The theory of electron-phonon superconductivity: does retardation really lead to a small Coulomb pseudopotential?

The theory of electron-phonon superconductivity depends on retardation drastically reducing the effects of the strong Coulomb repulsion. The standard theory only treats the lowest order diagram, which is an uncontrolled approximation. We study retardation in the Hubbard-Holstein model in a controlled way using perturbation theory and dynamical mean-field theory. We calculate second order results for the pseudopotential mu* analytically and demonstrate the validity up to intermediate couplings by comparison with non-perturbative results. Retardation effects are still operative, but less efficient, leading to somewhat larger values of mu*. Therefore, our theory can help in the understanding of situations where the standard theory yields overestimates for T-c.