Nonadiabatic superconductivity. I. Vertex corrections for the electron-phonon interactions

High-temperature superconductors, including the fullerene compounds, are all characterized by a very small value of the Fermi energy (${\mathit{E}}_{\mathit{F}}$), of the order of the Debye phonon frequency (${\mathrm{\omega }}_{\mathit{D}}$). This implies a breakdown of Migdal’s theorem for the electron-phonon interactions and it requires a generalization of the many-body theory of superconductivity. In this and in the following paper we consider the first steps of this generalization in a perturbative scheme with respect to the parameter (λ${\mathrm{\omega }}_{\mathit{D}}$/${\mathit{E}}_{\mathit{F}}$). Here we discuss in detail the vertex correction function and the self-energy for ${\mathrm{\omega }}_{\mathit{D}}$/${\mathit{E}}_{\mathit{F}}$≠0. The main result is that the vertex function shows a complex behavior with respect to the momentum (q) and frequency (ω) of the exchanged phonon. In particular vertex corrections are positive for small values of q. We discuss that the small q region may be favored by electronic correlations and by density of states effects. In such a situation it is possible to obtain a strong enhancement of ${\mathit{T}}_{\mathit{c}}$ with respect to the usual theory. In addition vertex corrections and other effects due to the breakdown of Migdal’s theorem should have important consequences on other properties like the isotope effect, transport properties, phonon frequencies, tunneling and photoemission data. The essential results of this and the next paper should also apply to cases in which the attractive interaction of the electron pairs is due to a bosonic excitation different from phonons.