Two-Loop Four-Fermion Scattering Amplitude in QED

We present the first fully analytic evaluation of the transition amplitude for the scattering of a massless into a massive pair of fermions at the two-loop level in quantum electrodynamics. Our result is an essential ingredient for the determination of the electromagnetic coupling within scattering reactions, beyond the currently known accuracy, which has a crucial impact on the evaluation of the anomalous magnetic moment of the muon. It will allow, in particular, for a precise determination of the leading hadronic contribution to the $(g-2{)}_{\mu }$ in the MUonE experiment at CERN, and therefore can be used to shed light on the current discrepancy between the standard model prediction and the experimental measurement for this important physical observable.