Hadronic effects in M{\o}ller scattering at NNLO

Two-loop electroweak corrections to polarized Moller scattering are studied in two different schemes at low energies. We find the finite $Q^2$ corrections to be well under control. The hadronic and perturbative QCD corrections to the $\gamma Z$ two-point function are incorporated through the weak mixing angle at low energies, which introduce an error of $0.08\times {10}^{-3}$ in the weak charge of the electron $Q_W^e$. Furthermore, by studying the scheme dependence, we obtain an estimate of the current perturbative electroweak uncertainty, $\delta Q_W^e\approx 0.23\times {10}^{-3}$, which is five times smaller than the precision estimated for the MOLLER experiment $(\delta Q_W^e=1.1\times10^{-3})$. Future work is possible to reduce the theory error further.