Retardation effects and the Coulomb pseudopotential in the theory of superconductivity

In the theory of electron-phonon superconductivity both the magnitude of the electron-phonon coupling lambda and the Coulomb pseudopotential mu* are important to determine the transition temperature T-c and other properties. We calculate corrections to the conventional result for the Coulomb pseudopotential. Our calculations are based on the Hubbard-Holstein model, where electron-electron and electron-phonon interactions are local. We develop a perturbation expansion, which accounts for the important renormalization effects for the electrons, the phonons, and the electron-phonon vertex. We show that retardation effects are still operative for higher order corrections, but less efficient due to a reduction of the effective bandwidth. This can lead to larger values of the pseudopotential and reduced values of T-c. The conclusions from the perturbative calculations are corroborated up to intermediate couplings by comparison with nonperturbative dynamical mean-field results. DOI: 10.1103/PhysRevB.87.054507