The Amplituhedron and the One-loop Grassmannian Measure

TL;DR

This paper explores the geometric structure of the one-loop amplituhedron in supersymmetric Yang-Mills theory, introducing a new measure formula and linking it to on-shell diagrams and polygonal geometry in projective space.

Contribution

It presents a novel formula for the one-loop Grassmannian measure and establishes a correspondence between amplituhedron cells, on-shell diagrams, and residues.

Findings

New one-loop Grassmannian measure formula

Correspondence between cells, diagrams, and residues

Polygonal geometric interpretation for NMHV case

Abstract

All-loop planar scattering amplitudes in maximally supersymmetric Yang-Mills theory can be formulated geometrically in terms of the "amplituhedron". We study the mathematical structures of the one-loop amplituhedron, and present a new formula for its canonical measure, or the one-loop Grassmannian measure formula. Using the recently proposed momentum-twistor diagrams, we show that there is a correspondence between the cells of one-loop amplituhedron, BCFW terms or equivalently on-shell diagrams, and residues of the one-loop Grassmannian formula. In particular, for the first non-trivial case of one-loop NMHV, these structures are naturally associated with a nice geometric picture as polygons in projective space, as we discuss in various illustrative examples.