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We evaluate the master integrals for the two-loop, planar box-diagramscontributing to the elastic scattering of muons and electrons atnext-to-next-to leading-order in QED. We adopt the method of differentialequations and the Magnus exponential series to determine a canonical set ofintegrals, finally expressed as a Taylor series around four space-timedimensions, with coefficients written as combination of generalisedpolylogarithms. The electron is treated as massless, while we retain fulldependence on the muon mass. The considered integrals are also relevant forcrossing-related processes, such as di-muon production at $e^+ e^-$-colliders,as well as for the QCD corrections to $top$-pair production at hadroncolliders.

Master integrals for the NNLO virtual corrections to $\mu e$ scattering in QED: the planar graphs