Abstract
We report on our calculation of the order α_s^4 axial singlet contributions for the decay rates of the Z-boson as well as the vector singlet contribution to the cross section for electron-positron annihilation into hadrons. Together with recently finished O(α_s^4) calculations of the non-signlet corrections [1,2], the new results directly lead us to the first complete O(α_s^4) predictions for the total hadronic decay rate of the Z-boson and the ratio R(s).
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