Annihilation Rates of Heavy $1^{--}$ S-wave Quarkonia in Salpeter Method

The annihilation rates of vector $1^{--}$ charmonium and bottomonium ${}^3{S}_1$ states $V\to e^{+}{e}^{-}$ and $V\to 3\gamma$, $V \rightarrow \gamma gg$and$V \rightarrow 3g$ are estimated in the relativistic Salpeter method. We obtained $\mathrm{\Gamma }(J\mathrm{/}\psi \to 3\gamma )=6.8\times {10}^{-4}$ keV, $\mathrm{\Gamma }(\psi (2S)\to 3\gamma )=2.5\times {10}^{-4}$ keV, $\mathrm{\Gamma }(\psi (3S)\to 3\gamma )=1.7\times {10}^{-4}$ keV, $\mathrm{\Gamma }(\mathrm{{\rm Y}}(1S)\to 3\gamma )=1.5\times {10}^{-5}$ keV, $\mathrm{\Gamma }(\mathrm{{\rm Y}}(2S)\to 3\gamma )=5.7\times {10}^{-6}$ keV, $\mathrm{\Gamma }(\mathrm{{\rm Y}}(3S)\to 3\gamma )=3.5\times {10}^{-6}$ keV and $\mathrm{\Gamma }(\mathrm{{\rm Y}}(4S)\to 3\gamma )=2.6\times {10}^{-6}$ keV. In our calculations, special attention is paid to the relativistic correction, which is important and can not be ignored for excited $2S$, $3S$ and higher excited states.