Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Colour-Kinematics Duality for One-Loop Rational Amplitudes

Rutger H. Boels, Reinke Sven Isermann, Ricardo Monteiro, Donal O'Connell

TL;DR

This work identifies two infinite classes of one-loop Yang–Mills amplitudes (all-plus and one-minus) that admit colour–kinematics dual numerators derived from the self-dual sector, and shows how the gravity amplitudes emerge through the double copy. It provides explicit constructions and checks at low points, analyzes how CK duality persists after integration to yield relations among partial amplitudes, and extends the framework to colour-dual form factors. The results illuminate the algebraic structure underlying perturbative gauge theories and their gravitational counterparts, with potential implications for ultraviolet behavior and connections to supersymmetric theories via the double copy. Overall, the paper advances the understanding of CK duality at loop level and offers practical, dual representations for a broad class of rational amplitudes and observables.

Abstract

Colour-kinematics duality is the conjecture of a group theory-like structure for the kinematic dependence of scattering amplitudes in gauge theory and gravity. This structure has been verified at tree level in various ways, but similar progress has been lacking at loop level, where the power of the duality would be most significant. Here we explore colour-kinematics duality at one loop using the self-dual sector as a starting point. The duality is shown to exist in pure Yang-Mills theory for two infinite classes of amplitudes: amplitudes with any number of particles either all of the same helicity or with one particle helicity opposite the rest. We provide a simple Lagrangian-based argument in favour of the double copy relation between gauge theory and gravity amplitudes in these classes, and provide some explicit examples. We further discuss aspects of the duality which persist after integration, leading to relations among partial amplitudes. Finally, we describe form factors in the self-dual theory at tree level which also satisfy the duality.

Colour-Kinematics Duality for One-Loop Rational Amplitudes

TL;DR

This work identifies two infinite classes of one-loop Yang–Mills amplitudes (all-plus and one-minus) that admit colour–kinematics dual numerators derived from the self-dual sector, and shows how the gravity amplitudes emerge through the double copy. It provides explicit constructions and checks at low points, analyzes how CK duality persists after integration to yield relations among partial amplitudes, and extends the framework to colour-dual form factors. The results illuminate the algebraic structure underlying perturbative gauge theories and their gravitational counterparts, with potential implications for ultraviolet behavior and connections to supersymmetric theories via the double copy. Overall, the paper advances the understanding of CK duality at loop level and offers practical, dual representations for a broad class of rational amplitudes and observables.

Abstract

Colour-kinematics duality is the conjecture of a group theory-like structure for the kinematic dependence of scattering amplitudes in gauge theory and gravity. This structure has been verified at tree level in various ways, but similar progress has been lacking at loop level, where the power of the duality would be most significant. Here we explore colour-kinematics duality at one loop using the self-dual sector as a starting point. The duality is shown to exist in pure Yang-Mills theory for two infinite classes of amplitudes: amplitudes with any number of particles either all of the same helicity or with one particle helicity opposite the rest. We provide a simple Lagrangian-based argument in favour of the double copy relation between gauge theory and gravity amplitudes in these classes, and provide some explicit examples. We further discuss aspects of the duality which persist after integration, leading to relations among partial amplitudes. Finally, we describe form factors in the self-dual theory at tree level which also satisfy the duality.

Paper Structure

This paper contains 12 sections, 118 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Review and formalism
  3. Generalities of colour-kinematics duality
  4. Colour-kinematics duality in the self-dual sector and MHV amplitudes
  5. Manifestly dual integrands at one loop
  6. Self-dual gauge theory and one-loop all-plus amplitudes
  7. One-loop one-minus amplitudes
  8. Double copy
  9. Examples
  10. Colour-kinematics duality after integration
  11. A series of colour-dual form factors
  12. Discussion and conclusions

Figures (5)

  • Figure 1: An all-plus diagram.
  • Figure 2: The one-loop $(+++)$ current of Yang-Mills theory. Leg 3 is off-shell.
  • Figure 3: The one-loop $(-++)$ current of Yang-Mills theory. Leg 3 is off-shell.
  • Figure 4: A box cut of an $n$ point amplitude.
  • Figure 5: This subamplitude is part of the pentagon contribution to the box cut.