  We introduce a diagrammatic multi-scale approach to the Hubbard model based
on the interaction-irreducible (multi-boson) vertex of a small cluster embedded
in a self-consistent medium. The vertex captures short-ranged correlations up
to the length scale of the cluster, while long-ranged correlations are
recovered from a set of diagrammatic equations for the Hedin three-leg vertex.
By virtue of the crossing symmetry, the Fierz decoupling ambiguity of the
Hubbard interaction is resolved exactly. Our benchmarks for the half-filled
Hubbard model on the square lattice are in very good agreement with numerically
exact diagrammatic Monte Carlo simulations.
