Simplex-like Structures of Maximally Supersymmetric Scattering Amplitudes

TL;DR

This paper explores the simplex-like geometric structures underlying tree-level scattering amplitudes in maximally supersymmetric Yang-Mills theory, revealing a refined, cell-based description linked to the positive Grassmannian.

Contribution

It introduces a novel geometric framework using fully-spanning cells and positive Grassmannian structures to efficiently describe tree amplitudes in N=4 SYM.

Findings

Tree amplitudes are characterized by fully-spanning cells.

A finite set of cells captures amplitudes with fixed negative helicities.

The approach simplifies the understanding of large external particle numbers.

Abstract

We elaborate the two-fold simplex-like structures of tree amplitudes in planar maximally supersymmetric Yang-Mills (N=4 SYM), through its connection to a mathematical structure known as the positive Grassmannian. Exploiting the reduced Grassmannian geometry and the matrix form of on-shell recursion relation in terms of super momentum twistors, we manifest that tree amplitudes can be remarkably refined via the essential building blocks named as fully-spanning cells. For a fixed number of negative helicities, an amplitude can be entirely captured by finite, compact information of the relevant fully-spanning cells up to an arbitrarily large number of external particles.