Kubo formulas for the shear and bulk viscosity relaxation times and the scalar field theory shear ${\tau }_{\pi }$ calculation

In this paper we provide a quantum field theoretical study on the shear and bulk relaxation times. First, we find Kubo formulas for the shear and the bulk relaxation times, respectively. They are found by examining response functions of the stress-energy tensor. We use general properties of correlation functions and the gravitational Ward identity to parametrize analytical structures of the Green functions describing both sound and diffusion mode. We find that the hydrodynamic limits of the real parts of the respective energy-momentum tensor correlation functions provide us with the method of computing both the shear and bulk viscosity relaxation times. Next, we calculate the shear viscosity relaxation time using the diagrammatic approach in the Keldysh basis for the massless $\lambda {\varphi }^4$ theory. We derive a respective integral equation which enables us to compute $\eta {\tau }_{\pi }$ and then we extract the shear relaxation time. The relaxation time is shown to be inversely related to the thermal width as it should be.

Figure 1

One-loop diagram of the free scalar field theory representing the contribution to the retarded Green function of the energy-momentum tensor operator.

Figure 2

Resummation of ladder diagrams. The insertions of the energy-momentum tensor operator ${\widehat{T}}^{xy}$ is denoted by the crossed dots and black dots are the vertices with the coupling constant $\lambda$.

Figure 3

Four-point Green function.

Figure 4

Integral equation for the effective vertex.

Figure 5

Shear relaxation time as a function of $m_{\mathrm{th}}/T$. One-loop result $({\tau }_{\pi 0}$, red dashed curve) and multiloop resummation $({\tau }_{\pi }$, blue solid curve) are presented.

Figure 6

The ratio $\langle \epsilon +P\rangle {\tau }_{\pi }/\eta$ as a function of $m_{\mathrm{th}}/T$. Evaluation in the one-loop limit (red dashed curve) and multiloop resummation (blue solid curve) are presented.