In quantum field theories (to be relativistic, (non-)relativistic statistical or whatever), we have the powerful diagrammatic approach at our disposal. Most of the time we can not sum up all the diagrams, only a sub-class of them. I was thinking of the Hartree-Fock ones to give an example, but they are several classes we can re-sum exactly (skeleton, ladder, ...). Somehow this re-summation of diagrams is exact. I will keep italics to refer to this exactness.
Clearly the solution we find that way is not the full solution of the problem we considered at the beginning. By "problem" I obviously mean the eigenstates/eigenvalues problem of a secular equation given by a Hamiltonian/Lagrangian.
My question is the following: do we know the problem we exactly solved summing only partial class(es) of diagram ? I mean: can we write out an effective Hamiltonian/Lagrangian that our exact summation is the exact solution of ? Can we at least know the exact sub-space (eigenstates and/or eigenvalues), the solutions we found exactly belong to ?