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  • Open Access

Rotating the Color Glass Condensate

Renaud Boussarie1,*, Paul Caucal2,†, Piotr Korcyl3,‡, and Yacine Mehtar-Tani4,§

  • *Contact author: renaud.boussarie@polytechnique.edu
  • †Contact author: caucal@subatech.in2p3.fr
  • ‡Contact author: piotr.korcyl@uj.edu.pl
  • §Contact author: mehtartani@bnl.gov
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Phys. Rev. Lett. 137, 121903 – Published 15 September, 2026

DOI: https://doi.org/10.1103/k59p-5nb6

Abstract

High-energy QCD evolution beyond leading order suffers from instabilities driven by large collinear logarithms. We present a framework, consistent with the standard high-energy operator product expansion, that restores perturbative stability order by order. The method involves a change of basis in the space of high-energy operators, which modifies both the evolution kernel and the coefficient functions while leaving physical observables invariant. Within this factorization scheme, we derive a next-to-leading-order renormalization-group equation whose numerical solution exhibits stable evolution up to large rapidities, paving the way for a systematic framework for precision studies of gluon saturation at current and future colliders.

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Physics Subject Headings (PhySH)

See Also

Collinear structure of nonlinear small-x evolution

Renaud Boussarie, Paul Caucal, and Yacine Mehtar-Tani
Phys. Rev. D 114, 054029 (2026)

Article Text

Supplemental Material

References (72)

  1. R. P. Feynman, Phys. Rev. Lett. 23, 1415 (1969).
  2. J. D. Bjorken and E. A. Paschos, Phys. Rev. 185, 1975 (1969).
  3. L. V. Gribov, E. M. Levin, and M. G. Ryskin, Phys. Rep. 100, 1 (1983).
  4. A. H. Mueller and J.-w. Qiu, Nucl. Phys. B268, 427 (1986).
  5. L. D. McLerran and R. Venugopalan, Phys. Rev. D 49, 2233 (1994).
  6. A. Morreale and F. Salazar, Universe 7, 312 (2021).
  7. R. Abdul Khalek et al., Nucl. Phys. A1026, 122447 (2022).
  8. F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, Annu. Rev. Nucl. Part. Sci. 60, 463 (2010).
  9. I. Balitsky, Nucl. Phys. B463, 99 (1996).
  10. Y. V. Kovchegov, Phys. Rev. D 60, 034008 (1999).
  11. J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, Nucl. Phys. B504, 415 (1997).
  12. J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, Phys. Rev. D 59, 014014 (1998).
  13. A. Kovner, J. G. Milhano, and H. Weigert, Phys. Rev. D 62, 114005 (2000).
  14. E. Iancu, A. Leonidov, and L. D. McLerran, Nucl. Phys. A692, 583 (2001).
  15. E. Iancu, A. Leonidov, and L. D. McLerran, Phys. Lett. B 510, 133 (2001).
  16. E. Ferreiro, E. Iancu, A. Leonidov, and L. McLerran, Nucl. Phys. A703, 489 (2002).
  17. I. Balitsky and G. A. Chirilli, Phys. Rev. D 77, 014019 (2008).
  18. I. Balitsky and G. A. Chirilli, Nucl. Phys. B822, 45 (2009).
  19. T. Lappi and H. Mäntysaari, Phys. Rev. D 91, 074016 (2015).
  20. B. Andersson, G. Gustafson, and J. Samuelsson, Nucl. Phys. B467, 443 (1996).
  21. J. Kwiecinski, A. D. Martin, and P. J. Sutton, Z. Phys. C 71, 585 (1996).
  22. J. Kwiecinski, A. D. Martin, and A. M. Stasto, Phys. Rev. D 56, 3991 (1997).
  23. G. P. Salam, J. High Energy Phys. 07 (1998) 019.
  24. M. Ciafaloni and D. Colferai, Phys. Lett. B 452, 372 (1999).
  25. M. Ciafaloni, D. Colferai, and G. P. Salam, Phys. Rev. D 60, 114036 (1999).
  26. M. Ciafaloni, D. Colferai, G. P. Salam, and A. M. Stasto, Phys. Rev. D 68, 114003 (2003).
  27. A. Sabio Vera, Nucl. Phys. B722, 65 (2005).
  28. L. Motyka and A. M. Stasto, Phys. Rev. D 79, 085016 (2009).
  29. L. N. Lipatov, Sov. J. Nucl. Phys. 23, 338 (1976).
  30. E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, Sov. Phys. JETP 45, 199 (1977).
  31. I. I. Balitsky and L. N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978).
  32. V. S. Fadin and L. N. Lipatov, Phys. Lett. B 429, 127 (1998).
  33. M. Ciafaloni and G. Camici, Phys. Lett. B 430, 349 (1998).
  34. V. S. Fadin, R. Fiore, and A. Papa, Nucl. Phys. B769, 108 (2007).
  35. V. S. Fadin, R. Fiore, and A. Papa, Phys. Lett. B 647, 179 (2007).
  36. V. S. Fadin, R. Fiore, A. V. Grabovsky, and A. Papa, Nucl. Phys. B784, 49 (2007).
  37. G. Beuf, Phys. Rev. D 89, 074039 (2014).
  38. E. Iancu, J. D. Madrigal, A. H. Mueller, G. Soyez, and D. N. Triantafyllopoulos, Phys. Lett. B 744, 293 (2015).

  39. Also referred to as the next-to-leading logarithmic (NLL) accuracy.

  40. B. Ducloué, E. Iancu, A. H. Mueller, G. Soyez, and D. N. Triantafyllopoulos, J. High Energy Phys. 04 (2019) 081.
  41. T. Altinoluk, G. Beuf, and J. Penttala, J. High Energy Phys. 04 (2026) 125.
  42. T. Lappi and H. Mäntysaari, Phys. Rev. D 93, 094004 (2016).
  43. J. Cepila, J. G. Contreras, M. Matas, and M. Vaculciak, Phys. Rev. D 111, 096015 (2025).
  44. R. Boussarie, P. Caucal, and Y. Mehtar-Tani, companion paper, Phys. Rev. D 114, 054029 (2026).
  45. T. Regge, Nuovo Cimento 14, 951 (1959).

  46. An analog formulation was proposed in the context of jet physics [47, 48].

  47. T. Becher, M. Neubert, L. Rothen, and D. Y. Shao, J. High Energy Phys. 11 (2016) 019; 05 (2017) 154(E).
  48. S. Caron-Huot, J. High Energy Phys. 03 (2018) 036.
  49. M. Doi, J. Phys. A 9, 1465 (1976).
  50. L. Peliti, J. Phys. France 46, 1469 (1985).
  51. Y. V. Kovchegov and E. Levin, Quantum Chromodynamics at High Energy (Oxford University Press, New York, 2013), Vol. 33.
  52. I. Balitsky and G. A. Chirilli, Phys. Rev. D 83, 031502 (2011).
  53. G. Beuf, Phys. Rev. D 85, 034039 (2012).
  54. G. Beuf, Phys. Rev. D 94, 054016 (2016).
  55. G. Beuf, Phys. Rev. D 96, 074033 (2017).
  56. H. Hänninen, T. Lappi, and R. Paatelainen, Ann. Phys. (Amsterdam) 393, 358 (2018).
  57. G. Beuf, T. Lappi, and R. Paatelainen, Phys. Rev. D 104, 056032 (2021).
  58. G. Beuf, T. Lappi, and R. Paatelainen, Phys. Rev. D 106, 034013 (2022).
  59. G. Beuf, T. Lappi, and R. Paatelainen, Phys. Rev. Lett. 129, 072001 (2022).
  60. See Supplemental Material at http://link.aps.org/supplemental/10.1103/k59p-5nb6 for the explicit computation of the commutator [KLO,LLO] in Sec. 1; additional details of the techniques used in this calculation in Sec. 2; and a brief discussion of the most relevant tests in Sec. 3.
  61. E. Iancu, J. D. Madrigal, A. H. Mueller, G. Soyez, and D. N. Triantafyllopoulos, Phys. Lett. B 750, 643 (2015).
  62. A. Kovner, M. Lublinsky, V. V. Skokov, and Z. Zhao, J. High Energy Phys. 07 (2024) 148.
  63. L. McLerran and R. Venugopalan, Phys. Rev. D 49, 2233 (1994).
  64. L. McLerran and R. Venugopalan, Phys. Rev. D 49, 3352 (1994).
  65. I. Balitsky, Phys. Rev. D 75, 014001 (2007).
  66. E. Iancu, J. Madrigal, A. Mueller, G. Soyez, and D. Triantafyllopoulos, Phys. Lett. B 750, 643 (2015).
  67. K. J. Golec-Biernat and M. Wusthoff, Phys. Rev. D 59, 014017 (1998).
  68. S. J. Brodsky, G. P. Lepage, and P. B. Mackenzie, Phys. Rev. D 28, 228 (1983).
  69. I. Balitsky, Phys. Rev. D 75, 014001 (2007).
  70. H. Abramowicz et al. (H1 Collaboration and ZEUS Collaboration), Eur. Phys. J. C 75, 580 (2015).
  71. S. Caron-Huot and M. Herranen, J. High Energy Phys. 02 (2018) 058.
  72. G. Brunello, S. Caron-Huot, G. Crisanti, M. Giroux, and S. Smith, J. High Energy Phys. 11 (2025) 055.

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Phys. Rev. Lett. 137, 121903– Published 15 September, 2026
  • Received 21 November 2025
  • Revised 12 June 2026
  • Accepted 9 July 2026
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