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

The annihilation rates of vector $1^{--}$ charmonium and bottomonium $^3S_1$ states $V \rightarrow e^+e^-$ and $V\rightarrow 3γ$, $V \rightarrow γgg$ and $V \rightarrow 3g$ are estimated in the relativistic Salpeter method. We obtained $Γ(J/ψ\rightarrow 3γ)=6.8\times 10^{-4}$ keV, $Γ(ψ(2S)\rightarrow 3γ)=2.5\times 10^{-4}$ keV, $Γ(ψ(3S)\rightarrow 3γ)=1.7\times 10^{-4}$ keV, $Γ(Υ(1S)\rightarrow 3γ)=1.5\times 10^{-5}$ keV, $Γ(Υ(2S)\rightarrow 3γ)=5.7\times 10^{-6}$ keV, $Γ(Υ(3S)\rightarrow 3γ)=3.5\times 10^{-6}$ keV and $Γ(Υ(4S)\rightarrow 3γ)=2.6\times 10^{-6}$ 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.