Nonplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes

TL;DR

This paper establishes a geometric criterion for bipartite on-shell diagrams to produce rational top-forms in scattering amplitude calculations, enabling recursive derivation of higher-loop leading singularities.

Contribution

It proves that BCFW-decomposability corresponds to linear shifts in algebraic constraints, providing a geometric interpretation for constructing rational top-forms.

Findings

Rational top-forms are obtainable via linear shifts in geometric constraints.

All higher-loop leading singularities can be derived recursively for BCFW-decomposable diagrams.

A geometric interpretation in the Grassmannian manifold simplifies the understanding of on-shell diagrams.

Abstract

Bipartite on-shell diagrams are the latest tool in constructing scattering amplitudes. In this paper we prove that a Britto-Cachazo-Feng-Witten (BCFW)-decomposable on-shell diagram process a rational top-form if and only if the algebraic ideal comprised of the geometrical constraints is shifted linearly during successive BCFW integrations. With a proper geometric interpretation of the constraints in the Grassmannian manifold, the rational top-form integration contours can thus be obtained, and understood, in a straightforward way. All rational top-form integrands of arbitrary higher loops leading singularities can therefore be derived recursively, as long as the corresponding on-shell diagram is BCFW-decomposable.