Scattering Forms, Worldsheet Forms and Amplitudes from Subspaces

Abstract

We present a general construction of two types of differential forms, based on any $(n-3)$-dimensional subspace in the kinematic space of $n$ massless particles. The first type is the so-called projective, scattering forms in kinematic space, while the second is defined in the moduli space of $n$-punctured Riemann spheres which we call worldsheet forms. We show that the pushforward of worldsheet forms, by summing over solutions of scattering equations, gives the corresponding scattering forms, which generalizes the results of [1711.09102]. The pullback of scattering forms to subspaces can have natural interpretations as amplitudes in terms of Bern-Carrasco-Johansson double-copy construction or Cachazo-He-Yuan formula. As an application of our formalism, we construct in this way a large class of $d\log$ scattering forms and worldsheet forms, which are in one-to-one correspondence with non-planar MHV leading singularities in $\mathcal{N}=4$ super-Yang-Mills. For every leading singularity function, we present a new determinant formula in moduli space, as well as a (combinatoric) polytope and associated scattering form in kinematic space. These include the so-called Cayley cases, where in each case the scattering form is the canonical forms of a convex polytope in the subspace, and scattering equations admit elegant rewritings as a map from the moduli space to the subspace.

25+4 pages, a dozen figures, lots of polytopes

References in corpus (11)

• New Relations for Gauge-Theory Amplitudes
• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• Scattering Amplitudes from Intersection Theory
• Two-Loop Scattering Amplitudes from the Riemann Sphere
• Labelled tree graphs, Feynman diagrams and disk integrals
• Biadjoint scalar tree amplitudes and intersecting dual associahedra
• Properties of scattering forms and their relation to associahedra
• Canonical Bases for Permutohedral Plates
• Coxeter Complexes and Graph-Associahedra
• Combinatorial Factorization
• Cluster Algebras in Kinematic Space of Scattering Amplitudes

Cited by in corpus (14)

• Aspects of Scattering Amplitudes and Moduli Space Localization
• 1-loop Amplitudes from the Halohedron
• Notes on Scattering Amplitudes as Differential Forms
• Amplituhedron meets Jeffrey-Kirwan Residue
• Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals
• Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra
• One-loop diagrams with quadratic propagators from the worldsheet
• On Positive Geometry and Scattering Forms for Matter Particles
• Open associahedra and scattering forms
• Covariant color-kinematics duality, Hopf algebras and permutohedra
• One-loop integrand from generalised scattering equations
• Double Soft Theorem for Generalised Biadjoint Scalar Amplitudes
• Triangulations for ABHY Polytopes and Recursions for Tree and Loop Amplitudes
• CHY Theory for Several Fields