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)/d.o.f obtained from the fits are similar. Both the analytic and numerical calculations imply that the Balitsky-Kovchegov equation at the miming 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.