Systematic analysis of scaling properties in deep inelastic scattering

Using the ``quality factor'' method, we analyze the scaling properties of deep inelastic processes at the accelerator HERA and fixed target experiments for $x\ensuremath{\le}{10}^{\ensuremath{-}2}$. We look for scaling formulas of the form ${\ensuremath{\sigma}}^{{\ensuremath{\gamma}}^{*}p}(\ensuremath{\tau})$, where $\ensuremath{\tau}(L=\mathrm{log}{Q}^{2},Y)$ is a scaling variable suggested by the asymptotic properties of QCD evolution equations with rapidity $Y$. We consider four cases: ``fixed coupling,'' corresponding to the original geometric scaling proposal and motivated by the asymptotic properties of the Balitsky-Kovchegov equation with fixed QCD coupling constant; two versions, ``running coupling I, II,'' of the scaling suggested by the Balitsky-Kovchegov equation with running coupling; and ``diffusive scaling'' suggested by the QCD evolution equation with Pomeron loops. The quality factors, quantifying the phenomenological validity of the candidate scaling variables, are fitted on the total and deeply virtual Compton scattering cross-section data from HERA and predictions are made for the elastic vector meson and for the diffractive cross sections at fixed small ${x}_{\mathbb{P}}$ or $\ensuremath{\beta}$. The first three scaling formulas have comparably good quality factors while the fourth one is disfavored. Adjusting initial conditions gives a significant improvement of the running coupling II scaling.