Eliashberg equations of electrons on a square lattice: Interplay of strong electron-phonon interactions and electron correlations

We discuss vertex corrections to the Eliashberg equations of correlated electrons due to strong electron-phonon interactions. Local electron correlations have been taken into account in the mean-field approximation for slave-boson fields in the U-->infinity limit of the two-dimensional Hubbard model. The modification of the wave-function renormalization factor originating from vertex corrections has been evaluated in a self-consistent way. We have considered both ionic and covalent parts of the electron-phonon interaction as possible pairing mechanisms. For the ionic contribution at low doping (strong-coupling limit), inclusion of vertex corrections leads to a decrease of the superconducting transition temperature T-c for d-wave symmetry. This decrease agrees with the experimentally observed dependence of T-c on the concentration of holes and originates from the enhancement of the density of states near the van Hove singularities due to correlations. Another important effect originating from inclusion of vertex corrections is a pronounced reduction of the isotope shift exponent alpha at optimal doping. This supports the view that experimentally observed small values of alpha still allow for phonon-induced superconductivity in strongly correlated systems. We discuss the controversial question with respect to the sign with which the vertex corrections contribute to the pairing kernels of the Eliashberg equations of strongly correlated electron-phonon systems.