Hadronic contributions to $\alpha(Q^{2})$ and $\sin^{2}\theta_{W}(Q^{2})$ from spectral reconstruction of lattice-QCD data

We present preliminary results from a lattice-QCD study of the hadronic contributions to the running of the electromagnetic coupling, $\Delta\alpha(Q^2)$, and the electroweak mixing angle, $\Delta\sin^2\theta_{W}(Q^2)$. Using $N_f = 2+1+1$ HISQ ensembles at physical quark masses, we discuss the challenges posed by strong statistical correlations in the time-momentum representation and propose a spectral-reconstruction strategy to obtain controlled continuum-extrapolated results across the full energy range.