Transverse Momentum and Pseudorapidity Distributions of Charged Particles and Spatial Shapes of Interacting Events in Pb-Pb Collisions at 2 . 76 TeV

The transverse momentum and pseudorapidity distributions of charged particles produced in Pb-Pb collisions with different centrality intervals at center-of-mass energy per nucleon pair √sNN = 2.76TeV have been analyzed by using the improved multisource thermal model in which the whole interacting system and then the sources are described by the Tsallis statistics. The modelling results are in agreement with experimental data of the ALICE Collaboration. The rapidity distributions of charged particles are obtained according to the extracted parameter values.The shapes of interacting events (the dispersion plots of charged particles) are given in the momentum, rapidity, velocity, and coordinate spaces. Meanwhile, the event shapes in different spaces consisted by different transverse quantities and longitudinal quantities are presented.


Introduction
Because the Relativistic Heavy Ion Collider (RHIC) run successfully in 2000, the studies of high energy heavy ion collisions have been arriving in the collider era since then.Many experiments on heavy ion collisions in GeV energy region have been finished in the past years [1][2][3][4].The center-of-mass energy per nucleon pair (√  ) at the RHIC is superlatively 200 GeV.Presently, the most powerful heavy-ion collider in the world, the Large Hadron Collider (LHC), was built successfully in [2008][2009].Recently, the LHC does perform successful experiments on proton-proton collision at total energy of 7 TeV, proton-lead collisions at √  = 5.02 TeV, and lead-lead (Pb-Pb) collisions at √  = 2.76 TeV [5][6][7][8].
Many experiments do measure distributions and correlations of multiplicities, pseudorapidities (rapidities), transverse momentums, azimuths, and others due to that they are the "first day" measurement quantities on charged particles.Generally, these "first day" measurement quantities are studied in different centrality intervals (impacting parameters), sizes of interacting systems, center-of-mass energies, and other available dependent quantities.These charged particles are divided into positively charged particles, negatively charged particles, charged mesons, charged baryons, protons, antiprotons, and other identified charged particles according to different classifications.Even if for the transverse momentum and pseudorapidity (rapidity) distributions of charged particles, we have different interacting mechanisms and distribution regions which are considered in modelling analyses.
The transverse momentum and pseudorapidity (rapidity) distributions of charged particles contain abundant information.Studies of transverse momentum and pseudorapidity (rapidity) distributions have been persisting from fixed target experiments at accelerators to collider experiments at the RHIC and LHC.Recently, the ALICE Collaboration reported the centrality dependences of the transverse momentum and pseudorapidity distributions for charged particles in Pb-Pb collisions at √  = 2.76 TeV [8,9].We are interested in analyzing the ALICE data by using the current models 2 Advances in High Energy Physics (for a collection of many models, see [10]), particularly the multisource thermal model which was proposed by us some years ago [11][12][13].
In this paper, we analyze the ALICE data on transverse momentum and pseudorapidity distributions of charged particles produced in Pb-Pb collisions at √  = 2.76 TeV [8,9] by using the improved multisource thermal model [11][12][13] in which the Boltzmann distribution is replaced by the Tsallis statistics [14][15][16][17][18][19][20][21].The rapidity distributions of charged particles are obtained according to the extracted parameter values.The dispersion plots of charged particles, that is, the shapes of interacting events, are given in different spaces.

The Model and Method
In a given reference frame and in the rapidity space, in the framework of the multisource thermal model [11][12][13], many emission sources with different rapidities   are assumed to form in high energy collisions.According to the values of   , a target cylinder in rapidity interval [  min ,   max ], a projectile cylinder in rapidity interval [  min ,   max ], a leading target nucleon cylinder in rapidity interval [  min ,   max ], and a leading projectile nucleon cylinder in rapidity interval [  min ,   max ] are assumed to form.For a symmetric collision system, we have equal relations   min = −  max ,   max = −  min ,   min = −  max , and   max =   min .
In the original multisource thermal model [11][12][13], we have used in fact a multitemperature relativistic ideal gas model which expects a few local thermal equilibrium states existing in the collisions.Each local equilibrium state can be described by the relativistic ideal gas model (Boltzmann distribution) [22,23] with a given temperature, and the emission of particles in the rest frame of the considered source is isotropic.This multitemperature picture can be described by the Tsallis statistics [14][15][16][17][18][19][20][21] which describes the invariant particle momentum (  ) distribution in the rest frame of the considered source to be where  1 is the normalization constant of the distribution,  is the particle number,  is the temperature parameter of the source,  is a parameter (an entropic index) to characterize the degree of nonequilibrium, and   ,  0 , and  are, respectively, the energy, rest mass, and chemical potential of the considered particle.At LHC energies, the chemical potential can be neglected due to its small value.The real free parameters in (1) are then  and .Because of the introduction of the Tsallis statistics, the multisource thermal model is improved by us.
In the following section, except for the descriptions of transverse momentum and pseudorapidity distributions and the extraction of rapidity distributions, we shall present the event shapes in momentum space   (  ) −   , rapidity space  1 ( 2 )−, and velocity space   (  )−  (rescaled coordinate space ()/ 0 − / 0 ).Meanwhile, the event shapes in some transverse quantities [  ,   ,   (  / 0 )] and longitudinal quantities [  , ,   (/ 0 )] spaces are presented, too.We would like to point out that we can obtain separately rapidity and pseudorapidity distributions in the calculation.We do not need to do a conversion between the two distributions.Except for nucleus-nucleus collisions, the Tsallis statistics is also used to deal with proton-proton collision in literature [24][25][26].

Comparison and Extraction
The transverse momentum distributions of  + +  − ,  + +  − , and  +  produced in Pb-Pb collisions at √  = 2.76 TeV in different centrality intervals are shown in Figures 1(a)-1(c), respectively, where  EV denotes the number of events.From up data to low one in each panel, the corresponding centrality  80-90%  intervals are 0-5%, 5-10%, 10-20%, . .., and 80-90% scaled by multiplying 2 9 , 2 8 , 2 7 , . .., and 2 0 , respectively.The symbols represent the experimental data of the ALICE Collaboration [8] measured in the rapidity region || < 0.5 and the curves are results of the Tsallis statistics.The fitted parameter values are listed in Table 1 with the values of  2 /dof (per degree of freedom).To see clearly the dependences of parameters on centrality, the parameter values in Table 1 are also given in Figure 1(d).We can see that the temperature and nonequilibrium degree do not change in the centrality interval 0-40%.The temperature decreases and the nonequilibrium degree increases with increase of the centrality percentage in the interval 40-90%.Figure 2 presents the pseudorapidity distributions of charged particles produced in Pb-Pb collisions at √  = 2.76 TeV, where  ch denotes the number of charged particles.1), where the indexes  and  denote the target/projectile cylinders and the leading target/projectile nucleon cylinders, respectively.The values of  2 /dof are 0.120, 0.141, 0.110, and 0.109, respectively.In the calculation for Figure 2, only the contributions of  + +  − for the target/projectile cylinders and those of  +  for the leading target/projectile nucleon cylinders are considered due to the other contributions being small.We take  0 as the rest mass of a charged pion (or a proton) for the target/projectile cylinders (or the leading target/projectile nucleon cylinders).Another approximation is that the parameter values obtained in || < 0.5 (Figure 1) are used for a wide pseudorapidity range (Figure 2).The treatment for the target/projectile cylinders underestimates   by ∼0.01 GeV and overestimates   by ∼0.02.Contrarily, the second approximation overestimates   and underestimates   .In fact, the pseudorapidity distribution is not mainly determined by the temperature and degree of nonequilibrium but by the rapidity shifts and contribution ratio of leading nucleons.One can see that the model describes the pseudorapidity distributions of charged particles produced in Pb-Pb collisions at the LHC energy.The rapidity shifts, contribution ratio of leading nucleons, temperature, and degree of nonequilibrium do not depend obviously on the centrality percentage in the considered centrality interval 0-30%.
According to the parameter values extracted from the pseudorapidity distributions, we present the rapidity distributions of charged particles in Figure 3.For the purpose of comparison, the pseudorapidity distributions presented in Figure 2 are directly presented in Figure 3 by the dotteddashed cures.The meanings of other curves in Figure 3 are the same as those in Figure 2. One can see the difference between the rapidity and pseudorapidity distributions of respectively.If we rescale coordinates , , and  to / 0 , / 0 , and / 0 , respectively, Figure 6 is also the event shapes in (rescaled) coordinate space / 0 − / 0 (or / 0 − / 0 ).In the three figures, the closed circles, open circles, open squares, and closed squares correspond to the contributions of leading target nucleons, target cylinder, projectile cylinder, and leading projectile nucleons, respectively.One can see that the densities in the small momentum components, small  1 ( 2 ) and large , small   (  ) and large   , and small  () and large  regions are larger than those in other regions.
The event shapes (dispersion plots) in the momentum, rapidity, and velocity (coordinate) spaces are obviously different.The differences in event shapes (relative density distributions in the dispersion plots) for different centrality intervals are not obvious for the considered collisions.
To extract more information from the pseudorapidity distributions, we present the event shapes (dispersion plots) from 1000 charged particles in Pb-Pb collisions in different centrality intervals at √  = 2.76 TeV in some spaces consisted by different transverse quantities and different longitudinal quantities.Figures 7, 8 in different spaces consisted by different transverse quantities and different longitudinal quantities.The results in the same space for different centrality intervals are not obvious due to almost the same extracted parameters (except the total particle number).Although we have not compared more predicted results with experimental data in the present work, the improved multisource thermal model which adopts the Tsallis statistics can describe both the transverse momentum distribution and the pseudorapidity (rapidity) distribution in high energy collisions.The transverse momentum distribution is mainly determined by the temperature and nonequilibrium degree, and the pseudorapidity distribution is mainly determined by the rapidity shifts and contribution ratio of leading nucleons.
There are some fluctuations in our calculation by using the Monte Carlo method.For example, for an event with a given centrality, different sets of random numbers render different dispersion plots, which are small statistical fluctuations from event to event.The events with different centralities correspond to large statistical fluctuations from event to event.In some case, different events may have different interacting mechanisms, which render dynamical fluctuations from event to event.The present work concerns only the statistical fluctuations.

Conclusions
From the above discussions, we obtain following conclusions.
(a) The transverse momentum distributions of charged particles produced in Pb-Pb collisions with centrality intervals from 0-5% to 80-90% at √  = 2.76 TeV have been analyzed by using the improved multisource thermal model in which the whole interacting system and then the sources are described by the Tsallis statistics [14][15][16][17][18][19][20][21].The modelling results are in agreement with the experimental data of the ALICE Collaboration [8].In the centrality intervals from 0-5% to 30-40%, the temperature and nonequilibrium degree do not show a change.In the centrality intervals from 40-50% to 80-90%, the temperature decreases and the nonequilibrium degree increases with increase of the centrality percentage.
(b) The pseudorapidity distributions of charged particles produced in Pb-Pb collisions with centrality intervals from 0-5% to 20-30% at √  = 2.76 TeV have been analyzed by using the improved multisource thermal model, too.The modelling results are in agreement with the experimental data of the ALICE Advances in High Energy Physics Collaboration [9].The contributions of the leading target nucleons, target cylinder, projectile cylinder, and leading projectile nucleons are given in different regions in the rapidity space.The rapidity shifts and contribution ratio of leading nucleons do not depend obviously on the centrality percentage in the intervals from 0-5% to 20-30%.
(     obtained.The results for different centrality intervals are not obvious in the considered intervals from 0-5% to 20-30%.

Figure 5 :
Figure 5: The same as for Figure 4, but showing the results in the rapidity space  1 −  (or  2 − ).

Figure 7 :Figure 8 :Figure 9 :
Figure 7: The same as for Figure 4, but showing the results in the space   −   .

Figure 10 :
Figure10: The same as for Figure4, but showing the results in the space   − .

Figure 11 :Figure 12 :
Figure 11: The same as for Figure 4, but showing the results in the space   − .

Figure 13 :Figure 14 :Figure 15 :
Figure 13: The same as for Figure 4, but showing the results in the space   −   (or   − / 0 ).