Two-Loop Master Integrals for $γ^* \to 3$ Jets: The planar topologies
T. Gehrmann, E. Remiddi
TL;DR
The paper develops a differential-equation framework to compute all planar two-loop four-point master integrals with one external leg off-shell, which are essential for NNLO predictions of 3-jet observables in e+e− annihilation. Central to the method is the introduction of two-dimensional harmonic polylogarithms (2dHPL) to express ε-expanded results, with boundary conditions fixed by analyticity. The authors categorize integrals by one-, two-, and three-scale behavior, providing explicit ε-expansions and cross-checks against known results and alternative formalisms (e.g., Smirnov). This approach enables analytic continuation to different kinematic regions and paves the way for extending to non-planar topologies, significantly advancing precision QCD calculations for jet processes.
Abstract
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 $\e=(4-d)/2$, 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 in terms of Nielsen's polylogarithms are given. The analytic continuation of our results to other kinematical situations is outlined.