The Balitsky-Kovchegov equation for dipole gluon density in the momentum space

I present derivation the BK equation for the dipole gluon density in momentum space, starting from its standard formulation in coordinate space. I review the equation for both proton and nuclear targets, and I also discuss the resummed BK evolution.The purpose of this paper is to consolidate derivations and formulas scattered across the literature, to show in detail how the structure of the triple-Pomeron vertex emerges in the unfolded form of the nonlinear term, and to establish a consistent notation throughout.

💡 Research Summary

The manuscript presents a comprehensive derivation of the Balitsky‑Kovchegov (BK) evolution equation for the dipole gluon density in momentum space, starting from its well‑known formulation in coordinate space. After a brief historical introduction that situates the BK equation within the broader context of small‑x physics, saturation, and the Color Glass Condensate (CGC) effective theory, the author proceeds to the technical core of the work.

First, the dipole amplitude N(r,b,x) depending on dipole size r, impact parameter b, and Bjorken‑x is introduced. Two Fourier transforms are defined: the dipole gluon density F(x,k²) and an auxiliary quantity Φ(x,k²). The relationship between them is given by differential operators acting on Φ, which makes the nonlinear term of the BK equation local in Φ. By applying the Fourier transform to the coordinate‑space BK equation, the author obtains a momentum‑space evolution equation for Φ that contains a linear BFKL‑type kernel and a quadratic term proportional to Φ².

The linear kernel is expressed through the characteristic BFKL function χ(γ)=2ψ(1)−ψ(γ)−ψ(1−γ) and expanded as a power series around γ=0, allowing the author to write the linear part as a differential operator acting on Φ. The crucial step is the treatment of the nonlinear term: a Laplacian ∇²_k is applied to the integral operator that appears after the Fourier transform. This operation yields an “unfolded” form of the nonlinear term, explicitly displaying a double integral over transverse momenta multiplied by a logarithmic kernel. The resulting structure is identified with the triple‑Pomeron vertex, a well‑known object in high‑energy QCD, and is illustrated schematically in the paper’s figures.

To eliminate the explicit impact‑parameter dependence, the author introduces either a simple transverse area factor S⊥ or a normalized profile function S(b) (a step function with radius Rp). By imposing the normalization ∫d²b S(b)=1 and ∫d²b S²(b)=1/πRp², the nonlinear term acquires a factor 1/Rp², showing that the saturation correction is suppressed for a very large proton and enhanced for a dense nucleus. The nuclear case is treated by scaling the gluon density as AF and the radius as RA=Rp A^{1/3}, leading to an A^{1/3} enhancement of the nonlinear term.

The paper then moves to a resummed version of the BK equation. By performing a Mellin transform in x, separating resolved emissions (q²>μ²) from unresolved ones, and recombining virtual contributions, a Regge‑type form factor Δ_R(z,k,μ)=exp