  Coupled-cluster and Green's function theories are highly successful in
treating many-body electron correlation and there has been significant interest
in identifying and leveraging connections between them. Here we present a
diagrammatic definition of the irreducible coupled-cluster self-energy that
directly embeds coupled-cluster theory within the framework of many-body field
theory. The EOM-CC treatment emerges naturally from our definition via the
Dyson and Bethe-Salpeter equations, providing a unified description of RPA,
$GW$-BSE and CC theory for ground state and excitation energies. This clarifies
the origin of previously established connections between RPA, $GW$-BSE and
coupled-cluster theory, and exposes the relationship between vertex corrections
and the coupled-cluster amplitude equations.
