Improved Evaluation of the NNLO QCD Effects in the Tau Decay, e⁺e⁻ Annihilation into Hadrons and Deep-Inelastic Sum Rules

A systematic method is proposed for analyzing the renormalization scheme uncertainties in the next-next-to-leading order QCD predicitions, based on a condition which eliminates schemes that give rise to large cancellations in the expression for the characteristic scheme invariant combination of the expansion coefficients. Using this method it is shown that the QCD corrections to the tau lepton decay are rather stable with respect to change of the scheme, provided that an improved formula is used, which involves numerical evaluation of the contour integral in the complex energy plane with the Adler function under the integral. Optimized predictions for the tau decay corrections are given. It is shown that also in the case of the of QCD corrections to $e^{+}e^{-}$ annihilation into hadrons the conventional expansion has sizable scheme dependence, even at large energies. However, a considerable improvement is obtained when the QCD corrections are expressed as a contour integral, with the Adler function under the integral, resumming in this way the large $\pi^{2}$ contributions. In the case of the corrections to the Bjorken sum rule for polarized structure functions it is found that for $n_{f}=4$ they are insensitive to change of the scheme. However, the $n_{f}=3$ expression is found to be strongly scheme dependent at lower energies.