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We compute the two-loop QED helicity amplitudes for the scattering of alepton pair with an off-shell and an on-shell photon,$0\to\ell\bar\ell\gamma\gamma^*$, using the approximation of massless leptons.We express all master integrals relevant for the scattering of four masslessparticles with a single external off-shell leg up to two loops in a basis ofalgebraically independent multiple polylogarithms, which guarantees anefficient numerical evaluation and compact analytic representations of theamplitudes. Analytic forms of the amplitudes are reconstructed from numericalevaluations over finite fields. Our results complete the amplitude-levelingredients contributing to the N$^3$LO predictions of electron-muon scattering$e\mu\to e\mu$, which are required to meet the precision goal of the futureMUonE experiment.

Lepton-pair scattering with an off-shell and an on-shell photon at two loops in massless QED