Determination of the QCD $\mathrm{\Lambda }$-parameter and the accuracy of perturbation theory at high energies

We discuss the determination of the strong coupling ${\alpha }_{\overline{\mathrm{M}\mathrm{S}}}^{}(m_{\mathrm{Z}})$ or equivalently the QCD $\mathrm{\Lambda }$-parameter. Its determination requires the use of perturbation theory in ${\alpha }_s(\mu )$ in some scheme, $s$, and at some energy scale $\mu$. The higher the scale $\mu$ the more accurate perturbation theory becomes, owing to asymptotic freedom. As one step in our computation of the $\mathrm{\Lambda }$-parameter in three-flavor QCD, we perform lattice computations in a scheme which allows us to non-perturbatively reach very high energies, corresponding to $\alpha_s = 0.1$ and below. We find that (continuum) perturbation theory is very accurate there, yielding a three percent error in the $\mathrm{\Lambda }$-parameter, while data around ${\alpha }_s\approx 0.2$ is clearly insufficient to quote such a precision. It is important to realize that these findings are expected to be generic, as our scheme has advantageous properties regarding the applicability of perturbation theory.