Weak Decays of $J\mathrm{/}\psi$ and $\mathrm{{\rm Y}}(1S)$

In this paper we study the weak decays of $J\mathrm{/}\psi$ and $\mathrm{{\rm Y}}(1S)$. Using the Bethe-Salpeter method, we calculate the hadronic transition amplitude and give the form factors. We find that two new form factors $h_1$ and $h_2$, which do not appear in existing literature, have contributions in $1^{-}\to 1^{-}$ decays. They affect the branching ratios of semi-leptonic and non-leptonic decays by the rate of $3\mathrm{\%}\sim 6\mathrm{\%}$ and $2\mathrm{\%}\sim 14\mathrm{\%}$, respectively, so their contributions can not be ignored and should be considered. Our results show that, for the semi-leptonic decay modes, the largest branching ratios are of the order of ${10}^{-10}$ both for $J\mathrm{/}\psi$ and $\mathrm{{\rm Y}}(1S)$ decays, and the largest branching ratios of non-leptonic decays are of the order of ${10}^{-9}$ for $J\mathrm{/}\psi$ and ${10}^{-10}$ for $\mathrm{{\rm Y}}(1S)$.