General properties of logarithmically divergent one-loop lattice Feynman integrals

TL;DR

This paper proves that one-loop lattice Feynman integrals with logarithmic divergence have a universal form involving a logarithmic term and a finite part, crucial for understanding lattice QCD renormalization without infrared issues.

Contribution

It establishes the universal structure of logarithmically divergent one-loop lattice integrals without using Taylor expansion, simplifying previous proofs and ensuring universality in lattice QCD.

Findings

The coefficient f(p) matches the continuum integral result.

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

The approach avoids infrared divergences by not relying on momentum expansion.

Abstract

We prove that logarithmically divergent one-loop lattice Feynman integrals have the general form I(p,a) = f(p)log(aM)+g(p,M) up to terms which vanish for lattice spacing a -> 0. Here p denotes collectively the external momenta and M is an arbitrary mass scale. The f(p) is shown to be universal and to coincide with the analogous quantity in the corresponding continuum integral (regularized, e.g., by momentum cut-off). This is essential for universality of the lattice QCD beta-function and anomalous dimensions of renormalized lattice operators at one loop. The result and argument presented here are simplified versions of ones given in arXiv:0709.0781. A noteworthy feature of the argument here is that it does not involve Taylor expansion in external momenta, hence infra-red divergences associated with that expansion do not arise.