The Domain of Color–Kinematics Duality and Double Copy

Color–kinematics duality reorganizes gauge-theory amplitudes so that kinematic numerators obey Jacobi relations mirroring color factors; replacing color by a second numerator can then produce gravity amplitudes. The construction is exceptionally well tested, but “the double copy works” hides several inequivalent claims. Tree amplitudes in ordinary Yang–Mills have the firmest footing; loop-level, matter-coupled, off-shell, and observable-level extensions have theory-dependent evidence and no single all-purpose theorem.

Evidence cutoff. 11 August 2026.

Required background. Color–kinematics duality fixes the cubic-graph numerator problem; the double copy supplies the replacement rule and its state-content checks. Helpful background. Color decomposition separates group theory from partial amplitudes, three-point amplitudes supplies on-shell seeds, and symmetry, gauge redundancy, and duality prevents a representation choice from being mistaken for an observable symmetry.

Demonstrated domains of the duality

Normative question. Under which theories, representations, loop orders, and observable conditions is color–kinematics duality demonstrated rather than conjectured?

The scope is perturbative scattering in flat spacetime. “Demonstrated” means either proved for the stated amplitude class or represented by explicit numerators whose generalized unitarity cuts and integrated amplitude checks close. This page does not treat the classical double copy as automatically equivalent to the amplitude construction, claim a nonperturbative definition of quantum gravity, or infer a physical symmetry from one convenient integrand.

For tree-level pure Yang–Mills amplitudes, Bern, Carrasco, and Johansson identified kinematic Jacobi relations and the resulting amplitude relations (BCJ 2008). On-shell recursion and scattering-equation formulations subsequently supplied broad tree-level constructions. At this level, the existence of color-dual representations and the gravity double copy is established for standard adjoint Yang–Mills amplitudes, while the numerators themselves remain non-unique under generalized gauge transformations.

At loop level, the integrand is expanded schematically as

$$\mathcal A^{(L)}=\sum_i\int\!\frac{d^{LD}\ell}{(2\pi)^{LD}}\, \frac{c_i n_i}{S_i\prod_{\alpha_i}p_{\alpha_i}^2}, \qquad c_i+c_j+c_k=0\ \Longrightarrow\ n_i+n_j+n_k=0.$$

Many explicit integrands satisfy these relations and reproduce all generalized cuts; replacing $c_i$ by a second numerator then yields tested gravity integrands (BCJ 2010). But cut agreement, locality, graph completeness, regulator choice, and integrated counterterms must all be checked. A representation valid on cuts may require contact terms away from them, and an integrand identity can be obscured or altered by integration.

Claim class	Strongest support	Limitation
Tree-level adjoint Yang–Mills	General constructions, amplitude relations, factorization, and gravity matching	Numerator representation is not unique
Tree amplitudes with fundamental matter	Explicit QCD-like constructions and state projections	Matter representation and unwanted double-copy states require case-by-case control
Supersymmetric loop amplitudes	High-loop explicit integrands and generalized-cut verification	Evidence at selected multiplicities and loop orders is not an all-loop existence theorem
Nonsupersymmetric loop amplitudes	Numerous low-loop examples and counterterm analyses	Rational terms, regularization, anomalies, and contact terms are more delicate
Actions, curved backgrounds, and classical solutions	Manifest-duality actions or classical maps in bounded sectors	Not automatically equivalent to the full quantum S-matrix statement

Johansson and Ochirov constructed color-dual tree amplitudes with massive flavored matter, showing that fundamental representations are not intrinsically excluded (Johansson and Ochirov 2016). Conversely, a naïve double copy can produce gravity coupled to dilatons, antisymmetric tensors, ghosts, or extra matter. Obtaining pure Einstein gravity or a chosen supergravity requires projections or compensating matter whose unitarity must be checked.

Competing formulations and obstructions

There are three serious formulations: amplitude-level existence of Jacobi-satisfying numerators; algebraic or action-level kinematic Lie structures; and geometric or string-inspired origins such as Kawai, Lewellen, and Tye 1986, tree-level string relations. They overlap, but none currently proves every version of the others. The action-level construction of Borsten and collaborators argues for loop duality up to Jacobian counterterms (Borsten et al. 2021); the counterterm qualification is material, not cosmetic.

Known obstacles include the rapid growth of graph ansätze, locality versus manifest Jacobi identities, regularization dependence, evanescent operators, anomalous matter, and difficulty finding numerators that satisfy all cuts without nonlocal terms. Failure to find a representation is not a counterexample to existence. A genuine counterexample would prove incompatibility among locality, unitarity cuts, power counting, and the Jacobi system for a specified amplitude class.

Assessment. The question is partially resolved. Tree-level adjoint Yang–Mills and many defined extensions are demonstrated. Loop-level duality is strongly supported across important examples, especially supersymmetric ones, but remains conjectural as a universal statement. Classical, off-shell, curved-background, and nonperturbative versions must carry their own scope labels.

What would settle a claimed domain?

For a fixed theory and regulator, resolution requires either an all-multiplicity, all-loop construction whose generalized cuts, factorization, locality, state content, and counterterms are proved, or a no-go theorem exhibiting an incompatible cut or anomaly. A double-copy claim additionally needs a spectrum projection and equality to independently constructed gravity observables—not merely Jacobi-looking numerators.

Related routes include amplitudes and precision scattering, multiloop amplitudes and resummation, and holographic reconstruction and gravitational path integrals. The flat-space S-matrix limit is optional depth when comparing with AdS-derived amplitudes.

Evidence cutoff and source selection

The finite set emphasizes foundational constructions, representative matter and loop extensions, and explicit caveats about actions and counterterms. Targeted arXiv, INSPIRE, journal, and forward-citation searches were screened through 11 August 2026; applications lacking a full cut or observable check were not treated as independent demonstrations.

References

• Bern, Zvi, John Joseph M. Carrasco, and Henrik Johansson. “New Relations for Gauge-Theory Amplitudes.” Physical Review D 78 (2008): 085011. DOI.
• Bern, Zvi, John Joseph M. Carrasco, and Henrik Johansson. “Perturbative Quantum Gravity as a Double Copy of Gauge Theory.” Physical Review Letters 105 (2010): 061602. DOI.
• Borsten, Leron, et al. “Tree-Level Color–Kinematics Duality Implies Loop-Level Color–Kinematics Duality up to Counterterms.” arXiv:2108.03030 (2021). arXiv.
• Johansson, Henrik, and Alexander Ochirov. “Color–Kinematics Duality for QCD Amplitudes.” Journal of High Energy Physics 2016, no. 1 (2016): 170. DOI.
• Kawai, Hikaru, David C. Lewellen, and S.-H. Henry Tye. “A Relation Between Tree Amplitudes of Closed and Open Strings.” Nuclear Physics B 269 (1986): 1–23. DOI.