Next-to-leading and resummed BFKL evolution with saturation boundary

TL;DR

The paper tackles the instability of next-to-leading order BFKL evolution at small x by combining saturation boundaries with a renormalization-group–improved (RG–improved) resummation. It shows that nonlinear saturation cannot cure the large NLL corrections, and that a full resummation is necessary for a stable nonlinear evolution; this RG–improved framework delays the onset of saturation and significantly lowers the saturation scale $Q_s$ in many cases. The authors implement both fixed and running coupling analyses, finding strong sensitivity to scale choices and exposing preasymptotic effects like a dip in the splitting function that influence phenomenology. The work demonstrates that resummed small-$x$ evolution with saturation yields more reliable behavior and provides quantitative estimates for the saturation scale and front velocity, offering important guidance for interpreting data in high-energy hadronic processes.

Abstract

We investigate the effects of the saturation boundary on small-x evolution at the next-to-leading order accuracy and beyond. We demonstrate that the instabilities of the next-to-leading order BFKL evolution are not cured by the presence of the nonlinear saturation effects, and a resummation of the higher order corrections is therefore needed for the nonlinear evolution. The renormalization group improved resummed equation in the presence of the saturation boundary is investigated, and the corresponding saturation scale is extracted. A significant reduction of the saturation scale is found, and we observe that the onset of the saturation corrections is delayed to higher rapidities. This seems to be related to the characteristic feature of the resummed splitting function which at moderately small values of x possesses a minimum.

Figures (12)

• Figure 1: Regge-type factorization formula for the cross section. The slashed gluon lines indicate reggeized gluons.
• Figure 2: Solutions to the leading and next-to-leading BFKL equations for a fixed coupling $\bar{\alpha}_s = 0.2$ and for $Y=2,6,10,14$: NLL BFKL with asymmetric scale choice (solid red), NLL BFKL with symmetric scale choice (solid blue), LL BFKL (dotted blue).
• Figure 3: Comparison of the NLL BFKL solution with (solid red) and without (dotted blue) the absorptive saturation boundary \ref{['absorpbound']} for fixed $\bar{\alpha}_s = 0.2$, and $Y=2,6,10,14$.
• Figure 4: Left: Comparison of the NLL BFKL solution with (solid red) and without (dotted blue) the frozen saturation boundary \ref{['frozenbound']} for $Y=2,6,10,14$. Right: Comparison of the NLL solution with the frozen (solid red) saturation boundary \ref{['frozenbound']} versus the absorptive (dotted blue) saturation boundary for $Y=2,6,10,14, 20$. Simulations done for the fixed coupling $\bar{\alpha}_s = 0.2$.
• Figure 5: The saturation scale $Q_s^2(Y)$ extracted from the evolution using the boundary \ref{['frozenbound']}, for LL BFKL (red, upper circles) and NLL BFKL (blue, lower circles), with fixed coupling $\bar{\alpha}_s = 0.2$.