- 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
  • Open Access

Two-loop spacelike splitting amplitude for N=4 Super-Yang-Mills theory

Johannes Henn1, Rourou Ma2, Yongqun Xu2, Kai Yan4,5, Yang Zhang2,3,7, and Hua Xing Zhu6,7

PDF

Phys. Rev. D 112, 076003 – Published 3 October, 2025

DOI: https://doi.org/10.1103/5qrc-4vq1

Abstract

The study of collinear behavior for gauge theories in the spacelike region is of great phenomenological and theoretical importance. We analytically calculate the two-loop spacelike splitting amplitude for the full color N=4 super-Yang-Mills theory. The result is derived by two complementary methods starting from the known amplitude: one is based on a discontinuity analysis, while the other one is based on calculating Feynman integrals analytically in the collinear limit. Our result explicitly shows terms that violate naive factorization. However we show that factorization is restored at the level of color-summed unpolarized squared amplitudes at next-to-next-to-next-to leading order. We conjecture that the two-loop tripole terms in the generalized splitting amplitudes in quantum chromodynamics are identical to what we obtain in N=4 super Yang-Mills theory.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (79)

  1. M. Czakon, A. Mitov, and R. Poncelet, Phys. Rev. Lett. 127, 152001 (2021); 129, 119901(E) (2022).
  2. B. Agarwal, F. Buccioni, A. von Manteuffel, and L. Tancredi, J. High Energy Phys. 04 (2021) 201.
  3. H. A. Chawdhry, M. Czakon, A. Mitov, and R. Poncelet, J. High Energy Phys. 09 (2021) 093.
  4. S. Kallweit, V. Sotnikov, and M. Wiesemann, Phys. Lett. B 812, 136013 (2021).
  5. S. BadgerM. Czakon, H. B. Hartanto, R. Moodie, T. Peraro, R. Poncelet, and S. Zoia, J. High Energy Phys. 10 (2023) 071.
  6. S. Badger, T. Gehrmann, M. Marcoli, and R. Moodie, Phys. Lett. B 824, 136802 (2022).
  7. C. Anastasiou, C. Duhr, F. Dulat, F. Herzog, and B. Mistlberger, Phys. Rev. Lett. 114, 212001 (2015).
  8. B. Mistlberger, J. High Energy Phys. 05 (2018) 028.
  9. F. Dulat, B. Mistlberger, and A. Pelloni, Phys. Rev. D 99, 034004 (2019).
  10. L. Cieri, X. Chen, T. Gehrmann, E. W. N. Glover, and A. Huss, J. High Energy Phys. 02 (2019) 096.
  11. F. A. Dreyer and A. Karlberg, Phys. Rev. D 98, 114016 (2018).
  12. C. Duhr, F. Dulat, and B. Mistlberger, Phys. Rev. Lett. 125, 051804 (2020).
  13. L.-B. Chen, H. T. Li, H.-S. Shao, and J. Wang, Phys. Lett. B 803, 135292 (2020).
  14. C. Duhr, F. Dulat, and B. Mistlberger, J. High Energy Phys. 11 (2020) 143.
  15. C. Duhr, F. Dulat, and B. Mistlberger, Phys. Rev. Lett. 125, 172001 (2020).
  16. X. ChenT. Gehrmann, N. Glover, A. Huss, T.-Z. YangH. X. Zhu, Phys. Rev. Lett. 128, 052001 (2022).
  17. G. Billis, B. Dehnadi, M. A. Ebert, J. K. L. Michel, and F. J. Tackmann, Phys. Rev. Lett. 127, 072001 (2021).
  18. X. Chen, T. Gehrmann, E.  W.  N. Glover, A. Huss, B. Mistlberger, A. Pelloni, Phys. Rev. Lett. 127, 072002 (2021).
  19. S. Camarda, L. Cieri, and G. Ferrera, Phys. Rev. D 104, L111503 (2021).
  20. X. Chen, T. Gehrmann, N. Glover, A. Huss, P. F. Monni, E. Re, L. Rottoli, and P. Torrielli , Phys. Rev. Lett. 128, 252001 (2022).
  21. X. Chen, T. Gehrmann, N. Glover, A. Huss, T.-Z. Yang, and H. X. Zhu, Phys. Lett. B 840, 137876 (2023).
  22. S. Frixione and M. Grazzini, J. High Energy Phys. 06 (2005) 010.
  23. A. Gehrmann-De Ridder, T. Gehrmann, and E. W. N. Glover, J. High Energy Phys. 09 (2005) 056.
  24. G. Somogyi, Z. Trocsanyi, and V. Del Duca, J. High Energy Phys. 06 (2005) 024.
  25. S. Catani and M. Grazzini, Phys. Rev. Lett. 98, 222002 (2007).
  26. M. Czakon, Phys. Lett. B 693, 259 (2010).
  27. C. Anastasiou, K. Melnikov, and F. Petriello, Phys. Rev. D 69, 076010 (2004).
  28. F. Caola, K. Melnikov, and R. Röntsch, Eur. Phys. J. C 77, 248 (2017).
  29. R. Boughezal, K. Melnikov, and F. Petriello, Phys. Rev. D 85, 034025 (2012).
  30. R. Boughezal, C. Focke, X. Liu, and F. Petriello, Phys. Rev. Lett. 115, 062002 (2015).
  31. J. Gaunt, M. Stahlhofen, F. J. Tackmann, and J. R. Walsh, J. High Energy Phys. 09 (2015) 058.
  32. M. Cacciari, F. A. Dreyer, A. Karlberg, G. P. Salam, and G. Zanderighi, Phys. Rev. Lett. 115, 082002 (2015); 120, 139901(E) (2018).
  33. G. F. R. Sborlini, F. Driencourt-Mangin, and G. Rodrigo, J. High Energy Phys. 10 (2016) 162.
  34. L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli, and S. Uccirati , J. High Energy Phys. 12 (2018) 107; 06 (2019) 13.
  35. N. Agarwal, L. Magnea, C. Signorile-Signorile, and A. Tripathi, Phys. Rep. 994, 1 (2023).
  36. J. C. Collins, D. E. Soper, and G. F. Sterman, Nucl. Phys. B261, 104 (1985).
  37. J. C. Collins, D. E. Soper, and G. F. Sterman, Adv. Ser. Dir. High Energy Phys. 5, 1 (1989).
  38. A. Bacchetta, C. J. Bomhof, P. J. Mulders, and F. Pijlman, Phys. Rev. D 72, 034030 (2005).
  39. C. J. Bomhof, P. J. Mulders, and F. Pijlman, Phys. Lett. B 596, 277 (2004).
  40. C. J. Bomhof, P. J. Mulders, and F. Pijlman, Eur. Phys. J. C 47, 147 (2006).
  41. J. Collins and J.-W. Qiu, Phys. Rev. D 75, 114014 (2007).
  42. J. R. Gaunt, J. High Energy Phys. 07 (2014) 110.
  43. M. Zeng, J. High Energy Phys. 10 (2015) 189.
  44. S. Catani, D. de Florian, and G. Rodrigo, J. High Energy Phys. 07 (2012) 026.
  45. I. Z. Rothstein and I. W. Stewart, J. High Energy Phys. 08 (2016) 025.
  46. J. R. Forshaw, M. H. Seymour, and A. Siodmok, J. High Energy Phys. 11 (2012) 066.
  47. M. D. Schwartz, K. Yan, and H. X. Zhu, Phys. Rev. D 96, 056005 (2017).
  48. X. Guan, F. Herzog, Y. Ma, B. Mistlberger, and A. Suresh, J. High Energy Phys. 01 (2025) 090.
  49. L. J. Dixon, E. Herrmann, K. Yan, and H. X. Zhu, J. High Energy Phys. 05 (2020) 135.
  50. L. Cieri, P. K. Dhani, and G. Rodrigo, arXiv:2402.14749.
  51. D. Chicherin, T. Gehrmann, J.  M. Henn, P. Wasser, Y. Zhang, and S. Zoia, Phys. Rev. Lett. 123, 041603 (2019).
  52. D. Chicherin and V. Sotnikov, J. High Energy Phys. 20 (2020) 167.
  53. J. Henn, T. Peraro, Y. Xu, and Y. Zhang, J. High Energy Phys. 03 (2022) 056.
  54. J. M. Henn, A. Matijašić, J. Miczajka, T. Peraro, Y. Xu, and Y. Zhang, J. High Energy Phys. 08 (2024) 027.
  55. B. Agarwal, F. Buccioni, F. Devoto, G. Gambuti, A. von Manteuff, and L. Tancredi, Phys. Rev. D 109, 094025 (2024).
  56. G. De Laurentis, H. Ita, M. Klinkert, and V. Sotnikov, Phys. Rev. D 109, 094023 (2024).
  57. A. B. Goncharov, Math. Res. Lett. 5, 497 (1998).
  58. S. Abreu, L. J. Dixon, E. Herrmann, B. Page, and M. Zeng, Phys. Rev. Lett. 122, 121603 (2019).
  59. T. Gehrmann, J. M. Henn, and N. A. Lo Presti, J. High Energy Phys. 10 (2018) 103.
  60. D. Chicherin, T. Gehrmann, J. M. Henn, N. A. Lo Presti, V. Mitev, and P. Wasser, J. High Energy Phys. 03 (2019) 042.
  61. D. Chicherin, J. Henn, and V. Mitev, J. High Energy Phys. 05 (2018) 164.
  62. See Supplemental Material at http://link.aps.org/supplemental/10.1103/5qrc-4vq1 for the analytic expression for two-loop five-point N=4 super-Yang-Mills amplitude in the space-like collinear limit, and the intermediate results of one integral family to demonstrate the analytic computation steps.
  63. J. J. M. Carrasco and H. Johansson, Phys. Rev. D 85, 025006 (2012).
  64. A. C. Edison and S. G. Naculich, Nucl. Phys. B858, 488 (2012).
  65. D. Chicherin, T. Gehrmann, J.  M. Henn, P. Wasser, Y. Zhang, and S. Zoia, Phys. Rev. Lett. 122, 121602 (2019).
  66. J. M. Henn and B. Mistlberger, Phys. Rev. Lett. 117, 171601 (2016).
  67. C. Duhr and F. Dulat, J. High Energy Phys. 08 (2019) 135.
  68. Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, Nucl. Phys. B425, 217 (1994).
  69. Z. Bern and G. Chalmers, Nucl. Phys. B447, 465 (1995).
  70. D. A. Kosower, Nucl. Phys. B552, 319 (1999).
  71. I. Feige and M. D. Schwartz, Phys. Rev. D 90, 105020 (2014).
  72. Z. Bern, V. Del Duca, and C. R. Schmidt, Phys. Lett. B 445, 168 (1998).
  73. Z. Bern, V. Del Duca, W. B. Kilgore, and C. R. Schmidt, Phys. Rev. D 60, 116001 (1999).
  74. Z. Bern, L. J. Dixon, and D. A. Kosower, J. High Energy Phys. 08 (2004) 012.
  75. S. D. Badger and E. W. N. Glover, J. High Energy Phys. 07 (2004) 040.
  76. C. Duhr, J. High Energy Phys. 08 (2012) 043.
  77. S. Catani and M. H. Seymour, Nucl. Phys. B485, 291 (1997); B510, 503(E) (1998).

  78. A typo was corrected in the updated version of [49].

  79. A. V. Kotikov and L. N. Lipatov, Nucl. Phys. B661, 19 (2003); B685, 405(E) (2004).

Outline

Information

Phys. Rev. D 112, 076003– Published 3 October, 2025
  • Received 6 August 2024
  • Accepted 9 September 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