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Gluonic and leptonic decays of heavy quarkonia and the determination of $α_s(m_c)$ and $α_s(m_b)$

QCD running coupling constant $α_s(m_c)$ and $α_s(m_b)$ are determined from heavy quarkonia $c\overline{c}$ and $b\overline{b}$ decays. The decay rates of $V\rightarrow 3g$ and $V\rightarrow e^+ e^-$ for $V=J/ψ$ and $Υ$ 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 $V\rightarrow 3g$ 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 {Γ(V\longrightarrow 3g)}% {Γ(V\longrightarrow e^{+}e^{-})}\approx 10,~32 $ for $V=J/ψ, ~Υ$ respectively, and the calculated widths , we find $α_{s}(m_c)=0.29\pm 0.02 $ and $α_s(m_b)=0.20\pm 0.02$. 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 $Λ_{\overline {MS}}^{(4)}% \approx 200MeV$. We also find that these results are mainly due to kinematic corrections and not sensitive to the dynamical models.