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N >= 4 Supergravity Amplitudes from Gauge Theory at One Loop

Z. Bern, C. Boucher-Veronneau, H. Johansson

TL;DR

This work investigates the conjectured BCJ color-kinematics duality and its gravity double-copy at loop level, focusing on one-loop four- and five-point amplitudes in ${\cal N} \ge 4$ supergravity. By arranging gauge-theory numerators to satisfy the duality and employing the double-copy construction, the authors express integrated gravity amplitudes as simple linear combinations of $D$-dimensional gauge-theory amplitudes, with explicit results for ${\cal N}=8,6,4$ at four and five points. Key contributions include dimensionally extended, duality-respecting representations that yield compact, testable formulas and demonstrate cancellations of infrared and ultraviolet divergences in several cases, aligning with known results such as Dunbar–Norridge. The findings provide concrete loop-level evidence for the duality and offer a practical route to constructing higher-point and higher-loop gravity amplitudes via gauge theory inputs, reinforcing the deep gauge–gravity connection and suggesting avenues for further extension.

Abstract

We expose simple and practical relations between the integrated four- and five-point one-loop amplitudes of N >= 4 supergravity and the corresponding (super-)Yang-Mills amplitudes. The link between the amplitudes is simply understood using the recently uncovered duality between color and kinematics that leads to a double-copy structure for gravity. These examples provide additional direct confirmations of the duality and double-copy properties at loop level for a sample of different theories.

N >= 4 Supergravity Amplitudes from Gauge Theory at One Loop

TL;DR

This work investigates the conjectured BCJ color-kinematics duality and its gravity double-copy at loop level, focusing on one-loop four- and five-point amplitudes in supergravity. By arranging gauge-theory numerators to satisfy the duality and employing the double-copy construction, the authors express integrated gravity amplitudes as simple linear combinations of -dimensional gauge-theory amplitudes, with explicit results for at four and five points. Key contributions include dimensionally extended, duality-respecting representations that yield compact, testable formulas and demonstrate cancellations of infrared and ultraviolet divergences in several cases, aligning with known results such as Dunbar–Norridge. The findings provide concrete loop-level evidence for the duality and offer a practical route to constructing higher-point and higher-loop gravity amplitudes via gauge theory inputs, reinforcing the deep gauge–gravity connection and suggesting avenues for further extension.

Abstract

We expose simple and practical relations between the integrated four- and five-point one-loop amplitudes of N >= 4 supergravity and the corresponding (super-)Yang-Mills amplitudes. The link between the amplitudes is simply understood using the recently uncovered duality between color and kinematics that leads to a double-copy structure for gravity. These examples provide additional direct confirmations of the duality and double-copy properties at loop level for a sample of different theories.

Paper Structure

This paper contains 13 sections, 71 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review
  3. Duality between color and kinematics
  4. Gravity as a double copy of gauge theory
  5. Decomposing one-loop ${\cal N} \ge 4$ supergravity amplitudes.
  6. Implications of the duality at one loop
  7. Implications for generic one-loop amplitudes
  8. Four-point one-loop ${\cal N} \ge 4$ supergravity amplitudes
  9. Five-point one-loop ${\cal N} \ge 4$ supergravity amplitudes
  10. Comments on two loops
  11. Conclusions
  12. The one-loop five-point Yang-Mills amplitudes
  13. Integrals

Figures (4)

  • Figure 1: the one-loop $m$-gon master diagram for the cyclic ordering $1,2, \ldots, m$.
  • Figure 2: The basic Jacobi relation between three one-loop graphs that can be used to express any color factor or kinematic numerator factor for any one-loop graph in terms of the parent $m$-gons.
  • Figure 3: Pentagon and box integrals appearing in the ${{\cal N}=4}$ super-Yang-Mills five-point one-loop amplitudes. The complete set of such integrals is generated by permuting external legs and removing overcounts.
  • Figure 4: The two-loop cubic diagrams appearing in the two-loop four-point ${{\cal N}=4}$ and ${{\cal N}=8}$ supergravity amplitudes.