Two-Loop Scattering Amplitudes from the Riemann Sphere

TL;DR

Problem: extend CHY/ambitwistor scattering equations to two loops. Method: reduce genus-two ambitwistor-string expressions to a doubly-nodal Riemann sphere carrying loop momenta, derive two-loop scattering equations with off-shell terms, and check via explicit 4-point amplitudes. Findings: factorization constraints fix the off-shell parameter and signs, yielding correct two-loop integrands for maximal supergravity and super-Yang-Mills, including planar and non-planar sectors; supersymmetric amplitudes come from regular solutions with degenerate contributions vanishing, while non-supersymmetric cases may require handling degenerate solutions. Significance: provides a concrete nodal-sphere framework for two-loop amplitudes and paves the way for higher loops and broader theories, with potential links to vertex-operator formulations and full string theory.

Abstract

The scattering equations give striking formulae for massless scattering amplitudes at tree level and, as shown recently, at one loop. The progress at loop level was based on ambitwistor string theory, which naturally yields the scattering equations. We proposed that, for ambitwistor strings, the standard loop expansion in terms of the genus of the worldsheet is equivalent to an expansion in terms of nodes of a Riemann sphere, with the nodes carrying the loop momenta. In this paper, we show how to obtain two-loop scattering equations with the correct factorization properties. We adapt genus-two integrands from the ambitwistor string to the nodal Riemann sphere and show that these yield correct answers, by matching standard results for the four-point two-loop amplitudes of maximal supergravity and super-Yang-Mills theory. In the Yang-Mills case, this requires the loop analogue of the Parke-Taylor factor carrying the colour dependence, which includes non-planar contributions.

Figures (1)

• Figure 1: Different possible worldsheet degenerations.