Valence quark distributions in mesons in generalized QCD sum rules

Abstract

The method for calculation of valence quark distributions at intermediate $x$ is presented. The imaginary part of the virtual photon forward scattering amplitude on the quark current with meson quantum number is considered. 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 operator product expansion (OPE) in $p^2_1, p^2_2$ up to dimension 6 operators is performed. Double dispersion representations in $p^2_1, p^2_2$ of the amplitude in terms of physical states contributions are used. Equalling them to those calculated in QCD by OPE the desired sum rules for quark distributions in mesons are found. The double Borel transformations are applied to the sum rules, killing non-diagonal transition terms, which deteriorated the accuracy in the previous calculations of quark distributions in nucleon. Leading order perturbative corrections are accounted. Valence quark distributions in pion, longitudinally and transversally polarized $\rho$-mesons are calculated at intermediate $x$, $0.2 \la x \la 0.7$ and normalization points $Q^2 = 2-4 GeV^2$ with no fitting parameters. The use of the Regge behaviour at small $x$ and quark counting rules at large $x$ allows one to find the first and the second moments of valence quark distributions. The obtained quark distributions may be used as an input for evolution equations. In the case of pion the quark distribution is in agreement with those found from the data on the Drell-Yan process.