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Two-Dimensional Transverse-Momentum Subtraction and Semi-Inclusive Deep-Inelastic Scattering at Next-to-Next-to-Next-to-Leading Order in QCD

Liang Dong1,*, Shen Fang2,†, Jun Gao1,‡, Hai Tao Li3,§, Ding Yu Shao2,4,5,6,∥, Hua Xing Zhu7,5,6,¶, and Yu Jiao Zhu8,**

  • 1State Key Laboratory of Dark Matter Physics, Shanghai Key Laboratory for Particle Physics and Cosmology, Key Laboratory for Particle Astrophysics and Cosmology (MOE), School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
  • 2Department of Physics, Center for Field Theory and Particle Physics, and Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Fudan University, Shanghai, 200433, China
  • 3School of Physics, Shandong University, Jinan, Shandong 250100, China
  • 4Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai 200438, China
  • 5Center for High Energy Physics, Peking University, Beijing 100871, China
  • 6Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences, Huizhou 516000, Guangdong Province, China
  • 7School of Physics, Peking University, Beijing 100871, China
  • 8Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Boltzmannstrasse 8, 85748 Garching, Germany

  • *Contact author: liang.dong@sjtu.edu.cn
  • †Contact author: sfang23@m.fudan.edu.cn
  • ‡Contact author: jung49@sjtu.edu.cn
  • §Contact author: haitao.li@sdu.edu.cn
  • ∥Contact author: dyshao@fudan.edu.cn
  • Contact author: zhuhx@pku.edu.cn
  • **Contact author: yzhu@mpp.mpg.de
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Phys. Rev. Lett. 137, 151901 – Published 6 October, 2026

DOI: https://doi.org/10.1103/gnk4-8p43

Abstract

Identified hadron production is essential for the study of nucleon structure and QCD hadronization at high energies. We present the first calculation of unpolarized semi-inclusive deep-inelastic scattering at next-to-next-to-next-to-leading order in perturbative QCD. Our calculation is based on a novel method of two-dimensional transverse-momentum subtraction motivated by QCD factorization of soft and collinear singularities. The next-to-next-to-next-to-leading order corrections are moderate in general but can be significant in threshold regions, and exhibit excellent perturbative convergence and reduced scale variations. The fully differential framework allows for arbitrary selection cuts and directly enables precision nucleon tomography at the upcoming Electron-Ion Collider, establishing the theory foundation needed to match the anticipated experimental accuracy. Generalization of the method to calculations of polarized semi-inclusive deep-inelastic scattering is also feasible.

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References (119)

  1. C. Anastasiou, C. Duhr, F. Dulat, F. Herzog, and B. Mistlberger, Phys. Rev. Lett. 114, 212001 (2015).
  2. C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog, A. Lazopoulos, and B. Mistlberger, J. High Energy Phys. 05 (2016) 058.
  3. F. A. Dreyer and A. Karlberg, Phys. Rev. Lett. 117, 072001 (2016).
  4. B. Mistlberger, J. High Energy Phys. 05 (2018) 028.
  5. C. Duhr, F. Dulat, and B. Mistlberger, Phys. Rev. Lett. 125, 051804 (2020).
  6. C. Duhr, F. Dulat, V. Hirschi, and B. Mistlberger, J. High Energy Phys. 08 (2020) 017.
  7. J. Baglio, C. Duhr, B. Mistlberger, and R. Szafron, J. High Energy Phys. 12 (2022) 066.
  8. C. Duhr, F. Dulat, and B. Mistlberger, Phys. Rev. Lett. 125, 172001 (2020).
  9. C. Duhr, F. Dulat, and B. Mistlberger, J. High Energy Phys. 11 (2020) 143.
  10. C. Duhr and B. Mistlberger, J. High Energy Phys. 03 (2022) 116.
  11. X. Chen, T. Gehrmann, E. W. N. Glover, A. Huss, P. F. Monni, E. Re, L. Rottoli, and P. Torrielli, Phys. Rev. Lett. 128, 252001 (2022).
  12. L.-B. Chen, H. T. Li, H.-S. Shao, and J. Wang, J. High Energy Phys. 03 (2020) 072.
  13. L.-B. Chen, H. T. Li, H.-S. Shao, and J. Wang, Phys. Lett. B 803, 135292 (2020).
  14. L. Chen, X. Chen, X. Guan, and Y.-Q. Ma, Phys. Rev. D 114, 034005 (2026).
  15. L.-B. Chen, H. T. Li, Z. Li, J. Wang, Y. Wang, and Q.-f. Wu, Phys. Rev. D 109, L071503 (2024).
  16. M. Abele, D. de Florian, and W. Vogelsang, Phys. Rev. D 104, 094046 (2021).
  17. S. Goyal, S.-O. Moch, V. Pathak, N. Rana, and V. Ravindran, Phys. Rev. Lett. 132, 251902 (2024).
  18. L. Bonino, T. Gehrmann, and G. Stagnitto, Phys. Rev. Lett. 132, 251901 (2024).
  19. L. Bonino, T. Gehrmann, M. Löchner, K. Schönwald, and G. Stagnitto, Phys. Rev. Lett. 133, 211904 (2024).
  20. S. Goyal, R. N. Lee, S.-O. Moch, V. Pathak, N. Rana, and V. Ravindran, Phys. Rev. Lett. 133, 211905 (2024).
  21. S. Goyal, R. N. Lee, S.-O. Moch, V. Pathak, N. Rana, and V. Ravindran, Phys. Rev. D 111, 094007 (2025).
  22. L. Bonino, T. Gehrmann, M. Löchner, K. Schönwald, and G. Stagnitto, Phys. Rev. Lett. 135, 211902 (2025).
  23. J. Haug and F. Wunder, Phys. Rev. D 112, 114036 (2025).
  24. L. Bonino, T. Gehrmann, M. Löchner, K. Schönwald, and G. Stagnitto, J. High Energy Phys. 10 (2025) 016.
  25. B. Zhou and J. Gao, Phys. Rev. D 112, 074016 (2025).
  26. L. Bonino, T. Gehrmann, M. Löchner, K. Schönwald, and G. Stagnitto, J. High Energy Phys. 03 (2026) 109.
  27. S. Goyal, R. N. Lee, S.-O. Moch, V. Pathak, and V. Ravindran, arXiv:2510.18872.
  28. S. Goyal, S.-O. Moch, V. Pathak, N. Rana, and V. Ravindran, Phys. Rev. D 113, 034004 (2026).
  29. V. Bertone, S. Carrazza, N. P. Hartland, E. R. Nocera, and J. Rojo (NNPDF Collaboration), Eur. Phys. J. C 77, 516 (2017).
  30. M. Soleymaninia, M. Goharipour, and H. Khanpour, Phys. Rev. D 98, 074002 (2018).
  31. E. Moffat, W. Melnitchouk, T. C. Rogers, and N. Sato (Jefferson Lab Angular Momentum (JAM) Collaboration), Phys. Rev. D 104, 016015 (2021).
  32. R. Abdul Khalek, V. Bertone, A. Khoudli, and E. R. Nocera (MAP (Multi-dimensional Analyses of Partonic distributions) Collaboration), Phys. Lett. B 834, 137456 (2022).
  33. I. Borsa, R. Sassot, D. de Florian, M. Stratmann, and W. Vogelsang, Phys. Rev. Lett. 129, 012002 (2022).
  34. J. Gao, C. Liu, X. Shen, H. Xing, and Y. Zhao, Phys. Rev. Lett. 132, 261903 (2024).
  35. J. Gao, C. Liu, X. Shen, H. Xing, and Y. Zhao, Phys. Rev. D 110, 114019 (2024).
  36. J. Gao, X. Shen, H. Xing, Y. Zhao, and B. Zhou, Phys. Rev. Lett. 135, 041902 (2025).
  37. J. Gao, C. Liu, and B. Zhou, Chin. Phys. C 50, 053101 (2026).
  38. J. Gao, C. Liu, M. Li, X. Shen, H. Xing, Y. Zhao, and Y. Zhou, Phys. Rev. D 112, 054045 (2025).
  39. V. Bertone, A. Chiefa, and E. R. Nocera (MAP (Multi-dimensional Analyses of Partonic distributions) Collaboration), Phys. Lett. B 865, 139497 (2025).
  40. I. Borsa, M. Stratmann, W. Vogelsang, D. de Florian, and R. Sassot, Phys. Rev. Lett. 133, 151901 (2024).
  41. J. Cruz-Martinez, T. Hasenack, F. Hekhorn, G. Magni, E. R. Nocera, T. R. Rabemananjara, J. Rojo, T. Sharma, and G. van Seeventer, J. High Energy Phys. 07 (2025) 168.
  42. C. Cocuzza, N. T. Hunt-Smith, W. Melnitchouk, N. Sato, and A. W. Thomas (JAM Collaboration (Spin PDF Analysis Group), Phys. Rev. D 112, 114017 (2025).
  43. M. Procura and I. W. Stewart, Phys. Rev. D 81, 074009 (2010); 83, 039902(E) (2011).
  44. A. Jain, M. Procura, and W. J. Waalewijn, J. High Energy Phys. 05 (2011) 035.
  45. H.-M. Chang, M. Procura, J. Thaler, and W. J. Waalewijn, Phys. Rev. Lett. 111, 102002 (2013).
  46. M. Ritzmann and W. J. Waalewijn, Phys. Rev. D 90, 054029 (2014).
  47. T. Kaufmann, A. Mukherjee, and W. Vogelsang, Phys. Rev. D 92, 054015 (2015); 101, 079901(E) (2020).
  48. L. Dai, C. Kim, and A. K. Leibovich, Phys. Rev. D 94, 114023 (2016).
  49. Y.-T. Chien and I. Vitev, Phys. Rev. Lett. 119, 112301 (2017).
  50. Z.-B. Kang, F. Ringer, and I. Vitev, J. High Energy Phys. 11 (2016) 155.
  51. Z.-B. Kang, X. Liu, F. Ringer, and H. Xing, J. High Energy Phys. 11 (2017) 068.
  52. D. P. Anderle, T. Kaufmann, M. Stratmann, F. Ringer, and I. Vitev, Phys. Rev. D 96, 034028 (2017).
  53. Z.-B. Kang, K. Lee, J. Terry, and H. Xing, Phys. Lett. B 798, 134978 (2019).
  54. Z.-B. Kang, K. Lee, D. Y. Shao, and F. Zhao, J. High Energy Phys. 11 (2021) 005.
  55. S. Caletti, A. J. Larkoski, S. Marzani, and D. Reichelt, J. High Energy Phys. 10 (2022) 158.
  56. L. J. Dixon, I. Moult, and H. X. Zhu, Phys. Rev. D 100, 014009 (2019).
  57. H. Chen, I. Moult, X. Zhang, and H. X. Zhu, Phys. Rev. D 102, 054012 (2020).
  58. R.-J. Fu, R. Rahn, D. Y. Shao, W. J. Waalewijn, and B. Wu, Phys. Rev. Lett. 135, 171903 (2025).
  59. M. Czakon, T. Generet, A. Mitov, and R. Poncelet, Phys. Rev. Lett. 135, 171902 (2025).
  60. T. Generet, R. Poncelet, and M. Muškinja, J. High Energy Phys. 02 (2026) 023.
  61. J. Gao, H. T. Li, H. X. Zhu, and Y. J. Zhu, Sci. China Phys. Mech. Astron. (2026), https://www.sciengine.com/SCPMA/doi/10.1360/s11433-026-3121-1.
  62. L. Bonino, A. Gehrmann-De Ridder, T. Gehrmann, A. Huss, F. Merlotti, and G. Stagnitto, Phys. Rev. Lett. 136, 251903 (2026).
  63. L. Dong, S. Fang, J. Gao, H. T. Li, D. Y. Shao, and Y. J. Zhu, Phys. Rev. Lett. 137, 081902 (2026).
  64. C.-Q. He, H. Xing, T.-Z. Yang, and H. X. Zhu, Phys. Rev. Lett. 135, 101901 (2025).
  65. R. Abdul Khalek et al., Nucl. Phys. A1026, 122447 (2022).
  66. R. Abdul Khalek et al., arXiv:2203.13199.
  67. M. Dong et al. (CEPC Study Group), arXiv:1811.10545.
  68. A. Abada et al. (FCC Collaboration), Eur. Phys. J. Special Topics 228, 261 (2019).
  69. S. Catani and M. Grazzini, Phys. Rev. Lett. 98, 222002 (2007).
  70. S. Catani, L. Cieri, G. Ferrera, D. de Florian, and M. Grazzini, Phys. Rev. Lett. 103, 082001 (2009).
  71. S. Catani, G. Ferrera, and M. Grazzini, J. High Energy Phys. 05 (2010) 006.
  72. S. Catani, L. Cieri, D. de Florian, G. Ferrera, and M. Grazzini, Phys. Rev. Lett. 108, 072001 (2012); 117, 089901(E) (2016).
  73. C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart, Phys. Rev. D 63, 114020 (2001).
  74. C. W. Bauer and I. W. Stewart, Phys. Lett. B 516, 134 (2001).
  75. C. W. Bauer, D. Pirjol, and I. W. Stewart, Phys. Rev. D 65, 054022 (2002).
  76. G. Salam, Et∞ scheme (unpublished).
  77. D. Bertolini, T. Chan, and J. Thaler, J. High Energy Phys. 04 (2014) 013.
  78. C. Liu, X. Shen, B. Zhou, and J. Gao, J. High Energy Phys. 09 (2023) 108.
  79. Z. Bern, L. J. Dixon, and D. A. Kosower, Nucl. Phys. B513, 3 (1998).
  80. J. M. Campbell and R. K. Ellis, Phys. Rev. D 65, 113007 (2002).
  81. J. M. Campbell, R. K. Ellis, and D. L. Rainwater, Phys. Rev. D 68, 094021 (2003).
  82. J. M. Campbell and R. K. Ellis, Nucl. Phys. B, Proc. Suppl. 205–206, 10 (2010).
  83. T. Becher, M. Neubert, and B. D. Pecjak, J. High Energy Phys. 01 (2007) 076.
  84. S. Moch, J. A. M. Vermaseren, and A. Vogt, Phys. Lett. B 625, 245 (2005).
  85. S. Moch, J. A. M. Vermaseren, and A. Vogt, J. High Energy Phys. 08 (2005) 049.
  86. P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Phys. Rev. Lett. 102, 212002 (2009).
  87. T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, and C. Studerus, J. High Energy Phys. 06 (2010) 094.
  88. T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, and C. Studerus, J. High Energy Phys. 11 (2010) 102.
  89. R. N. Lee, A. von Manteuffel, R. M. Schabinger, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Phys. Rev. Lett. 128, 212002 (2022).
  90. M.-x. Luo, T.-Z. Yang, H. X. Zhu, and Y. J. Zhu, Phys. Rev. Lett. 124, 092001 (2020).
  91. M.-x. Luo, T.-Z. Yang, H. X. Zhu, and Y. J. Zhu, J. High Energy Phys. 06 (2021) 115.
  92. M. A. Ebert, B. Mistlberger, and G. Vita, J. High Energy Phys. 09 (2020) 146.
  93. Y. Li and H. X. Zhu, Phys. Rev. Lett. 118, 022004 (2017).
  94. Y. J. Zhu, J. High Energy Phys. 02 (2026) 018.
  95. Y.-T. Chien, R. Rahn, S. Schrijnder van Velzen, D. Y. Shao, W. J. Waalewijn, and B. Wu, Phys. Lett. B 815, 136124 (2021).
  96. Y.-T. Chien, R. Rahn, D. Y. Shao, W. J. Waalewijn, and B. Wu, J. High Energy Phys. 02 (2023) 256.
  97. R.-J. Fu, R. Rahn, D. Y. Shao, W. J. Waalewijn, and B. Wu, J. High Energy Phys. 08 (2026) 105.
  98. D. Gutierrez-Reyes, I. Scimemi, W. J. Waalewijn, and L. Zoppi, J. High Energy Phys. 10 (2019) 031.
  99. S. Fang, M.-S. Gao, H. T. Li, and D. Y. Shao, J. High Energy Phys. 01 (2025) 029.
  100. G. Bell, K. Brune, G. Das, and M. Wald, SciPost Phys. Proc. 7, 021 (2022).
  101. K. M. Brune, Automation of jet function calculations in soft-collinear effective theory, Ph.D. thesis, Siegen University, 2022.
  102. L. Buonocore, M. Grazzini, F. Guadagni, J. Haag, and L. Rottoli, Eur. Phys. J. C 85, 1290 (2025).
  103. M. G. Echevarria, I. Scimemi, and A. Vladimirov, Phys. Rev. D 93, 054004 (2016).
  104. T. Lübbert, J. Oredsson, and M. Stahlhofen, J. High Energy Phys. 03 (2016) 168.
  105. A. Gao, H. T. Li, I. Moult, and H. X. Zhu, Phys. Rev. Lett. 123, 062001 (2019).
  106. A. Gao, H. T. Li, I. Moult, and H. X. Zhu, J. High Energy Phys. 09 (2024) 072.
  107. M. A. Ebert, I. Moult, I. W. Stewart, F. J. Tackmann, G. Vita, and H. X. Zhu, J. High Energy Phys. 04 (2019) 123.
  108. S. Camarda, L. Cieri, and G. Ferrera, Eur. Phys. J. C 82, 575 (2022).
  109. L. Buonocore, S. Kallweit, L. Rottoli, and M. Wiesemann, Phys. Lett. B 829, 137118 (2022).
  110. T.-J. Hou et al., Phys. Rev. D 103, 014013 (2021).
  111. M. Grazzini, S. Kallweit, and M. Wiesemann, Eur. Phys. J. C 78, 537 (2018).
  112. M. A. Ebert and F. J. Tackmann, J. High Energy Phys. 03 (2020) 158.
  113. See Supplemental Material at http://link.aps.org/supplemental/10.1103/gnk4-8p43 for details of the slicing-cutoff dependence and additional z- and x-differential cross sections at s=45  GeV.
  114. M.-X. Luo, X. Wang, X. Xu, L. L. Yang, T.-Z. Yang, and H. X. Zhu, J. High Energy Phys. 10 (2019) 083.
  115. M.-X. Luo, T.-Z. Yang, H. X. Zhu, and Y. J. Zhu, J. High Energy Phys. 01 (2020) 040.
  116. Y. Hu, C. Sun, X.-M. Shen, and J. Gao, J. High Energy Phys. 08 (2021) 122.
  117. P. A. Baikov and K. G. Chetyrkin, Phys. Rev. Lett. 97, 061803 (2006).
  118. M. Abele, D. de Florian, and W. Vogelsang, Phys. Rev. D 106, 014015 (2022).
  119. G. D. Alexeev et al. (COMPASS Collaboration), Phys. Rev. D 112, 012002 (2025).

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Phys. Rev. Lett. 137, 151901– Published 6 October, 2026
  • Received 12 April 2026
  • Revised 3 August 2026
  • Accepted 9 September 2026
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