On the fulfillment of Ward identities in the functional renormalization group approach

TL;DR

This paper examines how well Ward identities are maintained in the functional renormalization group approach, showing that accuracy depends on the loop order and proposing a self-consistent two-loop scheme that better preserves conservation laws.

Contribution

It introduces a self-consistent two-loop RG scheme that improves the fulfillment of Ward identities, especially in the ladder approximation, reducing errors from neglected terms.

Findings

Ward identities are fulfilled within the order of neglected two-loop terms.

The proposed self-consistent scheme exactly satisfies Ward identities in the ladder approximation.

Errors in Ward identities decrease with the improved two-loop RG equations.

Abstract

I consider the fulfillment of conservation laws and Ward identities in the one- and two-loop functional renormalization group approach. It is shown that in a one-particle irreducible scheme of this approach Ward identities are fulfilled only with the accuracy of the neglected two-loop terms O(V^3) at one-loop order, and with the accuracy O(V^4) at two-loop order (V is the effective interaction vertex at scale \Lambda). The one-particle self-consistent version of the two-loop RG equations which leads to smaller errors in Ward identities due to the absence of the terms with non-overlapping loops, is proposed. In particular, these modified equations exactly satisfy Ward identities in the ladder approximation.