Structure of logarithmically divergent one-loop lattice Feynman integrals

TL;DR

This paper proves the universal structure of logarithmically divergent one-loop lattice Feynman integrals, establishing their relation to continuum integrals and ensuring consistency in lattice QCD renormalization.

Contribution

It demonstrates the universal form of one-loop lattice integrals and their correspondence with continuum regularization methods, crucial for lattice QCD consistency.

Findings

I(p,a) = f(p)log(aM)+g(p)+h(p,M) up to vanishing terms as a -> 0

f(p) and h(p,M) are universal and match continuum integrals with regularization

g(p) is a homogeneous polynomial of the same degree as f(p)

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 cut-off or dimensional regularization. The non-universal 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.