Diffusion into infra-red and unitarization of the BFKL pomeron

TL;DR

The paper demonstrates that the Balitsky-Kovchegov equation unitarizes the BFKL pomeron by generating a rapidity-dependent saturation scale $Q_s(Y)$, which suppresses diffusion into the infrared and induces geometric scaling in gluon distributions. Through numerical studies and analytic Mellin-based insights, it shows a universal scaling behavior with $Q_s(Y) o Q_0 e^{ ext{λ}Y}$ and identifies three evolution regimes (linear, saturated, and transition). It further investigates subleading corrections—running coupling and a kinematic constraint—finding that they modulate but do not destroy the saturation picture: RC reduces IR sensitivity and keeps evolution perturbative-dominated at high $Y$, while KC lowers the growth rate of $Q_s(Y)$ and the density rise. These results underscore the importance of including such corrections for reliable high-energy QCD phenomenology.

Abstract

The BFKL pomeron in perturbative QCD is plagued by the lack of unitarity and diffusion into the infra-red region of gluon virtualities. These two problems are intimately related. We perform numerical studies of the evolution equation proposed by Balitsky and Kovchegov which unitarizes the BFKL pomeron. We show how diffusion into the infra-red region is suppressed due to the emergence of a saturation scale and scaling behaviour. We study universality of this phenomenon as well as its dependence on subleading corrections to the BFKL pomeron such as the running coupling and kinematic constraint. These corrections are very important for phenomenological applications.

Figures (10)

• Figure 1: Kinematics of the gluons in the BFKL ladder. $x$ and $x/z$ are the longitudinal momentum fractions, and $k$, $k^\prime$ and $q=k-k^\prime$ are transverse momenta of the gluons.
• Figure 2: The functions $k\phi(k,Y)$ constructed from solutions to the BFKL and the Balitsky-Kovchegov equations with the input (\ref{['eq:start1']}) for different values of the evolution parameter $Y=\ln(1/x)$ ranging from $1$ to $10$. The coupling constant $\alpha_s=0.2$.
• Figure 3: The re-normalized solutions $\Psi(k,Y)$ from the input (\ref{['eq:start1']}) for the BFKL and the Balitsky-Kovchegov equations as a function of the rapidity $Y$ and $\eta=\ln(k/k_0)$ with $k_0=10^{-10} \; \rm GeV$
• Figure 4: The lines of constant values of the BFKL and the BK re-normalized solutions $\Psi(k,Y)$ ($Y=\ln (1/x)$) in the $(\log_{10}(k),\log_{10}(1/x))$-plane.
• Figure 5: Scaling condition (\ref{['eq:crticond']}) as a function of $Y$.