Fully-strange tetraquark $ss\bar{s}\bar{s}$ spectrum and possible experimental evidence

In this work we construct 36 tetraquark configurations for the $1S$-, $1P$-, and $2S$-wave states, and make a prediction of the mass spectrum for the tetraquark $ss\bar{s}\bar{s}$ system in the framework of a nonrelativistic potential quark model without the diquark-antidiquark approximation. The model parameters are well determined by our previous study of the strangeonium spectrum. We find that the resonances $f_0(2200)$ and $f_2(2340)$ may favor the assignments of ground states $T_{(ss\bar{s}\bar{s})0^{++}}(2218)$ and $T_{(ss\bar{s}\bar{s})2^{++}}(2378)$, respectively, and the newly observed $X(2500)$ at BESIII may be a candidate of the lowest mass $1P$-wave $0^{-+}$ state $T_{(ss\bar{s}\bar{s})0^{-+}}(2481)$. Signals for the other $0^{++}$ ground state $T_{(ss\bar{s}\bar{s})0^{++}}(2440)$ may also have been observed in the $\phi\phi$ invariant mass spectrum in $J/\psi\to\gamma\phi\phi$ at BESIII. The masses of the $J^{PC}=1^{--}$ $T_{ss\bar{s}\bar{s}}$ states are predicted to be in the range of $\sim 2.44-2.99$ GeV, which indicates that the $\phi(2170)$ resonance may not be a good candidate of the $T_{ss\bar{s}\bar{s}}$ state. This study may provide a useful guidance for searching for the $T_{ss\bar{s}\bar{s}}$ states in experiments.