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Inspired by the recent measurement of the process $e^+e^-\rightarrow\Lambda\bar{\Lambda}$, we calculate the mass spectrum of the $\phi$ meson withthe GI model. For the excited vector strangeonium states $\phi(3S,~4S,~5S,~6S)$and $\phi(2D,~3D,~4D,~5D)$, we further investigate the electronic decay widthwith the Van Royen-Weisskopf formula, and the partial widths of the$\Lambda\bar{\Lambda}$, $\Xi^{-(*)}\bar{\Xi}^+$, and$\Sigma^{+(*)}\bar{\Sigma}^{-(*)}$ decay modes with the extended quark paircreation model. We find that the electronic decay width of the $D$-wave vectorstrangeonium is about $3\sim8$ times larger than that of the $S$-wave vectorstrangeonium. Around 2232 MeV the partial decay width of the$\Lambda\bar{\Lambda}$ mode can reach up to several MeV for $\phi(3^3S_1)$,while the partial $\Lambda\bar{\Lambda}$ decay width of $\phi(2^3D_1)$ is$\mathcal{O}(10^{-3})$ keV. If the threshold enhancement reported by the BESIIICollaboration arises from the strangeonium meson, this state is very likely tobe the $\phi(3^3S_1)$ state. We also note that the $\Lambda\bar{\Lambda}$ and$\Sigma^{+}\bar{\Sigma}^{-}$ partial decay widths of the states $\phi(3^3D_1)$and $\phi(4^3S_1)$ are about several MeV, respectively, which are enough to beobserved in future experiments.

A possible explanation of the threshold enhancement in the process $e^+e^-\rightarrow \Lambda\bar{\Lambda}$