Relations between electroweak parameters to order $α_s^3$

Abstract: Relations between electroweak parameters can be measured with ever-increasing precision and a future lepton collider will require major advances in the theoretical predictions. We derive top-quark mediated four-loop QCD corrections to $Δr$ and $Δκ$, that is to the relations between the weak boson masses, the electroweak couplings, and the weak mixing angle. To this end, we expand the vacuum polarization functions of charged and neutral gauge bosons in $M_Z² \ll m_t<sup>2$, finding a negligible truncation error after six terms. Our results imply a shift of a few hundred keV in the W boson mass, which will be numerically relevant for measurements at the FCC-ee.