Heavy quarkonia mass-splittings in QCD: test of the $1/m$-expansion and estimates of $<{\alpha }_s{G}^2>$ and ${\alpha }_s$

Abstract

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. We show that at the $c$ and $b$ quark mass scales the $1/m$-expansion does not converge for these quarkonia channels, while a connection of our mass and width formulae, with the potential model ones is done. Using the present value of the QCD coupling ${\alpha }_s$, we deduce the value: $<{\alpha }_s{G}^2>=(7.5\pm 2.5)\times 10^{-2}$ GeV${}^4$ of the gluon condensate from $M_{\psi }-M_{{\eta }_c}$ and $M_{{\chi }_b}-M_{{\rm Y}}$, which we compare with the ones from different fits of the heavy and light quark channels. We also find that $M_{{\chi }_c(P_1^1)}-M_{{\chi }_c(P_1^3)}$ is gouverned by the radiative corrections and gives ${\alpha }_s$(1.3 GeV) = 0.64${}_{-0.18}^{+0.36}$ for 4 flavours, implying ${\alpha }_s(M_Z)=0.127\pm 0.011$. Our predictions for the splittings of different heavy quarkonia states are summarized in Table 2, where, in particular, we find $M_{{\rm Y}}-M_{{\eta }_b}\approx 63_{+51}^{-29}$ MeV implying the possible observation of the ${\eta }_b$ in the ${\rm Y}$-radiative decay.

Comments: Version to appear in Phys. Lett. B which contains more discussions on the estimate of the gluon condensate and ${\alpha }_s$