Massive Spin One and Renormalizable Gauges

Abstract

For many decades of the last century, physicists were struggling to define consistent (renormalizable and unitarity preserving) models for spin-one massive particles (Proca fields). As we know, this was beautifully achieved by Weinberg, Salam and Glashow in 1967 when they proposed an electroweak unified theory which we now call the Standard Model. The electroweak symmetry breaking mechanism, among other things, generates mass terms for the W and Z bosons, while preserving renormalizability and unitarity. The longitudinal degrees of freedom of the massive spin-one particles are given by the Goldostone bosons. Choosing one gauge or another might seem just a matter of convenience and in most cases the unitary gauge is preferred. Here we will show, with an explicit example, that when performing loop calculations the unitary gauge is not really a good choice and that some Green functions are not renormalizable in this gauge. We will also show that working in the so-called renormalizable gauges completely fixes the problem. As it is a non-trivial task we shall also explicitly perform the renormalization of the W boson sector of the Standard Model and check the Ward identities with a simple one-loop example.