# Determination of the QCD coupling from the static energy and the free energy

## TL;DR

This paper provides two precise determinations of the strong coupling constant lpha_s at the Z boson mass scale using static energy and free energy lattice data, with detailed error analysis and updated results.

## Contribution

It offers new lattice-based measurements of lpha_s with improved accuracy and shorter distance data, updating previous determinations and including comprehensive error estimates.

## Key findings

- lpha_s(M_Z)=0.11660^{+0.00110}_{-0.00056} from static energy
- lpha_s(M_Z)=0.11638^{+0.00095}_{-0.00087} from free energy data
- Inclusion of new lattice data at smaller spacings and distances

## Abstract

We present two determinations of the strong coupling \(\alpha_s\). The first one is from the static energy at three-loop accuracy, and may be considered an update of earlier determinations by some of us. The new analysis includes new lattice data at smaller lattice spacings, and reaches distances as short as \(0.0237\,{\rm fm}\). We present a comprehensive and detailed estimate of the error sources that contribute to the uncertainty of the final result, $\alpha_s(M_Z)=0.11660^{+0.00110}_{-0.00056}$. The second determination is based on lattice data for the singlet free energy at finite temperature up to distances as small as \(0.0081\,{\rm fm}\), from which we obtain $\alpha_s(M_Z)=0.11638^{+0.0009 5}_{-0.00087}$.