Gauge Invariant Description of the $SU(2)$ Higgs model: Ward identities and Renormalization

TL;DR

This paper investigates the renormalization and symmetry properties of gauge-invariant operators in the $SU(2)$ Higgs model, revealing new Ward identities and confirming non-renormalization theorems in a non-Abelian context.

Contribution

It generalizes previous $U(1)$ Higgs model results to the $SU(2)$ case, deriving a new Ward identity related to custodial symmetry and analyzing the renormalization of gauge-invariant operators.

Findings

Gauge-invariant vector operators are conserved Noether currents.

These operators do not undergo renormalization at quantum level.

The ghost-antighost-vector boson vertex non-renormalization theorem holds in the Higgs context.

Abstract

The renormalization properties of two local gauge invariant composite operators $(O,R^a_{\mu})$ corresponding, respectively, to the gauge invariant description of the Higgs particle and of the massive gauge vector boson, are analyzed to all orders in perturbation theory by means of the algebraic renormalization in the $SU(2)$ Higgs model, with a single scalar in the fundamental representation, when quantized in the Landau gauge in Euclidean space-time. The present analysis generalizes earlier results presented in the case of the $U(1)$ Higgs model. A powerful global Ward identity, related to an exact custodial symmetry, is derived for the first time, with deep consequences at the quantum level. In particular, the gauge invariant vector operators $R^a_{\mu}$ turn out to be the conserved Noether currents of the above-mentioned custodial symmetry. As such, these composite operators do not renormalize, as expressed by the fact that the renormalization $Z$-factors of the corresponding external sources, needed to define the operators $R^a_{\mu}$ at the quantum level, do not receive any quantum corrections. Another consistency feature of our analysis is that the longitudinal component of the two-point correlation function $\langle R^a_\mu(p) R^b_\nu(-p) \rangle$ exhibits only a tree level non-vanishing contribution which, moreover, is momentum independent, being thus not associated to any physical propagating mode. Finally, we point out that the renowned non-renormalization theorem for the ghost-antighost-vector boson vertex in Landau gauge remains true to all orders, also in presence of the Higgs field.