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

D Djukanovic,J Gegelia,Ulf-G Meißner

doi:10.1016/j.physletb.2018.09.016

D Djukanovic, J Gegelia + Show 1 more

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https://doi.org/10.1016/j.physletb.2018.09.016

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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.

Highlights

The standard model (SM) is widely accepted as the established consistent theory of the strong, electromagnetic and weak interactions [1]
In the current work we revisited the problem of the uniqueness of a theory with spontaneously broken gauge symmetry as a consistent framework for describing the electroweak interactions
Following the modern point of view of the Standard Model being the leading order approximation of an effective field theory we analyzed the most general Lorentz-invariant leading order effective Lagrangian of massive vector bosons interacting with a massive scalar field

Summary

Introduction

The standard model (SM) is widely accepted as the established consistent theory of the strong, electromagnetic and weak interactions [1]. [2,3,4,5] This result could be considered as a (more or less) satisfactory answer to the above raised question, the modern point of view considers the SM as an effective field theory (EFT) [1] which inevitably violates the tree-order unitarity condition at sufficiently high energies. The Lagrangian of an EFT consists of an infinite number of terms, the contributions of non-renormalizable interactions in physical quantities are suppressed for energies much lower than some large scale. In what follows we analyze the constraint structure and the conditions of perturbative renormalizability and scale separation for the most general Lorentz-invariant effective Lagrangian of massive vector bosons interacting with a scalar field.

Starting assumptions and required constraints

Perturbative renormalizability

Summary and discussions