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

摘要

We show that QCD Minkowski observables such as the $e^{+}e^{-}$ R-ratio andthe hadronic tau decay $R_{\tau}$ are completely determined by the effectivecharge (EC) beta-function, $\rho(x)$, corresponding to the Euclidean QCD vacuumpolarization Adler D-function, together with the next-to-leading order (NLO)perturbative coefficient of D. An efficient numerical algorithm is given forevaluating 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 betruncated at next-to-NLO (NNLO) using the known exact perturbative calculationor the uncalculated N^3 LO and higher terms can be approximated by the portioncontaining the highest power of b, the first QCD beta-function coefficient. Thedifference between the R, $R_{\tau}$ constructed using the NNLO and "leading-b"resummed versions of $\rho(x)$ provides an estimate of the uncertainty due tothe uncalculated higher order corrections. Simple numerical parametrizationsare 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 uncertaintyis much less than rather pessimistic estimates by other authors based onanalogous all-orders resummations, which we demonstrate to be extremelydependent on the chosen renormalization scheme, and hence misleading.