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 and Bose particles including the How large can be the modifcation in �?Teanswerto relevant degrees of freedom and the interactions resulting this question is conditioned, for instance, to the capability in formation of resonances and well describing the particle of the HRG model with fnite-volumed constituents to production in high-energy collisions. Under these assump- reproduce frst-principle lattice QCD simulations. At radius 1 tions, the sum over the single-particle partition functions �ℎ �>0.2fm, it was found that the disagreement with reliable Advances in High Energy Physics 3 lattice QCD calculations becomes more and more larger [29]. the Bose-Einstein condensation of lowest-lying Nambu- It was concluded that such an excluded volume-correction Goldstone bosons could be studied from the partition func- becomespracticallyirrelevant,asitcausesnegligibleefectsat tion [39] low temperatures [29]. But on the other hand, a remarkable � �3� deviation from the lattice QCD calculations is noticed at high �(�,� )= (�2 −�2)�2 −�∫ ln � � 3 �. � (2�) (7) Te repulsive interactions between hadrons are consid- � −(�−� )/� −(�+� )/� ⋅[ + (1 − � � )+ (1 − � � )] , ered as a phenomenological extension of the HRG model. � ln ln Exclusively, this is based on van der Waals excluded volume [31–34]. Intensive theoretical works have been devoted to where � is a parameter carrying full infrared characters of the estimation of the excluded volume and its efects on the the scalar feld. Tis can be treated as a variational parameter particle production and the fuctuations [35], for instance. It relating to the charge of condensed boson particle. At |��|< is conjectured that the hard-core radius of the hadrons can �, (2) can obviously be recovered. When the volume element 3 be related to the multiplicity fuctuations of the produced � � is expressed in ��,rapidity� and azimuthal angle � as 3 particles [36]. In the present work, we simply assume that � �=���� cosh(�)��� �� �� and the energy becomes �= thehadronsarespheresandallhavethesameradius.On 2 2 1/2 �� cosh(�),thenat�� �→ � ,where�� =(�� +� ) is the the other hand, the assumption that the radii would be pion transverse mass, the transverse momentum spectrum of depending on the hadron masses and sizes could come up pions is given as with a very small improvement. Various types of interactions have been assumed, as well [37, 38]. For the sake of possible 1 �2� � =� comparison with existing literature, we focus on the van der 2��� ����� Waals repulsive interaction. By replacing the system volume � by the actual one, ����, the van der Waals excluded volume (�) (�) �� cosh (�) �� cosh (�) − �� can be deduced [31] ⋅ {exp [ ] (2�)3 �

� =�−∑V � , −1 ��� ℎ ℎ (3) ℎ −1} +∑� (8) reson.�→�

V =4(4��3/3) � �(�����.) (�) �(�����.) (�) − � where ℎ ℎ is volume and ℎ is the number of ⋅ � cosh { [ � cosh � ] 3 exp each constituent hadron. �ℎ is the corresponding hard sphere (2�) � radius of ℎ-th particle. Te procedure encoded in (3) leads to −1 modifcation in the chemical potentials �̃ℎ =�ℎ − Vℎ �,where the thermodynamic pressure � is self-consistently expressed −1} ×�reson.�→�, �� as ∑ℎ �ℎ (�, �̃ℎ) and

where �reson.�→� is the branching ratio of resonances decaying �� into pions. ∑ℎ �ℎ (�, �̃ℎ) �= , (4) Te pion ��-spectrum is calculated from direct pions 1+∑ V ��� (�, �̃ ) ℎ ℎ ℎ ℎ plus all contributions stemming from the heavier hadron �� resonances decaying into pions, in which the corresponding ∑ℎ �ℎ (�, �̃ℎ) �= , (5) branching ratio should be taken into consideration (8) 1+∑ V ��� (�, �̃ ) ℎ ℎ ℎ ℎ �total 2 � 1 � �� � �� ̃ � ∑ℎ �ℎ (�, �ℎ) 2�� �� �� � �= , (6) � � �� 1+∑ V ��� (�, �̃ ) ℎ ℎ ℎ ℎ � 2 � 1 � �� � = � 2�� �� �� � where the superscript �� refers to thermodynamic quantities � � �� (9) 2 � calculated in HRG model with point-like constituents, i.e., 1 � � � ideal gas. +∑ reson. � 2�� �� �� � In the section that follows, we work out out-of- reson.�→� � � �reson.�→� � equilibrium � spectra of the positive pions, where fnite pion ×� . chemical potential shall be ad hoc inserted in. reson.�→� In the results shown in Figure 1, we distinguish between 3. Out-of-Equilibrium ��-Spectra of Pions the hadron resonances with the given mass cut-of and that without sigma states. In both cases, we also distinguish In U(1) global symmetry, where the scalar feld �(�) has between results at chemical equilibrium of pion production, a unitary transformation by the phase factor exp(−��), i.e., �� =0, and that at out-of-equilibrium, i.e., �� =0̸ . 4 Advances in High Energy Physics ] ] ] −2 −2 −2 )[＇？６ )[＇？６ )[＇？６ Ｓ Ｓ Ｓ ＞Ｊ ＞Ｊ 1.0e+03 ＞Ｊ 1.0e+03 1.0e+03 ４ ４ ４ ＞Ｊ ＞Ｊ ＞Ｊ ４ ４ ４ 1.0e+02 1.0e+02 1.0e+02 dN/ (2  Ｊ dN/ (2  Ｊ dN/ (2  Ｊ ) ) 1.0e+01 ) 1.0e+01 1.0e+01 ？Ｐ ？Ｐ 0 0.2 0.4 0.6 0.8 1 ？Ｐ 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

(1/． Ｊ [GeV] (1/． Ｊ４ [GeV] (1/． Ｊ４ [GeV] ４

+ ALICE, (0-5%)-PbPb, 2760 GeV + ALICE, (0-5%)-PbPb, 2760 GeV + ALICE, (0-5%)-PbPb, 2760 GeV  HRG no  =0 PLB243,181 (1990)  = 0.12 GeV HRG  =120－？６  HRG no  =120－？６ PLB243,181 (1990)  = 0.0 HRG  =0.0  HRG no  =120－？６Ｌ = 0.2＠Ｇ HRG  =120－？６ＬＢ =0.2＠Ｇ  Ｂ HRG no  =120－？６Ｌ = 0.3＠Ｇ HRG  =120－？６ＬＢ =0.4＠Ｇ  Ｂ

Figure 1: Number of positive pions per transverse momentum per rapidity is given dependent on the transverse momentum. Te symbols refer to the ALICE measurements in the most central (0−5%) Pb+Pb collisions at 2.67 TeV. Te lef-hand panel shows a comparison with (10) at vanishing �� =0and fnite �� =0.12GeV [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 5. Results

In this section, we recall another thermal distribution at out- Figure 1 shows ��-spectra of positive pions in dependence on of-equilibrium [13], inspired by the proposal of Bogolubov to the transverse momentum. Te symbols refer to the ALICE explaining the phenomenon of superfuidity as degeneracy of measurements in the most central (0−1%) Pb+Pb collisions a nonperfect Bose-Einstein gas and determine a general form at 2.67 TeV [41]. Te curves represent the present calculations. of energy spectra, a kind of ingenious application of second Te calculations from (10) at vanishing (dash-double- quantization. dotted curve) and fnite �� =0.12GeV (solid curve) are In the approach introduced in [13], the ��-spectra of depicted in the lef panel. It is obvious that fnite �� leads negatively chargedbosonsmeasuredfrom200 AGeVO+Au to excellent agreement with the experimental results. Te and S+S collisions by NA35 experiment have been success- middle and right panels show the impacts for the excluded fully reproduced. It was assumed that a cylindrical tube of volume corrections and the scalar sigma states, respectively. matter with radius � expands, longitudinally, but without As done in the lef panel, the contributions added by transverse fow ]� =�/�.Whenreplacing� in (2), for fnite �� are examined in the middle panel. We fnd that the instance, by the azimuthal angle � and the covariant form results from ideal HRG model at vanishing �� underestimate � �� � (four-momentum and velocity) and then integrating the experimental results. Finite �� and fnite �ℎ improve the over the freeze-out time ��� =�,the��-distribution could reproduction of the experimental results. be determined. It intends to investigate the importance of these sigma Ten, at fnite rapidity �=0 ̸ ,thepion��-spectrum reads states in reproducing ��-spectra of pions at LHC energies, right panel. Various sigma states are to be in(ex)cluded: ∗ 2 1 � � � (�) �=1/2, �0 (800) known as �,whichwasexcludedfrom � =(��2� ) � cosh �� 3 2014 particle data group and from our calculations as well, 2��� ����� (2�) ∗ and � (1430), �=1, �0(980) and �0(1450) and �=0, (10) 0 ∞ �0(500) widely known as �, �0(980), �0(1370), �0(1500),and �+1 �� �� ⋅ ∑ (±) (� )� [� (�)] . �0(1710). exp � 1 � cosh �=1 Te HRG calculations (8) with and without sigma states are presented in the right panels. Also, here we distinguish Te contributions likely added by heavy resonances are con- between vanishing (dash-double-dotted curve) and fnite tributing straightforwardly to (10). In Figure 1, we distinguish �� =0.12GeV(solidcurve).Asthecasewithout-of- between results at chemical equilibrium of pion production, equilibrium thermal distributions, almost the same conclu- i.e., �� =0and that at out-of-equilibrium, i.e., �� =0̸ . sion can be drawn here. Furthermore, the impacts of the Accordingly, we can examine the possibility of proposing a inclusion of scalar sigma states are analyzed. It is obvious plausible explanation for the long-standing baryon-to-meson that these states seem to enhance the out-of-equilibrium, ratios, such as proton-to-pion ratios, especially at top RHIC especially at large ��. Te excluded volume corrections are and LHC energies [40]. We propose that this would be due to taken into consideration in the way that all stable hadrons anomaly in the pion production. and resonances with masses <2GeV are assumed to equally Advances in High Energy Physics 5

have fnite volume, �ℎ. As discussed earlier, at �ℎ =0.2 especiallyattopRHICandLHCenergies[40].Inreproducing fm, the corrections seem very small. Tere is no such a the measured ��-spectra of positive pions at LHC energies, large variety to increase �ℎ and simultaneously conserve the we have decided in favor for another alternative thermal thermodynamics consistency. A systematic analysis has been approach assuming an out-of-equilibrium production of discussed in [29]; see Figure 1 and the related text. pions, (�� =0̸ ). In addition to this, we have introduced out- Te small diference when �ℎ is increased from 0.2 to of-equilibrium production of pions to the well-known HRG 0.4 fm can basically be understood from the corresponding model. In ancillary to the baryon, the strangeness chemical freeze-out temperature, with the inclusion of scalar sigma potential and the electric charge potential, we have imposed states ��ℎ ≃ 151 MeV, at �ℎ =0.2fm. But at �ℎ =0.4fm, �� =0̸ ,aswell. � ≃ 167 �ℎ becomes MeV. Tese results are confrmed when We have shortly highlighted the incompleteness of � ≃ 159 � =0.2 scalar sigma states are removed, �ℎ MeV at ℎ nonextensive Tsallis statistics, especially when confronted to fm. Here, it was not possible to increase �ℎ to 0.4 fm due to bulk matter created at relativistic energies. Te basic idea divergence in the thermodynamic quantities, at the chemical of implementing an additive resonance gas even with Tsallis � ∼1 freeze-out, for instance, �ℎ should jump to GeV to fulfll algebra, where both exponential and logarithm functions the freeze-out conditions [20–22]! Terefore we have checked are properly replaced by Tsallis algebra, seems not at all �ℎ =0.3fm. Te corresponding ��ℎ ≃ 164 MeV. It seems in modifying the extensivity. Te latter obviously contradicts the order to highlight that these results are limited to the validity intention of taking into account out-of-equilibrium particle of the approximation that all hadrons are conjectured to have production. the same radius, �ℎ =0.3fm. � � We conclude that �-spectra of positive pions produced We fnd that at vanishing �, the results from (8) with in most central (0−5%) Pb+Pb collisions at 2.67 TeV are well and without scalar sigma states are almost identical. But at � =0.12 � reproduced at � GeV.Tisisthecaseintwodiferent fnite �, the HRG calculations with and without scalar sigma thermal approaches. Te frst one is based on degeneracy of states become distinguishable. Tis can be understood due a nonperfect Bose-Einstein gas or an ingenious application to the remarkable characteristics and the great experimental of second quantization known as Bogolubov superfuidity. abundances of the sigma states, which would appear as Te second one is the well-known HRG model, in which two-pion scalar-isoscalar resonances and are regarded as �� =0̸ was ad hoc introduced. Te baryon, strangeness, excitations of the scalar condensates, i.e., playing similar roles �� and electric charge chemical potentials, etc. can also be taken as the Higgs boson does for strong interactions [42]. Teir - into consideration. Tis approach seems being successful in channel interaction is a maximally attractive one. Tis could reproducing ��-spectra of positive pions, at LHC energies, be so strong that it breaks spontaneously the chiral symmetry when PDG sigma states are taken into account. and produces a quark condensate. Instanton-induced ’t Hoof Future works shall be devoted to a systematic comparison interaction is also conjectured as a mechanism for that with other approaches and to investigating the impacts that attraction [42]. the PDG sigma states in reproducing ��-spectra of other In order to elaborate more about the reasons why we hadrons. Furthermore, ��-spectra of other bosons shall have chosen another alternative and thought of confronting be extended to understand whether they require out-of- measured ��-spectra of pions to Bogolubov superfuidity equilibrium processes, as the positive pions do. Te behavior of Bose-Einstein gas and/or a thermal approach for out-of- of the pion production at high energies, where the pro- equilibrium production of pions, i.e., fnite ��,someremarks duction of kaons, protons, and antiprotons seem requiring are now in order. Interpreting out-of-equilibrium in heavy- anomalously large contributions from the exponential term ion collisions at LHC in terms of the nonextensive Tsallis to describe the shape of their transverse momenta, could statistics [43] should be a subject of a fundamental revision. be a subject of a similar out-of-equilibrium analysis, as Tis has been discussed in great detail in [9–11]. On the one well. 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, Data Availability where both exponential and logarithm functions are merely Te availability data are available. replaced with the corresponding Tsallis counterpart function. On the other hand, the ��-spectra of positive pions from the most central (0−5%) Pb+Pb collisions at 2.67 TeV are Conflicts of Interest not reproducible by (20) in[43](notdrawninFigure1). But an excellent reproduction of these ��-spectra is achieved Te author declares that he has no conficts of interest. at �� =0.12GeV, Section 4. Te excellent agreement � apparently covers the entire range of �,lefpanelof Acknowledgments Figure 1. TeauthorwouldliketothankDavidBlaschkeandErnst- 6. Conclusions and Outlook Michael Ilgenfritz for the stimulating discussions on the nonequilibrium pion production at high energies. Te Te standard statistical thermal approach was reported as author acknowledges the fnancial support by DAAD- not being able to reproduce various baryon-to-boson ratios, Wiedereinladung no. 57378442. 6 Advances in High Energy Physics

References Particles, Fields, Gravitation and Cosmology,vol.71,ArticleID 054502, 2005. [1] N. Gyulassy and L. McLerran, “New forms of QCD matter [20] A. Tawfk, “Auniversal description for the freezeout parameters discovered at RHIC,” Nuclear Physics A,vol.750,no.1,pp.30– in heavy-ion collisions,” Nuclear Physics A,vol.764,pp.387–392, 63, 2005. 2006. [2]W.BauerandG.D.Westfall,Advances in Nuclear Dynamics 5, [21] A. Tawfk, “Condition driving chemical freeze-out,” Europhysics Springer, New York, NY, USA, 1999. Letters,vol.75,no.3,p.420,2006. [3] C. Y. Wong and G. Wilk, “Tsallis fts to p� spectra and multiple [22] A. Tawfk, “Infuence of strange quarks on the QCD phase hard scattering in pp collisions at the LHC,” Physical Review D: diagram and chemical freeze-out,” Journal of Physics G: Nuclear Particles, Fields, Gravitation and Cosmology,vol.87,ArticleID and Particle Physics, vol. 31, no. 6, pp. S1105–S1110, 2005. 114007, 2013. [23] R. Venugopalan and M. Prakash, “Termal properties of inter- [4] B.Andersson,G.Gustafson,andB.Nilsson-Almqvist,“Amodel acting hadrons,” Nuclear Physics A,vol.546,no.4,pp.718–760, for low-pT hadronic reactions with generalizations to hadron- 1992. nucleus and nucleus-nucleus collisions,” Nuclear Physics B,vol. [24] A. N. Tawfk, “Equilibrium statistical-thermal models in high- 281, no. 1-2, pp. 289–309, 1987. energy physics,” International Journal of Modern Physics A,vol. [5] C.-Y. Wong and G. Gatof, “Te transverse profle of a color fux 29,no.17,ArticleID1430021,2014. tube,” Physics Reports,vol.242,no.4-6,pp.489–494,1994. [25] M. Tanabashi, K. Hagiwara, K. Hikasa et al., “Review of particle [6] I. Kraus, J. Cleymans, H. Oeschler, and K. Redlich, “Particle physics,” Physical Review D: Particles, Fields, Gravitation and − √ production in p p collisions and predictions for s=14 TeV at Cosmology,vol.98,ArticleID030001,2018. the CERN Large Hadron Collider (LHC),” Physical Review C: [26] R. Hagedorn, “Statistical thermodynamics of strong interac- Nuclear Physics,vol.79,no.1,ArticleID014901,2009. tions at high energies,” Nuovo Cimento Supplemento,vol.3,pp. [7] T.Wibig, “Journal of Physics G: Nuclear and Particle PhysicsTe 147–186, 1965. non-extensivity parameter of a thermodynamical model of [27] R. Hagedorn, “Large-angle cross-section p+p�→ A+B and hadronic interactions at LHC energies,” Journal of Physics G: �+p�→ A+B at high energies predicted by the statistical model,” Nuclear and Particle Physics, vol. 37, no. 11, Article ID 115009, Nuovo Cimento Supplemento,vol.35,pp.216–226,1965. 2010. [28] A. Majumder and B. M¨uller, “Hadron mass spectrum from [8]S.J.Brodsky,G.deT´eramond, and M. Karliner, “Puzzles in lattice QCD,” Physical Review Letters,vol.105,ArticleID hadronic physics and novel quantum chromodynamics phe- 252002, p. 25, 2002. nomenology,” Annual Review of Nuclear and Particle Science, [29] A. Tawfk, “Constant-trace anomaly as a universal condition for vol.62,pp.1–35,2012. the chemical freeze-out,” Physical Review C, vol. 88, Article ID [9] A. Tawfk, H. Yassin, and E. R. Abo Elyazeed, “Chemical freeze- 035203, 2013. out parameters within generic nonextensive statistics,” Indian [30] M. Floris, “Hadron yields and the phase diagram of strongly Journal of Physics,vol.92,no.10,pp.1325–1335,2018. interacting matter,” Nuclear Physics A,vol.931,pp.103–112,2014. [10] A. N. Tawfk, “Lattice QCD thermodynamics and RHIC- [31] D. H. Rischke, M. I. Gorenstein, H. St¨ocker, and W. Greiner, BES particle production within generic nonextensive statistics,” “Excludedvolumeefectforthenuclearmatterequationof Physics of Particles and Nuclei Letters,vol.15,no.3,pp.199–209, state,” Zeitschrif fur¨ Physik C Particles and Fields,vol.51,no.3, 2018. pp. 485–489, 1991. [11] A. N. Tawfk, “Axiomatic nonextensive statistics at NICA ener- [32]J.Cleymans,M.I.Gorenstein,J.Stalnacke,andE.Suhonen, gies,” Te European Physical Journal A,vol.52,p.253,2016. “Te hadronisation of a quark-gluon plasma,” Zeitschrif fur¨ ⊥ [12] A. Bialas, “Tsallis p distribution from statistical clusters,” Physik C Particles and Fields,vol.58,no.2,pp.347–355,1993. Physics Letters B,vol.747,pp.190–192,2015. [33]G.D.Yen,M.I.Gorenstein,W.Greiner,andS.N.Yang, [13]M.KatajaandP.V.Ruuskanen,“Non-zerochemicalpotential “Excluded volume hadron gas model for particle number ratios and the shape of the pT-distribution of hadrons in heavy-ion in A + A collisions,” Physical Review C: Nuclear Physics,vol.56, collisions,” Physics Letters B,vol.243,no.3,pp.181–184,1990. no. 4, pp. 2210–2218, 1997. [14] N. Bogolubov, “On the theory of superfuidity,” Journal of [34] M. I. Gorenstein, M. Gazdzicki,´ and W. Greiner, “Critical line Physics,vol.XI,no.1,p.23,1947. of the deconfnement phase transitions,” Physical Review C: [15] L. Landau, “On the theory of superfuidity,” Physical Review A: Nuclear Physics,vol.72,no.2,ArticleID024909,2005. Atomic, Molecular and Optical Physics,vol.75,no.5,pp.884- [35] M. I. Gorenstein, M. Hauer, and D. O. Nikolajenko, “Particle 885, 1949. number fuctuations in nuclear collisions within an excluded [16] F. Karsch, K. Redlich, and A. Tawfk, “Hadron resonance mass volume hadron gas model,” Physical Review C: Nuclear Physics, spectrum and lattice QCD thermodynamics,” Te European vol. 76, no. 2, Article ID 024901, 2007. Physical Journal C,vol.29,no.4,pp.549–556,2003. [36] M. I. Gorenstein, M. Hauer, and O. N. Moroz, “Viscosity in the [17] F. Karsch, K. Redlich, and A. Tawfk, “Termodynamics at excluded volume hadron gas model,” Physical Review C: Nuclear non-zero Baryon number density: a comparison of lattice and Physics,vol.77,no.2,ArticleID024911,2008. Hadron resonance gas model calculations,” Physics Letters B, [37] V. Vovchenko, M. I. Gorenstein, and H. Stoecker, “Modeling vol. 571, no. 1-2, pp. 67–74, 2003. baryonic interactions with the Clausius-type equation of state,” [18] K. Redlich, F. Karsch, and A. Tawfk, “Heavy-ion collisions Te European Physical Journal A,vol.54,p.16,2018. and lattice QCD at fnite baryon density,” Journal of Physics G: [38] V.Vovchenko, M. I. Gorenstein, and H. Stoecker, “van der Waals Nuclear and Particle Physics,vol.30,no.8,p.S1271,2004. interactions in hadron resonance gas: from nuclear matter to [19] A. Tawfk, “QCD phase diagram: a comparison of lattice and lattice QCD,” Physical Review Letters,vol.118,no.18,ArticleID hadron resonance gas model calculations,” Physical Review D: 182301, 2017. Advances in High Energy Physics 7

[39] J. I. Kapusta, Finite-Temperature Field Teory,Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, UK, 1989. [40] B. Abelev, J. Adam, D. Adamov´a et al., “Production of charged pions, kaons and protons at large transverse momenta in pp and Pb–Pb collisions at √s�� =2.76 TeV,” Physics Letters B,vol.736, pp.196–207,2014. [41] B. Abelev, J. Adam, D. Adamov´a et al., “Centrality dependence of �,K,andpproductioninPb-Pbcollisionsat√s�� =2.76 TeV,” Physical Review C,vol.88,ArticleID044910,2013. [42] V. V. Flambaum and E. V. Shuryak, “Dependence of hadronic properties on quark masses and constraints on their cosmo- logical variation,” Physical Review D, vol. 67, Ar ticle ID 0835 07, 2003. [43] M. D. Azmi and J. Cleymans, “Transverse momentum distri- butions in proton-proton collisions at LHC energies and Tsallis thermodynamics,” Journal of Physics G: Nuclear and Particle Physics,vol.41,no.6,ArticleID065001,2014. Journal of International Journal of Advances in The Scientifc Applied Bionics Chemistry Engineering World Journal Geophysics and Biomechanics Hindawi Hindawi Hindawi Publishing Corporation Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 http://www.hindawi.comwww.hindawi.com Volume 20182013 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

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