# Running coupling effect in next-to-leading order Balitsky-Kovchegov evolution equations

• Balitsky-Kovchegov equations in projectile and target rapidity representations are analytically solved for fixed and running coupling cases in the saturation domain. Interestingly, we find that the respective analytic S-matrices in the two rapidity representations have almost the same rapidity dependence in the exponent in the running coupling case, which provides a method to explain why the equally good fits to HERA data were obtained when using three different Balitsky-Kovchegov equations formulated in the two representations. To test the analytic outcomes, we solve the Balitsky-Kovchegov equations and numerically compute the ratios between these dipole amplitudes in the saturation region. The ratios are close to one, which confirms the analytic results. Moreover, the running coupling, collinearly-improved, and extended full collinearly-improved Balitsky-Kovchegov equations are used to fit the HERA data. We find that all of them provide high quality descriptions of the data, and the $\chi^2/\mathrm{d.o.f}$ obtained from the fits are similar. Both the analytic and numerical calculations imply that the Balitsky-Kovchegov equation at the running coupling level is robust and has a sufficiently strong predictive power at HERA energies; however, higher order corrections could be significant for future experiments, such as those at the EIC or LHeC.
•  [1] F. Gelis, E. Iancu, J. Jalilian-Marian et al., Ann. Rev. Nucl. Part. Sci. 60, 463 (2010) doi: 10.1146/annurev.nucl.010909.083629 [2] J. Jalilian-Marian, A. Kovner, A. Leonidov et al., Nucl. Phys. B 504, 415 (1997) [3] J. Jalilian-Marian, A. Kovner, A. Leonidov et al., Phys. Rev. D 59, 014014 (1998) [4] E. Iancu, A. Leonidov, and L. McLerran, Nucl. Phys. A 692, 583 (2001) [5] E. Ferreiro, E. Iancu, A. Leonidov et al., Nucl. Phys. 703, 489 (2002) [6] I. Balitsky, Nucl. Phys. B 463, 99 (1996) [7] Y. Kovchegov, Phys. Rev. D 60, 034008 (1999); ibid. 61, 074018 (1999) [8] A. Stasto, K. Golec-Biernat, and J. Kwiecinski, Phys. Rev. Lett. 86, 596 (2001) doi: 10.1103/PhysRevLett.86.596 [9] E Iancu, K. Itakura, and S. Munier, Phys. Lett. B 590, 199 (2004) [10] M. Kozlov, A. Shoshi, and W. Xiang, JHEP 0710, 020 (2007) [11] D. Kkarzeev and E. Levin, Phys. Lett. B 523, 79 (2001) [12] A. Dumitru, A. Hayashigaki, and J. Jalilian-Marian, Nucl. Phys. A 765, 464 (2006) [13] J. Albacete, N. Armesto, J. Milhano et al., Phys. Rev. D 80, 034031 (2009) [14] J. Albacete, N. Armesto, J. Milhano et al., Eur. Phys. J. C 71, 1705 (2011) [15] G. Chirilli, B. Xiao, and F. Yuan, Phys. Rev. Lett. 108, 122301 (2012) doi: 10.1103/PhysRevLett.108.122301 [16] A. Stasto, B. Xiao, and D. Zaslavsky, Phys. Rev. Lett. 112, 012302 (2014) doi: 10.1103/PhysRevLett.112.012302 [17] B. Ducloue, T. Lappi, and Y. Zhu, Phys. Rev. D 95, 114007 (2017) [18] E. Levin and A. Rezaeian, Phys. Rev. D 82, 074016 (2010) [19] J. Albacete and C. Marquet, Phys. Lett. B 687, 174 (2010) [20] I. Balitsky, Phys. Rev. D 75, 014001 (2007) [21] Y. Kovchegov and H. Weigert, Nucl. Phys. A 784, 188 (2007) [22] J. Albacete and Yu.V. Kovchegov, Phys. Rev. D 75, 125021 (2007) [23] I. Balitsky and G. Chirilli, Phys. Rev. D 77, 014019 (2008) [24] G. Beuf, Phys. Rev. D 89, 074039 (2014) [25] E. Iancu, J. Madrigal, A. Mueller et al., Phys. Lett. B 744, 293 (2015) [26] B. Ducloue, E. Iancu, A. Mueller et al., JHEP 1904, 081 (2019) [27] B. Ducloue, E. Iancu, G. Soyez et al., Phys. Lett. B 803, 135305 (2020) [28] D. Zheng and J. Zhou, JHEP 1911, 177 (2019) [29] W. Xiang, Y. Cai, M. Wang et al., Phys. Rev. D 104, 016018 (2021) [30] G. Beuf, H. Hanninen, T. Lappi et al., Phys. Rev. D 102, 074028 (2020) [31] W. Xiang, Phys. Rev. D 79, 014012 (2009) [32] W. Xiang, S. Cai, and D. Zhou, Phys. Rev. D 95, 116009 (2017) [33] W. Xiang, Y. Cai, M. Wang et al., Phys. Rev. D 101, 076005 (2020) [34] A. Mueller, hep-ph/0111244. [35] E. Levin and K. Tuchin, Nucl. Phys. B 573, 83 (2000) [36] C. Contreras, E. Levin, R. Meneses et al., Phys. Rev. D 94, 114028 (2016) [37] Y. Cai, W. Xiang, M. Wang et al., Chin. Phys. C 44, 074110 (2020) [38] W. Xiang, M. Wang, Y. Cai et al., Chin. Phys. C 45, 014103 (2021) [39] E. Iancu, K. Itakura, and L. McLerran, Nucl. Phys. A 708, 327 (2002) [40] T. Lappi and H. Mäntysaari, Phys. Rev. D 91, 074016 (2015) [41] T. Lappi and H. Mäntysaari, Phys. Rev. D 93, 094004 (2016) [42] E. Iancu, J. Madrigal, A. Mueller et al., Phys. Lett. B 750, 643 (2015) [43] L. McLerran and R. Venugopalan, Phys. Rev. D 49, 2233 (1994) [44] F. Aaron et al., JHEP 1001, 109 (2010) [45] K. Golec-Biernat and M. Wusthoff, Phys. Rev. D 59, 014017 (1999)

Figures(3) / Tables(1)

Get Citation
Wenchang Xiang, Mengliang Wang, Yanbing Cai and Daicui Zhou. On the running coupling effect in next-to-leading order Balitsky-Kovchegov evolution equation[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac4ee9
Wenchang Xiang, Mengliang Wang, Yanbing Cai and Daicui Zhou. On the running coupling effect in next-to-leading order Balitsky-Kovchegov evolution equation[J]. Chinese Physics C.
Milestone
Article Metric

Article Views(265)
Cited by(0)
Policy on re-use
To reuse of subscription content published by CPC, the users need to request permission from CPC, unless the content was published under an Open Access license which automatically permits that type of reuse.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

Title:
Email:

## Running coupling effect in next-to-leading order Balitsky-Kovchegov evolution equations

###### Corresponding author: Daicui Zhou, dczhou@mail.ccnu.edu.cn
• 1. Guizhou Key Laboratory of Big Data Statistic Analysis, and Guizhou Key Laboratory in Physics and Related Areas, Guizhou University of Finance and Economics, Guiyang 550025, China
• 2. Key Laboratory of Quark and Lepton Physics (MOE), and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China

Abstract: Balitsky-Kovchegov equations in projectile and target rapidity representations are analytically solved for fixed and running coupling cases in the saturation domain. Interestingly, we find that the respective analytic S-matrices in the two rapidity representations have almost the same rapidity dependence in the exponent in the running coupling case, which provides a method to explain why the equally good fits to HERA data were obtained when using three different Balitsky-Kovchegov equations formulated in the two representations. To test the analytic outcomes, we solve the Balitsky-Kovchegov equations and numerically compute the ratios between these dipole amplitudes in the saturation region. The ratios are close to one, which confirms the analytic results. Moreover, the running coupling, collinearly-improved, and extended full collinearly-improved Balitsky-Kovchegov equations are used to fit the HERA data. We find that all of them provide high quality descriptions of the data, and the $\chi^2/\mathrm{d.o.f}$ obtained from the fits are similar. Both the analytic and numerical calculations imply that the Balitsky-Kovchegov equation at the running coupling level is robust and has a sufficiently strong predictive power at HERA energies; however, higher order corrections could be significant for future experiments, such as those at the EIC or LHeC.

Reference (45)

/