Highly excited and exotic fully-strange tetraquark states

Some hadrons have the exotic quantum numbers that the traditional $\bar{q}q$ mesons and $qqq$ baryons can not reach, such as $J^{PC}=0^{--}\mathrm{/}{0}^{+-}\mathrm{/}{1}^{-+}\mathrm{/}{2}^{+-}\mathrm{/}{3}^{-+}\mathrm{/}{4}^{+-}$, etc. We investigate for the first time the exotic quantum number $J^{PC}=4^{+-}$, and study the fully-strange tetraquark states with such an exotic quantum number. We systematically construct all the diquark-antidiquark interpolating currents, and apply the method of QCD sum rules to calculate both the diagonal and off-diagonal correlation functions. The obtained results are used to construct three mixing currents that are nearly non-correlated, and we use one of them to extract the mass of the lowest-lying state to be $2.85_{-0.22}^{+0.19}$ GeV. We apply the Fierz rearrangement to transform this mixing current to be the combination of three meson-meson currents, and the obtained Fierz identity suggests that this state dominantly decays into the $P$-wave $\phi (1020)f_2^{\mathrm{\prime }}(1525)$ channel. This fully-strange tetraquark state of $J^{PC}=4^{+-}$ is a purely exotic hadron to be potentially observed in future particle experiments.