Transport coefficients in Polyakov quark meson coupling model: a relaxation time approximation

We compute the transport coefficients, namely, the coefficients of shear and bulk viscosities as well as thermal conductivity for hot and dense matter. The calculations are performed within the Polyakov quark meson model. The estimation of the transport coefficients is made using the Boltzmann kinetic equation within the relaxation time approximation. The energy dependent relaxation time is estimated from meson meson scattering, quark meson scattering and quark quark scattering within the model. In our calculations, the shear viscosity to entropy ratio and the coefficient of thermal conductivity show a minimum at the critical temperature, while the ratio of bulk viscosity to entropy density exhibits a peak at this transition point.The effect of confinement modelled through a Polyakov loop potential plays an important role in the estimation of these dissipative coefficients both below and above the critical temperature.


I. INTRODUCTION
Transport coefficients of matter under extreme conditions of temperature, density or external fields are interesting for several reasons. In the context of relativistic heavy ion collisions, these properties enter as dissipative coefficients in the hydrodynamic evolution of the quark gluon plasma that is produced following the collision [1][2][3][4][5]. Indeed, an extremely low value of the shear viscosity-to-entropy ratio (η/s) is needed to successfully describe the collective dynamics of the quark gluon matter at high temperature and vanishing chemical potential to explain the elliptic flow data [6,7]. At intermediate densities, near the chiral phase transition, which is being probed at the Facility for antiproton and Ion Research(FAIR) program at Geselleschaft fuer Schwerionenforschung(GSI)- [8] and the Nuclotron-based Ion Collider fAcility(NICA) program at Joint Institute for Nuclear Research(JINR)- [9] motivates us to understand the behavior of transport coefficients at finite chemical potential and temperature. motivates us to understand the behavior of transport coefficients at finite chemical potential and temperature. Further, in the low temperature and high-density regime, the matter could be in one of the possible types of color superconducting phases of which transport properties also need to be understood [10,11]. The cooling of neutron stars at short time scales constrains the thermal conductivity [12,13] while the cooling through neutrino emission on a much larger time scales constrains the phase of the matter in the interior of the compact star [14,15]. Further, the observable regarding the viscosity of the such matter is the r-mode instability. In the absence of viscous damping, the fluid in the rotating star becomes unstable to a mode that is coupled to gravity and radiates away the angular momentum of the star [16,17]. Apart from the wide variety of applications of the transport coefficients of strongly interacting matter, their temperature and chemical potential dependence may also be indicative of a phase transition [18].
Transport coefficients for QCD matter in principle can be calculated using Kubo formulation [19]. However, QCD is strongly interacting for both at energies accessible in heavy ion collision experiments as well as for the densities expected to be there in the core of the neutron stars making the perturbative estimations unreliable. Calculations using lattice QCD simulations at finite chemical potential is also challenging and is limited only to the equilibrium thermodynamic properties at small chemical potentials.
The understanding of the elliptic flow in relativistic heavy ion collisions using hydrodynamics with a low (η/s) and its connection to the conjectured lower bound (η/s > 1/4π) using ADS/CFT correspondence [20] stimulated extensive investigation of this ratio for QCD matter. These have been studied using perturbative QCD [21], transport simulations of the Boltzmann equation [22,23], relaxation time approximation for solving the Boltzmann equations [24][25][26][27][28] and lattice simulation of QCD [29]. Most of these calculations have been performed at vanishing baryon density. The general variation of this ratio with temperature in most of these studies shows a minimum at the transition temperature. The numerical value of η at the minimum, however, differs by orders of magnitude. For example, Ref. [30,31], Refs. [32][33][34] have predicted η of order 0.001 GeV 3 , η=0.002-0.003 GeV 3 while Ref. [35] predicts a value of η ≃ 0.4 GeV 3 . Further, the behavior of η/s shows a monotonic decrease with temperature in the Nambu-Jona-Lasinio (NJL) model in Ref. [46].
The bulk viscosity coefficient ζ has also been estimated in various effective models as well as in lattice QCD. The rise of the bulk viscosity coefficient near the transition temperature has been observed in these effective models such as chiral perturbation theory [36], quasiparticle models [57], linear sigma model [37], and the Nambu-Jona-Lasinio model [24,26]. Large bulk viscosity of matter produced in relativistic heavy ion collisions can give rise to different interesting phenomenon such as cavitation where pressure vanishes and hydrodynamic description of evolution becomes invalid [38]. Here, again, the numerical value of the bulk viscosity coefficients vary widely from 10 −5 GeV 3 [39] to 10 −2 GeV 3 [24].
The other transport coefficient that is important at finite baryon density is the coefficient of thermal conductivity λ [40][41][42]. The effects of thermal conductivity in relativistic hydrodynamics has been discussed recently in Refs. [42,43]. This coefficient has been evaluated in various effective models like the Nambu-Jona-Lasinio model using the Green-Kubo approach [44], relaxation time approximation [26] and the instanton liquid model [45]. The results, however, vary over a wide range of values, with λ = 0.008 GeV −2 as in Ref. [32] to λ ∼ 10 GeV −2 as in Ref. [46] for a range of temperatures (0.12 GeV <T< 0.17 GeV), which has been nicely tabulated in Ref. [47].
We shall attempt here to estimate these transport coefficients within an effective model of strong interaction, the Polyakov loop extended quark meson (PQM) model. It has become quite popular during last few years due to its close relationship with the linear sigma model that captures the chiral symmetry breaking aspect while being in agreement with the lattice QCD results for thermodynamics at vanishing baryon density. The physics of confinement is taken care of at least partially by coupling the quark field to the Polyakov loops so that quark excitations are suppressed below the transition temperature. Let us note that the transport coefficients like bulk viscosity apart from the distribution functions also depend upon the bulk thermodynamic quantities like velocity of sound. We wish to explore the effects of such nonperturbative properties on the transport coefficients.
The transport coefficients are evaluated within the relaxation time approximation of Boltzmann equation. The relaxation time is calculated by evaluating the scattering rates of the particles in the model, namely, the quarks and pion and sigma mesons, with their respective medium-dependent masses. The scattering processes considered here are meson scatterings as considered in Ref. [37], quark scattering through meson exchanges as in Refs. [24,26,46], and quark-meson scatterings. As we shall see in the following, each of these processes brings out distinct features for the transport coefficients. We would like to mention here that these coefficients have also been estimated using Kubo formulation through one-loop self-energies for quarks and mesons in a separate work [48].
We organize the present investigation as follows. In the following section, we discuss the two-flavor PQM model thermodynamics. The reason is that the expressions for transport coefficients involve meson masses which are medium dependent. Further, some transport coefficients like the bulk viscosity involves bulk thermodynamical properties such as energy density, pressure and the velocity of sound. As the order parameters for chiral and confinementdeconfinement transitions are coupled, this leads to nontrivial relations for derivatives of the thermodynamic potential with respect to external parameters like chemical potential or temperature as the mean fields themselves are also medium-dependent. Furthermore, the implicit dependence of these mean fields/ order parameters are calculated here analytically to avoid possible numerical errors. In Sec. III, we give the expressions for the transport coefficients in terms of relaxation time and estimate them to finally give the results for these coefficients. We also compare them with the same obtained with alternate approaches like the NJL model so that the effects of confinement-deconfinement transition modeled through Polyakov loop potential is explicitly seen. Finally, we summarize and draw the conclusions of the present investigation in section IV.

II. THERMODYNAMICS OF PQM MODEL
We shall adopt here an effective model that captures two important features of QCD, namely, chiral symmetry breaking and its restoration at high temperature and/densities as well as the confinement-deconfinement transitions. Two such effective models have become popular recently-the Polyakov loop extended Nambu-Jona-Lasinio (PNJL) model and the Polyakov loop extended quark meson coupling model (PQM). These models are extensions respectively of NJL model and linear sigma model that captures various aspects of chiral symmetry breaking pattern of strong interaction physics. Explicitly, the Lagrangian of the PQM model is given by [49][50][51][52][53] In the above, the first term is the kinetic and interaction term for the quark doublet ψ = (u, d) interacting with the scalar (σ) and the isovector pseudoscalar pion (π) field. The scalar field σ and the pion field π together form a SU(2) isovector field. The quark field is also coupled to a spatially constant temporal gauge field A 0 through the covariant The mesonic potential U χ (σ, π) essentially describes the chiral symmetry breaking pattern in strong interaction and is given by The last term in the Lagrangian in Eq. (1) is responsible for including the physics of color confinement in terms of a potential energy for the expectation value of the Polyakov loop φ andφ which are defined in terms of the Polyakov loop operator which is a Wilson loop in the temporal direction In the Polyakov gauge A 0 is time independent and is in the Cartan subalgebra i.e. A a 0 = A 3 0 λ 3 + A 8 0 λ 8 . One can perform the integration over the time variable trivially as path ordering becomes irrelevant so that P(x) = exp(βA 0 ). The Polyakov loop variable φ and its Hermitian conjugateφ are defined as In the limit of heavy quark mass, the confining phase is center symmetric and therefore φ = 0, while for deconfined phase φ = 0. Finite quark masses break this symmetry explicitly. The explicit form of the potential U p (φ,φ) is not known from first principle calculations. The common strategy is to choose a functional form of the potential that reproduces the pure gauge lattice simulation thermodynamic results. Several forms of this potential has been suggested in literature. We shall use here the polynomial parametrization [49] with the temperature-dependent coefficient b 2 given as The numerical values of the parameters are a 0 = 6.75, a 1 = −1.95, a 2 = 2.625, The parameter T 0 corresponds to the transition temperature of Yang-Mills theory. However, for the full dynamical QCD, there is a flavor dependence on T 0 (N f ). For two flavors we take it to be T 0 (2) = 192 MeV as in Ref. [49]. The Lagrangian in Eq. (1) is invariant under SU (2) L ×SU (2) R transformation when the explicit symmetry breaking term cσ vanishes in the potential U χ in Eq. (2). The parameters of the potential U χ are chosen such that the chiral symmetry is spontaneously broken in the vacuum. The expectation values of the meson fields in vacuum are σ = f π and π = 0. Here f π = 93 MeV is the pion decay constant. The coefficient of the symmetry breaking linear term is decided from the partial conservation of axial vector current as c = f π m 2 π , m π = 138 MeV, being the pion mass. Then minimizing the potential one has v 2 = f 2 π − m 2 π /λ. The quartic coupling for the meson, λ is determined from the mass of the sigma meson given as m 2 σ = m 2 π + 2λf 2 π . In the present work we take m σ = 600MeV which gives λ=19.7. The coupling g σ is fixed here from the constituent quark mass in vacuum M q = g q f π which has to be about one-third of the nucleon mass that leads to g σ = 3.3 [54].
To calculate the bulk thermodynamical properties of the system we use a mean field approximation for the meson and the Polyakov fields while retaining the quantum and thermal fluctuations of the quark fields. The thermodynamic potential can then be written as The fermionic part of the thermodynamic potential is given as modulo a divergent vacuum part. In the above, ω ∓ = E p ∓ µ, with the single particle quark/antiquark energy E p = p 2 + M 2 . The constituent quark/antiquark mass is defined to be The divergent vacuum part arises from the negative energy states of the Dirac sea. Using standard renormalization, it can be partly absorbed in the coupling λ and v 2 . However, a logarithmic correction from the renormalization scale remains, and we neglect it in the calculations that follow [54]. The mean fields are obtained by minimizing Ω with respect to σ, φ,φ, and π. Extremizing the effective potential with respect to the σ field leads to where, the scalar density ρ s = − ψ ψ is given by In the above, f ∓ (p) are the distribution functions for the quarks and anti quarks given as and, The condition ∂Ω ∂φ = 0 leads to where , Similarly, ∂Ω ∂φ = 0 leads to with, Finally, minimization of the effective potential with respect to π fields leads to where, the pseudoscalar density can be expressed as The σ and π masses are determined by the curvature of Ω at the global minimum These equations lead to the masses for the σ and pions given as Explicitly, using Eq. (13), The derivatives of the distribution functions with respect to to the scalar field σ are given as and, Similarly, using Eq. (21) In the above we have set the expectation value of pion field to be zero, i.e. π = 0 so that constituent quark mass is The net quark density is given by, The energy density ǫ = Ω − T ∂Ω/∂T + µρ q is given by In Fig. 1(a), we have plotted the constituent quark mass, and the meson masses as given in Eq. (23) and Eq. (24) as a function of temperature for vanishing baryon density. In the chirally broken phase, the pion mass, being the mass of an approximate Goldstone mode is protected and varies weakly with temperature. On the other hand, the mass of σ , M σ , which is approximately twice the constituent quark mass,M drops significantly near the crossover temperature. At high temperature, being chiral partners, the masses of σ and π mesons become degenerate and increase linearly with temperature. In Fig. 1(b), we have plotted the order parameters σ and φ as a function of temperature for vanishing quark chemical potential. We also note that for µ = 0, the order parameters φ andφ are the same. Because of the approximate chiral symmetry, the chiral order parameter decreases with temperatures to small values but never vanishes. The Polyakov loop parameter on the other hand grows from φ = 0 at zero temperature to about φ = 1 at high temperatures. We might mention here that at very high temperature the value of polyakov loop parameter exceeds unity, the value in the infinite quark mass limit.
Next, in Fig. 2, we show the dependence of the trace anomaly (ǫ−3p)/T 4 on temperature. The conformal symmetry is broken maximally at the critical temperature. Further finite chemical potential enhances this breaking as it breaks scale symmetry explicitly. As we shall see later, this will have its implication on the bulk viscosity coefficients.
Next, to discuss critical behavior as well as to calculate different thermodynamic quantities one has to take derivatives of the thermodynamic potential with respect to the mean fields as well as the parameters like temperature and the chemical potential. Vanishing of the first order derivatives of the thermodynamic potential with respect to the order parameters leads to the values of the order parameters satisfying the coupled gap equations as shown. However, to calculate many different thermodynamic quantities one also has to take into account the implicit dependence of the order parameters on temperature as well as chemical potential. One can do a numerical differentiation of the order parameters after solving for them from the gap equation. However, this can be numerically less accurate particularly for the higher-order derivatives. We shall use here a semianalytic approach to calculate the implicit contributions to the extent of taking the differentiation of the expressions analytically [55]. Only the values of the final expressions so obtained are computed numerically. For example, to calculate the derivative of the order parameter X, (X = σ, φ,φ) with respect to temperature is given by the equation Φ, σ with, Ω ab = ∂ 2 Ω ∂a∂b where a, b stand for σ, φ andφ. Similarly, to calculate the derivatives with respect to chemical potential, the coefficient matrix A remains the same while the matrix B will involve derivatives of the thermodynamic potential involving explicit dependence on the chemical potential.
Solving Eq. (31) this way, we have plotted the derivatives of the order parameters in Fig. 3. The critical temperature is defined by the position of the peaks of these derivatives of the order parameters. At zero chemical potential this occurs at T C ≃ 176 MeV. Let us note that at T C , the quark mass is m q = g σ σ = 134MeV, while the Polyakov loop variable φ ∼= 0.5. Thus at the critical temperature the effect of interaction is significant. As chemical potential for the quarks increase the critical temperature decreases. With finite chemical potential the peaks also become sharper and at higher chemical potential the transition becomes a first order one. The critical point within this model occurs at (T c , µ c ) = (155, 163) MeV.
The other thermodynamic quantity that enters into the transport coefficient calculation is the velocity of sound. The same at constant density is defined as where, P ,the pressure, is the negative of the thermodynamic potential given in Eq. (9). Further, s = − ∂Ω ∂T is the entropy density and the susceptibilities are defined as χ xy = − ∂ 2 Ω ∂x∂y . The velocity of sound shows a minimum near the crossover temperature. Within the model, at low temperature when the constituent quarks start contributing to the pressure, their contribution to the energy density is significant compared to their contribution to the pressure leading to decreasing behavior of the velocity of sound until the crossover temperature, beyond which it increases as the quarks become light and approach the massless limit of c 2 s = 1 3 . Such a dip in the velocity of sound is also observed in lattice simulation [56]. As we shall observe later, this behavior will have important consequences for the behavior of bulk viscosity as a function of temperature. We might mention here that such a dip for the sound velocity was not observed for two-flavor NJL [26]. For the linear sigma model calculations such a dip was observed only for a large sigma meson mass [37].

III. TRANSPORT COEFFICIENTS IN RELAXATION TIME APPROXIMATION
We shall attempt here to estimate the transport coefficients in the relaxation time approximation where the particle masses are medium dependent. Such attempts were made earlier for the σ-model [37] as well as in the NJL model to compute the shear and bulk viscosity coefficients. Such an approach was also made to estimate the viscosity coefficients of pure gluon matter [60]. In all these attempts, the expressions for the viscosity coefficients were derived for vanishing chemical potential. Several attempts were made to estimate these coefficients with finite chemical potential with different Ansatze. These expressions were put on firmer ground by deriving the expressions when there are mean fields and medium-dependent masses in a quasiparticle picture [58]. The resulting expressions for the transport coefficients were manifestly positive definite as they should be. These expressions were derived explicitly for the NJL model [26]. We use the same expressions here for the transport coefficients. The shear viscosity coefficient is given by where, the sum is over all the different species contributing to the viscosity coefficients including the antiparticles, and, τ a is the energy-dependent relaxation time that we define in the following subsection. The coefficient of bulk viscosity is given by The thermal conductivity on the other hand is given by In the above expressions, f 0 a is the equilibrium fermion/boson distribution functions depending upon the statistics with (1 ± f 0 a ) being the Bose enhancement/ Fermi suppression factors and t a = +1, 1 and 0 for the quark, antiquark, meson respectively. Further, c 2 s = ∂p ∂ǫ n is the velocity of sound at constant density and w = ǫ + p is the enthalpy density.

A. Relaxation time estimation-meson scatterings
As may be noted, the expressions for the transport coefficients as in Eqs. (34,35,36), depend not only on bulk thermodynamic properties like energy density, pressure, velocity of sound but also on the energy-dependent relaxation time τ (E). In the following we shall first estimate the relaxation times involving meson exchanges similar to Ref. [37].
Using the Lagrangian Eq. (1), we calculate the relaxation time in PQM model by taking into account the following scattering amplitudes with the corresponding matrix elements being given as The terms involving the propagators yield divergent integrals due to the poles in s and u channel which is known in the literature [37]. To regulate these integrals one can include a width for the mesons as evaluated in the next subsection (Eq. (54)). However, such a substitution violates crossing symmetry. Further, these terms are generated from the three-point vertices which are not taken into account in the mean field approximation used in solving the gap equations and the resulting equation of state. Hence, to be consistent with equation of state while maintaining crossing symmetry for the scattering amplitudes, we approximate the above scattering amplitudes by their limits when s, t and u are taken to be infinity and the scattering amplitudes reduce to constants [37]. Thus, the scattering amplitudes essentially reduce to constants. This allows us to compare our results with earlier work of [37] and study the effect of Polyakov loop and quarks within similar approximation.
The energy-dependent interaction frequency ω a (E a ) for the particle specie 'a' arising from a scattering process a, b → c, d, which is also the inverse of the energy-dependent relaxation time τ (E a ) is given by, with In the above, the summation is over all the particles except the species a with a, b as the initial state. The quantity W ab is dimensionless, Lorentz-invariant, and depends only on the Mandelstam variable s and is given by In the above, we have included the Bose enhancement factors for the meson scattering. The quantity W ab (s) is related to the cross section by noting that, with t as the Mandelstam variable t = (p a − p c ) 2 , where, p ab (s) = 1/(2 √ s) λ(s, m 2 a , m 2 b ), and the kinematic function λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2yz − 2zx, is the magnitude of the 3-momentum of the incoming particle in the c.m. frame. In the c.m. frame, using the energy momentum-conserving delta function and integrating over the final momenta, we have where, In the limit of constant |M | 2 , Eq. (44) reduces to and, the transition frequency or the inverse relaxation time is given as In the above, To calculate e.g. the π + relaxation time (τ π + ), we consider the scattering processes π + + π i → π + + π i (i = +, −, 0) and, π + + σ → π + + σ.
To get an order of magnitude of the average relaxation time, one can also calculate an energy averaged mean interaction frequency for a given species asω a ≡τ −1 a as ω a = 1 n a dp (2π) 3 with n a = dp (2π) 3 f a (E a ).

B. Relaxation time estimation-Quark scatterings
We next consider the quark scattering within the model through the exchange of pion and sigma meson resonances. The approach is similar to Refs. [25,26,46] performed within NJL model to estimate the corresponding relaxation time for the quarks and antiquarks. The transition frequency is again given by Eq. (41), with the corresponding W ab given as where, with the corresponding suppression factors appropriate for fermions. For the quark scatterings, in the present case for two flavors we consider the following scattering processes: One can use i-spin symmetry, charge conjugation symmetry and crossing symmetry to relate the matrix element square for the above 12 processes to get them related to one another and one has to evaluate only two independent matrix elements to evaluate all the 12 processes. We choose these, as in Ref. [25], to be the processes uū → uū and ud → ud and use the symmetry conditions to calculate the rest. We note, however, that, while the matrix elements are related, the thermal-averaged rates are not, as they involve also the thermal distribution functions for the initial states as well as the Pauli blocking factors for the final states. We also write down the square of the matrix elements for these two processes explicitly [25,26]-.
Similarly, the same for the process ud → ud is given as [25] |M The meson propagators D a ( √ s, 0), (a = σ, π) is given by In the above, the masses of the mesons are given by Eqs. (23) and (24) determined by the curvature of the thermodynamic potential. Further, in Eq. (53), ImΠ( √ s, 0) which is related to the width of the resonance as Γ a = ImΠ a /M a is given as [25] ImΠ a (ω, with ǫ a = 0 for pions and ǫ a = 2m for sigma mesons.
With the squared matrix elements for the quark scatterings given as above the transition frequency for the quark of a given species is C. Quark pion scattering and relaxation time Next, we compute the contribution of quark meson scattering to the relaxation times for both mesons as well as quarks. One can argue that the dominant contribution comes from pions as their number is large compared to the sigma mesons both below and above T c . Therefore, in the following we consider the quark-pion scattering only. The Lorentz-invariant scattering matrix element can be written asŪ (p 2 )T ba U (p 1 ), withŪ U = 2m q and with p 1 , p 2 denoting the initial and the final quark momenta, respectively, and q 1 , q 2 , being the momenta of the pions. where, and Averaging over the spin and isospin factors, the matrix element square for the quark-pion scattering is given by The corresponding transition frequency is given by where, In the above p 2 0 = (s + m 2 q − m 2 π ) 2 /(4s) − m 2 q . The scattering will contribute to both the quark relaxation time as well as to the pion relaxation time using Eq. (60) with appropriate modification for the initial state.
Let us note that there are poles in the u channel in the quark pion scattering term beyond the critical temperature when the pion mass become larger than the quark mass. However, this is taken care of once we include the imaginary part of the quark self-energy in the propagators for the quarks in the calculation of the amplitude in Eqs. (57)- (58). The quark self-energy due to scattering with mesons can be written as [31] so that the quark propagators get modified as The imaginary part of the dimensionless functions Σ j , (j = 0, 3, 4),i is given as In the above, E b = E f + p 0 , p 0 = p 2 + m 2 and f ± are the distribution functions for the quarks/antiquarks, f b is the meson distribution functions,and, C j s are the weight factors given as The integration limits are given by Further, the degeneracy factors d 3,4 are 3 for pions and 1 for sigma while d 0 is -3 for pions and 1 for the sigma meson. To calculate the total relaxation time for a quark of species 'a', we compute the total interaction frequency as ω total q (E a ) = ω(E a ) + ω qπ (E a ). One can define an average relaxation time for the quarks similar to Eq. (47) as IV. RESULTS

A. Meson scatterings
Let us first discuss the results arising from meson scattering alone. Using Eqs. (46), with constant |M | 2 as discussed, we have plotted the average relaxation times for the σ-meson and π mesons in Fig. 5. The relaxation times are minimum at the transition temperature. Because of larger mass of σ-mesons below the transition temperature, τ σ is much larger as compared toτ π . They become almost degenerate after the chiral transition as may be expected from the behavior of their masses beyond the transition temperature. We may comment here that the particle with larger relaxation time dominates the viscosities as it can transport energy and momentum to larger distances before interacting. In Fig. 6 we have shown the behavior of the specific viscosities (normalized to entropy density) as a function of temperature. In Fig. 6(a), we have plotted the temperature dependence of the ratio η/s for µ = 0. The behavior of this ratio is essentially determined by the behavior of the relaxation time. Similar to Fig. 5, η/s shows a minimum at the crossover temperature and the value at the minimum is about η/s ∼ 0.053 which is slightly lower than the KSS bound of 1/4π. We note that we have considered here only the contributions from meson scatterings. As we shall see later, inclusion of quark degrees of freedom increases the ratio. We have also compared with linear sigma model calculations [37] in which the quark as well as Polyakov loop contributions are not taken into account. The general behavior of the present calculations is similar to earlier calculations in the sense of having a minimum at the chiral crossover temperature. However, the magnitude of the ratio at the critical temperature is smaller compared to [37]. This is probably due to the fact that, the entropy density in the present calculations has contributions including those of gluon included through the Polyakov loop potential. The large entropy density, we believe, decreases the magnitude of the ratio.
In Fig. 6(b) the ratio of bulk viscosity to entropy is plotted which shows a maximum at the transition temperature. We have also plotted in the same figure the results without quarks and Polyakov loop potential. The present results show a distinct peak structure in the ζ/s ratio at the crossover temperature. Let us note that such a peak is expected as an effect of large conformality violation at the transition temperature as indicated in lattice simulations [29,64]. In Ref. [37], a peak structure is seen for a heavier sigma meson (m σ = 900MeV) which was interpreted as an effect of stronger self-coupling λ for higher M σ . However, in the present case, this arises with quark and polyakov loop degrees of freedom even with a lighter M σ = 600 MeV. The other characteristic feature of the present calculation is that, beyond the critical temperature the ratio ζ/s falls at a slower rate as compared to results of previous calculations. This has to do with the fact that velocity of sound approaches the ideal gas limit slowly as the effect of Polyakov loops on the quark distribution function remains significant beyond the critical temperature. In fact, at the transition temperature the value of the Polyakov loop remains about half its value of the ideal limit. Apart from this, the masses of mesons also get affected by the quark distribution functions significantly beyond the critical temperature. These non ideal effects lead to a slower decrease of the ratio beyond the critical temperature.

B. Quark scatterings
Next, we discuss quark scattering. In Fig. 7 we show the behavior of average relaxation time for quark scattering. The quark scattering through exchange of mesons is shown by the solid line in the figure. Let us recall that the average relaxation time is inversely proportional to the transition rate which is related to the cross section. The dominant contribution here comes from the quark-antiquark scattering from the s channels through propagation of the resonance states, the pions and the sigma mesons. The masses of the sigma meson decrease with temperature, becoming a minimum at the transition temperature, leading to an enhancement of the cross section. Beyond this, the cross section decreases due to the increase in the masses of the mesons. This, in turn, leads to a minimum in the relaxation time.
The average relaxation time for quarks including the quark meson scattering along with the quark scattering is shown as the dashed curve in Fig. 7. This curve lies below the quark quark scattering curve as there is additional contribution to the transition rate from the quark meson scattering. Below the critical temperature, the quark meson scattering dominates over the quark quark scattering due to the smaller mass of the pions as compared to the massive constituent quarks. Beyond the critical temperature, one would have expected the quark meson scattering contribution to be negligible because of the suppression due to the large meson masses. However, as was noted earlier, beyond the critical temperature, there are poles in the scattering amplitude in the u-channel for quark-pion scattering as the pion mass becomes larger than the quark masses. This is, however, regulated by the finite width of the quarks as calculated in Eq. (62). Nonetheless, the contribution of the quark pion scattering to the total quark interaction frequency ω qπ (E) is non-negligible beyond the critical temperature. We next discuss the contribution of different scatterings to the specific shear viscosity η/s. The same is shown in Fig. 8(a) for vanishing chemical potential. The contribution from the mesons to the shear viscosity is arising from the meson scatterings only is shown by the green dashed curve while the effect of including the meson-quark scattering is shown by the maroon dotted curve. Similarly the quark contribution to this ratio η/s arising from quark quark scattering only is shown by the red solid line while the total contributions including the quark-pion scattering is shown by the blue dotted line. This also demonstrates the importance of the scattering of quarks and mesons to the total viscosity coefficient. The total contributions from both the quarks and mesons is shown as the black dashed curve in Fig. 8.
In a similar manner, various contributions to the specific bulk viscosity (ζ/s) coefficient are shown in Fig. 8(b). As may be observed, while no peak structure is seen for this coefficient from the contributions arising from quarks scatterings only, such a structure is seen only when one includes the quark meson scattering. The total effect is shown as black dashed curve in Fig. 8(b).
In Fig. 9, we compare the present results with earlier works on the NJL model. As may be noted, in general, the behavior is similar regarding the shear viscosity-to-entropy ratio. Both NJL as well as the present calculations of the PQM model show the similiar behavior of having a minimum at the transition temperature as in Refs. [24,26]. The results of Ref. [46], on the other hand, show a monotonic decrease with temperature. The bulk viscosity-to-entropy ratio, here however shows a much faster rise as the temperature is lowered below the critical temperature. In fact, both the specific viscosities rise much faster compared to NJL models below the critical temperature in the PQM model considered here. The reason could be due to the fact that the entropy density for PQM model is smaller compared to NJL models. The Polyakov loop decreases as temperature is lowered which leads to a suppression of quark distribution functions leading to decrease of entropy density at a faster rate as compared to NJL model. Moreover, within the present approximation pions do not contribute to the thermodynamics here. Further, for temerature larger than the critical temperature, the bulk viscosity vanishes slowly with increase in temperature as compared to NJL model. This is due to the fact that the Polyakov loop variable takes its asymptotic values only at very high temperatures.
Next, we discuss about effect of finite chemical potential on the transport coefficients. To begin with let us note that the average relaxation timeτ a as in Eq. (67) depends both on the transition rate and the density of the particles in the initial state. To this end, let us discuss the case of T>T c . Here, the quark densities are larger than those of antiquarks. Further, the dominant contribution in this range of temperatures arises from ud → ud scatterings. As there are fewer antiquarks to scatter off, the average transition frequency of quark-antiquark scattering decreases. This leads toτ q (µ) >τ q (µ = 0). On the other hand, for the antiquarks, there are more quarks to scatter off than compared to the case of µ = 0. Hence, this leads toτq(µ) <τq(µ = 0). This expected behavior is seen in Fig. 10. Next, let us consider the case T<T c . In this case, the antiquark density is heavily suppressed due to constituent quark mass and the chemical potential and dominant contribution for quark relaxation time, therefore arises from quark-quark scatterings. This leads toτ q (µ) <τ q (µ = 0). On the other hand, for the antiquarks, though their number density is smaller, their interaction frequency is enhanced both by the larger amplitude for M ud→ud scattering and the larger number of quarks as compared to case at µ=0. This leads toτq(µ) <τ q (µ = 0) <τq(µ = 0). This general behavior is reflected in the average relaxation time dependence on T in Fig. 10 below the critical temperature.
In Fig. 11, we have shown the results for the viscosities at µ = 100 MeV. Fig. 11 (a) shows the variation of the specific shear viscosity (η/s) as a function of temperature for zero and finite chemical potential. The behavior of shear viscosity essentially follows that of the behavior of the relaxation time. η/s has a minimum at the critical temperature with η/s| min ∼ 0.23 (µ = 0) due to suppression of the scattering cross section at higher temperature. At finite µ, the ratio is little higher as compared to the value at vanishing µ. This is due to two reasons. Firstly, the relaxation time at nonzero chemical potential is larger and, moreover, the quark density also becomes larger at finite chemical potential. At temperatures below the critical temperature and near the critical temperature,η/s(µ) < η/s(µ = 0) as the relaxation time is lower. However, at lower temperatures, the meson scattering becomes significant and η/s for finite chemical potential becomes similar to that at vanishing chemical potential as is observed in the figure. In Fig. 11(b), we have plotted the bulk viscosity-to-entropy ratio for µ = 0 MeV and µ = 100 MeV. It turns out that at finite µ the specific bulk viscosity is smaller than the value at µ = 0 MeV. The reason for it is the fact that the dominating contribution to the finite µ arise from the term M 2 − T M dM dT − µM dM dµ in the expression for ζ/s in Eq. (35). This is due to the sharp variations of the order parameters at finite chemical potential as may be observed in Fig. 3. As this term contributes negatively to the expression for ζ, the specific bulk viscosity at finite µ is lower than that at µ = 0 MeV.
In Fig. 12 , we have shown the results for thermal conductivity. We have plotted here the dimensionless quantity λ/T 2 as a function of temperature. We have plotted the results for µ = 100MeV. As is well known, thermal conduction which involves the relative flow of energy and baryon number vanishes at zero baryon density. In fact, λ diverges as 1/n 2 as may be expected from the expression given in Eq. (36). However, in the dissipative current, the conductivity occurs as λn 2 [61,62] and the heat conduction vanishes for µ = 0 [63]. On the other hand, in some cases, such as when pion number is conserved, heat conduction can be sustained by pions. In presece of a pionic chemical potential corresponding to a conserved pion number, thermal conductivity can be nonzero at vanishing baryonic chemical potential. This has been the basis for estimation of thermal conductivity at zero baryon density but finite pion density [32,39,47]. However, in the present case, we consider the case of vanishing pion chemical potential and show only the contribution of quarks to thermal conductivity.
As expected from the behavior of the relaxation time, the specific thermal conductivity has a minimum at the critical temperature similar to Ref. [26] for the NJL model. The sharp rise of λ/T 2 can be understood by performing a dimensional argument to show that at very high temperature when chiral symmetry is is restored the integral increases as T 3 while the prefactor w/(nT ) grows as T 2 for small chemical potentials. Apart from this kinematic consideration, the integrand further is multiplied by τ (E) which itself is an increasing function of temperature beyond T c . This leads to the sharp rise of the ratio λ/T 2 beyond the critical temperature. Below, the critical temperature, however, the ratio decreases which is in contrast to NJL results of Ref. [26]. The reason is twofold. First, the The results of Bulk viscosity to entropy ratio compared with other results in NJL models. The notation is similar to of (a). magnitude of the relaxation time decreases when quark meson scattering is included as compared to quark-quark scattering as shown in Fig. 7. Apart from this, in the integrand,the distribution functions are suppressed by Polyakov loops as compared to NJL model. As the antiquark densities are suppressed compared to quark densities at finite chemical potential, the high-temperature behavior is decided by the quark-quark scattering.

SUMMARY
Transport coefficients of hot and dense matter are important inputs for the hydrodynamic evolution of the plasma that is produced following a heavy ion collision. In the present study, we have investigated these cofficients taking into account the the nonperturbative effects related to chiral symmetry breaking as well as confinement properties of strong interaction physics within an effective model, the Polyakov loop extended quark meson coupling model. These coefficients are estimated using the relaxation time approximation for the solutions of the Boltzman kinetic equation.
We first calculated the medium-dependent masses of the mesons and quarks within a mean field approximation. The contribution of the mesons to the transport coefficients has been calculated through estimating the relaxation time for the mesons arising both from meson-meson scattering and meson-quark scattering. The contribution to the transport coefficients arises mostly from the meson scatterings at temperatures below the critical temperature, while above the critical temperature, the contributions arising from the quark scatterings become dominant. In particular, quark-meson scattering contributes significantly to the relaxation time for the quarks both below and above the critical temperature. The quark-pion scattering above the critical temperature gives significant contribution due to the pole structure of the corresponding scattering amplitude.
One important approximation in the present analysis is that the kinetic terms for the mesons are not modified at finite temperature and meson dispersion relation remains similar to those at the zero-temperature relativistic dispersion relation. The only temperature effect that remains in the meson dispersion lies in the temperature-dependent meson masses obtained through the curvature of the effective potential [54]. A more realistic approach would be to use effective field theory to have different dispersion relations for the mesons [59] depending upon their velocities and calculate the scattering processes to estimate the viscosities. However, such an approach is beyond the scope of present work in which we have restricted ourselves to thermal and density effects included in the masses and widths for the mesons.
In general, the effect of Polyakov loops lies in suppressing the quark contribution below the critical temperature. This leads to, in particular, the suppression of thermal conductivity at lower temperature arising from quark scattering. The effect of Polyakov loop also is significant near and above the critical temperature. Indeed, both the quark masses as well as Polyakov loop order parameter remain significantly different from their asymptotic values near the critical temperature. It will be interesting to examine the consequences of such nonperturbative features on the transport coefficients of heavy quarks as well as on the collective modes of QGP above and near the critical temperature. Some of these works are in progress and will be reported elsewhere.