QCD Corrections from the Top Quark to Relations between Electroweak Parameters up to Orderαs2

The vacuum polarization functions $\Pi \left(q^2\right)$ of charged and neutral gauge bosons that arise from top and bottom quark loops lead to important shifts in relations between electroweak parameters that can be measured with ever-increasing precision. The large mass of the top quark allows approximation of these functions through the first two terms of an expansion in $M_Z^2{/M}_t^2$. The first three terms of the Taylor series of $\Pi \left(q^2\right)$ are evaluated analytically up to order ${\alpha }_s^2$. Results for the subleading contributions to $\Delta r$ and the effective mixing angle ${sin}^2\overline{\Theta }$ are presented.