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

Parton model contributions as next-to-eikonal corrections to the dipole factorization of DIS and SIDIS at low xBj

Tolga Altinoluk1, Guillaume Beuf1, and Swaleha Mulani1,2,3

Phys. Rev. D 113, 034011 – Published 9 February, 2026

DOI: https://doi.org/10.1103/15qz-1krh

Abstract

We compute the next-to-eikonal (NEik) power corrections to inclusive deep inelastic scattering (DIS) and semi-inclusive deep inelastic scattering (SIDIS) at low x beyond dipole factorization, which represent the eikonal result. The analysis is restricted to contributions arising from t-channel quark exchanges, thereby probing the quark background field of the target. For a transversely polarized virtual photon, the NEik corrections to inclusive DIS are expressed in terms of quark and antiquark collinear parton distribution functions, while the corresponding corrections to SIDIS are formulated in terms of the unpolarized quark transverse-momentum-dependent distribution. In contrast, for a longitudinally polarized photon, the NEik corrections to both inclusive DIS and SIDIS vanish at lowest order in αs.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (120)

  1. J. C. Collins, D. E. Soper, and G. F. Sterman, Adv. Ser. Dir. High Energy Phys. 5, 1 (1989).
  2. V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. 15, 438 (1972).
  3. G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977).
  4. Y. L. Dokshitzer, Sov. Phys. JETP 46, 641 (1977).
  5. L. N. Lipatov, Sov. J. Nucl. Phys. 23, 338 (1976).
  6. E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, Sov. Phys. JETP 45, 199 (1977).
  7. I. I. Balitsky and L. N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978).
  8. F. D. Aaron et al. (H1 Collaboration and ZEUS Collaboration), J. High Energy Phys. 01 (2010) 109.
  9. F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, Annu. Rev. Nucl. Part. Sci. 60, 463 (2010).
  10. J. L. Albacete and C. Marquet, Prog. Part. Nucl. Phys. 76, 1 (2014).
  11. J.-P. Blaizot, Rep. Prog. Phys. 80, 032301 (2017).
  12. I. Balitsky, Nucl. Phys. B463, 99 (1996).
  13. Y. V. Kovchegov, Phys. Rev. D 60, 034008 (1999).
  14. Y. V. Kovchegov, Phys. Rev. D 61, 074018 (2000).
  15. J. Jalilian-Marian, A. Kovner, L. D. McLerran, and H. Weigert, Phys. Rev. D 55, 5414 (1997).
  16. J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, Nucl. Phys. B504, 415 (1997).
  17. J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, Phys. Rev. D 59, 014014 (1998).
  18. J. Jalilian-Marian, A. Kovner, and H. Weigert, Phys. Rev. D 59, 014015 (1998).
  19. A. Kovner, J. G. Milhano, and H. Weigert, Phys. Rev. D 62, 114005 (2000).
  20. H. Weigert, Nucl. Phys. A703, 823 (2002).
  21. E. Iancu, A. Leonidov, and L. D. McLerran, Nucl. Phys. A692, 583 (2001).
  22. E. Iancu, A. Leonidov, and L. D. McLerran, Phys. Lett. B 510, 133 (2001).
  23. E. Ferreiro, E. Iancu, A. Leonidov, and L. McLerran, Nucl. Phys. A703, 489 (2002).
  24. J. D. Bjorken, J. B. Kogut, and D. E. Soper, Phys. Rev. D 3, 1382 (1971).
  25. N. N. Nikolaev and B. G. Zakharov, Z. Phys. C 49, 607 (1991).
  26. N. Nikolaev and B. G. Zakharov, Z. Phys. C 53, 331 (1992).
  27. J. Collins, Foundations of Perturbative QCD, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology, Vol. 32 (Cambridge University Press, Cambridge, England, 2023).
  28. R. Boussarie et al., arXiv:2304.03302.
  29. C. Marquet, B.-W. Xiao, and F. Yuan, Phys. Lett. B 682, 207 (2009).
  30. S. Catani and F. Hautmann, Nucl. Phys. B427, 475 (1994).
  31. X.-d. Ji, J.-P. Ma, and F. Yuan, J. High Energy Phys. 07 (2005) 020.
  32. F. Dominguez, C. Marquet, B.-W. Xiao, and F. Yuan, Phys. Rev. D 83, 105005 (2011).
  33. B.-W. Xiao, F. Yuan, and J. Zhou, Nucl. Phys. B921, 104 (2017).
  34. M. Hentschinski, A. Kusina, K. Kutak, and M. Serino, Eur. Phys. J. C 78, 174 (2018).
  35. P. Caucal, M. G. Morales, E. Iancu, F. Salazar, and F. Yuan, arXiv:2503.16162.
  36. A. Accardi et al., Eur. Phys. J. A 52, 268 (2016).
  37. J. L. Abelleira Fernandez et al. (LHeC Study Group), J. Phys. G 39, 075001 (2012).
  38. I. Balitsky and G. A. Chirilli, Phys. Rev. D 83, 031502 (2011).
  39. I. Balitsky and G. A. Chirilli, Phys. Rev. D 87, 014013 (2013).
  40. G. Beuf, Phys. Rev. D 85, 034039 (2012).
  41. G. Beuf, Phys. Rev. D 94, 054016 (2016).
  42. G. Beuf, Phys. Rev. D 96, 074033 (2017).
  43. B. Ducloué, H. Hänninen, T. Lappi, and Y. Zhu, Phys. Rev. D 96, 094017 (2017).
  44. H. Hänninen, T. Lappi, and R. Paatelainen, Ann. Phys. (Amsterdam) 393, 358 (2018).
  45. G. Beuf, H. Hänninen, T. Lappi, and H. Mäntysaari, Phys. Rev. D 102, 074028 (2020).
  46. G. Beuf, T. Lappi, and R. Paatelainen, Phys. Rev. D 104, 056032 (2021).
  47. G. Beuf, T. Lappi, and R. Paatelainen, Phys. Rev. Lett. 129, 072001 (2022).
  48. G. Beuf, T. Lappi, and R. Paatelainen, Phys. Rev. D 106, 034013 (2022).
  49. H. Hänninen, H. Mäntysaari, R. Paatelainen, and J. Penttala, Phys. Rev. Lett. 130, 192301 (2023).
  50. C. Casuga, H. Hänninen, and H. Mäntysaari, Phys. Rev. D 112, 034003 (2025).
  51. P. Caucal, E. Ferrand, and F. Salazar, J. High Energy Phys. 05 (2024) 110.
  52. F. Bergabo and J. Jalilian-Marian, Phys. Rev. D 109, 074011 (2024).
  53. F. Bergabo and J. Jalilian-Marian, J. High Energy Phys. 01 (2023) 095.
  54. T. Altinoluk, F. Bergabo, J. Jalilian-Marian, C. Marquet, and Y. Shi, Phys. Rev. D 112, 054020 (2025).
  55. P. Caucal, E. Iancu, A. H. Mueller, and F. Yuan, Phys. Rev. Lett. 134, 061903 (2025).
  56. T. Altinoluk, N. Armesto, G. Beuf, M. Martínez, and C. A. Salgado, J. High Energy Phys. 07 (2014) 068.
  57. T. Altinoluk, N. Armesto, G. Beuf, and A. Moscoso, J. High Energy Phys. 01 (2016) 114.
  58. T. Altinoluk and A. Dumitru, Phys. Rev. D 94, 074032 (2016).
  59. P. Agostini, T. Altinoluk, and N. Armesto, Eur. Phys. J. C 79, 600 (2019).
  60. P. Agostini, T. Altinoluk, and N. Armesto, Eur. Phys. J. C 79, 790 (2019).
  61. P. Agostini, T. Altinoluk, N. Armesto, F. Dominguez, and J. G. Milhano, Eur. Phys. J. C 82, 1001 (2022).
  62. P. Agostini, T. Altinoluk, and N. Armesto, Phys. Lett. B 840, 137892 (2023).
  63. T. Altinoluk, G. Beuf, A. Czajka, and A. Tymowska, Phys. Rev. D 104, 014019 (2021).
  64. T. Altinoluk and G. Beuf, Phys. Rev. D 105, 074026 (2022).
  65. T. Altinoluk, G. Beuf, A. Czajka, and A. Tymowska, Phys. Rev. D 107, 074016 (2023).
  66. P. Agostini, T. Altinoluk, and N. Armesto, J. High Energy Phys. 07 (2024) 137.
  67. T. Altinoluk, N. Armesto, and G. Beuf, Phys. Rev. D 108, 074023 (2023).
  68. T. Altinoluk, G. Beuf, A. Czajka, and C. Marquet, Phys. Rev. D 111, 014010 (2025).
  69. T. Altinoluk, G. Beuf, and S. Mulani, Phys. Rev. D 111, 034028 (2025).
  70. T. Altinoluk, G. Beuf, E. Blanco, and S. Mulani, J. High Energy Phys. 06 (2025) 097.
  71. Y. V. Kovchegov, D. Pitonyak, and M. D. Sievert, J. High Energy Phys. 01 (2016) 072; 10 (2016) 148(E).
  72. Y. V. Kovchegov, D. Pitonyak, and M. D. Sievert, Phys. Rev. D 95, 014033 (2017).
  73. Y. V. Kovchegov, D. Pitonyak, and M. D. Sievert, Phys. Rev. Lett. 118, 052001 (2017).
  74. Y. V. Kovchegov, D. Pitonyak, and M. D. Sievert, Phys. Lett. B 772, 136 (2017).
  75. Y. V. Kovchegov, D. Pitonyak, and M. D. Sievert, J. High Energy Phys. 10 (2017) 198.
  76. Y. V. Kovchegov and M. D. Sievert, Phys. Rev. D 99, 054032 (2019).
  77. Y. V. Kovchegov and M. D. Sievert, Phys. Rev. D 99, 054033 (2019).
  78. Y. V. Kovchegov and M. G. Santiago, Phys. Rev. D 102, 014022 (2020).
  79. Y. V. Kovchegov and Y. Tawabutr, J. High Energy Phys. 08 (2020) 014.
  80. D. Adamiak, Y. V. Kovchegov, W. Melnitchouk, D. Pitonyak, N. Sato, and M. D. Sievert (Jefferson Lab Angular Momentum Collaboration), Phys. Rev. D 104, L031501 (2021).
  81. Y. V. Kovchegov, A. Tarasov, and Y. Tawabutr, J. High Energy Phys. 03 (2022) 184.
  82. Y. V. Kovchegov and M. G. Santiago, J. High Energy Phys. 11 (2021) 200; 09 (2022) 186(E).
  83. F. Cougoulic, Y. V. Kovchegov, A. Tarasov, and Y. Tawabutr, J. High Energy Phys. 07 (2022) 095.
  84. Y. V. Kovchegov and M. G. Santiago, J. High Energy Phys. 11 (2022) 098.
  85. J. Borden and Y. V. Kovchegov, Phys. Rev. D 108, 014001 (2023).
  86. Y. V. Kovchegov and M. Li, J. High Energy Phys. 05 (2024) 177.
  87. J. Borden, Y. V. Kovchegov, and M. Li, J. High Energy Phys. 09 (2024) 037.
  88. D. Adamiak, N. Baldonado, Y. V. Kovchegov, M. Li, W. Melnitchouk, D. Pitonyak, N. Sato, M. D. Sievert, A. Tarasov, and Y. Tawabutr, Phys. Rev. D 112, 094032 (2025).
  89. Y. V. Kovchegov and M. Li, J. High Energy Phys. 08 (2025) 206.
  90. J. Borden and Y. V. Kovchegov, arXiv:2508.00195.
  91. F. Cougoulic and Y. V. Kovchegov, Phys. Rev. D 100, 114020 (2019).
  92. F. Cougoulic and Y. V. Kovchegov, Nucl. Phys. A1004, 122051 (2020).
  93. I. Balitsky and A. Tarasov, J. High Energy Phys. 10 (2015) 017.
  94. I. Balitsky and A. Tarasov, J. High Energy Phys. 06 (2016) 164.
  95. I. Balitsky and A. Tarasov, J. High Energy Phys. 07 (2017) 095.
  96. R. Boussarie and Y. Mehtar-Tani, Phys. Lett. B 831, 137125 (2022).
  97. R. Boussarie and Y. Mehtar-Tani, J. High Energy Phys. 07 (2022) 080.
  98. R. Boussarie and Y. Mehtar-Tani, J. High Energy Phys. 10 (2024) 056.
  99. G. A. Chirilli, J. High Energy Phys. 01 (2019) 118.
  100. G. A. Chirilli, J. High Energy Phys. 06 (2021) 096.
  101. M. Li, J. High Energy Phys. 07 (2023) 158.
  102. M. Li, Phys. Rev. Lett. 133, 021902 (2024).
  103. M. Li, Phys. Rev. D 111, 034027 (2025).
  104. J. Jalilian-Marian, Phys. Rev. D 96, 074020 (2017).
  105. J. Jalilian-Marian, Phys. Rev. D 99, 014043 (2019).
  106. J. Jalilian-Marian, Phys. Rev. D 102, 014008 (2020).
  107. Y. Hatta, Y. Nakagawa, F. Yuan, Y. Zhao, and B. Xiao, Phys. Rev. D 95, 114032 (2017).
  108. Y. V. Kovchegov, J. High Energy Phys. 03 (2019) 174.
  109. R. Boussarie, Y. Hatta, and F. Yuan, Phys. Lett. B 797, 134817 (2019).
  110. Y. V. Kovchegov and B. Manley, J. High Energy Phys. 02 (2024) 060.
  111. Y. V. Kovchegov and B. Manley, Phys. Rev. D 111, 054017 (2025).
  112. L. D. McLerran and R. Venugopalan, Phys. Rev. D 49, 2233 (1994).
  113. L. D. McLerran and R. Venugopalan, Phys. Rev. D 49, 3352 (1994).
  114. L. D. McLerran and R. Venugopalan, Phys. Rev. D 50, 2225 (1994).
  115. A. V. Belitsky, X. Ji, and F. Yuan, Nucl. Phys. B656, 165 (2003).
  116. C. Marquet, E. Petreska, and C. Roiesnel, J. High Energy Phys. 10 (2016) 065.
  117. T. Altinoluk and R. Boussarie, J. High Energy Phys. 10 (2019) 208.
  118. H. Mäntysaari and P. Zurita, Phys. Rev. D 98, 036002 (2018).
  119. E. Iancu, A. H. Mueller, D. N. Triantafyllopoulos, and S. Y. Wei, J. High Energy Phys. 07 (2021) 196.
  120. T. Altinoluk, J. Jalilian-Marian, and C. Marquet, Phys. Rev. D 110, 094056 (2024).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation