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Supertranslations from Scattering Amplitudes

Asaad Elkhidir1, Donal O’Connell1, and Radu Roiban2

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Phys. Rev. Lett. 135, 151601 – Published 8 October, 2025

DOI: https://doi.org/10.1103/fxdk-5qwc

Abstract

On shell methods have found a new application to local observables such as asymptotic radiation fields and gravitational waveforms. While these observables are invariant under small gauge transformations, they are known to depend on a choice of asymptotic gauge; in gravity on asymptotically Minkowski spacetimes, this is a choice of Bondi-Van der Burg-Metzner-Sachs frame. In this Letter, we provide a method for capturing these supertranslations, to all orders in perturbation theory, using the on shell framework of scattering amplitudes.

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References (81)

  1. A. Cristofoli, R. Gonzo, D. A. Kosower, and D. O’Connell, Waveforms from amplitudes, Phys. Rev. D 106, 056007 (2022).
  2. P. Di Vecchia, C. Heissenberg, and R. Russo, Angular momentum of zero-frequency gravitons, J. High Energy Phys. 08 (2022) 172.
  3. D. A. Kosower, R. Monteiro, and D. O’Connell, The SAGEX review on scattering amplitudes Chapter 14: Classical gravity from scattering amplitudes, J. Phys. A 55, 443015 (2022).
  4. A. Herderschee, R. Roiban, and F. Teng, The sub-leading scattering waveform from amplitudes, J. High Energy Phys. 06 (2023) 004.
  5. A. Brandhuber, G. R. Brown, G. Chen, S. De Angelis, J. Gowdy, and G. Travaglini, One-loop gravitational bremsstrahlung and waveforms from a heavy-mass effective field theory, J. High Energy Phys. 06 (2023) 048.
  6. A. Georgoudis, C. Heissenberg, and I. Vazquez-Holm, Inelastic exponentiation and classical gravitational scattering at one loop, J. High Energy Phys. 06 (2023) 126.
  7. A. Elkhidir, D. O’Connell, M. Sergola, and I. A. Vazquez-Holm, Radiation and reaction at one loop, J. High Energy Phys. 07 (2024) 272.
  8. S. De Angelis, P. P. Novichkov, and R. Gonzo, Spinning waveforms from the Kosower-Maybee-O’Connell formalism at leading order, Phys. Rev. D 110, L041502 (2024).
  9. A. Brandhuber, G. R. Brown, G. Chen, J. Gowdy, and G. Travaglini, Resummed spinning waveforms from five-point amplitudes, J. High Energy Phys. 02 (2024) 026.
  10. R. Aoude, K. Haddad, C. Heissenberg, and A. Helset, Leading-order gravitational radiation to all spin orders, Phys. Rev. D 109, 036007 (2024).
  11. A. Georgoudis, C. Heissenberg, and R. Russo, An eikonal-inspired approach to the gravitational scattering waveform, J. High Energy Phys. 03 (2024) 089.
  12. L. Bohnenblust, H. Ita, M. Kraus, and J. Schlenk, Gravitational Bremsstrahlung in black-hole scattering at O(G3): Linear-in-spin effects, J. High Energy Phys. 11 (2024) 109.
  13. K. Fernandes and F.-L. Lin, Next-to-eikonal corrected double graviton dressing and gravitational wave observables at O(G2), J. High Energy Phys. 06 (2024) 015.
  14. D. Bini, T. Damour, S. De Angelis, A. Geralico, A. Herderschee, R. Roiban, and F. Teng, Gravitational waveforms: A tale of two formalisms, Phys. Rev. D 109, 125008 (2024).
  15. A. Georgoudis, C. Heissenberg, and R. Russo, Post-Newtonian multipoles from the next-to-leading post-Minkowskian gravitational waveform, Phys. Rev. D 109, 106020 (2024).
  16. K. Aoki and A. Cristofoli, Cosmology and the classical limit of the S-matrix, J. High Energy Phys. 06 (2024) 032.
  17. G. Brunello and S. De Angelis, An improved framework for computing waveforms, J. High Energy Phys. 07 (2024) 062.
  18. C. Heissenberg and R. Russo, Revisiting gravitational angular momentum and mass dipole losses in the eikonal framework, Classical Quantum Gravity 42, 045014 (2025).
  19. A. Falkowski and P. Marinellis, Spinning waveforms of scalar radiation in quadratic modified gravity, Eur. Phys. J. C 85, 74 (2025).
  20. A. Brandhuber, G. R. Brown, G. Chen, G. Travaglini, and P. Vives Matasan, Spinning waveforms in cubic effective field theories of gravity, J. High Energy Phys. 12 (2024) 039.
  21. F. Fucito, J. F. Morales, and R. Russo, Gravitational wave forms for extreme mass ratio collisions from supersymmetric gauge theories, Phys. Rev. D 111, 044054 (2025).
  22. J. Collins, A new approach to the LSZ reduction formula, arXiv:1904.10923.
  23. S. Caron-Huot, M. Giroux, H. S. Hannesdottir, and S. Mizera, What can be measured asymptotically?, J. High Energy Phys. 01 (2024) 139.
  24. R. Monteiro, D. O’Connell, D. Peinador Veiga, and M. Sergola, Classical solutions and their double copy in split signature, J. High Energy Phys. 05 (2021) 268.
  25. R. Monteiro, S. Nagy, D. O’Connell, D. Peinador Veiga, and M. Sergola, NS-NS spacetimes from amplitudes, J. High Energy Phys. 06 (2022) 021.
  26. R. Akhoury, R. Saotome, and G. Sterman, Collinear and soft divergences in perturbative quantum gravity, Phys. Rev. D 84, 104040 (2011).

  27. The projector Π=∑ηeη⊗eη* arises because of the identification of the helicity labels in the perturbative graviton operator and the soft charge due to the commutation relations of the graviton creation and annihilation operators.

  28. G. Veneziano and G. A. Vilkovisky, Angular momentum loss in gravitational scattering, radiation reaction, and the Bondi gauge ambiguity, Phys. Lett. B 834, 137419 (2022).
  29. E. E. Flanagan and D. A. Nichols, Conserved charges of the extended Bondi-Metzner-Sachs algebra, Phys. Rev. D 95, 044002 (2017); 108, 069902(E) (2023).
  30. D. Bini, T. Damour, and A. Geralico, Comparing one-loop gravitational bremsstrahlung amplitudes to the multipolar-post-Minkowskian waveform, Phys. Rev. D 108, 124052 (2023).
  31. A. Cristofoli, A. Elkhidir, A. Ilderton, and D. O’Connell, Large gauge effects and the structure of amplitudes, J. High Energy Phys. 06 (2023) 204.

  32. We define f∂μ↔g≡12[f(∂g/∂kμ)−(∂f/∂kμ)g].

  33. T. Damour, Radiative contribution to classical gravitational scattering at the third order in G, Phys. Rev. D 102, 124008 (2020).
  34. A. V. Manohar, A. K. Ridgway, and C.-H. Shen, Radiated angular momentum and dissipative effects in classical scattering, Phys. Rev. Lett. 129, 121601 (2022).
  35. D. Christodoulou, Nonlinear nature of gravitation and gravitational wave experiments, Phys. Rev. Lett. 67, 1486 (1991).
  36. A. G. Wiseman and C. M. Will, Christodoulou’s nonlinear gravitational wave memory: Evaluation in the quadrupole approximation, Phys. Rev. D 44, R2945 (1991).
  37. K. S. Thorne, Gravitational-wave bursts with memory: The Christodoulou effect, Phys. Rev. D 45, 520 (1992).

  38. We work in a mostly-minus signature.

  39. A. Cristofoli, R. Gonzo, N. Moynihan, D. O’Connell, A. Ross, M. Sergola, and C. D. White, The uncertainty principle and classical amplitudes, J. High Energy Phys. 06 (2024) 181.
  40. M. Campiglia and A. Laddha, Asymptotic symmetries of gravity and soft theorems for massive particles, J. High Energy Phys. 12 (2015) 094.
  41. M. Campiglia, Null to time-like infinity Green’s functions for asymptotic symmetries in Minkowski spacetime, J. High Energy Phys. 11 (2015) 160.
  42. M. Campiglia and A. Laddha, Asymptotic symmetries of QED and Weinberg’s soft photon theorem, J. High Energy Phys. 07 (2015) 115.
  43. A. Strominger, On BMS invariance of gravitational scattering, J. High Energy Phys. 07 (2014) 152.
  44. M. Campiglia and A. Laddha, New symmetries for the gravitational S-matrix, J. High Energy Phys. 04 (2015) 076.
  45. T. He, V. Lysov, P. Mitra, and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, J. High Energy Phys. 05 (2015) 151.
  46. J. Ware, R. Saotome, and R. Akhoury, Construction of an asymptotic S matrix for perturbative quantum gravity, J. High Energy Phys. 10 (2013) 159.
  47. T. Adamo, E. Casali, and D. Skinner, Perturbative gravity at null infinity, Classical Quantum Gravity 31, 225008 (2014).
  48. A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory (Princeton University Press, 2017), https://press.princeton.edu/books/hardcover/9780691179506/lectures-on-the-infrared-structure-of-gravity-and-gauge-theory.
  49. S. Choi, U. Kol, and R. Akhoury, Asymptotic dynamics in perturbative quantum gravity and BMS supertranslations, J. High Energy Phys. 01 (2018) 142.
  50. S. Choi and R. Akhoury, BMS supertranslation symmetry implies Faddeev-Kulish amplitudes, J. High Energy Phys. 02 (2018) 171.
  51. J. de Boer and S. N. Solodukhin, A holographic reduction of Minkowski space-time, Nucl. Phys. B665, 545 (2003).
  52. G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105, 111103 (2010).
  53. G. Barnich and C. Troessaert, Supertranslations call for superrotations, Proc. Sci. CNCFG2010 (2010) 010 [arXiv:1102.4632].
  54. D. Kapec, V. Lysov, S. Pasterski, and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity S-matrix, J. High Energy Phys. 08 (2014) 058.
  55. M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D 90, 124028 (2014).
  56. A. Laddha and A. Sen, Logarithmic terms in the soft expansion in four dimensions, J. High Energy Phys. 10 (2018) 056.
  57. A. Laddha and A. Sen, Observational signature of the logarithmic terms in the soft Graviton theorem, Phys. Rev. D 100, 024009 (2019).
  58. B. Sahoo and A. Sen, Classical and quantum results on logarithmic terms in the soft theorem in four dimensions, J. High Energy Phys. 02 (2019) 086.
  59. A. P. Saha, B. Sahoo, and A. Sen, Proof of the classical soft graviton theorem in D=4, J. High Energy Phys. 06 (2020) 153.
  60. B. Sahoo, Classical Sub-subleading soft photon and soft graviton theorems in four spacetime dimensions, J. High Energy Phys. 12 (2020) 070.
  61. T. C. Quinn, Axiomatic approach to radiation reaction of scalar point particles in curved space-time, Phys. Rev. D 62, 064029 (2000).
  62. S. E. Gralla and K. Lobo, Self-force effects in post-Minkowskian scattering, Classical Quantum Gravity 39, 095001 (2022); 41, 179501(E) (2024).

  63. See Refs. [64, 65] for a discussion of the asymptotic symmetries of massless scalars.

  64. M. Campiglia, L. Coito, and S. Mizera, Can scalars have asymptotic symmetries?, Phys. Rev. D 97, 046002 (2018).
  65. M. Campiglia, L. Freidel, F. Hopfmueller, and R. M. Soni, Scalar asymptotic charges and dual large gauge transformations, J. High Energy Phys. 04 (2019) 003.
  66. R. Penrose, Republication of: Conformal treatment of infinity, Gen. Relativ. Gravit. 43, 901 (2011).
  67. E. T. Newman and R. Penrose, Note on the Bondi-Metzner-Sachs group, J. Math. Phys. (N.Y.) 7, 863 (1966).
  68. A. Ashtekar and M. Streubel, On angular momentum of stationary gravitating systems, J. Math. Phys. (N.Y.) 20, 1362 (1979).
  69. A. Ashtekar, T. De Lorenzo, and N. Khera, Compact binary coalescences: The subtle issue of angular momentum, Phys. Rev. D 101, 044005 (2020).
  70. G. Compère, R. Oliveri, and A. Seraj, The Poincaré and BMS flux-balance laws with application to binary systems, J. High Energy Phys. 10 (2020) 116; 06 (2024) 45.
  71. P.-N. Chen, M.-T. Wang, Y.-K. Wang, and S.-T. Yau, BMS charges without supertranslation smbiguity, Commun. Math. Phys. 393, 1411 (2022).
  72. P.-N. Chen, M.-T. Wang, Y.-K. Wang, and S.-T. Yau, Supertranslation invariance of angular momentum, Adv. Theor. Math. Phys. 25, 777 (2021).
  73. R. Javadinezhad and M. Porrati, Supertranslation-invariant formula for the angular momentum flux in gravitational scattering, Phys. Rev. Lett. 130, 011401 (2023).
  74. R. Javadinezhad, U. Kol, and M. Porrati, Supertranslation-invariant dressed Lorentz charges, J. High Energy Phys. 04 (2022) 069.
  75. P. Mao, J.-B. Wu, and X. Wu, Angular momentum and memory effect, Phys. Rev. D 107, L101501 (2023).
  76. M. M. Riva, F. Vernizzi, and L. K. Wong, Angular momentum balance in gravitational two-body scattering: Flux, memory, and supertranslation invariance, Phys. Rev. D 108, 104052 (2023).
  77. R. Javadinezhad and M. Porrati, Three puzzles with covariance and supertranslation invariance of angular momentum flux and their solutions, Phys. Rev. Lett. 132, 151604 (2024).
  78. P. Mao and B. Zeng, Supertranslation ambiguity in post-Minkowskian expansion, Phys. Rev. D 111, L021502 (2025).
  79. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, Classical gravitational observables from the Eikonal operator, Phys. Lett. B 843, 138049 (2023).
  80. B. Bonga and E. Poisson, Coulombic contribution to angular momentum flux in general relativity, Phys. Rev. D 99, 064024 (2019).

  81. This is analogous to reassigning the rephasing in Eq. (3) to the S matrix rather than to the external states.

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Phys. Rev. Lett. 135, 151601– Published 8 October, 2025
  • Received 24 April 2025
  • Revised 10 July 2025
  • Accepted 8 August 2025
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