Gluon condensate in charmonium sum rules with three-loop corrections

Charmonium sum rules are analyzed with the primary goal to obtain the restrictions on the value of the dimension 4 gluon condensate. The moments Mn(Q2) of the polarization operator of the vector charm currents are calculated and compared with the experimental data. The three-loop (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha_{\mathrm{s}}^2$\end{document}) perturbative corrections, the contribution of the gluon condensate with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha_{\mathrm{s}}$\end{document} corrections and the contribution of the dimension 6 operator G3 are accounted. It is shown that the sum rules for the moments do not work at Q2 = 0, where the perturbation series diverges and the G3 contribution is large. The domain in the (n, Q2) plane where the sum rules are legitimate is found. A strong correlation of the values of gluon condensate and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline{\mathrm{MS}}$\end{document} charm quark mass is determined. The absolute limits are found to be for the gluon condensate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\langle ({\alpha_{\mathrm{s}}} /{\pi}) G^2 \rangle = 0.009\pm 0.007{\mathrm{GeV}}^4$\end{document} and for the charm quark mass \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\bar m}({\bar m}) = 1.275\pm 0.015 {\mathrm{GeV}}$\end{document} in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline{\mathrm{MS}}$\end{document} scheme.