A phase cell cluster expansion for Φ34

Abstract

We complete the work begun by Battle and Federbush in their approach to the φ34 Euclidean field theory. In their papers, they applied a cluster expansion to a φ34 interaction using a diagonal covariance Cij = δij. In our model we take the usual covariance C = (−Δ + M2) 1. Since this is no longer diagonal we must interpolate this covariance to decouple terms. Complications in our model come from an extra term that arises during this interpolation and also from the couplings in integration-by-parts formulas between variables at different scales. The extra term that arises in interpolating the covariance leads to a chain that must sometimes be attached to different places in the tree structure of the expansion. We find more factorials from the additional attachments created doing this and use some of these to cancel a part of the number divergence that arises from the nondiagonal covariance terms in our integration-by-parts formulas.