Bulk and shear viscosities of hot and dense hadron gas
aa r X i v : . [ h e p - ph ] A ug Bulk and shear viscosities of hot and dense hadron gas
Guru Prakash Kadam ∗ and Hiranmaya Mishra † Theory Division, Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India (Dated: September 5, 2018)We estimate bulk and shear viscosity at finite temperature and baryon densities of hadronicmatter within hadron resonance gas model. For bulk viscosity we use low energy theorems of QCDfor the energy momentum tensor correlators. For shear viscosity coefficient, we estimate the sameusing molecular kinetic theory to relate the shear viscosity coefficient to average momentum ofthe hadrons in the hot and dense hadron gas. The bulk viscosity to entropy ratio increases withchemical potential and is related to the reduction of velocity of sound at nonzero chemical potential.The shear viscosity to entropy ratio on the other hand, shows a nontrivial behavior with the ratiodecreasing with chemical potential for small temperatures but increasing with chemical potentialat high temperatures and is related to decrease of entropy density with chemical potential at hightemperature due to finite volume of the hadrons.
PACS numbers: 12.38.Mh, 24.10.Pa,24.85.+p,25.75.Dw ∗ [email protected] † [email protected] I. INTRODUCTION
Recently, the transport properties of hot and dense matter has attracted lot of attention in the context of relativisticheavy ion collisions[1] as well as cosmology[2]. Such properties enter in the hydrodynamical evolution and thereforeessential for studying the near equilibrium evolution of a thermodynamic system. In the context of heavy ion collisions,the coefficients of shear viscosity perhaps has been the mostly studied transport coefficient. The spatial anisotropy ina nuclear collision gets converted to a momentum anisotropy through a hydrodynamic evolution. The equilibrationof momentum anisotropy is mainly controlled by shear viscosity. Indeed, elliptic flow measurement at RHIC led to ηs , the ratio of shear viscosity ( η ) to the entropy density s , close to π which is the smallest for any known liquid innature [3]. Indeed, arguments based on ADS/CFT correspondence suggest that the ratio ηs cannot be lower than this’Kovtun-Son-Starinets’ (KSS) bound [4]. Thus the quark gluon plasma(QGP) formed in the heavy ion collision is themost perfect fluid.Apart from shear viscosity, the transport coefficient that relates the momentum flux with a velocity gradient is thebulk viscosity. Generally, it was believed that the bulk viscosity does not play any significant role in the hydrodynamicevolution of the matter produced in heavy ion collision experiments. The argument being that the bulk viscosity ζ scales like ǫ − p and therefore will not play any significant role as the matter might be following the ideal gas equationof state. However, in course of the expansion of the fire ball the temperature can be near the critical temperature T c where ǫ − p can be large as expected from the lattice QCD simulations[5, 6] leading to large value of bulk viscosity.This in turn can give rise to phenomena of cavitation when the pressure vanishes and the hydrodynamic descriptionfor the evolution breaks down[7].There has been various attempts to estimate bulk viscosity for strongly interacting matter. The rise of bulkviscosity near the transition temperature has been observed in various effective models of strong interaction. Theseinclude chiral perturbation theory[8], quasi particle models [9] as well as Nambu-Jona-Lasinio model [10]. One ofthe interesting way to extract this is using symmetry properties of QCD once one realizes that the bulk viscositycharacterizes the response to conformal transformation. This was attempted in Ref.[11]. Based on Kubo formula forthe ζ and the low energy theorems [12] bulk viscosity gets related to thermodynamic properties of strongly interactingsystem.It may be noted that, it is also of both practical and fundamental importance to know the transport coefficients inthe hadron phase to distinguish the signatures of QGP matter and hadronic matter. The computation of the transportcoefficient of the hadronic mixture is not an easy task. There have been various attempt on this field over last fewyears involving various approximations like relaxation time approximation, Chapman-Enscog as well as Green Kuboapproach to estimate the shear viscosity to entropy ratio using different effective models for hadronic interactions[10, 13–16].In a different approach, ηs has also been calculated within a hadron resonance gas model in an excluded volumeapproximation [17] with a molecular kinetic theory approach to relate shear viscosity coefficient to the average mo-mentum transfer. This was used later to include the effects of rapidly rising hadronic density of states near the criticaltemperature modeled by hagedorn type exponential rise of density of states [18]. Such a description could describethe lattice data and indicated that the hadronic matter could become almost a perfect fluid where ηs could approachthe KKS bound. Later lattice data [6] which indicated a lower pseudocritical temperature about 160 MeV led to theassertion that the hot hadronic matter described through hadron resonance gas is far from being a perfect fluid[20].All these studies have been done at zero baryon density.It has been also known that the basic features of hadronization in heavy ion collisions are well described by thehadron resonance gas models. The multiplicities of particle abundances of various hadrons in these experiments showgood agreement with the corresponding thermal abundances calculated in HRG model with appropriately chosentemperature and chemical potentials [19]. In the present work, we generalize the above approach of [20] for studyingviscosity coefficients within the ambit of hadron resonance gas model to include finite chemical potential effects. Thiscan possibly have some relevance on the current and planned experiments with heavy ion collisions at beam energyscan at RHIC [21], compressed baryonic matter at GSI [22] and neuclotron-based ion collider facility (NICA) at Dubna[23].The shear viscosity to entropy ratio at finite baryon density has been estimated using relativistic Boltzmannequations for pion nucleon system using phenomenological scattering amplitude[24, 25]. This leads to the ratio asa decreasing function of chemical potential in the T- µ plane. Further, this ratio as a function of chemical potentialshows a valley structure at low temperature which was interpreted as a signature of liquid gas phase transition[24, 25].The bulk viscosity at finite chemical potential using low energy theorems of QCD has been studied in Ref.[26].This was estimated using a Schwinger Dyson approach to calculate the dressed quark propagator at finite chemicalpotential to use it for calculation of thermodynamical quantities needed to estimate bulk viscosity. As mentioned, weshall estimate these viscosity coefficients within the ambit of hadron resonance gas which can be a complimentary tothe above approaches.We organize the paper as follows. In the following section we recapitulate the results of Ref.[11] for bulk viscositycoefficient as related to the thermodynamic quantities using Kubo formula and low energy theorems generalized toinclude finite chemical potential terms. We also note down here the expression for the shear viscosity using quantummolecular dynamics method modified appropriately for relativistic system. In section III we spell out the the hadronresonance gas model including a hagedorn spectrum above a cut off and the resulting thermodynamics. We estimatethe quark condensates in a thermal, dense medium of hadron gas in a subsection here. In section IV we discuss theresults. Finally, we summarize and conclude in section V. II. BULK AND SHEAR VISCOSITY COEFFICIENTS AT FINITE T AND µ Bulk viscosity corresponds to the response of the system to conformal transformations and can be written as perKubo formula as a bilocal correlation function[11] ζ = lim ω → ω Z ∞ dt Z d x exp( iωt ) (cid:2) θ µµ ( x ) , θ µµ (0) (cid:3) ≡ Z d x iG R ( x ) (1)with G R ( x ) being the retarded function for the trace of energy momentum tensor. One can introduce a spectralfunction ρ ( ω, p ) = − (1 /π ) ImG ( ω, p ) to write a dispersion relation for the G R ( ω, p ). Assuming an ansatz for thespectral function at low energy[11] as ρ ( ω, ) /ω = (9 ζ/π )( ω / ( ω + ω ), where, ω is a scale at which perturbationtheory becomes valid , the bulk viscosity can be written as9 ζω = 2 Z ∞ du ρ ( u, u du = Z d x h θ µµ ( x ) θ µµ (0) i ≡ Π (2)The stress energy tensor for QCD is given as θ µµ = m ¯ qq + β ( g )2 g G aµν G aµν ≡ θ q + θ g (3)In the above g is the strong coupling and β ( g ) is the QCD beta function that decides the running of the QCD coupling.Thus the evaluation of the bulk viscosity reduces to evaluation of the stress energy correlator. This is done by usingthe low energy theorems of QCD generalized to finite temperature and density according to which for any operatorˆ O , its correlator with the gluonic part of the stress tensor θ g is given as Z d x h θ g ( x ) ˆ O ) i = ( ˆ D − d ) h ˆ O i ( T, µ ) , (4)where, ˆ D = T ∂/∂T + µ∂/∂µ − d , with d being the canonical dimension of the operator ˆ O . Using Eq.(4) in Eq.(2) onehas Π = ( ˆ D − h θ µµ i + ( ˆ D − h θ qµµ i = 16 | ǫ gvac | + 6( f π m π + f k m k )+ T S ( 1 c s −
3) + ( µ ∂∂µ − ǫ ∗ − p ∗ ) + ( ˆ D − m q h ¯ qq i ∗ (5)In the above we have used h θ µµ i = ǫ − p and the thermodynamic relations c v = ∂ǫ/∂T , ∂p/∂T = s and c s = S/c v for the velocity of sound of the medium. We have also separated the contributions to the correlators in terms of thevacuum and the medium. In Eq.(5) we have neglected terms quadratic in the current quark masses and have usedPCAC relations to express vacuum condensates to the masses and decay widths of pions and kaons. It is trivial tocheck that for µ = 0 Eq.(5) reduces to the main results of Ref.[11]. For T=0 and µ = 0, one can simplify Eq.(5)andEq.(2) reduces to 9 ζ ( µ ) ω = 16 P ( µ ) − µρ + µ ∂ρ∂µ + ( µ ∂∂µ − m h ¯ qq i (6)We might note here that the above expression differs from the same given in Ref.[26].This, however, matches withthe expression given in Ref.[27] in the appropriate limit, where, bulk viscosity was computed including the effects ofmagnetic field at finite baryon densities and temperature.Thus the coefficients of bulk viscosity gets related to the vacuum properties of QCD as well as to the equilibriumthermodynamic system parameters of QCD like the velocity of sound, non-ideality and the in medium quark conden-sates. These thermodynamic quantities shall be estimated within hadron resonance gas model which we shall spellout in the next section.Next, we consider the shear viscosity coefficient η for the hadronic medium. It is known that hadrons interact invarious channels and there is possibility of attractive and repulsive interactions. Within the hadron resonance model,the attractive channels are effectively included by including the resonances and the repulsive channels can be modeledin a simple manner through and excluded volume correction [28–30]. The shear viscosity in a relativistic gas of multicomponent hard core spheres can be written as [17, 20] η = 564 √ r X i h| p |i n i n (7)where, h| p |i is the average momentum of the i-th species particles and r corresponds to hard core radius of eachhadron . Further, in the above, n i is the number density of the i-th particle species and n = P i n i . III. HADRON RESONANCE GAS MODEL
The central quantity in the hadron resonance gas models (HRGM) is the thermodynamic potential which is thatof a free boson or fermion gas and is given aslog(
Z, β, µ, ) = Z dm ( ρ M ( m ) log Z b ( m, v, β, µ ) + ρ B ( m ) log Z f ( m, v, β, µ )) (8)where, the gas of hadrons is contained in a volume V, at a temperature β − and chemical potential µ . Z b , Z f are thepartition functions of boson and fermions respectively with mass m . Further, ρ M and ρ B are the spectral densitiesof bosons and fermions respectively. Using Eq.(8), one can calculate the energy density ǫ by taking derivative withrespect to β , pressure p, by taking a derivative with respect to V , number density ρ by taking a derivative withrespect to µ . One can also find out the trace anomaly ǫ − p , entropy density, specific heat as well as the speed ofsound from these quantities.Hadron properties enter in these models through the spectral densities ρ B/M ( m ). One common approach in HRGMsis taking all the hadrons and their resonances up to a mass cutoff Λ and write ρ B/M ( m ) = M i < Λ X i g i δ ( m − M i ) (9)where, the sum is over all the baryons or meson states up to a mass that is less than the cut off Λ. M i are the massesof the known hadrons and g i is the degeneracy factor (spin, isospin). On the other hand, an exponentially increasingdensity of state was necessary to explain the rapid increase in entropy density near the transition region in latticeQCD simulation [31]. Such exponential rise of density of states has also been used to study observables like dileptonproduction [32] as well as chemical equilibration[33]. Motivated by such observations we take the modified spectralfunction as[18, 34, 35] ρ B/M ( m ) = M i < Λ X i g i δ ( m − M i ) + ρ HS ( m ) (10)where ρ HS ( m ) is the spectral density for the heavier Hagedorn states(HS). To describe the much needed large densityof states, one can take an exponentially rising density of state [36] for ρ HS beyond the cut-off Λ which implies anunderlying string picture for hadrons. On the other hand, one can also consider a simple power law form introducedin Ref.[38] as a nice alternative to describe the rise of the hadronic mass spectrum [38]. We shall consider here boththe forms for the continuum part of the spectral density given as ρ exp = A ( m + m ) e mTH (11) ρ power = AT H (cid:18) mT H (cid:19) α (12)where parametrization of the two spectral forms is given in table below.spectral density T H ( GeV ) A m ( GeV ) αρ exp ρ power A and m for ρ exp as in Ref. [31] and taken a different value for T H so as to fit thelattice data of Ref.[39]. Similarly the parameters α and T H for ρ power is taken so as to fit the lattice data of Ref.[39]while keeping the parameter A same as taken in Ref.[20].With the ansatz for the spectral densities, the pressure P = P M + P B arising from mesons and baryons respectivelyare given by P M = 12 π (cid:20) − X i g i Z k dk log (1 − exp( − βǫ i ))+ Z ∞ Λ ρ HS ( m ) dm m β K ( βm ) (cid:21) (13) P B = 12 π (cid:20) − X i g i Z k dk (cid:18) log (cid:0) − exp( − β ( ǫ i − µ )) (cid:1) + log (1 − exp( − β ( ǫ i + µ ))) (cid:19) + 2 Z ∞ Λ ρ H ( m ) dm m β K ( βm ) cosh( βµ ) (cid:21) (14)Here, K n ( x ) is the modified Bessel function of order n . Similarly, the energy density ǫ = − β ∂∂β ( βp ) + µ ∂∂µ p = ǫ M + ǫ B , with the energy density of mesons ǫ M given as ǫ M = 12 π (cid:20) X i g i Z k dk ǫ i exp( βǫ i ) − Z ∞ Λ ρ HS ( m ) dmm (cid:18) β m K ( βm ) + 1 βm K ( βm ) (cid:19) (cid:21) (15)and, the contribution of the baryons to the energy density ǫ B is given as ǫ B = 12 π (cid:20) X i g i Z k dkǫ i (cid:18) β ( ǫ i − µ )) + 1 + 1exp( β ( ǫ i + µ )) + 1 (cid:19) + Z ∞ Λ ρ H ( m ) dmm (cid:18) β m K ( βm ) + 1 βm K ( βm ) (cid:19) (cid:21) (16)The baryon number density is given by n B = 12 π (cid:20) g i Z k dk (cid:18) β ( ǫ i − µ )) + 1 − β ( ǫ i + µ )) + 1 (cid:19) + 2 Z ∞ Λ ρ H ( m ) dm m β K ( βm ) (cid:21) (17)Using these quantities one can calculate the other quantities like the interaction measure ǫ − p , entropy density s = ( ∂p∂T ) as needed for the estimation of bulk viscosity. A. quark condensates in the hadronic medium
The other quantity we need to know is the quark condensates in the medium to estimate the bulk viscosity. Toestimate this within the framework of HRGM, it is necessary to know the dependence of hadron masses on the currentquark masses. The chiral condensate is given in terms of the thermodynamic potential (negative of the pressure) as h ¯ qq i = − ∂p∂m q which leads to h ¯ qq i = h ¯ qq i + X mesons σ M m q n M + X baryons σ B m q n B (18)where n M and n B are the scalar densities of mesons and baryons given respectively as n M = g i π Z k dk m M ǫ M βǫ M ) − , (19) n B = g i π Z k dk m B ǫ B (cid:18) β ( ǫ B − µ B )) + 1 + 1exp( β ( ǫ B + µ B )) + 1 (cid:19) (20)Further, the σ M/B is the hadronic sigma term i.e. the response of hadronic masses to the changes of the current quarkmasses σ M/Bq = m q ∂M M/B ∂m q (21)Thus computing the behavior of in-medium condensate within HRGM reduces to the problem of calculating the σ -terms of the hadrons. We do this in a manner similar to given in Ref.[40]. For the pseudoscalar bosons, we use theGell Mann-Oakes-Renner (GOR) relation to have ∂m π ∂m q = − h ¯ qq i f π (cid:18) κ m p i f π (cid:19) (22) ∂m K ∂m q,s = − h ¯ qq i + h ¯ ss i f K (cid:18) κ m K f π (cid:19) (23)with the parameter κ = 0 . ± .
008 [41]. here, we have taken m q = m u = m d = 5 . M eV , m S = 138 M eV , f π = 92 . M eV , f K = 113 M eV h ¯ uu i = h ¯ dd i = h ¯ qq i = ( − M eV ) , h ¯ ss i = 0 . h ¯ qq i . For the other hadrons we usea model based on valence quark structure as in Ref. [42]. Here the masses of the baryons (B) or mesons (M) scale as m B = (3 − N s ) M q + N s M s + κ B (24) m M = (2 − n s ) M q + N s M s + κ m (25)In the above, M q , M s are the constituent quark masses for the light and strange quarks respectively, κ B/M areconstants depending upon the hadronic state but not on current quark masses and N s is the numbers of strangequarks. The constituent quarks M q and M s partially account for the strong interaction dynamics. For computationof the σ term one meeds to know the variation of the constituent quarks with current quark masses. This dependenceis taken from Nambu-Jona-Lasinio model [43] where, the dynamical mass changes by 14 MeV as the current quarkmass is changed from 0 to 5.5 MeV. Similarly for strange quark the mass change is about 235.5 MeV as current quarkmass is varied from 0 to 140.7 MeV. This e.g. results in σ terms for nucleons and Λ hyperon as 42 MeV and 263.5MeV respectively. MeV (HG)MeV (HG+HS)(HG) P T T ρ exp µ = 0 µ = 0 µ = 300MeV µ = 300MeV(HG+HS) MeV (HG)MeV (HG+HS)(HG) P T T ρ power µ = 0 µ = 0 µ = 300MeV µ = 300MeV(HG+HS) (1 a) (1 b)FIG. 1. Thermodynamics of hadron resonance gas. Left panel (Fig. 1 a) shows pressure as a function of temperature for µ b = 0(blue) and µ B = 300 MeV (red) with the hagedorn spectrum ρ = ρ exp as in Eq.(11). The dotted line correspond totaking discrete spectrum for hadron resonance gas. The right panel shows the same quantities but with the spectral function ρ = ρ power as given in Eq.(12). The data points are from the lattice simulation results taken from Ref. [39]. IV. RESULTS AND DISCUSSIONS
Let us first discuss the thermodynamics of hadron resonance gas specified by the spectral density as given byEq.(10). To estimate different thermodynamic quantities, for the discrete part of spectrum , we have taken all thehadrons and their resonances with mass less than 2 GeV [44]. For the Hagedorn part, consider both the forms ofspectral density given by Eq. (11) and Eq.(12).In Fig.1 we have plotted the pressure in units of T for two different chemical potentials, µ = 0 MeV and µ = 300MeV. The lattice points with the error bars have been taken from the table 4 of the Ref. [39] corresponding to thecontinuum extrapolation. The dotted lines in Fig.1 correspond to considering only the discrete part of the spectraldensity in Eq.(9) . Left panel correspond to exponential form of spectral density for continuum part while right panelcorresponds to power law form of spectral density in Eq.(10). As can be noted in this figure, the discrete spectrumcoupled with continuum spectrum describe the lattice data quite well up to T = 170 M eV with the parametrizationgiven in table 1 within the error bars of the lattice simulations.In fig. 2 we have plotted the dimensionless scale anomaly ( ǫ − p ) /T as a function of temperature at two differentchemical potentials. As can be noted from both the Fig.s (2a) and (2b), the discrete part of the spectral densitydoes not give a good fit to the lattice data beyond 140 M eV , but when coupled with continuum part as in Eq.(10)gives good fit to lattice data up to 150
M eV even at µ = 300 M eV reasonably well. We have taken a higher T H valuecompared to [20] that was required to fit the lattice data[39]. This is because, in Ref.[20], the lattice data was takenfor N t = 10 lattice data of Ref.[6] while we have fitted with the continuum extrapolation of for µ = 0 the lattice datain Ref.[39].Fig 3 shows speed of sound squared ( C s ) as a function of temperature at fixed values of chemical potential alongwith the lattice simulation results of Ref.[39]. As can be noted from the figure, keeping only the discrete part ofthe spectral density, does not fit the lattice results although the same could fit the lattice result for pressure andthe scale anomaly results of Ref.[39]. On the other hand the power law parametrization for the continuum part ofspectral density along with the discrete part leads to a reasonable fit to lattice data up to 150 M eV both at µ = 0and µ = 300 M eV . The initial rise in sound velocity with temperature is reflection of the fact that the light degreesof freedom are excited easily at low temperature and contribute to pressure and energy. But at larger temperatureswhen baryons are excited, they contribute significantly to energy density but almost nothing to pressure. This leadsto decrease of sound velocity with temperature seen at higher temperatures (
T >
MeV (HG)MeV (HG+HS)(HG) ǫ − P T T ρ exp µ = 0 µ = 0 µ = 300MeV µ = 300MeV(HG+HS) MeV (HG)MeV (HG+HS)(HG) ǫ − P T T ρ power µ = 0 µ = 0 µ = 300MeV µ = 300MeV(HG+HS) (2 a) (2 b)FIG. 2. Scale anomaly as a function of temperature for exponential spectral density (2 a) and power law spectral densityfunction (2 b). MeV (HG)MeV (HG+HS)(HG) C s T ρ exp µ = 0 µ = 0 µ = 300MeV µ = 300MeV(HG+HS) MeV (HG)MeV (HG+HS)(HG) C s T ρ power µ = 0 µ = 0 µ = 300MeV µ = 300MeV(HG+HS) (3 a) (3 b)FIG. 3. Square of sound velocity as a function of temperature for exponential spectral density (3 a) and power law spectraldensity function (3 b). nothing to pressure. This leads to lower values of C s as the chemical potential is increased.We have also plotted speed of sound for isentropic situation in figure 4. To get the chemical potential for a giventemperature, we vary chemical potential so that ratio S/N is constant. Resulting isentropic trajectories in the µ − T phase space is shown in fig. (4a). S/N = 30 and
S/N = 45 corresponds to AGS and SPS [45]. As expected from theresults for constant chemical potential (Fig.3), sound velocity is lower for lower
S/N .We next use these thermodynamic results for the hadron resonance gas to Eq.(2) and Eq.(5) to estimate the bulkviscosity. We also include here the contributions from the quark condensates in the discrete part of the spectrumusing Eq.(18). Contribution of these terms to ζ/s turns out to be only few percent of the the total contribution. Theresulting behavior of ζ/s as a function of temperature is shown in Fig.5 for different values of the baryon chemicalpotential. In general, the ratio decrease with temperature at low temperature followed by a sharp increase and finally T ( G e V ) µ ρ power s/n = 30 s/n = 45 C s T ρ power s/n = 30 s/n = 45 (4 a) (4 b)FIG. 4. Velocity of sound at constant entropy per baryon rations. Left panel (4a) shows trajectories of constant entropy perbaryon in the phase diagram. velocity of sound for constant entropy per baryon is plotted in Fig. 4b. (MeV)(MeV)(MeV)(MeV)(MeV) ζ / s T ρ exp µ = 0 µ = 100 µ = 200 µ = 300 µ = 500 (MeV)(MeV)(MeV)(MeV)(MeV) ζ / s T ρ power µ = 0 µ = 100 µ = 200 µ = 300 µ = 500 (5 a) (5 b)FIG. 5. Bulk viscosity to entropy ratio as a function of temperature for different chemical potentials. Left panel is withexponential hagedorn spectrum and the right panel is with power law hagedorn spectrum flattens out at temperatures around 160 MeV. This behavior is connected with the behavior of velocity of sound withtemperature through Eq.(5). The initial decrease of ζ/s with temperature is due to increase of sound velocity at lowtemperature due to excitation of light hadrons. At temperature T > r factors arising from the finite size of the hadrons asin Eq.(7). We also retain here the finite volume corrections to the entropy density s as in Ref.[46]. We might mentionhere that the thermodynamic quantities are not sensitive to the value of r for r > . (MeV)(MeV)(MeV)(MeV)(MeV) η / s T ρ exp µ = 0 µ = 100 µ = 200 µ = 300 µ = 500 (MeV)(MeV)(MeV)(MeV)(MeV) η / s T ρ power µ = 0 µ = 100 µ = 200 µ = 300 µ = 500 (6 a) (6 b)FIG. 6. Shear viscosity to entropy ratio in the hadronic phase. Left panel (6 a) shows ηs as a function of temperature fordifferent chemical potential with the exponential hagedorn spectrum . The right panel shows the same with the power lawhagedorn spectrum. -1
200 400 600 800 1000 (MeV) (MeV)(MeV)(MeV)(MeV) η / s µ ρ exp T = 10 T = 20 T = 25 T = 30 (MeV) (MeV)(MeV)(MeV)(MeV) η / s µ ρ exp T = 10 T = 20 T = 25 T = 30 (7 a) (7 b)FIG. 7. Shear viscosity to entropy ratio as a function of chemical potential. Ref.[20]. We have taken here a uniform size of r = 0 . f m for all mesons and r = 0 . f m for all baryons [17, 47]. For µ = 0 the minimum reaches about ηs = 0 . ηs = π .As the chemical potential is increased, temperature dependence is similar to that at µ = 0. On the other hand, thebehavior of the ratio η/s is nontrivial. It decreases with chemical potential for temperature less than about 130 MeV,beyond which increases with µ . The initial decrease of η with respect to µ can be understood as enhancement of thehardcore cross section with nucleon number density . However, the entropy density σ starts decreasing with increasein chemical potential at higher temperature. This is due to the fact that the volume corrections proportional to thedensity of the particles enter in the denominator for the entropy density [46]. This in turn makes a larger value forthe ratio ηs We also looked into the behavior of ηs at low temperatures as a function of µ where it shows a valley structure1which is plotted in Fig.7. Such an observation was also made in Ref.s[24, 25, 49]. The existence of a minimum of η/s was interpreted in these references indicative of a liquid gas phase transition. This is due to the fact that aminimum in the ratio η/s as a function of the controlling parameter of thermodynamics like temperature or chemicalpotential could be indicative of a phase transition [24, 25, 48]. As the temperature increase, the valley structurebecome shallower as is clearly shown in Fig.7 b, possibly suggestive of the phase transition. However, on the otherhand, the corresponding entropy does not show such a structure. Further, the corresponding nucleon number densityhere (0.07/ f m ), however, turns out to be about half the nuclear matter density. V. SUMMARY
We have here tried to estimate the bulk and shear viscosity to entropy ratio in a hadronic medium modeling thesame as a hadron resonance gas. Apart from including all the hadrons below a cutoff of 2 GeV, we have also includeda continuum density of state beyond 2GeV. Such a description of hadronic model gives a good fit to the lattice databoth at zero and finite chemical potential [39]. The thermodynamic quantities so obtained is used to estimate thebulk viscosity of hadron gas at finite chemical potential using the method as outlined in Ref.[11] for finite temperatureand zero chemical potential. At finite chemical potential, the ζs become higher as compared to µ = 0 and is related tothe fact that the velocity of sound becomes smaller due to finite chemical potential with excitation of heavier baryonscontributing more to the energy density as compared to the pressure.This approach has already been used to estimate η/s for hadronic medium in Ref. [18] to obtain η/s reaching theviscosity bound of π at temperature of about T = 190MeV using the lattice data available at that time. However,later lattice data pointed to a lower critical temperature giving rise to indicate that the hadron resonance gas canlead to η/s being about an order of magnitude higher than the viscosity bound. We observe that at finite chemicalpotential ηs increases with temperature with its magnitude increasing with chemical potential. For low temperatures( T < M eV ) and high baryon chemical potential, we observed a valley structure for this ratio which can have aconnection with liquid gas phase transition in nuclear matter. [1] U. Heinz and R. Snellings, Annu. Rev. Nucl. Part. Sci. 63, 123-151, 2013[2] A. Bstero-Gil, A. Berera and R. Ramos, JCAP1107, 030 (2011).[3] P. Romatschke and U. Romatschke, Phys. Rev. Lett. ,172301, (2007); T. Hirano and M. Gyulassy, Nucl. Phys. A 769 ,71, (2006).[4] P. Kovtun, D.T. Son and A.O. Starinets, Phys. Rev. Lett. , 111601, (2005).[5] A. Bazavov etal , e-print:arXiv:1407.6387.[6] S. Borsonyi etal , JHEP1011, 077 (2010).[7] K. Rajagopal and N. Trupuraneni, JHEP1003, 018(2010); J. Bhatt, H. Mishra and V. Sreekanth, JHEP 1011,106,(2010); ibid Phys. Lett. B704, 486 (2011); ibid
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