Estimates of the higher-order QCD corrections: Theory and Applications

Abstract

We consider the further development of the formalism of the estimates of higher-order perturbative corrections in the Euclidean region, which is based on the application of the scheme-invariant methods, namely the principle of minimal sensitivity and the effective charges approach. We present the estimates of the order $O({\alpha }_s^4)$ QCD corrections to the Euclidean quantities: the $e^{+}{e}^{-}$-annihilation $D$-function and the deep inelastic scattering sum rules, namely the non-polarized and polarized Bjorken sum rules and to the Gross--Llewellyn Smith sum rule. The results for the $D$-function are further applied to estimate the $O({\alpha }_s^4)$ QCD corrections to the Minkowskian quantities $R(s)={\sigma }_{tot}(e^{+}{e}^{-}\to \mathrm{hadrons})/\sigma (e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-})$ and $R_{\tau }=\Gamma (\tau \to {\nu }_{\tau }+\mathrm{hadrons})/\Gamma (\tau \to {\nu }_{\tau }{\overline{\nu }}_{e}e)$. The problem of the fixation of the uncertainties due to the $O({\alpha }_s^5)$ corrections to the considered quantities is also discussed.

revised version and improved version of CERN.TH-7400/94, LATEX 10 pages, six-loop estimates for R(s) in Table 2 are revised, thanks to J. Ellis for pointing numerical shortcomings (general formulae are non-affected). Details of derivations of six-loop estimates for R_tau are presented

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