The steps essentially involved in the evaluation of transport coefficients in linear response theory using Kubo formulas are to relate the defining

One of the most interesting results from experiments at the Relativistic Heavy Ion Collider (RHIC) is the surprisingly large magnitude of the elliptic flow of the emitted hadrons. Viscous hydrodynamic simulations of heavy ion collisions require a rather small value of

This finding has led to widespread interest in the study of nonequilibrium dynamics, especially in the microscopic evaluation of the transport coefficients of both partonic as well as hadronic forms of strongly interacting matter. In the literature one comes across basically two approaches that have been used to determine these quantities. One is the kinetic theory method in which the nonequilibrium distribution function which appears in the transport equation is expanded in terms of the gradients of the flow velocity field. The coefficients of this expansion which are related to the transport coefficients are then perturbatively determined using various approximation methods. The other approach is based on response theory in which the nonequilibrium transport coefficients are related by Kubo formulas to equilibrium correlation functions. They are then perturbatively evaluated using the techniques of thermal field theory. Alternatively, the Kubo formulas can be directly evaluated on the lattice [

Thermal quantum field theory has been formulated in the imaginary as well as real-time [

A difficulty with the real-time formulation is, however, that all two-point functions take the form of

In the literature transport coefficients are evaluated using the imaginary time formulation [

We also calculate the viscous coefficients in a kinetic theory framework by solving the transport equation in the Chapman-Enskog approximation to the leading order. This approach being computationally more efficient [

In Section

In this section we review the real-time formulation of equilibrium thermal field theory leading to the spectral representations of bosonic two-point functions [

The contour

Let a general bosonic interacting field in the Heisenberg representation be denoted by

The thermal expectation value of the product

In just the same way, we can work out the Fourier transform of

We next introduce the

With the previous ingredients, we can build the spectral representations for the two types of thermal propagators. First consider the

As

Back to real-time, we can work out the usual temporal Fourier transform of the components of the matrix to get

The matrix

Looking back at the spectral functions

We next consider the

We now use the linear response approach to arrive at expressions of the transport coefficients as integrals of retarded Green’s functions over space. We follow the method proposed by Zubarev [

The expression (

In order to expand

Applying this formula to the energy-momentum tensor, we get its response to the thermodynamical forces as [

We now use (

Again, starting with the pressure

Recall that

Clearly the spectral forms and their interrelations derived in Section

The first term in the so-called skeleton expansion of the two-point function. Heavy lines denote full propagators.

To work out the

Having obtained the real and imaginary parts of

Returning to the expression (

As shown in [

It turns out that the integral over

Then

Proceeding analogously as mentioned above, the lowest order contribution to the bulk viscosity can be obtained as [

The width

The kinetic theory approach is suitable for studying transport properties of dilute systems. Here one assumes that the system is characterized by a distribution function which gives the phase space probability density of the particles making up the fluid. Except during collisions, these (on-shell) particles are assumed to propagate classically with well-defined position, momenta, and energy. It is possible to obtain the nonequilibrium distribution function by solving the transport equation in the hydrodynamic regime by expanding the distribution function in a local equilibrium part along with nonequilibrium corrections. This expansion in terms of gradients of the velocity field is used to linearize the transport equation. The coefficients of expansion which are related to the transport coefficients satisfy linear integral equations. The standard method of solution involves the use of polynomial functions to reduce these integral equations to algebraic ones.

The evolution of the phase space distribution of the pions is governed by the (transport) equation:

For small deviation from local equilibrium, we write, in the first Chapman-Enskog approximation,

and the functions

Note that the differential cross-section which appears in the denominator is the dynamical input in the expressions for

The strong interaction dynamics of the pions enters the collision integrals through the cross-section. In Figure

The

To get a handle on the dynamics, we now evaluate the

The integrated cross-section, after ignoring the

After this normalisation to data, we now turn to the in-medium cross-section by introducing the effective propagator for the

As discussed in Section

The in-medium cross-section is now obtained by using the full

We plot

The shear viscosity as a function of temperature in the Chapman-Enskog approximation. The dash-dotted line indicates use of the vacuum cross-section, and the dashed and solid lines correspond to in-medium cross-section for the

We noted in Section

The shear viscosity as a function of temperature in the relaxation time approximation. The dash-dotted and solid lines correspond to the use of in-medium cross-sections in (

In Figure

The bulk viscosity as a function of temperature for a chemically frozen pion gas. The dashed and solid lines correspond to the use of in-medium cross-sections in (

To summarize, we have calculated the shear viscosity coefficient of a pion gas in the real-time version of thermal field theory. It is simpler to the imaginary version in that we do not have to continue to imaginary time at any stage of the calculation. As an element in the theory of linear response, a transport coefficient is defined in terms of a retarded thermal two-point function of the components of the energy-momentum tensor. We derive Källen-Lehmann representation for any (bosonic) two-point function of both time-ordered and retarded types to get the relation between them. Once this relation is obtained, we can calculate the retarded function in the Feynman-Dyson framework of the perturbation theory.

Clearly the method is not restricted to transport coefficients. Any linear response leads to a retarded two-point function, which can be calculated in this way. Also quadratic response formulae have been derived in the real-time formulation [

We have also evaluated the viscous coefficients in the kinetic theory approach to leading order in the Chapman-Enskog expansion. Here we have incorporated an in-medium

The viscous coefficients and their temperature dependence could affect the quantitative estimates of signals of heavy ion collisions particularly where hydrodynamic simulations are involved. For example, it has been argued in [

The author gratefully acknowledges the contribution from his collaborators S. Mallik, S. Ghosh, and S. Mitra to various topics presented here.