Two-Loop Master Integrals for γ ∗ → 3 Jets: The planar topologies

The calculation of the two-loop corrections to the three jet production rate and to event shapes in electron-positron annihilation requires the computation of a number of up to now unknown two-loop four-point master integrals with one off-shell and three on-shell legs. In this paper, we compute those master integrals which correspond to planar topologies by solving differential equations in the external invariants which are fulfilled by the master integrals. We obtain the master integrals as expansions in ǫ = (4 − d)/2, where d is the space-time dimension. The results are expressed in terms of newly introduced two-dimensional harmonic polylogarithms, whose properties are shortly discussed. For all two-dimensional harmonic polylogarithms appearing in the divergent parts of the integrals, expressions The calculation of perturbative next-to-next-to-leading order corrections to 2 → 2 scattering or 1 → 3 decay processes is a yet outstanding task for many precision applications in particle physics phenomenology. Progress in this field was up to very recently hampered by difficulties in the calculation of the virtual two-loop corrections to the corresponding Feynman amplitudes.