The $X(2239)$ and $\eta(2225)$ as hidden-strange molecular states from $\Lambda\bar{\Lambda}$ interaction

In this work, we propose a possible assignment of the newly observed $X(2239)$, as well as the $\eta(2225)$, as a molecular state from the interaction of a baryon $\Lambda$ and an antibaryon $\bar{\Lambda}$. With the help of effective Lagrangians, the $\Lambda\bar{\Lambda}$ interaction is described within the one-boson-exchange model with $\eta$, $\eta'$, $\omega$, $\phi$, and $\sigma$ exchanges considered. After inserting the potential kernel into the quasipotential Bethe-Salpeter equation, the bound states from the $\Lambda\bar{\Lambda}$ interaction can be studied by searching for the pole of the scattering amplitude. Two loosely bound states with spin parities $I^G(J^{PC})=0^+(0^{-+})$ and $0^-(1^{--})$ appear near the threshold almost with the same parameter. The $0^-(1^{--})$ state can be assigned to the $X(2239)$ observed at BESIII, which is very close to the $\Lambda\bar{\Lambda}$ threshold. The scalar meson $\eta(2225)$ can be interpreted as a $0^+(0^{-+})$ state from the $\Lambda\bar{\Lambda}$ interaction. The annihilation effect is also discussed through a coupled-channel calculation plus a phenomenological optical potential. It provides large widths to two bound states produced from the $\Lambda\bar{\Lambda}$ interaction. The mass of the $1^-$ state is a little larger than the mass of the $0^-$ state after including the annihilation effect, which is consistent with our assignment of these two states as $X(2239)$ and $\eta(2225)$, respectively. The results suggest that further investigation is expected to understand the structures near the $\Lambda\bar{\Lambda}$ threshold, such as $X(2239)$, $\eta(2225)$, and $X(2175)$.