Generalized Goldstone theorem: Automatic imposition of the Higgs mechanism and application to scale and conformal symmetry breaking

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., if the expectation value of the scalar field is nonzero then the gauge bosons must be 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.