We conduct systematic calculations of the S-wave fully strange systems with ``normal" $\left(J^{ P C}=0^{ ++}, 1^{ +-}, 2^{ ++}\right)$ and ``exotic" $\left(J^{ P C}=0^{ +-}, 1^{ ++}, 2^{ +-}\right)$ C-parities, which are the strange analogue of the fully charmed tetraquark state $X(6900)$. Within a constituent quark potential model, we employ the Gaussian expansion method to solve the four-body Schr\"odinger equation and the complex scaling method to identify resonant states. We obtain a series of resonant states and zero-width states in the mass range of 2.7 to 3.3 GeV, with their widths ranging from less than 1 MeV to about 50 MeV. Their rms radii strongly indicate that they are compact tetraquark states. Among these states, the $T_{ 4s,2^{ ++}}(2714)$ may be the most likely one to be observed experimentally. We urge the experimental exploration of the $2^{ ++}$ $s s \bar{ s} \bar{ s}$ state around 2.7 GeV in the $\phi\phi$ channel. Since the lowest S-wave $s s \bar{ s} \bar{ s}$ state is around 2.7 GeV, the compact P-wave $s s \bar{ s} \bar{ s}$ states are expected to be heavier. Hence, $\phi(2170)$ and $X(2370)$ are unlikely to be compact tetraquark states.
Categories: hep-ph hep-ex hep-lat nucl-th