Annihilation Rate of $2^{++}$ Charmonium and Bottomonium

Two-photon annihilation rates of $2^+$ tensor charmonium and bottomonium up to third radial excited states are estimated in the relativistic Salpeter method. Full Salpeter equation for $2^+$ tensor state is solved with a well defined relativistic wave function and we calculated the annihilation amplitude using the Mandelstam formalism. Our estimates of the decay widths are: $\Gamma(\chi_{c2} \to 2\gamma)=501$ eV, $\Gamma(\chi'_{c2} \to 2\gamma)=534$ eV, $\Gamma(\chi_{b2} \to 2\gamma)=7.4$ eV and $\Gamma(\chi'_{b2} \to 2\gamma)=7.7$ eV. We also give total decay widths of the lowest states estimated by the two-gluon decay rates, and the results are: $\Gamma_{tot}(\chi_{c2})=2.64$ MeV, $\Gamma_{tot}(\chi_{b2})=0.220$ MeV and $\Gamma_{tot}(\chi'_{b2})=0.248$ MeV.