In this paper, we have employed Fractal formalism to calculate the Fracture Functions of the Leading neutron produced in [Formula: see text] collisions. The Fractal concept describes the self-similar behavior of the proton structure at Leading neutron production of semi-inclusive Deep Inelastic Scattering at the low values of the fractional momentum variable [Formula: see text]. The Fracture Functions (FFs) parameterized the non-perturbative part of the fragmentation process at the initial scale of Q[Formula: see text]. In this analysis, we benefit from the Leading neutron (Ln) experimental data published by H1 Collaboration and in order to estimate the uncertainty of neutron FFs and corresponding observables, we used the Hessian method. As a consequence of this Next-to-Leading order QCD analysis, we achieve a nice agreement between the prediction of our model for neutron FFs and experimental data. It seems that the Fractal approach based on the self-similar behavior of these conditional parton distribution functions can nicely describe the experimental data at low [Formula: see text].

In this article, we have employed fractal formalism to calculate the Fracture Functions of the Leading neutron produced in ep collisions. The fractal concept describes the selfsimilar behavior of the proton structure at Leading neutron production of semi-inclusive Deep Inelastic Scattering at the low values of the fractional momentum variable β. The Fracture Functions (FFs) parameterized the non-perturbative part of the fragmentation process at the initial scale of Q 2 0 . In this analysis, we benefit from the Leading neutron (Ln) experimental data published by H1 Collaboration and in order to estimate the uncertainty of neutron FFs and corresponding observables, we used the Hessian method. As a consequence of this Next-to-Leading order QCD analysis, we achieve a nice agreement between the prediction of our model for neutron FFs and experimental data. It seems that the fractal approach based on the self-similar behavior of these conditional parton distribution functions at low x can nicely describe the experimental data. Contents I. Introduction II. Theory setup A. Fracture Functions and the Leading neutron Production B. Fractals C. The Validity test of using the Fractal Model In Leading neutron Production Mechanism III. Essential parts of the Fractal Neutron FFs and their uncertainties A. Neutron FFs in the Fractal approach B. Experimental data C. χ 2 minimization and nFFs Uncertainties

In this paper, we present the extraction of the Parton Distribution Functions (PDFs) at small momentum fractions x and at the next-to-leading order (NLO) accuracy in perturbative QCD. We show that the "sea quark distribution functions" have "Fractal" or self-similar behaviour with fixed exponent at x < 0.01. To this end, a simple parametrization for the Parton Distribution Functions and especially for the sea quarks PDFs based on the "Fractal" approach is considered. The small x experimental datasets on electron-proton (e − p) and positron-proton (e + p) in DIS processes at HERA for the range of 1.5 < Q 2 < 650 (GeV 2 ) and x < 0.01 are included in this analysis. The estimations of the uncertainty in the present analysis are carried out using the standard "Hessian" method. In total, considering the overall value of χ 2 /dof and theory/data comparisons, the results indicate nice agreements between the experimental datasets and the theory predictions at small momentum fractionsx. Finally, we present detailed comparisons between predictions for the relevant small-x observables obtained with various recent models of proton PDFs available in literature. Contents

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