Heavy quarkonia mass-splittings in QCD: Gluon condensate, alpha(s) and 1/m-expansion

New double ratios of exponential moments of different two-point functions, which are less sensitive to the heavy quark mass and to the continuum effects than the commonly used ratio of moments, are presented for a more refined analysis of the mass-splittings between the different heavy quarkonia states. Using the present value of the QCD coupling alpha(s), we deduce the value: [alpha(s)G(2)] = (7.5 +/- 2.5) x 10(-2) GeV4 of the gluon condensate from M(psi) - M(eta c) and M(chi b) - M(Y), which combined with the recent estimate from tau-like decay sum rule in e(+)e(-) --> I = 1 hadrons data, leads to the update average value [alpha(s)G(2)] = (7.1 +/- 0.9) x 10(-2) GeV4 from the light and heavy quark systems. We also find that M(chi c(P11)) - M(chi c(P13)) is governed by the radiative corrections and gives alpha(s) (1.3 GeV) similar or equal to 0.64(-0.18)(+0.36), +/- 0.02 for 4 flavours, leading to alpha(s)(M(z)) similar or equal to 0.127 +/- 0.011, which is compatible with the one from LEP and tau-decay data. Our predictions for the splittings of different heavy quarkonia states are summarized in Table 2, where, in particular, we find M(Y) - M(eta b) approximate to 63(+51)(-29) MeV implying the possible observation of the eta(b) in the Y radiative decay. The validity of the 1/m-expansion, including finite mass corrections is tested, while a connection of our mass and width formulae, with the potential model ones is done.