Shear viscosity in $φ^4$ theory from an extended ladder resummation

We study shear viscosity in weakly coupled hot $\phi^4$ theory using the CTP formalism . We show that the viscosity can be obtained as the integral of a three-point function. Non-perturbative corrections to the bare one-loop result can be obtained by solving a decoupled Schwinger-Dyson type integral equation for this vertex. This integral equation represents the resummation of an infinite series of ladder diagrams which contribute to the leading order result. It can be shown that this integral equation has exactly the same form as the Boltzmann equation. We show that the integral equation for the viscosity can be reexpressed by writing the vertex as a combination of polarization tensors. An expression for this polarization tensor can be obtained by solving another Schwinger-Dyson type integral equation. This procedure results in an expression for the viscosity that represents a non-perturbative resummation of contributions to the viscosity which includes certain non-ladder graphs, as well as the usual ladders. We discuss the motivation for this resummation. We show that these resummations can also be obtained by writing the viscosity as an integral equation involving a single four-point function. Finally, we show that when the viscosity is expressed in terms of a four-point function, it is possible to further extend the set of graphs included in the resummation by treating vertex and propagator corrections self-consistently. We discuss the significance of such a self-consistent resummation and show that the integral equation contains cancellations between vertex and propagator corrections.