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  • Open Access

Energy correlators resolving proton spin

Jun Gao*

Hai Tao Li†

Yu Jiao Zhu‡

  • State 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

  • *Contact author: jung49@sjtu.edu.cn
  • †Contact author: haitao.li@sdu.edu.cn
  • ‡Contact author: yzhu@mpp.mpg.de
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Phys. Rev. D 113, 034028 – Published 24 February, 2026

DOI: https://doi.org/10.1103/b8gn-wlg2

Abstract

We investigate the partonic origin of the proton longitudinal spin using spin-dependent energy correlators measured in lepton-hadron collisions with longitudinally polarized proton beams. These observables encode angular correlations in energy flow and are sensitive to the spin-momentum structure of confined partons. Using soft-collinear effective theory, we analyze the correlation patterns in both nearly back-to-back and forward limits, which establishes a direct correspondence with longitudinally polarized transverse momentum-dependent distributions (TMDs) and nucleon energy correlators (NECs). The TMDs and NECs allow consistent matching onto hard radiation regions and provide a comprehensive description of the transition from perturbative parton branching to nonperturbative confinement. Using renormalization group evolution, we obtain joint next-to-next-to-next-to-leading and next-to-next-to-leading logarithmic quantitative predictions for spin-dependent energy correlation patterns in the current and target fragmentation regions. The framework provides new theoretical insight into how the internal motion and spin of partons contribute to the formation of the proton longitudinal spin and offers an experimental paradigm for probing the interplay between color confinement and spin dynamics at the forthcoming Electron-Ion Collider.

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References (193)

  1. J. R. Ellis and R. L. Jaffe, Phys. Rev. D 9, 1444 (1974); 10, 1669(E) (1974).
  2. J. Ashman et al. (European Muon Collaboration), Phys. Lett. B 206, 364 (1988).
  3. J. Ashman et al. (European Muon Collaboration), Nucl. Phys. B328, 1 (1989).
  4. X. Ji, F. Yuan, and Y. Zhao, Nat. Rev. Phys. 3, 27 (2021).
  5. A. Airapetian et al. (HERMES Collaboration), Phys. Rev. D 71, 012003 (2005).
  6. M. Alekseev et al. (COMPASS Collaboration), Phys. Lett. B 680, 217 (2009).
  7. E.-C. Aschenauer et al. (RHIC SPIN Collaboration), arXiv:2302.00605.
  8. C. A. Aidala, S. D. Bass, D. Hasch, and G. K. Mallot, Rev. Mod. Phys. 85, 655 (2013).
  9. A. Deur, S. J. Brodsky, and G. F. De Téramond, Rep. Prog. Phys. 82, 076201 (2019).
  10. I. Borsa, M. Stratmann, W. Vogelsang, D. de Florian, and R. Sassot, Phys. Rev. Lett. 133, 151901 (2024).
  11. V. Bertone, A. Chiefa, and E. R. Nocera (MAP [Multi-dimensional Analyses of Partonic distributions] Collaboration), Phys. Lett. B 865, 139497 (2025).
  12. 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.
  13. R. Abdul Khalek et al., Nucl. Phys. A1026, 122447 (2022).
  14. R. Abdul Khalek et al., arXiv:2203.13199.
  15. X.-D. Ji, Phys. Rev. Lett. 78, 610 (1997).
  16. X.-D. Ji, Phys. Rev. D 58, 056003 (1998).
  17. R. L. Jaffe and A. Manohar, Nucl. Phys. B337, 509 (1990).
  18. S. Moch, J. A. M. Vermaseren, and A. Vogt, Nucl. Phys. B889, 351 (2014).
  19. S. Moch, J. A. M. Vermaseren, and A. Vogt, Phys. Lett. B 748, 432 (2015).
  20. J. Blümlein, P. Marquard, C. Schneider, and K. Schönwald, J. High Energy Phys. 11 (2022) 156.
  21. J. Blümlein, P. Marquard, C. Schneider, and K. Schönwald, Nucl. Phys. B971, 115542 (2021).
  22. J. Blümlein, P. Marquard, C. Schneider, and K. Schönwald, J. High Energy Phys. 01 (2022) 193.
  23. Y. J. Zhu, arXiv:2509.01655.
  24. J. She, J. Zhu, and B.-Q. Ma, Phys. Rev. D 79, 054008 (2009).
  25. H. Avakian, A. V. Efremov, P. Schweitzer, O. V. Teryaev, F. Yuan, and P. Zavada, Mod. Phys. Lett. A 24, 2995 (2009).
  26. H. Avakian, A. V. Efremov, P. Schweitzer, and F. Yuan, Phys. Rev. D 81, 074035 (2010).
  27. X. Ji, J.-H. Zhang, and Y. Zhao, Phys. Lett. B 743, 180 (2015).
  28. Y. Zhao, K.-F. Liu, and Y. Yang, Phys. Rev. D 93, 054006 (2016).
  29. C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath, K. Hadjiyiannakou, K. Jansen, G. Koutsou, H. Panagopoulos, and G. Spanoudes, Phys. Rev. D 101, 094513 (2020).
  30. C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, A. Vaquero Avilés-Casco, and C. Wiese, Phys. Rev. Lett. 119, 142002 (2017).
  31. Y.-B. Yang, R. S. Sufian, A. Alexandru, T. Draper, M. J. Glatzmaier, K.-F. Liu, and Y. Zhao, Phys. Rev. Lett. 118, 102001 (2017).
  32. Y. V. Kovchegov, D. Pitonyak, and M. D. Sievert, Phys. Rev. Lett. 118, 052001 (2017).
  33. Y. V. Kovchegov, D. Pitonyak, and M. D. Sievert, J. High Energy Phys. 01 (2016) 072; 10 (2016) 148(E).
  34. Y. V. Kovchegov, A. Tarasov, and Y. Tawabutr, J. High Energy Phys. 03 (2022) 184.
  35. F. Cougoulic, Y. V. Kovchegov, A. Tarasov, and Y. Tawabutr, J. High Energy Phys. 07 (2022) 095; 09 (2024) 052(E).
  36. V. Ravindran, J. Smith, and W. L. van Neerven, Nucl. Phys. B682, 421 (2004).
  37. R. Boughezal, H. T. Li, and F. Petriello, Phys. Lett. B 817, 136333 (2021).
  38. I. Borsa, D. de Florian, and I. Pedron, Phys. Rev. Lett. 125, 082001 (2020).
  39. I. Borsa, D. de Florian, and I. Pedron, Phys. Rev. D 103, 014008 (2021).
  40. I. Borsa, D. de Florian, and I. Pedron, Eur. Phys. J. C 82, 1167 (2022).
  41. I. Borsa, D. de Florian, and I. Pedron, Phys. Rev. D 107, 054027 (2023).
  42. C. L. Basham, L. S. Brown, S. D. Ellis, and S. T. Love, Phys. Rev. D 17, 2298 (1978).
  43. C. L. Basham, L. S. Brown, S. D. Ellis, and S. T. Love, Phys. Rev. Lett. 41, 1585 (1978).
  44. C. L. Basham, L. S. Brown, S. D. Ellis, and S. T. Love, Phys. Rev. D 19, 2018 (1979).
  45. C. L. Basham, L. S. Brown, S. D. Ellis, and S. T. Love, Phys. Lett. 85B, 297 (1979).
  46. J. C. Collins and D. E. Soper, Phys. Rev. Lett. 48, 655 (1982).
  47. D. de Florian and M. Grazzini, Nucl. Phys. B704, 387 (2005).
  48. I. Moult and H. X. Zhu, J. High Energy Phys. 08 (2018) 160.
  49. C. Duhr, B. Mistlberger, and G. Vita, Phys. Rev. Lett. 129, 162001 (2022).
  50. U. G. Aglietti and G. Ferrera, Phys. Rev. D 110, 114004 (2024).
  51. G. P. Korchemsky, J. High Energy Phys. 01 (2020) 008.
  52. H. Chen, X. Zhou, and H. X. Zhu, J. High Energy Phys. 10 (2023) 132.
  53. I. Moult, G. Vita, and K. Yan, J. High Energy Phys. 07 (2020) 005.
  54. A. Kardos, S. Kluth, G. Somogyi, Z. Tulipánt, and A. Verbytskyi, Eur. Phys. J. C 78, 498 (2018).
  55. Z. Tulipánt, A. Kardos, and G. Somogyi, Eur. Phys. J. C 77, 749 (2017).
  56. R. von Kuk, J. K. L. Michel, and Z. Sun, J. High Energy Phys. 07 (2024) 129.
  57. U. G. Aglietti and G. Ferrera, Eur. Phys. J. C 85, 272 (2025).
  58. Z.-B. Kang, J. Penttala, and C. Zhang, arXiv:2410.21435.
  59. A. Gao, H. T. Li, I. Moult, and H. X. Zhu, Phys. Rev. Lett. 123, 062001 (2019).
  60. A. Gao, H. T. Li, I. Moult, and H. X. Zhu, J. High Energy Phys. 09 (2024) 072.
  61. K. Lee, I. Moult, and X. Zhang, J. High Energy Phys. 05 (2025) 129.
  62. K. Lee, I. Moult, and X. Zhang, arXiv:2410.01902.
  63. X. Liu, W. Vogelsang, F. Yuan, and H. X. Zhu, Phys. Rev. Lett. 134, 151901 (2025).
  64. Y. L. Dokshitzer, G. Marchesini, and B. R. Webber, J. High Energy Phys. 07 (1999) 012.
  65. K. Lee, A. Pathak, I. W. Stewart, and Z. Sun, Phys. Rev. Lett. 133, 231902 (2024).
  66. H. Chen, P. F. Monni, Z. Xu, and H. X. Zhu, Phys. Rev. Lett. 133, 231901 (2024).
  67. K. Konishi, A. Ukawa, and G. Veneziano, Phys. Lett. 80B, 259 (1979).
  68. K. Konishi, A. Ukawa, and G. Veneziano, Nucl. Phys. B157, 45 (1979).
  69. K. Konishi, A. Ukawa, and G. Veneziano, Phys. Lett. 78B, 243 (1978).
  70. L. J. Dixon, I. Moult, and H. X. Zhu, Phys. Rev. D 100, 014009 (2019).
  71. H. Chen, M.-X. Luo, I. Moult, T.-Z. Yang, X. Zhang, and H. X. Zhu, J. High Energy Phys. 08 (2020) 028.
  72. H. Chen, J. High Energy Phys. 01 (2024) 035.
  73. E. Craft, K. Lee, B. Meçaj, and I. Moult, arXiv:2210.09311.
  74. H. Chen, I. Moult, and H. X. Zhu, J. High Energy Phys. 08 (2022) 233.
  75. H. Chen, I. Moult, and H. X. Zhu, Phys. Rev. Lett. 126, 112003 (2021).
  76. H. Chen, H. Ruan, and H. X. Zhu, J. High Energy Phys. 12 (2025) 168.
  77. M.-S. Gao, Z.-B. Kang, W. Li, and D. Y. Shao, arXiv:2509.15809.
  78. W. Chen, J. Gao, Y. Li, Z. Xu, X. Zhang, and H. X. Zhu, J. High Energy Phys. 05 (2024) 043.
  79. M. Jaarsma, Y. Li, I. Moult, W. J. Waalewijn, and H. X. Zhu, J. High Energy Phys. 12 (2023) 087.
  80. A. Budhraja, H. Chen, and W. J. Waalewijn, Phys. Lett. B 861, 139239 (2025).
  81. Y.-K. Song, S.-Y. Wei, L. Yang, and J. Zhou, arXiv:2509.14960.
  82. H. T. Li, I. Vitev, and Y. J. Zhu, J. High Energy Phys. 11 (2020) 051.
  83. A. Ali, G. Li, W. Wang, and Z.-P. Xing, Eur. Phys. J. C 80, 1096 (2020).
  84. H. T. Li, Y. Makris, and I. Vitev, Phys. Rev. D 103, 094005 (2021).
  85. H. Cao, H. T. Li, and Z. Mi, Phys. Rev. D 109, 096004 (2024).
  86. Z.-B. Kang, K. Lee, D. Y. Shao, and F. Zhao, J. High Energy Phys. 03 (2024) 153.
  87. I. Moult and H. X. Zhu, arXiv:2506.09119.
  88. J. Collins, Foundations of Perturbative QCD (Cambridge University Press, Cambridge, England, 2011), Vol. 32.
  89. I. I. Balitsky and L. N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978).
  90. E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, Sov. Phys. JETP 45, 199 (1977).
  91. I. Feige and M. D. Schwartz, Phys. Rev. D 90, 105020 (2014).
  92. C. Lee and G. F. Sterman, Phys. Rev. D 75, 014022 (2007).
  93. C. W. Bauer, S. Fleming, and M. E. Luke, Phys. Rev. D 63, 014006 (2000).
  94. C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart, Phys. Rev. D 63, 114020 (2001).
  95. C. W. Bauer and I. W. Stewart, Phys. Lett. B 516, 134 (2001).
  96. C. W. Bauer, D. Pirjol, and I. W. Stewart, Phys. Rev. D 65, 054022 (2002).
  97. G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B44, 189 (1972).
  98. I. Moult, I. W. Stewart, F. J. Tackmann, and W. J. Waalewijn, Phys. Rev. D 93, 094003 (2016).
  99. Y. Li, D. Neill, and H. X. Zhu, Nucl. Phys. B960, 115193 (2020).
  100. A. V. Manohar and I. W. Stewart, Phys. Rev. D 76, 074002 (2007).
  101. R.-b. Meng, F. I. Olness, and D. E. Soper, Nucl. Phys. B371, 79 (1992).
  102. R. Meng, F. I. Olness, and D. E. Soper, Phys. Rev. D 54, 1919 (1996).
  103. P. M. Nadolsky, D. R. Stump, and C. P. Yuan, Phys. Rev. D 61, 014003 (2000); 64, 059903(E) (2001).
  104. X.-d. Ji, J.-P. Ma, and F. Yuan, Phys. Lett. B 597, 299 (2004).
  105. Y. Koike, J. Nagashima, and W. Vogelsang, Nucl. Phys. B744, 59 (2006).
  106. T. Becher, M. Neubert, and B. D. Pecjak, J. High Energy Phys. 01 (2007) 076.
  107. S. Moch, J. A. M. Vermaseren, and A. Vogt, Phys. Lett. B 625, 245 (2005).
  108. S. Moch, J. A. M. Vermaseren, and A. Vogt, J. High Energy Phys. 08 (2005) 049.
  109. T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, and C. Studerus, J. High Energy Phys. 06 (2010) 094.
  110. P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Phys. Rev. Lett. 102, 212002 (2009).
  111. T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, and C. Studerus, J. High Energy Phys. 11 (2010) 102.
  112. R. N. Lee, A. von Manteuffel, R. M. Schabinger, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Phys. Rev. Lett. 128, 212002 (2022).
  113. Y. Li and H. X. Zhu, Phys. Rev. Lett. 118, 022004 (2017).
  114. I. Moult, H. X. Zhu, and Y. J. Zhu, J. High Energy Phys. 08 (2022) 280.
  115. A. B. Cuerpo, I. Scimemi, and A. Vladimirov, J. High Energy Phys. 11 (2025) 013.
  116. J. C. Collins and D. E. Soper, Nucl. Phys. B193, 381 (1981); B213, 545(E) (1983).
  117. J. C. Collins and D. E. Soper, Annu. Rev. Nucl. Part. Sci. 37, 383 (1987).
  118. R. A. Brandt, F. Neri, and M.-A. Sato, Phys. Rev. D 24, 879 (1981).
  119. R. A. Brandt, A. Gocksch, M. A. Sato, and F. Neri, Phys. Rev. D 26, 3611 (1982).
  120. M.-H. Chu et al. (Lattice Parton (LPC) Collaboration), Phys. Rev. D 106, 034509 (2022).
  121. P. Shanahan, M. Wagman, and Y. Zhao, Phys. Rev. D 102, 014511 (2020).
  122. P. Shanahan, M. Wagman, and Y. Zhao, Phys. Rev. D 104, 114502 (2021).
  123. A. Avkhadiev, P. E. Shanahan, M. L. Wagman, and Y. Zhao, Phys. Rev. Lett. 132, 231901 (2024).
  124. M. Schlemmer, A. Vladimirov, C. Zimmermann, M. Engelhardt, and A. Schäfer, J. High Energy Phys. 08 (2021) 004.
  125. Y. Li et al., Phys. Rev. Lett. 128, 062002 (2022).
  126. Q.-A. Zhang et al. (Lattice Parton Collaboration), Phys. Rev. Lett. 125, 192001 (2020).
  127. J.-X. Tan et al., arXiv:2511.22547.
  128. J.-y. Chiu, A. Jain, D. Neill, and I. Z. Rothstein, Phys. Rev. Lett. 108, 151601 (2012).
  129. J.-Y. Chiu, A. Jain, D. Neill, and I. Z. Rothstein, J. High Energy Phys. 05 (2012) 084.
  130. P. Sun, J. Isaacson, C. P. Yuan, and F. Yuan, Int. J. Mod. Phys. A 33, 1841006 (2018).
  131. M.-X. Luo, X. Wang, X. Xu, L. L. Yang, T.-Z. Yang, and H. X. Zhu, J. High Energy Phys. 10 (2019) 083.
  132. M.-X. Luo, T.-Z. Yang, H. X. Zhu, and Y. J. Zhu, J. High Energy Phys. 01 (2020) 040.
  133. M.-x. Luo, T.-Z. Yang, H. X. Zhu, and Y. J. Zhu, J. High Energy Phys. 06 (2021) 115.
  134. M. A. Ebert, B. Mistlberger, and G. Vita, J. High Energy Phys. 07 (2021) 121.
  135. M. A. Ebert, B. Mistlberger, and G. Vita, J. High Energy Phys. 09 (2020) 146.
  136. J. Collins and T. Rogers, Phys. Rev. D 91, 074020 (2015).
  137. M. Bury, F. Hautmann, S. Leal-Gomez, I. Scimemi, A. Vladimirov, and P. Zurita, J. High Energy Phys. 10 (2022) 118.
  138. A. Buckley, J. Ferrando, S. Lloyd, K. Nordström, B. Page, M. Rüfenacht, M. Schönherr, and G. Watt, Eur. Phys. J. C 75, 132 (2015).
  139. S. Dulat, T.-J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, and C. P. Yuan, Phys. Rev. D 93, 033006 (2016).
  140. E. R. Nocera, R. D. Ball, S. Forte, G. Ridolfi, and J. Rojo (NNPDF Collaboration), Nucl. Phys. B887, 276 (2014).
  141. C. Liu, X. Shen, B. Zhou, and J. Gao, J. High Energy Phys. 09 (2023) 108.
  142. J. Gao, C. Liu, X. Shen, H. Xing, and Y. Zhao, Phys. Rev. D 110, 114019 (2024).
  143. L. Trentadue and G. Veneziano, Phys. Lett. B 323, 201 (1994).
  144. A. Berera and D. E. Soper, Phys. Rev. D 53, 6162 (1996).
  145. M. Grazzini, L. Trentadue, and G. Veneziano, Nucl. Phys. B519, 394 (1998).
  146. D. de Florian and R. Sassot, Phys. Rev. D 56, 426 (1997).
  147. F. A. Ceccopieri and L. Trentadue, Phys. Lett. B 655, 15 (2007).
  148. M. Anselmino, V. Barone, and A. Kotzinian, Phys. Lett. B 699, 108 (2011).
  149. X. Liu and H. X. Zhu, Phys. Rev. Lett. 130, 091901 (2023).
  150. H. Cao, X. Liu, and H. X. Zhu, Phys. Rev. D 107, 114008 (2023).
  151. K.-B. Chen, J.-P. Ma, and X.-B. Tong, J. High Energy Phys. 08 (2024) 227.
  152. C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein, and I. W. Stewart, Phys. Rev. D 66, 014017 (2002).
  153. J. A. M. Vermaseren, A. Vogt, and S. Moch, Nucl. Phys. B724, 3 (2005).
  154. S. Moch, M. Rogal, and A. Vogt, Nucl. Phys. B790, 317 (2008).
  155. S. Moch, J. A. M. Vermaseren, and A. Vogt, Nucl. Phys. B813, 220 (2009).
  156. E. B. Zijlstra and W. L. van Neerven, Nucl. Phys. B417, 61 (1994); B426, 245(E) (1994); B773, 105(E) (2007); B501, 599(E) (1997).
  157. A. Vogt, S. Moch, M. Rogal, and J. A. M. Vermaseren, Nucl. Phys. B, Proc. Suppl. 183, 155 (2008).
  158. E. B. Zijlstra and W. L. van Neerven, Nucl. Phys. B383, 525 (1992).
  159. S. Moch and J. A. M. Vermaseren, Nucl. Phys. B573, 853 (2000).
  160. V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. 15, 438 (1972).
  161. L. N. Lipatov, Yad. Fiz. 20, 181 (1974).
  162. G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977).
  163. A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, and K. Schönwald, arXiv:2512.13508.
  164. J. G. Korner, D. Kreimer, and K. Schilcher, Z. Phys. C 54, 503 (1992).
  165. D. Kreimer, arXiv:hep-ph/9401354.
  166. Y. Matiounine, J. Smith, and W. L. van Neerven, Phys. Rev. D 58, 076002 (1998).
  167. P. J. Rijken and W. L. van Neerven, Nucl. Phys. B523, 245 (1998).
  168. P. Breitenlohner and D. Maison, Commun. Math. Phys. 52, 11 (1977).
  169. S. A. Larin and J. A. M. Vermaseren, Phys. Lett. B 259, 345 (1991).
  170. E. B. Zijlstra and W. L. van Neerven, Phys. Lett. B 297, 377 (1992).
  171. S. A. Larin, Phys. Lett. B 303, 113 (1993).
  172. D. Gutiérrez-Reyes, I. Scimemi, and A. A. Vladimirov, Phys. Lett. B 769, 84 (2017).
  173. S. L. Adler and W. A. Bardeen, Phys. Rev. 182, 1517 (1969).
  174. L. Chen, J. High Energy Phys. 05 (2025) 109.
  175. W. Vogelsang, Nucl. Phys. B475, 47 (1996).
  176. W. Vogelsang, Phys. Rev. D 54, 2023 (1996).
  177. See Supplemental Material at http://link.aps.org/supplemental/10.1103/b8gn-wlg2 for the NNLO twist-2 matching for polarized nucleon energy correlators.
  178. C. Duhr and F. Dulat, J. High Energy Phys. 08 (2019) 135.
  179. J. C. Collins, Phys. Rev. D 57, 3051 (1998); 61, 019902(E) (2000).
  180. I. Z. Rothstein and I. W. Stewart, J. High Energy Phys. 08 (2016) 025.
  181. M. D. Schwartz, K. Yan, and H. X. Zhu, Phys. Rev. D 97, 096017 (2018).
  182. L. N. Lipatov and M. I. Vyazovsky, Nucl. Phys. B597, 399 (2001).
  183. I. Moult, M. P. Solon, I. W. Stewart, and G. Vita, J. High Energy Phys. 02 (2018) 134.
  184. Y. V. Kovchegov and M. D. Sievert, Phys. Rev. D 99, 054032 (2019).
  185. X. Liu and H. X. Zhu, arXiv:2403.08874.
  186. D. Boer and P. J. Mulders, Phys. Rev. D 57, 5780 (1998).
  187. D. W. Sivers, Phys. Rev. D 41, 83 (1990).
  188. J. C. Collins, Phys. Lett. B 536, 43 (2002).
  189. S. J. Brodsky, D. S. Hwang, and I. Schmidt, Nucl. Phys. 642, 344 (2002).
  190. S. J. Brodsky, D. S. Hwang, and I. Schmidt, Phys. Lett. B 530, 99 (2002).
  191. S. J. Brodsky, D. S. Hwang, Y. V. Kovchegov, I. Schmidt, and M. D. Sievert, Phys. Rev. D 88, 014032 (2013).
  192. A. V. Belitsky, X. Ji, and F. Yuan, Nucl. Phys. B656, 165 (2003).
  193. J. Gao, H. T. Li, and Y. J. Zhu, Zenodo (2026), 10.5281/zenodo.18136246.

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Phys. Rev. D 113, 034028– Published 24 February, 2026
  • Received 16 October 2025
  • Accepted 22 January 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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