How tropical are seven- and eight-particle amplitudes?
Niklas Henke, Georgios Papathanasiou
Abstract
A bstract We study tropical Grassmanians Tr( k, n ) in relation to cluster algebras, and assess their applicability to n -particle amplitudes for n = 7 , 8. In $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 super Yang-Mills theory, we first show that while the totally positive part of Tr(4 , 7) may encompass the iterated discontinuity structure of the seven-point Maximally Helicity Violating (MHV) amplitude, it is too small for the Next-to-MHV helicity configuration. Then, using Tr(4 , 8) we propose a finite set of 356 cluster $$ \mathcal{A} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> -coordinates expected to contain the rational symbol letters of the eight-particle MHV amplitude, and discuss how the remaining square-root letters may be obtained from limits of infinite mutation sequences. Finally, we use a triangulation of the totally positive part of Tr(3 , 8) to obtain the associated generalised biadjoint scalar amplitude in a form containing a near-minimal amount of spurious poles.