The uncertainty in $α_{s}(M_Z^2)$ determined from hadronic tau decay measurements
We show that QCD Minkowski observables such as the $e^{+}e^{-}$ R-ratio and the hadronic tau decay $R_τ$ are completely determined by the effective charge (EC) beta-function, $ρ(x)$, corresponding to the Euclidean QCD vacuum polarization Adler D-function, together with the next-to-leading order (NLO) perturbative coefficient of D. An efficient numerical algorithm is given for evaluating R, $R_τ$ from a weighted contour integration of $D(se^{iθ})$ around a circle in the complex squared energy s-plane, with $ρ(x)$ used to evolve in s around the contour. The EC beta-function can be truncated at next-to-NLO (NNLO) using the known exact perturbative calculation or the uncalculated N^3 LO and higher terms can be approximated by the portion containing the highest power of b, the first QCD beta-function coefficient. The difference between the R, $R_τ$ constructed using the NNLO and "leading-b" resummed versions of $ρ(x)$ provides an estimate of the uncertainty due to the uncalculated higher order corrections. Simple numerical parametrizations are given to facilitate these fits. For $R_τ$ we estimate an uncertainty $δα_{s}(m_τ^{2})\simeq0.01$, corresponding to $δα_{s}(M_{Z}^{2})\simeq0.002$. This encouragingly small uncertainty is much less than rather pessimistic estimates by other authors based on analogous all-orders resummations, which we demonstrate to be extremely dependent on the chosen renormalization scheme, and hence misleading.