Abstract
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.
Highlights
It is known that the energy evolution of the high energy dipole-hadron scattering amplitude is governed by the non-linear Balitsky-JIMWLK1[1,2,3,4,5] hierarchy and its mean field approximation known as the BalitskyKovchegov (BK) equation[1, 6]
A pioneer work on the next-to-leading order (NLO) corrections to the BK equation was done by Balitsky in Ref.[14] and Kovchegov-Weigert in Ref.[15], in which the NLO corrections associated with the QCD coupling are resumed to all orders leading to the running coupling BK equation
We find that the analytic solutions of the non-local collinearly improved Balitsky-Kovchegov (ciBK) and suppressed Balitsky-Kovchegov (SSBK) in η with the fixed coupling Eqs.(37) and (44) are similar to the one gained at leading order (LO) BK in Y Eq(9), except that the coefficients in the exponent are different
Summary
It is known that the energy evolution of the high energy dipole-hadron scattering amplitude is governed by the non-linear Balitsky-JIMWLK1[1,2,3,4,5] hierarchy and its mean field approximation known as the BalitskyKovchegov (BK) equation[1, 6]. The other is to resum the double logarithmic corrections to all orders giving rise to a local collinearly improved Balitsky-Kovchegov (ciBK) equation in Y [21] These two approaches are equivalent to each other in the leading double logarithmic level, they bring the modifications to the structure of the evolution equation. Soon after the non-local ciBK equation was established, it was found that there are important corrections to the evolution kernel from the sub-leading double logarithms located beyond the strong time-ordering region, it was shown that the sub-leading double logarithms arise from the incomplete cancellation between real and virtual corrections and are the typical Sudakov type ones[27] When these double logarithms are resummed to all orders, a Sudakov suppressed Balitsky-Kovchegov (SSBK) equation in the evolution of the target rapidity is obtained[27]. The review shall give us a chance to introduce notations and explain the kinematics of color dipoles
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