CGC and saturation approach: Impact-parameter dependent model of perturbative QCD and combined HERA data

In this paper we confront the CGC/saturation approach of Ref.~\cite{CLMS} with the experimental combined HERA data and obtain its parameters. The model includes two features that are in accordance with our theoretical knowledge of deep inelastic scattering. These consist of: \textit{i}) the use of analytical solution for the non-linear Balitsky-Kovchegov (BK) evolution equation and \textit{ii}) the exponential behavior of the saturation momentum on the impact parameter $b$-dependence, characterized by $Q_s$ $\propto\exp(-mb)$ which reproduces the correct behavior of the scattering amplitude at large $b$ in accord with Froissart theorem. The model results are then compared to data at small-$x$ for the structure-function of the proton $F_{2}$, the longitudinal structure function $F_{L}$, the charm structure function $F_2^{c\bar{c}}$, the exclusive vector meson ($J/\psi,\phi,\rho$) production and Deeply Virtual Compton Scattering (DVCS). We obtain a good agreement for the processes in a wide kinematic range of $Q^2$ at small-$x$. Our results provide a strong guide for finding an approach, based on Color Glass Condensate/saturation effective theory for high energy QCD, to make reliable predictions from first principles and for forthcoming experiments like the Electron-Ion Collider and the LHeC.