temperature of the interesting system.Traditionally, temperature can be extracted from measurements of spectrum slopes or double isotopic ratios at lower energies [5,6].In some cases, we cannot obtain absolute values of concerned temperature parameters, but relative values corresponding to different particle spectrums.
Multicomponent Erlang distribution derived from multisource thermal model [7,8] has been applied to collisions in relatively low energy region comparing to RHIC and LHC energies.Energy spectrum of nuclear fragments, multiplicity distribution of charged particles, neutron number distribution of isotope in nuclear fragments, transverse momentum (mass) spectrum of relativistic particles, and so forth were described by the multicomponent Erlang distribution.The Lévy distribution has been also applied to transverse momentum spectrums in high-energy collisions [9][10][11].We can study transverse momentum spectrums by 2 Advances in High Energy Physics using the multicomponent Erlang distribution [7,8] or the Lévy distribution [9][10][11] to extract temperature parameters.
In this paper, the transverse momentum spectrums of different final-state products produced in nucleus-nucleus and proton-proton collisions at RHIC and LHC energies are studied with the two distributions mentioned above.Temperature parameters are then obtained from fitting experimental data of the STAR, CMS, and ALICE Collaborations.

Formalism
The multicomponent Erlang distribution can be derived from the multisource thermal model [7,8].In the model, many emission sources of particles are assumed to form in high energy collisions.According to different interaction mechanisms, geometrical relations, selected conditions, or other factors, the emission sources are divided into  groups.Source number in the th group is assumed to be   .Each source contributes final-state distribution to be an exponential function.We have the transverse momentum   spectrum contributed by the th source in the th group to be where  denotes number of final-state particles and is mean transverse momentum contributed by the sources in the th group.The transverse momentum   spectrum contributed by the th group is the fold of   exponential functions; that is, This is an Erlang distribution.In final state, the   spectrum contributed by the  groups can be written as where   is the relative weight contributed by the th group.It is a multicomponent Erlang distribution.
Considering relative contribution of the th group, we have the mean transverse momentum of final-state particles to be Generally, ⟨  ⟩ reflects the mean excitation degree of the emission sources and can be used to describe the source temperature parameter   .As in the ideal gas model in which   obeys Rayleigh distribution, we have where  0 denotes rest mass and γ is mean Lorentz factor of considered particles.Further, where Ē and ⟨⟩ are mean energy and mean momentum of considered particles, respectively.On other hand, as the inverse slope parameter, ⟨  ⟩ can be used to describe excitation degree of the emission sources.We define as a new temperature parameter.
The Lévy distributions appear in many branches of physics, mathematics, biology, economy, computer science, and other areas, where the distribution forms may be different in different branches and the scale of fluctuations may be characterized by long tails and an asymptotic power-lawlike behavior.The Lévy distributions are a generalization of the Gaussian distribution.They are similar to the Gaussian distribution and remain stable under the convolution.In fact the Lévy distributions are quite general distributions which contain Gaussian and Cauchy distributions as special cases [12].
Let  be the nonextensive parameter.As a probability distribution, the Lévy distribution is commonly the following power-like distribution [9]: which is just a one-parameter generation of the Boltzmann-Gibbs exponential formula with 1 ≤  < 1.5, where   is the normalization constant and  is in the range from 0 to infinity.For the transverse momentum distributions in highenergy collisions, we use directly the function form of Lévy distribution [10]: where   is the slope parameter and  represents the scale of possible fluctuation in   .The parameter   can be regarded as the temperature parameter in the Lévy distribution.

Comparisons with Experimental Data
The transverse momentum spectrums of final-state particles produced in Cu-Cu and Au-Au collisions at RHIC energy (√  = 0.2 TeV) are shown in Figure 1.The symbols represent experimental data of the STAR Collaboration [11].The solid and dashed curves represent results calculated by the multicomponent Erlang distribution with  = 1 or 2 and the Lévy distribution, respectively.The results for different centralities (0-10%, 20-30%, and 40-60% in Cu + Cu, as well as 0-5%, 20-40%, and 60-80% in Au + Au) and also for different particles ( 0  , Λ, Ξ, and Ω + Ω in Cu + Cu, as well as  0  and Λ in Au + Au) in central rapidity range (|| < 0.5) are displayed in different panels.For the sake of  1 along with values of  2 per degree of freedom ( 2 /dof) and extracted temperatures.One can see that the concerned experimental data are described approximately by the two distributions.Light particles correspond to a lower temperature comparing with the heavy particles.The multicomponent Erlang distribution seems to give a better description than the Lévy distribution.We can use the new distribution, the multicomponent Erlang distribution, to describe the transverse momentum spectrums.
In Figure 2, we give the transverse momentum spectrums of leading and subleading jets produced in Pb-Pb and pp collisions at the LHC energy (√  or √ = 2.76 TeV), where the selections of leading and subleading jets can be found in experimental material [13].The symbols represent experimental data of the CMS Collaboration [13].The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively.Figures 2(a), 2(b), and 2(c) correspond to different selected conditions shown in the panels, where ∫ , , anti-  , , and Flow denote the integral luminosity, azimuth, sequential recombination algorithm for high-  particle, resolution parameter, and particle flow, respectively.The parameter values used in the calculations are shown in Table 2 with values of  2 /dof and extracted temperatures.
It is again observed that the two distributions describe approximately the concerned experimental data.
In the Lévy distribution, we need to know the rest mass of final-state product.However, the rest mass of jet is uncertain.In fact, we regarded  0 as a parameter in Figure 2. To see dependence of jet   spectrum on  0 in the Lévy distribution, we redraw the Lévy distribution curves for different  0 values in Figure 3, where the same experimental data [13] as those cited in Figure 2 3. We see that the temperature extracted from a given jet spectrum decreases with increase of the jet mass and is greater than that extracted from particle spectrums.It should be noticed that the jet mass is the total mass of particles in the jet.For a jet with a given total transverse momentum, a larger mass corresponds to more particle number.Then, the transverse momentum per particle will be smaller, which renders a lower temperature.
In Figure 4, another data sample on   spectrums of leading and subleading jets produced in Pb-Pb collisions at √  = 2.76 TeV is analyzed.The symbols represent experimental data of the CMS Collaboration [13].The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively.The values of all the parameters along with the values of  2 /dof are given in Table 2.We see that except for a few points the two distributions describe approximately the experimental data.Different spectrums corresponding to different   (dijet imbalance parameter) ranges can be described by the same distribution which reflects a common law in the spectrums.
The   spectrums of charged jets produced in Pb-Pb collisions at √  = 2.76 TeV is given in Figure 5.The symbols represent experimental data of the ALICE Collaboration [14].The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively.All the parameter values along with values of  2 /dof and extracted temperatures are given in Table 2.One can see that both the distributions describe approximately the experimental data, and the former one gives a better description than the latter one.
The   spectrums of charged particles (which can be approximately regarded as  ± ) produced in √  = 2.76 TeV Pb-Pb collisions in different centrality bins with different multiplications are shown in Figure 6(a).Meanwhile,  2, 4, and 5.The values of  2 /dof and extracted temperatures are given.The abbreviations LJ and SJ represent leading and subleading jets, respectively.The errors for  1 ,  2 , and  can be neglected, and the relative errors for other parameters are less than 10%.

Figures
Types the   spectrums of  − ,  0  ,  − , and p produced in central (0-5%) Pb-Pb collisions at the same energy are shown in Figure 6(b).The symbols represent experimental data of the ALICE Collaboration [14,15] measured in the pseudorapidity range of || < 0.8.The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively.Corresponding to Figures 6(a) and 6(b), the parameter values with values of  2 /dof and extracted temperatures are given in Tables 4 and  5, respectively.One can see that the multicomponent Erlang distribution describes well the   spectrums in all the cases.The Lévy distribution describes well the spectrums in some cases, and in other cases it describes approximately the mean trends of the spectrums.
Figures 7(a), 7(b), and 7(c) show, respectively,   spectrums of final-state particles  + +  − ,  0 , and p produced in √  = 2.76 TeV Pb-Pb collisions in different centrality bins with different multiplications.Selected condition for p is rapidity being in the range of || < 0.5.For the sake of comparison, the results for  + +  − and  0 produced in 2.76 TeV p-p collisions are also given in Figures 7(a  The transverse momentum spectrums of Ξ and Ω as well as inclusive electrons produced in inelastic p-p collision at 7 TeV are given in Figures 8(a) and 8(b), respectively.Experimental data measured by the ALICE Collaboration [15,18] are shown by the symbols.Results calculated by using the multicomponent Erlang distributions and the Lévy distributions are shown by the solid and dashed curves, respectively.The parameter values used in the calculation are listed in Table 4.We see that both distributions describe approximately the experimental data.
The transverse momentum spectrums of  + ,  + , and ;  − ,  − , and p ;  0  , Λ, Λ, , and Ξ − + Ξ + produced in p-p collisions at 0.9 TeV are displayed in Figures 9(a), 9(b), and 9(c), respectively.The symbols represent experimental data of the ALICE Collaboration [19,20].The solid and dashed curves represent results calculated by using the multicomponent Erlang distribution and the Lévy distribution, respectively.The related parameter values are given in Table 5.One can see that both the two distributions describe approximately the experimental data.
In Figure 10, the transverse momentum spectrum of charged particles (which can be approximately regarded as  ± ) produced in nonsingle diffractive (NSD) p-p collisions at 0.9 TeV is presented.The symbols represent experimental data measured in the pseudorapidity range of || < 0.8 by the ALICE Collaboration [19].The solid and dashed curves  represent results of the multicomponent Erlang distribution and the Lévy distribution, respectively.The related parameter values are given in Table 4.One can see that both the two distributions describe approximately the experimental data.
To see dependences of temperature  (  and   ) on centrality and √  , in Figures 11 and 12, we plot different values of   and   taken from Tables 1-6.The related impacting types, √  , centralities, and final-state products are shown in the figures.Figures 11(a), 11(b), 11(c) and 11(d) as well as 11(e) and 11(f) correspond to dependence on centrality for particle productions at 0.2 and 2.76 TeV and jet production at 2.76 TeV, respectively.Figure 12 corresponds to dependence on √  for particle productions at RHIC and LHC energies.One can see that the extracted temperature for light particles is less than that for heavy particles.Central collisions or high √  correspond to a relative high temperature.The multicomponent Erlang distribution Advances in High Energy Physics extracts a relatively high temperature comparing to the Lévy distribution.Besides, from the parameter tables (Tables 1, 2, and 4-6) and ( 8), one can easily obtain values of   which show similar behaviors as those of   .

Conclusions and Discussions
The transverse momentum spectrums of final-state products produced in high-energy collisions are analysed by using the multicomponent Erlang distribution and the Lévy distribution.In most cases, both the distributions are approximately in agreement with experimental data at RHIC and LHC energies.The multicomponent Erlang distribution seems to give a better description as compared to the Lévy distribution.Although the Lévy distribution is well known to give the transverse momentum spectrums, the multicomponent Erlang distribution gives a new method to describe the transverse momentum spectrums.The temperature parameters of interacting system corresponding to different types of final-state products are extracted from transverse momentum spectrums.Light particles correspond to a low temperature emission, and heavy Figure 9: The   spectrums of (a)  + ,  + , and ; (b)  − ,  − , and p ; and (c)  0  , Λ, Λ, , and Ξ − + Ξ + produced in p-p collisions at 0.9 TeV.The symbols represent experimental data of the ALICE Collaboration [19,20].The solid and dashed curves represent results calculated by using the multicomponent Erlang distribution and the Lévy distribution, respectively.particles correspond to a high temperature emission.For a jet with a given transverse momentum, larger mass corresponds to larger particle number and lesser transverse momentum per particle, which renders a lower temperature.Central collisions or high √  correspond to a relative high temperature.The multicomponent Erlang distribution extracts a relatively high temperature comparing with the Lévy distribution.
System size of the hadronic spectrums is well described by the two modeling distributions in the present   work.We see some correlations between the parameter values and system size.Particularly, the extracted temperature increases with increase of the system size from p-p collision to Cu-Cu and Au-Au (Pb-Pb) collisions at the same √  .This renders that the excitation degree of the interacting system increases with increase of the system size.Comparing with light nuclear collisions, a participant nucleon in heavy nuclear collisions takes part in more binary collisions, and more energy per nucleon deposits in heavy nuclear collisions.
It is well known that most of the hadrons in low transverse momentum region are produced in the process dominated by soft interaction, whereas the hadrons with high transverse momentums are produced in the process dominated by hard parton-parton scattering.According to the discussions in the present work, the first group of sources in the multicomponent Erlang distribution corresponds generally to the soft interaction, and the second or third group of sources corresponds to the hard scattering.The Lévy distribution does not distinguish the transverse momentum regions of soft interaction and hard scattering.
Although there are more or less differences in both the modeling distributions for the observed transverse momentum spectrums, the multicomponent Erlang distribution and the Lévy distribution describing approximately the transverse momentum spectrums in different systems render that there are some common laws or universality in multihadron production [21,22], even in general probability distributions.For example, the multicomponent Erlang distribution is also used to describe the probability distributions of some plant seed masses and sizes [23], and the Lévy distribution has more   other applications [12,24].We are interested in searching new applications of the two distributions.

Figure 1 :
Figure 1: Transverse momentum spectrums of final-state particles produced in Cu-Cu and Au-Au collisions at √  = 0.2 TeV.The symbols represent experimental data of the STAR Collaboration [11].The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively, (a), (b), (c), (d), (e), and (f) correspond to different final-state particles and collisions.
are used.Different values of  0 correspond to different results shown in the figure by different types of curves.All the parameter values with values of  2 /dof are given in Table

Figure 2 :
Figure 2: Transverse momentum spectrums of leading and subleading jets produced in Pb-Pb and p-p collisions at √  or √ = 2.76 TeV.The symbols represent experimental data of the CMS Collaboration [13].The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively, (a), (b), and (c) correspond to different selected conditions.
) and 7(b), respectively.The symbols represent experimental data

Figure 3 :Figure 4 :
Figure 3: Dependence of jet   spectrum on  0 in the Lévy distribution.The same experimental data [13] as those cited in Figure 2 are used.Different values of  0 correspond to different results shown in the figure by different types of curves.The unit of  0 is GeV/c 2 .

Figure 5 :
Figure 5: The   spectrums of charged jets produced in Pb-Pb collisions at √  = 2.76 TeV.The symbols represent experimental data of the ALICE Collaboration [14].The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively.

Figure 6 :
Figure 6: The   spectrums of (a) charged and (b) identified particles produced in Pb-Pb collisions at √  = 2.76 TeV.The symbols represent experimental data of the ALICE Collaboration [14, 15].The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively.

Figure 7 :
Figure7: The   spectrums of (a)  + +  − , (b)  0 , and (c) p produced in √  = 2.76 TeV Pb-Pb collisions in different centrality bins.For the sake of comparison, the results for  + +  − and  0 produced in 2.76 TeV p-p collisions are also given.The symbols represent experimental data of the ALICE Collaboration[16,17].The solid and dashed curves represent results calculated by the multicomponent Erlang distribution and the Lévy distribution, respectively.

Figure 8 :
Figure 8: The   spectrums of (a) Ξ and Ω as well as (b) inclusive electrons produced in inelastic p-p collision at 7 TeV.Experimental data measured by the ALICE Collaboration [15, 18] are shown by the symbols.Results calculated by using the multicomponent Erlang distributions and the Lévy distributions are shown by the solid and dashed curves, respectively.

Figure 11 :
Figure 11: Dependences of temperatures   and   on centrality.(a), (b), (c), and (d) as well as (e) and (f) correspond to dependence on centrality for particle productions at 0.2 and 2.76 TeV and jet production at 2.76 TeV, respectively.

Table 1 :
Parameter values for the two kinds of curves in Figure1.The values of  2 /dof and extracted temperatures are given.The errors for  1 ,  2 , and  can be neglected, and the relative errors for other parameters are less than 10%.
convenience, the spectrums are for various centrality bins, with each being scaled by the amount indicated in the legend.The parameter values used in the calculations are shown in Table

Table 2 :
Parameter values for the two kinds of curves in Figures

Table 3 :
Parameter values for different curves of the Lévy distributions in Figure3.The values of  2 /dof and extracted temperatures are given.The little marks LJ and SJ represent leading and subleading jets, respectively.The relative errors for the parameters are less than 10%.

Table 4 :
Parameter values for the two kinds of curves in Figures6(a), 8, and 10.The values of  2 /dof and extracted temperatures are given.The errors for  1,2,3,4 and  can be neglected, and the relative errors for other parameters are less than 10%.

Table 5 :
Parameter values for the two kinds of curves in Figures 6(b), 7(b), 7(c), and 9.The values of  2 /dof and extracted temperatures are given.The errors for  1 ,  2 , and  can be neglected, and the relative errors for other parameters are less than 10%.

Table 6 :
Parameter values for the two kinds of curves in Figure7(a).The values of  2 /dof and extracted temperatures are given.The errors for  1,2,3 and  can be neglected, and the relative errors for other parameters are less than 10%.