A Note on Polytopes for Scattering Amplitudes

TL;DR

Arkani-Hamed, Bourjaily, Cachazo, Hodges, and Trnka propose a geometric polytope framework for perturbative scattering amplitudes in planar ${\cal N}=4$ SYM, where amplitudes are volumes of polytopes in extended momentum-twistor spaces. They show that NMHV tree amplitudes and the 1-loop MHV integrand can be understood as polytope volumes, with standard BCFW/CSW representations arising from triangulations and new, simpler cyclic/local expressions emerging from alternative triangulations. The NMHV amplitude is realized as the volume of a polytope ${\cal P}_n=\tfrac12 L_n\otimes L_n$ in ${\mathbb{CP}}^4$, whose boundary is given by $R$-invariants, while the 1-loop MHV integrand decomposes into products of CP$^2$ areas plus a cross-term, all organized to make cyclicity and locality manifest. The authors also relate these geometric constructions to Grassmannian residues and the dual Wilson-loop picture, arguing for a deeper invariant structure that could extend to all amplitudes and loops, and hinting at connections to spin-chain formulations and potential applications beyond supersymmetric theories.

Abstract

In this note we continue the exploration of the polytope picture for scattering amplitudes, where amplitudes are associated with the volumes of polytopes in generalized momentum-twistor spaces. After a quick warm-up example illustrating the essential ideas with the elementary geometry of polygons in CP^2, we interpret the 1-loop MHV integrand as the volume of a polytope in CP^3x CP^3, which can be thought of as the space obtained by taking the geometric dual of the Wilson loop in each CP^3 of the product. We then review the polytope picture for the NMHV tree amplitude and give it a more direct and intrinsic definition as the geometric dual of a canonical "square" of the Wilson-Loop polygon, living in a certain extension of momentum-twistor space into CP^4. In both cases, one natural class of triangulations of the polytope produces the BCFW/CSW representations of the amplitudes; another class of triangulations leads to a striking new form, which is both remarkably simple as well as manifestly cyclic and local.