$O(\alpha \alpha_s^2)$ correction to the electroweak $\rho$ parameter

Abstract

The three-loop QCD contributions to the vacuum polarization functions of the $Z$ and $W$ bosons at zero momentum are calculated. The top quark is considered to be massive and the other quarks massless. Using these results, we calculate the correction to the electroweak $\rho$ parameter. All computations are done in the framework of dimensional regularization as well as regularization by dimensional reduction. We use recurrence relations obtained by the method of integration by parts to reduce all integrals to a small set of master integrals. A comparison of the two-loop and three-loop QCD corrections to the $\rho$ parameter is performed.