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

Abstract

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 $D(s{e}^{i\theta })$ 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 $\delta {\alpha }_s(m_{\tau }^2)\simeq 0.01$, corresponding to $\delta {\alpha }_s(M_Z^2)\simeq 0.002$. 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.

Comments: 25 pages, uses LaTeX, 10 Postscript figures, epsfig.sty