Renormalization of the Higgs model: Minimizers, propagators and the stability of mean field theory

Abstract

We study the effective actions S ( k ) obtained by k iterations of a renormalization transformation of the U(1) Higgs model in d =2 or 3 spacetime dimensions. We identify a quadratic approximation S Q ( k ) to S ( k ) which we call mean field theory, and which will serve as the starting point for a convergent expansion of the Green's functions, uniformly in the lattice spacing. Here we show how the approximations S Q ( k ) arise and how to handle gauge fixing, necessary for the analysis of the continuum limit. We also establish stability bounds on S Q ( k ) , uniformly in k . This is an essential step toward proving the existence of a gap in the mass spectrum and exponential decay of gauge invariant correlations.