The Two-Loop Remainder Function for Eight and Nine Particles

TL;DR

This paper computes the complete two-loop remainder functions for eight- and nine-particle MHV amplitudes in planar ${ m N}=4$ SYM, revealing their cluster algebraic structure and expressing them in terms of polylogarithms.

Contribution

It advances the understanding of multi-particle amplitudes by explicitly constructing their analytic forms using cluster algebra and polylogarithms, extending previous partial results.

Findings

Complete analytic forms of the two-loop eight- and nine-particle amplitudes.

Identification of a unique $A_3$ cluster polylogarithm decomposition.

Numerical analysis of the remainder function in the positive region.

Abstract

Two-loop MHV amplitudes in planar ${\cal N} = 4$ 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 $A_3$ cluster polylogarithm, and explore the numerical behavior of the remainder function along lines in the positive region.