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Massive tree-level splitting functions beyond kinematical limits

Stefan Höche1, Matt LeBlanc2, Jennifer Roloff2, and Grant Whitman2

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Phys. Rev. D 113, 054009 – Published 4 March, 2026

DOI: https://doi.org/10.1103/7jj6-qfqh

Abstract

We present a compact form of the massive 1→3 tree-level QCD splitting functions and discuss a decomposition of the results in terms of lower-order expressions, scalar dipole antenna functions and pure higher-order remainders. The two-gluon radiator functions introduced in this context are novel and generalize expressions obtained from the double-soft approximation. Our results are obtained without reference to soft or quasicollinear limits.

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References (68)

  1. ATLAS and CMS Collaborations, ATL-PHYS-PUB-2025-018, 2025.
  2. H. Qu, C. Li, and S. Qian, Proc. Mach. Learn. Res. 162, 18281 (2022).
  3. ATLAS Collaboration, Nat. Commun. 17, 541 (2026).
  4. ATLAS Collaboration, ATL-PHYS-PUB-2025-029, 2025.
  5. CMS Collaboration, CMS-DP-2024-024, 2024.
  6. CMS Collaboration, CMS-DP-2022-050, 2022.
  7. ATLAS Collaboration, ATL-PHYS-PUB-2025-006, 2025.
  8. ATLAS Collaboration, Phys. Rev. Lett. 121, 092001 (2018).
  9. CMS Collaboration, J. High Energy Phys. 11 (2018) 113.
  10. CMS Collaboration, J. High Energy Phys. 01 (2022) 188.
  11. ATLAS Collaboration, Phys. Rev. Lett. 124, 222002 (2020).
  12. CMS Collaboration, J. High Energy Phys. 05 (2024) 116.
  13. ATLAS Collaboration, Phys. Lett. B 859, 139090 (2024).
  14. ALICE Collaboration, Nature (London) 605, 440 (2022); ALICE Collaboration607, E22 (2022).
  15. ATLAS Collaboration, Phys. Lett. B 868, 139680 (2025).
  16. ALICE Collaboration, arXiv:2409.12687.
  17. CMS Collaboration, Phys. Rev. Lett. 133, 071903 (2024).
  18. ATLAS Collaboration, Phys. Rev. D 101, 052007 (2020).
  19. ALICE Collaboration, arXiv:2504.03431.
  20. A. Buckley et al., Phys. Rep. 504, 145 (2011).
  21. J. M. Campbell et al., SciPost Phys. 16, 130 (2024).
  22. ATLAS Collaboration, J. Instrum. 19, P08018 (2024).
  23. R. Gambhir, M. LeBlanc, and Y. Zhou, arXiv:2509.19431.
  24. A. Garcia Caffaro, I. Moult, and C. Shimmin, arXiv:2510.06314.
  25. P. K. Dhani, G. Rodrigo, and G. F. R. Sborlini, J. High Energy Phys. 12 (2023) 188.
  26. E. Craft, M. Gonzalez, K. Lee, B. Mecaj, and I. Moult, J. High Energy Phys. 07 (2024) 080.
  27. G. Heinrich, Phys. Rep. 922, 1 (2021).
  28. A. Huss, J. Huston, S. Jones, and M. Pellen, J. Phys. G 50, 043001 (2023).
  29. D. Gaggero, A. Ghira, S. Marzani, and G. Ridolfi, J. High Energy Phys. 09 (2022) 058.
  30. A. Ghira, S. Marzani, and G. Ridolfi, J. High Energy Phys. 11 (2023) 120.
  31. A. Ghira, L. Mai, and S. Marzani, Eur. Phys. J. C 85, 294 (2025).
  32. S. Coleman and R. E. Norton, Nuovo Cimento 38, 438 (1965).
  33. G. F. Sterman, in TASI 95: QCD and Beyond (1995), pp. 327–408, arXiv:hep-ph/9606312.
  34. J. Frenkel and J. C. Taylor, Nucl. Phys. B109, 439 (1976); B117, 546(E) (1976); B155, 544(E) (1979).
  35. Y. L. Dokshitzer, D. Diakonov, and S. I. Troian, Phys. Rep. 58, 269 (1980).
  36. R. K. Ellis, H. Georgi, M. Machacek, H. D. Politzer, and G. G. Ross, Phys. Lett. 78B, 281 (1978).
  37. R. K. Ellis, H. Georgi, M. Machacek, H. D. Politzer, and G. G. Ross, Nucl. Phys. B152, 285 (1979).
  38. B. Humpert and W. L. van Neerven, Phys. Lett. 101B, 101 (1981).
  39. M. Gell-Mann and M. L. Goldberger, Phys. Rev. 96, 1433 (1954).
  40. L. S. Brown and R. L. Goble, Phys. Rev. 173, 1505 (1968).
  41. J. M. Campbell, S. Höche, M. Knobbe, C. T. Preuss, and D. Reichelt, arXiv:2505.10408.
  42. A. Bassetto, M. Ciafaloni, and G. Marchesini, Phys. Rep. 100, 201 (1983).
  43. W. Gordon, Z. Phys. 140, 630 (1928).
  44. G. Marchesini and B. R. Webber, Nucl. Phys. B330, 261 (1990).
  45. Y. L. Dokshitzer, V. A. Khoze, and S. I. Troian, J. Phys. G 17, 1602 (1991).
  46. S. Catani, S. Dittmaier, M. H. Seymour, and Z. Trocsanyi, Nucl. Phys. B627, 189 (2002).
  47. C. Duhr and F. Maltoni, J. High Energy Phys. 11 (2008) 002.
  48. T. Cohen and M. Riembau, J. High Energy Phys. 10 (2024) 132.
  49. D. Amati, R. Petronzio, and G. Veneziano, Nucl. Phys. B140, 54 (1978).
  50. D. Amati, R. Petronzio, and G. Veneziano, Nucl. Phys. B146, 29 (1978).
  51. J. Kalinowski, K. Konishi, and T. R. Taylor, Nucl. Phys. B181, 221 (1981).
  52. J. Kalinowski, K. Konishi, P. N. Scharbach, and T. R. Taylor, Nucl. Phys. B181, 253 (1981).
  53. S. Catani and M. Grazzini, Nucl. Phys. B570, 287 (2000).
  54. R. L. Arnowitt and S. I. Fickler, Phys. Rev. B127, 1821 (1962).
  55. W. Konetschny and W. Kummer, Nucl. Phys. 100, 106 (1975).
  56. W. Konetschny and W. Kummer, Nucl. Phys. B108, 397 (1976).
  57. J. Frenkel, Phys. Rev. D 13, 2325 (1976).
  58. W. Konetschny, Phys. Lett. 90B, 263 (1980); 115B, 503(E) (1982).
  59. F. A. Berends and W. T. Giele, Nucl. Phys. B306, 759 (1988).
  60. F. A. Berends, W. T. Giele, and H. Kuijf, Nucl. Phys. B321, 39 (1989).
  61. F. A. Berends, H. Kuijf, B. Tausk, and W. T. Giele, Nucl. Phys. B357, 32 (1991).
  62. C. Duhr, S. Hoeche, and F. Maltoni, J. High Energy Phys. 08 (2006) 062.
  63. V. V. Sudakov, Sov. Phys. JETP 3, 65 (1956).
  64. J. M. Campbell and E. W. N. Glover, Nucl. Phys. B527, 264 (1998).
  65. S. Catani and M. H. Seymour, Nucl. Phys. B485, 291 (1997); B510, 503(E) (1998).
  66. M. Czakon, Nucl. Phys. B849, 250 (2011).
  67. S. Catani, D. Colferai, and A. Torrini, J. High Energy Phys. 01 (2020) 118.
  68. R. D. Field, Applications of Perturbative QCD, Frontiers in Physics, Lecture Note Series (Addison-Wesley Publishing Co., Inc., Reading, MA, 1989).

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Phys. Rev. D 113, 054009– Published 4 March, 2026
  • Received 12 December 2025
  • Accepted 11 February 2026
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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.

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