One-loop amplitudes on the Riemann sphere

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.

49 pages + appendices, 3 figures. v2: further comments on singular solutions. v3: improved factorisation section, published version

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• The Kinematic Algebra From the Self-Dual Sector
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• Ambitwistor strings in 4-dimensions
• Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles
• Gravity from Rational Curves
• A Worldsheet Theory for Supergravity
• Scattering equations, generating functions and all massless five point tree amplitudes

Cited by in corpus (106)

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• Two-Loop Scattering Amplitudes from Ambitwistor Strings: from Genus Two to the Nodal Riemann Sphere
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• Towards the n-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure
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