Prescriptive Unitarity with Elliptic Leading Singularities

We investigate the consequences of elliptic leading singularities for the unitarity-based representations of two-loop amplitudes in planar, maximally supersymmetric Yang-Mills theory. We show that diagonalizing with respect to these leading singularities ensures that the integrand basis is term-wise pure (suitably generalized, to the elliptic multiple polylogarithms, as necessary). We also investigate an alternative strategy based on diagonalizing a basis of integrands on differential forms; this strategy, while neither term-wise Yangian-invariant nor pure, offers several advantages in terms of complexity.