Notes on Scattering Amplitudes as Differential Forms

Abstract

Inspired by the idea of viewing amplitudes in $\mathcal{N}=4$ SYM as differential forms on momentum twistor space, we introduce differential forms on the space of spinor variables, which combine helicity amplitudes in any four-dimensional gauge theory as a single object. In this note we focus on such differential forms in $\mathcal{N}=4$ SYM, which can also be thought of as "bosonizing" superamplitudes in non-chiral superspace. Remarkably all tree-level amplitudes in $\mathcal{N}=4$ SYM combine to a $d\log$ form in spinor variables, which is given by pushforward of canonical forms of Grassmannian cells, the tree forms can also be obtained using BCFW or inverse-soft construction, and we present all-multiplicity expression for MHV and NMHV forms to illustrate their simplicity. Similarly all-loop planar integrands can be naturally written as $d\log$ forms in the Grassmannian/on-shell-diagram picture, and we expect the same to hold beyond the planar limit. Just as the form in momentum twistor space reveals underlying positive geometry of the amplituhedron, the form in terms of spinor variables strongly suggests an "amplituhedron in momentum space". We initiate the study of its geometry by connecting it to the moduli space of Witten's twistor-string theory, which provides a pushforward formula for tree forms in $\mathcal{N}=4$ SYM.

24 pages, a dozen figures, references added

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Cited by in corpus (25)

• The Momentum Amplituhedron
• Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• The momentum amplituhedron of SYM and ABJM from twistor-string maps
• The orthogonal momentum amplituhedron and ABJM amplitudes
• On-Shell Electroweak Sector and the Higgs Mechanism
• A Note on Letters of Yangian Invariants
• 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
• Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra
• All-loop cuts from the Amplituhedron
• Momentum Amplituhedron meets Kinematic Associahedron
• Gravity loop integrands from the ultraviolet
• Non-planar BCFW Grassmannian Geometries
• Kleiss-Kuijf Relations from Momentum Amplituhedron Geometry
• Open associahedra and scattering forms
• Bootstrapping octagons in reduced kinematics from $A_2$ cluster algebras
• $\Delta$-Algebra and Scattering Amplitudes
• Bootstrapping solutions of scattering equations
• Emergent unitarity from the amplituhedron
• CHY Theory for Several Fields
• Scattering amplitudes as multi-particle higher-spin charges in the correspondence space