The Master Ward Identity for scalar QED

TL;DR

This paper demonstrates that the Master Ward Identity (MWI) for scalar QED, which accounts for derivative couplings, can be consistently fulfilled through finite renormalizations, offering advantages over the naive Ward Identity.

Contribution

It rigorously relates the naive Ward Identity to the MWI in scalar QED and shows how to implement the MWI in all orders of perturbation theory via renormalization.

Findings

The MWI is related to the naive WI by finite renormalization.

The MWI can be fulfilled at all perturbative orders.

The MWI has advantages in proof and formulation.

Abstract

It is emphasized that for interactions with derivative couplings, the Ward Identity (WI) securing the preservation of a global U(1) symmetry should be modified. Scalar QED is taken as an explicit example. More precisely, it is rigorously shown in scalar QED that the naive WI and the improved Ward Identity ("Master Ward Identity", MWI) are related to each other by a finite renormalization of the time-ordered product ("T-product") for the derivative fields; and we point out that the MWI has advantages over the naive WI - in particular with regard to the proof of the MWI. We show that the MWI can be fulfilled in all orders of perturbation theory by an appropriate renormalization of the T-product, without conflict with other standard renormalization conditions. Relations with other recent formulations of the MWI are established.