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

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Out-Of-Equilibrium Transverse Momentum Spectra of Pions at LHC Energies Hindawi 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 Abdel Nasser Tawfik 1,2 1 Nile University, Egyptian Center for Teoretical 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 Tawfk; [email protected] Received 9 March 2019; Revised 10 May 2019; Accepted 20 May 2019; Published 2 June 2019 Guest Editor: Sakina Fakhraddin Copyright © 2019 Abdel Nasser Tawfk. Tis 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. 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 fnite pion chemical potential. Te 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 �� =0.12GeV and this covers the entire range of ��. Te 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 attopRHICandLHCenergies. 1. Introduction be well applied to low ��-regime (below a few GeV/c) [3]. Fragmentation of QCD string [4], parton wave functions in Te collective properties of strongly interacting matter (radial fux tube [5], parton thermodynamics [6], and parton recom- fow, for instance) and dynamics of colliding hadrons can bination [7] are examples on underlying physics. At high-��, be explored from the study of transverse momentum dis- hard-scattering cross-section from QCD perturbative calcu- � tributions ( �) of produced particles. RHIC results on well- lations, parton distribution functions, and parton-to-hadron � identifed particles produced at low �, especially pions, have fragmentation functions have been successfully utilized in shown that the bulk matter created can be well described by reproducing ��-spectra of various produced particles [8]. hydrodynamics [1]. It should be emphasized that the high- It is worth mentioning that there is no well-defned line � � spectra, especially for the lowest-lying Nambu-Goldstone separating nonperturbative from perturbative ��-regimes bosons, pions, likely manifest dynamics and interactions [3]. Even the various theoretical studies are not distinguishing of partons and jets created in the earliest stage of nuclear sharply between both of them. For instance, when con- collisions [2]. For instance, the collective expansion in form structing partition functions, extensive and nonextensive of radial fow might be caused by internal pressure gradients. statistical approaches are frequently misconducted [9–11]. For � Furthermore, the �-distributions are assumed to determine instance, the claim that high ��-spectra of diferent produced conditions, such as temperature and fow velocity, gaining particles are to be reproduced by Tsallis statistics seems being dominance during the late eras of the evolution of the incomplete [11, 12]. Tis simply inspires a great contradiction high-energy collision which is generically well-described as between nonperturbative and perturbative QCD [12]. Te kinetic freeze-out, where the elastic interactions are ceased, statistical cluster decay could be scaled as power laws very conclusively. similar to the ones of Tsallis statistics. Te earlier is conjec- From the theoretical point of view, the ��-spectra are tured to cover a wide range of ��, while the latter is limited to excellent measurements enabling us a better understanding acertain��-regime. Tis would lead to an undesired mixing of the QCD interactions. Sof nonperturbative QCD can up that the observed power laws might be stemming from 2 Advances in High Energy Physics the statistical cluster decay and interpreted as a Tsallis-type of existing hadrons and their resonances introduces dynamics of nonextensivity. to the partition function, In addition to the proposal of utilizing a generic �(�,� ,�) (non)extensive statistical approach [9–11], we want here to ln ℎ recall another theoretical framework based on an ad hoc � ∞ � −� (2) =�∑± ℎ ∫ �2�� {1 ± [ ℎ ℎ ]} , physically motivated assumption that the pion production 2�2 ln exp � might be interpreted due an out-of-equilibrium process [13]. ℎ 0 Such an approach is stemming from the pioneering works of � =(�2 +�2)1/2 ℎ Bogolubov devoted to an explanation for the phenomenon where ℎ ℎ is the dispersion relation of -th of superfuidity on the basis of degeneracy of a non-perfect particle, �ℎ is spin-isospin degeneracy factor, and ± stands for Bose-Einstein gas [14] and determining the general form fermions and bosons, respectively. of the energy spectrum, an ingenious application of the In the present work, we include hadron resonances with second quantization [15]. Finite pion chemical potential masses ≤2GeV compiling by the particle data group (PDG) recalls Bogolubov dispersion relation for low-lying elemen- 2018 [25]. Tis mass cut-of is assumed to defne the validity of tary excitations of pion fuid. In this case, degenerate state the HRG model in characterizing the hadron phase [26, 27]. of statistical equilibrium is removed through inserting a Te inclusion of hadron resonances with heavier masses leads noninvariant term to the Hamiltonian, e.g., pion chemical to divergences in all thermodynamic quantities expected potential. at temperatures larger than the Hagedorn temperature [16, Afer a short review of the thermal approach, the Hadron 17]. In addition to these aspects, there are fundamental Resonance Gas (HRG) model in equilibrium is introduced in reasons (will be elaborated in forthcoming sections) favoring Section 2. A discussion on how to drive it towards nonequi- the utilization of even ideal HRG model in predicting the librium through inclusion of repulsive interactions is added. hadron abundances and their thermodynamics. For the sake In Section 3, we elaborate modifcations carried out towards of completeness, we highlight that the hadronic resonances implementing nonperfect Bose-Einstein gas based on a which are not yet measured, including missing ones, can be proposal of pion superfuidity. Another out-of-equilibrium parameterized as a spectral function [28]. thermal approach is outlined in Section 4, where fnite pion As given earlier, we assume that the constituents of the chemical potential is ad hoc imposed. Te results shall be HRG are free (collisionless) particles. Some authors prefer discussed in Section 5. Te conclusions are given in Section 6. taking into account the repulsive (electromagnetic)vander Waals interactions in order to partly compensate strong interactions in the hadronic medium [29] and/or to drif the 2. A Short Review on Equilibrium Resonance system towards even partial nonequilibrium. Accordingly, Gas with Van der Waals each constituent is allowed to have an eigenvolume and the hadronic system of interest becomes thermodynamically Te hadron resonances treated as a noninteracting gas [16– partially out-of-equilibrium (how does a statistical thermal 22] are conjectured to determine the equilibrium thermo- system, like HRG, become out-of-equilibrium? To answer dynamic pressure of QCD matter below chiral and decon- this question, one might need to recall the main parameters fnement critical temperature, i.e., hadron phase. It has been describing particle production in equilibrium. Tese are �, shown that the thermodynamics of a strongly interacting �ℎ,and� [30]. In the present work, we frst focus on the system can also be approximated as an ideal gas composed of third parameter and therefore describe this as a partial out- hadron resonances with masses ≲2GeV [19, 23]. Interested of-equilibrium process. Te volume �, the normalization readers are kindly advised to consult the most recent review parameter typically constrained by pions, becomes a subject article [24]. Te resonances added in contribution with of modifcation through van der Waals repulsive interactions, the degrees of freedom needed to characterize the hadron for instance. Furthermore, it should be also noticed that the phase. chemical potentials �ℎ should be modifed, as well, at least in Te grand canonical partition function can be con- connection with the modifcation in �. Tis would explain structed as that taking into account van der Waals repulsive interactions, known as excluded volume corrections, contributes to deriv- ing the system of interest towards nonequilibrium.). Tus, (� N−H)/T �(�,�,�)=Tr [exp ], (1) the total volume of HRG constituents should be subtracted from the freball volume or that of heat bath. Considerable modifcations in thermodynamics of HRG including energy, where �, �,and� are the Hamiltonian, the temperature, entropy, and number densities should be taken into consid- and the chemical potential of the system, respectively. Te eration. It should be highlighted that the hard-core radius of Hamiltonian can be given by as summation of the kinetic hadron nuclei can be related to the multiplicity fuctuations. energies of relativistic Fermi
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