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

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