Rotating the Color Glass Condensate

TL;DR

The paper addresses perturbative instabilities in high-energy QCD evolution at next-to-leading order due to large collinear logarithms in the CGC framework. They introduce a scheme transformation realized as a rotation in the space of high-energy Wilson-line operators, formalized by $|\bar{S}(\zeta,\mu^2)) = e^{-\bar{\alpha}_s L} |S(\zeta))$, which preserves physical observables while shifting collinear logs into coefficient functions and adjusting both the kernel and operators. The resulting NLO BK equation in this rotated scheme exhibits numerical stability up to large rapidities, demonstrated with a MV-like initial condition and fixed coupling. This provides a consistent, OPE-faithful foundation for precision small-$x$ phenomenology, with potential extension to NNLO and running coupling, and applicability to other observables.

Abstract

High-energy QCD evolution beyond leading order suffers from instabilities driven by large collinear logarithms. We present a framework, consistent with the standard high-energy operator product expansion (OPE), that restores perturbative stability order by order. The method involves a change of basis in the space of high-energy operators, which modifies both the evolution kernel and the coefficient functions while leaving physical observables invariant. Within this factorization scheme, we derive a next-to-leading-order renormalization-group equation whose numerical solution exhibits stable evolution up to large rapidities, thereby establishing a solid foundation for precision studies of gluon saturation at current and future colliders.

Figures (4)

• Figure 1: Evolution of $\bar{N}_{12}$ from the initial condition at $Y=0$ (black dashed line) to $Y=5$ (blue line) and $Y=10$ (red line) using Eq. \ref{['eq:composite-dipole-RG']}. The abscissa axis is $Q_0x_{12}=Q_0|{\boldsymbol x}_{12}|$. The inset shows the evolution from the same initial condition using the standard NLO BK equation.
• Figure 2: The evolution of $\bar{N}_{12}$ from the initial condition up to $Y=10.0$ with $N=256$. Black dashed curve denotes the initial condition, while the red, blue, and black data correspond to the Euler, midpoint, and Heun methods, respectively. The difference between the Euler method and the other two is largest in the small region around $Q_0 x_{12} \approx 0.02$, but remains negligible overall.
• Figure 3: The evolution of $\bar{N}_{12}$ from the initial condition up to $Y=10.0$ with $N=256$. Plotted is the difference between the results obtained with $\delta Y=0.025$ and $\delta Y=0.0125$. As expected, the convergence of the Euler method is slower compared to the other methods. The maximal difference for the midpoint method is of the order of $10^{-5}$.
• Figure 4: The evolution of $\bar{N}_{12}$ from the initial condition up to $Y=10.0$ using the Euler method and two different grid sizes: $N=256$ (red data) and $N=384$ (blue data). The largest deviation of the order of $0.002$ is observed around $Q_0 x_{12} \approx 0.1$, but again remains negligible overall.