Advances in High Energy Physics

Volume 2019, Article ID 4604608, 7 pages

https://doi.org/10.1155/2019/4604608

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

^{1}Nile University, Egyptian Center for Theoretical Physics, Juhayna Square of 26th-July-Corridor, 12588 Giza, Egypt^{2}World 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 [9–11]. 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 [9–11], 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 [16–22] 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* eigen*volume 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 [31–34]. 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., .