Study of the renormalization of BRST invariant local composite operators in the $U(1)$ Higgs model

TL;DR

This paper investigates the one-loop renormalization of BRST invariant local composite operators in the $U(1)$ Higgs model, revealing operator mixing and establishing algebraic methods for calculating renormalization factors.

Contribution

It provides an explicit algebraic renormalization analysis of BRST invariant operators, including their mixing and Ward identities, in the $U(1)$ Higgs model at one-loop order.

Findings

Operators $O$ and $V_mu$ mix with gauge invariant operators.

Two Ward identities determine all renormalization constants.

The framework enables perturbative computation of correlation functions.

Abstract

The renormalization properties of two local BRST invariant composite operators, $(O,V_\mu)$, corresponding respectively to the gauge invariant description of the Higgs particle and of the massive gauge vector boson, are scrutinized in the $U(1)$ Higgs model by means of the algebraic renormalization setup. Their renormalization $Z$'s factors are explicitly evaluated at one-loop order in the $\overline{\text{MS}}$ scheme by taking into due account the mixing with other gauge invariant operators. In particular, it turns out that the operator $V_\mu$ mixes with the gauge invariant quantity $\partial_\nu F_{\mu\nu}$, which has the same quantum numbers, giving rise to a $2 \times 2$ mixing matrix. Moreover, two additional powerful Ward identities exist which enable us to determine the whole set of $Z$'s factors entering the $2 \times 2$ mixing matrix as well as the $Z$ factor of the operator $O$ in a purely algebraic way. An explicit check of these Ward identities is provided. The final setup obtained allows for computing perturbatively the full renormalized result for any $n$-point correlation function of the scalar and vector composite operators.