Bern–Carrasco–Johansson colour–kinematics duality conjecture

Let $\scA_{n,L}$ be a scattering amplitude represented using trivalent diagrams, with colour numerators $\sfc_i$ and kinematic numerators $\sfn_i$. A triple of trivalent diagrams $(i,j,k)$ has colour numerators obeying a Jacobi identity when

$$\sfc_i+\sfc_j+\sfc_k=0.$$

Bern–Carrasco–Johansson colour–kinematics duality conjecture. There exists a choice of kinematic numerators for the trivalent diagrams entering $\scA_{n,L}$ such that whenever a triple $(i,j,k)$ has colour numerators obeying a Jacobi identity, the corresponding kinematic numerators obey the same identity,

$$\sfn_i+\sfn_j+\sfn_k=0,$$

and if, in any individual diagram, interchanging two legs gives $\sfc_i\mapsto-\sfc_i$, then simultaneously $\sfn_i\mapsto-\sfn_i$.

This conjecture asserts a colour–kinematics correspondence for gauge-theory amplitudes and underlies the double-copy construction of gravity amplitudes. It was quickly established at tree level, while its validity at loop level remains conjectural.