  Hedin's equations provide an elegant route to compute the exact one-body
Green's function (or propagator) via the self-consistent iteration of a set of
non-linear equations.
  Its first-order approximation, known as $GW$, corresponds to a resummation of
ring diagrams and has shown to be extremely successful in physics and
chemistry. Systematic improvement is possible, although challenging, via the
introduction of vertex corrections.
  Considering anomalous propagators and an external pairing potential, we
derive a new self-consistent set of closed equations equivalent to the famous
Hedin equations but having as a first-order approximation the particle-particle
(pp) $T$-matrix approximation where one performs a resummation of the ladder
diagrams.
  This pp version of Hedin's equations offers a way to go systematically beyond
the $T$-matrix approximation by accounting for low-order pp vertex corrections.
