The Balitsky-Kovchegov equation in full momentum space

TL;DR

The paper formulates the Balitsky-Kovchegov equation in full momentum space by a Fourier transform of the dipole amplitude, and solves the resulting integro-differential equation numerically. It confirms traveling-wave solutions and geometric scaling in the regime $|k| \gg |q|$, with a saturation scale that scales as $Q_s^2(Y) \sim q^2 \Omega_s^2(Y)$ and grows as $Q_s(Y) = \max(q, Q_T)$ depending on the relation between $q$ and the target scale $Q_T$. The results show how saturation dynamics depend on the momentum transfer $q$ while avoiding unphysical tails in $q$ and demonstrate nonlinear damping of BFKL growth. The momentum-space formulation provides a natural framework for analyzing saturation phenomena in processes with nonzero momentum transfer and lays groundwork for future work including running coupling, fluctuations, and odderon effects.

Abstract

We analyse the Balitsky-Kovchegov (BK) saturation equation in momentum space and solve it numerically. We confirm that, in the limit where the transverse momentum of the incident particle k is much bigger than the momentum transfer q, the equation admits travelling-wave solutions. We extract the q dependence of the saturation scale Q_s(Y) and verify that Q_s(Y=cste) scales as max(q,Q_T), where Q_T is the scale caracterizing the target.

Figures (7)

• Figure 1: Linear contributions to the evolution of the dipole density in momentum space: (a) real gluon emission, (b) virtual gluon emission.
• Figure 2: Non-linear contribution to the evolution of the dipole density in momentum space. This correspond to the resummation of fan diagrams.
• Figure 3: Rapidity evolution of the dipole density as a function of $p$ for different values of $q$. For each plot, we show the amplitude for $Y$ varying between 0 and 25 by steps of 2.5.
• Figure 4: This figure presents the results concerning the saturation scale: (a) shows that the evolution of the saturation scale with $Y$ has the predicted behaviour and (b) represents the $q$-evolution of the saturation scale (see text for details).
• Figure 5: This figure shows the $Y$ dependence of the amplitude $\tilde{\cal N}(\mathbf{p};\mathbf{q})$. In the left plot we have fixed $\log(Rq)=-3$, while $\log(Rq)=0$ in the centre plot and $\log(Rq)=3$ in the right one.