Planar Six-Point Feynman Integrals for Four-Dimensional Gauge Theories

We compute all planar two-loop six-point Feynman integrals entering scattering observables in massless gauge theories such as QCD. A central result of this paper is the formulation of the differential-equations method under the algebraic constraints stemming from four-dimensional kinematics, which in this case leaves only 8 independent scales. We show that these constraints imply that one must compute topologies with only up to 8 propagators, instead of the expected 9. This leads to the decoupling of entire classes of integrals that do not contribute to scattering amplitudes in four dimensional gauge theories. We construct a pure basis and derive their canonical differential equations, of which we discuss the numerical solution. This work marks an important step towards the calculation of massless $2\to 4$ scattering processes at two loops.