- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Conservative Black Hole Scattering at Fifth Post-Minkowskian and Second Self-Force Order

Mathias Driesse1, Gustav Uhre Jakobsen1,2, Gustav Mogull1,2,3, Christoph Nega2,4, Jan Plefka1, Benjamin Sauer1, and Johann Usovitsch1

PDF

Phys. Rev. Lett. 137, 081402 – Published 21 August, 2026

DOI: https://doi.org/10.1103/4dvk-nglx

Abstract

Using the worldline quantum field theory formalism, we compute conservative contributions to the scattering angle and impulse for classical black hole scattering at fifth post-Minkowskian and second self-force order. This four-loop calculation involves nonplanar Feynman integrals and requires advanced integration-by-parts reduction, novel differential-equation strategies, and efficient boundary-integral algorithms to solve a system of hundreds of master integrals in four integral families on high-performance computing systems. The resulting function space includes multiple polylogarithms as well as iterated integrals with a K3 period, which generate a spurious velocity divergence at v/c=8/3, γ=3. This divergence is present in the potential region and must be canceled by contributions from the radiative memory region, while its dimensional-regularization pole should cancel against the radiative tail region. As the standard use of Feynman propagators fails to ensure this cancellation, we instead propose a “(γ−3)” conservative prescription that realizes both cancellations, leading to a physically sensible answer. All available low-velocity checks of our result against the post-Newtonian literature are satisfied.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (169)

  1. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102 (2016).
  2. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW170817: Observation of gravitational waves from a binary neutron star inspiral, Phys. Rev. Lett. 119, 161101 (2017).
  3. R. Abbott et al. (KAGRA, Virgo, and LIGO Scientific Collaborations), GWTC-3: Compact binary coalescences observed by LIGO and Virgo during the second part of the third observing run, Phys. Rev. X 13, 041039 (2023).
  4. A. G. Abac et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), GWTC-4.0: Updating the gravitational-wave transient catalog with observations from the first part of the fourth LIGO-Virgo-KAGRA observing run, Astrophys. J. Lett. 1004, L22 (2026).
  5. P. Amaro-Seoane et al. (LISA Collaboration), Laser Interferometer Space Antenna, arXiv:1702.00786.
  6. M. Punturo et al., The Einstein Telescope: A third-generation gravitational wave observatory, Classical Quantum Gravity 27, 194002 (2010).
  7. S. W. Ballmer et al., Snowmass2021 cosmic frontier white paper: Future gravitational-wave detector facilities, in Snowmass 2021 (2022), arXiv:2203.08228.
  8. L. Blanchet, Gravitational radiation from post-Newtonian sources and inspiralling compact binaries, Living Rev. Relativity 17, 2 (2014).
  9. R. A. Porto, The effective field theorist’s approach to gravitational dynamics, Phys. Rep. 633, 1 (2016).
  10. M. Levi, Effective field theories of post-Newtonian gravity: A comprehensive review, Rep. Prog. Phys. 83, 075901 (2020).
  11. G. Brunello, M. K. Mandal, P. Mastrolia, R. Patil, M. Pegorin, J. Ronca et al., Six-loop gravitational interactions at the sixth post-Newtonian order, arXiv:2512.19498.
  12. 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).
  13. N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Plante, and P. Vanhove, The SAGEX review on scattering amplitudes Chapter 13: Post-Minkowskian expansion from scattering amplitudes, J. Phys. A 55, 443014 (2022).
  14. A. Buonanno, M. Khalil, D. O’Connell, R. Roiban, M. P. Solon, and M. Zeng, Snowmass white paper: Gravitational waves and scattering amplitudes, in Snowmass 2021 (2022), arXiv:2204.05194.
  15. 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).
  16. G. U. Jakobsen, Gravitational scattering of compact bodies from worldline quantum field theory, Ph.D. thesis, Humboldt-University Berlin, 2023.
  17. Y. Mino, M. Sasaki, and T. Tanaka, Gravitational radiation reaction to a particle motion, Phys. Rev. D 55, 3457 (1997).
  18. E. Poisson, A. Pound, and I. Vega, The motion of point particles in curved spacetime, Living Rev. Relativity 14, 7 (2011).
  19. L. Barack and A. Pound, Self-force and radiation reaction in general relativity, Rep. Prog. Phys. 82, 016904 (2019).
  20. S. E. Gralla and K. Lobo, Self-force effects in post-Minkowskian scattering, Classical Quantum Gravity 39, 095001 (2022).
  21. F. Pretorius, Evolution of binary black hole spacetimes, Phys. Rev. Lett. 95, 121101 (2005).
  22. M. Boyle et al., The SXS Collaboration catalog of binary black hole simulations, Classical Quantum Gravity 36, 195006 (2019).
  23. T. Damour, F. Guercilena, I. Hinder, S. Hopper, A. Nagar, and L. Rezzolla, Strong-field scattering of two black holes: Numerics versus analytics, Phys. Rev. D 89, 081503 (2014).
  24. 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).
  25. S. J. Kovacs and K. S. Thorne, The generation of gravitational waves. 4. Bremsstrahlung, Astrophys. J. 224, 62 (1978).
  26. K. Westpfahl and M. Goller, Gravitational scattering of two relativistic particles in postlinear approximation, Lett. Nuovo Cimento 26, 573 (1979).
  27. L. Bel, T. Damour, N. Deruelle, J. Ibanez, and J. Martin, Poincaré-invariant gravitational field and equations of motion of two pointlike objects: The postlinear approximation of general relativity, Gen. Relativ. Gravit. 13, 963 (1981).
  28. T. Damour, High-energy gravitational scattering and the general relativistic two-body problem, Phys. Rev. D 97, 044038 (2018).
  29. S. Hopper, A. Nagar, and P. Rettegno, Strong-field scattering of two spinning black holes: Numerics versus analytics, Phys. Rev. D 107, 124034 (2023).
  30. W. D. Goldberger and I. Z. Rothstein, An effective field theory of gravity for extended objects, Phys. Rev. D 73, 104029 (2006).
  31. G. Kälin and R. A. Porto, Post-Minkowskian effective field theory for conservative binary dynamics, J. High Energy Phys. 11 (2020) 106.
  32. 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).
  33. G. Kälin, Z. Liu, and R. A. Porto, Conservative tidal effects in compact binary systems to next-to-leading post-Minkowskian order, Phys. Rev. D 102, 124025 (2020).
  34. G. Mogull, J. Plefka, and J. Steinhoff, Classical black hole scattering from a worldline quantum field theory, J. High Energy Phys. 02 (2021) 048.
  35. 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).
  36. 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).
  37. 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).
  38. S. Mougiakakos, M. M. Riva, and F. Vernizzi, Gravitational bremsstrahlung in the post-Minkowskian effective field theory, Phys. Rev. D 104, 024041 (2021).
  39. M. M. Riva and F. Vernizzi, Radiated momentum in the post-Minkowskian worldline approach via reverse unitarity, J. High Energy Phys. 11 (2021) 228.
  40. 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).
  41. C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Bootstrapping the relativistic two-body problem, J. High Energy Phys. 08 (2023) 109.
  42. Z. Liu, R. A. Porto, and Z. Yang, Spin effects in the effective field theory approach to post-Minkowskian conservative dynamics, J. High Energy Phys. 06 (2021) 012.
  43. S. Mougiakakos, M. M. Riva, and F. Vernizzi, Gravitational bremsstrahlung with tidal effects in the post-Minkowskian expansion, Phys. Rev. Lett. 129, 121101 (2022).
  44. M. M. Riva, F. Vernizzi, and L. K. Wong, Gravitational bremsstrahlung from spinning binaries in the post-Minkowskian expansion, Phys. Rev. D 106, 044013 (2022).
  45. G. U. Jakobsen, G. Mogull, J. Plefka, and J. Steinhoff, Gravitational bremsstrahlung and hidden supersymmetry of spinning bodies, Phys. Rev. Lett. 128, 011101 (2022).
  46. G. U. Jakobsen, G. Mogull, J. Plefka, and J. Steinhoff, SUSY in the sky with gravitons, J. High Energy Phys. 01 (2022) 027.
  47. G. U. Jakobsen and G. Mogull, Conservative and radiative dynamics of spinning bodies at third post-Minkowskian order using worldline quantum field theory, Phys. Rev. Lett. 128, 141102 (2022).
  48. G. U. Jakobsen and G. Mogull, Linear response, Hamiltonian, and radiative spinning two-body dynamics, Phys. Rev. D 107, 044033 (2023).
  49. G. U. Jakobsen, G. Mogull, J. Plefka, and B. Sauer, All things retarded: Radiation-reaction in worldline quantum field theory, J. High Energy Phys. 10 (2022) 128.
  50. C. Shi and J. Plefka, Classical double copy of worldline quantum field theory, Phys. Rev. D 105, 026007 (2022).
  51. F. Bastianelli, F. Comberiati, and L. de la Cruz, Light bending from eikonal in worldline quantum field theory, J. High Energy Phys. 02 (2022) 209.
  52. F. Comberiati and C. Shi, Classical double copy of spinning worldline quantum field theory, J. High Energy Phys. 04 (2023) 008.
  53. T. Wang, Binary dynamics from worldline QFT for scalar QED, Phys. Rev. D 107, 085011 (2023).
  54. M. Ben-Shahar, Scattering of spinning compact objects from a worldline EFT, J. High Energy Phys. 03 (2024) 108.
  55. A. Bhattacharyya, D. Ghosh, S. Ghosh, and S. Pal, Observables from classical black hole scattering in scalar-tensor theory of gravity from worldline quantum field theory, J. High Energy Phys. 04 (2024) 015.
  56. G. U. Jakobsen, G. Mogull, J. Plefka, B. Sauer, and Y. Xu, Conservative scattering of spinning black holes at fourth post-Minkowskian order, Phys. Rev. Lett. 131, 151401 (2023).
  57. G. U. Jakobsen, G. Mogull, J. Plefka, and B. Sauer, Dissipative scattering of spinning black holes at fourth post-Minkowskian order, Phys. Rev. Lett. 131, 241402 (2023).
  58. G. U. Jakobsen, G. Mogull, J. Plefka, and B. Sauer, Tidal effects and renormalization at fourth post-Minkowskian order, Phys. Rev. D 109, L041504 (2024).
  59. S. Caron-Huot, M. Correia, G. Isabella, and M. Solon, Gravitational wave scattering via the born series: Scalar tidal matching to O(G7) and beyond, Phys. Rev. Lett. 135, 191601 (2025).
  60. M. M. Ivanov, Y.-Z. Li, J. Parra-Martinez, and Z. Zhou, Resummation of universal tails in gravitational waveforms, Phys. Rev. Lett. 135, 141401 (2025).
  61. O. Combaluzier-Szteinsznaider, D. Glazer, A. Joyce, M. J. Rodriguez, and L. Santoni, Dynamical tidal response of Schwarzschild black holes, J. High Energy Phys. 06 (2026) 032.
  62. D. Neill and I. Z. Rothstein, Classical space-times from the S matrix, Nucl. Phys. B877, 177 (2013).
  63. A. Luna, I. Nicholson, D. O’Connell, and C. D. White, Inelastic black hole scattering from charged scalar amplitudes, J. High Energy Phys. 03 (2018) 044.
  64. D. A. Kosower, B. Maybee, and D. O’Connell, Amplitudes, observables, and classical scattering, J. High Energy Phys. 02 (2019) 137.
  65. A. Cristofoli, R. Gonzo, D. A. Kosower, and D. O’Connell, Waveforms from amplitudes, Phys. Rev. D 106, 056007 (2022).
  66. N. E. J. Bjerrum-Bohr, J. F. Donoghue, and P. Vanhove, On-shell techniques and universal results in quantum gravity, J. High Energy Phys. 02 (2014) 111.
  67. 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).
  68. 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).
  69. 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.
  70. N. E. J. Bjerrum-Bohr, L. Planté, and P. Vanhove, Post-Minkowskian radial action from soft limits and velocity cuts, J. High Energy Phys. 03 (2022) 071.
  71. C. Cheung and M. P. Solon, Classical gravitational scattering at O(G3) from Feynman diagrams, J. High Energy Phys. 06 (2020) 144.
  72. 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.
  73. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, Universality of ultra-relativistic gravitational scattering, Phys. Lett. B 811, 135924 (2020).
  74. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, The eikonal approach to gravitational scattering and radiation at O(G3), J. High Energy Phys. 07 (2021) 169.
  75. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, Radiation reaction from soft theorems, Phys. Lett. B 818, 136379 (2021).
  76. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, Classical gravitational observables from the Eikonal operator, Phys. Lett. B 843, 138049 (2023).
  77. C. Heissenberg, Angular momentum loss due to tidal effects in the post-Minkowskian expansion, Phys. Rev. Lett. 131, 011603 (2023).
  78. T. Damour, Radiative contribution to classical gravitational scattering at the third order in G, Phys. Rev. D 102, 124008 (2020).
  79. 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.
  80. P. H. Damgaard, K. Haddad, and A. Helset, Heavy black hole effective theory, J. High Energy Phys. 11 (2019) 070.
  81. P. H. Damgaard, L. Plante, and P. Vanhove, On an exponential representation of the gravitational S-matrix, J. High Energy Phys. 11 (2021) 213.
  82. P. H. Damgaard, E. R. Hansen, L. Planté, and P. Vanhove, The relation between KMOC and worldline formalisms for classical gravity, J. High Energy Phys. 09 (2023) 059.
  83. R. Aoude, K. Haddad, and A. Helset, On-shell heavy particle effective theories, J. High Energy Phys. 05 (2020) 051.
  84. M. Accettulli Huber, A. Brandhuber, S. De Angelis, and G. Travaglini, From amplitudes to gravitational radiation with cubic interactions and tidal effects, Phys. Rev. D 103, 045015 (2021).
  85. A. Brandhuber, G. Chen, G. Travaglini, and C. Wen, Classical gravitational scattering from a gauge-invariant double copy, J. High Energy Phys. 10 (2021) 118.
  86. 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).
  87. 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).
  88. Z. Bern, D. Kosmopoulos, A. Luna, R. Roiban, and F. Teng, Binary dynamics through the fifth power of spin at O(G2), Phys. Rev. Lett. 130, 201402 (2023).
  89. Z. Bern, D. Kosmopoulos, A. Luna, R. Roiban, T. Scheopner, F. Teng, and J. Vines, Quantum field theory, worldline theory, and spin magnitude change in orbital evolution, Phys. Rev. D 109, 045011 (2024).
  90. 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.
  91. 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.
  92. 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.
  93. 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).
  94. A. Herderschee, R. Roiban, and F. Teng, The sub-leading scattering waveform from amplitudes, J. High Energy Phys. 06 (2023) 004.
  95. S. Caron-Huot, M. Giroux, H. S. Hannesdottir, and S. Mizera, What can be measured asymptotically?, J. High Energy Phys. 01 (2024) 139.
  96. F. Febres Cordero, M. Kraus, G. Lin, M. S. Ruf, and M. Zeng, Conservative binary dynamics with a spinning black hole at O(G3) from scattering amplitudes, Phys. Rev. Lett. 130, 021601 (2023).
  97. 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.
  98. K. Haddad, G. U. Jakobsen, G. Mogull, and J. Plefka, Spinning bodies in general relativity from bosonic worldline oscillators, J. High Energy Phys. 02 (2025) 019.
  99. 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).
  100. O. Long, H. P. Pfeiffer, L. E. Kidder, and M. A. Scheel, Black-hole scattering with numerical relativity: Self-force extraction and post-Minkowskian validation, Phys. Rev. D 112, 124038 (2025).
  101. N. Warburton, Gravitational radiation from hyperbolic orbits: Comparison between self-force, post-Minkowskian, post-Newtonian, and numerical relativity results, Phys. Rev. D 113, 084059 (2026).
  102. Z. Bern, E. Herrmann, R. Roiban, M. S. Ruf, A. V. Smirnov, V. A. Smirnov, and M. Zeng, Second-order self-force potential-region binary dynamics at O(G5) in supergravity, Phys. Rev. Lett. 136, 081401 (2026).
  103. Z. Bern, E. Herrmann, R. Roiban, M. S. Ruf, A. V. Smirnov, S. Smith, and M. Zeng, Scattering amplitudes and conservative binary dynamics at O(G5) without self-force truncation, Phys. Rev. Lett. 137, 021402 (2026).
  104. G. Kälin and R. A. Porto, From boundary data to bound states, J. High Energy Phys. 01 (2020) 072.
  105. G. Kälin and R. A. Porto, From boundary data to bound states. Part II. Scattering angle to dynamical invariants (with twist), J. High Energy Phys. 02 (2020) 120.
  106. G. Cho, G. Kälin, and R. A. Porto, From boundary data to bound states. Part III. Radiative effects, J. High Energy Phys. 04 (2022) 154.
  107. C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Local in time conservative binary dynamics at fourth post-Minkowskian order, Phys. Rev. Lett. 132, 221401 (2024).
  108. C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Local-in-time conservative binary dynamics at fifth post-Minkowskian and first self-force orders, Phys. Rev. Lett. 135, 251401 (2025).
  109. 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.
  110. See the Supplemental Material at http://link.aps.org/supplemental/10.1103/4dvk-nglx for details on the integral families, IBP reduction, boundary integrala and the novel γ=3 prescription.
  111. L. Blanchet and T. Damour, Hereditary effects in gravitational radiation, Phys. Rev. D 46, 4304 (1992).
  112. R. A. Porto, M. M. Riva, and Z. Yang, Nonlinear gravitational radiation reaction: Failed tail, memories & squares, J. High Energy Phys. 04 (2025) 050.
  113. K. Cunningham, C. Kavanagh, A. Pound, D. Trestini, N. Warburton, and J. Neef, Gravitational memory: New results from post-Newtonian and self-force theory, Classical Quantum Gravity 42, 135009 (2025).
  114. A. Georgoudis, V. Goncalves, C. Heissenberg, and J. Parra-Martinez, Nonlinear gravitational memory in the post-Minkowskian expansion, Phys. Rev. Lett. 136, 121401 (2026).
  115. P. Maierhöfer, J. Usovitsch, and P. Uwer, kira—A Feynman integral reduction program, Comput. Phys. Commun. 230, 99 (2018).
  116. J. Klappert, F. Lange, P. Maierhöfer, and J. Usovitsch, Integral reduction with kira 2.0 and finite field methods, Comput. Phys. Commun. 266, 108024 (2021).
  117. F. Lange, P. Maierhöfer, and J. Usovitsch, Developments since kira 2.0, SciPost Phys. Proc. 7, 017 (2022).
  118. F. Lange, J. Usovitsch, and Z. Wu, kira 3: Integral reduction with efficient seeding and optimized equation selection, Comput. Phys. Commun. 322, 109999 (2026).
  119. J. Klappert and F. Lange, Reconstructing rational functions with firefly, Comput. Phys. Commun. 247, 106951 (2020).
  120. J. Klappert, S. Y. Klein, and F. Lange, Interpolation of dense and sparse rational functions and other improvements in firefly, Comput. Phys. Commun. 264, 107968 (2021).
  121. A. V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254, 158 (1991).
  122. E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cimento Soc. Ital. Fis. 110A, 1435 (1997).
  123. T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B580, 485 (2000).
  124. J. M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110, 251601 (2013).
  125. L. Görges, C. Nega, L. Tancredi, and F. J. Wagner, On a procedure to derive ε-factorized differential equations beyond polylogarithms, J. High Energy Phys. 07 (2023) 206.
  126. C. Duhr, S. Maggio, C. Nega, B. Sauer, L. Tancredi, and F. J. Wagner, Aspects of canonical differential equations for Calabi-Yau geometries and beyond, J. High Energy Phys. 06 (2025) 128.
  127. H. Frellesvig, R. Morales, and M. Wilhelm, Calabi-Yau meets gravity: A Calabi-Yau threefold at fifth post-Minkowskian order, Phys. Rev. Lett. 132, 201602 (2024).
  128. A. Klemm, C. Nega, B. Sauer, and J. Plefka, Calabi-Yau periods for black hole scattering in classical general relativity, Phys. Rev. D 109, 124046 (2024).
  129. H. Frellesvig, R. Morales, and M. Wilhelm, Classifying post-Minkowskian geometries for gravitational waves via loop-by-loop Baikov, J. High Energy Phys. 08 (2024) 243.
  130. H. Frellesvig, R. Morales, S. Pögel, S. Weinzierl, and M. Wilhelm, Calabi-Yau Feynman integrals in gravity: ϵ-factorized form for apparent singularities, J. High Energy Phys. 02 (2025) 209.
  131. S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampére equation, I, Commun. Pure Appl. Math. 31, 339 (1978).
  132. P. Candelas, G. T. Horowitz, A. Strominger, and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B258, 46 (1985).
  133. A. Klemm, The B-model approach to topological string theory on Calabi-Yau n-folds, in B-model Gromov-Witten Theory, Trends Math.(Birkhäuser/Springer, Cham, 2018), pp. 79–397.
  134. P. Vanhove, The physics and the mixed Hodge structure of Feynman integrals, Proc. Symp. Pure Math. 88, 161 (2014).
  135. J. L. Bourjaily, A. J. McLeod, M. von Hippel, and M. Wilhelm, Bounded collection of Feynman integral Calabi-Yau geometries, Phys. Rev. Lett. 122, 031601 (2019).
  136. A. Klemm, C. Nega, and R. Safari, The l-loop banana amplitude from GKZ systems and relative Calabi-Yau periods, J. High Energy Phys. 04 (2020) 088.
  137. K. Bönisch, F. Fischbach, A. Klemm, C. Nega, and R. Safari, Analytic structure of all loop banana integrals, J. High Energy Phys. 05 (2021) 066.
  138. K. Bönisch, C. Duhr, F. Fischbach, A. Klemm, and C. Nega, Feynman integrals in dimensional regularization and extensions of Calabi-Yau motives, J. High Energy Phys. 09 (2022) 156.
  139. J. L. Bourjaily et al., Functions beyond multiple polylogarithms for precision collider physics, in Snowmass 2021 (2022), arXiv:2203.07088.
  140. C. Duhr, A. Klemm, F. Loebbert, C. Nega, and F. Porkert, Yangian-invariant fishnet integrals in two dimensions as volumes of Calabi-Yau varieties, Phys. Rev. Lett. 130, 041602 (2023).
  141. D. Huybrechts, Lectures on K3 Surfaces, Vol. 158 of Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, England, 2016), 10.1017/CBO9781316594193.
  142. R. Apéry, Irrationalité de ζ2 et ζ3, in Journées Arithmétiques de Luminy (Société Mathématique de France, Paris, 1979), No. 61, pp. 11–13.
  143. D. Zagier, The arithmetic and topology of differential equations, in European Congress of Mathematics (European Mathematical Society, Zürich, 2018), pp. 717–776.
  144. C. Meyer, Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA, Comput. Phys. Commun. 222, 295 (2018).
  145. H. Frellesvig, The loop-by-loop Baikov representation—Strategies and implementation, J. High Energy Phys. 04 (2025) 111.
  146. C. Dlapa, J. Henn, and K. Yan, Deriving canonical differential equations for Feynman integrals from a single uniform weight integral, J. High Energy Phys. 05 (2020) 025.
  147. T. Peraro, FiniteFlow: Multivariate functional reconstruction using finite fields and dataflow graphs, J. High Energy Phys. 07 (2019) 031.
  148. M. Heller and A. von Manteuffel, MultivariateApart: Generalized partial fractions, Comput. Phys. Commun. 271, 108174 (2022).
  149. M. Beneke and V. A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B522, 321 (1998).
  150. V. A. Smirnov, Analytic Tools for Feynman Integrals (Springer, Berlin, Heidelberg, 2012), Vol. 250, 10.1007/978-3-642-34886-0.
  151. T. Becher, A. Broggio, and A. Ferroglia, Introduction to Soft-Collinear Effective Theory (Springer, New York, 2015), Vol. 896, 10.1007/978-3-319-14848-9.
  152. L. Blanchet and T. Damour, Tail-transported temporal correlations in the dynamics of a gravitating system, Phys. Rev. D 37, 1410 (1988).
  153. M. Hidding and J. Usovitsch, Feynman parameter integration through differential equations, Phys. Rev. D 108, 036024 (2023).
  154. H. R. P. Ferguson, D. H. Bailey, and S. Arno, Analysis of PSLQ, an integer relation finding algorithm, Math. Comput. 68, 351 (1999).
  155. M. Driesse, G. U. Jakobsen, G. Mogull, C. Nega, J. Plefka, B. Sauer et al., Conservative black hole scattering at fifth post-Minkowskian and second self-force order, 10.5281/zenodo.18340333 (2026).
  156. D. Bini, T. Damour, and A. Geralico, Sixth post-Newtonian nonlocal-in-time dynamics of binary systems, Phys. Rev. D 102, 084047 (2020).
  157. J. Blümlein, A. Maier, P. Marquard, and G. Schäfer, Fourth post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach, Nucl. Phys. B955, 115041 (2020).
  158. D. Bini, T. Damour, and A. Geralico, Radiative contributions to gravitational scattering, Phys. Rev. D 104, 084031 (2021).
  159. 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).
  160. D. Bini and T. Damour, High precision black hole scattering: Tutti frutti vs worldline effective field theory, Phys. Rev. D 112 (2025) 044002.
  161. 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).
  162. R. A. Porto and M. M. Riva (to be published).
  163. D. Bini and T. Damour, Gravitational scattering of two black holes at the fourth post-Newtonian approximation, Phys. Rev. D 96, 064021 (2017).
  164. J.-H. Kim, J.-W. Kim, S. Kim, and S. Lee, Classical eikonal from Magnus expansion, J. High Energy Phys. 01 (2025) 111.
  165. F. Alessio, R. Gonzo, and C. Shi, Dirac brackets for classical radiative observables, Phys. Rev. D 112, 104060 (2025).
  166. J.-W. Kim, R. Patil, T. Scheopner, and J. Steinhoff, Magnusian: Relating the eikonal phase, the on-shell action, and the scattering generator, J. High Energy Phys. 03 (2026) 241.
  167. K. Haddad, G. U. Jakobsen, G. Mogull, and J. Plefka, Unitarity and the on-shell action of worldline quantum field theory, J. High Energy Phys. 02 (2026) 008.
  168. A. Brandhuber, G. R. Brown, P. Pichini, G. Travaglini, and P. Vives Matasan, The Magnus expansion in relativistic quantum field theory, arXiv:2512.05017.
  169. http://www.nhr-verein.de/en/our-partners.

Outline

Information

Phys. Rev. Lett. 137, 081402– Published 21 August, 2026
  • Received 30 January 2026
  • Revised 22 April 2026
  • Accepted 10 July 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation