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

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-Q2 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 (∝1/ε) of the various diagrams regulated in the dimensional-regularization scheme. Using the explicit results on the renormalization constants, we extract the β and γ functions which govern the behavior of the theory through the RG equation as Q2→∞. We then solve explicitly the coupled RG equations, obtaining the result that the vacuum expectation value v2(Q2) vanishes as Q2→∞ (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