# Solution to the Sudakov suppressed Balitsky-Kovchegov equation and its application to the HERA data

• We analytically solve the Sudakov suppressed Balitsky-Kovchegov evolution equation with the fixed and running coupling constants in the saturation region. The analytic solution of the S-matrix shows the $\exp(-{\cal{O}}(\eta^2))$ rapidity dependence of the solution with the fixed coupling constant is replaced by $\exp(-{\cal{O}}(\eta^{3/2}))$ dependence in the smallest dipole running coupling case rather than obeying the law found in our previous publication, in which all the solutions of the next-to-leading order evolution equations comply with $\exp(-{\cal{O}}(\eta))$ 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 part of the evolution decrease of the dipole amplitude made by running coupling effect. To test the analytic findings, we calculate the numerical solutions of the Sudakov suppressed evolution equation, the numerical results confirm the analytic outcomes. Moreover, we use the numerical solutions of the evolution equation to fit the HERA data. It shows that the Sudakov suppressed evolution equation can give good quality fit to the data.
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Wenchang Xiang, Mengliang Wang, Yanbing Cai and Daicui Zhou. Solution to the Sudakov suppressed Balitsky-Kovchegov equation and its application to the HERA data[J]. Chinese Physics C.
Wenchang Xiang, Mengliang Wang, Yanbing Cai and Daicui Zhou. Solution to the Sudakov suppressed Balitsky-Kovchegov equation and its application to the HERA data[J]. Chinese Physics C.
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## Solution to the Sudakov suppressed Balitsky-Kovchegov equation and its application to the HERA data

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

Abstract: We analytically solve the Sudakov suppressed Balitsky-Kovchegov evolution equation with the fixed and running coupling constants in the saturation region. The analytic solution of the S-matrix shows the $\exp(-{\cal{O}}(\eta^2))$ rapidity dependence of the solution with the fixed coupling constant is replaced by $\exp(-{\cal{O}}(\eta^{3/2}))$ dependence in the smallest dipole running coupling case rather than obeying the law found in our previous publication, in which all the solutions of the next-to-leading order evolution equations comply with $\exp(-{\cal{O}}(\eta))$ 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 part of the evolution decrease of the dipole amplitude made by running coupling effect. To test the analytic findings, we calculate the numerical solutions of the Sudakov suppressed evolution equation, the numerical results confirm the analytic outcomes. Moreover, we use the numerical solutions of the evolution equation to fit the HERA data. It shows that the Sudakov suppressed evolution equation can give good quality fit to the data.

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