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Heavy quarkonia mass-splittings in QCD: test of the $1/m$-expansion and estimates of $<α_s G^2>$ and $α_s$

I present a more refined analysis of the mass-splittings between the different heavy quarkonia states, using {\it new double ratios} of exponential moments of different two-point functions. Then, I test the validity of the $1/m$-expansion, extract $α_s(M_Z)=0.127\pm 0.011$ from $M_{χ_c(P^1_1)} - M_{χ_c(P^3_1)}$, and provide a new estimate of the gluon condensate from $M_ψ-M_{η_c}$ and $M_{χ_b}-M_Υ$, which combined with the recent estimate from the $τ$-like decay sum rules in $e^+e^-\rar I=1$ hadrons data, leads to the {\it update average value} $\laα_s G^2\ra = (7.1\pm 0.9)\times 10^{-2}$ GeV$^4$ from the light and heavy quark systems. I also find $M_Υ-M_{η_b}\approx 63^{-29}_{+51}$ MeV implying the possible observation of the $η_b$ in the $Υ$-radiative decay.