Prescriptive Unitarity from Positive Geometries

TL;DR

This paper introduces the momentum amplituhedron in split-signature space, providing a geometric framework that encodes scattering amplitudes in N=4 super Yang-Mills theory and naturally derives prescriptive unitarity.

Contribution

It defines a new geometric object, the momentum amplituhedron, that encodes scattering amplitudes and unifies geometric and unitarity-based approaches.

Findings

One-loop integrands are represented by a curvy polytope with vertices from maximal cuts.

The geometric formulae agree with prescriptive unitarity results.

The approach provides new explicit formulas for arbitrary multiplicity and helicity amplitudes.

Abstract

In this paper, we define the momentum amplituhedron in the four-dimensional split-signature space of dual momenta. It encodes scattering amplitudes at tree level and loop integrands for N=4 super Yang-Mills in the planar sector. In this description, every point in the tree-level geometry is specified by a null polygon. Using the null structure of this kinematic space, we find a geometry whose canonical differential form produces loop-amplitude integrands. Remarkably, at one loop it is a curvy version of a simple polytope, whose vertices are specified by maximal cuts of the amplitude. This construction allows us to find novel formulae for the one-loop integrands for amplitudes with any multiplicity and helicity. The formulae obtained in this way agree with the ones derived via prescriptive unitarity. It makes prescriptive unitarity naturally emerge from this geometric description.