Extended Massive Double-Copy Approach

• Extended Massive Double-Copy Approach is a generalization of the BCJ framework that incorporates massive fields, compactified extra dimensions, and higher-derivative effects.
• The approach employs shifted BCJ relations, Kaluza–Klein spectral constraints, and loop-level subtraction schemes to construct consistent gauge and gravity amplitudes.
• Implications include new methods for generating gravitational amplitudes in topologically massive theories, effective field theory regimes, and UV completions for both classical and quantum sectors.

The Extended Massive Double-Copy Approach generalizes the color-kinematics duality and double-copy paradigm to settings involving massive fields, compactified extra dimensions, nontrivial mass spectra, and higher-derivative completions. Recent developments span tree and loop amplitudes in gauge and gravity theories, consistent Kaluza-Klein reductions, topologically massive sectors, string theory origins, and resummed effective field theory expansions. This article provides a comprehensive survey of technical frameworks, algebraic mechanisms, and physical implications of the extended massive double-copy, referencing recent explicit constructions and theoretical advances.

1. Origins and Generalization of Color-Kinematics Duality

The original double-copy relation, rooted in the Bern–Carrasco–Johansson (BCJ) paradigm, establishes a structural equivalence between color factors and kinematic numerators in gauge-theory amplitudes, enabling the construction of gravitational amplitudes by replacing color structures with a second (possibly identical) copy of the kinematic numerators (Johansson et al., 2019). In pure-adjoint, massless gauge theories, amplitudes can be organized as

$$\mathcal{A}_n = \sum_{j \in \Gamma} \frac{c_j n_j}{D_j}$$

where $c_j$ are color factors, $n_j$ are local kinematic numerators, and $D_j$ are propagators. The fundamental requirement is that both $c_j$ and $n_j$ satisfy isomorphic Jacobi identities. The gravitational amplitude is then obtained as

$\mathcal{M}_n = \sum_j \frac{n_j \tilde n_j}{D_j}$

with, for instance, $\tilde n_j$ from a second copy of the Yang–Mills numerators.

Extensions to massive representations—such as fundamental fermions, massive scalars, and Kaluza–Klein towers—were developed by constructing color-kinematics dual representations at tree and loop level. For massive QCD with Dirac fermions or scalars, the same bootstrap and Jacobi relations enforce a color-dual structure, with additional technical tools to handle spinor chains and flavor indices (Carrasco et al., 2023, Johansson et al., 2019, Carrasco et al., 2020).

2. Kaluza–Klein Compactification and Mass-Spectral Constraints

The double-copy for massive Kaluza–Klein states arises via dimensional reduction (on tori or warped backgrounds), where both gauge and gravity amplitudes in $d$ dimensions inherit infinite towers of massive modes from a higher-dimensional theory (Li et al., 2022, Hang et al., 18 Jun 2024, Li et al., 2021). The organizing principle is the "shifting prescription": $s_{ij} \rightarrow s_{ij} - (n_i + n_j)^2 / R^2$ where $n_i$ is the KK level and $R$ the compactification radius. BCJ numerators constructed in higher dimensions can be directly mapped, yielding massive amplitudes that satisfy generalized Jacobi and spectral constraints. The massive BCJ relations and KLT-type double-copy at $N$ points require stringent spectral conditions, satisfied automatically for standard KK towers but generally violated for arbitrary mass spectra (Johnson et al., 2020, González et al., 2022).

In warped compactifications (with nonflat metrics), explicit alignment of gauge and gravity couplings, masses, and wavefunction overlap integrals is necessary. The equivalence theorems (GAET, GRET) relate amplitudes of physical vector/graviton longitudinal modes to the Goldstone sector and propagate from 3-point to $N$-point amplitudes at leading order in $E/M$ (Hang et al., 18 Jun 2024).

3. Loop-Level and Effective Field Theory Extensions

At loop level, the extended double-copy is realized through color-dual integrands for massive matter in the fundamental or adjoint and generalized unitarity cut bootstraps. The loop construction is formalized as a sum over cubic graphs, with numerator functionals constrained by maximal and next-to-maximal cuts, Jacobi relations, and functional ansätze adapted for fermionic/chiral/massive sectors (Carrasco et al., 2023, Carrasco et al., 2020, Carrasco et al., 2021).

Advanced frameworks extend this to effective field theory (EFT) completion. The Consistent Massive Resonance Double-Copy (CMRDC) employs resummed higher-derivative operators to exponentiate contact terms and generate infinite towers of massive poles in the UV, with all IR Wilson coefficients tied to the spectrum by double-copy consistency and bootstrapped via color-kinematics (Carrasco et al., 2023). The associated inverse problem of reconstructing the UV spectrum from finite IR data is tractable via Padé extrapolation.

In gravity, loop-level double-copy constructions for massive vector (Proca) or scalar loops are elegantly realized in the Landau–Lifshitz metric-density formalism. Here, master numerators factorize into products of purely scalar and vector structures and enable the construction of all-plus and mostly-plus gravity amplitudes in an "extended massive double-copy" form, valid to higher multiplicity and directly compatible with the single/double copy for Yang–Mills, gravity plus scalar, or gravity plus Proca sectors (Lopez-Arcos, 21 Oct 2025).

4. Extended Double Copy in Three Dimensions and Twistor Approaches

A distinct regime arises in three-dimensional topologically massive theories. The BCJ consistency requirement reduces to a single nontrivial relation at five points, circumventing the usual proliferation of spurious poles (González et al., 2021). Topologically Massive Yang–Mills (TMYM) and Topologically Massive Gravity (TMG) enjoy a fully consistent double-copy structure up to five points, with the mass parameter mapping identically from gauge to gravity.

A further extension is the twistor-based double-copy for topologically massive fields, formulated in minitwistor space in terms of massive Penrose transforms and cohomology products. Here, a general multiplicative structure among representatives $f_{-2n}$ in $H^1(\mathrm{MT}, \mathcal{O}(N, M))$ encodes the relationship between gauge and Cotton tensor sectors. While the position-space Cotton double copy is valid only for type N solutions, the twistor-space double copy is universally valid across algebraic types, provided the cohomology classes are fixed from scattering amplitudes (González et al., 2022).

5. String Theory, Worldline, and Nonlinear Sigma Models

The string-theory realization of the massive double-copy uses the explicit factorization between open-string (vector) and closed-string (massive spin-2) vertices under compactification (Lust et al., 2023). The three-point amplitudes for massive spin-2 closed-string states match those of ghost-free bimetric theory, with extended brane SUSY ensuring absence of higher-derivative pathologies.

Worldline quantum field theory (WQFT) furnishes a classical all-order double-copy, mapping observables (eikonal phase, impulse, radiation) sequentially from bi-adjoint scalar to Yang–Mills to dilaton gravity. The locality kernel of trivalent graphs is preserved, and the classical limit of quantum amplitudes reproduces WQFT results (Shi et al., 2021). The double copy for spinning and extended objects requires precise algebraic mapping of spins, tidal operators, and worldline actions, with systematic constraints on Wilson coefficients (Goldberger et al., 2019, Bautista et al., 2019).

In scalar field theories, the extended massive double-copy is realized by the Kaluza–Klein reduction of the nonlinear sigma model and special Galileon in five dimensions, or by the construction of infinite towers and cyclic-antisymmetric vertices satisfying BCJ duality (González et al., 2022). The resulting local double-copy structure mirrors the massless case once the mass-spectral condition is imposed.

6. Constraints, Subtleties, and UV Completion

A generic obstacle for extending the double copy to massive theories is the appearance of spurious singularities at $n\ge5$ unless strict spectral conditions are satisfied. For uniform mass spectra (as in massive Yang–Mills), the KLT kernel for the double copy develops nonphysical poles unless all four- and five-point mass conditions are imposed, which is generally feasible only for Kaluza–Klein-type spectra (Johnson et al., 2020, González et al., 2022).

Subtraction schemes, such as the projection onto the symmetric traceless subspace for gravitons, are applied at the level of loop integrands to systematically remove unwanted dilaton and axion states from the naive double copy, isolating pure Einstein-Hilbert contributions (Carrasco et al., 2021). In topologically massive theories, the amplitude-level double copy is only valid after nontrivial shifts of the numerators, and the eikonal exponentiation provides a route for consistent loop-level resummation in the high-energy regime (González et al., 2021).

7. Physical Applications and Future Directions

The extended massive double-copy framework supplies a general algorithmic procedure for constructing gravitational amplitudes with massive matter, external legs, and nontrivial spectra, including bimetric and dRGT gravities with fixed ghost-free parameters (Momeni et al., 2020, Lust et al., 2023). It underlies analytical computations in classical radiation (black hole encounters) and post-Minkowskian expansions. The string and twistor perspectives suggest further extensions to curved backgrounds, higher-spin sectors, and possible links with worldsheet and cohomological formulations.

Summary table illustrating the landscape:

Setting	Key double-copy feature	Reference(s)
Kaluza–Klein field theory	Shifted BCJ structure, spectral conds	(Li et al., 2022, Hang et al., 18 Jun 2024, Li et al., 2021)
Massive loop-level amplitudes	Color-dual functionals, subtraction	(Carrasco et al., 2023, Carrasco et al., 2020, Lopez-Arcos, 21 Oct 2025, Carrasco et al., 2021)
Topologically massive (3d)	Unique BCJ constraint, Gram-null	(González et al., 2021, González et al., 2021, González et al., 2022)
String theory (compactification)	Massive KLT via open-string factorization	(Lust et al., 2023, Li et al., 2021)
EFT effective completion	CMRDC, resummation, UV spectrum	(Carrasco et al., 2023)
Scalar field theory	Kaluza–Klein & cyclic 3-pt extends	(González et al., 2022)
Spinning/extended sources	Worldline double copy, coefficient constraints	(Goldberger et al., 2019, Bautista et al., 2019)

The theoretical structure underscores a unified algebraic mechanism (color-kinematics duality) that persists through mass deformations, compactifications, higher-loop generalizations, and effective completions, provided that the generalized BCJ relations, mass-spectral constraints, and kernel/integrand factorization are respected. Future research is poised to address systematic higher-point and higher-loop constructions, the implementation in curved and nontrivial string backgrounds, and the integration of extended double-copy methods with string and twistor-theoretic techniques.