Wilson loops to 20th order numerical stochastic perturbation theory

TL;DR

This paper computes Wilson loops up to 20 loops in SU(3) lattice gauge theory using numerical stochastic perturbation theory, revealing the series' behavior and estimating non-perturbative contributions.

Contribution

It provides high-order perturbative calculations of Wilson loops and introduces new methods to analyze series behavior and non-perturbative effects.

Findings

No factorial growth observed up to 20 loops.

Series can be modeled with hypergeometric functions.

Non-perturbative parts assessed using generalized ratios.

Abstract

We calculate Wilson loops of various sizes up to 20 loops in SU(3) pure lattice gauge theory at different lattice sizes for Wilson gauge action using the technique of numerical stochastic perturbation theory. This allows us to investigate the perturbative series for various Wilson loops at high loop orders. We observe differences in the behavior of those series as function of the loop order. Up to $n=20$ we do not find evidence for the factorial growth of the expansion coefficients often assumed to characterize an asymptotic series. Based on the actually observed behavior we sum the series in a model parametrized by hypergeometric functions. Alternatively we estimate the total series in boosted perturbation theory using information from the first 14 loops. We introduce generalized ratios of Wilson loops of different sizes. Together with the corresponding Wilson loops from standard Monte Carlo measurements they enable us to assess their non-perturbative parts.