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 $O(α^{4}_{s})$ 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 $O(α_s^4)$ QCD corrections to the Minkowskian quantities $R(s) = σ_{tot} (e^{+}e^{-} \to {\rm hadrons}) / σ(e^{+}e^{-} \to μ^{+} μ^{-})$ and $R_τ = Γ(τ\to ν_τ + {\rm hadrons}) / Γ(τ\to ν_τ \overlineν_{e} e)$. The problem of the fixation of the uncertainties due to the $O(α_s^5)$ corrections to the considered quantities is also discussed.