Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
Niklas Henke, Γεώργιος Παπαθανασίου
Abstract
A bstract 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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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.