  Within many-body perturbation theory, Hedin's formalism offers a systematic
way to iteratively compute the self-energy $\Sigma$ of any interacting system,
provided one can evaluate the interaction vertex $\Gamma$ exactly. This is
however impossible in general, for it involves the functional derivative of
$\Sigma$ with respect to the Green's function. Here, we analyze the structure
of this derivative, splitting it into four contributions and outlining the type
of quasiparticle interactions that each of them generate. Moreover we show how,
in the implementation of self-consistency, the action of these contributions
can be classified into two: a quantitative renormalization of previously
included interaction terms, and the inclusion of qualitatively novel
interaction terms through successive functional derivatives of $\Gamma$ itself,
neglected until now. Implementing this latter type of self-consistency can
extend the validity of Hedin's equations towards the high interaction limit, as
we show in the example of the Hubbard dimer. Our analysis also provides a
unifying perspective on the perturbation theory landscape, showing how the
T-matrix approach is completely contained in Hedin's formalism.
