Third-order mixed electroweak-QCD corrections to the W-boson mass prediction from the muon lifetime

We present the calculation of the so far missing $\mathcal{O}({\alpha }^2{\alpha }_{\mathrm{s}})$ corrections to the quantity $\mathrm{\Delta }r$, which relates the Fermi constant to the W-boson mass, and enables precision predictions of the latter. While the $\mathcal{O}({\alpha }^2{\alpha }_{\mathrm{s}})$ corrections from diagrams with two closed fermion loops are already known, we here focus on the subset with one closed fermion loop, which is a substantially more complex problem. The calculation has been carried out through a combination of analytical and numerical techniques for the three-loop integrals and the on-shell renormalization. The impact of the new corrections is numerically significant, raising the Standard Model prediction for the W-boson mass by more than 3 MeV.