Scattering Forms, Worldsheet Forms and Amplitudes from Subspaces

TL;DR

This paper develops a unified geometric framework connecting scattering forms, worldsheet forms, and amplitudes, revealing new determinant formulas and polytope structures related to non-planar MHV singularities in super-Yang-Mills theory.

Contribution

It introduces a general construction of scattering and worldsheet forms from subspaces, generalizes the pushforward of worldsheet forms, and links these to amplitude calculations and polytope geometries.

Findings

Constructed a large class of $d\log$ scattering forms and worldsheet forms.

Derived new determinant formulas for leading singularities.

Identified convex polytopes corresponding to scattering forms in 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 ${\cal N}=4$ 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.