Chiral symmetry breaking in strongly coupled U(1) gauge theory coupled to scalar and fermion matter

We construct a transformation for scalar QED (U(1) gauge theory coupled with a charged scalar field) which leads to a dual effective gauge theory having a perturbation theory expansion in inverse powers of the (electric charge) coupling constant. The dual formulation is applied to the special case of scalar QED coupled to a massless charged fermion. A four-fermion interaction term arises naturally under the duality mapping. As an extension to discussions initiated by Miransky, Bardeen and Kondo and their collaborators in the late 1980's, we use this formulation to explore the possibility of a dynamical breakdown of chiral symmetry for the strongly coupled U(1) gauge theory interacting with both scalar and fermion degrees of freedom. Assuming that scalar QED is realized in the Higgs phase, we examine the Schwinger–Dyson equation approach to the fermion propagator in next-to-leading order of the strong coupling expansion (quenched approximation, i.e. no fermion loops). We construct a consistent renormalization procedure to deal with the ultraviolet infinities. For fixed scalar field vacuum expectation values, we obtain in the plane of the two important parameters of the theory (electric charge and four-fermion coupling constant) critical transition lines to a chirality broken phase.