Ward Identity Constraints on Ladder Kernels in Transport Coefficient Calculations

Using diagrammatic methods, we show how the Ward identity can be used to constrain the ladder kernel in transport coefficients calculations. More specifically, we use the Ward identity to determine the necessary diagrams that must be resummed (using the usual integral equation). Our main result is an equation relating the kernel of the integral equation with functional derivatives of the full (imaginary) self-energy; it is similar to what is obtained with 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 (LPM) effect. In this study we restrict our discussion to electrical conductivity and shear viscosity in QED, but our method can in principles be generalized to other transport coefficients and other theories.