Towards a Non-Perturbative Classical Double Copy

Abstract: We construct an exact solution-generating correspondence between restricted ansatz sectors of quartic biadjoint scalar theory, complexified $SU(2)$ Yang--Mills theory and four-dimensional general relativity with conformally flat metrics and traceless energy-momentum. A factorised biadjoint field, a Corrigan--Fairlie--'t Hooft--Wilczek gauge potential and a conformal metric are generated by a common scalar seed. Their equations of motion reduce to the same equation of motion under the identification of couplings, which retains the nonlinear dynamics on all three legs of the correspondence. The scalar equation captures only the trace of the Einstein equations; deriving the trace-free part of the Einstein equation additionally yields a source relation between Gravity and Yang-Mills. From this, we construct a non-perturbative classical double copy for the restricted ansatz sectors between quartic biadjoint scalar theory, $SU(2)$ Yang--Mills theory and gravity. We illustrate the dictionary of mappable solutions with rational and constant profiles representing $AdS_4$, planar $dS_4$ and Minkowski space, and with Jacobi-elliptic profiles generating a positive-energy recollapsing branch for negative cosmological constant and a negative-energy bouncing branch for positive cosmological constant. Using these results, we provide a new example of the Kerr-Schild double copy.

• The paper constructs an exact, fully nonlinear correspondence among quartic biadjoint scalar theory, complexified SU(2) Yang–Mills theory, and conformally flat gravity through the shared equation □φ + λφ³ = 0.
• The method identifies the coupling relation 4μ²λ₄ = λ = 2Λ/3 and derives the gravitational double-copy structure Kμν = Qμν/φ² by imposing the trace-free Einstein equations and matching rescaled energy-momentum tensors.
• The solution dictionary includes vacuum AdS₄, de Sitter and Minkowski branches, recollapsing and bouncing radiation-dominated FLRW cosmologies, Bertotti–Robinson spacetime, and a curved-background Kerr–Schild construction, while remaining restricted to special ansätze and SU(2)-type sectors.

The classical double copy has historically been restricted to solutions that linearise the gravitational field equations or confine the gauge sector to an effectively Abelian configuration. The paper "Towards a Non-Perturbative Classical Double Copy" (2608.14512) addresses this restriction by constructing an exact, fully nonlinear solution-generating correspondence between three theories: quartic biadjoint scalar theory on a $\mathfrak{su}(2)\times\mathfrak{su}(2)$ colour algebra, complexified $SU(2)$ Yang–Mills theory in four dimensions, and general relativity restricted to conformally flat metrics with traceless energy-momentum. All three sectors are generated from a single scalar seed $\phi$, and their equations of motion reduce to the same nonlinear equation $\Box\phi + \lambda\phi^3 = 0$ under appropriate ansätze.

The common scalar reduction

The first leg of the correspondence is purely quartic biadjoint scalar theory, obtained by setting the mass, cubic coupling and current to zero in the generalised biadjoint Lagrangian. Fixing both Lie algebras to $\mathfrak{su}(2)$ converts the structure constants to Levi-Civita symbols, and factorising the biadjoint field as $\Phi^{aa'} = \mu U^{aa'}\phi$ with $U \in O(3)$ reduces the equation of motion to

$\phi + 4\mu^2\lambda_4\,\phi^3 = 0,$

where the dimensional parameter $\mu$ (of mass dimension one) offsets the dimensionless quartic coupling $\lambda_4$.

On the gauge side, the Corrigan–Fairlie–'t Hooft–Wilczek ansatz $SU(2)$0 — with $SU(2)$1 built from the $SU(2)$2 Pauli matrices — reduces the full nonlinear $SU(2)$3 Yang–Mills equations to $SU(2)$4, where $SU(2)$5 is an integration constant of mass dimension two. A notable structural feature of this ansatz is that the resulting vector field generates volume-preserving diffeomorphisms ($SU(2)$6), inducing nontrivial colour–spacetime mixing; the author suggests this may connect to kinematic algebras, though this remains speculative. Identifying couplings via $SU(2)$7 yields a direct non-perturbative map between the two reduced sectors: every solution of the reduced Yang–Mills equation has a counterpart in quartic biadjoint scalar theory.

The gravitational sector and the source relation

For conformally flat metrics $SU(2)$8 with conformally invariant matter ($SU(2)$9), the trace of the Einstein equations gives $\phi$0, which reduces to

$\phi$1

matching the other two reductions under $\phi$2. Crucially, the paper is explicit that this trace equation alone does not determine a gravitational solution: the trace-free part of the Einstein equation must also be imposed. Doing so fixes the admissible traceless source,

$\phi$3

for one-parameter profiles $\phi$4 with $\phi$5, where $\phi$6 is the conserved quantity from the first integral of the reduced equation. Comparing with the Yang–Mills energy-momentum tensor for the same ansatz yields the conformal rescaling relation $\phi$7 between the gauge and gravitational sources. Encapsulating the left-hand side of the Einstein equations as $\phi$8, the result takes the double-copy form

$\phi$9

with $\Box\phi + \lambda\phi^3 = 0$0 a rescaled square of the Yang–Mills field strength. This differs structurally from prior Kerr–Schild-type constructions in three ways: it is non-perturbative on both sides, the denominator involves a solution of quartic rather than cubic biadjoint theory, and the scalar appears squared rather than unsquared. The explicit $\Box\phi + \lambda\phi^3 = 0$1 dependence cancels against the overall factor of $\Box\phi + \lambda\phi^3 = 0$2 in the evaluated Yang–Mills energy-momentum tensor, so the final relation carries no explicit coupling dependence.

Solution dictionary

The plane-wave-type solutions of the reduced equation are catalogued for rational and Jacobi-elliptic profiles:

Profile	Condition	Gravitational interpretation
$\Box\phi + \lambda\phi^3 = 0$3	$\Box\phi + \lambda\phi^3 = 0$4	Vacuum: $\Box\phi + \lambda\phi^3 = 0$5 (Poincaré patch), planar $\Box\phi + \lambda\phi^3 = 0$6, or Minkowski
$\Box\phi + \lambda\phi^3 = 0$7	$\Box\phi + \lambda\phi^3 = 0$8	Positive-radiation recollapsing FLRW, $\Box\phi + \lambda\phi^3 = 0$9
$\mathfrak{su}(2)$0	$\mathfrak{su}(2)$1	Negative-radiation bouncing FLRW, $\mathfrak{su}(2)$2
$\mathfrak{su}(2)$3	$\mathfrak{su}(2)$4	Related to $\mathfrak{su}(2)$5 by shift/scaling
$\mathfrak{su}(2)$6	$\mathfrak{su}(2)$7	Related to $\mathfrak{su}(2)$8 by shift/scaling

All elliptic solutions are complex-valued yet yield real energy-momentum tensors, analogous to classical electromagnetic waves. For the vacuum branch, the construction reproduces known results from a new angle: Minkowski space corresponds to a trivial gauge field $\mathfrak{su}(2)$9 and a constant biadjoint profile, re-deriving the standard dictionary entry non-perturbatively.

On the cosmological branch, the $\Phi^{aa'} = \mu U^{aa'}\phi$0 seed with $\Phi^{aa'} = \mu U^{aa'}\phi$1 gives a radiation-dominated universe ($\Phi^{aa'} = \mu U^{aa'}\phi$2) with positive energy density $\Phi^{aa'} = \mu U^{aa'}\phi$3 whose squared scale factor in cosmic time behaves as $\Phi^{aa'} = \mu U^{aa'}\phi$4: the universe expands to maximum size at $\Phi^{aa'} = \mu U^{aa'}\phi$5 and recollapses at $\Phi^{aa'} = \mu U^{aa'}\phi$6. Conversely, the $\Phi^{aa'} = \mu U^{aa'}\phi$7 seed with $\Phi^{aa'} = \mu U^{aa'}\phi$8 produces a bouncing cosmology with scale factor $\Phi^{aa'} = \mu U^{aa'}\phi$9, but at the cost of a strictly negative radiation energy density — a physically unpalatable feature the paper states plainly rather than resolving.

The Bertotti–Robinson spacetime ($U \in O(3)$0 electrovacuum) extends the dictionary beyond the single-variable parametrisation onto the harmonic/self-dual branch $U \in O(3)$1, with seed $U \in O(3)$2 mapping to a Coulombic Yang–Mills potential and a biadjoint field scaling as $U \in O(3)$3 while the field strength scales as $U \in O(3)$4.

Relation to the Kerr–Schild double copy

Using the Bertotti–Robinson metric as a background, the paper constructs a new Kerr–Schild-type correspondence. Because the background gauge field satisfies $U \in O(3)$5, any Abelian Kerr–Schild potential $U \in O(3)$6 can be embedded into a full $U \in O(3)$7 solution via $U \in O(3)$8, giving a background-plus-perturbation dictionary $U \in O(3)$9 and $\phi + 4\mu^2\lambda_4\,\phi^3 = 0,$0. This is claimed to be the first such correspondence between a gravitational perturbation about a curved background and solutions of the full $\phi + 4\mu^2\lambda_4\,\phi^3 = 0,$1 Yang–Mills equations. However, the author concedes that the effective dynamics reduce to a $\phi + 4\mu^2\lambda_4\,\phi^3 = 0,$2 subsector, so this example should not be regarded as fully nonlinear.

Limitations and open questions

Several restrictions bound the scope of the result. The correspondence holds only within the restricted ansatz sectors: factorised colour structure for the biadjoint field, the Corrigan–Fairlie–'t Hooft–Wilczek form for the gauge field, and conformal flatness (Petrov type O, vanishing Weyl tensor) for gravity. Because the Weyl tensor vanishes, the standard Weyl double copy cannot interpret these solutions, and the spinorial origin of the map — whether Ricci- and $\phi + 4\mu^2\lambda_4\,\phi^3 = 0,$3-driven spinor structures can play the role the Weyl spinor plays in the Kerr–Schild case — is left open. The embedding into larger gauge groups such as $\phi + 4\mu^2\lambda_4\,\phi^3 = 0,$4 via an $\phi + 4\mu^2\lambda_4\,\phi^3 = 0,$5 subgroup introduces no genuinely new colour dynamics; whether constant colour tensors exist whose contractions reproduce the scalar reduction without confinement to an $\phi + 4\mu^2\lambda_4\,\phi^3 = 0,$6 subalgebra is unresolved, and would determine whether the role of the Levi-Civita symbol and the $\phi + 4\mu^2\lambda_4\,\phi^3 = 0,$7 matrix is accidental or part of a broader algebraic mechanism. Finally, the presence of $\phi + 4\mu^2\lambda_4\,\phi^3 = 0,$8 in the solution family raises but does not establish a holographic connection: no boundary conditions, boundary data, or renormalised observables are specified, so no AdS/CFT dictionary follows from the bulk correspondence alone.

Conclusion

This paper establishes an exact three-way correspondence among restricted sectors of quartic biadjoint scalar theory, complexified $\phi + 4\mu^2\lambda_4\,\phi^3 = 0,$9 Yang–Mills theory, and conformally flat gravity with traceless sources, all governed by the shared nonlinear seed equation $\mu$0 under the coupling identification $\mu$1. The trace-free Einstein equation supplies the missing ingredient — a conformal relation between gauge and gravitational energy-momentum tensors — yielding a genuinely non-perturbative double-copy formula $\mu$2. The dictionary covers vacuum, bouncing and recollapsing FLRW, and Bertotti–Robinson geometries, and produces a novel Kerr–Schild copy about a curved background. Whether the common scalar seed reflects a deep structural principle or an artefact of these particular ansätze remains the central open question.