# Solving The High Energy Evolution Equation Including Running Coupling Corrections

## TL;DR

This paper numerically solves the nonlinear BK evolution equation with recent running coupling corrections, revealing a universal scaling behavior and emphasizing the importance of subtraction terms for accurate results.

## Contribution

It introduces a comprehensive numerical solution of the BK equation including the subtraction terms, clarifying their impact on the evolution and scaling behavior.

## Key findings

- Running coupling corrections significantly alter the shape of the scaling function.
- Subtraction terms are crucial for consistent evolution kernel calculations.
- Universal scaling behavior persists at large rapidity with full corrections.

## Abstract

We study the solution of the nonlinear BK evolution equation with the recently calculated running coupling corrections [hep-ph/0609105, hep-ph/0609090]. Performing a numerical solution we confirm the earlier result of [hep-ph/0408216] that the high energy evolution with the running coupling leads to a universal scaling behavior for the dipole scattering amplitude. The running coupling corrections calculated recently significantly change the shape of the scaling function as compared to the fixed coupling case leading to a considerable increase in the anomalous dimension and to a slow-down of the evolution with rapidity. The difference between the two recent calculations is due to an extra contribution to the evolution kernel, referred to as the subtraction term, which arises when running coupling corrections are included. These subtraction terms were neglected in both recent calculations. We evaluate numerically the subtraction terms for both calculations, and demonstrate that when the subtraction terms are added back to the evolution kernels obtained in the two works the resulting dipole amplitudes agree with each other! We then use the complete running coupling kernel including the subtraction term to find the numerical solution of the resulting full non-linear evolution equation with the running coupling corrections. Again the scaling regime is recovered at very large rapidity.