Master integrals for the NNLO virtual corrections to μe scattering in QED: the planar graphs

We evaluate the master integrals for the two-loop, planar box-diagrams contributing to the elastic scattering of muons and electrons at next-to-next-to leading-order in QED. We adopt the method of differential equations and the Magnus exponential series to determine a canonical set of integrals, finally expressed as a Taylor series around four space-time dimensions, with coefficients written as combination of generalised poly-logarithms. The electron is treated as massless, while we retain full dependence on the muon mass. The considered integrals are also relevant for crossing-related processes, such as di-muon production at e(+)e(-)-colliders, as well as for the QCD corrections to top-pair production at hadron colliders.