We estimate the decay rates of ${\eta }_c\to 2\gamma$, ${\eta }_c^{\prime }\to 2\gamma$, and $J/\psi \to e^{+}{e}^{-}$, ${\psi }^{\prime }\to e^{+}{e}^{-}$, by taking into account both relativistic and QCD radiative corrections. The decay amplitudes are derived in the Bethe-Salpeter formalism. The Bethe-Salpeter equation with a QCD-inspired interquark potential is used to calculate the wave functions and decay widths for these $c\overline{c}$ states. We find that the relativistic correction to the ratio $R\equiv \Gamma ({\eta }_c\to 2\gamma )/\Gamma (J/\psi \to e^{+}{e}^{-})$ is negative and tends to compensate the positive contribution from the QCD radiative correction. Our estimates give $\Gamma ({\eta }_c\to 2\gamma )=\left(6\mathrm{}-7\right)$ keV and $\Gamma ({\eta }_c^{\prime }\to 2\gamma )=2\mathrm{}$ keV, which are smaller than their nonrelativistic values. The hadronic widths $\Gamma ({\eta }_c\to 2g)=\left(17-23\right)$ MeV and $\Gamma ({\eta }_c^{\prime }\to 2g)=\left(5-7\right)$ MeV are then indicated accordingly to the first-order QCD radiative correction, if ${\alpha }_s{(m}_c)=0.26\mathrm{}-0.29$. The decay widths for the $b\overline{b}$ states are also estimated. We show that when making the assumption that the quarks are on their mass shells our expressions for the decay widths will become identical to that in the nonrelativistic QCD theory to the next to leading order of $v^2$ and ${\alpha }_s.$

• Received 22 January 1996

DOI:https://doi.org/10.1103/PhysRevD.56.368