Sign Flip Triangulations of the Amplituhedron

Abstract

We present new triangulations of the $m=4$ amplituhedron relevant for scattering amplitudes in planar $\mathcal{N}=4$ super-Yang-Mills, obtained directly from the combinatorial definition of the geometry. Using the "sign flip" characterization of the amplituhedron, we reproduce the canonical forms for the all-multiplicity next-to-maximally helicity violating (NMHV) and next-to-next-to-maximally helicity violating (${\text{N}}^2$MHV) tree-level as well as the NMHV one-loop cases, without using any input from traditional amplitudes methods. Our results provide strong evidence for the equivalence of the original definition of the amplituhedron \cite{Arkani-Hamed:2013jha} and the topological one \cite{Arkani-Hamed:2017vfh}, and suggest a new path forward for computing higher loop amplitudes geometrically. In particular, we realize the NMHV one-loop amplituhedron as the intersection of two amplituhedra of lower dimensionality, which is reflected in the novel structure of the corresponding canonical form.

arXiv comment38 pages, 5 figures

References in corpus (35)

• MHV Vertices And Tree Amplitudes In Gauge Theory
• The Amplituhedron
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• Into the Amplituhedron
• Positive Geometries and Canonical Forms
• Unwinding the Amplituhedron in Binary
• Twistor-Space Recursive Formulation of Gauge-Theory Amplitudes
• Positive Amplitudes In The Amplituhedron
• On the Singularity Structure of Maximally Supersymmetric Scattering Amplitudes
• A Note on Polytopes for Scattering Amplitudes
• The Momentum Amplituhedron
• MHV Diagrams in Momentum Twistor Space
• Landau Singularities from the Amplituhedron
• The Amplituhedron from Momentum Twistor Diagrams
• Cutting Deep Into The Amplituhedron
• Towards the Amplituhedron Volume
• Stokes Polytopes : The positive geometry for ${\varphi }^4$ interactions
• 1-loop Amplitudes from the Halohedron
• The Amplituhedron and the One-loop Grassmannian Measure
• On the Positive Geometry of Conformal Field Theory
• Notes on Scattering Amplitudes as Differential Forms
• Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
• On Positive Geometries of Quartic Interactions II : Stokes polytopes, Lower Forms on Associahedra and Worldsheet Forms
• All-Helicity Symbol Alphabets from Unwound Amplituhedra
• Cluster Adjacency for m=2 Yangian Invariants
• An Etude on Recursion Relations and Triangulations
• Amplituhedron meets Jeffrey-Kirwan Residue
• Hyperbolic Geometry and Amplituhedra in 1+2 dimensions
• All-loop cuts from the Amplituhedron
• Tree-level scattering amplitudes from the amplituhedron
• Triangulation of 2-loop MHV Amplituhedron from Sign Flips
• Emergent unitarity from the amplituhedron
• All-loop Mondrian Diagrammatics and 4-particle Amplituhedron
• 4-particle Amplituhedron at 3-loop and its Mondrian Diagrammatic Implication

Cited by in corpus (9)

• Positive geometry, local triangulations, and the dual of the Amplituhedron
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Amplituhedra, and Beyond
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Internal boundaries of the loop amplituhedron
• Non-planar BCFW Grassmannian Geometries
• Poles At Infinity in On-shell Diagrams
• Triangulation-free Trivialization of 2-loop MHV Amplituhedron
• Leading singularities and chambers of Correlahedron