# The three-loop beta function in SU(N) lattice gauge theories

## TL;DR

This paper computes the third coefficient of the lattice beta function in pure Yang-Mills theory, enhancing understanding of scaling behavior and confirming previous results with a novel computational approach.

## Contribution

It introduces a new method for calculating the third coefficient of the lattice beta function, confirming previous results and analyzing its impact on scaling functions.

## Key findings

- Results agree with previous calculations by Lüscher and Weisz.
- The third coefficient significantly affects the lattice scaling function.
- Asymptotic scaling is well achieved in the energy scheme.

## Abstract

We calculate the third coefficient of the lattice $\beta$ function in pure Yang-Mills theory. We make use of a computer code for solving perturbation theory analytically on the lattice. We compute the divergent integrals by using a method based on a Taylor expansion of the integrand in powers of the external momenta in $4 - \epsilon$ dimensions. Our results are in agreement with a previous calculation by M. L\"uscher and P. Weisz where the authors used a different technique. We also show how this new coefficient modifies the scaling function on the lattice in both the standard and energy schemes. In particular we show that asymptotic scaling is extremely well achieved in the energy scheme.