Subleading-N_c corrections in non-linear small-x evolution

TL;DR

This work analyzes subleading-Nc corrections to non-linear small-x evolution by comparing the full JIMWLK evolution with the BK truncation. It shows that saturation and group-theory coincidence limits strongly suppress $1/N_c$ corrections, yielding a ~0.1% difference for dipole amplitudes, and demonstrates that a minimal Gaussian truncation (GT) captures many of the remaining effects while maintaining the same dynamical content as BK. Numerical JIMWLK studies on large lattices reveal a modest 3–5% slowdown relative to BK, attributed to factorization violations, with running coupling expected to lessen this impact. Extending GT to include higher-order correlators hints at small Casimir-scaling violations and multi-reggeon contributions that remain subdominant for the dipole observable. Overall, the results reinforce BK’s robustness for predicting dipole evolution at small x while providing a framework to quantify and probe subleading $1/N_c$ dynamics.

Abstract

We explore the subleading-N_c corrections to the large-N_c Balitsky-Kovchegov (BK) evolution equation by comparing its solution to that of the all-N_c Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) equation. In earlier simulations it was observed that the difference between the solutions of JIMWLK and BK is unusually small for a quark dipole scattering amplitude, of the order of 0.1%, which is two orders of magnitude smaller than the naively expected 1/N_c^2 or 11%. In this paper we argue that this smallness is not accidental. We show that saturation effects and correlator coincidence limits fixed by group theory constraints conspire with the particular structure of the dipole kernel to suppress subleading-N_c corrections reducing the difference between the solutions of JIMWLK and BK to 0.1%. We solve the JIMWLK equation with improved numerical accuracy and verify that the remaining 1/N_c corrections, while small, still manage to slow down the rapidity-dependence of JIMWLK evolution compared to that of BK. We demonstrate that a truncation of JIMWLK evolution in the form of a minimal Gaussian generalization of the BK equation captures some of the remaining 1/N_c contributions leading to an even better agreement with JIMWLK evolution. As the 1/N_c corrections to BK include multi-reggeon exchanges one may conclude that the net effect of multi-reggeon exchanges on the dipole amplitude is rather small.

Figures (11)

• Figure 1: Contour plot of the correlators on the right-hand side of the BK equation (\ref{['eq:BK']}) (divided by $\bigl\langle \Hat S_{\bm{x y}}\bigr\rangle$ to normalize the large $\bm z$ asymptotics to $-1$) in the $\bm z'$-plane at different stages in the evolution, for fixed $|\bm x-\bm y|= R_s (Y_0)$. Here ${\bm z}' = (z'_1, z'_2)$. The left panel shows the initial conditions for the evolution, while the right panel displays the evolved distribution (see text). The dots mark $\bm x$ and $\bm y$, the locations of the parent $q$ and $\Bar q$. These points always fall on the boundaries between positive and negative contributions to the right-hand side of \ref{['eq:BK']} (marked by contour lines going through the dots). The half-rays denote the angles at which the factorization violations will be plotted in Fig. \ref{['fig:fact-viols-fixed-xmy']}.
• Figure 2: Factorization violations from JIMWLK evolution (scaled up by $N_c^2$) plotted against varying parent dipole size at fixed $|\bm z'| = 0.4\cdot R_s(Y)$. The angles $0^\circ$ (left), $45^\circ$ (middle), and $90^\circ$ (right) are the angles between $\bm z'$ and $\bm r$ and refer to rays in the $\bm z'$-plane as indicated in Fig \ref{['fig:z-plane']} for one fixed $|\bm r| = |{\bm x} - {\bm y}|$. Shown are three different rapidities each. One obtains a reduction by a factor of 10 compared to the natural size $\Delta^J \sim 1/N_c^2$ (or $\Delta^J \, N_c^2 \sim 1$). This was observed earlier in Rummukainen:2003ns, with any differences being due to the slightly different correlator geometries chosen here for ease of comparison with the discussion below. Shown are "typical" regions that contribute to evolution, see Sect. \ref{['sec:orig-fact']} for details. Note also that only the $0^\circ$ ray shows special structure since it contains strict coincidence limits (i.e. the limits where $\bm x$, $\bm y$ or $\bm z$ overlap), all other angles are qualitatively well represented by the $90^\circ$ case.
• Figure 3: Schematic representation of physically distinct cases for correlator configurations. The shaded areas denote regions of size $R_s$, the particle coordinates $\bm x$, $\bm z'$, $\bm y$ are at the corners of the triangles with the gluon position $\bm z'$ marked by a dot.
• Figure 4: Behavior of three point correlator as discussed in Eq. \ref{['eq:corr-regions']} taken from our numerical solution of the JIMWLK equation at some intermediate $Y$. $\bm z'$ is varied along rays of fixed angle with respect to $\bm r$ ($0^\circ$ and $90^\circ$), c.f. Fig. \ref{['fig:z-plane']}. The regions are labeled in correspondence to Fig. \ref{['fig:scalecases']}, the correlators display the generic behavior anticipated in Eq. \ref{['eq:corr-regions']}. Note that region "b" is only present near $0^\circ$ and completely disappears for $90^\circ$.
• Figure 5: Contour plots of the factorization violation $\Delta$ scaled up by $N_c^2$ at a fixed rapidity in JIMWLK (left) and in the Gaussian truncation (right). The plots scan parent dipole size $|\bm x-\bm y|$ and distance $|\bm z'|$ from the midpoint $(\bm x+\bm y)/2$ at $90^\circ$ (top row) and $0^\circ$ (bottom row) with respect to $\bm x-\bm y$. The $0^\circ$ case is special since it contains contributions where $\Delta$ strictly vanishes due to the coincidence limit $\bm z=\bm x$ or $\bm y$ which appears here as a line with $|\bm z'|=|\bm x-\bm y|/2$. The bulk of the contributions is similar to the $90^\circ$ case. As was the case for the three-point-functions of Fig. \ref{['fig:regions']}, the contributions along the axes $|\bm r|=0$ and $|\bm z'|=0$ are identical for all angles. Fig. \ref{['fig:fact-viols-fixed-xmy']} shows cuts along horizontal lines near the bottom of the two JIMWLK plots.