Impact-parameter-dependent solutions to the Balitsky-Kovchegov equation at next-to-leading order

TL;DR

This work delivers the first stable numerical solutions of the impact-parameter dependent next-to-leading-order Balitsky-Kovchegov equation, enabling a realistic treatment of high-energy QCD saturation that includes $N(Y,r,b)$. Using shifted, offset integration grids and a first-order Euler stepping in rapidity, the authors obtain robust solutions and compare them to the Collinearly Improved (CI) LO/BK framework, revealing a slower evolution and a near-elimination of Coulomb tails at NLO. They analyze the dipole amplitude evolution, anomalous dimension, and saturation scale, finding that the NLO evolution yields a more stable $\gamma(Y,r,b)$ and a distinct $Q_s(Y,b)$ trajectory, with strong suppression of large-dipole and large-$b$ contributions. The study highlights the necessity of proper resummation of collinear NLO contributions to reproduce physical large-distance behavior and provides a viable foundation for future observables at NLO precision in saturation physics, with data publicly available at CERN's Zenodo repository.

Abstract

A stable numerical solution of the impact-parameter-dependent next-to-leading order Balitsky-Kovchegov equation is presented for the first time. The rapidity evolution of the dipole amplitude is discussed in detail. Dipole amplitude properties, such as the evolution speed or anomalous dimension behaviour, are studied as a function of the impact parameter and the dipole size and compared to solutions of the impact-parameter-dependent leading-order Balitsky-Kovchegov equation with the collinearly improved kernel. The next-to-leading evolution also strongly suppresses the Coulomb tails compared to the collinearly improved and leading order solutions.

Figures (11)

• Figure 1: The transversal structure of the mother and daughter dipoles within the next-to-leading evolution. The leading order contribution is highlighted in blue, while the additional gluonic emission and the associated dipoles are in orange. The angle $\theta$ controls the tilting of the mother dipole with respect to the target, and $\varphi$ is the angle of the impact parameter. In this work, we assume an isotropic target and neglect the $\theta$ dependence, so the scattering amplitude is a function of rapidity $Y$, dipole size $r$, and impact parameter $b$ only.
• Figure 2: The evolution of dipole scattering amplitude $N(Y, r, b)$ according to the NLO BK equation is shown at two fixed values of rapidity in the top panels: $Y = 0$ (top left), and $Y = 10$ (top right). A detailed view of the rapidity evolution on $r$ and $b$ for fixed representative values of $b=0.01$ GeV$^{-1}$ and $r=4$ GeV$^{-1}$ is shown in the bottom left, respectively right, panels.
• Figure 3: Comparison of the dipole scattering amplitude obtained within the collinearly improved (CI) and next-to-leading order (NLO) evolutions for a fixed value of $b = 0.01$ GeV$^{-1}$ (left) and $r = 4$ GeV$^{-1}$ (right). Three different values of rapidity are shown.
• Figure 4: Anomalous dimension $\gamma$ as a function of the dipole size at a representative impact parameter $b = 0.01 \mathrm{~GeV}^{-1}$ and three points of the rapidity evolution. Dashed curves show the leading order results (CI), while the NLO results are plotted in full curves.
• Figure 5: The evolution speed as a function of the dipole size $r$ at a representative impact parameter $b = 0.01 \mathrm{~GeV}^{-1}$ and three steps of the rapidity evolution. The CI BK corresponds to the dashed curves, the NLO BK is shown in full.