Amplituhedron-like geometries

TL;DR

This paper introduces amplituhedron-like geometries with non-maximal winding, proposing their canonical forms relate to products of amplitudes and providing new geometric and algebraic characterizations.

Contribution

It defines and characterizes amplituhedron-like geometries with non-maximal winding, connecting their canonical forms to amplitude products and extending geometric definitions.

Findings

Canonical form corresponds to product of parity conjugate amplitudes.

Union of geometries defined by physical inequalities yields the amplitude square.

Introduces a star product in bosonised superspace for amplitude products.

Abstract

We consider amplituhedron-like geometries which are defined in a similar way to the intrinsic definition of the amplituhedron but with non-maximal winding number. We propose that for the cases with minimal number of points the canonical form of these geometries corresponds to the product of parity conjugate amplitudes at tree as well as loop level. The product of amplitudes in superspace lifts to a star product in bosonised superspace which we give a precise definition of. We give an alternative definition of amplituhedron-like geometries, analogous to the original amplituhedron definition, and also a characterisation as a sum over pairs of on-shell diagrams that we use to prove the conjecture at tree level. The union of all amplituhedron-like geometries has a very simple definition given by only physical inequalities. Although such a union does not give a positive geometry, a natural extension of the standard definition of canonical form, the globally oriented canonical form, acts on this union and gives the square of the amplitude.