Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry

• Main Authors: Niklas Henke, Georgios Papathanasiou
• Format: Article
• Language: English
• Published: SpringerOpen 2021-10-01
• Series: Journal of High Energy Physics
• Subjects: 1/N Expansion; Extended Supersymmetry; Scattering Amplitudes; Supersymmetric Gauge Theory
• Online Access: https://doi.org/10.1007/JHEP10(2021)007

Abstract We further exploit the relation between tropical Grassmannians and Gr(4, n) cluster algebras in order to make and refine predictions for the singularities of scattering amplitudes in planar N $$ \mathcal{N} $$ = 4 super Yang-Mills theory at higher multiplicity n ≥ 8. As a mathematical foundation that provides access to square-root symbol letters in principle for any n, we analyse infinite mutation sequences in cluster algebras with general coefficients. First specialising our analysis to the eight-particle amplitude, and comparing it with a recent, closely related approach based on scattering diagrams, we find that the only additional letters the latter provides are the two square roots associated to the four-mass box. In combination with a tropical rule for selecting a finite subset of variables of the infinite Gr(4, 9) cluster algebra, we then apply our results to obtain a collection of 3, 078 rational and 2, 349 square-root letters expected to appear in the nine-particle amplitude. In particular these contain the alphabet found in an explicit 2-loop NMHV symbol calculation at this multiplicity.