- 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
  • Letter
  • Open Access

Systematic integral evaluation for spin-resummed binary dynamics

Gang Chen1,*, Jung-Wook Kim (김정욱)2,†, and Tianheng Wang3,‡

  • *Contact author: gang.chen@nbi.ku.dk
  • †Contact author: jung-wook.kim@aei.mpg.de
  • ‡Contact author: tianhengwang@snu.ac.kr
PDF

Phys. Rev. D 111, L021701 – Published 6 January, 2025

DOI: https://doi.org/10.1103/PhysRevD.111.L021701

Abstract

Computation of spin-resummed observables in post-Minkowskian dynamics typically involve evaluation of Feynman integrals deformed by an exponential factor, where the exponent is a linear sum of the momenta being integrated. Such integrals can be viewed as tensor integral generating functions, which provide alternative approaches to tensor reduction of Feynman integrals. We develop a systematic method to evaluate tensor integral generating functions using conventional multiloop integration techniques. The spin-resummed aligned-spin eikonal at second post-Minkowskian order is considered as a phenomenologically relevant example where evaluation of tensor integral generating functions is necessary.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (138)

  1. Y. Iwasaki, Prog. Theor. Phys. 46, 1587 (1971).
  2. J. F. Donoghue, Phys. Rev. D 50, 3874 (1994).
  3. N. E. J. Bjerrum-Bohr, J. F. Donoghue, and B. R. Holstein, Phys. Rev. D 67, 084033 (2003); 71, 069903(E) (2005).
  4. B. R. Holstein and J. F. Donoghue, Phys. Rev. Lett. 93, 201602 (2004).
  5. B. R. Holstein and A. Ross, arXiv:0802.0716.
  6. D. Neill and I. Z. Rothstein, Nucl. Phys. B877, 177 (2013).
  7. N. E. J. Bjerrum-Bohr, J. F. Donoghue, and P. Vanhove, J. High Energy Phys. 02 (2014) 111.
  8. T. Damour, Phys. Rev. D 97, 044038 (2018).
  9. N. E. J. Bjerrum-Bohr, P. H. Damgaard, G. Festuccia, L. Plante, and P. Vanhove, Phys. Rev. Lett. 121, 171601 (2018).
  10. C. Cheung, I. Z. Rothstein, and M. P. Solon, Phys. Rev. Lett. 121, 251101 (2018).
  11. A. Cristofoli, N. E. J. Bjerrum-Bohr, P. H. Damgaard, and P. Vanhove, Phys. Rev. D 100, 084040 (2019).
  12. Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 122, 201603 (2019).
  13. Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, J. High Energy Phys. 10 (2019) 206.
  14. J. Parra-Martinez, M. S. Ruf, and M. Zeng, J. High Energy Phys. 11 (2020) 023.
  15. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, Phys. Lett. B 811, 135924 (2020).
  16. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, Phys. Lett. B 818, 136379 (2021).
  17. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, J. High Energy Phys. 07 (2021) 169.
  18. E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, J. High Energy Phys. 10 (2021) 148.
  19. N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Planté, and P. Vanhove, Phys. Rev. D 104, 026009 (2021).
  20. N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Planté, and P. Vanhove, J. High Energy Phys. 08 (2021) 172.
  21. P. H. Damgaard, L. Plante, and P. Vanhove, J. High Energy Phys. 11 (2021) 213.
  22. A. Brandhuber, G. Chen, G. Travaglini, and C. Wen, J. High Energy Phys. 10 (2021) 118.
  23. N. E. J. Bjerrum-Bohr, L. Planté, and P. Vanhove, J. High Energy Phys. 03 (2022) 071.
  24. N. E. J. Bjerrum-Bohr, L. Planté, and P. Vanhove, arXiv:2212.08957.
  25. Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 128, 161103 (2022).
  26. Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon, and M. Zeng, Proc. Sci., LL2022 (2022) 051.
  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. P. H. Damgaard, E. R. Hansen, L. Planté, and P. Vanhove, J. High Energy Phys. 09 (2023) 183.
  29. P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, Phys. Rep. 1083, 1 (2024).
  30. C. Heissenberg, Phys. Rev. D 108, 106003 (2023).
  31. T. Adamo and R. Gonzo, J. High Energy Phys. 05 (2023) 088.
  32. D. A. Kosower, B. Maybee, and D. O’Connell, J. High Energy Phys. 02 (2019) 137.
  33. A. Brandhuber, G. R. Brown, G. Chen, S. De Angelis, J. Gowdy, and G. Travaglini, J. High Energy Phys. 06 (2023) 048.
  34. A. Herderschee, R. Roiban, and F. Teng, J. High Energy Phys. 06 (2023) 004.
  35. A. Elkhidir, D. O’Connell, M. Sergola, and I. A. Vazquez-Holm, J. High Energy Phys. 07 (2024) 272.
  36. A. Georgoudis, C. Heissenberg, and I. Vazquez-Holm, J. High Energy Phys. 06 (2023) 126.
  37. S. Caron-Huot, M. Giroux, H. S. Hannesdottir, and S. Mizera, J. High Energy Phys. 01 (2024) 139.
  38. L. Bohnenblust, H. Ita, M. Kraus, and J. Schlenk, arXiv:2312.14859.
  39. D. Bini, T. Damour, and A. Geralico, Phys. Rev. D 108, 124052 (2023).
  40. A. Georgoudis, C. Heissenberg, and R. Russo, J. High Energy Phys. 03 (2024) 089.
  41. T. Adamo, R. Gonzo, and A. Ilderton, J. High Energy Phys. 05 (2024) 034.
  42. D. Bini, T. Damour, S. De Angelis, A. Geralico, A. Herderschee, R. Roiban, and F. Teng, Phys. Rev. D 109, 125008 (2024).
  43. F. Alessio and P. Di Vecchia, J. Phys. A 57, 475402 (2024).
  44. A. Georgoudis, C. Heissenberg, and R. Russo, Phys. Rev. D 109, 106020 (2024).
  45. G. Brunello and S. De Angelis, J. High Energy Phys. 07 (2024) 062.
  46. A. Guevara, J. High Energy Phys. 04 (2019) 033.
  47. N. Arkani-Hamed, T.-C. Huang, and Y.-t. Huang, J. High Energy Phys. 11 (2021) 070.
  48. A. Guevara, A. Ochirov, and J. Vines, J. High Energy Phys. 09 (2019) 056.
  49. M.-Z. Chung, Y.-T. Huang, J.-W. Kim, and S. Lee, J. High Energy Phys. 04 (2019) 156.
  50. A. Guevara, A. Ochirov, and J. Vines, Phys. Rev. D 100, 104024 (2019).
  51. N. Arkani-Hamed, Y.-t. Huang, and D. O’Connell, J. High Energy Phys. 01 (2020) 046.
  52. R. Aoude, K. Haddad, and A. Helset, J. High Energy Phys. 05 (2020) 051.
  53. M.-Z. Chung, Y.-t. Huang, J.-W. Kim, and S. Lee, J. High Energy Phys. 05 (2020) 105.
  54. A. Guevara, B. Maybee, A. Ochirov, D. O’connell, and J. Vines, J. High Energy Phys. 03 (2021) 201.
  55. W.-M. Chen, M.-Z. Chung, Y.-t. Huang, and J.-W. Kim, J. High Energy Phys. 08 (2022) 148.
  56. D. Kosmopoulos and A. Luna, J. High Energy Phys. 07 (2021) 037.
  57. M. Chiodaroli, H. Johansson, and P. Pichini, J. High Energy Phys. 02 (2022) 156.
  58. Y. F. Bautista, A. Guevara, C. Kavanagh, and J. Vines, J. High Energy Phys. 03 (2023) 136.
  59. L. Cangemi, M. Chiodaroli, H. Johansson, A. Ochirov, P. Pichini, and E. Skvortsov, Phys. Rev. Lett. 131, 221401 (2023).
  60. A. Ochirov and E. Skvortsov, Phys. Rev. Lett. 129, 241601 (2022).
  61. P. H. Damgaard, K. Haddad, and A. Helset, J. High Energy Phys. 11 (2019) 070.
  62. Z. Bern, A. Luna, R. Roiban, C.-H. Shen, and M. Zeng, Phys. Rev. D 104, 065014 (2021).
  63. F. Comberiati and L. de la Cruz, J. High Energy Phys. 03 (2023) 068.
  64. B. Maybee, D. O’Connell, and J. Vines, J. High Energy Phys. 12 (2019) 156.
  65. K. Haddad, Phys. Rev. D 105, 026004 (2022).
  66. W.-M. Chen, M.-Z. Chung, Y.-t. Huang, and J.-W. Kim, J. High Energy Phys. 12 (2022) 058.
  67. G. Menezes and M. Sergola, J. High Energy Phys. 10 (2022) 105.
  68. F. Febres Cordero, M. Kraus, G. Lin, M. S. Ruf, and M. Zeng, Phys. Rev. Lett. 130, 021601 (2023).
  69. F. Alessio and P. Di Vecchia, Phys. Lett. B 832, 137258 (2022).
  70. Z. Bern, D. Kosmopoulos, A. Luna, R. Roiban, and F. Teng, Phys. Rev. Lett. 130, 201402 (2023).
  71. R. Aoude, K. Haddad, and A. Helset, Phys. Rev. Lett. 129, 141102 (2022).
  72. R. Aoude, K. Haddad, and A. Helset, J. High Energy Phys. 07 (2022) 072.
  73. F. Alessio, J. High Energy Phys. 04 (2024) 058.
  74. R. Aoude, K. Haddad, and A. Helset, Phys. Rev. D 108, 024050 (2023).
  75. Y. F. Bautista, Phys. Rev. D 108, 084036 (2023).
  76. S. De Angelis, R. Gonzo, and P. P. Novichkov, Phys. Rev. D 110, L041502 (2024).
  77. A. Brandhuber, G. R. Brown, G. Chen, J. Gowdy, and G. Travaglini, J. High Energy Phys. 02 (2024) 026.
  78. R. Aoude, K. Haddad, C. Heissenberg, and A. Helset, Phys. Rev. D 109, 036007 (2024).
  79. M. M. Riva, F. Vernizzi, and L. K. Wong, Phys. Rev. D 106, 044013 (2022).
  80. A. Luna, N. Moynihan, D. O’Connell, and A. Ross, J. High Energy Phys. 08 (2024) 045.
  81. A. Antonelli, A. Buonanno, J. Steinhoff, M. van de Meent, and J. Vines, Phys. Rev. D 99, 104004 (2019).
  82. T. Damour, Phys. Rev. D 102, 124008 (2020).
  83. G. Kälin, Z. Liu, and R. A. Porto, Phys. Rev. Lett. 125, 261103 (2020).
  84. C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Phys. Lett. B 831, 137203 (2022).
  85. C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, Phys. Rev. Lett. 128, 161104 (2022).
  86. C. Dlapa, G. Kälin, Z. Liu, J. Neef, and R. A. Porto, Phys. Rev. Lett. 130, 101401 (2023).
  87. G. U. Jakobsen, G. Mogull, J. Plefka, B. Sauer, and Y. Xu, Phys. Rev. Lett. 131, 151401 (2023).
  88. G. U. Jakobsen, G. Mogull, J. Plefka, and J. Steinhoff, Phys. Rev. Lett. 126, 201103 (2021).
  89. S. Mougiakakos, M. M. Riva, and F. Vernizzi, Phys. Rev. D 104, 024041 (2021).
  90. G. U. Jakobsen, G. Mogull, J. Plefka, and J. Steinhoff, Phys. Rev. Lett. 128, 011101 (2022).
  91. G. Mogull, J. Plefka, and J. Steinhoff, J. High Energy Phys. 02 (2021) 048.
  92. G. U. Jakobsen, G. Mogull, J. Plefka, and J. Steinhoff, J. High Energy Phys. 01 (2022) 027.
  93. J. Vines, Classical Quantum Gravity 35, 084002 (2018).
  94. J. Vines, J. Steinhoff, and A. Buonanno, Phys. Rev. D 99, 064054 (2019).
  95. Z. Liu, R. A. Porto, and Z. Yang, J. High Energy Phys. 06 (2021) 012.
  96. G. U. Jakobsen and G. Mogull, Phys. Rev. Lett. 128, 141102 (2022).
  97. P. H. Damgaard, J. Hoogeveen, A. Luna, and J. Vines, Phys. Rev. D 106, 124030 (2022).
  98. M. Bianchi, C. Gambino, and F. Riccioni, J. High Energy Phys. 08 (2023) 188.
  99. R. Gonzo and C. Shi, arXiv:2405.09687.
  100. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 116, 061102 (2016).
  101. A. Buonanno, G. Mogull, R. Patil, and L. Pompili, Phys. Rev. Lett. 133, 211402 (2024).
  102. A. Smirnov, J. High Energy Phys. 10 (2008) 107.
  103. R. N. Lee, J. High Energy Phys. 04 (2015) 108.
  104. K. J. Larsen and Y. Zhang, Phys. Rev. D 93, 041701 (2016).
  105. T. Gehrmann and E. Remiddi, Nucl. Phys. B580, 485 (2000).
  106. A. V. Kotikov, Phys. Lett. B 254, 158 (1991).
  107. J. M. Henn, Phys. Rev. Lett. 110, 251601 (2013).
  108. H. Frellesvig, R. Morales, and M. Wilhelm, Phys. Rev. Lett. 132, 201602 (2024).
  109. M. Driesse, G. U. Jakobsen, G. Mogull, J. Plefka, B. Sauer, and J. Usovitsch, Phys. Rev. Lett. 132, 241402 (2024).
  110. A. Klemm, C. Nega, B. Sauer, and J. Plefka, Phys. Rev. D 109, 124046 (2024).
  111. H. Frellesvig, R. Morales, and M. Wilhelm, J. High Energy Phys. 08 (2024) 243.
  112. A. Dhani, S. Völkel, A. Buonanno, H. Estelles, J. Gair, H. P. Pfeiffer, L. Pompili, and A. Toubiana, arXiv:2404.05811.
  113. A. Buonanno and T. Damour, Phys. Rev. D 59, 084006 (1999).
  114. L. Pompili et al., Phys. Rev. D 108, 124035 (2023).
  115. A. Ramos-Buades, A. Buonanno, H. Estellés, M. Khalil, D. P. Mihaylov, S. Ossokine, L. Pompili, and M. Shiferaw, Phys. Rev. D 108, 124037 (2023).
  116. M. Khalil, A. Buonanno, H. Estelles, D. P. Mihaylov, S. Ossokine, L. Pompili, and A. Ramos-Buades, Phys. Rev. D 108, 124036 (2023).
  117. J.-H. Kim, J.-W. Kim, and S. Lee, J. High Energy Phys. 08 (2024) 080.
  118. E. T. Newman and A. I. Janis, J. Math. Phys. (N.Y.) 6, 915 (1965).
  119. B. Feng, Commun. Theor. Phys. 75, 025203 (2023).
  120. N. E. J. Bjerrum-Bohr, G. Chen, and M. Skowronek, Phys. Rev. Lett. 132, 191603 (2024).
  121. N. E. J. Bjerrum-Bohr, G. Chen, and M. Skowronek, J. High Energy Phys. 06 (2023) 170.
  122. Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, Nucl. Phys. B425, 217 (1994).
  123. Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, Nucl. Phys. B435, 59 (1995).

  124. Although generating functions are usually defined by real exponents, imaginary exponents were used to avoid factors of i when solving for the integrals. Such an analytic continuation is known in the probability literature as characteristic functions. This analytic continuation is allowed since Feynman integrals should be understood as generalized functions which may not be well defined by conventional definitions of integrals.

  125. The delta constraints can be converted to denominators through the distributional identity 2πi δ(x)=(x−i0+)−1−(x+i0+)−1 and its derivatives, therefore they can be considered as propagator factors.

  126. C. Hu, T. Li, J. Shen, and Y. Xu, J. High Energy Phys. 02 (2024) 158.
  127. B. Feng, C. Hu, J. Shen, and Y. Zhang, arXiv:2403.16040.
  128. T. Li, Y. Song, and L. Zhang, arXiv:2404.04644.
  129. G. Brunello, G. Crisanti, M. Giroux, P. Mastrolia, and S. Smith, Phys. Rev. D 109, 094047 (2024).
  130. G. Chen and T. Wang, arXiv:2406.09086.
  131. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevD.111.L021701 for the tree amplitude used.
  132. G. Chen, J.-W. Kim, and T. Wang, https://github.com/amplitudegravity/kerreikonal2pm (2024).

  133. We adopt Mathematica’s definition for the elliptic integrals.

  134. K. Cho and K. Matsumoto, Nagoya mathematical Journal 139, 67 (1995).
  135. P. Mastrolia and S. Mizera, J. High Energy Phys. 02 (2019) 139.
  136. H. Frellesvig, F. Gasparotto, S. Laporta, M. K. Mandal, P. Mastrolia, L. Mattiazzi, and S. Mizera, J. High Energy Phys. 05 (2019) 153.
  137. J. Broedel, C. Duhr, F. Dulat, and L. Tancredi, J. High Energy Phys. 05 (2018) 093.
  138. J. L. Bourjaily et al., arXiv:2203.07088.

Outline

Information

Phys. Rev. D 111, L021701– Published 6 January, 2025
  • Received 9 July 2024
  • Accepted 22 November 2024
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