We evaluate the effect of a virtual top quark on the coefficient of $\alpha_{ s}^{ 2}$ in the decay rates $\Gamma(Z \rightarrow { \rm hadrons})$ and $\Gamma(Z \rightarrow b\overline{ b} )$. 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 $\overline{ \rm MS}$ and Collins-Wilczek-Zee renormalization prescriptions. The results can be applied to the hadronic decays of the $\tau$-lepton.
Categories: hep-ph