Properties of scattering forms and their relation to associahedra

Abstract

We show that the half-integrands in the CHY representation of tree amplitudes give rise to the definition of differential forms -- the scattering forms -- on the moduli space of a Riemann sphere with $n$ marked points. These differential forms have some remarkable properties. We show that all singularities are on the divisor ${\overline{\mathcal{M}}}_{0,n}\backslash {\mathcal{M}}_{0,n}$. Each singularity is logarithmic and the residue factorises into two differential forms of lower points. In order for this to work, we provide a threefold generalisation of the CHY polarisation factor (also known as reduced Pfaffian) towards off-shell momenta, unphysical polarisations and away from the solutions of the scattering equations. We discuss explicitly the cases of bi-adjoint scalar amplitudes, Yang-Mills amplitudes and gravity amplitudes.

40 pages, version to be published

References in corpus (10)

• New Relations for Gauge-Theory Amplitudes
• Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations
• Combinatorics and Topology of Kawai-Lewellen-Tye Relations
• New Relations for Einstein-Yang-Mills Amplitudes
• Einstein-Yang-Mills from pure Yang-Mills amplitudes
• Manifesting Color-Kinematics Duality in the Scattering Equation Formalism
• The Amplituhedron from Momentum Twistor Diagrams
• On the solutions of the scattering equations
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• Labelled tree graphs, Feynman diagrams and disk integrals

Cited by in corpus (18)

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• Kinematic Jacobi Identity is a Residue Theorem: Geometry of Color-Kinematics Duality for Gauge and Gravity Amplitudes
• Classical Double Copy of Worldline Quantum Field Theory
• Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers
• On Positive Geometries of Quartic Interactions II : Stokes polytopes, Lower Forms on Associahedra and Worldsheet Forms
• UV cancelations in gravity loop integrands
• Scattering Forms, Worldsheet Forms and Amplitudes from Subspaces
• Resonant resurgent asymptotics from quantum field theory
• Color-kinematics duality, double copy and the unitarity method for higher-derivative QCD and quadratic gravity
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Hyperbolic Geometry and Amplituhedra in 1+2 dimensions
• A double copy from twisted (co)homology at genus one
• Flavour-kinematics duality for Goldstone modes
• The singularities of Selberg- and Dotsenko-Fateev-like integrals
• Off-Shell CHY Amplitudes and Feynman Graphs
• CHY Theory for Several Fields