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

TL;DR

This paper introduces a Ward identity for many-fermion perturbation theory that enables explicit, causal, and conserving approximations with diagrammatic control over two-particle functions.

Contribution

It presents a new Ward identity that allows for direct construction of self-energy from two-particle vertices, improving diagrammatic control and physical consistency.

Findings

The Ward identity ensures causality and conservation laws in approximations.

Explicit construction of self-energy from two-particle vertices is possible.

The approach controls singularities in dynamical two-particle functions.

Abstract

We analyze two-particle renormalizations within many-fermion perturbation expansion. We show that present diagrammatic theories suffer from lack of a 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 an explicit diagrammatic control of singularities in dynamical two-particle functions.