One-loop amplitudes on the Riemann sphere

TL;DR

This paper extends the scattering equations framework to derive one-loop amplitude formulas on the Riemann sphere for various theories, including non-supersymmetric cases, with systematic proofs and consistency checks.

Contribution

It introduces new one-loop integrand formulas for non-supersymmetric Yang-Mills and gravity theories derived from ambitwistor string theory and provides systematic proofs based on worldsheet factorization.

Findings

New formulas for non-supersymmetric one-loop integrands.

Validation of formulas through Q-cut decomposition.

Systematic proof using worldsheet factorization.

Abstract

The scattering equations provide a powerful framework for the study of scattering amplitudes in a variety of theories. Their derivation from ambitwistor string theory led to proposals for formulae at one loop on a torus for 10 dimensional supergravity, and we recently showed how these can be reduced to the Riemann sphere and checked in simple cases. We also proposed analogous formulae for other theories including maximal super-Yang-Mills theory and supergravity in other dimensions at one loop. We give further details of these results and extend them in two directions. Firstly, we propose new formulae for the one-loop integrands of Yang-Mills theory and gravity in the absence of supersymmetry. These follow from the identification of the states running in the loop as expressed in the ambitwistor-string correlator. Secondly, we give a systematic proof of the non-supersymmetric formulae using the worldsheet factorisation properties of the nodal Riemann sphere underlying the scattering equations at one loop. Our formulae have the same decomposition under the recently introduced Q-cuts as one-loop integrands and hence give the correct amplitudes.