The uncertainty in $\alpha_{s}(M_Z^2)$ determined from hadronic tau decay measurements

Nucl.Phys. B535 (1998) 19-40 We show that QCD Minkowski observables such as the $e^{+}e^{-}$ R-ratio and the hadronic tau decay $R_{\tau}$ are completely determined by the effective charge (EC) beta-function, $\rho(x)$, 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, $R_{\tau}$ from a weighted contour integration of $D(se^{i\theta})$ around a circle in the complex squared energy s-plane, with $\rho(x)$ 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, $R_{\tau}$ constructed using the NNLO and "leading-b" resummed versions of $\rho(x)$ provides an estimate of the uncertainty due to the uncalculated higher order corrections. Simple numerical parametrizations are given to facilitate these fits. For $R_{\tau}$ we estimate an uncertainty $\delta\alpha_{s}(m_{\tau}^{2})\simeq0.01$, corresponding to $\delta\alpha_{s}(M_{Z}^{2})\simeq0.002$. 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.