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: $I(p,a)=f(p)\mathrm{log}(aM)+g(p)+h(p,M)$ up to terms which vanish for lattice spacing $a\ensuremath{\rightarrow}0$. 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.