BRST Invariance and Renormalisability of the SU(2)$\times$U(1) Electroweak Theory with Massive W Z Bosons

TL;DR

This paper demonstrates that the SU(2)×U(1) electroweak theory with massive W and Z bosons remains renormalisable by extending BRST invariance and quantization techniques used for massive gauge theories, ensuring its consistency.

Contribution

It establishes the renormalisability of the electroweak theory with massive gauge bosons using BRST invariance and constraint quantization, extending previous results for massive SU(n) theories.

Findings

BRST invariance is maintained with mass constraints.

The renormalisability of the massive electroweak theory is proven.

The theory's divergence structure is characterized and shown to be manageable.

Abstract

Since the SU(n) gauge theory with massive gauge bosons has been proven to be renormalisable we reinvestigate the renormalisability of the SU$_L$(2) $\times$ U$_Y$(1) electroweak theory with massive W Z bosons. We expound that with the constraint conditions caused by the W Z mass term and the additional condition chosen by us we can performed the quantization and construct the ghost action in a way similar to that used for the massive SU(n) theory. We also show that when the $\delta-$ functions appearing in the path integral of the Green functions and representing the constraint conditions are rewritten as Fourier integrals with Lagrange multipliers $\lambda_a$ and $\lambda_y$, the BRST invariance is kept in the total effective action consisting of the Lagrange multipliers, ghost fields and the original fields. Furthermore, by comparing with the massless theory and with the massive SU(n) theory we find the general form of the divergent part of the generating functional for the regular vertex functions and prove the renormalisability of the theory. It is also clarified that the renormalisability of the theory with the W Z mass term is ensured by that of the massless theory and the massive SU(n) theory.