  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 instanton
radius 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 mass
m_c(m_c). Using a global analysis of selected ratios of moments at different
Q^2=0, 4m_c^2 and 8m_c^2 and taking < alpha_s G^2 > from 0.06 GeV^4, where the
estimate 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 the
errors) by the variation of < alpha_s G^2 >, where a common solution from
different 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 and
the previous value of < g^3f_{abc}G^3 >, we deduce: m_c(m_c)=1260(18) MeV and
m_b(m_b)=4173(10) MeV, where an estimate of the 4-loops contribution has been
included. Our analysis indicates that the errors in the determinations of the
charm quark mass without taking into account the ones of the gluon condensates
have been underestimated. To that accuracy, one can deduce the running light
and heavy quark masses and their ratios evaluated at M_Z, where it is
remarkable 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 quark
mass matrix in some Grand Unified Theories.
