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  • Open Access

Higher spin fields and the field strength multicopy

Graham R. Brown1,* and Bill Spence2,†

  • *Contact author: graham.brown@ed.ac.uk
  • †Contact author: w.j.spence@qmul.ac.uk
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Phys. Rev. D 113, 065017 – Published 23 March, 2026

DOI: https://doi.org/10.1103/6f9q-jbzn

Abstract

We discuss the generalization of the Weyl double copy to higher spin “multicopies,” showing how the natural linearized higher spin field strengths can be related to sums of powers of the Maxwell tensor. The tracelessness of the field strength involves the appropriate Fronsdal equations of motion for the higher spin field. We work with spacetimes admitting Kerr-Schild coordinates and give a number of examples in different dimensions. We note that the multicopy is particularly transparent in four dimensions if one uses spinor descriptions of the fields, relating this to the Penrose transform. The higher dimensional spinor multicopy is also explored and reveals some interesting new features arising from the little group based identification of higher spin field strengths and Maxwell tensor types. We then turn to the vector superspace formalism describing higher spin and “continuous” spin representations given by Schuster and Toro, based on symmetric tensor fields. Here the Kerr-Schild higher spin fields we have used earlier naturally package into a simple expression involving an arbitrary function, when the continuous spin scale ρ is set to zero. Further, we discuss the case of an anti–de Sitter background, where there is also a vector space formalism given by Segal and we clarify this approach using a different definition of the covariant derivative. We give a general solution of Kerr-Schild type and finally we describe some of the obstacles to a continuous spin formulation.

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References (75)

  1. Z. Bern, J. J. M. Carrasco, and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78, 085011 (2008).
  2. Z. Bern, J. J. M. Carrasco, and H. Johansson, Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett. 105, 061602 (2010).
  3. G. Travaglini et al., The SAGEX review on scattering amplitudes, J. Phys. A 55, 443001 (2022).
  4. C. D. White, The Classical Double Copy (World Scientific, Singapore, 2024).
  5. A. Luna, R. Monteiro, I. Nicholson, and D. O’Connell, Type D spacetimes and the Weyl double copy, Classical Quantum Gravity 36, 065003 (2019).
  6. M. Walker and R. Penrose, On quadratic first integrals of the geodesic equations for type [22] spacetimes, Commun. Math. Phys. 18, 265 (1970).
  7. L. Hughston, R. Penrose, P. Sommers, and M. Walker, On a quadratic first integral for the charged particle orbits in the charged Kerr solution, Commun. Math. Phys. 27, 303 (1972).
  8. W. Dietz and R. Rüdiger, Space-times admitting Killing-Yano tensors, Proc. R. Soc. A 375, 361 (1981).
  9. H. Godazgar, M. Godazgar, R. Monteiro, D. Peinador Veiga, and C. N. Pope, Weyl double copy for gravitational waves, Phys. Rev. Lett. 126, 101103 (2021).
  10. S. Sabharwal and J. W. Dalhuisen, Anti-self-dual spacetimes, gravitational instantons and knotted zeros of the Weyl tensor, J. High Energy Phys. 07 (2019) 004.
  11. R. Alawadhi, D. S. Berman, B. Spence, and D. Peinador Veiga, S-duality and the double copy, J. High Energy Phys. 03 (2020) 059.
  12. R. Alawadhi, D. S. Berman, and B. Spence, Weyl doubling, J. High Energy Phys. 09 (2020) 127.
  13. G. Elor, K. Farnsworth, M. L. Graesser, and G. Herczeg, The Newman-Penrose map and the classical double copy, J. High Energy Phys. 12 (2020) 121.
  14. E. Chacón, S. Nagy, and C. D. White, The Weyl double copy from twistor space, J. High Energy Phys. 05 (2021) 2239.
  15. E. Chacón, S. Nagy, and C. D. White, Alternative formulations of the twistor double copy, J. High Energy Phys. 03 (2022) 180.
  16. P. Schuster and N. Toro, Continuous-spin particle field theory with helicity correspondence, Phys. Rev. D 91, 025023 (2015).
  17. R. Kerr and A. Schild, A new class of vacuum solutions of the Einstein field equations, in Atti del convegno sulla relativita‘ generale; problemi dell’energia e onde gravitazionali", edited by G. Barbera (Springer, Berlin, 1965), p. 173.
  18. R. Kerr and A. Schild, Some algebraically degenerate solutions of Einstein’s gravitational field equations, edited by R. Finn, Proc. Sym. in Applied Math, Vol. XVII (American Mathematical Society, Providence Rhode Island, 1965), pp. 199–209.
  19. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact Solutions of Einstein’s Field Equations (Cambridge University Press, Cambridge, England, 2003).
  20. R. Monteiro, D. O’Connell, and C. D. White, Black holes and the double copy, J. High Energy Phys. 12 (2014) 056.
  21. A. Luna, R. Monteiro, D. O’Connell, and C. D. White, The classical double copy for Taub–NUT spacetime, Phys. Lett. B 750, 272 (2015).
  22. V. E. Didenko, A. S. Matveev, and M. A. Vasiliev, Unfolded description of AdS4 Kerr black hole, Phys. Lett. B 665, 284 (2008).
  23. V. E. Didenko and M. A. Vasiliev, Static BPS black hole in 4d higher-spin gauge theory, Phys. Lett. B 682, 305 (2009).
  24. V. E. Didenko and N. K. Dosmanbetov, Classical double copy and higher spin fields, Phys. Rev. Lett. 130, 071603 (2023).
  25. D. Ponomarev, Basic introduction to higher-spin theories, Int. J. Theor. Phys. 62, 146 (2023).
  26. D. Sorokin, Introduction to the classical theory of higher spins, AIP Conf. Proc. 767, 172 (2005).
  27. N. Bouatta, G. Compere, and A. Sagnotti, An Introduction to free higher-spin fields, in 1st Solvay Workshop on Higher Spin Gauge Theories (Springer, Brussels Belgium, 2004), pp. 79–99; arXiv:hep-th/0409068.
  28. P. de Medeiros and C. Hull, Exotic tensor gauge theory and duality, Commun. Math. Phys. 235, 255 (2003).
  29. P. de Medeiros and C. Hull, Geometric second order field equations for general tensor gauge fields, J. High Energy Phys. 05 (2003) 019.
  30. C. D. White, Twistorial foundation for the classical double copy, Phys. Rev. Lett. 126, 061602 (2021).
  31. K. Armstrong-Williams, N. Moynihan, and C. D. White, Deriving Weyl double copies with sources, J. High Energy Phys. 03 (2025) 121.
  32. V. E. Didenko and A. V. Korybut, Planar solutions of higher-spin theory. Part I. Free field level, J. High Energy Phys. 08 (2021) 144.
  33. R. Monteiro, I. Nicholson, and D. O’Connell, Spinor-helicity and the algebraic classification of higher-dimensional spacetimes, Classical Quantum Gravity 36, 065006 (2019).
  34. E. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. Math. 40, 149 (1939).
  35. V. O. Rivelles, Gauge theory formulations for continuous and higher spin fields, Phys. Rev. D 91, 125035 (2015).
  36. V. O. Rivelles, Remarks on a gauge theory for continuous spin particles, Eur. Phys. J. C 77, 433 (2017).
  37. X. Bekaert and E. D. Skvortsov, Elementary particles with continuous spin, Int. J. Mod. Phys. A 32, 1730019 (2017).
  38. M. Najafizadeh, Modified Wigner equations and continuous spin gauge field, Phys. Rev. D 97, 065009 (2018).
  39. P. Schuster and N. Toro, A new class of particle in 2+1 dimensions, Phys. Lett. B 743, 224 (2015).
  40. X. Bekaert, M. Najafizadeh, and M. R. Setare, A gauge field theory of fermionic continuous-spin particles, Phys. Lett. B 760, 320 (2016).
  41. M. Najafizadeh, Supersymmetric continuous spin gauge theory, J. High Energy Phys. 03 (2020) 027.
  42. M. Najafizadeh, Off-shell supersymmetric continuous spin gauge theory, J. High Energy Phys. 02 (2022) 038.
  43. P. Schuster and N. Toro, Quantum electrodynamics mediated by a photon with continuous spin, Phys. Rev. D 109, 096008 (2024).
  44. R. R. Metsaev, BRST-BV approach to continuous-spin field, Phys. Lett. B 781, 568 (2018).
  45. I. L. Buchbinder, V. A. Krykhtin, and H. Takata, BRST approach to Lagrangian construction for bosonic continuous spin field, Phys. Lett. B 785, 315 (2018).
  46. K. B. Alkalaev and M. A. Grigoriev, Continuous spin fields of mixed-symmetry type, J. High Energy Phys. 03 (2018) 030.
  47. P. Schuster, N. Toro, and K. Zhou, Interactions of particles with “continuous spin” fields, J. High Energy Phys. 04 (2023) 010.
  48. B. Bellazzini, S. De Angelis, and M. Romano, Continuous-spin particles, on shell, J. High Energy Phys. 05 (2025) 166.
  49. P. Schuster, G. Sundaresan, and N. Toro, Thermodynamics of continuous spin photons, Phys. Rev. D 111, 056019 (2025).
  50. A. Reilly, P. Schuster, and N. Toro, Probing “continuous spin” QED with rare atomic transitions, Phys. Rev. D 113, 015028 (2026).
  51. S. Kundu, P. Schuster, and N. Toro, A first look at “continuous spin” gravity—time delay signatures, arXiv:2503.03817.
  52. R. R. Metsaev, Interacting massive/massless continuous-spin fields and integer-spin fields, J. High Energy Phys. 09 (2025) 002.
  53. S. Kundu, A. Russo, P. Schuster, and N. Toro, Interactions of a continuous-spin field with a spin-1/2 particle, J. High Energy Phys. 11 (2025) 125.
  54. A. Reilly, A. Russo, P. Schuster, and N. Toro, Hydrogen 21 cm constraints on the photon’s spin scale, arXiv:2505.15890.
  55. R. R. Metsaev, Continuous spin gauge field in (A)dS space, Phys. Lett. B 767, 458 (2017).
  56. R. R. Metsaev, Fermionic continuous spin gauge field in (A)dS space, Phys. Lett. B 773, 135 (2017).
  57. R. R. Metsaev, Continuous-spin mixed-symmetry fields in AdS(5), J. Phys. A 51, 215401 (2018).
  58. M. V. Khabarov and Y. M. Zinoviev, Infinite (continuous) spin fields in the frame-like formalism, Nucl. Phys. B928, 182 (2018).
  59. R. R. Metsaev, Light-cone continuous-spin field in AdS space, Phys. Lett. B 793, 134 (2019).
  60. R. R. Metsaev, Mixed-symmetry continuous-spin fields in flat and AdS spaces, Phys. Lett. B 820, 136497 (2021).
  61. I. L. Buchbinder, S. A. Fedoruk, A. P. Isaev, and V. A. Krykhtin, Infinite (continuous) spin particle in constant curvature space, Phys. Lett. B 853, 138689 (2024).
  62. I. L. Buchbinder, S. A. Fedoruk, A. P. Isaev, and V. A. Krykhtin, BRST construction for infinite spin field on AdS4, Eur. Phys. J. Plus 139, 621 (2024).
  63. V. E. Didenko, Coordinate independent approach to 5d black holes, Classical Quantum Gravity 29, 025009 (2012).
  64. A. Y. Segal, A generating formulation for free higher spin massless fields, arXiv:hep-th/0103028.
  65. M. Najafizadeh, Local action for fermionic unconstrained higher spin gauge fields in AdS and dS spacetimes, Phys. Rev. D 98, 125012 (2018).
  66. M. Taronna, Higher-spin interactions: Three-point functions and beyond, Ph.D. thesis, Pisa, Scuola Normale Superiore, 2012, arXiv:1209.5755.
  67. T. Nutma and M. Taronna, On conformal higher spin wave operators, J. High Energy Phys. 06 (2014) 066.
  68. I. V. Gorbunov, S. L. Lyakhovich, and A. A. Sharapov, Wick quantization of cotangent bundles over Riemannian manifolds, J. Geom. Phys. 53, 98 (2005).
  69. K. P. Mok, Metrics and connections on the cotangent bundle, Kodai Math. Sem. Rep. 28, 226 (1977).
  70. V. E. Didenko and A. V. Korybut, Planar solutions of higher-spin theory. Nonlinear corrections, J. High Energy Phys. 01 (2022) 125.
  71. R. R. Metsaev, Light-cone vector superspace and continuous-spin field in AdS, Phys. Lett. B 868, 139778 (2025).
  72. I. L. Buchbinder, S. A. Fedoruk, A. P. Isaev, and M. A. Podoinitsyn, On the realization of infinite (continuous) spin field representations in AdS4 space, Phys. Lett. B 861, 139226 (2025).
  73. X. Bekaert, S. Cnockaert, C. Iazeolla, and M. A. Vasiliev, Nonlinear higher spin theories in various dimensions, in 1st Solvay Workshop on Higher Spin Gauge Theories (Springer, Brussels Belgium, 2004), pp. 132–197; arXiv:hep-th/0503128.
  74. A. Cannas da Silva, Lectures on Symplectic Geometry (Springer, Berlin, Heidelberg, 2001).
  75. S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10, 338 (1958).

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Phys. Rev. D 113, 065017– Published 23 March, 2026
  • Received 7 January 2026
  • Accepted 25 February 2026
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