- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

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
  • Editors' Suggestion
  • Open Access

Gravitational Compton scattering at the fourth post-Minkowskian order

Giacomo Brunello1,2,*, Mario Meo1,†, and Sid Smith3,4,5,‡

  • *Contact author: giacomo.brunello@sns.it
  • †Contact author: mario.meo@sns.it
  • ‡Contact author: sid.smith@ed.ac.uk
PDF

Phys. Rev. D 114, 085022 – Published 22 October, 2026

DOI: https://doi.org/10.1103/nbxl-68tz

Abstract

We compute the classical gravitational Compton amplitude at the fourth post-Minkowskian order, O(G4), within the Worldline Quantum Field Theory framework. We derive the associated N-matrix element, which provides the gravitational-wave scattering phase shift at the same order. As a nontrivial check, we show that our result agrees with black hole perturbation theory.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (128)

  1. T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957).
  2. F. J. Zerilli, Phys. Rev. Lett. 24, 737 (1970).
  3. S. A. Teukolsky, Astrophys. J. 185, 635 (1973).
  4. S. Mano, H. Suzuki, and E. Takasugi, Prog. Theor. Phys. 95, 1079 (1996).
  5. S. R. Dolan, Phys. Rev. D 76, 084001 (2007).
  6. W. D. Goldberger and I. Z. Rothstein, Phys. Rev. D 73, 104029 (2006).
  7. W. D. Goldberger and I. Z. Rothstein, Phys. Rev. D 73, 104030 (2006).
  8. S. Foffa and R. Sturani, Classical Quantum Gravity 31, 043001 (2014).
  9. S. Foffa, P. Mastrolia, R. Sturani, and C. Sturm, Phys. Rev. D 95, 104009 (2017).
  10. S. Foffa and R. Sturani, Phys. Rev. D 100, 024047 (2019).
  11. S. Foffa, R. A. Porto, I. Rothstein, and R. Sturani, Phys. Rev. D 100, 024048 (2019).
  12. S. Foffa, P. Mastrolia, R. Sturani, C. Sturm, and W. J. Torres Bobadilla, Phys. Rev. Lett. 122, 241605 (2019).
  13. R. A. Porto, M. M. Riva, and Z. Yang, J. High Energy Phys. 04 (2025) 050.
  14. R. A. Porto and M. M. Riva, arXiv:2604.09545.
  15. G. Brunello, M. K. Mandal, P. Mastrolia, R. Patil, M. Pegorin, J. Ronca, S. Smith, J. Steinhoff, and W. J. Torres Bobadilla, arXiv:2512.19498.
  16. G. Brunello, M. K. Mandal, P. Mastrolia, R. Patil, M. Pegorin, S. Smith, and J. Steinhoff, arXiv:2604.14134.
  17. C. Cheung, I. Z. Rothstein, and M. P. Solon, Phys. Rev. Lett. 121, 251101 (2018).
  18. N. E. J. Bjerrum-Bohr, P. H. Damgaard, G. Festuccia, L. Planté, and P. Vanhove, Phys. Rev. Lett. 121, 171601 (2018).
  19. Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 122, 201603 (2019).
  20. Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, J. High Energy Phys. 10 (2019) 206.
  21. J. Parra-Martinez, M. S. Ruf, and M. Zeng, J. High Energy Phys. 11 (2020) 023.
  22. C. Cheung and M. P. Solon, J. High Energy Phys. 06 (2020) 144.
  23. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, J. High Energy Phys. 07 (2021) 169.
  24. A. Brandhuber, G. Chen, G. Travaglini, and C. Wen, J. High Energy Phys. 10 (2021) 118.
  25. G. Kälin, Z. Liu, and R. A. Porto, Phys. Rev. Lett. 125, 261103 (2020).
  26. C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Phys. Lett. B 831, 137203 (2022).
  27. Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 126, 171601 (2021).
  28. Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon, and M. Zeng, Proc. Sci. LL2022 (2022) 051.
  29. C. Dlapa, G. Kälin, Z. Liu, J. Neef, and R. A. Porto, Phys. Rev. Lett. 130, 101401 (2023).
  30. G. U. Jakobsen, G. Mogull, J. Plefka, B. Sauer, and Y. Xu, Phys. Rev. Lett. 131, 151401 (2023).
  31. G. U. Jakobsen, G. Mogull, J. Plefka, and B. Sauer, Phys. Rev. Lett. 131, 241402 (2023).
  32. P. H. Damgaard, E. R. Hansen, L. Planté, and P. Vanhove, J. High Energy Phys. 09 (2023) 183.
  33. Z. Bern, E. Herrmann, R. Roiban, M. S. Ruf, A. V. Smirnov, V. A. Smirnov, and M. Zeng, Phys. Rev. Lett. 132, 251601 (2024).
  34. M. Driesse, G. U. Jakobsen, G. Mogull, J. Plefka, B. Sauer, and J. Usovitsch, Phys. Rev. Lett. 132, 241402 (2024).
  35. Z. Bern, E. Herrmann, R. Roiban, M. S. Ruf, A. V. Smirnov, V. A. Smirnov, and M. Zeng, J. High Energy Phys. 10 (2024) 023.
  36. M. Driesse, G. U. Jakobsen, A. Klemm, G. Mogull, C. Nega, J. Plefka, B. Sauer, and J. Usovitsch, Nature (London) 641, 603 (2025).
  37. Z. Bern, E. Herrmann, R. Roiban, M. S. Ruf, A. V. Smirnov, S. Smith, and M. Zeng, Phys. Rev. Lett. 137, 021402 (2026).
  38. M. Driesse, G. U. Jakobsen, G. Mogull, C. Nega, J. Plefka, B. Sauer, and J. Usovitsch, Phys. Rev. Lett. 137, 081402 (2026).
  39. C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Phys. Rev. D 114, 024029 (2026).
  40. D. A. Kosower, B. Maybee, and D. O’Connell, J. High Energy Phys. 02 (2019) 137.
  41. G. Mogull, J. Plefka, and J. Steinhoff, J. High Energy Phys. 02 (2021) 048.
  42. G. U. Jakobsen, G. Mogull, J. Plefka, and J. Steinhoff, Phys. Rev. Lett. 126, 201103 (2021).
  43. G. Kälin and R. A. Porto, J. High Energy Phys. 11 (2020) 106.
  44. G. Kälin, J. Neef, and R. A. Porto, J. High Energy Phys. 01 (2023) 140.
  45. S. Mougiakakos, M. M. Riva, and F. Vernizzi, Phys. Rev. D 104, 024041 (2021).
  46. G. U. Jakobsen, G. Mogull, J. Plefka, and J. Steinhoff, Phys. Rev. Lett. 128, 011101 (2022).
  47. S. De Angelis, P. P. Novichkov, and R. Gonzo, Phys. Rev. D 110, L041502 (2024).
  48. A. Brandhuber, G. R. Brown, G. Chen, J. Gowdy, and G. Travaglini, J. High Energy Phys. 02 (2024) 026.
  49. R. Aoude, K. Haddad, C. Heissenberg, and A. Helset, Phys. Rev. D 109, 036007 (2024).
  50. A. Falkowski and P. Marinellis, Eur. Phys. J. C 85, 74 (2025).
  51. G. Brunello, V. Chestnov, G. Crisanti, M. Giroux, and S. Smith, Phys. Rev. D 113, 085011 (2026).
  52. A. Brandhuber, G. R. Brown, G. Chen, S. De Angelis, J. Gowdy, and G. Travaglini, J. High Energy Phys. 06 (2023) 048.
  53. A. Herderschee, R. Roiban, and F. Teng, J. High Energy Phys. 06 (2023) 004.
  54. A. Georgoudis, C. Heissenberg, and I. Vazquez-Holm, J. High Energy Phys. 06 (2023) 126.
  55. A. Elkhidir, D. O’Connell, M. Sergola, and I. A. Vazquez-Holm, J. High Energy Phys. 07 (2024) 272.
  56. S. Caron-Huot, M. Giroux, H. S. Hannesdottir, and S. Mizera, J. High Energy Phys. 01 (2024) 139.
  57. L. Bohnenblust, H. Ita, M. Kraus, and J. Schlenk, J. High Energy Phys. 11 (2024) 109.
  58. G. Brunello and S. De Angelis, J. High Energy Phys. 07 (2024) 062.
  59. L. Bohnenblust, H. Ita, M. Kraus, and J. Schlenk, J. High Energy Phys. 12 (2025) 100.
  60. G. Brunello, S. De Angelis, and D. A. Kosower, Phys. Rev. D 114, 045004 (2026).
  61. N. E. J. Bjerrum-Bohr, G. Chen, C. J. Eriksen, and N. Shah, J. High Energy Phys. 10 (2025) 235.
  62. D. Akpinar, Phys. Rev. D 113, 045003 (2026).
  63. N. E. J. Bjerrum-Bohr, G. Chen, C. Jordan Eriksen, and N. Shah, arXiv:2602.06947.
  64. Y. F. Bautista, M. Driesse, K. Haddad, and G. U. Jakobsen, J. High Energy Phys. 05 (2026) 252.
  65. M. M. Ivanov, Y.-Z. Li, J. Parra-Martinez, and Z. Zhou, arXiv:2602.06951.
  66. P. H. Damgaard, L. Plante, and P. Vanhove, J. High Energy Phys. 11 (2021) 213.
  67. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, Phys. Rep. 1083, 1 (2024).
  68. J.-W. Kim, R. Patil, T. Scheopner, and J. Steinhoff, J. High Energy Phys. 03 (2026) 241.
  69. M. M. Ivanov and Z. Zhou, Phys. Rev. Lett. 130, 091403 (2023).
  70. M. V. S. Saketh, Z. Zhou, and M. M. Ivanov, Phys. Rev. D 109, 064058 (2024).
  71. M. M. Ivanov, Y.-Z. Li, J. Parra-Martinez, and Z. Zhou, Phys. Rev. Lett. 132, 131401 (2024); 134, 159901(E) (2025).
  72. S. Caron-Huot, M. Correia, G. Isabella, and M. Solon, Phys. Rev. Lett. 135, 191601 (2025).
  73. M. Correia and G. Isabella, J. High Energy Phys. 03 (2025) 144.
  74. G. U. Jakobsen, G. Mogull, J. Plefka, and B. Sauer, J. High Energy Phys. 10 (2022) 128.
  75. F. V. Tkachov, Phys. Lett. 100B, 65 (1981).
  76. K. G. Chetyrkin and F. V. Tkachov, Nucl. Phys. B192, 159 (1981).
  77. S. Laporta, Int. J. Mod. Phys. A 15, 5087 (2000).
  78. A. V. Kotikov, Phys. Lett. B 254, 158 (1991).
  79. A. V. Kotikov, Phys. Lett. B 267, 123 (1991); 295, 409(E) (1992).
  80. E. Remiddi, Nuovo Cimento A 110, 1435 (1997).
  81. T. Gehrmann and E. Remiddi, Nucl. Phys. B580, 485 (2000).
  82. M. Argeri and P. Mastrolia, Int. J. Mod. Phys. A 22, 4375 (2007).
  83. J. M. Henn, Phys. Rev. Lett. 110, 251601 (2013).
  84. M. Argeri, S. Di Vita, P. Mastrolia, E. Mirabella, J. Schlenk, U. Schubert, and L. Tancredi, J. High Energy Phys. 03 (2014) 082.
  85. J. S. Schwinger, J. Math. Phys. (N.Y.) 2, 407 (1961).
  86. L. V. Keldysh, Sov. Phys. JETP 20, 1018 (1965).
  87. F. A. Berends and W. T. Giele, Nucl. Phys. B306, 759 (1988).
  88. G. U. Jakobsen, Gravitational scattering of compact bodies from worldline quantum field theory, Ph.D. thesis, Humboldt U., Berlin, Humboldt U., Berlin (main), 2023, arXiv:2308.04388.
  89. J. Davies, T. Kaneko, C. Marinissen, T. Ueda, and J. A. M. Vermaseren, arXiv:2601.19982.
  90. C. Anastasiou, J. Karlen, and M. Vicini, J. High Energy Phys. 12 (2023) 169.
  91. G. Brunello, P. Mastrolia, J. Ronca, S. Smith, and M. Zeng (to be published).
  92. J. Gluza, K. Kajda, and D. A. Kosower, Phys. Rev. D 83, 045012 (2011).
  93. Z. Wu, J. Boehm, R. Ma, H. Xu, and Y. Zhang, Comput. Phys. Commun. 295, 108999 (2024).
  94. Z. Wu, J. Böhm, R. Ma, J. Usovitsch, Y. Xu, and Y. Zhang, Comput. Phys. Commun. 316, 109798 (2025).
  95. S. Smith and M. Zeng, J. High Energy Phys. 01 (2026) 102.
  96. F. Lange, J. Usovitsch, and Z. Wu, Comput. Phys. Commun. 322, 109999 (2026).
  97. T. Peraro, J. High Energy Phys. 07 (2019) 031.
  98. W. Flieger and W. J. Torres Bobadilla, Eur. Phys. J. Plus 139, 1022 (2024).
  99. L. Görges, C. Nega, L. Tancredi, and F. J. Wagner, J. High Energy Phys. 07 (2023) 206.
  100. C. Duhr, S. Maggio, C. Nega, B. Sauer, L. Tancredi, and F. J. Wagner, J. High Energy Phys. 06 (2025) 128.
  101. F. Forner, C. C. Mella, C. Nega, L. Tancredi, and F. J. Wagner, arXiv:2604.25270.
  102. C. Meyer, Comput. Phys. Commun. 222, 295 (2018).
  103. K.-T. Chen, Bull. Am. Math. Soc. 83, 831 (1977).
  104. E. Remiddi and J. A. M. Vermaseren, Int. J. Mod. Phys. A 15, 725 (2000).
  105. J. Broedel, C. Duhr, F. Dulat, B. Penante, and L. Tancredi, J. High Energy Phys. 01 (2019) 023.
  106. C. Duhr and F. Dulat, J. High Energy Phys. 08 (2019) 135.
  107. M. Beneke and V. A. Smirnov, Nucl. Phys. B522, 321 (1998).
  108. A. Pak and A. Smirnov, Eur. Phys. J. C 71, 1626 (2011).
  109. B. Jantzen, A. V. Smirnov, and V. A. Smirnov, Eur. Phys. J. C 72, 2139 (2012).
  110. P. Mastrolia, M. Passera, A. Primo, and U. Schubert, J. High Energy Phys. 11 (2017) 198.
  111. F. Dulat and B. Mistlberger, arXiv:1411.3586.
  112. V. Chestnov, S. J. Matsubara-Heo, H. J. Munch, and N. Takayama, J. High Energy Phys. 11 (2023) 202.
  113. S. Laporta, Phys. Lett. B 523, 95 (2001).
  114. R. N. Lee and V. A. Smirnov, J. High Energy Phys. 02 (2011) 102.
  115. K. G. Chetyrkin and A. G. Grozin, Nucl. Phys. B666, 289 (2003).
  116. C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, J. High Energy Phys. 08 (2023) 109.
  117. R. Jinno, G. Kälin, Z. Liu, and H. Rubira, J. High Energy Phys. 07 (2023) 181.
  118. X. Liu and Y.-Q. Ma, Comput. Phys. Commun. 283, 108565 (2023).
  119. S. Weinberg, Phys. Rev. 140, B516 (1965).
  120. S. Weinberg, The Quantum Theory of Fields. Vol. 1: Foundations (Cambridge University Press, Cambridge, England, 2005).
  121. S. Weinberg, The Quantum Theory of Fields. Vol. 2: Modern Applications (Cambridge University Press, Cambridge, England, 2013).
  122. A. Brandhuber, G. R. Brown, P. Pichini, G. Travaglini, and P. Vives Matasan, J. High Energy Phys. 07 (2026) 151.
  123. C. Anastasiou and K. Melnikov, Nucl. Phys. B646, 220 (2002).
  124. C. Anastasiou, K. Melnikov, and F. Petriello, Phys. Rev. D 69, 076010 (2004).
  125. Black Hole Perturbation Toolkit, https://bhptoolkit.org.
  126. J. Markovic and M. M. Ivanov, Phys. Rev. D 114, 024002 (2026).
  127. Y. F. Bautista, M. Driesse, K. Haddad, and G. U. Jakobsen, arXiv:2606.27544.
  128. V. A. Smirnov, Analytic Tools for Feynman Integrals (Springer, Berlin, Heidelberg, 2012), Vol. 250.

Outline

Information

Phys. Rev. D 114, 085022– Published 22 October, 2026
  • Received 14 July 2026
  • Accepted 6 August 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. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation