# Algebraic algorithm for the computation of one-loop Feynman diagrams in lattice QCD with Wilson fermions

## TL;DR

This paper introduces an algebraic algorithm that simplifies the calculation of one-loop Feynman diagrams in lattice QCD with Wilson fermions, enabling high-precision computation of integrals and extending to higher loops.

## Contribution

The paper presents a novel algebraic method to express one-loop lattice integrals in terms of basic constants, applicable to any space dimension and useful for higher-loop calculations.

## Key findings

- Expresses one-loop integrals with basic constants
- Provides a method for lattice propagator in coordinate space
- Enables application of higher-loop techniques to fermions

## Abstract

We describe an algebraic algorithm which allows to express every one-loop lattice integral with gluon or Wilson-fermion propagators in terms of a small number of basic constants which can be computed with arbitrary high precision. Although the presentation is restricted to four dimensions the technique can be generalized to every space dimension. Various examples are given, including the one-loop self-energies of the quarks and gluons and the renormalization constants for some dimension-three and dimension-four lattice operators. We also give a method to express the lattice free propagator for Wilson fermions in coordinate space as a linear function of its values in eight points near the origin. This is an essential step in order to apply the recent methods of L\"{u}scher and Weisz to higher-loop integrals with fermions.