QCD sum rule studies on the $ss\bar{s}\bar{s}$ tetraquark states of $J^{PC}=0^{-+}$

We apply the method of QCD sum rules to study the $ss\bar{s}\bar{s}$ tetraquark states of $J^{PC}=0^{-+}$. We construct all the relevant $s s \bar s \bar s$ tetraquark currents, and find that there are only two independent ones. We use them to further construct two weakly-correlated mixed currents. One of them leads to reliable QCD sum rule results and the mass is extracted to be $2.51_{-0.12}^{+0.15}$ GeV, suggesting that the $X(2370)$ or the $X(2500)$ can be explained as the $ss\bar{s}\bar{s}$ tetraquark state of $J^{PC}=0^{-+}$. To verify this interpretation, we propose to further study the $\pi\pi/K \bar K$invariantmassspectraofthe$J/\psi \to \gamma \pi \pi \eta^\prime/\gamma K \bar K \eta^\prime$decaysinBESIIItoexaminewhetherthereexiststhe$f_0(980)$ resonance.