Solution to the Balitsky-Kovchegov equation in the saturation domain

Abstract

The solution to the Balitsky-Kovchegov equation is found in the deep saturation domain. The controversy between different approaches regarding the asymptotic behaviour of the scattering amplitude is solved. It is shown that the dipole amplitude behaves as $ 1 - \exp (- z + \ln z)$ with $ z = \ln (r^2 Q^2_s)$ ($ r$ -size of the dipole, $Q_s$ is the saturation scale) in the deep saturation region. This solution is developed from the scaling solution to the homogeneous Balitsky-Kovchegov equation. The dangers associated with making simplifications in the BFKL kernel, to investigate the asymptotic behaviour of the scattering amplitude, is pointed out . In particular, the fact that the Balitsky-Kovchegov equation belongs to the Fisher-Kolmogorov-Petrovsky-Piscounov -type of equation, needs further careful investigation.