Compatibility of phenomenological dipole cross sections with the Balitsky-Kovchegov equation

Abstract

Phenomenological models of the dipole cross section that enters in the description of for instance deep inelastic scattering at very high energies have had considerable success in describing the available small-$x$ data in both the saturation region and the so-called extended geometric scaling (EGS) region. We investigate to what extent such models are compatible with the numerical solutions of the Balitsky-Kovchegov (BK) equation which is expected to describe the nonlinear evolution in $x$ of the dipole cross section in these momentum regions. We find that in the EGS region the BK equation yields results that are qualitatively different from those of phenomenological studies. In particular, geometric scaling around the saturation scale is only obtained at asymptotic rapidities. We find that in this limit the anomalous dimension $\ensuremath{\gamma}(r,x)$ of phenomenological models approaches a limiting function that is universal for a large range of initial conditions. At the saturation scale, this function equals approximately 0.44, in contrast to the value 0.628 commonly used in the models. We further investigate the dependence of these results on the starting distribution, the small-$r$ limit of the anomalous dimension for fixed rapidities and the $x$-dependence of the saturation scale.