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

Abstract

The annihilation rates of vector $1^{--}$ charmonium and bottomonium ${}^3{S}_1$ states $V\to e^{+}{e}^{-}$ and $V\to 3\gamma$, $V\to \gamma gg$ and $V\to 3g$ are estimated in the relativistic Salpeter method. We obtained $\Gamma (J/\psi \to 3\gamma )=6.8\times 10^{-4}$ keV, $\Gamma (\psi (2S)\to 3\gamma )=2.5\times 10^{-4}$ keV, $\Gamma (\psi (3S)\to 3\gamma )=1.7\times 10^{-4}$ keV, $\Gamma ({\rm Y}(1S)\to 3\gamma )=1.5\times 10^{-5}$ keV, $\Gamma ({\rm Y}(2S)\to 3\gamma )=5.7\times 10^{-6}$ keV, $\Gamma ({\rm Y}(3S)\to 3\gamma )=3.5\times 10^{-6}$ keV and $\Gamma ({\rm Y}(4S)\to 3\gamma )=2.6\times 10^{-6}$ 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.

Comments: 10 pages,2 figures, 5 tables