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Augmented Recursion For One-loop Gravity Amplitudes

David C. Dunbar, James H. Ettle, Warren B. Perkins

TL;DR

This work tackles the challenge of computing the rational part $R$ of one-loop gravity amplitudes, where standard recursion is obstructed by double-pole structures under complex shifts. It introduces an augmented recursion based on axial-gauge diagrammatics to determine the pole-under-the-pole residues and leverages KLT relations to connect gravity substructures to Yang-Mills amplitudes. The authors explicitly construct $M^{\mathrm{1-loop}}(1^-,2^+,3^+,4^+,5^+)$ and outline the six-point extension, validating the results against string-based rules and providing public code. The approach provides a practical framework for systematic, high-point gravity amplitude calculations and facilitates extraction of purely rational terms in gravity.

Abstract

We present a semi-recursive method for calculating the rational parts of one-loop gravity amplitudes which utilises axial gauge diagrams to determine the non-factorising pieces of the amplitude. This method is used to compute the one-loop amplitudes M(1-,2+,3+,4+,5+) and M(1-,2+,3+,4+,5+,6+).

Augmented Recursion For One-loop Gravity Amplitudes

TL;DR

This work tackles the challenge of computing the rational part of one-loop gravity amplitudes, where standard recursion is obstructed by double-pole structures under complex shifts. It introduces an augmented recursion based on axial-gauge diagrammatics to determine the pole-under-the-pole residues and leverages KLT relations to connect gravity substructures to Yang-Mills amplitudes. The authors explicitly construct and outline the six-point extension, validating the results against string-based rules and providing public code. The approach provides a practical framework for systematic, high-point gravity amplitude calculations and facilitates extraction of purely rational terms in gravity.

Abstract

We present a semi-recursive method for calculating the rational parts of one-loop gravity amplitudes which utilises axial gauge diagrams to determine the non-factorising pieces of the amplitude. This method is used to compute the one-loop amplitudes M(1-,2+,3+,4+,5+) and M(1-,2+,3+,4+,5+,6+).

Paper Structure

This paper contains 10 sections, 69 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Recursion
  3. Axial gauge diagrammatics
  4. The graviton scattering amplitude $M^\text{1-loop}(1^-,2^+,3^+,4^+,5^+)$
  5. Conclusions and remarks
  6. Six-point single-minus amplitude
  7. Graviton scattering amplitudes
  8. String-based rules calculation of $M^\text{1-loop}(1^-,2^+,3^+,4^+,5^+)$
  9. Summary of the string-based rules for gravity amplitudes
  10. Application to $M^\text{1-loop}(1^-,2^+,3^+,4^+,5^+)$

Figures (6)

  • Figure 1: Diagrams contributing to the recursive construction of $A(1^-, 2^+, 3^+, 4^+, 5^+)$ with legs $1$ and $5$ shifted in the manner of (2.5). The diagram (c) contains the one-loop vertex $V^\text{1-loop}(\hat{K}^+, 4^+, \hat{5}^+)$ that contributes the double-pole.
  • Figure 2: The form of the tree insertion that augments the recursion in order to construct the double pole and its underlying single pole.
  • Figure 3: With the constraints that (1) the negative-helicity leg enters via an MHV three-point vertex and (2) the four-point vertices vanish, we only have non-vanishing diagrams with a single three-point MHV vertex with the remaining vertices three-point $\overline{\text{MHV}}$ with internal helicities organised as shown in these sample diagrams.
  • Figure 4: Singularities in $s_{bc}$ and $s_{ay}$ arise in integrations over the terms shown.
  • Figure 5: The three possible helicity structures of figure 4a.
  • ...and 1 more figures