# The strong coupling from a nonperturbative determination of the $\Lambda$ parameter in three-flavor QCD

## TL;DR

This paper nonperturbatively determines the QCD $
m{ extLambda}$ parameter and strong coupling at the Z boson mass by lattice calculations, ensuring controlled systematic effects across a wide energy range.

## Contribution

It provides a precise nonperturbative lattice calculation of the $
m{ extLambda}$ parameter in three-flavor QCD and the strong coupling at the Z mass, improving the understanding of QCD at high energies.

## Key findings

- $
m{ extLambda}_{
m{ar{MS}}}^{(3)}=341(12)$ MeV
- $
m{ extalpha}_{
m{ar{MS}}}^{(5)}(m_Z)=0.11852(84)$
- Nonperturbative running reduces systematic uncertainties

## Abstract

We present a lattice determination of the $\Lambda$ parameter in three-flavor QCD and the strong coupling at the Z pole mass. Computing the nonperturbative running of the coupling in the range from $0.2\,$GeV to $70\,$GeV, and using experimental input values for the masses and decay constants of the pion and the kaon, we obtain $\Lambda_{\overline{\rm MS}}^{(3)}=341(12)\,$MeV. The nonperturbative running up to very high energies guarantees that systematic effects associated with perturbation theory are well under control. Using the four-loop prediction for $\Lambda_{\overline{\rm MS}}^{(5)}/\Lambda_{\overline{\rm MS}}^{(3)}$ yields $\alpha^{(5)}_{\overline{\rm MS}}(m_{\rm Z}) = 0.11852(84)$.