Quark contribution to the small-$x$ evolution of color dipole

TL;DR

This work analyzes NLO corrections to the small-$x$ evolution of color dipoles in QCD by computing the quark-loop contribution to the BK kernel in a shock-wave background. It demonstrates that no new Wilson-line operators are generated at one loop and that the running coupling in the BK equation is governed by the parent dipole size, not the produced dipoles, with the gluon-loop sector left for future work. The results provide a consistency bridge to NLO BFKL and clarify the operator content and renormalization structure of the quark contribution. Together with the planned gluon-loop calculations, these findings support a robust Wilson-line description of high-energy scattering at NLO and guide future phenomenology of running coupling in small-$x$ evolution.

Abstract

The small-$x$ deep inelastic scattering in the saturation region is governed by the non-linear evolution of Wilson-lines operators. In the leading logarithmic approximation it is given by the BK equation for the evolution of color dipoles. In the NLO the nonlinear equation gets contributions from quark and gluon loops. In this paper I calculate the quark-loop contribution to small-x evolution of Wilson lines in the NLO. It turns out that there are no new operators at the one-loop level - just as at the tree level, the high-energy scattering can be described in terms of Wilson lines. In addition, from the analysis of quark loops I find that the argument of coupling constant in the BK equation is determined by the size of the parent dipole rather than by the size of produced dipoles. These results are to be supported by future calculation of gluon loops.

Figures (11)

• Figure 1: Leading-order diagrams for the small-$x$ evolution of color dipole.
• Figure 2: Quark-loop contribution to the gluon propagator in a shock-wave background .
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