Gluonic and leptonic decays of heavy quarkonia and the determination of ${\alpha }_s(m_c)$ and ${\alpha }_s(m_b)$

Abstract

QCD running coupling constant ${\alpha }_s(m_c)$ and ${\alpha }_s(m_b)$ are determined from heavy quarkonia $c\overline{c}$ and $b\overline{b}$ decays. The decay rates of $V\to 3g$ and $V\to e^{+}{e}^{-}$ for $V=J/\psi$ and ${\rm Y}$ are estimated by taking into account both relativistic and QCD radiative corrections. The decay amplitudes are derived in the Bethe-Salpeter formalism, and the decay rates are estimated by using the meson wavefunctions which are obtained with a QCD-inspired inter-quark potential. For the $V\to 3g$ decay we find the relativistic correction to be very large and to severely suppress the decay rate. Using the experimental values of ratio R_g\equiv \frac {\Gamma (V\longrightarrow 3g)}% {\Gamma (V\longrightarrow e^{+}e^{-})}\approx 10,~32 for $V=J/\psi ,{\rm Y}$ respectively, and the calculated widths , we find ${\alpha }_s(m_c)=0.29\pm 0.02$ and ${\alpha }_s(m_b)=0.20\pm 0.02$. These values for the QCD running coupling constant are substantially enhanced, as compared with the ones obtained without relativistic corrections, and are consistent with the QCD scale parameter ${\Lambda }_{\overline{MS}}^{(4)}$. We also find that these results are mainly due to kinematic corrections and not sensitive to the dynamical models.

Comments: 15 pages in Latex