title: Loop Integrands for Scattering Amplitudes from the Riemann Sphere.
abstract: The scattering equations on the Riemann sphere give rise to remarkable formulas 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 a wide application, based on off-shell scattering equations that depend on the loop momentum. We present new formulas, 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.
authors.affiliation: Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom.
authors.forename: Yvonne
authors.initials: Y
authors.lastname: Geyer
authors.affiliation: Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom.
authors.forename: Lionel
authors.initials: L
authors.lastname: Mason
authors.affiliation: DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom.
authors.forename: Ricardo
authors.initials: R
authors.lastname: Monteiro
authors.affiliation: DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom.
authors.forename: Piotr
authors.initials: P
authors.lastname: Tourkine