Derivation of spontaneously broken gauge symmetry from the consistency of effective field theory I: Massive vector bosons coupled to a scalar field

TL;DR

This paper explores how local gauge invariance with spontaneous symmetry breaking naturally emerges from the consistency conditions of an effective field theory involving massive vector bosons and a scalar field, extending beyond tree-level unitarity.

Contribution

It demonstrates that imposing constraints like renormalizability and scale separation leads to gauge-invariant Lagrangians with spontaneous symmetry breaking within an effective field theory framework.

Findings

Restrictions on interaction coupling constants derived

Recovery of gauge-invariant Lagrangian in the unitary gauge

Conditions applicable at one-loop order

Abstract

We revisit the problem of deriving local gauge invariance with spontaneous symmetry breaking in the context of an effective field theory. Previous derivations were based on the condition of tree-order unitarity. However, the modern point of view considers the Standard Model as the leading order approximation to an effective field theory. As tree-order unitarity is in any case violated by higher-order terms in an effective field theory, it is instructive to investigate a formalism which can be also applied to analyze higher-order interactions. In the current work we consider an effective field theory of massive vector bosons interacting with a massive scalar field. We impose the conditions of generating the right number of constraints for systems with spin-one particles and perturbative renormalizability as well as the separation of scales at one-loop order. We find that the above conditions impose severe restrictions on the coupling constants of the interaction terms. Except for the strengths of the self-interactions of the scalar field, that can not be determined at this order from the analysis of three- and four-point functions, we recover the gauge-invariant Lagrangian with spontaneous symmetry breaking taken in the unitary gauge as the leading order approximation to an effective field theory. We also outline the additional work that is required to finish this program.