Local color-kinematics duality for one-loop pure Yang-Mills amplitudes
In plain words
Even at one loop it is unclear whether gluon amplitudes of ordinary Yang-Mills theory, with no supersymmetry, can be put in the special form using simple polynomial numerators.
Precise statement
Determine for which multiplicities n the one-loop n-gluon amplitude of pure Yang-Mills admits BCJ numerators that are local (polynomial in momenta and polarization vectors) and valid in arbitrary spacetime dimension D. An answer is the set of n for which such numerators exist, with constructions and obstructions, and a statement of whether dimension-specific (for example 4D) numerators escape any obstruction.
What would settle it
Explicit local numerators for all n, or a verified obstruction at a specific n.
Status in the literature
Unverified note
A September 2026 preprint (Sung, arXiv:2609.10693) reports that no dimension-independent, parity-even polynomial numerators multilinear in polarizations exist at $n = 6$; unrefereed and not independently checked; dimension-specific numerators are not excluded.