Weak Decays of \(J/\psi\) and \(\Upsilon(1S)\)

In this paper we study the weak decays of \(J/\psi\) and \(\Upsilon(1S)\). 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 \(3\%\sim6\%\) and \(2\%\sim14\%\), 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 \(J/\psi\) and \(\Upsilon(1S)\) decays, and the largest branching ratios of non-leptonic decays are of the order of \(10^{-9}\) for \(J/\psi\) and \(10^{-10}\) for \(\Upsilon(1S)\).