We calculate the rare fluctuations of the S-matrix on top of the full next-to-leading order corrections in the center of mass frame. The relevant result in the saturation regime shows that the exponential factor of the S-matrix is √ 2 as large as the result which emerges when the rare fluctuation effects are taken into account. We find that the factor of √ 2 change of the exponential factor is induced by the gluon loop corrections which compensate part of rapidity decrease of the S-matrix made by quark loops and lead to the rare fluctuations becoming important again. To ensure the relevant results of the S-matrix are independent of the frame choice, the rare fluctuations of the S-matrix are also derived in a general frame. It is found that all the results are consistent with each other in both frames. * Electronic address:

We analytically solve the Sudakov suppressed Balitsky-Kovchegov evolution equation with fixed and running coupling constants in the saturation region. The analytic solution of the S-matrix shows that the rapidity dependence of the solution with the fixed coupling constant is replaced by the dependence in the smallest dipole running coupling case, as opposed to obeying the law found in our previous publication, where all the solutions of the next-to-leading order evolution equations comply with rapidity dependence once the QCD coupling is switched from the fixed coupling to the smallest dipole running coupling prescription. This finding indicates that the corrections of the sub-leading double logarithms in the Sudakov suppressed evolution equation are significant, which compensate for a part of the evolution decrease of the dipole amplitude introduced by the running coupling effect. To test the analytic findings, we calculate the numerical solutions of the Sudakov suppressed evolution equation, and the numerical results confirm the analytic outcomes. Moreover, we use the numerical solutions of the evolution equationto fit the HERA data. This demonstrates that the Sudakov suppressed evolution equation can achieve a good quality fit to the data.

We extend the hotspot model to include the virtuality dependence and use it to study the exclusive and dissociative production combined with the dipole amplitude in the target rapidity representation. We determined that virtuality takes effect on a number of hotspots, thus providing a better description of the production data at HERA. The collinear improved Balitsky-Kovchegove equation in the target rapidity representation is numerically solved and used to fit the experimental data with a series of hotspot sizes. We infer that virtuality significantly influences the number and size of hotspots. The expression resulting from the fit with the collinear improved dipole amplitude in the target rapidity representation is more reasonable than the corresponding originating from the leading order fit, which indicates that the collinear improved evolution equation in the target rapidity representation can provide a relatively good depiction of the exclusive and dissociative HERA data.

The Balitsky-Kovchegov equations in the projectile and target rapidity representations are analytically solved in fixed and running coupling cases in the saturation domain. We find an interesting outcome that the respective analytic S-matrixes in the two rapidity representations have almost the same rapidity dependence in the exponent in running coupling case, which can provide a way to explain a surprised result appeared in Ref.[1] where 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 compute the ratios between these dipole amplitudes numerically in saturation region. The ratios are close to one, which confirm the analytic results. Moreover, the running coupling, collinearly-improved, and extend full collinearly-improved Balitsky-Kovchegov equations are used to fit the HERA data. We find that all of them give high quality description of the data, and the χ2/d.o.f obtained from the fits are close to each other. Both the analytic and numerical calculations imply that the Balitsky-Kovchegov equation at running coupling level is robust and has a strong enough predictive power at HERA energies, although other higher order corrections could be significant for future experiments like EIC or LHeC.

To get a reasonable description of the hadron production at the LHC energies, the impact pa rameter dependent saturation model is modified by inclusion of an anomalous dimension, γ, which controls the slope of the scattering amplitude in the transition from the dilute to saturation regions. We calculate the transverse momentum distribution and nuclear modification factor of the π0 and charged hadrons with the improved model, the results show rather good consistent with the mea surements at the LHC. Moreover, we also use the original impact parameter dependent model to study the aforementioned measurements at the LHC by adjusting its parameters. We find that the improved model is more favored by the experimental data than the original one, since the anomalous dimension takes a significant role in the suppression of the evolution of the scattering amplitude.