Maintaining Gauge Symmetry in Renormalizing Chiral Gauge Theories

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.