High energy asymptotic behavior of the $S$-matrix in the saturation region with the smallest dipole running coupling prescription

Abstract

We present results from analytic solutions to the running coupling, full next-to-leading order, and collinearly improved next-to-leading order Balitsky-Kovchegov equations in the saturation region with the smallest dipole size QCD running coupling prescription. The analytic results of the $S$-matrix of the latter two equations show that the $\exp(-\mathcal{O}(Y^{3/2}))$ rapidity dependence of the solutions are replaced by $\exp(-\mathcal{O}(Y))$ dependence once the running coupling prescription is switched from parent dipole to the smallest dipole prescription, which indicate that the $S$-matrix has a strong dependence on the choice of running coupling prescription. We compute the numerical solutions of these Balitsky-Kovchegov equations with the smallest and parent dipole running coupling prescriptions, the numerical results confirm the analytic outcomes. The rare fluctuations of the $S$-matrix on top of next-to-leading order corrections are also studied under the smallest dipole running coupling prescription in the center of mass frame. It shows that the rare fluctuations are strongly suppressed and less important in the smallest dipole running coupling prescription case as compared to the parent dipole running coupling prescription case.