Abstract
Two-loop MHV amplitudes in planar mathcal{N} = 4 supersymmetric Yang Mills theory are known to exhibit many intriguing forms of cluster-algebraic structure. We leverage this structure to upgrade the symbols of the eight- and nine-particle amplitudes to complete analytic functions. This is done by systematically projecting onto the components of these amplitudes that take different functional forms, and matching each component to an ansatz of multiple polylogarithms with negative cluster-coordinate arguments. The remaining additive constant can be determined analytically by comparing the collinear limit of each amplitude to known lower-multiplicity results. We also observe that the nonclassical part of each of these amplitudes admits a unique decomposition in terms of a specific A3 cluster polylogarithm, and explore the numerical behavior of the remainder function along lines in the positive region.
Highlights
Maximally-helicity-violating (MHV) and next-to-MHV (NMHV) amplitudes, which can be computed at any multiplicity using the methods of [15,16,17], special kinematic configurations involving large numbers of particles have been studied
We find here that the eight-particle remainder function can be decomposed in terms of the same A5 function that appeared at seven points, and that both the eight- and nine-particle remainder functions admit a unique decomposition in terms of the function fA+3− that was defined in [21]
We focus on the dihedrally-symmetric point that can be found in this region at each particle multiplicity, and plot the behavior of the remainder function along lines that move away from these symmetric points into successive collinear limits, all the way down to five-point kinematics
Summary
While the remainder function is expected to depend on increasingly complicated types of functions at large multiplicities and high loop orders [84,85,86,87,88,89,90,91], it turns out that it can be expressed in terms of just multiple polylogarithms at two loops [15]. Multiple polylogarithms generalize classical polylogarithms to iterated integrals over more general logarithmic kernels They often appear in the notation z dt Ga1,...,aw (z) = 0 t − a1 Ga2,...,aw (z) ,.
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