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Pion Spectra in Collisions as a Function of and Event Multiplicity
We study the charged pion transverse momentum spectra in collisions as a function of collision energy and event multiplicity using Tsallis distribution. This study gives an insight of the pion production process in collisions. The study covers pion spectra measured in collisions at SPS energies (6.27–17.27 GeV), RHIC energies (62.4 GeV and 200 GeV), and LHC energies (900 GeV, 2.76 TeV, and 7 TeV). The Tsallis parameters have been obtained and parameterized as a function of . The study suggests that as we move to higher energy more and more hard processes contribute to the spectra. We also study the charged pion spectra for different event multiplicities in collisions for LHC energies using Tsallis distribution. The variation of the Tsallis parameters as a function of event multiplicity has been obtained and their behavior is found to be independent of collision energy.
The particle spectra measured in hadronic collisions are of utmost interest because of their fundamental nature and simplicity, which allow verifying pQCD [1, 2] calculations and also help to make comprehensive phenomenological studies. The ratios of the particle yields obtained from the measured spectra allow getting the chemical freeze-out conditions, whereas the spectra themselves reflect the conditions at the kinetic freeze-out. The particle spectra provide useful information about the collision dynamics. The low region of the spectrum corresponds to the particles originating from low momentum transfer and multiscattering processes (nonperturbative QCD), whereas the high region comes from the hard-parton-scattering (pQCD) among the initial partons. The transition of this nonperturbative to perturbative dynamics has no sharp boundary, though one can have an estimate from the “” . Extensive [4, 5] and nonextensive [6–10] statistical approaches have been used to characterize particle spectra in terms of thermodynamic variables. Extensive statistics assume thermal and chemical equilibrium of the system at hadronic phase which lead to an exponential distribution of the particle spectra. In experiments, the particle spectra show a power-law behavior at high . This behavior is reproduced by the nonextensive approach with an additional parameter. In recent times, the Tsallis  statistical approach is widely used to describe the particle spectra obtained in high-energy collisions with only two parameters, the temperature and , known as nonextensivity parameter which is a measure of temperature fluctuations or degree of nonequilibrium in the system.
The Tsallis distribution gives an excellent description of spectra of all identified mesons measured in collisions at GeV . In a recent work [12, 13], the Tsallis distribution has been used to describe the spectra of identified charged hadrons measured in collisions at RHIC and at LHC energies. Such an approach has also been applied to the inclusive charged hadron data in recent publications [14, 15]. It has been shown in [12, 16] that the functional form of the Tsallis distribution with thermodynamic origin is of the same form as the QCD-inspired Hagedorn formula [17, 18]. This could be the reason of success of Tsallis distribution in collisions which is a power law typical of QCD hard scatterings. The hardness of the spectra is thus related to and the parameter governs the contribution from soft collisions.
Using the Tsallis phenomenological function, we review and study the charged pion spectra in collisions in a large energy regime, spanning from SPS  (6.27 GeV–17.27 GeV) and RHIC  (62.4 and 200 GeV) to LHC  (900 GeV, 2.76 TeV and 7 TeV) energies. The object of the present work is to study the behaviour of the Tsallis parameters as a function of collision energy. We also study the charged pion spectra for different event multiplicities in collisions for LHC energies. Among all hadrons, pions are chosen because of their abundance in collisions, simple quark structure, and availability of the data at different energies.
The transverse momentum spectra of hadrons, obtained from different fixed and collider experiments, have shown that the high region of the spectra can be described successfully by the power law, where is the normalization constant and is the power which determines the shape of the spectra at high . However, the low region of the particle spectra shows an exponential shape and can be described by the Boltzmann-Gibbs [22, 23] statistical approach, where is the normalization constant, is the particle energy, and is the temperature of the system.
In the early 80s, Hagedorn  proposed a phenomenological function which describes the particle spectra for both the higher and lower regions: where , , and are the fit parameters. The above equation describes an exponential behavior for low and a power-law behavior for high . Consider The parameter in this equation is often related to the “power” in the “QCD-inspired” quark interchange model .
In the late 80s, Tsallis  introduced the idea of the nonextensive statistics in place of thermal Boltzmann-Gibbs statistics. This approach includes a parameter , known as nonextensive parameter which quantifies the temperature fluctuation  in the system as . The nonextensive statistics assume Boltzmann-Gibbs form in the limit . In Tsallis approach, the Boltzmann-Gibbs distribution takes the form where is the normalization factor. One can use the relation = at mid-rapidity and in (5) to obtain where is the normalization factor. Equation (6) can be rewritten as The value of can be obtained by integrating (7) over momentum space: Here the quantity is the integrated yield. Equation (6) with the normalization constant takes the form 
Larger values of correspond to smaller values of which imply dominant hard QCD point-like scattering. Both and have been interchangeably used in Tsallis distribution [7, 11, 20, 25, 26]. The Tsallis interpretation of parameters as temperature and as nonextensivity parameter is more suited for heavy ion collisions while for collisions Hagedorn interpretation in terms of power and a parameter which controls soft physics processes is more meaningful. Phenomenological studies suggest that, for quark-quark point scattering, [27, 28] and when multiple scattering centers are involved, grows larger.
3. Results and Discussions
All the studies are performed with (9) and the fit parameters , , and are obtained. The different experiments, energies, rapidity ranges, and particles used in the analysis are summarized in Table 1. For SPS energies only available data is for measured by NA61 Collaboration ; for RHIC and LHC energies we use . All the data used are measured in mid-rapidity and are given for unit rapidity. The difference in rapidity range is not expected to affect the behaviour of the spectra. CMS experiment presented  transverse momentum spectra for different events classified on the basis of number of true tracks here referred to as track multiplicity of event or simply multiplicity. Each multiplicity class is represented by average number of tracks ().
3.1. Tsallis Parameters as a Function of in System
In this analysis all the Tsallis parameters are obtained for charged pion spectra as a function of in system for SPS , RHIC , and LHC  energies. Similar study is available in  using RHIC and LHC data and in  for SPS and LHC data.
The pion spectra measured in collisions at different are shown in Figure 1 along with with Tsallis fits (9) shown by solid lines. The spectra are scaled by arbitrary factors (given in figure) for visual clarity. In case of RHIC data, we restrict the range to 1.7 GeV/ to have similar range at all energies. It can be noticed that the spectra become harder with increase in which is depictive of occurrence of harder scatterings at higher collision energy. The per degree of freedom values for all the fits are given in Table 2. The values are 1, which is indicative of good fit quality.
The parameters and obtained from this analysis are shown in Figures 2 and 3, respectively, as a function of . The variation of as a function of is shown in Figure 4. The parameter decreases with increasing and starts saturating at LHC energies. The value of also reduces slowly from SPS energies to LHC energies. The integrated yield increases 10 times when going from SPS to highest LHC energy.
Larger value of (also larger value of ) suggests that the spectra have contribution from processes involving small momentum transfer arising due to the rescattering, recombination of partons, fragmentation from strings, and so forth, whereas smaller values of are indicative of harder processes being involved in particle production. Thus the spectra at SPS energies have large softer contribution and as the collision energy increases more and more contribution from hard processes is added.
All the three parameters can be parametrized by a function of type Here , , and for , , , and for , and , , and for . Using the parameterizations for by (10) we get in the limit . The extrapolated values for , and for TeV are , MeV, and .
3.2. Tsallis Parameters as a Function of Multiplicity () for LHC Energies
The Tsallis parameters for charged pion spectra are studied as a function of event multiplicity for different LHC energies 900 GeV and 2.76 and 7 TeV. The event multiplicity data was also studied in a recent work  but our analysis and interpretations are different.
The invariant yield spectra corresponding to different multiplicities are fitted with (9) and are shown by the solid black lines in Figure 5 for 900 GeV, in Figure 6 for 2.76 TeV, and in Figure 7 for 7 TeV center of mass energy. The spectra are scaled up for distinctness. The Tsallis distribution describes all the spectra well, shown by the values given in Table 3. The values are little higher for some of the lower multiplicities due to the deviation of first data point in spectra with the curve.
The parameters and obtained from the fits are shown in Figures 8 and 9, respectively, as a function of . The circles, squares, and triangles correspond to the parameter values obtained from data at 900 GeV, 2.76 TeV, and 7 TeV, respectively. It is seen that both the parameters and decrease rapidly and then start saturating with the increase of for all three energies. This variation (of and ) is very similar to the variation which we find as a function of . It means that events with higher multiplicity have larger contribution from hard processes. The value of for high multiplicity events at 7 TeV is which is depictive of production from point quark-quark scattering. The variation of and as a function of can be described by the same curve given in the figure for all three energies and are parameterized by Here , , and for and , , and for .
The integrated pion yield distribution in different multiplicity classes is shown in Figure 10 for the three LHC energies. The total integrated pion yield for each energy can be obtained by integrating the above distributions over all multiplicity classes. It is noticed that as the energy increases more and more high mutliplicity events are added in the sample with mean of the distribution shifting towards higher .
This work presented the study of the transverse momentum spectra of the charged pions for different collisional energies and also for different event multiplicities (at LHC energies) using Tsallis distribution. The Tsallis parameter decreases with increasing and starts saturating at LHC energies. The value of also reduces slowly from SPS energies to LHC energies. It means that the spectra at SPS energies have large softer contribution and as the collision energy increases more and more contribution from hard processes is added. The integrated pion yield increases with increasing and becomes 10 times when going from SPS to highest LHC energy. The Tsallis parameters are also obtained as a function of event multiplicity for all three LHC energies which can be described by the same curve. The variation of and as a function of multiplicity is very similar to the variation which we find as a function of . It means that events with higher multiplicity have larger contribution from hard processes. The value of for high multiplicity events at 7 TeV is which is depictive of production from point quark-quark scattering. The integrated pion yield distribution for the three LHC energies shows that as the energy increases, more and more high mutliplicity events are added in the sample with mean of the distribution shifting towards higher multiplicity.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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