Overcoming Obstacles to Colour-Kinematics Duality at Two Loops

TL;DR

This paper explores the challenges in constructing colour-kinematics dual numerators at two loops for five-gluon amplitudes, overcoming obstacles by adding loop momentum and leveraging symmetries.

Contribution

It introduces a novel method to build dual numerators at two loops by incorporating additional loop momentum and symmetry considerations.

Findings

Successfully constructed dual numerators with increased loop momentum powers.

Identified symmetry properties that constrain numerator construction.

Demonstrated the approach on five-gluon all-positive helicity amplitude.

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.