We extract (for the first time) the ratio of the gluon condensate 〈g3fabcG3〉/〈αsG2〉 expressed in terms of the liquid instanton radius ρc from charmonium moments sum rules by examining the effects of 〈αsG2〉 in the determinations of both ρc and the running MS mass mc(mc). Using a global analysis of selected ratios of moments at different Q2=0, 4mc2 and 8mc2 and keeping 〈αsG2〉 from 0.06 GeV4, where the estimate of ρc is almost independent of 〈αsG2〉, we deduce: ρc=0.98(21) GeV-1 corresponding to 〈g3fabcG3〉=(31±13) GeV2〈αsG2〉. The value of mc(mc) is less affected (within the errors) by the variation of 〈g3fabcG3〉, where a common solution from different moments are reached for 〈αsG2〉≥0.02 GeV4. Using the values of 〈αsG2〉=0.06(2) GeV4 from some other channels and the previous value of 〈g3fabcG3〉, we deduce: m̄c(mc)=1261(18) MeV and mb(mb)=4232(10) MeV, where an estimate of the 4-loops (O(αs3)) contribution has been included. Our analysis indicates that the errors in the determinations of the charm quark mass and of αs 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 MZ, where it is remarkable to notice the approximate equalities: ms/mu≈mb/ms≈mt/mb≈51(9), which might reveal some eventual underlying novel symmetry of the quark mass matrix in some Grand Unified Theories. © 2010 Elsevier B.V.
Narison, S. (2010). Gluon condensates and c, b quark masses from quarkonia ratios of moments. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 693(5), 559–566. https://doi.org/10.1016/j.physletb.2010.09.007