Effective action and cluster properties of the abelian Higgs model

We continue our program to establish the Higgs mechanism and mass gap for the abelian Higgs model in two and three dimensions. We develop a multiscale cluster expansion for the high frequency modes of the theory, within a framework of iterated renormalization group transformations. The expansions yield decoupling properties needed for a proof of exponential decay of correlations. The result of this analysis is a gauge invariant unit lattice theory with a deep Higgs potential of the shape required to exhibit the Higgs mechanism.Research partially supported by the National Science Foundation under Grant DMS-8602207 and by the Air Force Office of Scientific Research under Grant AFOSR-86-0229Alfred P. Sloan Research Fellow. Research partially supported by the National Science Foundation under Grants PHY-84-13285 and PHY-85-13554Research partially supported by the National Science Foundation under Grant PHY-85-13554