The BFKL Equation from the Wilson Renormalization Group

TL;DR

The paper develops a Wilson renormalization group framework for the low-$x$ effective action in QCD and demonstrates that, in the linear weak-field limit, the RG flow reproduces the BFKL equation for the unintegrated gluon density $\varphi(y,\mathbf{k})$ with $y=\ln(1/x)$. The kernel decomposes into virtual and real parts, $K_{\mathrm{virt}}$ and $K_{\mathrm{re}}$, whose sum yields the standard BFKL evolution, thereby connecting the semiclassical MV/JKMW approach to the conventional BFKL formalism and clarifying its relation to Lipatov's high-energy effective action. The discussion contrasts the imaginary color-charge weight $F[\rho]$ in this framework with Lipatov's reggeon-based action and outlines the path toward nonlinear saturation via RG flow and potential fixed points, setting the stage for future work on nonlinear low-$x$ dynamics and unitarization.

Abstract

We discuss the Wilson renormalization group approach to the effective action for low $x$ physics. It is shown that in the linearized, weak field regime the RG equation reduces to the BFKL equation for the evolution of the unintegrated gluon density. We discuss the relation of this approach with that of Lipatov.