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The NNLO non-singlet QCD analysis of parton distributions based on Bernstein polynomials

A non-singlet QCD analysis of the structure function $xF_3$ up to NNLO is performed based on the Bernstein polynomials approach. We use recently calculated NNLO anomalous dimension coefficients for the moments of the $xF_3$ structure function in $νN$ scattering. In the fitting procedure, Bernstein polynomial method is used to construct experimental moments from the $xF_3$ data of the CCFR collaboration in the region of $x$ which is inaccessible experimentally. We also consider Bernstein averages to obtain some unknown parameters which exist in the valence quark densities in a wide range of $x$ and $Q^2$. The results of valence quark distributions up to NNLO are in good agreement with the available theoretical models. In the analysis we determined the QCD-scale $Λ^ {\bar{MS}}_{QCD, N_{f}=4}=211$ MeV (LO), 259 MeV (NLO) and 230 MeV (NNLO), corresponding to $α_s(M_Z^2)=0.1291$ LO, $α_s(M_Z^2)=0.1150$ NLO and $α_s(M_Z^2)=0.1142$ NNLO. We compare our results for the QCD scale and the $α_s(M_Z^2)$ with those obtained from deep inelastic scattering processes.