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N >= 4 Supergravity Amplitudes from Gauge Theory at Two Loops

C. Boucher-Veronneau, L. J. Dixon

TL;DR

This work computes the full two-loop four-graviton amplitudes for ${ m N}=4,5,6$ supergravity using the BCJ color-kinematics duality and the gravity double-copy construction. By combining the ${ m N}=4$ sYM amplitude with ${ m N}_{ m YM}=0,1,2$ sYM data, the authors obtain explicit gravity amplitudes, verify the infrared structure (divergences given by the square of the one-loop amplitudes), and present compact finite remainders, including a pure-function representation with a simple symbol for ${ m N}=8$. The results extend the double-copy approach beyond maximal supersymmetry, provide detailed expressions in terms of planar and nonplanar double-box integrals, and analyze forward-scattering behavior, offering a foundation for potential three-loop investigations in non-maximal supergravity.

Abstract

We present the full two-loop four-graviton amplitudes in N=4,5,6 supergravity. These results were obtained using the double-copy structure of gravity, which follows from the recently conjectured color-kinematics duality in gauge theory. The two-loop four-gluon scattering amplitudes in N=0,1,2 supersymmetric gauge theory are a second essential ingredient. The gravity amplitudes have the expected infrared behavior: the two-loop divergences are given in terms of the squares of the corresponding one-loop amplitudes. The finite remainders are presented in a compact form. The finite remainder for N=8 supergravity is also presented, in a form that utilizes a pure function with a very simple symbol.

N >= 4 Supergravity Amplitudes from Gauge Theory at Two Loops

TL;DR

This work computes the full two-loop four-graviton amplitudes for supergravity using the BCJ color-kinematics duality and the gravity double-copy construction. By combining the sYM amplitude with sYM data, the authors obtain explicit gravity amplitudes, verify the infrared structure (divergences given by the square of the one-loop amplitudes), and present compact finite remainders, including a pure-function representation with a simple symbol for . The results extend the double-copy approach beyond maximal supersymmetry, provide detailed expressions in terms of planar and nonplanar double-box integrals, and analyze forward-scattering behavior, offering a foundation for potential three-loop investigations in non-maximal supergravity.

Abstract

We present the full two-loop four-graviton amplitudes in N=4,5,6 supergravity. These results were obtained using the double-copy structure of gravity, which follows from the recently conjectured color-kinematics duality in gauge theory. The two-loop four-gluon scattering amplitudes in N=0,1,2 supersymmetric gauge theory are a second essential ingredient. The gravity amplitudes have the expected infrared behavior: the two-loop divergences are given in terms of the squares of the corresponding one-loop amplitudes. The finite remainders are presented in a compact form. The finite remainder for N=8 supergravity is also presented, in a form that utilizes a pure function with a very simple symbol.

Paper Structure

This paper contains 8 sections, 70 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of the BCJ duality and squaring relations
  3. Two-loop ${{\cal N}=8}$ supergravity
  4. Two-Loop $4 \le {\cal N} < 8$ Supergravity
  5. Infrared poles and finite remainders
  6. Forward-scattering limit of the amplitudes
  7. Conclusions
  8. One-loop expressions

Figures (2)

  • Figure 1: The planar and nonplanar cubic diagrams at two loops. The marked (colored) propagators in the planar diagram are used in the text to describe different color and kinematic Jacobi identities.
  • Figure 2: Two-loop diagrams related by a Jacobi identity. The Jacobi identity is applied to the four-point tree-level subdiagram that contains the (light blue) intermediate line marked $a$. The rest of the diagram is unchanged.