Definition of the Electromagnetic Field in the Broken-Symmetry Phase of the Electroweak Theory

Abstract

In the broken-symmetry phase of the electroweak theory there is no unique definition of the electromagnetic field tensor in cases where the magnitude of the Higgs field differs from a constant value. The meaning of the electromagnetic field is therefore dubious near defects and during non-equilibrium stages of the electroweak phase transition. Nevertheless, by imposing a minimal set of natural requirements one is led to a specific, gauge-invariant definition that retains the familiar properties of an electromagnetic field. An electromagnetic vector potential is constructed whose curl (exterior derivative) in any gauge gives the electromagnetic field tensor. As is required, this vector potential transforms at most by a pure gradient under arbitrary SU(2)xU(1) gauge transformations. The flux of the magnetic field is expressed as a gauge-invariant line integral. Curiously, this provides a definition for magnetic flux in cases where the spatial region with broken symmetry is not simply connected and the magnetic field itself is not everywhere defined.