The Momentum Amplituhedron

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.

23 pages

References in corpus (21)

• Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory
• Perturbative Gauge Theory As A String Theory In Twistor Space
• New Recursion Relations for Tree Amplitudes of Gluons
• The Amplituhedron
• A Duality For The S Matrix
• Dual Superconformal Invariance, Momentum Twistors and Grassmannians
• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• On the Tree-Level S-Matrix of Yang-Mills Theory
• Positive Geometries and Canonical Forms
• Unwinding the Amplituhedron in Binary
• Positive Amplitudes In The Amplituhedron
• Cosmological Polytopes and the Wavefunction of the Universe
• The Correlahedron
• Towards the Amplituhedron Volume
• Positroids, Plabic Graphs, and Scattering Amplitudes in Mathematica
• On the Positive Geometry of Conformal Field Theory
• The S Matrix of 6D Super Yang-Mills and Maximal Supergravity from Rational Maps
• Notes on Scattering Amplitudes as Differential Forms
• The polarized scattering equations for 6d superamplitudes
• M5-Brane and D-Brane Scattering Amplitudes
• All Tree Amplitudes of 6D $(2,0)$ Supergravity: Interacting Tensor Multiplets and the $K3$ Moduli Space

Cited by in corpus (38)

• Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• The momentum amplituhedron of SYM and ABJM from twistor-string maps
• From Momentum Amplituhedron Boundaries to Amplitude Singularities and Back
• Propagators, BCFW Recursion and New Scattering Equations at One Loop
• The orthogonal momentum amplituhedron and ABJM amplitudes
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron
• Positive Geometries and Differential Forms with Non-Logarithmic Singularities I
• Sign Flip Triangulations of the Amplituhedron
• Amplituhedra, and Beyond
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Boundaries of the Amplituhedron with amplituhedronBoundaries
• Integrated negative geometries in ABJM
• Multi-spin soft bootstrap and scalar-vector Galileon
• The Loop Momentum Amplituhedron
• Amplituhedron-like geometries
• A Soft Theorem for the Tropical Grassmannian
• Momentum Amplituhedron meets Kinematic Associahedron
• Gravity loop integrands from the ultraviolet
• Internal boundaries of the loop amplituhedron
• Emergent unitarity, all-loop cuts and integrations from the ABJM amplituhedron
• Kleiss-Kuijf Relations from Momentum Amplituhedron Geometry
• Non-planar BCFW Grassmannian Geometries
• The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers
• Recursion and worldsheet formulae for 6d superamplitudes
• Open associahedra and scattering forms
• On the geometry of the orthogonal momentum amplituhedron
• Emergent unitarity from the amplituhedron
• Poles At Infinity in On-shell Diagrams
• Triangulation-free Trivialization of 2-loop MHV Amplituhedron
• Pushforwards via Scattering Equations with Applications to Positive Geometries
• The hypersimplex canonical forms and the momentum amplituhedron-like logarithmic forms
• Positive Geometry for Stringy Scalar Amplitudes
• Extended DBI and its generalizations from graded soft theorems
• BCFW like recursion for Deformed Associahedron
• The Geometry of BCFW for ABJM Loop Integrands
• Plabic Tangles and Cluster Promotion Maps