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

Understand

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.