- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

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
  • Access by Xinjiang University

Dipole-dipole scattering: Summing large Pomeron loops in the nonlinear evolution with a leading twist kernel

Eugene Levin*

  • *Contact author: leving@tauex.tau.ac.il
PDF

Phys. Rev. D 114, 036005 – Published 4 August, 2026

DOI: https://doi.org/10.1103/n1jz-dfyx

Abstract

It is shown in this paper that the QCD equations for dipole density have the natural solution: the “fan” diagrams of the Pomeron calculus. We found the dipole densities comparing the analytic solution to the Balitsky-Kovchegov (BK) equation for the simplified leading twist kernel with the t channel unitarity. Using these densities we calculate the contributions of large Pomeron loops to dipole-dipole scattering at high energies. Applying the Abramovsky, Gribov and Kancheli cutting rules we found that the produced gluons are distributed accordingly the Koba, Nielsen and Olesen (KNO) law which leads to the entropy SE=ln(xG(x,Q2)) in an agreement with Kharzeev-Levin predictions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (72)

  1. E. Levin, Phys. Rev. D 107, 054012 (2023).
  2. E. Levin, Phys. Rev. D 110, 116021 (2024).
  3. E. Levin, Phys. Rev. D 111, 096004 (2025).
  4. E. Levin, Phys. Rev. D 112, 054033 (2025).
  5. V. S. Fadin, E. A. Kuraev, and L. N. Lipatov, Phys. Lett. 60B, 50 (1975); E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, Zh. Eksp. Teor. Fiz. 72, 377 (1977) [Sov. Phys. JETP 45, 199 (1977)]; I. I. Balitsky and L. N. Lipatov, Yad. Fiz. 28, 1597 (1978) [Sov. J. Nucl. Phys. 28, 822 (1978)].
  6. I. Balitsky, Phys. Rev. D 60, 014020 (1999).
  7. Y. V. Kovchegov, Phys. Rev. D 60, 034008 (1999).
  8. E. Levin and K. Tuchin, Nucl. Phys. B573, 833 (2000); A691, 779 (2001). 693, 787 (2001).
  9. Yuri V. Kovchegov and Eugene Levin, Quantum Chromodynamics at High Energies, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology (Cambridge University Press, Cambridge, England, 2012).
  10. A. Kovner, E. Levin, and M. Lublinsky, J. High Energy Phys. 10 (2024) 127.
  11. A. H. Mueller and G. P. Salam, Nucl. Phys. B475, 293 (1996) G. P. Salam, B461, 512 (1996).
  12. E. Levin and M. Lublinsky, Phys. Lett. B 607, 131 (2005); Nucl. Phys. A763, 172 (2005).
  13. L. N. Lipatov, Zh. Eksp. Teor. Fiz. 90, 1536 (1986) [Sov. Phys. JETP 63, 904 (1986)].
  14. Y. V. Kovchegov, Phys. Rev. D 72, 094009 (2005).
  15. E. Levin, Nucl. Phys. A763, 140 (2005).
  16. P. Rembiesa and A. M. Stasto, Nucl. Phys. B725, 251 (2005).
  17. A. Kovner and M. Lublinsky, Nucl. Phys. A767, 171 (2006).
  18. A. I. Shoshi and B. W. Xiao, Phys. Rev. D 73, 094014 (2006).
  19. M. Kozlov and E. Levin, Nucl. Phys. A779, 142 (2006).
  20. N. Armesto, S. Bondarenko, J. G. Milhano, and P. Quiroga, J. High Energy Phys. 05 (2008) 103.
  21. E. Levin and A. Prygarin, Eur. Phys. J. C 53, 385 (2008).
  22. A. D. Le, A. H. Mueller, and S. Munier, Phys. Rev. D 104, 034026 (2021).
  23. E. Levin, Phys. Rev. D 104, 056025 (2021).
  24. E. Iancu and A. Mueller, Nucl. Phys. A730, 494 (2004).
  25. E. Iancu and A. Mueller, Nucl. Phys. A730, 460 (2004).
  26. A. H. Mueller and D. N. Triantafyllopoulos, Nucl. Phys. B640, 331 (2002).
  27. E. Iancu, K. Itakura, and L. McLerran, Nucl. Phys. A708, 327 (2002).
  28. L. N. Lipatov, Phys. Rep. 286, 131 (1997).
  29. L. N. Lipatov, Nucl. Phys. B365, 614 (1991); B452, 369 (1995); R. Kirschner, L. N. Lipatov, and L. Szymanowski, B425, 579 (1994); Phys. Rev. D 51, 838 (1995).
  30. L. V. Gribov, E. M. Levin, and M. G. Ryskin, Phys. Rep. 100, 1 (1983).
  31. E. M. Levin and M. G. Ryskin, Phys. Rep. 189, 267 (1990).
  32. A. H. Mueller and J. Qiu, Nucl. Phys. B268, 427 (1986).
  33. A. H. Mueller, Nucl. Phys. B415, 373 (1994); B437, 107 (1995); A. H. Mueller and B. Patel, B425, 471 (1994).
  34. G. P. Salam, Nucl. Phys. B461, 512 (1996).
  35. H. Navelet and R. B. Peschanski, Nucl. Phys. B507, 353 (1997).
  36. J. Bartels, Z. Phys. C 60, 471 (1993); J. Bartels and M. Wusthoff, 66, 157 (1995); J. Bartels and C. Ewerz, J. High Energy Phys. 09 (1999) 026; C. Ewerz, 04 (2001) 031.
  37. J. Bartels, Nucl. Phys. B175, 365 (1980); J. Kwiecinski and M. Praszalowicz, Phys. Lett. 94B, 413 (1980).
  38. L. McLerran and R. Venugopalan, Phys. Rev. D 49, 2233 (1994); 50, 2225 (1994); 59, 094019 (1999).
  39. Y. V. Kovchegov and E. Levin, Nucl. Phys. B577, 221 (2000).
  40. M. A. Braun, Eur. Phys. J. C 16, 337 (2000); M. A. Braun and G. P. Vacca, 6, 147 (1999); J. Bartels, M. Braun, and G. P. Vacca, 40, 419 (2005); J. Bartels, L. N. Lipatov, and G. P. Vacca, Nucl. Phys. B706, 391 (2005).
  41. M. A. Braun, Phys. Lett. B 483, 115 (2000). Eur. Phys. J. C 33, 113 (2004); Phys. Lett. B 632, 297 (2006).
  42. A. Kovner and M. Lublinsky, J. High Energy Phys. 02 (2007) 058.
  43. J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, Nucl. Phys. B504, 415 (1997).
  44. J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, Phys. Rev. D 59, 014014 (1998).
  45. A. Kovner, J. G. Milhano, and H. Weigert, Phys. Rev. D 62, 114005 (2000).
  46. E. Iancu, A. Leonidov, and L. D. McLerran, Nucl. Phys. A692, 583 (2001).
  47. E. Iancu, A. Leonidov, and L. D. McLerran, Phys. Lett. B 510, 133 (2001).
  48. E. Ferreiro, E. Iancu, A. Leonidov, and L. McLerran, Nucl. Phys. A703, 489 (2002).
  49. H. Weigert, Nucl. Phys. A703, 823 (2002).
  50. A. Kovner and J. G. Milhano, Phys. Rev. D 61, 014012 (2000).
  51. T. Altinoluk, A. Kovner, E. Levin, and M. Lublinsky, J. High Energy Phys. 04 (2014) 075.
  52. A. Kovner and M. Lublinsky, Phys. Rev. D 71, 085004 (2005).
  53. A. Kovner and M. Lublinsky, Phys. Rev. Lett. 94, 181603 (2005).
  54. I. Balitsky, Phys. Rev. D 72, 074027 (2005).
  55. Y. Hatta, E. Iancu, L. McLerran, A. Stasto, and D. N. Triantafyllopoulos, Nucl. Phys. A764, 423 (2006).
  56. A. Kovner, M. Lublinsky, and U. Wiedemann, J. High Energy Phys. 06 (2007) 075; T. Altinoluk, A. Kovner, M. Lublinsky, and J. Peressutti, 03 (2009) 109.
  57. A. Kovner, E. Levin, M. Li, and M. Lublinsky, J. High Energy Phys. 09 (2020) 199.
  58. A. Kovner, E. Levin, M. Li, and M. Lublinsky, J. High Energy Phys. 10 (2020) 185.
  59. A. Kovner and M. Lublinsky, Nucl. Phys. A767, 171 (2006).
  60. A. Kormilitzin, E. Levin, and A. Prygarin, Nucl. Phys. A813, 1 (2008); E. Levin and A. Prygarin, Eur. Phys. J. C 53, 385 (2008).
  61. S. Munier and R. B. Peschanski, Phys. Rev. D 69, 034008 (2004); Phys. Rev. Lett. 91, 232001 (2003).
  62. I. Gradstein and I. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic Press, London, 1994).
  63. J. Bartels and E. Levin, Nucl. Phys. B387, 617 (1992);  A. M. Stasto, K. J. Golec-Biernat, and J. Kwiecinski, Phys. Rev. Lett. 86, 596 (2001); L. McLerran and M. Praszalowicz, Acta Phys. Pol. B 42, 99 (2011).
  64. D. E. Kharzeev and E. M. Levin, Phys. Rev. D 95, 114008 (2017).
  65. A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations (Chapman and Hall/CRC Press, Boca Raton, New York, London, Tokyo, 2004).
  66. Bruce Shawyer and Bruce Watson, Borel’s Method of Summability, Theory and Application (Clarendon Press, Oxford, 1994); Ovidiu Costin, Asymptotics and Borel Summability, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics (CRC Press, Taylor & Frencis Group, Boca Raton, London, New York, 2009).
  67. E. Levin, arXiv:2403.10364.
  68. V. A. Abramovsky, V. N. Gribov, and O. V. Kancheli, Yad. Fiz. 18, 595 (1973) [Sov. J. Nucl. Phys. 18, 308 (1974))].
  69. A. M. Polyakov, Zh. Eksp. Teor. Fiz. 59, 542 (1970); Z. Koba, H. B. Nielsen, and P. Olesen, Nucl. Phys. B40, 317 (1972); Z. Koba, in Proceedings of the 1973 CERN School of Physics, CERN Yellow Report CERN-73-12 (1973) p. 171.
  70. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (United States Department of Commerce, National Bureau of Standards, 1964).
  71. E. Levin, Eur. Phys. J. C 84, 662 (2024).
  72. E. Levin, Phys. Rev. D 111, 016019 (2025).

Outline

Information

Phys. Rev. D 114, 036005– Published 4 August, 2026
  • Received 11 February 2026
  • Accepted 16 July 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation