Maintaining Gauge Symmetry in Renormalizing Chiral Gauge Theories

TL;DR

This paper introduces a simple method using the rightmost gamma_5 scheme in dimensional regularization to efficiently compute finite counter terms needed for gauge symmetry restoration in chiral gauge theories at 1-loop order.

Contribution

It presents a straightforward approach to determine finite counter terms in dimensional regularization by moving gamma_5 to the rightmost position before continuation, simplifying gauge symmetry restoration.

Findings

The method simplifies the calculation of finite counter terms at 1-loop.

Differences between schemes match the required finite counter terms for gauge invariance.

Applicable to both Abelian and non-Abelian chiral gauge theories.

Abstract

It is known that the $\gamma_{5}$ scheme of Breitenlohner and Maison (BM) in dimensional regularization requires finite counter-term renormalization to restore gauge symmetry and implementing this finite renormalization in practical calculation is a daunting task even at 1-loop order. In this paper, we show that there is a simple and straightforward method to obtain these finite counter terms by using the rightmost $\gamma_{5}$ scheme in which we move all the $\gamma_{5}$ matrices to the rightmost position before analytically continuing the dimension. For any 1-loop Feynman diagram, the difference between the amplitude regularized in the rightmost $\gamma_{5}$ scheme and the amplitude regularized in the BM scheme can be easily calculated. The differences for all 1-loop diagrams in the chiral Abelian-Higgs gauge theory and in the chiral non-Abelian gauge theory are shown to be the same as the amplitudes due to the finite counter terms that are required to restore gauge symmetry.