$O(N_f{\alpha }^2)$ Radiative Corrections in Low-Energy Electroweak Processes

Of the the three best-measured electroweak observables, $\alpha$, $G_{\mu }$ and $M_Z$, the first two are extracted from low-energy processes. Both $G_{\mu }$ and $M_Z$ are now known to an accuracy of about 2 parts in ${10}^5$ and there is a proposal to improve the measurement of the muon lifetime by a factor of 10 in an experiment at Brookhaven. Yet calculations of electroweak radiative corrections currently do no better than a few parts in ${10}^3$–${10}^4$ and therefore cannot exploit to available experimental precision. We report on the calculation of the $\mathcal{O}(N_f{\alpha }^2)$ corrections to Thomson scattering and the muon lifetime from which $\alpha$ and $G_{\mu }$, respectively, are obtained. The $\mathcal{O}(N_f{\alpha }^2)$ corrections are expected to be a dominant gauge-invariant subset of 2-loop corrections.