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Advances in High Energy Physics
Volume 2019, Article ID 4604608, 7 pages
https://doi.org/10.1155/2019/4604608
Research Article

Out-Of-Equilibrium Transverse Momentum Spectra of Pions at LHC Energies

1Nile University, Egyptian Center for Theoretical Physics, Juhayna Square of 26th-July-Corridor, 12588 Giza, Egypt
2World Laboratory for Cosmology And Particle Physics (WLCAPP), 11571 Cairo, Egypt

Correspondence should be addressed to Abdel Nasser Tawfik; ge.ude.un@kifwata

Received 9 March 2019; Revised 10 May 2019; Accepted 20 May 2019; Published 2 June 2019

Guest Editor: Sakina Fakhraddin

Copyright © 2019 Abdel Nasser Tawfik. 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.

Abstract

In order to characterize the transverse momentum spectra () of positive pions measured in the ALICE experiment, two thermal approaches are utilized; one is based on degeneracy of nonperfect Bose-Einstein gas and the other imposes an ad hoc finite pion chemical potential. The inclusion of missing hadron states and the out-of-equilibrium contribute greatly to the excellent characterization of pion production. An excellent reproduction of these -spectra is achieved at GeV and this covers the entire range of . The excellent agreement with the experimental results can be understood as a manifestation of not-yet-regarded anomalous pion production, which likely contributes to the long-standing debate on “anomalous” proton-to-pion ratios at top RHIC and LHC energies.

1. Introduction

The collective properties of strongly interacting matter (radial flow, for instance) and dynamics of colliding hadrons can be explored from the study of transverse momentum distributions () of produced particles. RHIC results on well-identified particles produced at low , especially pions, have shown that the bulk matter created can be well described by hydrodynamics [1]. It should be emphasized that the high- spectra, especially for the lowest-lying Nambu-Goldstone bosons, pions, likely manifest dynamics and interactions of partons and jets created in the earliest stage of nuclear collisions [2]. For instance, the collective expansion in form of radial flow might be caused by internal pressure gradients. Furthermore, the -distributions are assumed to determine conditions, such as temperature and flow velocity, gaining dominance during the late eras of the evolution of the high-energy collision which is generically well-described as kinetic freeze-out, where the elastic interactions are ceased, conclusively.

From the theoretical point of view, the -spectra are excellent measurements enabling us a better understanding of the QCD interactions. Soft nonperturbative QCD can be well applied to low -regime (below a few GeV/c) [3]. Fragmentation of QCD string [4], parton wave functions in flux tube [5], parton thermodynamics [6], and parton recombination [7] are examples on underlying physics. At high-, hard-scattering cross-section from QCD perturbative calculations, parton distribution functions, and parton-to-hadron fragmentation functions have been successfully utilized in reproducing -spectra of various produced particles [8].

It is worth mentioning that there is no well-defined line separating nonperturbative from perturbative -regimes [3]. Even the various theoretical studies are not distinguishing sharply between both of them. For instance, when constructing partition functions, extensive and nonextensive statistical approaches are frequently misconducted [911]. For instance, the claim that high -spectra of different produced particles are to be reproduced by Tsallis statistics seems being incomplete [11, 12]. This simply inspires a great contradiction between nonperturbative and perturbative QCD [12]. The statistical cluster decay could be scaled as power laws very similar to the ones of Tsallis statistics. The earlier is conjectured to cover a wide range of , while the latter is limited to a certain -regime. This would lead to an undesired mixing up that the observed power laws might be stemming from the statistical cluster decay and interpreted as a Tsallis-type of nonextensivity.

In addition to the proposal of utilizing a generic (non)extensive statistical approach [911], we want here to recall another theoretical framework based on an ad hoc physically motivated assumption that the pion production might be interpreted due an out-of-equilibrium process [13]. Such an approach is stemming from the pioneering works of Bogolubov devoted to an explanation for the phenomenon of superfluidity on the basis of degeneracy of a non-perfect Bose-Einstein gas [14] and determining the general form of the energy spectrum, an ingenious application of the second quantization [15]. Finite pion chemical potential recalls Bogolubov dispersion relation for low-lying elementary excitations of pion fluid. In this case, degenerate state of statistical equilibrium is removed through inserting a noninvariant term to the Hamiltonian, e.g., pion chemical potential.

After a short review of the thermal approach, the Hadron Resonance Gas (HRG) model in equilibrium is introduced in Section 2. A discussion on how to drive it towards nonequilibrium through inclusion of repulsive interactions is added. In Section 3, we elaborate modifications carried out towards implementing nonperfect Bose-Einstein gas based on a proposal of pion superfluidity. Another out-of-equilibrium thermal approach is outlined in Section 4, where finite pion chemical potential is ad hoc imposed. The results shall be discussed in Section 5. The conclusions are given in Section 6.

2. A Short Review on Equilibrium Resonance Gas with Van der Waals

The hadron resonances treated as a noninteracting gas [1622] are conjectured to determine the equilibrium thermodynamic pressure of QCD matter below chiral and deconfinement critical temperature, i.e., hadron phase. It has been shown that the thermodynamics of a strongly interacting system can also be approximated as an ideal gas composed of hadron resonances with masses GeV [19, 23]. Interested readers are kindly advised to consult the most recent review article [24]. The resonances added in contribution with the degrees of freedom needed to characterize the hadron phase.

The grand canonical partition function can be constructed as where , , and are the Hamiltonian, the temperature, and the chemical potential of the system, respectively. The Hamiltonian can be given by as summation of the kinetic energies of relativistic Fermi and Bose particles including the relevant degrees of freedom and the interactions resulting in formation of resonances and well describing the particle production in high-energy collisions. Under these assumptions, the sum over the single-particle partition functions of existing hadrons and their resonances introduces dynamics to the partition function,where is the dispersion relation of -th particle, is spin-isospin degeneracy factor, and stands for fermions and bosons, respectively.

In the present work, we include hadron resonances with masses GeV compiling by the particle data group (PDG) 2018 [25]. This mass cut-off is assumed to define the validity of the HRG model in characterizing the hadron phase [26, 27]. The inclusion of hadron resonances with heavier masses leads to divergences in all thermodynamic quantities expected at temperatures larger than the Hagedorn temperature [16, 17]. In addition to these aspects, there are fundamental reasons (will be elaborated in forthcoming sections) favoring the utilization of even ideal HRG model in predicting the hadron abundances and their thermodynamics. For the sake of completeness, we highlight that the hadronic resonances which are not yet measured, including missing ones, can be parameterized as a spectral function [28].

As given earlier, we assume that the constituents of the HRG are free (collisionless) particles. Some authors prefer taking into account the repulsive (electromagnetic) van der Waals interactions in order to partly compensate strong interactions in the hadronic medium [29] and/or to drift the system towards even partial nonequilibrium. Accordingly, each constituent is allowed to have an eigenvolume and the hadronic system of interest becomes thermodynamically partially out-of-equilibrium (how does a statistical thermal system, like HRG, become out-of-equilibrium? To answer this question, one might need to recall the main parameters describing particle production in equilibrium. These are , , and [30]. In the present work, we first focus on the third parameter and therefore describe this as a partial out-of-equilibrium process. The volume , the normalization parameter typically constrained by pions, becomes a subject of modification through van der Waals repulsive interactions, for instance. Furthermore, it should be also noticed that the chemical potentials should be modified, as well, at least in connection with the modification in . This would explain that taking into account van der Waals repulsive interactions, known as excluded volume corrections, contributes to deriving the system of interest towards nonequilibrium.). Thus, the total volume of HRG constituents should be subtracted from the fireball volume or that of heat bath. Considerable modifications in thermodynamics of HRG including energy, entropy, and number densities should be taken into consideration. It should be highlighted that the hard-core radius of hadron nuclei can be related to the multiplicity fluctuations.

How large can be the modification in ? The answer to this question is conditioned, for instance, to the capability of the HRG model with finite-volumed constituents to reproduce first-principle lattice QCD simulations. At radius fm, it was found that the disagreement with reliable lattice QCD calculations becomes more and more larger [29]. It was concluded that such an excluded volume-correction becomes practically irrelevant, as it causes negligible effects at low temperatures [29]. But on the other hand, a remarkable deviation from the lattice QCD calculations is noticed at high .

The repulsive interactions between hadrons are considered as a phenomenological extension of the HRG model. Exclusively, this is based on van der Waals excluded volume [3134]. Intensive theoretical works have been devoted to the estimation of the excluded volume and its effects on the particle production and the fluctuations [35], for instance. It is conjectured that the hard-core radius of the hadrons can be related to the multiplicity fluctuations of the produced particles [36]. In the present work, we simply assume that the hadrons are spheres and all have the same radius. On the other hand, the assumption that the radii would be depending on the hadron masses and sizes could come up with a very small improvement. Various types of interactions have been assumed, as well [37, 38]. For the sake of possible comparison with existing literature, we focus on the van der Waals repulsive interaction. By replacing the system volume by the actual one, , the van der Waals excluded volume can be deduced [31]where is volume and is the number of each constituent hadron. is the corresponding hard sphere radius of -th particle. The procedure encoded in (3) leads to modification in the chemical potentials , where the thermodynamic pressure is self-consistently expressed as and where the superscript refers to thermodynamic quantities calculated in HRG model with point-like constituents, i.e., ideal gas.

In the section that follows, we work out out-of-equilibrium spectra of the positive pions, where finite pion chemical potential shall be ad hoc inserted in.

3. Out-of-Equilibrium -Spectra of Pions

In U global symmetry, where the scalar field has a unitary transformation by the phase factor , the Bose-Einstein condensation of lowest-lying Nambu-Goldstone bosons could be studied from the partition function [39] where is a parameter carrying full infrared characters of the scalar field. This can be treated as a variational parameter relating to the charge of condensed boson particle. At , (2) can obviously be recovered. When the volume element is expressed in , rapidity and azimuthal angle as and the energy becomes , then at , where is the pion transverse mass, the transverse momentum spectrum of pions is given as where is the branching ratio of resonances decaying into pions.

The pion -spectrum is calculated from direct pions plus all contributions stemming from the heavier hadron resonances decaying into pions, in which the corresponding branching ratio should be taken into consideration (8)

In the results shown in Figure 1, we distinguish between the hadron resonances with the given mass cut-off and that without sigma states. In both cases, we also distinguish between results at chemical equilibrium of pion production, i.e., , and that at out-of-equilibrium, i.e., .

Figure 1: Number of positive pions per transverse momentum per rapidity is given dependent on the transverse momentum. The symbols refer to the ALICE measurements in the most central () Pb+Pb collisions at TeV. The left-hand panel shows a comparison with (10) at vanishing and finite GeV [41]. In middle and right-hand panels, HRG calculations (10), with and without scalar states and excluded volume corrections are confronted to the experimental measurements, as well.

4. Out-of-Equilibrium Thermal Distribution

In this section, we recall another thermal distribution at out-of-equilibrium [13], inspired by the proposal of Bogolubov to explaining the phenomenon of superfluidity as degeneracy of a nonperfect Bose-Einstein gas and determine a general form of energy spectra, a kind of ingenious application of second quantization.

In the approach introduced in [13], the -spectra of negatively charged bosons measured from A GeV O+Au and S+S collisions by NA35 experiment have been successfully reproduced. It was assumed that a cylindrical tube of matter with radius expands, longitudinally, but without transverse flow . When replacing in (2), for instance, by the azimuthal angle and the covariant form (four-momentum and velocity) and then integrating over the freeze-out time , the -distribution could be determined.

Then, at finite rapidity , the pion -spectrum reads The contributions likely added by heavy resonances are contributing straightforwardly to (10). In Figure 1, we distinguish between results at chemical equilibrium of pion production, i.e., and that at out-of-equilibrium, i.e., . Accordingly, we can examine the possibility of proposing a plausible explanation for the long-standing baryon-to-meson ratios, such as proton-to-pion ratios, especially at top RHIC and LHC energies [40]. We propose that this would be due to anomaly in the pion production.

5. Results

Figure 1 shows -spectra of positive pions in dependence on the transverse momentum. The symbols refer to the ALICE measurements in the most central () Pb+Pb collisions at TeV [41]. The curves represent the present calculations.

The calculations from (10) at vanishing (dash-double-dotted curve) and finite GeV (solid curve) are depicted in the left panel. It is obvious that finite leads to excellent agreement with the experimental results. The middle and right panels show the impacts for the excluded volume corrections and the scalar sigma states, respectively.

As done in the left panel, the contributions added by finite are examined in the middle panel. We find that the results from ideal HRG model at vanishing underestimate the experimental results. Finite and finite improve the reproduction of the experimental results.

It intends to investigate the importance of these sigma states in reproducing -spectra of pions at LHC energies, right panel. Various sigma states are to be in(ex)cluded: , known as , which was excluded from 2014 particle data group and from our calculations as well, and , , and and , widely known as , , , , and .

The HRG calculations (8) with and without sigma states are presented in the right panels. Also, here we distinguish between vanishing (dash-double-dotted curve) and finite GeV (solid curve). As the case with out-of-equilibrium thermal distributions, almost the same conclusion can be drawn here. Furthermore, the impacts of the inclusion of scalar sigma states are analyzed. It is obvious that these states seem to enhance the out-of-equilibrium, especially at large . The excluded volume corrections are taken into consideration in the way that all stable hadrons and resonances with masses GeV are assumed to equally have finite volume, . As discussed earlier, at fm, the corrections seem very small. There is no such a large variety to increase and simultaneously conserve the thermodynamics consistency. A systematic analysis has been discussed in [29]; see Figure 1 and the related text.

The small difference when is increased from to fm can basically be understood from the corresponding freeze-out temperature, with the inclusion of scalar sigma states MeV, at fm. But at fm, becomes MeV. These results are confirmed when scalar sigma states are removed, MeV at fm. Here, it was not possible to increase to fm due to divergence in the thermodynamic quantities, at the chemical freeze-out, for instance, should jump to GeV to fulfill the freeze-out conditions [2022]! Therefore we have checked fm. The corresponding MeV. It seems in order to highlight that these results are limited to the validity of the approximation that all hadrons are conjectured to have the same radius, fm.

We find that at vanishing , the results from (8) with and without scalar sigma states are almost identical. But at finite , the HRG calculations with and without scalar sigma states become distinguishable. This can be understood due to the remarkable characteristics and the great experimental abundances of the sigma states, which would appear as two-pion scalar-isoscalar resonances and are regarded as excitations of the scalar condensates, i.e., playing similar roles as the Higgs boson does for strong interactions [42]. Their -channel interaction is a maximally attractive one. This could be so strong that it breaks spontaneously the chiral symmetry and produces a quark condensate. Instanton-induced ’t Hooft interaction is also conjectured as a mechanism for that attraction [42].

In order to elaborate more about the reasons why we have chosen another alternative and thought of confronting measured -spectra of pions to Bogolubov superfluidity of Bose-Einstein gas and/or a thermal approach for out-of-equilibrium production of pions, i.e., finite , some remarks are now in order. Interpreting out-of-equilibrium in heavy-ion collisions at LHC in terms of the nonextensive Tsallis statistics [43] should be a subject of a fundamental revision. This has been discussed in great detail in [911]. On the one hand, the basic idea behind the thermal approach such as the HRG model is apparently additivity. Obviously, the Tsallis-type approach applies extensive statistics to the additive HRG, where both exponential and logarithm functions are merely replaced with the corresponding Tsallis counterpart function. On the other hand, the -spectra of positive pions from the most central () Pb+Pb collisions at TeV are not reproducible by in [43] (not drawn in Figure 1). But an excellent reproduction of these -spectra is achieved at GeV, Section 4. The excellent agreement apparently covers the entire range of , left panel of Figure 1.

6. Conclusions and Outlook

The standard statistical thermal approach was reported as not being able to reproduce various baryon-to-boson ratios, especially at top RHIC and LHC energies [40]. In reproducing the measured -spectra of positive pions at LHC energies, we have decided in favor for another alternative thermal approach assuming an out-of-equilibrium production of pions, (). In addition to this, we have introduced out-of-equilibrium production of pions to the well-known HRG model. In ancillary to the baryon, the strangeness chemical potential and the electric charge potential, we have imposed , as well.

We have shortly highlighted the incompleteness of nonextensive Tsallis statistics, especially when confronted to bulk matter created at relativistic energies. The basic idea of implementing an additive resonance gas even with Tsallis algebra, where both exponential and logarithm functions are properly replaced by Tsallis algebra, seems not at all modifying the extensivity. The latter obviously contradicts the intention of taking into account out-of-equilibrium particle production.

We conclude that -spectra of positive pions produced in most central () Pb+Pb collisions at TeV are well reproduced at GeV. This is the case in two different thermal approaches. The first one is based on degeneracy of a nonperfect Bose-Einstein gas or an ingenious application of second quantization known as Bogolubov superfluidity. The second one is the well-known HRG model, in which was ad hoc introduced. The baryon, strangeness, and electric charge chemical potentials, etc. can also be taken into consideration. This approach seems being successful in reproducing -spectra of positive pions, at LHC energies, when PDG sigma states are taken into account.

Future works shall be devoted to a systematic comparison with other approaches and to investigating the impacts that the PDG sigma states in reproducing -spectra of other hadrons. Furthermore, -spectra of other bosons shall be extended to understand whether they require out-of-equilibrium processes, as the positive pions do. The behavior of the pion production at high energies, where the production of kaons, protons, and antiprotons seem requiring anomalously large contributions from the exponential term to describe the shape of their transverse momenta, could be a subject of a similar out-of-equilibrium analysis, as well.

Data Availability

The availability data are available.

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

The author would like to thank David Blaschke and Ernst-Michael Ilgenfritz for the stimulating discussions on the nonequilibrium pion production at high energies. The author acknowledges the financial support by DAAD-Wiedereinladung no. 57378442.

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