Two-Loop Scattering Amplitudes from the Riemann Sphere

TL;DR

This paper develops two-loop scattering equations on a nodal Riemann sphere, enabling calculation of two-loop amplitudes in supergravity and super-Yang-Mills theories with correct factorization and non-planar contributions.

Contribution

It introduces a method to derive two-loop scattering equations from ambitwistor string theory on a nodal Riemann sphere, extending previous one-loop results.

Findings

Successfully derived two-loop scattering equations with proper factorization.

Matched results for four-point two-loop amplitudes in supergravity and super-Yang-Mills.

Included non-planar contributions via a loop analogue of the Parke-Taylor factor.

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.