Valence quark distributions in nucleon at low $Q^2$ in QCD

Valence u- and d-quarks distributions in proton are calculated in QCD at low $Q^2$ and intermediate x, basing on the operator product expansion (OPE). The imaginary part of the virtual photon scattering amplitude on quark current with proton quantum numbers is considered. The initial and final virtualities $p^2_1$ and $p^2_2$ of the currents are assumed to be large, negative and different, $p^2_1 \not= p^2_2$. The OPE in $p^2_1$, $p^2_2$ up to dimension 6 operators was performed. Double dispersion representations in $p^2_1, p^2_2$ of the amplitudes in terms of physical states contributions are used. Putting them to be equal to those calculated in QCD, the sum rules for quark distributions are found. The double Borel transformations are applied to the sum rules. Leading order perturbative corrections are accounted. Valence quark distributions are found: $u(x)_v$ at $0.15<x<65$,~ $d(x)_v$ at $0.25<x<0.55$ with an accuracy $\sim 30$ in the middles and $\sim 50$ at the ends of these intervals. The quark distributions obtained are in agreement with those found from the analysis of hard processes data.