Violation of the geometric scaling behaviour of the amplitude for   running QCD coupling in the saturation region
  In this paper we show that the intuitive guess that the geometric scaling behaviour should be violated in the case of the running QCD coupling, turns out to be correct. The scattering amplitude of the dipole with the size $r$ depends on new dimensional scale: $\Lambda_{QCD}$, even at large values $Y = \ln(1/x)$ and $l\,=\, \ln\Lb \as\Lb r^2\Rb/\as\Lb 1/Q^2_s\Rb\Rb$. However, in this region we found a new scaling behaviour: the amplitude is a function of $\zeta = Y\,l$. We state that only in the vicinity of the saturation scale $Q_s$ ($\as(Q^2_s)\,\ln \Lb r^2 Q^2_s\Rb\,\leq \, 1$), the amplitude shows the geometric scaling behaviour. Based on these finding the geometric scaling behavior that has been seen experimentally, stems from either we have not probed the proton at HERA and the LHC deeply inside the saturation region or that there exists the mechanism of freezing of the QCD coupling constant at $r^2 \approx 1/Q^2_s$. 
Bastian Diaz Saez and Eugene Levin
hep-ph