Gluon condensate in charmonium sum rules with 3-loop corrections
  Charmonium sum rules are analyzed with the primary goal to obtain the restrictions on the value of dimension 4 gluon condensate. The moments M_n(Q^2) of the polarization operator of vector charm currents are calculated and compared with experimental data. The 3-loop (alpha_s^2) perturbative corrections, the gluon condensate contribution with alpha_s corrections and dimension 6 operator G^3 contribution are accounted. It is shown that the sum rules for the moments do not work at Q^2=0, where the perturbation series diverges and G^3 contribution is large. The domain in the plane (n, Q^2), where the sum rules are legitimate, is found. Strong correlation of the values of gluon condensate and MS-bar charm quark mass quark is determined. The absolute limits are found to be for gluon condensate <(alpha_s/pi)G^2>=0.009+-0.007 GeV^4 and for charm quark mass m(m)=1.275+-0.015 GeV in MS-bar scheme. 
B.L. Ioffe, K.N. Zyablyuk