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High precision black hole scattering: Tutti frutti vs worldline effective field theory

Donato Bini1 and Thibault Damour2

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Phys. Rev. D 112, 044002 – Published 1 August, 2025

DOI: https://doi.org/10.1103/8ks7-2blq

Abstract

We consider black hole scattering up to the fifth post-Minkowskian (G5) order and compare the predictions of the tutti frutti formalism to the results obtained within two different versions of worldline effective field theory. At the G4 order we highlight the complete agreement between tutti frutti results and the results of [C. Dlapa et al., [Phys. Rev. Lett. 130, 101401 (2023)] ], and show how the tutti frutti approach allows one to extract the O(G3) angular momentum loss from the O(G4) impulse. We compare the sixth post-Newtonian (6PN) accurate tutti frutti predictions to the recent results of [M. Driesse et al., [Nature 641, 603 (2025)] ], which are at the G5 order, and at the leading order in the two mass ratios, finding complete agreement. We highlight that this agreement involves the presence at the 5.5PN level of a nonlocal tail-of-tail contribution to the scattering (first computed in [D. Bini et al., [Phys. Rev. D 102, 084047 (2020)] ]), and involves, at the 6PN level, the presence of a O(G4) contribution to the angular momentum loss [C. Heissenberg, [Phys. Rev. D 111, 126012 (2025)] ]. At the second order in the mass ratios of the O(G5) order we predict two independent gauge-invariant observables to high-PN accuracy.

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References (71)

  1. L. Blanchet, Post-Newtonian theory for gravitational waves, Living Rev. Relativity 17, 2 (2014).
  2. G. Schäfer and P. Jaranowski, Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries, Living Rev. Relativity 21, 7 (2018).
  3. L. Blanchet and T. Damour, Radiative gravitational fields in general relativity I. General structure of the field outside the source, Phil. Trans. R. Soc. A 320, 379 (1986).
  4. L. Blanchet and T. Damour, Post-Newtonian generation of gravitational waves, Ann. Inst. Henri Poincare Phys. Theor. 50, 377 (1989).
  5. T. Damour and B. R. Iyer, Multipole analysis for electromagnetism and linearized gravity with irreducible Cartesian tensors, Phys. Rev. D 43, 3259 (1991).
  6. W. D. Goldberger and I. Z. Rothstein, An effective field theory of gravity for extended objects, Phys. Rev. D 73, 104029 (2006).
  7. S. Foffa and R. Sturani, Effective field theory calculation of conservative binary dynamics at third post-Newtonian order, Phys. Rev. D 84, 044031 (2011).
  8. S. Foffa, P. Mastrolia, R. Sturani, C. Sturm, and W. J. Torres Bobadilla, Static two-body potential at fifth post-Newtonian order, Phys. Rev. Lett. 122, 241605 (2019).
  9. R. A. Porto, The effective field theorist’s approach to gravitational dynamics, Phys. Rep. 633, 1 (2016).
  10. W. D. Goldberger and A. Ross, Gravitational radiative corrections from effective field theory, Phys. Rev. D 81, 124015 (2010).
  11. L. Barack and A. Pound, Self-force and radiation reaction in general relativity, Rep. Prog. Phys. 82, 016904 (2019).
  12. D. Bini, T. Damour, and A. Geralico, Novel approach to binary dynamics: Application to the fifth post-Newtonian level, Phys. Rev. Lett. 123, 231104 (2019).
  13. D. Bini, T. Damour, and A. Geralico, Sixth post-Newtonian nonlocal-in-time dynamics of binary systems, Phys. Rev. D 102, 084047 (2020).
  14. D. Bini, T. Damour, and A. Geralico, Binary dynamics at the fifth and fifth-and-a-half post-Newtonian orders, Phys. Rev. D 102, 024062 (2020).
  15. D. Bini, T. Damour, A. Geralico, S. Laporta, and P. Mastrolia, Gravitational scattering at the seventh order in G: Nonlocal contribution at the sixth post-Newtonian accuracy, Phys. Rev. D 103, 044038 (2021).
  16. D. Bini, T. Damour, and A. Geralico, Sixth post-Newtonian local-in-time dynamics of binary systems, Phys. Rev. D 102, 024061 (2020).
  17. D. Amati, M. Ciafaloni, and G. Veneziano, Higher order gravitational deflection and soft bremsstrahlung in Planckian energy superstring collisions, Nucl. Phys. B347, 550 (1990).
  18. T. Damour, High-energy gravitational scattering and the general relativistic two-body problem, Phys. Rev. D 97, 044038 (2018).
  19. N. E. J. Bjerrum-Bohr, P. H. Damgaard, G. Festuccia, L. Planté, and P. Vanhove, General relativity from scattering amplitudes, Phys. Rev. Lett. 121, 171601 (2018).
  20. D. A. Kosower, B. Maybee, and D. O’Connell, Amplitudes, observables, and classical scattering, J. High Energy Phys. 02 (2019) 137.
  21. C. Cheung, I. Z. Rothstein, and M. P. Solon, From scattering amplitudes to classical potentials in the post-Minkowskian expansion, Phys. Rev. Lett. 121, 251101 (2018).
  22. Z. Bern, C. Cheung, R. Roiban, C. H. Shen, M. P. Solon, and M. Zeng, Scattering amplitudes and the conservative Hamiltonian for binary systems at third post-Minkowskian order, Phys. Rev. Lett. 122, 201603 (2019).
  23. Z. Bern, C. Cheung, R. Roiban, C. H. Shen, M. P. Solon, and M. Zeng, Black hole binary dynamics from the double copy and effective theory, J. High Energy Phys. 10 (2019) 206.
  24. N. E. J. Bjerrum-Bohr, A. Cristofoli, and P. H. Damgaard, Post-Minkowskian scattering angle in Einstein gravity, J. High Energy Phys. 08 (2020) 038.
  25. N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Planté, and P. Vanhove, The amplitude for classical gravitational scattering at third Post-Minkowskian order, J. High Energy Phys. 08 (2021) 172.
  26. Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C. H. Shen, M. P. Solon, and M. Zeng, Scattering amplitudes and conservative binary dynamics at O(G4), Phys. Rev. Lett. 126, 171601 (2021).
  27. Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C. H. Shen, M. P. Solon, and M. Zeng, Scattering amplitudes, the tail effect, and conservative binary dynamics at O(G4), Phys. Rev. Lett. 128, 161103 (2022).
  28. G. Kälin and R. A. Porto, Post-Minkowskian effective field theory for conservative binary dynamics, J. High Energy Phys. 11 (2020) 106.
  29. G. Kälin, Z. Liu, and R. A. Porto, Conservative dynamics of binary systems to third post-Minkowskian order from the effective field theory approach, Phys. Rev. Lett. 125, 261103 (2020).
  30. C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Dynamics of binary systems to fourth post-Minkowskian order from the effective field theory approach, Phys. Lett. B 831, 137203 (2022).
  31. C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Conservative dynamics of binary systems at fourth post-Minkowskian order in the large-eccentricity expansion, Phys. Rev. Lett. 128, 161104 (2022).
  32. G. Kälin, J. Neef, and R. A. Porto, Radiation-reaction in the effective field theory approach to post-Minkowskian dynamics, J. High Energy Phys. 01 (2023) 140.
  33. G. Mogull, J. Plefka, and J. Steinhoff, Classical black hole scattering from a worldline quantum field theory, J. High Energy Phys. 02 (2021) 048.
  34. M. V. S. Saketh, J. Vines, J. Steinhoff, and A. Buonanno, Conservative and radiative dynamics in classical relativistic scattering and bound systems, Phys. Rev. Res. 4, 013127 (2022).
  35. M. Khalil, A. Buonanno, J. Steinhoff, and J. Vines, Energetics and scattering of gravitational two-body systems at fourth post-Minkowskian order, Phys. Rev. D 106, 024042 (2022).
  36. S. Mougiakakos, M. M. Riva, and F. Vernizzi, Gravitational Bremsstrahlung in the post-Minkowskian effective field theory, Phys. Rev. D 104, 024041 (2021).
  37. M. M. Riva and F. Vernizzi, Radiated momentum in the post-Minkowskian worldline approach via reverse unitarity, J. High Energy Phys. 11 (2021) 228.
  38. G. U. Jakobsen, G. Mogull, J. Plefka, and J. Steinhoff, Classical gravitational bremsstrahlung from a worldline quantum field theory, Phys. Rev. Lett. 126, 201103 (2021).
  39. E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, Radiative classical gravitational observables at O(G3) from scattering amplitudes, J. High Energy Phys. 10 (2021) 148.
  40. 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).
  41. C. Dlapa, G. Kälin, Z. Liu, J. Neef, and R. A. Porto, Radiation reaction and gravitational waves at fourth post-Minkowskian order, Phys. Rev. Lett. 130, 101401 (2023).
  42. M. Driesse, G. U. Jakobsen, G. Mogull, J. Plefka, B. Sauer, and J. Usovitsch, Conservative black hole scattering at fifth post-Minkowskian and first self-force order, Phys. Rev. Lett. 132, 241402 (2024).
  43. M. Driesse, G. U. Jakobsen, A. Klemm, G. Mogull, C. Nega, J. Plefka, B. Sauer, and J. Usovitsch, Emergence of Calabi–Yau manifolds in high-precision black-hole scattering, Nature (London) 641, 603 (2025).
  44. D. Bini, T. Damour, and A. Geralico, Radiative contributions to gravitational scattering, Phys. Rev. D 104, 084031 (2021).
  45. D. Bini, T. Damour, and A. Geralico, Radiated momentum and radiation reaction in gravitational two-body scattering including time-asymmetric effects, Phys. Rev. D 107, 024012 (2023).
  46. T. Damour, Radiative contribution to classical gravitational scattering at the third order in G, Phys. Rev. D 102, 124008 (2020).
  47. D. Bini and T. Damour, Radiation-reaction and angular momentum loss at the second post-Minkowskian order, Phys. Rev. D 106, 124049 (2022).
  48. E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, Gravitational bremsstrahlung from reverse unitarity, Phys. Rev. Lett. 126, 201602 (2021).
  49. M. Ciafaloni and D. Colferai, Rescattering corrections and self-consistent metric in Planckian scattering, J. High Energy Phys. 10 (2014) 085.
  50. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, The gravitational eikonal: From particle, string and brane collisions to black-hole encounters, Phys. Rep. 1083, 1 (2024).
  51. D. Bini and T. Damour, Gravitational radiation reaction along general orbits in the effective one-body formalism, Phys. Rev. D 86, 124012 (2012).
  52. N. E. J. Bjerrum-Bohr, L. Planté, and P. Vanhove, Effective field theory and applications: Weak field observables from scattering amplitudes in quantum field theory, arXiv:2212.08957.
  53. P. H. Damgaard, E. R. Hansen, L. Planté, and P. Vanhove, Classical observables from the exponential representation of the gravitational S-matrix, J. High Energy Phys. 09 (2023) 183.
  54. L. Blanchet and T. Damour, Tail transported temporal correlations in the dynamics of a gravitating system, Phys. Rev. D 37, 1410 (1988).
  55. L. Blanchet and T. Damour, Hereditary effects in gravitational radiation, Phys. Rev. D 46, 4304 (1992).
  56. T. Damour, P. Jaranowski, and G. Schäfer, Nonlocal-in-time action for the fourth post-Newtonian conservative dynamics of two-body systems, Phys. Rev. D 89, 064058 (2014).
  57. S. Foffa and R. Sturani, Tail terms in gravitational radiation reaction via effective field theory, Phys. Rev. D 87, 044056 (2013).
  58. L. Blanchet, Gravitational wave tails of tails, Classical Quantum Gravity 15, 113 (1998); 22, 3381(E) (2005).
  59. D. Bini and T. Damour, High-order post-Newtonian contributions to the two-body gravitational interaction potential from analytical gravitational self-force calculations, Phys. Rev. D 89, 064063 (2014).
  60. A. G. Shah, J. L. Friedman, and B. F. Whiting, Finding high-order analytic post-Newtonian parameters from a high-precision numerical self-force calculation, Phys. Rev. D 89, 064042 (2014).
  61. L. Blanchet, G. Faye, and B. F. Whiting, High-order half-integral conservative post-Newtonian coefficients in the redshift factor of black hole binaries, Phys. Rev. D 90, 044017 (2014).
  62. L. Blanchet, G. Faye, and B. F. Whiting, Half-integral conservative post-Newtonian approximations in the redshift factor of black hole binaries, Phys. Rev. D 89, 064026 (2014).
  63. T. Damour, P. Jaranowski, and G. Schäfer, Fourth post-Newtonian effective one-body dynamics, Phys. Rev. D 91, 084024 (2015).
  64. A. Edison and M. Levi, A tale of tails through generalized unitarity, Phys. Lett. B 837, 137634 (2023).
  65. A. Edison and M. Levi, Higher-order tails and RG flows due to scattering of gravitational radiation from binary inspirals, J. High Energy Phys. 08 (2024) 161.
  66. 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).
  67. M. M. Ivanov, Y. Z. Li, J. Parra-Martinez, and Z. Zhou, Resummation of universal tails in gravitational waveforms, arXiv:2504.07862.
  68. C. Heissenberg, Radiation-reaction and angular momentum loss at O(G4), Phys. Rev. D 111, 126012 (2025).
  69. J. Blümlein, A. Maier, P. Marquard, and G. Schäfer, The fifth-order post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach: Potential contributions, Nucl. Phys. B965, 115352 (2021).
  70. J. Blümlein, A. Maier, P. Marquard, and G. Schäfer, The fifth-order post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach, Nucl. Phys. B983, 115900 (2022); B985, 115991(E) (2022).
  71. See Supplemental Material at http://link.aps.org/supplemental/10.1103/8ks7-2blq included both in the arxiv version and in the published version of the present paper contain all data that support the findings of this article.

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Phys. Rev. D 112, 044002– Published 1 August, 2025
  • Received 5 May 2025
  • Accepted 6 July 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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