Two-particle renormalizations in many-fermion perturbation theory: the importance of the Ward identity

We analyse two-particle renormalizations within the many-fermion perturbation expansion. We show that present diagrammatic theories suffer from a lack of direct diagrammatic control over the physical two-particle functions. To rectify this, we introduce and prove a Ward identity enabling an explicit construction of the self-energy from a given two-particle irreducible vertex. Approximations constructed in this way are causal, obey conservation laws, and offer explicit diagrammatic control of singularities in dynamical two-particle functions.