On the Precision of the Computation of the QCD Corrections to Electroweak Vacuum Polarizations

We demonstrate that the dispersive computation of the threshold enhancements to heavy quark vacuum polarizations is unstable. Because of the slow convergence of the dispersion relations the result critically depends on the intermediate energy region where the non-relativistic approximation, intrinsic to threshold calculations, is invalid. We discuss other ambiguities precluding a reliable calculation of the threshold contribution to the vacuum polarizations. In the absence of a solution prudence should force one to assign an error to the radiative corrections not far below the level of the pertubative O($\alpha \alpha_s$) contributions. This may preclude the extraction of the Higgs mass from precision measurements.