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.
Highlights
In hadronic collisions at high energies, large gluon densities are created by the emission of soft gluons carrying a small fraction of the longitudinal momentum of the parent [1]
In the color glass condensate (CGC) framework, cross sections for various scattering processes can be expressed in terms of correlators of Wilson lines
It is usually convenient to work directly in terms of the Wilson line correlators, and to solve instead the Balitsky-Kovchegov (BK) equation [7,8] for the dipole operator, which can be obtained from the Jalilian-Marian–Iancu–McLerran–Weigert–Leonidov– Kovner (JIMWLK) equation in the large-Nc limit
Summary
In hadronic collisions at high energies, large gluon densities are created by the emission of soft gluons carrying a small fraction of the longitudinal momentum of the parent [1]. In the CGC framework, cross sections for various scattering processes can be expressed in terms of correlators of Wilson lines. A Wilson line describes the eikonal propagation of a parton in the strong color field of the target. The energy dependence of the target color fields, and cross sections, is obtained by solving the socalled Jalilian-Marian–Iancu–McLerran–Weigert–Leonidov– Kovner (JIMWLK) equation [3,4,5,6]. This is a perturbative evolution equation that describes the Bjorken-x dependence of a Wilson line. It is usually convenient to work directly in terms of the Wilson line correlators, and to solve instead the Balitsky-Kovchegov (BK) equation [7,8] for the dipole operator (correlator of two Wilson lines), which can be obtained from the JIMWLK equation in the large-Nc limit
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