Improved evaluation of the next-next-to-leading order QCD corrections toe+e−annihilation into hadrons

The next-next-to-leading order QCD corrections to $e^{+}{e}^{-}$ annihilation into hadrons are considered. The stability of the predictions with respect to change of the renormalization scheme is discussed in detail for the case of five, four, and three active quark flavors. The analysis is based on the recently proposed condition for selecting renormalization schemes according to the degree of cancellation that they introduce in the expression for the scheme-invariant combination of the expansion coefficients. It is demonstrated that the scheme dependence ambiguity in the predictions obtained with the conventional expansion is substantial, particularly at lower energies. It is shown, however, that the stability of the predictions is greatly improved when QCD corrections are evaluated in a more precise way, by utilizing the contour integral representation and calculating numerically the contour integral.