Determination of the QCD $Lambda$-parameter and the accuracy of perturbation theory at high energies

We discuss the determination of the strong coupling $alpha_mathrm{overline{MS}}^{}(m_mathrm{Z})$ or equivalently the QCD $Lambda$-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 $Lambda$-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 $Lambda$-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.