Leading-order calculation of electric conductivity in hot quantum electrodynamics from diagrammatic methods

Using diagrammatic methods, we show how the Ward identity can be used to constrain the ladder kernel in transport coefficient calculations. More specifically, we use the Ward identity to determine the necessary diagrams that must be resummed using an integral equation. One of our main results is an equation relating the kernel of the integral equation with functional derivatives of the full self-energy; it is similar to what is obtained with two-particle irreducible (2PI) effective action methods. However, since we use the Ward identity as our starting point, gauge invariance is preserved. Using power counting arguments, we also show which self-energies must be included in the resummation at leading order, including 2 to 2 scatterings and 1 to 2 collinear scatterings with the Landau-Pomeranchuk-Migdal effect. We show that our quantum field theory result is equivalent to the one of Arnold, Moore, and Yaffe obtained using effective kinetic theory. In this paper we restrict our discussion to electrical conductivity in hot QED, but our method can in principle be generalized to other transport coefficients and other theories.