Power Counting and Renormalization in Lattice Field Theory

Abstract

In an Euclidean quantum field theory a space-time lattice provides a quite natural non-perturbative ultraviolet cutoff. This has lead to a variety of non-perturbative methods. Observable physics has to be searched for in the continuum limit. However, this limit is very difficult to be treated. Very often one is forced to use perturbative methods, and this leads to well known diagrammatic expansions. I will briefly outline the very specific structure of momentum space Feynman integrals with a lattice cutoff and discuss the continuum limit behavior. The well known power counting theorems of Weinberg [l] and Hahn, Zimmermann [2] which state sufficient conditions for the convergence of Feynman integrals do not apply in presence of a lattice cutoff. Nevertheless, a power counting theorem can be given for a wide class of lattice field theories where a new kind of an ultraviolet divergence degree is used [4].