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We extract (for the first time) the ratio of the gluon condensate <g^3f_{abc}G^3 >/< alpha_s G^2 > expressed in terms of the liquid instantonradius rho_c from charmonium moments sum rules by examining the effects of <alpha_s G^2 > in the determinations of both rho_c and the running MS massm_c(m_c). Using a global analysis of selected ratios of moments at differentQ^2=0, 4m_c^2 and 8m_c^2 and taking < alpha_s G^2 > from 0.06 GeV^4, where theestimate of rho_c is almost independent of < alpha_s G^2 >, we deduce:rho_c=0.98(21) GeV^{-1} which corresponds to < g^3f_{abc}G^3 > = (31+- 13)GeV^2 < alpha_s G^2 >. The value of m_c(m_c) is less affected (within theerrors) by the variation of < alpha_s G^2 >, where a common solution fromdifferent moments are reached for < alpha_s G^2 > greater than 0.02 GeV^4.Using the values of < alpha_s G^2 >=0.06(2) GeV^4 from some other channels andthe previous value of < g^3f_{abc}G^3 >, we deduce: m_c(m_c)=1260(18) MeV andm_b(m_b)=4173(10) MeV, where an estimate of the 4-loops contribution has beenincluded. Our analysis indicates that the errors in the determinations of thecharm quark mass without taking into account the ones of the gluon condensateshave been underestimated. To that accuracy, one can deduce the running lightand heavy quark masses and their ratios evaluated at M_Z, where it isremarkable to notice the approximate equalities: m_s/m_u= m_b/m_s= m_t/m_b=51(4), which might reveal some eventual underlying novel symmetry of the quarkmass matrix in some Grand Unified Theories.

Gluon condensates and c, b quark masses from quarkonia ratios of moments