Conserving approximations for response functions of the Fermi gas in a random potential

One- and two-electron Green functions are simultaneously needed to determine the responsefunctions of the electron gas in a random potential. Reliable approximations must retainconsistency between the two types of Green functions expressed via Ward identities so thattheir output is compliant with macroscopic symmetries and conservation laws. Such aconsistency is not directly guaranteed when summing nonlocal corrections to the local(dynamical) mean field. We analyze the reasons for this failure and show how the full Wardidentity can generically be implemented in the diagrammatic approach to the vertexfunctions without breaking the analytic properties of the self-energy. We use thelow-energy asymptotics of the conserving two-particle vertex determining the singular partof response and correlation functions to derive an exact representation of the diffusionconstant in terms of Green functions of the perturbation theory. We then calculateexplicitly the leading vertex corrections to the mean-field diffusion constant due tomaximally-crossed diagrams.