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Infrared behaviour of the one-loop scattering equations and supergravity integrands

Eduardo Casali, Piotr Tourkine

TL;DR

The paper develops an analytic infrared (IR) solution to the one-loop ambitwistor-string scattering equations for the four-point case and demonstrates that the resulting ambitwistor integrand reproduces the IR limit of the four-graviton amplitude. By exploiting holomorphic propagators and theta-function identities, it shows the four-point numerator reduces to the supersymmetric $t_8 t_8 R^4$ structure and that the IR Jacobian matches the IR-leading scalar-box content, with the construction extending qualitatively to $n$ points. It also establishes a precise link between the ambitwistor one-loop saddle and the Gross & Mende saddle, clarifying normalization conventions and the role of the loop momentum, supported by an electrostatics analogy. The results provide a consistent, dimension-independent IR description and offer a path to understanding higher-point and heterotic extensions within the ambitwistor framework.

Abstract

The recently introduced ambitwistor string led to a striking proposal for one-loop maximal supergravity amplitudes, localised on the solutions of the ambitwistor one-loop scattering equations. However, these amplitudes have not yet been explicitly analysed due to the apparent complexity of the equations that determine the localisation. In this paper we propose an analytic solution to the four-point one-loop scattering equations in the infrared (IR) regime of the amplitude. Using this solution, we compute the ambitwistor integrand and demonstrate that it correctly reproduces the four-graviton integrand, in the IR regime. This solution qualitatively extends to n points. To conclude, we explain that the ambitwistor one-loop scattering equations actually correspond to the standard Gross & Mende saddle point.

Infrared behaviour of the one-loop scattering equations and supergravity integrands

TL;DR

The paper develops an analytic infrared (IR) solution to the one-loop ambitwistor-string scattering equations for the four-point case and demonstrates that the resulting ambitwistor integrand reproduces the IR limit of the four-graviton amplitude. By exploiting holomorphic propagators and theta-function identities, it shows the four-point numerator reduces to the supersymmetric structure and that the IR Jacobian matches the IR-leading scalar-box content, with the construction extending qualitatively to points. It also establishes a precise link between the ambitwistor one-loop saddle and the Gross & Mende saddle, clarifying normalization conventions and the role of the loop momentum, supported by an electrostatics analogy. The results provide a consistent, dimension-independent IR description and offer a path to understanding higher-point and heterotic extensions within the ambitwistor framework.

Abstract

The recently introduced ambitwistor string led to a striking proposal for one-loop maximal supergravity amplitudes, localised on the solutions of the ambitwistor one-loop scattering equations. However, these amplitudes have not yet been explicitly analysed due to the apparent complexity of the equations that determine the localisation. In this paper we propose an analytic solution to the four-point one-loop scattering equations in the infrared (IR) regime of the amplitude. Using this solution, we compute the ambitwistor integrand and demonstrate that it correctly reproduces the four-graviton integrand, in the IR regime. This solution qualitatively extends to n points. To conclude, we explain that the ambitwistor one-loop scattering equations actually correspond to the standard Gross & Mende saddle point.

Paper Structure

This paper contains 18 sections, 62 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of the one-loop ambitwistor string amplitude
  3. The amplitude
  4. The scattering equations
  5. The Jacobian
  6. Infrared Behaviour at One Loop
  7. Boundary behaviour of the ambitwistor amplitude
  8. Three propagators on-shell
  9. IR solution
  10. Numerator structure
  11. IR solution to the four-point one-loop scattering equations
  12. Computation of the Jacobian
  13. Extension to $n$ points
  14. Relation to Gross & Mende
  15. Changing the normalisation
  16. ...and 3 more sections

Figures (5)

  • Figure 1: An $SL(2,\mathbb{Z})$ fundamental domain of the torus.
  • Figure 2: 4-point pinched torus creates a 6-point sphere with two back-to-back momenta.
  • Figure 3: Typical IR divergences in theories of gravity.
  • Figure 4: The four boxes that contribute to the IR divergence
  • Figure 5: Gross & Mende equilibrium; the charges should be placed at half-periods of the lattice.