Structure of logarithmically divergent one-loop lattice Feynman integrals

Abstract

For logarithmically divergent one-loop lattice Feynman integrals I(p,a) , subject to mild general conditions, we prove the following expected and crucial structural result: I(p,a)=f(p)log(aM)+g(p)+h(p,M) up to terms which vanish for lattice spacing a→0 . Here p denotes collectively the external momenta and M is a mass scale which may be chosen arbitrarily. The f(p) and h(p,M) are shown to be universal and coincide with analogous quantities in the corresponding continuum integral when the latter is regularized either by momentum cutoff or dimensional regularization. The nonuniversal term g(p) is shown to be a homogeneous polynomial in p of the same degree as f(p) . This structure is essential for consistency between renormalized lattice and continuum formulations of QCD at one loop.