Generalized Goldstone Theorem: Automatic Imposition of the Higgs Mechanism and Application to 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.