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Overcoming Obstacles to Colour-Kinematics Duality at Two Loops

Gustav Mogull, Donal O'Connell

TL;DR

This work tackles the challenge of constructing colour-kinematics dual numerators at two loops for the five-point all-plus Yang-Mills amplitude. By starting from the Carrasco–Johansson N=4 pentabox and introducing a nonlocal, symmetry-constrained function $X$, the authors overcome obstructions that arise under conventional power counting, yielding numerators with $12$ loop-momentum powers. The construction hinges on enforcing strong automorphism conditions, analyzing spanning cuts, and embracing nonlocal kinematic structures, ultimately producing a valid BCJ representation that informs the gravity double-copy and related amplitudes. The results illuminate how obstructions can be managed via symmetry-guided, higher-momenta ansätze and motivate alternative Jacobi strategies based on spanning cuts.

Abstract

The discovery of colour-kinematics duality has allowed great progress in our understanding of the UV structure of gravity. However, it has proven difficult to find numerators which satisfy colour-kinematics duality in certain cases. We discuss obstacles to building a set of such numerators in the context of the five-gluon amplitude with all helicities positive at two loops. We are able to overcome the obstacles by adding more loop momentum to our numerator to accommodate tension between the values of certain cuts and the symmetries of certain diagrams. At the same time, we maintain control over the size of our ansatz by identifying a highly constraining but desirable symmetry property of our master numerator. The resulting numerators have twelve powers of loop momenta rather than the seven one would expect from the Feynman rules.

Overcoming Obstacles to Colour-Kinematics Duality at Two Loops

TL;DR

This work tackles the challenge of constructing colour-kinematics dual numerators at two loops for the five-point all-plus Yang-Mills amplitude. By starting from the Carrasco–Johansson N=4 pentabox and introducing a nonlocal, symmetry-constrained function , the authors overcome obstructions that arise under conventional power counting, yielding numerators with loop-momentum powers. The construction hinges on enforcing strong automorphism conditions, analyzing spanning cuts, and embracing nonlocal kinematic structures, ultimately producing a valid BCJ representation that informs the gravity double-copy and related amplitudes. The results illuminate how obstructions can be managed via symmetry-guided, higher-momenta ansätze and motivate alternative Jacobi strategies based on spanning cuts.

Abstract

The discovery of colour-kinematics duality has allowed great progress in our understanding of the UV structure of gravity. However, it has proven difficult to find numerators which satisfy colour-kinematics duality in certain cases. We discuss obstacles to building a set of such numerators in the context of the five-gluon amplitude with all helicities positive at two loops. We are able to overcome the obstacles by adding more loop momentum to our numerator to accommodate tension between the values of certain cuts and the symmetries of certain diagrams. At the same time, we maintain control over the size of our ansatz by identifying a highly constraining but desirable symmetry property of our master numerator. The resulting numerators have twelve powers of loop momenta rather than the seven one would expect from the Feynman rules.

Paper Structure

This paper contains 12 sections, 51 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Four-Point, Two-Loop BCJ System
  3. BCJ master numerators
  4. The maximally supersymmetric subsector
  5. Spanning cuts
  6. The Five-Point, Two-Loop BCJ System
  7. Symmetries and automorphisms
  8. Spanning cuts
  9. Nonlocality properties
  10. Attempting a minimal solution
  11. Solving the ansatz
  12. Conclusions

Figures (2)

  • Figure 1: The three two-loop spanning cuts required at four points. The first is a butterfly cut; the other two are satisfied automatically by the $\mathcal{N}=4$ subsector.
  • Figure 2: A spanning set of cuts at two loops and five points. The first two are butterfly cuts; the rest are satisfied automatically by the $\mathcal{N}=4$ subsector.