From Momentum Amplituhedron Boundaries to Amplitude Singularities and Back

Abstract

The momentum amplituhedron is a positive geometry encoding tree-level scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills directly in spinor-helicity space. In this paper we classify all boundaries of the momentum amplituhedron ${\mathcal{M}}_{n,k}$ and explain how these boundaries are related to the expected factorization channels, and soft and collinear limits of tree amplitudes. Conversely, all physical singularities of tree amplitudes are encoded in this boundary stratification. Finally, we find that the momentum amplituhedron ${\mathcal{M}}_{n,k}$ has Euler characteristic equal to one, which provides a first step towards proving that it is homeomorphic to a ball.

arXiv comment20 pages, 7 figures

References in corpus (4)

• The Amplituhedron
• Positive Geometries and Canonical Forms
• The Momentum Amplituhedron
• The m=1 amplituhedron and cyclic hyperplane arrangements

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• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• Amplituhedra, and Beyond
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• The Loop Momentum Amplituhedron
• Internal boundaries of the loop amplituhedron
• Momentum Amplituhedron meets Kinematic Associahedron
• Non-planar BCFW Grassmannian Geometries
• Kleiss-Kuijf Relations from Momentum Amplituhedron Geometry
• On the geometry of the orthogonal momentum amplituhedron
• Open associahedra and scattering forms
• Poles At Infinity in On-shell Diagrams
• Soft and Collinear Limits in $\mathcal{N}=8$ Supergravity using Double Copy Formalism
• Triangulation-free Trivialization of 2-loop MHV Amplituhedron
• Positive Geometry for Stringy Scalar Amplitudes