Running of the Electroweak Gauge Couplings from First Principles.

We present a high-precision calculation of the hadronic running of electroweak gauge couplings from first principles. Employing lattice QCD in the low-energy regime, we achieve permille precision for virtualities Q^{2}≲12  GeV^{2}. At Q^{2}≃1  GeV^{2}, our determination deviates by up to 7σ from estimates based on e^{+}e^{-} measurements. Combining lattice QCD with perturbative QCD via the Euclidean split technique, we obtain for the electromagnetic coupling Δα_{had}^{(5)}(M_{Z}^{2})=0.027821(34)_{lat}(35)_{pQCD}, which is more than twice as precise as recent phenomenological determinations. We assess improvement scenarios by which the precision target for next-generation electroweak measurements could be reached.