Effective gluon mass and the determination of ${\alpha }_s$ from $\frac{J}{\psi }$ and $\Upsilon$ branching ratios

The phase space modification associated with a nonvanishing effective mass for the primary gluons, $M_g=0.66\mathrm{}\pm 0.08$ GeV for the $\frac{J}{\psi }$ and $M_g=1.17\mathrm{}\pm 0.08$ GeV for the $\Upsilon$, is shown to be crucial for a consistent description of the photon spectrum from their radiative decays and for the determination of ${\alpha }_s$ from the recent, precise quarkonia decay branching ratios. In this approach, the role of the relativistic corrections is marginal and, after applying the $M_g$-dependent corrections, a good agreement is obtained with the relative perturbative running, ${\alpha }_s\left(m_c\right)\sim 0.30\pm 0.02$ and ${\alpha }_s\left(m_b\right)\sim 0.21\pm 0.01$, and with the extrapolation from deep inelastic scattering. On the other hand, for $M_g=0\mathrm{}$, the analysis of all experimental $c\overline{c}$ and $b\overline{b}$ quarkonia branching ratios is consistent with the same effective value of the strong interaction coupling constant ${\alpha }_s^{\mathrm{eff}}\sim 0.185\pm 0.010$. By assuming a "genuine," i.e., process-independent, gluon mass (∼ 1.2 GeV or higher) to be dynamically generated one predicts a strong suppression of the gluon splitting process at the $\frac{J}{\psi }$ and the hadronic final states should be mainly produced via gluon fusion into light $q\overline{q}$ pairs thus effectively reducing the fitted value of $M_g$ from the photon spectrum in $\frac{J}{\psi }\to \gamma +X$. The gluon fusion mechanism allows us to explain the structure of the hadronic final states observed in $\frac{J}{\psi }$ decays and their close similarity to the continuum $e^{+}{e}^{-}\to$ hadrons annihilation at comparable center of mass energies.