Proton structure functions at low x: The fractal distributions

This paper presents a method for extracting the parton distribution functions (PDFs) at small momentum fractions [Formula: see text] and at the next-to-leading order (NLO) accuracy in perturbative quantum chromodynamics. It turns out that the PDFs can be described by the “Fractal” or self-similar distributions at low [Formula: see text] below [Formula: see text]. To this end, a simple parametrization for the unintegrated PDFs based on the “Fractal” approach is considered. These functions have self-similar behavior at low [Formula: see text] and [Formula: see text] for sea quarks and have self-similar behavior at low [Formula: see text] for gluon distribution. By integration from these transverse momentum distributions (TMDs), the initial input densities are obtained and the model’s free parameters are then calculated using the experimental data released by the HERA experiment. The small-[Formula: see text] experimental datasets on electron–proton ([Formula: see text]) and positron–proton ([Formula: see text]) for natural current (NC) interactions in deep inelastic scattering (DIS) processes at HERA for the range of [Formula: see text] (GeV 2 ) and [Formula: see text] are included in this analysis. The uncertainty estimations in this analysis are carried out using the standard “Hessian” method. Considering the overall value of [Formula: see text] and theory-to-data comparisons, the results indicate good agreements between the experimental datasets and the theoretical predictions. A detailed comparison is also presented between the model’s predictions for the relevant small-[Formula: see text] observable and recent parametrizations for the PDFs.