Generalized Goldstone Theorem: Automatic Imposition of the Higgs Mechanism and Application to Scale and Conformal Symmetry Breaking

TL;DR

This paper generalizes Goldstone's theorem to include arbitrary field distributions, automatically deriving the Higgs mechanism and analyzing spontaneous breaking of scale and conformal symmetries.

Contribution

It introduces a generalized Goldstone's theorem that accounts for non-constant fields and spacetime-dependent transformations, unifying the Higgs mechanism and symmetry breaking analysis.

Findings

Automatically imposes the Higgs mechanism for gauge bosons.

Shows the disappearance of Goldstone bosons in gauge theories.

Analyzes spontaneous scale and conformal symmetry breaking.

Abstract

Standard discussions of Goldstone's theorem based on a symmetry of the action assume constant fields and global transformations, i.e., transformations which are independent of spacetime coordinates. By allowing for arbitrary field distributions in a general representation of the symmetry we derive a generalization of the standard Goldstone's theorem. When applied to gauge bosons coupled to scalars with a spontaneously broken symmetry the generalized theorem automatically imposes the Higgs mechanism, i.e. the gauge bosons become massive. The other aspect of the Higgs mechanism, the disappearance of the would-be Goldstone boson, follows directly from the generalized symmetry condition itself. We also use our generalized Goldstone's theorem to analyze the case of a system in which scale and conformal symmetries are both spontaneously broken.