  We use a Luttinger-Ward functional approach to study the problem of
phonon-mediated superconductivity in electron systems with strong
electron-electron interactions (EEIs). Our derivation does not rely on an
expansion in skeleton diagrams for the EEI and the resulting theory is
therefore nonperturbative in the strength of the latter. We show that one of
the building blocks of the theory is the irreducible six-leg vertex related to
EEIs. Diagrammatically, this implies five contributions (one of the Fock and
four of the Hartree type) to the electronic self-energy, which, to the best of
our knowledge, have never been discussed in the literature. Our approach is
applicable to (and in fact designed to tackle superconductivity in) strongly
correlated electron systems described by generic lattice models, as long as the
glue for electron pairing is provided by phonons.
