Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond

We compute the leading-color (planar) three-loop four-point amplitude of $N=4$ supersymmetric Yang-Mills theory in $4\ensuremath{-}2ϵ$ dimensions, as a Laurent expansion about $ϵ=0$ including the finite terms. The amplitude was constructed previously via the unitarity method, in terms of two Feynman loop integrals, one of which has been evaluated already. Here we use the Mellin-Barnes integration technique to evaluate the Laurent expansion of the second integral. Strikingly, the amplitude is expressible, through the finite terms, in terms of the corresponding one- and two-loop amplitudes, which provides strong evidence for a previous conjecture that higher-loop planar $N=4$ amplitudes have an iterative…