# The Momentum Amplituhedron

## TL;DR

This paper introduces the momentum amplituhedron, a positive geometric structure in spinor helicity space that encodes tree-level scattering amplitudes in N=4 super Yang-Mills theory, extending the amplituhedron concept.

## Contribution

The paper defines the momentum amplituhedron as a new positive geometry for scattering amplitudes, using bosonized variables and positive Grassmannian mappings.

## Key findings

- Defines the momentum amplituhedron in spinor helicity space.
- Shows how to extract scattering amplitudes from the canonical form.
- Establishes a geometric framework for tree-level amplitudes in N=4 SYM.

## Abstract

In this paper we define a new object, the momentum amplituhedron, which is the long sought-after positive geometry for tree-level scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory in spinor helicity space. Inspired by the construction of the ordinary amplituhedron, we introduce bosonized spinor helicity variables to represent our external kinematical data, and restrict them to a particular positive region. The momentum amplituhedron $\mathcal{M}_{n,k}$ is then the image of the positive Grassmannian via a map determined by such kinematics. The scattering amplitudes are extracted from the canonical form with logarithmic singularities on the boundaries of this geometry.