Hadron resonance gas with repulsive mean field interaction: Thermodynamics and transport properties
aa r X i v : . [ h e p - ph ] O c t Hadron resonance gas with repulsive mean field interaction:Thermodynamics and transport properties
Guruprakash Kadam ∗ and Hiranmaya Mishra † Department of Physics, Shivaji University, Kolhapur, Maharashtra-416004, India Theory Division, Physical Research Laboratory,Navarangpura, Ahmedabad - 380 009, India (Dated: October 9, 2019)We discuss the interacting hadron resonance gas model to describe the thermo-dynamics of hadronic matter. While the attractive interaction between hadrons istaken care of by including all the resonances with zero width, the repulsive interac-tions between them are described by a density-dependent mean field potential. Thebulk thermodynamic quantities are confronted with the lattice quantum chromody-namics simulation results at zero as well as at finite baryon chemical potential. Wefurther estimate the shear and bulk viscosity coefficients of hot and dense hadronicmatter within the ambit of this interacting hadron resonance gas model.
PACS numbers: 12.38.Mh, 12.39.-x, 11.30.Rd, 11.30.ErKeywords:
I. INTRODUCTION
Understanding the phase diagram of strongly interacting matter is one of the importantand challenging topics of current research in strong interaction physics- both theoreticallyand experimentally. The theoretical framework describing a nuclear matter at a fundamentallevel is quantum chromodynamics (QCD). At low temperature ( T ) and low baryon chemicalpotential ( µ ) the fundamental degrees of freedom of QCD are colorless hadrons while athigh temperature and high baryon density the fundamental degrees of freedom are coloredquarks and gluons. Lattice quantum chromodynamics (LQCD) simulations at zero chemicalpotential and finite temperature suggest a crossover transition for QCD matter from ahadronic phase to a quark-gluon-plasma (QGP) phase[1–6]. At zero chemical potential, thechiral crossover temperature is estimated to be T c ∼
156 MeV[7]. While LQCD simulationsat vanishing chemical potential has been quite successful, LQCD simulations at finite µ havebeen rather challenging particularly at high µ leading to large uncertainties in estimatingthe transition line in the T − µ plane of QCD phase diagram [8]. At small µ , however, precisecomputation of the transition line has been carried out recently [9–11].The low energy effective models of QCD, viz.,the Nambu-Jona-Lasinio model[12, 13], thequarak-meson coupling model[14] etc., provide a reasonable theoretical framework to explorestrongly interacting matter below the QCD transition temperature, T c . These models arebased on certain symmetries of QCD and they are tremendously successful in describingmany features of the QCD phase diagram at zero as well at finite baryon density. Apart from ∗ Electronic address: [email protected] † Electronic address: [email protected] these symmetry-based models another model that has also been tremendously successful indescribing the low temperature hadronic phase of QCD is the hadron resonance gas model(HRG). The hadron resonance gas model is the statistical model of QCD describing thelow temperature hadronic phase of quantum chromodynamics. This model is based on theso-called Dashen-Ma-Bernstein theorem which allows us to compute the partition functionof the interacting system of hadrons in terms of scattering matrix[15]. Using this S-matrixformulation of statistical mechanics it can be shown that if the dynamics of thermodynamicsystem of hadrons is dominated by narrow-resonance formation then the resulting systemessentially behaves like a noninteracting system of hadrons and resonances[16–18]. This idealHRG model, despite its success in describing hadron multiplicities in heavy-ion collisions[19–27], fails to account for the short-range repulsive interactions between hadrons. It hasbeen shown that the repulsive interactions modeled via excluded volume can have significanteffect on thermodynamic observables, especially higher order fluctuations[28–31] as well asin the context of statistical hadronization[32]. One possible way to include these repulsiveinteractions is through van der Waals excluded volume procedure[33, 34]. Another approachis to treat the repulsive interactions in mean field way[35, 36]. Recently, the relativistic meanfield approach has been used to calculate the fluctuations of conserved charges [37]. Thiswork discussed the repulsive mean field interactions which are present only at finite baryondensity. They showed the deviations of higher order fluctuations estimated using the idealHRG can be accounted by means of repulsive interactions treated in mean field way. Thefailure of ideal HRG model to explain the thermodynamical observable can be attributedto the fact that at high temperature and density the relativistic virial expansion up to thesecond-order virial coefficient cannot be a reasonable approximation and the validity of theHRG model needs to be checked against its agreement with LQCD results.In the past few decades relativistic and ultrarelativistic heavy-ion collision experimentshave provided a unique opportunity to study the phase diagram of QCD. The relativis-tic hydrodynamics has been tremendously successful in simulating the evolution of mattercreated in HIC experiments [38–46]. In the relativistic hydrodynamic simulations the coef-ficients of shear and bulk viscosities influence various observables, viz.,the flow coefficients,the transverse momentum distribution of produced particles. In fact, it has been foundthat a finite but very small shear viscosity-to-entropy ratio ( η/s ) should be included in thehydrodynamic description to explain elliptic flow data[47, 48]. Further, η/s obtained us-ing AdS/CFT correspondence [49] has put the lower bound on its value equal to π calledthe Kovtun-Son-Starinets (KSS) bound. This interesting finding has motivated many the-oretical investigations to understand and rigorously derive this ratio from a microscopictheory [50, 51, 53–65]. The bulk viscosity coefficient ( ζ ) has also been realized to be im-portant to be included the dissipative hydrodynamics. During the expansion of the fireball,when the temperature approaches the critical temperature, ζ can be large and give rise todifferent interesting phenomena like cavitation when the pressure vanishes and the hydrody-namic description breaks down [66, 67]. The effect of bulk viscosity on the particle spectraand flow coefficients have been investigated [68–70] while the interplay of shear and bulkviscosity coefficients have been studied in Refs. [71–73]. The coefficient of bulk viscosity hasbeen estimated for both the hadronic and the partonic systems [74–108].Hydrodynamic simulation of the matter created in HIC collision requires informationregarding equation of state (EoS) as well as the transport coefficients. In this work weanalyze the QCD equation of state of hadronic matter at finite baryon chemical potential. Weemploy the hadron resonance gas model to estimate all the thermodynamic quantities. Whilethe attractive interactions between hadrons are accounted for by including all the resonancestates up to 2 .
25 GeV, the short range repulsive interaction among hadrons are treated inthe mean field approach. We call this model relativistic mean field hadron resonance gasmodel (RMFHRG). The RMFHRG differs from the Walecka type mean field models in thesense that in former the repulsive mean field interactions are present even at zero baryondensity unlike the later case. We will also estimate the shear and bulk viscosity coefficientof hadronic matter within the ambit of RMFHRG.We organize the paper as follows. In Sec. II we compute the pressure and other bulkthermodynamic quantities for the interacting hadron resonance gas with a repulsive meanfield interaction. In Sec III, we discuss the results for the thermodynamics and confrontthem with the lattice simulation results both at zero and finite chemical potential. We thenestimate the viscosity coefficients for hot and dense hadronic matter within the ambit of theHRG model with mean field interactions. Finally, in Sec IV, we summarize the findings ofthe present investigation and give a possible outlook.
II. HADRON RESONANCE GAS MODEL WITH A REPULSIVE MEAN FIELDPOTENTIAL
Thermodynamic properties of hadron resonance gas model can be deduced from the grandcanonical partition function defined as Z ( V, T, µ ) = Z dm [ ρ b ( m ) ln Z b ( m, V, T, µ ) + ρ f ( m ) ln Z f ( m, V, T, µ )] (1)where ρ b and ρ f are the mass spectrum of the bosons and fermions respectively. We assumethat the hadron mass spectrum is given by ρ ( m ) = Λ X a g a δ ( m − m a ) θ (Λ − m ) (2)where g a is the degeneracy and m a is the mass of the a -th hadronic species. This discretemass spectrum consists of all the experimentally known hadrons with cutoff Λ. One can setdifferent cutoff values for baryons and mesons.To include the effect of a repulsive interaction among hadrons, we use a repulsive mean fieldapproach as was used in Refs. [35, 36] and more recently in the case of baryons in Ref.[37].In this approach, it is assumed that the repulsive interactions lead to a shift in the singleparticle energy and is given by ε a = p p + m a + U ( n ) = E a + U ( n ) (3)where E a = p p + m a and n is the total hadron number density. The potential energy U represents repulsive interaction between hadrons, and it is taken to be a function oftotal hadron density n . For any arbitrary interhadron potential V ( r ), the potential energyis U ( n ) = Kn . Here, the phenomenological parameter K is given by the integral of thepotential V ( r ) over the spatial volume [35, 36].In this work we assign different repulsive interaction parameter for baryons and mesons.We denote the mean field parameter for baryons ( B ) and anti-baryons ( ¯ B ) by K B , while formesons we denote it by K M . Thus, for baryons (antibaryons) U ( n B { ¯ B } ) = K B n B { ¯ B } (4)and for mesons U ( n M ) = K M n M (5)The total hadron number density is n ( T, µ ) = X a n a = n B + n ¯ B + n M (6)where n a is the number density of a -th hadronic species. Note that n B , n ¯ B and n M aretotal baryon, antibaryon and meson number densities respectively. Explicitly, for baryons, n B = X a ∈ B Z d Γ a e ( Ea − µa eff T ) + 1 (7)where the sum is over all the baryons. Here, d Γ a ≡ g a d p (2 π ) , and, µ a eff = q a µ − U ( n B ) is abaryon effective chemical potential, with q a being the baryonic charge of a -th baryon and µ the baryon chemical potential. Similarly, for antibaryons n ¯ B = X a ∈ ¯ B Z d Γ a e ( Ea − ¯ µa eff T ) + 1 (8)where ¯ µ a eff = (¯ q a µ − U ( n ¯ B )) is an antibaryon effective chemical potential with ¯ q a = − q a ,which is the corresponding baryonic charge. For mesons, n M = X a ∈ M Z d Γ a e ( Ea − KM nM ) T − µ = 0 for mesons since the baryon chargeis zero for them. In the Boltzmann approximation momentum integration can be readilyperformed and one can obtain much simpler expressions for the number density. For baryonswe get n B = X a ∈ B g a π m a T K (cid:18) m a T (cid:19) e µa eff T (10) n ¯ B = X a ∈ ¯ B g a π m a T K (cid:18) m a T (cid:19) e ¯ µa eff T (11)where K n ( z ) is the modified Bessel function of order n . For mesons we get n M = X a ∈ M g a π m a T K (cid:18) m a T (cid:19) e − KM nMT (12)Equations (10 1112) are self-consistent equations for number density which can be solvednumerically.The total baryon (antibaryon) energy density is ǫ B { ¯ B } = X a ∈ B { ¯ B } Z d Γ a ε a e [ Ea − µa eff { ¯ µ eff } ] T + 1 + φ B { ¯ B } ( n B { ¯ B } ) (13)and for mesons ǫ M = X a ∈ M Z d Γ a ε a e εaT − φ M ( n M ) (14)where φ ( n ) represents the correction to the energy density in order to avoid double countingthe potential. It can be determined using the condition that ε a = ∂ǫ∂n a . Taking the derivativeof baryon energy density and using Eq. (4) we get ∂φ B { ¯ B } ∂n B { ¯ B } = − K B n B { ¯ B } (15)and hence φ B ( n B { ¯ B } ) = − K B n B { ¯ B } (16)Similarly for mesons one can obtain φ M ( n M ) = − K M n M (17)Pressure of the gas can now be readily obtained. For baryons P B { ¯ B } ( T, µ ) = T X a ∈ B { ¯ B } Z d Γ a ln (cid:20) e − ( Ea − µa eff { ¯ µa eff } T ) (cid:21) − φ B { ¯ B } ( n B { ¯ B } ) (18)and for mesons P M ( T ) = T X a ∈ M Z d Γ a ln (cid:20) e − ( εaT ) (cid:21) − φ M ( n M ) (19)Finally, entropy density can be obtained from the fundamental thermodynamic relation s = ( ǫ + P − µn ) /T . It is worth noting that the effective interaction model we are consideringis different from the relativistic Lagrangian model. In the latter case the repulsive meanfields are present only at nonzero baryon density, while in the former case the repulsiveinteractions are present even at zero baryon density. III. RESULTS AND DISCUSSION
In the hadron resonance gas model it is customary to include all the hadrons and res-onances up to a certain cutoff Λ. We include all the mesons and baryons up to Λ = 2 . K is the spatially integratedvalue of the interhadron repulsive potential. In Ref.[35] the potential was taken to be thesame for all hadrons, i.e., for all the baryons and the mesons. In the present work we havetaken this parameter different for mesons and baryons. For baryons we have taken it tobe the same for all the baryons and the value is taken as in Ref. [37], namely, K B =450MeV f m for all baryons. Although different lattice calculations as well as chiral effectivetheories indicate that, the strength can be different for nucleon-nucleon, hyperon-hyperonor nucleon-hyperon interactions, there is not enough information about hadrons to have amore realistic and sophisticated mean field model. For the mesons, we have taken a muchsmaller value for the repulsion parameter K M =50 MeV f m in the present study. The moti-vation in choosing these two phenomenological parameters has been that the lattice results (MeV)(a) P / T T RMFHRGHRG (MeV)(b) ( ǫ − P ) / T T HRGRMFHRG
FIG. 1: Scaled pressure (left panel) and the interaction measure (right panel) in RMFHRG andideal HRG for µ = 0. The lattice data is taken from Ref.[8]. are reasonably reproduced regarding thermodynamics and then use them to estimate theviscosity parameters also at finite density.Figure 1 shows the scaled pressure and the interaction measure estimated within theambit of RMFHRG (blue solid curve) at vanishing baryon chemical potential. The dashedcurve corresponds to ideal HRG results while the circles with error bars correspond tolattice QCD simulation results[8]. We note that the effect of including the repulsive meanfield interaction is to suppress the thermodynamical quantities compared to their ideal HRGestimation counterpart (dashed magenta curve). While the HRG pressure [Fig.2(a)] startsto deviate from the lattice results at T ∼
160 MeV, the RMFHRG estimation agrees withthe lattice results all the way up to 190 MeV. It is not reasonable to push the HRG modelresults above the QCD transition temperature ( T c ) which LQCD estimates to lie in therange 155 −
160 MeV[7]. The reason for this is twofold. First, the HRG approximationof the hadronic matter might break down at high density near and above T c . Second, thehadrons do not exist above T c . But a recent study[29] shows that the hadrons do not meltquickly as one would expect on the basis of ideal HRG model. In this study the authors haveanalyzed the possible improvement of the ideal hadron resonance gas model by includingthe repulsive interactions between baryons. If one includes the attractive and the repulsiveinteractions between the baryons through van der Waals parameters, while keeping themeson gas ideal, the pressure of the hadron gas agrees with the LQCD data all the wayabove T c . We may similarly conclude that the inclusion of repulsive mean fields might pushthe validity of the HRG model well above T c . Nonetheless, we do not have any other strongreason to believe this except for the apparent agreement with the LQCD results.Unlike pressure the interaction measure is somewhat below LQCD results above T = 150MeV. It is an established fact that the socalled heavy Hagedorn states which are missingin our model contribute significantly to the energy density. The rapid rise of the energydensity can be explained by extending ideal HRG model by including continuum Hagedornstates alongwith the discrete states above the cutoff Λ in the density of states[79]. (MeV)(a) P / T T µ = MeV µ = MeV (MeV)(b) ( ǫ − P ) / T T µ = MeV µ = MeV
FIG. 2: Scaled pressure (left panel) and the interaction measure (right panel) in the RMFHRGmodel at finite baryon chemical potential. The lattice data is taken from Ref.[8].
Figure 2 shows the scaled pressure and interaction measure at finite baryon chemicalpotentials estimated within the ambit of RMFHRG. We note that the RMFHRG is in rea-sonable agreement with LQCD results even at finite baryon density. Further, the interactionmeasure is in better agreement with the LQCD results at finite density than in the µ = 0case. However, while making this observation, we have to keep in mind that the lattice dataof Ref.[8] is estimated at order µ . Figure 3 shows the adiabatic speed of sound at finitebaryon density. The RMFHRG estimations of C s are within the errorbars of LQCD results.Further, the C s has a minimum at T = 155 MeV for µ = 0 and at T = 140 MeV for µ = 300MeV which is in very close agreement with the LQCD results.The coefficients of shear and bulk viscosities can be extracted from the relativistic Boltz-mann equation. These have been derived in Refs. [50, 51] in the absence of any mean fields.While various authors have used different types of mean fields to include medium effects aswell as interactions, and have derived the transport coefficients, a rigorous, thermodynami-cally consistent derivation for the expressions for different transport coefficients was derivedin Ref.[52], both in the presence of a scalar and vector mean field. The scalar mean fieldaffect the mass while the repulsive vector mean field affects the chemical potential. Thepotential considered here does not affect the masses of the hadrons and is like a repulsivevector field, its effect is manifested in the effective chemical potential. In the relaxation timeapproximation of the Boltzmann equation, the shear ( η ) and bulk viscosity ( ζ ) coefficientsare given by [50–52] η = 115 T X a Z d p (2 π ) p E a ( τ a f a + ¯ τ a ¯ f a ) (20) (MeV) C s T µ = MeV µ = MeV
FIG. 3: Speed of sound in RMFHRG model at finite baryon density. The lattice data is takenfrom Ref.[8]. ζ = 1 T X a Z d p (2 π ) (cid:26) τ a f a (cid:20) E a C n B + (cid:18) ∂P∂n B (cid:19) ǫ − p E a (cid:21) + ¯ τ a ¯ f a (cid:20) E a C n B − (cid:18) ∂P∂n B (cid:19) ǫ a − p E a (cid:21) (cid:27) (21)where f is the equilibrium distribution function with an effective chemical potential in-cluding the mean field and C n B is the speed of sound at constant baryon number density.Further, in Eqs. (21) and (20), τ a is the relaxation time for a -th hadronic particle species,while the barred quantities corresponds to that of antiparticles. In this work we use thethermally averaged relaxation time which for a given species a is given by τ − a = X b n b h σ ab v ab i . (22)In the above, the sum is over all particles ( b ) other than the particle a with which thescattering takes place; σ ab is the total scattering cross section for the process a ( p a ) + b ( p b ) → c ( p c ) + d ( p d ) and v ab is the relative velocity given by v ab = p ( p a · p b ) − m a m b E a E b (23)Further, n b is the number density for particle species b given, with g b as the correspondingdegeneracy factor, as n b = g b (2 π ) Z d p f b ( p ) ≃ g b T π ( βm ) K ( βm ) exp( βµ beff ) (24)where the last step is written down in the Boltzmann approximation and µ beff = µ − K B n B for baryons, µ beff = ¯ µ − K B n ¯ B for antibaryons, and µ beff = − K M n M for mesons.Finally, the thermal average cross section h σ ab v ab i is given as h σ ab v ab i = R d p a d p b σ ab v ab f a ( p a ) f b ( p b ) R d p a d p b f a ( p a ) f b ( p b ) . (25)The only unknown quantity in Eq.(25) is the total cross section. We estimate it as follows.In Born approximation, the scattering amplitude f ( θ, φ ) for a particle with mass m thatencounters a scattering potential V ( r ) is given by [110] f ( θ, φ ) = − m π Z d r V ( r ) = − mK (2 π ) (26)and thus the cross section is given by σ = 4 π (cid:18) mK π (cid:19) (27)Then the thermal averaged cross section can be written as[111, 112] h σ ab v ab i = σ m a m b K ( βm a ) K ( βm b ) Z ∞ ( m a + m b ) dS [ S − ( m a − m b ) ] √ S [ S − ( m a + m b ) ] K ( β √ S )(28)where √ S is the centre-of-mass energy. Clearly, we have suppressed the baryon/meson indexin the expression for the cross section for the parameter K in Eq.(27). It may be relevanthere to mention that, while the cross section is independent of temperature and chemicalpotential, the thermal averaged cross section h σv i , in general, depends upon temperatureand chemical potential. However, in the Boltzmann approximation h σv i is independent of µ . After evaluating the thermal averaged relaxation time for each species, we estimate theviscosity coefficients using Eqs. (20) and (21).Figure 4 shows the ratio of shear viscosity to entropy density as a function of tempera-ture. We have compared the ratio η/s estimated within the ambit of RMFHRG with variousother model calculations [49, 59, 77, 96, 97]. The red dashed curve corresponds to Chapman-Enscog method with constant cross sections[59]. The dashed green curve corresponds to therelativistic Boltzmann equation in the relaxation time approximation. The thermodynamicquantities in this model have been estimated using the scaled hadron masses and coupling(SHMC) model[97]. the brown dashed curve corresponds to estimations made using rela-tivistic Boltzmann equation in RTA. The thermodynamic quantities are estimated withinthe excluded volume hadron resonance gas model (EHRG)[96]. The dot-dashed orchid curvecorresponds to the η/s of meson gas estimated using chiral perturbation theory[77]. Whilethe ratio η/s in our model is relatively large at low temperature as compared to other modelsit rapidly falls and approaches to the Kovtun-Son-Starinets (KSS) bound at T ∼
170 MeV.0 (MeV)(a) η / s T KSSRMFHRGDenicol et al.Khvorostukhin et al.Fernand´ez-Fraile et al.EHRG ( r h = 0 . fm) (MeV)(b) η / s T µ = MeV µ = MeV
FIG. 4: The left panel shows shear viscosity to entropy density ratio estimated within RMFHRGand compared with other model estimations. These figures correspond to µ = 0. The right panelshows η/s for two different baryon chemical potentials. Figure 5 shows the ratio of bulk viscosity to entropy density as a function of temperature.In Fig.5(a) the blue solid curve corresponds to the RMFHRG compared with that of theEHRG model (dashed magenta curve) [96] and the SHMC model[97]. Note that the ratio ζ /s is smaller when the repulsive interactions are treated in a mean field way. Figure5(a)shows the ratio ζ /s at finite baryon chemical potential. At low temperature the ratio islarger at finite µ ; it drops below that of µ = 0 case at high temperature. This observationmay be attributed to the fact that the entropy density rises much faster than that of ζ itselfat finite baryon density as compared to that of zero baryon density case.In the context of heavy nucleon-nucleon (NN) collision experiments viscosity coefficientscan be estimated along freezeout curve by finding the beam energy ( √ S NN ) dependence ofthe temperature and chemical potential. This is extracted from a statistical thermal modeldescription of the particle yield at various √ S NN [113–115]. We use the parametrization ofthe freezeout curve T ( µ B ) given in Ref.[114] as T ( p S NN ) = c + ( T + T p S NN ) + c − (cid:18) T lim0 + T √ S NN (cid:19) (29) µ ( p S NN ) = a b √ S NN (30)where, T = − . T = 30 . T = − . T lim0 = 161 . a = 1481 . b = 0 .
365 GeV − . The functions c + and c − smoothly connectthe different behaviors of centre-of-mass energies.Figure(6) shows viscosity coefficients, η/s and ζ /s along the freeze-out curve. It can benoted that the fluidity measure rapidly falls at low √ S and then it remains almost constant athigher √ S values. This indicates that the matter produced in heavy-ion collision experimentswith wide range of collision energies can exhibit substantial elliptic flow.1 (MeV)(a) ζ / s T RMFHRGKhvorostukhin et al.EHRG (MeV)(b) ζ / s T µ = MeV µ = MeV
FIG. 5: The left panel shows bulk viscosity to entropy density ratio estimated within RMFHRGand compared with other model estimations. These results are for µ =0. The right panel shows ζ/s at two different baryon chemical potentials. (GeV)(a) η / s √ S NN (GeV)(b) ζ / s √ S NN FIG. 6: Viscosity coefficients along the freeze-out curve. The freeze-out parametrization is takenfrom Ref.[114].
IV. SUMMARY
In this paper we confronted the RMFHRG model with LQCD at zero as well as finitedensity. The repulsive interaction between the hadrons is treated using the mean fieldapproach. The thermodynamic quantities estimated within RMFHRG are found to be in2reasonable agreement with LQCD at zero as well as finite chemical potential. The agreementof interaction measure ǫ − P/T estimated within RMFHRG is rather poor above T = 145MeV. In fact the interaction measure rises very rapidly near T c ∼ η/s estimated within the RMFHRG is large at low temperature ascompared to other calculations. This behavior is due to the smaller cross section of mesonsin our model. But η/s estimated in our calculation rapidly drops at high temperature andapproaches the KSS bound at T ∼
170 MeV. We further found that η/s at finite chemical issmaller in magnitude as compared to that of zero chemical potential but the overall behavioras a function of temperature do not change. We also found the reasonable agreement of theratio ζ /s with previous results. Finally, we have estimated viscosity coefficients along thefreeze-out line. We found that both the ratios, η/s and ζ /s , attain constant values at high √ S values. This indicates that the matter produced in heavy-ion collision experiments witha wide range of collision energies can exhibit substantial elliptic flow. V. ACKNOWLEDGEMENT
GK is financially supported by the DST-INSPIRE faculty award under Grant No.DST/INSPIRE/04/2017/002293. [1] Y. Aoki, Z. Fodor, S. D. Katz and K. K. Szabo, Phys. Lett. B , 46 (2006)doi:10.1016/j.physletb.2006.10.021 [hep-lat/0609068].[2] S. Borsanyi et al. [Wuppertal-Budapest Collaboration], JHEP , 073 (2010)doi:10.1007/JHEP09(2010)073 [arXiv:1005.3508 [hep-lat]].[3] S. Borsanyi et al. , Phys. Rev. D , no. 1, 014505 (2015) doi:10.1103/PhysRevD.92.014505[arXiv:1504.03676 [hep-lat]].[4] P. Petreczky, AIP Conf. Proc. , 103 (2013). doi:10.1063/1.4795947[5] H. T. Ding, F. Karsch and S. Mukherjee, Int. J. Mod. Phys. E , no. 10, 1530007 (2015)doi:10.1142/S0218301315300076 [arXiv:1504.05274 [hep-lat]].[6] B. Friman, C. Hohne, J. Knoll, S. Leupold, J. Randrup, R. Rapp and P. Senger, Lect. NotesPhys. , pp.1 (2011). doi:10.1007/978-3-642-13293-3[7] A. Bazavov et al. [HotQCD Collaboration], Phys. Lett. B (2019) 15doi:10.1016/j.physletb.2019.05.013 [arXiv:1812.08235 [hep-lat]].[8] S. Borsanyi, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti and K. K. Szabo, JHEP , 053 (2012) doi:10.1007/JHEP08(2012)053 [arXiv:1204.6710 [hep-lat]].[9] A. Bazavov et al. , Phys. Rev. D , no. 5, 054504 (2017) doi:10.1103/PhysRevD.95.054504[arXiv:1701.04325 [hep-lat]].[10] S. Datta, R. V. Gavai and S. Gupta, Phys. Rev. D , no. 5, 054512 (2017)doi:10.1103/PhysRevD.95.054512 [arXiv:1612.06673 [hep-lat]].[11] S. Datta, R. V. Gavai and S. Gupta, PoS LATTICE , 202 (2014).doi:10.22323/1.187.0202 [12] S. P. Klevansky, Rev. Mod. Phys. , 649 (1992). doi:10.1103/RevModPhys.64.649[13] T. Hatsuda and T. Kunihiro, Phys. Rept. , 221 (1994) doi:10.1016/0370-1573(94)90022-1[hep-ph/9401310].[14] B. J. Schaefer, J. M. Pawlowski and J. Wambach, Phys. Rev. D , 074023 (2007)doi:10.1103/PhysRevD.76.074023 [arXiv:0704.3234 [hep-ph]].[15] R. Dashen, S. K. Ma and H. J. Bernstein, Phys. Rev. , 345 (1969).[16] R. F. Dashen and R. Rajaraman, Phys. Rev. D , 694 (1974). doi:10.1103/PhysRevD.10.694[17] G. M. Welke, R. Venugopalan and M. Prakash, Phys. Lett. B , no. 2, 137 (1990).doi:10.1016/0370-2693(90)90123-N[18] R. Venugopalan and M. Prakash, Nucl. Phys. A , 718 (1992). doi:10.1016/0375-9474(92)90005-5[19] P. Braun-Munzinger, J. Stachel, J. P. Wessels and N. Xu, Phys. Lett. B , 1 (1996).[20] G. D. Yen and M. I. Gorenstein, Phys. Rev. C , 2788 (1999).[21] F. Becattini, J. Cleymans, A. Keranen, E. Suhonen and K. Redlich, Phys. Rev. C , 024901(2001).[22] J. Cleymans and H. Satz, Z. Phys. C , 135 (1993) doi:10.1007/BF01555746 [hep-ph/9207204].[23] P. Braun-Munzinger, D. Magestro, K. Redlich and J. Stachel, Phys. Lett. B , 41 (2001)doi:10.1016/S0370-2693(01)01069-3 [hep-ph/0105229].[24] J. Rafelski and J. Letessier, Nucl. Phys. A , 98 (2003) doi:10.1016/S0375-9474(02)01418-5[nucl-th/0209084].[25] A. Andronic, P. Braun-Munzinger and J. Stachel, Nucl. Phys. A , 167 (2006)doi:10.1016/j.nuclphysa.2006.03.012 [nucl-th/0511071].[26] S. Chatterjee, R. M. Godbole and S. Gupta, Phys. Lett. B , 554 (2013)doi:10.1016/j.physletb.2013.11.008 [arXiv:1306.2006 [nucl-th]].[27] S. Chatterjee, S. Das, L. Kumar, D. Mishra, B. Mohanty, R. Sahoo and N. Sharma, Adv.High Energy Phys. , 349013 (2015). doi:10.1155/2015/349013[28] M. Albright, J. Kapusta and C. Young, Phys. Rev. C , no. 4, 044904 (2015)doi:10.1103/PhysRevC.92.044904 [arXiv:1506.03408 [nucl-th]].[29] V. Vovchenko, M. I. Gorenstein and H. Stoecker, Phys. Rev. Lett. , no. 18, 182301 (2017)doi:10.1103/PhysRevLett.118.182301 [arXiv:1609.03975 [hep-ph]].[30] V. Vovchenko, A. Motornenko, P. Alba, M. I. Gorenstein, L. M. Satarov and H. Stoecker,Phys. Rev. C , no. 4, 045202 (2017) doi:10.1103/PhysRevC.96.045202 [arXiv:1707.09215[nucl-th]].[31] P. Alba, W. M. Alberico, A. Nada, M. Panero and H. Stcker, Phys. Rev. D , no. 9, 094511(2017) doi:10.1103/PhysRevD.95.094511 [arXiv:1611.05872 [hep-lat]].[32] P. Braun-Munzinger, I. Heppe and J. Stachel, Phys. Lett. B , 15 (1999)doi:10.1016/S0370-2693(99)01076-X [nucl-th/9903010].[33] D. H. Rischke, J. Schaffner, M. I. Gorenstein, A. Schaefer, H. Stoecker and W. Greiner, Z.Phys. C , 325 (1992). doi:10.1007/BF01555532[34] C. P. Singh, B. K. Patra and K. K. Singh, Phys. Lett. B , 680 (1996). doi:10.1016/0370-2693(96)01117-3[35] J. I. Kapusta and K. A. Olive, Nucl. Phys. A , 478 (1983). doi:10.1016/0375-9474(83)90241-5[36] K. A. Olive, Nucl. Phys. B , 483 (1981). doi:10.1016/0550-3213(81)90444-2[37] P. Huovinen and P. Petreczky, Phys. Lett. B , 125 (2018) doi:10.1016/j.physletb.2017.12.001 [arXiv:1708.00879 [hep-ph]].[38] C. Gale, S. Jeon and B. Schenke, Int. J. Mod. Phys. A , 1340011 (2013).[39] B. Schenke, J. Phys. G , 124009 (2011).[40] C. Shen and U. Heinz, Phys. Rev. C , 054902 (2012).[41] P. F. Kolb and U. W. Heinz, In *Hwa, R.C. (ed.) et al.: Quark gluon plasma* 634-714[nucl-th/0305084].[42] D. Teaney, J. Lauret and E. V. Shuryak, Phys. Rev. Lett. , 4783 (2001).[43] L. Del Zanna et al. , Eur. Phys. J. C , 2524 (2013).[44] I. Karpenko, P. Huovinen and M. Bleicher, Comput. Phys. Commun. , 3016 (2014).[45] H. Holopainen, H. Niemi and K. J. Eskola, J. Phys. G , 124164 (2011).[46] A. Jaiswal, B. Friman and K. Redlich, Phys. Lett. B , 548 (2015).[47] M. Gyulassy and L. McLerran, Nucl. Phys. A , 30 (2005).[48] L. P. Csernai, J. I. Kapusta and L. D. McLerran, Phys. Rev. Lett. , 152303 (2006).[49] P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. , 111601 (2005).[50] S. Gavin, Nucl. Phys. A , 826 (1985).[51] A. Hosoya and K. Kajantie, Nucl. Phys. B , 666 (1985).[52] M. Albright, J. Kapusta, Phys. Rev. C , 014903 (2016).[53] M. Prakash, M. Prakash, R. Venugopalan and G. Welke, Phys. Rept. , 321 (1993).[54] A. Dobado and F. J. Llanes-Estrada, Phys. Rev. D , 116004 (2004).[55] J. W. Chen, Y. H. Li, Y. F. Liu and E. Nakano, Phys. Rev. D , 114011 (2007).[56] K. Itakura, O. Morimatsu and H. Otomo, Phys. Rev. D , 014014 (2008).[57] A. Dobado, F. J. Llanes-Estrada and J. M. Torres-Rincon, Phys. Rev. D , 114015 (2009).[58] N. Demir and S. A. Bass, Phys. Rev. Lett. , 172302 (2009).[59] G. S. Denicol, C. Gale, S. Jeon and J. Noronha, Phys. Rev. C , no. 6, 064901 (2013)doi:10.1103/PhysRevC.88.064901 [arXiv:1308.1923 [nucl-th]].[60] A. Puglisi, S. Plumari and V. Greco, Phys. Lett. B , 326 (2015).[61] L. Thakur, P. K. Srivastava, G. P. Kadam, M. George and H. Mishra, Phys. Rev. D ,096009 (2017).[62] G. Kadam, S. Pawar and H. Mishra, J. Phys. G , no. 1, 015102 (2019) doi:10.1088/1361-6471/aaeba2 [arXiv:1807.05370 [nucl-th]].[63] G. Kadam and S. Pawar, Adv. High Energy Phys. , 6795041 (2019)doi:10.1155/2019/6795041 [arXiv:1802.01942 [hep-ph]].[64] M. I. Gorenstein, M. Hauer and O. N. Moroz, Phys. Rev. C , 024911 (2008).[65] J. Noronha-Hostler, J. Noronha and C. Greiner, Phys. Rev. C , 024913 (2012).[66] K. Rajagopal and N. Tripuraneni, JHEP , 018 (2010).[67] J. R. Bhatt, H. Mishra and V. Sreekanth, Phys. Lett. B , 486 (2011); Nucl. Phys. A ,181 (2012); Phys. Rev. C , 054906 (2009).[68] A. Monnai and T. Hirano, Phys. Rev. C , 054906 (2009).[69] G. S. Denicol, T. Kodama, T. Koide and P. Mota, Phys. Rev. C , 064901 (2009).[70] K. Dusling and T. Schfer, Phys. Rev. C , 044909 (2012).[71] H. Song and U. W. Heinz, Phys. Rev. C , 024905 (2010).[72] J. Noronha-Hostler, G. S. Denicol, J. Noronha, R. P. G. Andrade and F. Grassi, Phys. Rev.C , no. 4, 044916 (2013).[73] J. Noronha-Hostler, J. Noronha and F. Grassi, Phys. Rev. C , no. 3, 034907 (2014).[74] A. Dobado and S. N. Santalla, Phys. Rev. D , 096011 (2002).[75] D. Davesne, Phys. Rev. C , 3069 (1996). [76] D. Kharzeev and K. Tuchin, JHEP , 093 (2008).[77] D. Fernandez-Fraile and A. Gomez Nicola, Eur. Phys. J. C , 37 (2009)[78] J. W. Chen and J. Wang, Phys. Rev. C , 044913 (2009).[79] J. Noronha-Hostler, J. Noronha and C. Greiner, Phys. Rev. Lett. , 172302 (2009) ; Phys.Rev. C (2012) 024913[80] C. Sasaki and K. Redlich, Nucl. Phys. A , 62 (2010); Phys. Rev. C (2009) 055207.[81] D. Fernandez-Fraile and A. Gomez Nicola, Phys. Rev. Lett. , 121601 (2009).[82] A. Dobado and J. M. Torres-Rincon, Phys. Rev. D , 074021 (2012).[83] V. Ozvenchuk, O. Linnyk, M. I. Gorenstein, E. L. Bratkovskaya and W. Cassing, Phys. Rev.C , no. 6, 064903 (2013).[84] U. Gangopadhyaya, S. Ghosh, S. Sarkar and S. Mitra, Phys. Rev. C , no. 4, 044914 (2016).[85] P. Chakraborty and J. I. Kapusta, Phys. Rev. C , 014906 (2011).[86] C. Sasaki and K. Redlich, Phys. Rev. C , 055207 (2009).[87] H. Berrehrah, E. Bratkovskaya, W. Cassing and R. Marty, J. Phys. Conf. Ser. , no. 1,012050 (2015).[88] R. Marty, E. Bratkovskaya, W. Cassing, J. Aichelin and H. Berrehrah, Phys. Rev. C ,045204 (2013).[89] S. Samanta, S. Ghosh and B. Mohanty, J. Phys. G , no. 7, 075101 (2018) doi:10.1088/1361-6471/aac621 [arXiv:1706.07709 [hep-ph]].[90] K. Saha, S. Ghosh, S. Upadhaya and S. Maity, Phys. Rev. D , no. 11, 116020 (2018)doi:10.1103/PhysRevD.97.116020 [arXiv:1711.10169 [nucl-th]].[91] P. Singha, A. Abhishek, G. Kadam, S. Ghosh and H. Mishra, J. Phys. G , no. 1, 015201(2019) doi:10.1088/1361-6471/aaf256 [arXiv:1705.03084 [nucl-th]].[92] D. Fernandez-Fraile and A. Gomez Nicola, Int. J. Mod. Phys. E (2007) 3010.[93] S. Sarkar, Adv. High Energy Phys. (2013) 627137.[94] P. Deb, G. P. Kadam and H. Mishra, Phys. Rev. D , no. 9, 094002 (2016).[95] A. Abhishek, H. Mishra and S. Ghosh, arXiv:1709.08013 [hep-ph].[96] G. P. Kadam and H. Mishra, Phys. Rev. C , no. 3, 035203 (2015).[97] A. S. Khvorostukhin, V. D. Toneev and D. N. Voskresensky, Nucl. Phys. A , 106 (2010)doi:10.1016/j.nuclphysa.2010.05.058 [arXiv:1003.3531 [nucl-th]].[98] A. Dash, S. Samanta and B. Mohanty, arXiv:1905.07130 [nucl-th].[99] H. X. Zhang, J. W. Kang and B. W. Zhang, arXiv:1905.08146 [hep-ph].[100] V. Mykhaylova, M. Bluhm, K. Redlich and C. Sasaki, arXiv:1906.01697 [hep-ph].[101] C. A. Islam, J. Dey and S. Ghosh, arXiv:1901.09543 [nucl-th].[102] S. Ghosh, F. E. Serna, A. Abhishek, G. Krein and H. Mishra, Phys. Rev. D , no. 1, 014004(2019) doi:10.1103/PhysRevD.99.014004 [arXiv:1809.07594 [nucl-th]].[103] S. Ghosh, S. Ghosh and S. Bhattacharyya, Phys. Rev. C , no. 4, 045202 (2018)doi:10.1103/PhysRevC.98.045202 [arXiv:1807.03188 [hep-ph]].[104] F. Gao and Y. x. Liu, Phys. Rev. D , no. 5, 056011 (2018) doi:10.1103/PhysRevD.97.056011[arXiv:1702.01420 [hep-ph]].[105] S. Ghosh, S. Chatterjee and B. Mohanty, Phys. Rev. C , no. 4, 045208 (2016)doi:10.1103/PhysRevC.94.045208 [arXiv:1607.04779 [nucl-th]].[106] M. Attems, J. Casalderrey-Solana, D. Mateos, D. Santos-Olivn, C. F. Sopuerta, M. Trianaand M. Zilho, JHEP , 026 (2017) doi:10.1007/JHEP01(2017)026 [arXiv:1604.06439 [hep-th]].[107] M. Attems, J. Casalderrey-Solana, D. Mateos, D. Santos-Olivn, C. F. Sopuerta, M. Triana and M. Zilho, JHEP , 154 (2017) doi:10.1007/JHEP06(2017)154 [arXiv:1703.09681 [hep-th]].[108] W. Florkowski, E. Maksymiuk and R. Ryblewski, Phys. Rev. C , no. 2, 024915 (2018)doi:10.1103/PhysRevC.97.024915 [arXiv:1710.07095 [hep-ph]].[109] M. Tanabashi et al. [Particle Data Group], Phys. Rev. D , no. 3, 030001 (2018).doi:10.1103/PhysRevD.98.030001[110] David J. Griffiths, Introduction to Quantum Mechanics ,Second Ed., Pearson Education (sin-gapore) 2005.[111] M. Cannoni, Phys. Rev. D , no. 10, 103533 (2014).[112] P. Gondolo and G. Gelmini, Nucl. Phys. B , 145 (1991).[113] J. Cleymans, H. Oeschler, K. Redlich and S. Wheaton, Phys. Rev. C , 034905 (2006).[114] K. A. Bugaev, A. I. Ivanytskyi, D. R. Oliinychenko, E. G. Nikonov, V. V. Sagun and G. M. Zi-novjev, Ukr. J. Phys. , 181 (2015) [arXiv:1312.4367 [nucl-th]].[115] A. Tawfik, Nucl. Phys. A922