Pith. sign in

REVIEW 1 major objections 4 minor 90 references

Time-dependent solutions of biadjoint scalar field theories

T0 review · 1 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Biadjoint scalar field theory admits exact time-dependent plane-wave solutions, including bounded oscillatory and non-oscillatory profiles, obtained by reducing the field equations to a single ordinary differential equation.

desk verdict First time-dependent biadjoint solutions are real and checkable, but eq. (7) as printed has a quartic-index typo that must be fixed before the paper is usable. read the letter →

arxiv 2502.01294 v1 pith:DULSIWSP submitted 2025-02-03 hep-th

Kymani Armstrong-Williams , Chris D. White This is my paper · ORCID
classification hep-th
keywords biadjointscalarfieldtheoryexactclassicalsolutionstravellingwavesdoublecopysolitonsWeierstrassellipticfunctionsJacobinon-perturbativemethods
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Biadjoint scalar field theory describes a field valued in two Lie algebras and appears as the zeroth-copy building block in the double copy between gauge theory and gravity. The paper establishes that, once mass, quartic interactions, and a constant background current are added, this theory has exact time-dependent solutions of plane-wave form: travelling waves with a nontrivial profile along the direction of motion. The key step is a colour-diagonal ansatz \$Phi^{{aa'}}$=\$delta^{{aa'}}$f(p\cdot x) with identical Lie algebras in the two sectors, which makes the coupled nonlinear field equations collapse into a single ordinary differential equation. Solving that equation yields analytic profiles, including bounded solutions, which all previously known exact biadjoint solutions lacked. These exact non-linear waves give a concrete arena for probing non-perturbative aspects of the classical double copy.

What carries the argument

The machinery is the colour-diagonal plane-wave ansatz \$Phi^{{aa'}}$(x)=\$delta^{{aa'}}$f(p\cdot x), with \xi=p\cdot x, identical Lie algebras in both sectors, and a constant diagonal current J\$delta^{{aa'}}$. It does two jobs: it creates cross-talk between the two colour sectors (a factorised ansatz \$Phi^{{aa'}}$=\chi^a\$eta^{{a'}}$ makes all non-linear terms vanish), and it converts the partial differential equation into an autonomous second-order ODE equivalent to energy conservation for a point particle moving in the one-dimensional potential V(f). The resulting energy integral is what makes the solutions analytically tractable.

What would settle it

Choose two Lie algebras whose structure constants are not proportional and substitute the diagonal plane-wave ansatz into the field equation: the cubic colour factor $f^{{abc}}$\tilde $f^{{a'bc}}$ is not proportional to \$delta^{{aa'}}$, so the equation does not reduce to the ordinary differential equation used in the paper and the listed profiles are not solutions. A reader could verify this by direct substitution into eq. (7).

Watch

Extended reading notes

Core claim

The central claim is that the generalised biadjoint scalar equation (\$partial^{2}$+$m^{2}$)\$Phi^{{aa'}}$+y $f^{{abc}}$\tilde $f^{{a'b'c'}}$\$Phi^{{bb'}}$\$Phi^{{cc'}}$+\$\lambda$ $f^{{ebc}}$\tilde $f^{{a'b'c'}}$$f^{{eda}}$\tilde $f^{{e'd'a'}}$\$Phi^{{bb'}}$\$Phi^{{cc'}}$\$Phi^{{dd'}}$=$J^{{aa'}}$ admits exact plane-wave solutions. Taking identical Lie algebras and substituting \$Phi^{{aa'}}$=\$delta^{{aa'}}$f(p\cdot x) with constant current $J^{{aa'}}$=J\$delta^{{aa'}}$ reduces the equation to $p^{2}$ f''+$m^{2}$ f+yT_A $f^{2}$+\$\lambda$ $T_A^{2}$ $f^{3}$=J, where T_A is the adjoint Casimir $f^{{abc}}$$f^{{a'bc}}$=T_A\$delta^{{aa'}}$. This autonomous ODE integrates once to \frac12 $p^{2}$ (f')^2+V(f)=\varepsilon, with V(f)=\frac{$m^{2}$$f^{2}$}{2}+\frac{$yT_Af^{3}$}{3}+\frac{\$\lambda$ $T_A^{2}$$f^{4}$}{4}-Jf. Depending on the couplings, the sign of $m^{2}$, and the integration constant \varepsilon, the profile f is a Weierstrass elliptic function, a Jacobi elliptic function, a kink-type hyperbolic profile, or the rational bounded form f(\xi)=-\frac{$12p^{2}$y}{9\$\lambda$ $p^{2}$T_A+$2T_Ay^{2}$(\xi-c)^2} arising when cubic and quartic terms coexist with m=J=\varepsilon=0. These are genuinely time-dependent, non-linear generalisations of plane waves, and some of them are bounded and non-oscillatory.

Load-bearing premise

The construction rests on the field being colour-diagonal, \$Phi^{{aa'}}$=\$delta^{{aa'}}$f(p\cdot x), with the same Lie algebra in both sectors and a constant diagonal current J\$delta^{{aa'}}$; if any of those fail, the nonlinear terms do not collapse into the single ordinary differential equation that produces all the solutions.

Editorial extensions

If this is right

  • The catalogue of exact biadjoint solutions expands from static monopole-like objects, wires, and Euclidean instantons to genuinely time-dependent travelling waves.
  • Bounded solutions now exist in biadjoint scalar theory: oscillatory waves generated by a constant current or negative mass-squared, and a non-oscillatory rational profile when cubic and quartic couplings coexist.
  • Quartic biadjoint theory embeds well-known scalar solitons, including Jacobi elliptic functions and a kink-type profile, as special cases, so those known solutions become biadjoint solutions as well.
  • Because biadjoint scalar theory is the zeroth copy of gauge theory, these exact non-linear waves provide test objects for whether classical solutions double-copy to gauge and gravity solutions beyond perturbation theory.
  • The mapping between the field equation and a particle-potential problem supplies a simple organising principle for classifying future exact solutions by the shape of V(f) and the value of \varepsilon.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: The same reduction should work for any single-field potential V(f), so biadjoint scalar theory may inherit every exact travelling-wave solution of ordinary scalar field theory; the paper treats only cubic and quartic potentials.
  • Inference: The bounded rational profile is a natural candidate for zeroth-copy matching to a gauge-theory or gravity lump; checking whether its double-copied counterpart is a known solution would test whether the classical double copy extends to wave-like non-perturbative objects.
  • Inference: A stability analysis of the oscillatory waves against non-planar perturbations would show whether they are robust enough to describe collective excitations in condensed-matter analogues; the paper does not perform such an analysis.
  • Inference: The presence of a constant current J is essential for some bounded oscillatory solutions, and relaxing it to a spacetime-dependent current may produce further integrable reductions with richer profile shapes.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper considers a generalised biadjoint scalar field theory with mass, cubic and quartic interactions and a constant external current. It introduces a colour-diagonal plane-wave ansatz Phi^{aa'} = delta^{aa'} f(p dot x), which reduces the matrix-valued equation of motion to a single ODE for f. The authors derive exact solutions in various limits: a Weierstrass elliptic solution for the massless cubic theory, Jacobi elliptic and tanh solutions for the quartic theory with vanishing cubic coupling, and a rational non-oscillatory bounded solution when both cubic and quartic couplings are present. The solutions are interpreted via a classical particle analogy, and connections to the double copy are discussed.

Significance. If the central derivation is correct, the paper provides exact time-dependent solutions of biadjoint scalar field theory, including bounded solutions not previously known in this context. The presentation is mostly self-contained: the reduction to the ODE (12), the first integral (16), and the closed-form solutions are explicit and checkable by substitution, which is a notable strength. The results are relevant to the non-perturbative double-copy program and to the catalogue of exact solutions in scalar field theories. However, the printed equation of motion (7) contains an index typo that must be corrected before the claims can be accepted as stated.

major comments (1)
  1. [Eq. (7)] The quartic term in Eq. (7) has an inconsistent colour-index structure: as printed it reads f^{ebc} tilde f^{a'b'c'} f^{eda} tilde f^{e'd'a'} Phi^{bb'}Phi^{cc'}Phi^{dd'}, which leaves free indices a and e' (with a' summed twice) and therefore cannot be equated to J^{aa'}. Varying the quartic term of the Lagrangian (3) gives the correct contraction f^{ebc} tilde f^{e'b'c'} f^{eda} tilde f^{e'd'a'} Phi^{bb'}Phi^{cc'}Phi^{dd'}, and it is this corrected term that, together with the ansatz (9)-(10), yields Eq. (12) after using f^{abc} f^{a'bc} = T_A delta^{aa'}. Consequently, the displayed solutions (18)-(24) are solutions of the corrected field equation but not of Eq. (7) as written. The authors should correct the typo in Eq. (7) and re-verify the derivation; the central chain otherwise appears sound.
minor comments (4)
  1. [Figure 2 caption] The text states that Figure 2(c) corresponds to epsilon = 1 and J = 2.5, while the caption gives epsilon = -1 and J = 2.5; please reconcile this discrepancy.
  2. [Eq. (22)] The argument of the tanh in Eq. (22) is typeset as m(z-c)p/sqrt(2 p^2), which appears to contain a spurious 'p' and should read m(xi-c)/sqrt(2 p^2).
  3. [Conclusion] The statement that 'the addition of a quartic term in the Lagrangian makes the energy of the theory bounded from below' should be qualified: the quartic double-bracket term is positive semidefinite but vanishes for factorized configurations of the form (8), so the claim does not hold universally.
  4. [Throughout] There are several typographical errors, e.g., 'cospondence' in the Conclusion and the misuse of 'instigated' in Section 2; a careful proofread is recommended.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the solutions are obtained by direct substitution into the field equation and exact integration, with prior-work citations used only for context and not as load-bearing inputs.

full rationale

The paper's central derivation is self-contained: substituting the diagonal plane-wave ansatz of eqs. (9)-(11) into the biadjoint field equation gives the scalar ODE (12), whose first integral yields the energy-type equation (16). Every displayed solution then follows by explicit integration or algebraic solution of (16), with ε an integration constant rather than a fitted parameter. The comparison with the Weierstrass equation in eq. (17) uses an independent mathematical definition, and the bounded solution (24) is verified by substitution into (16). The citations to earlier work by the same group, e.g. refs. [27,64,65], provide the form of the ansatz and the context of known static solutions, but the validity of the ansatz is checked in the present paper, so the self-citations are not load-bearing. The only substantive concern is a correctness issue rather than circularity: the quartic term as printed in eq. (7) has inconsistent colour indices (the factor appears to carry free indices a and e′ with a′ summed, so it cannot be equated to J^{aa′}), meaning that as printed it does not reduce to eq. (12). The reduction works for the index structure derived from the Lagrangian in eq. (3), so this is a typographical/derivation-consistency defect, not a circular reuse of the claimed result. No fitted input is renamed as a prediction, and no central claim reduces to a self-citation chain. Hence the circularity score is minimal.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the plane-wave ansatz, the common-Lie-algebra diagonal-colour reduction, and the chosen quartic vertex. All are stated explicitly. No data fitting is involved; the only free constants are integration constants and the wave-vector norm. No new particles, forces, or dimensions are introduced.

free parameters (3)
  • p^2 (wave-vector norm squared) = arbitrary; sign chosen per branch, e.g. p^2 < 0 for the real kink in eq. (22)
    Constant 4-vector p in ansatz (10) sets the kinetic coefficient in ODE (12) and must be negative in eq. (22) to obtain the real tanh solution.
  • epsilon (first-integral constant) = arbitrary; special value -m^4/(4 lambda T_A^2) chosen in eq. (20)
    Integration constant in eq. (16) that selects bounded versus unbounded profiles. A special value is imposed to recover the tanh kink.
  • Integration shifts c, c1, c2 = arbitrary real constants
    Translation invariance in xi leads to constant shifts in eqs. (18), (22)-(24).
assumptions (5)
  • standard math Existence of a common Lie algebra with adjoint normalization f^{abc} f^{a'bc} = T_A delta^{aa'}.
    Needed for the diagonal colour ansatz (eqs. 9 and 13) to produce the scalar ODE (12).
  • domain assumption The travelling-wave and colour-diagonal ansatz Phi^{aa'} = delta^{aa'} f(p dot x) with J^{aa'} = J delta^{aa'} is a valid restriction of the field configuration space.
    If the two Lie algebras differ or the colour structure is non-diagonal, the nonlinear cross-talk vanishes and the construction fails; the paper does not show these are the only or most general time-dependent solutions.
  • standard math The first integral of the autonomous ODE is conserved, so epsilon is constant.
    Standard reduction of an autonomous second-order ODE, used to obtain eq. (16).
  • standard math Known identities for Weierstrass and Jacobi elliptic functions are taken as given.
    Eqs. (17) and (23) invoke these identities without proof.
  • domain assumption The quartic self-interaction is defined by the specific colour contractions in eqs. (1) and (3).
    The paper notes this explicit Lagrangian has not appeared before, so the quartic solutions apply to this particular choice of vertex.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Time-dependent solutions of biadjoint scalar field theories." pith.science (2026). https://pith.science/paper/DULSIWSP

@misc{pith2026250201294,
  author       = {Pith},
  title        = {Pith review of: Time-dependent solutions of biadjoint scalar field theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DULSIWSP}},
  note         = {Machine review of arXiv:2502.01294}
}
read the original abstract

Biadjoint scalar field theories appear in the study of scattering amplitudes and classical solutions in gauge, gravity and related theories. In this paper, we present new exact solutions of biadjoint scalar field theory, showing that time-dependent solutions are possible and analytically tractable. We generalise the theory to include mass and / or quartic terms, and also a coupling to a constant background field. This allows for more exact solutions, which make contact with previous soliton literature. We also find bounded solutions, in contrast to all known previous examples. Our results may be useful for the study of non-perturbative aspects of the double copy between gauge theories and gravity. We also speculate as to their possible practical applications.

Figures

Figures reproduced from arXiv: 2502.01294 by the authors.

Figure 1
Figure 1. (a) Potential energy V (f) corresponding to massless cubic biadjoint scalar field theory, for: (a) the vacuum case; (b) the non-vacuum case with a constant current density J > 0. In the latter case, oscillatory solutions are possible if the energy of the particle is less than the local maximum of the potential curve, shown in orange. For the non-vacuum case, a non-zero positive current density leads to a local minim… view at source ↗
Figure 2
Figure 2. Possible solutions for the wave profile f(ξ) in cubic massless biadjoint scalar field theory for: (a) J = 0 and ε = −4; (b) J = 0 and ε = 4; (c) J = 2.5 and ε = −1. with m2 < 0 to generate the appropriate minimum in the potential, similar to the case of a Higgs potential. It is then not possible to solve eq. (16) analytically, although numerical solutions can be straightforwardly obtained, with an example shown in f… view at source ↗
Figure 3
Figure 3. (a) Potential energy curve V (f) for massive cubic biadjoint scalar theory, with p 2 = y = 1 and m2 = −1; (b) oscillatory solution obtained numerically, with f(0) = 3 and f ′ (0) = 0. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: (a) Potential energy curve for massive quartic biadjoint scalar field theory with [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Bounded solution for the wave profile f(ξ) in massless biadjoint theory (for y = λ = 1), when cubic and quartic interactions are both present. solution of quartic scalar field theory. More generally for vanishing cubic term one finds a solution in terms of Jacobi ellip…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

90 extracted references · 5 canonical work pages

  1. [1]

    A Relation Between Tree Amplitudes of Closed and Open Strings,

    H. Kawai, D. Lewellen, and S. Tye, “A Relation Between Tree Amplitudes of Closed and Open Strings,” Nucl.Phys. B269 (1986) 1

    1986

  2. [2]

    New Relations for Gauge-Theory Amplitudes,

    Z. Bern, J. Carrasco, and H. Johansson, “New Relations for Gauge-Theory Amplitudes,” Phys.Rev. D78 (2008) 085011, 0805.3993

    arXiv 2008

  3. [3]

    Perturbative Quantum Gravity as a Double Copy of Gauge Theory,

    Z. Bern, J. J. M. Carrasco, and H. Johansson, “Perturbative Quantum Gravity as a Double Copy of Gauge Theory,” Phys.Rev.Lett. 105 (2010) 061602, 1004.0476

    arXiv 2010

  4. [4]

    Gravity as the Square of Gauge Theory,

    Z. Bern, T. Dennen, Y.-t. Huang, and M. Kiermaier, “Gravity as the Square of Gauge Theory,” Phys.Rev. D82 (2010) 065003, 1004.0693

    arXiv 2010

  5. [5]

    Black holes and the double copy,

    R. Monteiro, D. O’Connell, and C. D. White, “Black holes and the double copy,” JHEP 1412 (2014) 056, 1410.0239. 9

    arXiv 2014

  6. [6]

    The classical double copy for Taub-NUT spacetime,

    A. Luna, R. Monteiro, D. O’Connell, and C. D. White, “The classical double copy for Taub-NUT spacetime,” Phys. Lett. B750 (2015) 272–277, 1507.01869

    arXiv 2015

  7. [7]

    Static Spherically Symmetric Kerr-Schild Metrics and Implications for the Classical Double Copy,

    A. K. Ridgway and M. B. Wise, “Static Spherically Symmetric Kerr-Schild Metrics and Implications for the Classical Double Copy,” Phys. Rev. D94 (2016), no. 4, 044023, 1512.02243

    arXiv 2016

  8. [8]

    The Kerr-Schild double copy in curved spacetime,

    N. Bahjat-Abbas, A. Luna, and C. D. White, “The Kerr-Schild double copy in curved spacetime,” JHEP 12 (2017) 004, 1710.01953

    arXiv 2017

Show all 90 references
  1. [9]

    The classical double copy in maximally symmetric spacetimes,

    M. Carrillo-Gonz´ alez, R. Penco, and M. Trodden, “The classical double copy in maximally symmetric spacetimes,” JHEP 04 (2018) 028, 1711.01296

    2018 arXiv

  2. [10]

    The classical double copy in three spacetime dimensions,

    M. Carrillo Gonz´ alez, B. Melcher, K. Ratliff, S. Watson, and C. D. White, “The classical double copy in three spacetime dimensions,” JHEP 07 (2019) 167, 1904.11001

    2019 arXiv

  3. [11]

    Kerr-Schild Double Copy and Complex Worldlines,

    I. Bah, R. Dempsey, and P. Weck, “Kerr-Schild Double Copy and Complex Worldlines,” JHEP 02 (2020) 180, 1910.04197

    2020 arXiv

  4. [12]

    Kerr-Schild double copy of the Coulomb solution in three dimensions,

    G. Alkac, M. K. Gumus, and M. A. Olpak, “Kerr-Schild double copy of the Coulomb solution in three dimensions,” Phys. Rev. D 104 (2021), no. 4, 044034, 2105.11550

    2021 arXiv

  5. [13]

    Generalized black holes in 3D Kerr-Schild double copy,

    G. Alkac, M. K. Gumus, and M. A. Olpak, “Generalized black holes in 3D Kerr-Schild double copy,” Phys. Rev. D 106 (2022), no. 2, 026013, 2205.08503

    2022 arXiv

  6. [14]

    Type D Spacetimes and the Weyl Double Copy,

    A. Luna, R. Monteiro, I. Nicholson, and D. O’Connell, “Type D Spacetimes and the Weyl Double Copy,” Class. Quant. Grav. 36 (2019) 065003, 1810.08183

    2019 arXiv

  7. [15]

    Anti-Self-Dual Spacetimes, Gravitational Instantons and Knotted Zeros of the Weyl Tensor,

    S. Sabharwal and J. W. Dalhuisen, “Anti-Self-Dual Spacetimes, Gravitational Instantons and Knotted Zeros of the Weyl Tensor,” JHEP 07 (2019) 004, 1904.06030

    2019 arXiv

  8. [16]

    Weyl doubling,

    R. Alawadhi, D. S. Berman, and B. Spence, “Weyl doubling,” JHEP 09 (2020) 127, 2007.03264

    2020 arXiv

  9. [17]

    Weyl Double Copy for Gravitational Waves,

    H. Godazgar, M. Godazgar, R. Monteiro, D. Peinador Veiga, and C. N. Pope, “Weyl Double Copy for Gravitational Waves,” Phys. Rev. Lett. 126 (2021), no. 10, 101103, 2010.02925

    2021

  10. [18]

    Twistorial Foundation for the Classical Double Copy,

    C. D. White, “Twistorial Foundation for the Classical Double Copy,” Phys. Rev. Lett. 126 (2021), no. 6, 061602, 2012.02479

    2021 arXiv

  11. [19]

    New heavenly double copies,

    E. Chac´ on, H. Garc ´ ıa-Compe´ an, A. Luna, R. Monteiro, and C. D. White, “New heavenly double copies,” JHEP 03 (2021) 247, 2008.09603

    2021 arXiv

  12. [20]

    The Weyl double copy from twistor space,

    E. Chac´ on, S. Nagy, and C. D. White, “The Weyl double copy from twistor space,” JHEP 05 (2021) 2239, 2103.16441

    2021 arXiv

  13. [21]

    Double copy of the multipole expansion,

    E. Chac´ on, A. Luna, and C. D. White, “Double copy of the multipole expansion,” Phys. Rev. D 106 (2022), no. 8, 086020, 2108.07702

    2022 arXiv

  14. [22]

    Alternative formulations of the twistor double copy,

    E. Chac´ on, S. Nagy, and C. D. White, “Alternative formulations of the twistor double copy,” JHEP 03 (2022) 180, 2112.06764. 10

    2022 arXiv

  15. [23]

    Compactifying the Kerr-Schild double copy,

    R. Dempsey and P. Weck, “Compactifying the Kerr-Schild double copy,” JHEP 05 (2023) 198, 2211.14327

    2023 arXiv

  16. [24]

    Einstein-Maxwell theory and the Weyl double copy,

    D. A. Easson, T. Manton, and A. Svesko, “Einstein-Maxwell theory and the Weyl double copy,” Phys. Rev. D 107 (2023), no. 4, 044063, 2210.16339

    2023 arXiv

  17. [25]

    Aligned fields double copy to Kerr-NUT-(A)dS,

    S. Chawla and C. Keeler, “Aligned fields double copy to Kerr-NUT-(A)dS,” JHEP 04 (2023) 005, 2209.09275

    2023 arXiv

  18. [26]

    The Weyl double copy in vacuum spacetimes with a cosmological constant,

    S. Han, “The Weyl double copy in vacuum spacetimes with a cosmological constant,” JHEP 09 (2022) 238, 2205.08654

    2022 arXiv

  19. [27]

    Non-perturbative aspects of the self-dual double copy,

    K. Armstrong-Williams, C. D. White, and S. Wikeley, “Non-perturbative aspects of the self-dual double copy,” JHEP 08 (2022) 160, 2205.02136

    2022 arXiv

  20. [28]

    Weyl double copy and massless free-fields in curved spacetimes,

    S. Han, “Weyl double copy and massless free-fields in curved spacetimes,” Class. Quant. Grav. 39 (2022), no. 22, 225009, 2204.01907

    2022 arXiv

  21. [29]

    The Newman-Penrose Map and the Classical Double Copy,

    G. Elor, K. Farnsworth, M. L. Graesser, and G. Herczeg, “The Newman-Penrose Map and the Classical Double Copy,” JHEP 12 (2020) 121, 2006.08630

    2020 arXiv

  22. [30]

    Twistor space origins of the Newman-Penrose map,

    K. Farnsworth, M. L. Graesser, and G. Herczeg, “Twistor space origins of the Newman-Penrose map,” SciPost Phys. 13 (2022), no. 4, 099, 2104.09525

    2022 arXiv

  23. [31]

    Yang-Mills origin of gravitational symmetries,

    A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes, and S. Nagy, “Yang-Mills origin of gravitational symmetries,” Phys. Rev. Lett. 113 (2014), no. 23, 231606, 1408.4434

    2014 arXiv

  24. [32]

    Comments on the double copy construction for gravitational theories,

    G. Lopes Cardoso, G. Inverso, S. Nagy, and S. Nampuri, “Comments on the double copy construction for gravitational theories,” in 17th Hellenic School and Workshops on Elementary Particle Physics and Gravity (CORFU2017) Corfu, Greece, September 2-28,

  25. [33]

    Gravity as Gauge Theory Squared: A Ghost Story,

    A. Anastasiou, L. Borsten, M. J. Duff, S. Nagy, and M. Zoccali, “Gravity as Gauge Theory Squared: A Ghost Story,” Phys. Rev. Lett. 121 (2018), no. 21, 211601, 1807.02486

    2018 arXiv

  26. [34]

    The convolutional double copy: a case study with a point,

    A. Luna, S. Nagy, and C. White, “The convolutional double copy: a case study with a point,” JHEP 09 (2020) 062, 2004.11254

    2020 arXiv

  27. [35]

    The pure BRST Einstein-Hilbert Lagrangian from the double-copy to cubic order,

    L. Borsten and S. Nagy, “The pure BRST Einstein-Hilbert Lagrangian from the double-copy to cubic order,” JHEP 07 (2020) 093, 2004.14945

    2020 arXiv

  28. [36]

    Becchi-Rouet-Stora-Tyutin-Lagrangian Double Copy of Yang-Mills Theory,

    L. Borsten, B. Jurco, H. Kim, T. Macrelli, C. Saemann, and M. Wolf, “Becchi-Rouet-Stora-Tyutin-Lagrangian Double Copy of Yang-Mills Theory,” Phys. Rev. Lett. 126 (2021), no. 19, 191601, 2007.13803

    2021 arXiv

  29. [37]

    Classical gluon and graviton radiation from the bi-adjoint scalar double copy,

    W. D. Goldberger, S. G. Prabhu, and J. O. Thompson, “Classical gluon and graviton radiation from the bi-adjoint scalar double copy,” Phys. Rev. D96 (2017), no. 6, 065009, 1705.09263

    2017 arXiv

  30. [38]

    Bound states and the classical double copy,

    W. D. Goldberger and A. K. Ridgway, “Bound states and the classical double copy,” Phys. Rev. D97 (2018), no. 8, 085019, 1711.09493. 11

    2018 arXiv

  31. [39]

    Spinning particles, axion radiation, and the classical double copy,

    W. D. Goldberger, J. Li, and S. G. Prabhu, “Spinning particles, axion radiation, and the classical double copy,” Phys. Rev. D97 (2018), no. 10, 105018, 1712.09250

    2018 arXiv

  32. [40]

    Strings, extended objects, and the classical double copy,

    W. D. Goldberger and J. Li, “Strings, extended objects, and the classical double copy,” JHEP 02 (2020) 092, 1912.01650

    2020 arXiv

  33. [41]

    Radiation and the classical double copy for color charges,

    W. D. Goldberger and A. K. Ridgway, “Radiation and the classical double copy for color charges,” Phys. Rev. D95 (2017), no. 12, 125010, 1611.03493

    2017 arXiv

  34. [42]

    The classical double copy in curved spacetimes: perturbative Yang-Mills from the bi-adjoint scalar,

    S. G. Prabhu, “The classical double copy in curved spacetimes: perturbative Yang-Mills from the bi-adjoint scalar,” JHEP 05 (2024) 117, 2011.06588

    2024 arXiv

  35. [43]

    Perturbative spacetimes from Yang-Mills theory,

    A. Luna, R. Monteiro, I. Nicholson, A. Ochirov, D. O’Connell, N. Westerberg, and C. D. White, “Perturbative spacetimes from Yang-Mills theory,” JHEP 04 (2017) 069, 1611.07508

    2017 arXiv

  36. [44]

    Inelastic Black Hole Scattering from Charged Scalar Amplitudes,

    A. Luna, I. Nicholson, D. O’Connell, and C. D. White, “Inelastic Black Hole Scattering from Charged Scalar Amplitudes,” JHEP 03 (2018) 044, 1711.03901

    2018 arXiv

  37. [45]

    Symmetry for Flavor-Kinematics Duality from an Action,

    C. Cheung and C.-H. Shen, “Symmetry for Flavor-Kinematics Duality from an Action,” Phys. Rev. Lett. 118 (2017), no. 12, 121601, 1612.00868

    2017 arXiv

  38. [46]

    Covariant color-kinematics duality,

    C. Cheung and J. Mangan, “Covariant color-kinematics duality,” JHEP 11 (2021) 069, 2108.02276

    2021 arXiv

  39. [47]

    Geometry-kinematics duality,

    C. Cheung, A. Helset, and J. Parra-Martinez, “Geometry-kinematics duality,” Phys. Rev. D 106 (2022), no. 4, 045016, 2202.06972

    2022 arXiv

  40. [48]

    Non-perturbative Double Copy in Flatland,

    C. Cheung, J. Mangan, J. Parra-Martinez, and N. Shah, “Non-perturbative Double Copy in Flatland,” Phys. Rev. Lett. 129 (2022), no. 22, 221602, 2204.07130

    2022 arXiv

  41. [49]

    The Penrose limit of the Weyl double copy,

    S. Chawla, K. Fransen, and C. Keeler, “The Penrose limit of the Weyl double copy,” Class. Quant. Grav. 41 (2024), no. 24, 245015, 2406.14601

    2024 arXiv

  42. [50]

    On type-II Spacetimes and the Double Copy for Fluids Metrics,

    C. Keeler and N. Monga, “On type-II Spacetimes and the Double Copy for Fluids Metrics,” 2404.03195

    arXiv

  43. [51]

    Black hole horizons from the double copy,

    S. Chawla and C. Keeler, “Black hole horizons from the double copy,” Class. Quant. Grav. 40 (2023), no. 22, 225004, 2306.02417

    2023 arXiv

  44. [52]

    Classical double copy of nonsingular black holes,

    D. A. Easson, C. Keeler, and T. Manton, “Classical double copy of nonsingular black holes,” Phys. Rev. D 102 (2020), no. 8, 086015, 2007.16186

    2020 arXiv

  45. [53]

    Deriving Weyl double copies with sources,

    K. Armstrong-Williams, N. Moynihan, and C. D. White, “Deriving Weyl double copies with sources,” 2407.18107

    arXiv

  46. [54]

    A spinorial double copy for N = 0 supergravity,

    K. Armstrong-Williams and C. D. White, “A spinorial double copy for N = 0 supergravity,” JHEP 05 (2023) 047, 2303.04631

    2023 arXiv

  47. [55]

    Double Kerr-Schild spacetimes and the Newman-Penrose map,

    K. Farnsworth, M. L. Graesser, and G. Herczeg, “Double Kerr-Schild spacetimes and the Newman-Penrose map,” JHEP 10 (2023) 010, 2306.16445. 12

    2023 arXiv

  48. [56]

    The Kinematic Algebra From the Self-Dual Sector,

    R. Monteiro and D. O’Connell, “The Kinematic Algebra From the Self-Dual Sector,” JHEP 1107 (2011) 007, 1105.2565

    2011 arXiv

  49. [57]

    Double Copy from Homotopy Algebras,

    L. Borsten, H. Kim, B. Jurco, T. Macrelli, C. Saemann, and M. Wolf, “Double Copy from Homotopy Algebras,” Fortsch. Phys. 69 (2021), no. 8-9, 2100075, 2102.11390

    2021 arXiv

  50. [58]

    S-duality and the double copy,

    R. Alawadhi, D. S. Berman, B. Spence, and D. Peinador Veiga, “S-duality and the double copy,” JHEP 03 (2020) 059, 1911.06797

    2020 arXiv

  51. [59]

    Ehlers as EM duality in the double copy,

    A. Banerjee, E. O. Colg´ ain, J. A. Rosabal, and H. Yavartanoo, “Ehlers as EM duality in the double copy,” Phys. Rev. D 102 (2020) 126017, 1912.02597

    2020 arXiv

  52. [60]

    Double copy of electric-magnetic duality,

    Y.-T. Huang, U. Kol, and D. O’Connell, “Double copy of electric-magnetic duality,” Phys. Rev. D 102 (2020), no. 4, 046005, 1911.06318

    2020 arXiv

  53. [61]

    The self-dual classical double copy, and the Eguchi-Hanson instanton,

    D. S. Berman, E. Chac´ on, A. Luna, and C. D. White, “The self-dual classical double copy, and the Eguchi-Hanson instanton,” JHEP 01 (2019) 107, 1809.04063

    2019 arXiv

  54. [62]

    Topology and Wilson lines: global aspects of the double copy,

    L. Alfonsi, C. D. White, and S. Wikeley, “Topology and Wilson lines: global aspects of the double copy,” JHEP 07 (2020) 091, 2004.07181

    2020 arXiv

  55. [63]

    The single copy of the gravitational holonomy,

    R. Alawadhi, D. S. Berman, C. D. White, and S. Wikeley, “The single copy of the gravitational holonomy,” JHEP 10 (2021) 229, 2107.01114

    2021 arXiv

  56. [64]

    Exact solutions for the biadjoint scalar field,

    C. D. White, “Exact solutions for the biadjoint scalar field,” Phys. Lett. B763 (2016) 365–369, 1606.04724

    2016 arXiv

  57. [65]

    Extended solutions for the biadjoint scalar field,

    P.-J. De Smet and C. D. White, “Extended solutions for the biadjoint scalar field,” Phys. Lett. B775 (2017) 163–167, 1708.01103

    2017 arXiv

  58. [66]

    Biadjoint wires,

    N. Bahjat-Abbas, R. Stark-Much˜ ao, and C. D. White, “Biadjoint wires,” Phys. Lett. B788 (2019) 274–279, 1810.08118

    2019 arXiv

  59. [67]

    Massive covariant colour-kinematics in 3D,

    N. Moynihan, “Massive covariant colour-kinematics in 3D,” JHEP 05 (2024) 310, 2110.02209

    2024 arXiv

  60. [68]

    Kinematic Lie Algebras from Twistor Spaces,

    L. Borsten, B. Jurco, H. Kim, T. Macrelli, C. Saemann, and M. Wolf, “Kinematic Lie Algebras from Twistor Spaces,” Phys. Rev. Lett. 131 (2023), no. 4, 041603, 2211.13261

    2023 arXiv

  61. [69]

    Gravity as the square of gauge theory: a review,

    L. Borsten, “Gravity as the square of gauge theory: a review,” Riv. Nuovo Cim. 43 (2020), no. 3, 97–186

    2020

  62. [70]

    The Duality Between Color and Kinematics and its Applications,

    Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson, and R. Roiban, “The Duality Between Color and Kinematics and its Applications,” 1909.01358

    1909 arXiv

  63. [71]

    Snowmass White Paper: the Double Copy and its Applications,

    T. Adamo, J. J. M. Carrasco, M. Carrillo-Gonz´ alez, M. Chiodaroli, H. Elvang, H. Johansson, D. O’Connell, R. Roiban, and O. Schlotterer, “Snowmass White Paper: the Double Copy and its Applications,” in 2022 Snowmass Summer Study . 4, 2022. 2204.06547

    2022 arXiv

  64. [72]

    Chapter 2: An invitation to color-kinematics duality and the double copy,

    Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson, and R. Roiban, “Chapter 2: An invitation to color-kinematics duality and the double copy,” J. Phys. A 55 (2022), no. 44, 443003, 2203.13013. 13

    2022 arXiv

  65. [73]

    Double copy—from optics to quantum gravity: tutorial,

    C. D. White, “Double copy—from optics to quantum gravity: tutorial,” J. Opt. Soc. Am. B 38 (2021), no. 11, 3319–3330, 2105.06809

    2021 arXiv

  66. [74]

    C. D. White, The Classical Double Copy . World Scientific, 5, 2024

    2024

  67. [75]

    Positive Geometries and Canonical Forms,

    N. Arkani-Hamed, Y. Bai, and T. Lam, “Positive Geometries and Canonical Forms,” JHEP 11 (2017) 039, 1703.04541

    2017 arXiv

  68. [76]

    Unwinding the Amplituhedron in Binary,

    N. Arkani-Hamed, H. Thomas, and J. Trnka, “Unwinding the Amplituhedron in Binary,” JHEP 01 (2018) 016, 1704.05069

    2018 arXiv

  69. [77]

    Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet,

    N. Arkani-Hamed, Y. Bai, S. He, and G. Yan, “Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet,” JHEP 05 (2018) 096, 1711.09102

    2018 arXiv

  70. [78]

    Stokes polytopes: the positive geometry for ϕ4 interactions,

    P. Banerjee, A. Laddha, and P. Raman, “Stokes polytopes: the positive geometry for ϕ4 interactions,” JHEP 08 (2019) 067, 1811.05904

    2019 arXiv

  71. [79]

    Stokes Polytopes and Intersection Theory,

    N. Kalyanapuram, “Stokes Polytopes and Intersection Theory,” Phys. Rev. D 101 (2020), no. 10, 105010, 1910.12195

    2020 arXiv

  72. [80]

    On positive geometries of quartic interactions: Stokes polytopes, lower forms on associahedra and world-sheet forms,

    P. B. Aneesh, P. Banerjee, M. Jagadale, R. Rajan, A. Laddha, and S. Mahato, “On positive geometries of quartic interactions: Stokes polytopes, lower forms on associahedra and world-sheet forms,” JHEP 04 (2020) 149, 1911.06008

    2020 arXiv

  73. [81]

    Constraining the weights of Stokes polytopes using BCFW recursions for ϕ4,

    I. Srivastava, “Constraining the weights of Stokes polytopes using BCFW recursions for ϕ4,” JHEP 04 (2021) 064, 2005.12886

    2021 arXiv

  74. [82]

    Towards Positive Geometries of Massive Scalar field theories,

    M. Jagadale and A. Laddha, “Towards Positive Geometries of Massive Scalar field theories,” 2206.07979

    arXiv

  75. [83]

    E. J. Weinberg, Classical solutions in quantum field theory . Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2012

    2012

  76. [84]

    N. S. Manton and P. Sutcliffe, Topological solitons. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2004

    2004

  77. [85]

    Belinski and E

    V. Belinski and E. Verdaguer, Gravitational solitons. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2005

    2005

  78. [86]

    Monopoles, shockwaves and the classical double copy,

    N. Bahjat-Abbas, R. Stark-Much˜ ao, and C. D. White, “Monopoles, shockwaves and the classical double copy,” JHEP 04 (2020) 102, 2001.09918

    2020 arXiv

  79. [87]

    Exact solutions of classical scalar field equations,

    M. Frasca, “Exact solutions of classical scalar field equations,” J. Nonlin. Math. Phys. 18 (2011), no. 2, 291–297, 0907.4053

    2011 arXiv

  80. [88]

    Four Lectures on Weierstrass Elliptic Function and Applications in Classical and Quantum Mechanics,

    G. Pastras, “Four Lectures on Weierstrass Elliptic Function and Applications in Classical and Quantum Mechanics,” 6, 2017. 1706.07371

    2017 arXiv

  81. [89]

    New Classes of Solutions for Euclidean Scalar Field Theories,

    C. M. Bender and S. Sarkar, “New Classes of Solutions for Euclidean Scalar Field Theories,” Universe 10 (2024), no. 2, 72, 2304.11629

    2024 arXiv

  82. [90]

    Strong-Weak Bi-Adjoints, Gluon-W resonances, and new asymmetric LHC production processes,

    L. M. Carpenter and K. Schwind, “Strong-Weak Bi-Adjoints, Gluon-W resonances, and new asymmetric LHC production processes,” 2412.19896. 14

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.