Loop Integrands for Scattering Amplitudes from the Riemann Sphere

TL;DR

This paper develops a new framework for loop integrands in scattering amplitudes using off-shell scattering equations on the Riemann sphere, extending to arbitrary loop orders with applications to supergravity and super-Yang-Mills theories.

Contribution

It introduces a novel method to derive loop integrands from scattering equations on the Riemann sphere, extending previous conjectures to all loop orders.

Findings

Derived explicit one-loop supergravity and super-Yang-Mills integrands.

Extended scattering equations to arbitrary loop order.

Proposed all-loop integrands for supergravity and planar super-Yang-Mills.

Abstract

The scattering equations on the Riemann sphere give rise to remarkable formulae for tree-level gauge theory and gravity amplitudes. Adamo, Casali and Skinner conjectured a one-loop formula for supergravity amplitudes based on scattering equations on a torus. We use a residue theorem to transform this into a formula on the Riemann sphere. What emerges is a framework for loop integrands on the Riemann sphere that promises to have wide application, based on off-shell scattering equations that depend on the loop momentum. We present new formulae, checked explicitly at low points, for supergravity and super-Yang-Mills amplitudes and for n-gon integrands at one loop. Finally, we show that the off-shell scattering equations naturally extend to arbitrary loop order, and we give a proposal for the all-loop integrands for supergravity and planar super-Yang-Mills theory.