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One-loop amplitudes on the Riemann sphere

Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr Tourkine

TL;DR

This work extends the ambitwistor-string framework by deriving new one-loop, non-supersymmetric Yang–Mills and gravity integrands from sphere-based scattering equations and providing a systematic proof via nodal-sphere factorisation. It shows that these integrands decompose under Q-cuts exactly as known one-loop representations, and it develops a detailed factorisation analysis that matches the residues of the proposed loop formulas. The results unify supersymmetric and non-supersymmetric cases within a common sphere-based formalism, clarifying the role of GSO sectors, Pfaffian structures, and Parke–Taylor factors, and establishing robust UV and factorisation behavior. The findings pave the way for applying tree-level CHY techniques to loop amplitudes across theories and dimensions, with potential extensions to higher loops through Q-cut methods. Overall, the paper provides new, verifiable tools for constructing and validating one-loop amplitudes in YM and gravity from ambitwistor-string origins.

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.

One-loop amplitudes on the Riemann sphere

TL;DR

This work extends the ambitwistor-string framework by deriving new one-loop, non-supersymmetric Yang–Mills and gravity integrands from sphere-based scattering equations and providing a systematic proof via nodal-sphere factorisation. It shows that these integrands decompose under Q-cuts exactly as known one-loop representations, and it develops a detailed factorisation analysis that matches the residues of the proposed loop formulas. The results unify supersymmetric and non-supersymmetric cases within a common sphere-based formalism, clarifying the role of GSO sectors, Pfaffian structures, and Parke–Taylor factors, and establishing robust UV and factorisation behavior. The findings pave the way for applying tree-level CHY techniques to loop amplitudes across theories and dimensions, with potential extensions to higher loops through Q-cut methods. Overall, the paper provides new, verifiable tools for constructing and validating one-loop amplitudes in YM and gravity from ambitwistor-string origins.

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.

Paper Structure

This paper contains 32 sections, 4 theorems, 176 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background.
  3. Summary of this paper.
  4. Review
  5. The scattering equations on a torus
  6. From a torus to a Riemann sphere
  7. The n-gon conjecture, partial fractions and shifts
  8. Supersymmetric theories
  9. Supergravity
  10. Super-Yang-Mills theory
  11. Non-supersymmetric theories
  12. General form of the one-loop scattering equations
  13. Contributions of GSO sectors and the NS Pfaffian
  14. Pure YM and gravity amplitudes.
  15. Pfaffian structure of the new amplitudes.
  16. ...and 17 more sections

Key Result

Theorem 1

Consider a subset $I\subset\{1,\ldots, n, \ell^+,\ell^-\}$ with one fixed point in $I$ and two in its complement $\bar{I}$ and suppose that we have solutions to the scattering equations such that $\sigma_i= \sigma_I + \varepsilon x_i+\mathcal{O}(\varepsilon^2)$ for $i\in I$ with $x_i=O(1)$, $\sigma_ Our 1-loop formulae $\mathcal{M}^{(1)}$ on the Riemann sphere for n-gons, bi-adjoint scalar theory,

Figures (3)

  • Figure 1: Residue theorem in the fundamental domain.
  • Figure 2: Representation of the amplitude as a sum over Q-cuts.
  • Figure 3: Different possible worldsheet degenerations.

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 4.1: Factorisation Lemma Dolan:2013isa