Top Quark Contribution to Hadronic Decays of theZBoson atO(αs2)in QCD

We evaluate the effect of a virtual top quark on the coefficient of ${\alpha }_s^2$ in the decay rates $\Gamma (Z\to \mathrm{hadrons})$ and $\Gamma (Z\to b\overline{b})$. We treat the dependence on the top quark mass exactly instead of using a large mass expansion. The present work completes the evaluation of the ${\alpha }_s^2$ contributions to these quantities. The calculation uses both the modified minimal subtraction and Collins-Wilczek-Zee renormalization prescriptions. The results can be applied to the hadronic decays of the $\tau$ lepton.