Quark mass corrections to theZboson decay rates

The results of a perturbative QCD evaluation of the $\sim \frac{m_f^2}{M_Z^2}$ contributions to ${\Gamma }_{Z\to b\overline{b}}$ and ${\Gamma }_{Z\to \mathrm{hadrons}}$ for the quark masses $m_f\ll M_Z$ are presented. The recent results due to the combination of renormalization group constraints and the results of several other calculations are independently confirmed by direct computation. Some existing confusion in the literature is clarified. In addition, the calculated $O\left({\alpha }_s^2\right)$ correction to the correlation function in the axial channel is a necessary ingredient for the yet uncalculated axial part of the $O\left({\alpha }_s^3\right)$ mass correction to the $Z$ decay rates. The results can be applied to the $\tau$ hadronic width.