Large-Q2 behavior of the Higgs mechanism as derived from one-loop diagrams and the renormalization-group equation.

Abstract

The renormalization-group (RG) approach generalized by Lee and Weisberger to the case of gauge theories with spontaneous symmetry breaking is adopted in order to study the large-${Q}^{2}$ behavior of the Higgs mechanism. The approach amounts to a generalization of the well-known concept of "running coupling constants" to all the couplings involved in the theory. To simplify our consideration, we have considered scalar electrodynamics (QED) in which a U(1) gauge field interacts with both fermions and scalar fields (the latter responsible for spontaneous symmetry breaking). We first carry out a perturbative calculation including all one-loop diagrams with the explicit results on the various renormalization constants obtained by separating out pole parts ($\ensuremath{\propto}\frac{1}{\ensuremath{\epsilon}}$) of the various diagrams regulated in the dimensional-regularization scheme. Using the explicit results on the renormalization constants, we extract the $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ functions which govern the behavior of the theory through the RG equation as ${Q}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$. We then solve explicitly the coupled RG equations, obtaining the result that the vacuum expectation value ${v}^{2}({Q}^{2})$ vanishes as ${Q}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ (i.e., the broken symmetry is restored) but the gauge sector (consisting of the gauge field and fermions) does not decouple from the strongly self-interacting Higgs sector. In the case of non-Abelian gauge theories, we suggest looking for theories which allow for an eventual decoupling between the strongly interacting Higgs sector and the asymptotically free gauge sector.