Quantitative Study of Different Forms of Geometrical Scaling in Deep Inelastic Scattering at HERA

We use recently proposed method of ratios to assess the quality of geometrical scaling in deep inelastic scattering for different forms of the saturation scale. We consider original form of geometrical scaling (motivated by the Balitski-Kovchegov (BK) equation with fixed coupling) studied in more detail in our previous paper, and four new hypotheses: phenomenologically motivated case with $Q^2$ dependent exponent $\lambda$ that governs small $x$ dependence of the saturation scale, two versions of scaling (running coupling 1 and 2) that follow from the BK equation with running coupling, and diffusive scaling suggested by the QCD evolution equation beyond mean field approximation. It turns out that more sophisticated scenarios: running coupling scaling and diffusive scaling are disfavored by the combined HERA data on $e^+p$ deep inelastic structure function $F_2$.