Next-to-leading order Balitsky-Kovchegov equation beyond large $N_\mathrm{c}$

TL;DR

This paper computes finite-$N_c$ corrections to the NLO BK equation, providing analytical expressions and numerical analysis showing these corrections are generally small, around 10%, and have limited impact on the evolution of the amplitude.

Contribution

It introduces a method to analytically evaluate finite-$N_c$ corrections to the NLO BK equation using Gaussian approximation and matrix diagonalization.

Findings

Finite-$N_c$ corrections are typically smaller than 10%.

Corrections have limited impact on the shape of the amplitude.

Evolution speed is only slightly affected by these corrections.

Abstract

We calculate finite-$N_\mathrm{c}$ corrections to the next-to-leading order (NLO) Balitsky-Kovchegov (BK) equation. We find analytical expressions for the necessary correlators of six Wilson lines in terms of the two-point function using the Gaussian approximation. In a suitable basis, the problem reduces from the diagonalization of a six-by-six matrix to the diagonalization of a three-by-three matrix, which can easily be done analytically. We study numerically the effects of these finite-$N_\mathrm{c}$ corrections on the NLO BK equation. In general, we find that the finite-$N_\mathrm{c}$ corrections are smaller than the expected $1/N_\mathrm{c}^2 \sim 10\%$. The corrections may be large for individual correlators, but have less of an influence on the shape of the amplitude as a function of the dipole size. They have an even smaller effect on the evolution speed as a function of rapidity.