We derive the three-loop dilatation operator of the flavor $SU(2)$ subsector of $\mathcal{N}=4$ supersymmetric Yang-Mills theory in the planar limit by a direct Feynman diagram calculation in $\mathcal{N}=1$ superspace. The transcendentality three contributions which appear in intermediate steps cancel among each other, leaving a rational result which confirms the predictions from integrability. We derive finiteness conditions that allow us to avoid the explicit evaluation of entire classes of Feynman graphs and also yield constraints on the D-algebra manipulations. Based on these results, we discover universal cancellation mechanisms. As a check for the consistency of our result, we verify the cancellation of all higher-order poles.

• Received 1 June 2011

DOI:https://doi.org/10.1103/PhysRevD.84.045014