Loop Integrands for Scattering Amplitudes from the Riemann Sphere

TL;DR

This work develops a sphere-based framework for loop integrands in gauge and gravity theories starting from scattering equations originally formulated on a torus. By applying a residue argument in the modular parameter, the authors degenerate the elliptic curve to a nodal Riemann sphere and derive off-shell scattering equations that depend on the loop momentum $\ell$, yielding a rational, theta-function–free 1-loop integrand $\mathcal{M}^{(1)}_{SG}$ and a corresponding structure for Yang–Mills via Parke–Taylor factors. They establish the $n=4$ case, showing equivalence to the standard box representation through partial fractions and loop-momentum shifts, and provide explicit 5-point results with pentagon and box contributions that satisfy color-kinematics duality, matching the off-shell equations numerically. Finally, the paper outlines all-loop extensions by introducing a $g$-nodal degeneration and proposes all-loop integrands for supergravity and planar Yang–Mills, with analogous constructions for biadjoint scalars, suggesting a broad, modular-invariant method to compute loop amplitudes beyond tree level. The approach promises wide applicability to other theories by replacing worldsheet correlators with theory-dependent building blocks and provides a concrete path toward theta-function–free, globally defined loop integrands.

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.

Figures (1)

• Figure 1: Contour argument in the fundamental domain.