Strong decays of the $\phi(2170)$ as a fully-strange tetraquark state

We study strong decays of the $\phi(2170)$, along with its possible partner $X(2436)$, as two fully-strange tetraquark states of $J^{PC} = 1^{--}$. We consider seven decay channels: $\phi \eta$, $\phi \eta^\prime$, $\phi f_0(980)$, $\phi f_1(1420)$, $h_1(1415) \eta$, $h_1(1415) \eta^\prime$, and $h_1(1415) f_1(1420)$. Some of these channels are kinematically possible, and we calculate their relative branching ratios through the Fierz rearrangement. Future experimental measurements on these ratios can be useful in determining the nature of the $\phi(2170)$ and $X(2436)$. The $\phi(2170)$ has been observed in the $\phi f_0(980)$, $\phi \eta$, and $\phi \eta^\prime$ channels, and we propose to further examine it in the $h_1(1415) \eta$ channel. Evidences of the $X(2436)$ have been observed in the $\phi f_0(980)$ channel, and we propose to verify whether this structure exists or not in the $\phi \eta$, $\phi \eta^\prime$, $h_1(1415) \eta$, and $h_1(1415) \eta^\prime$ channels.