Determination of the QCD $\Lambda$ Parameter and the Accuracy of Perturbation Theory at High Energies

We discuss the determination of the strong coupling $\alpha_{\bar{MS}}(m_{Z})$ or, equivalently, the QCD $\Lambda$ parameter. Its determination requires the use of perturbation theory in $\alpha_{s}$(μ) in some scheme s and at some energy scale μ. The higher the scale μ, 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 that allows us to nonperturbatively reach very high energies, corresponding to $\alpha_{s}$ = 0.1 and below. We find that (continuum) perturbation theory is very accurate there, yielding a 3% error in the $\Lambda$ parameter, while data around $\alpha_{s}$ ≈ 0.2 are 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