Estimates of the higher-order QCD corrections: theory and applications

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^{4}_{s}) 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 {\rm hadrons}) / \sigma (e^{+}e^{-} \to \mu^{+} \mu^{-}) and R_{\tau} = \Gamma (\tau \to \nu_{\tau} + {\rm 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.