- 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

Analytic computation of three-loop five-point Feynman integrals

Yuanche Liu1,*, Antonela Matijašić2,†, Julian Miczajka3,‡, Yingxuan Xu4,§, Yongqun Xu5,∥, and Yang Zhang5,6,7,¶

  • *Contact author: liuyuanche@mail.ustc.edu.cn
  • †Contact author: amatijas@uni-mainz.de
  • ‡Contact author: julian.m@gmx.net
  • §Contact author: yingxu@physik.hu-berlin.de
  • ∥Contact author: yongqunxu@mail.ustc.edu.cn
  • Contact author: yzhphy@ustc.edu.cn
PDF

Phys. Rev. D 112, 016021 – Published 23 July, 2025

DOI: https://doi.org/10.1103/qrk2-cym5

Abstract

We evaluate the three-loop five-point pentagon-box-box massless integral family in the dimensional regularization scheme, via canonical differential equation. We use tools from computational algebraic geometry to enable the necessary integral reductions. The boundary values of the differential equation are determined analytically in the Euclidean region. To express the final result, we introduce a new representation of weight six functions in terms of one-fold integrals over the product of weight-three functions with weight-two kernels that are derived from the differential equation. Our work paves the way to the analytic computation of three-loop multileg Feynman integrals.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (82)

  1. I. Zurbano Fernandez et al., Report No. CERN-2020-010, 2020, 10.23731/CYRM-2020-0010.
  2. F. Zimmermann et al. (FCC Collaboration), in Proceedings of the IPAC2024 (JACoW, Geneva, Switzerland, 2024), pp. 2493–2496, WEPR14.
  3. G. Aarons et al. (ILC Collaboration), arXiv:0709.1893.
  4. W. Abdallah et al. (CEPC Study Group), Radiat. Detect. Technol. Methods 8, 1 (2024).
  5. M. Begel et al., arXiv:2209.14872.
  6. A. Huss, J. Huston, S. Jones, and M. Pellen, J. Phys. G 50, 043001 (2023).
  7. F. Caola et al., The path forward to N3LO, in Snowmass 2021 (2022), arXiv:2203.06730.
  8. J. Andersen et al., in Les Houches 2023: Physics at TeV Colliders: Standard Model Working Group Report (Physics of the TeV Scale and Beyond the Standard Model: Intensifying the Quest for New Physics) (2024), arXiv:2406.00708.
  9. Particle Data Group, S. Navas et al., Phys. Rev. D 110, 030001 (2024).
  10. A. V. Kotikov, Phys. Lett. B 254, 158 (1991).
  11. A. V. Kotikov, Phys. Lett. B 267, 123 (1991); 295, 409(E) (1992).
  12. J. M. Henn, Phys. Rev. Lett. 110, 251601 (2013).
  13. S. Borowka, G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, J. Schlenk, and T. Zirke, Comput. Phys. Commun. 222, 313 (2018).
  14. X. Liu and Y.-Q. Ma, Comput. Phys. Commun. 283, 108565 (2023).
  15. X. Liu, Y.-Q. Ma, and C.-Y. Wang, Phys. Lett. B 779, 353 (2018).
  16. R. N. Lee and V. A. Smirnov, J. High Energy Phys. 12 (2012) 104.
  17. D. Chicherin, T. Gehrmann, J.  M. Henn, P. Wasser, Y. Zhang, and S. Zoia, Phys. Rev. Lett. 123, 041603 (2019).
  18. S. Abreu, B. Page, and M. Zeng, J. High Energy Phys. 01 (2019) 006.
  19. D. Chicherin and V. Sotnikov, J. High Energy Phys. 12 (2020) 167.
  20. S. Abreu, H. Ita, F. Moriello, B. Page, W. Tschernow, and M. Zeng, J. High Energy Phys. 11 (2020) 117.
  21. D. D. Canko, C. G. Papadopoulos, and N. Syrrakos, J. High Energy Phys. 01 (2021) 199.
  22. S. Abreu, H. Ita, B. Page, and W. Tschernow, J. High Energy Phys. 03 (2022) 182.
  23. D. Chicherin, V. Sotnikov, and S. Zoia, J. High Energy Phys. 01 (2022) 096.
  24. S. Abreu, D. Chicherin, H. Ita, Ben Page, V. Sotnikov, W. Tschernow, and S. Zoia, Phys. Rev. Lett. 132, 141601 (2024).
  25. S. Abreu, D. Chicherin, V. Sotnikov, and S. Zoia, J. High Energy Phys. 10 (2024) 167.
  26. X. Jiang, J. Liu, X. Xu, and L. L. Yang, Phys. Lett. B 864, 139443 (2025).
  27. J. M. Henn, J. Lim, and W. J. Torres Bobadilla, J. High Energy Phys. 05 (2023) 026.
  28. S. Di Vita, P. Mastrolia, U. Schubert, and V. Yundin, J. High Energy Phys. 09 (2014) 148.
  29. M.-M. Long, J. High Energy Phys. 01 (2025) 018.
  30. J. Henn, T. Peraro, Y. Xu, and Y. Zhang, J. High Energy Phys. 03 (2022) 056.
  31. J. M. Henn, A. Matijašić, J. Miczajka, T. Peraro, Y. Xu, and Y. Zhang, J. High Energy Phys. 08 (2024) 027.
  32. L.-B. Chen, L. Dong, H. T. Li, Z. Li, J. Wang, and Y. Wang, Phys. Rev. D 106, 096029 (2022).
  33. L.-B. Chen, L. Dong, H. T. Li, Z. Li, J. Wang, and Y. Wang, J. High Energy Phys. 07 (2023) 089.
  34. J. Klappert, F. Lange, P. Maierhöfer, and J. Usovitsch, Comput. Phys. Commun. 266, 108024 (2021).
  35. A. V. Smirnov and F. S. Chukharev, Comput. Phys. Commun. 247, 106877 (2020).
  36. Z. Wu, J. Boehm, R. Ma, H. Xu, and Y. Zhang, Comput. Phys. Commun. 295, 108999 (2024).
  37. J. Chen, X. Jiang, C. Ma, X. Xu, and L. L. Yang, J. High Energy Phys. 07 (2022) 066.
  38. C. Dlapa, J. M. Henn, and F. J. Wagner, J. High Energy Phys. 08 (2023) 120.
  39. J. Henn, B. Mistlberger, V. A. Smirnov, and P. Wasser, J. High Energy Phys. 04 (2020) 167.
  40. Z. Li, J. Wang, Q.-S. Yan, and X. Zhao, Chin. Phys. C 40, 033103 (2016).
  41. S. Borowka, G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, and J. Schlenk, Comput. Phys. Commun. 240, 120 (2019).
  42. J. Gluza, K. Kajda, and D. A. Kosower, Phys. Rev. D 83, 045012 (2011).
  43. A. Georgoudis, K. J. Larsen, and Y. Zhang, Comput. Phys. Commun. 221, 203 (2017).
  44. S. Badger, H. Frellesvig, and Y. Zhang, J. Phys. Conf. Ser. 523, 012061 (2014).
  45. Y. Zhang, J. High Energy Phys. 09 (2012) 042.
  46. J. Boehm et al., Proc. Sci. MA2019 (2022) 004 [arXiv:2010.06895].
  47. J. Böhm, A. Georgoudis, K. J. Larsen, H. Schönemann, and Y. Zhang, J. High Energy Phys. 09 (2018) 024.
  48. K. J. Larsen and Y. Zhang, Phys. Rev. D 93, 041701 (2016).
  49. Z. Wu and Y. Zhang, arXiv:2406.20016.
  50. S. Laporta, Int. J. Mod. Phys. A 15, 5087 (2000).
  51. V. Magerya, arXiv:2211.03572.
  52. T. Peraro, J. High Energy Phys. 07 (2019) 031.
  53. D. D. Canko and N. Syrrakos, J. High Energy Phys. 04 (2022) 134.
  54. V. A. Smirnov, Analytic Tools for Feynman Integrals (Springer, New York, 2013).
  55. S. Abreu, L. J. Dixon, E. Herrmann, B. Page, and M. Zeng, Phys. Rev. Lett. 122, 121603 (2019).
  56. W. Magnus, Commun. Pure Appl. Math. 7, 649 (1954).
  57. M. Argeri, S. D. Vita, P. Mastrolia, E. Mirabella, J. Schlenk, U. Schubert, and L. Tancredi, J. High Energy Phys. 03 (2014) 082.
  58. S. He, Z. Li, R. Ma, Z. Wu, Q. Yang, and Y. Zhang, J. High Energy Phys. 10 (2022) 165.
  59. C. Meyer, J. High Energy Phys. 04 (2017) 006.
  60. C. Dlapa, X. Li, and Y. Zhang, J. High Energy Phys. 07 (2021) 227.
  61. J. M. Henn, A. V. Smirnov, and V. A. Smirnov, J. High Energy Phys. 07 (2013) 128.
  62. D. Chicherin, J. Henn, and V. Mitev, J. High Energy Phys. 05 (2018) 164.
  63. D. Chicherin, J. Henn, J. Trnka, and S.-Q. Zhang, J. High Energy Phys. 04 (2025) 022.
  64. T. Gehrmann, J. M. Henn, and N. A. Lo Presti, Phys. Rev. Lett. 116, 062001 (2016); 116, 189903(E) (2016).
  65. T. Gehrmann, J. M. Henn, and N. A. Lo Presti, J. High Energy Phys. 10 (2018) 103.
  66. R. N. Lee, arXiv:2407.12503.
  67. B. A. Kniehl and O. V. Tarasov, Nucl. Phys. B833, 298 (2010).
  68. M. G. Kozlov and R. N. Lee, J. High Energy Phys. 02 (2016) 021.
  69. N. Syrrakos, J. High Energy Phys. 06 (2021) 037.
  70. C. Duhr, H. Gangl, and J. R. Rhodes, J. High Energy Phys. 10 (2012) 075.
  71. M. Heller, A. von Manteuffel, and R. M. Schabinger, Phys. Rev. D 102, 016025 (2020).
  72. S. Zoia, Modern Analytic Methods for Scattering Amplitudes, With an Application to Two-Loop Five-Particle Processes, Springer Theses (Springer, Cham, 2021), 10.1007/978-3-031-01945-6.
  73. C. Duhr and F. Dulat, J. High Energy Phys. 08 (2019) 135.
  74. H. Frellesvig, D. Tommasini, and C. Wever, J. High Energy Phys. 03 (2016) 189.
  75. V. Del Duca, C. Duhr, E. W. Nigel Glover, and V. A. Smirnov, J. High Energy Phys. 01 (2010) 042.
  76. T. Gehrmann and E. Remiddi, Nucl. Phys. B640, 379 (2002).
  77. J. M. Henn, K. Melnikov, and V. A. Smirnov, J. High Energy Phys. 05 (2014) 090.
  78. D. Chicherin, T. Gehrmann, J. M. Henn, N. A. Lo Presti, V. Mitev, and P. Wasser, J. High Energy Phys. 03 (2019) 042.
  79. Z. Li, Multiple-polylogarithm, https://github.com/munuxi/Multiple-Polylogarithm (2021).
  80. S. Caron-Huot and J. M. Henn, J. High Energy Phys. 06 (2014) 114.
  81. J. Wang and Y. Wang, J. High Energy Phys. 02 (2023) 127.
  82. https://github.com/YongqunXu/3L5P_PBB

Outline

Information

Phys. Rev. D 112, 016021– Published 23 July, 2025
  • Received 15 January 2025
  • Accepted 28 May 2025
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