Fully strange tetraquark resonant states as the cousins of $X(6900)$

We conduct systematic calculations of the S-wave fully strange systems with ``normal" (JP C=0⁺⁺, 1⁺⁻, 2⁺⁺) and ``exotic" (JP C=0⁺⁻, 1⁺⁺, 2⁺⁻) 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 s̄ s̄ state around 2.7 GeV in the φφ channel. Since the lowest S-wave s s s̄ s̄ state is around 2.7 GeV, the compact P-wave s s s̄ s̄ states are expected to be heavier. Hence, φ(2170) and $X(2370)$ are unlikely to be compact tetraquark states.