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

We study the effective actionsS ^(k) obtained byk iterations of a renormalization transformation of the U(1) Higgs model ind=2 or 3 spacetime dimensions. We identify a quadratic approximationS _Q ^(k) toS ^(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 approximationsS _Q ^(k) arise and how to handle gauge fixing, necessary for the analysis of the continuum limit. We also establish stability bounds onS _Q ^(k) , uniformly ink. This is an essential step toward proving the existence of a gap in the mass spectrum and exponential decay of gauge invariant correlations.Dedicated to the memory of Kurt SymanzikSupported in part by the National Science Foundation under Grant PHY 82-03669