A hybrid nonet with $J^{PC}=1^{-+}$ or a tetraquark 81-plet

Confirming the existence of hybrid states remains challenging due to their experimental indistinguishability from tightly bound tetraquarks and loosely bound molecules. To address this issue, we employ QCD sum rules to systematically investigate the $\pi_1(1600)$ and $\eta_1(1855)$ as candidate tetraquark states with exotic quantum numbers $J^{PC} = 1^{-+}$. Within the hybrid framework, an $SU(3)$ flavor nonet is expected, featuring two isoscalar configurations, $q\bar{q}g$ and $s\bar{s}g$, where $q = u/d$. In contrast, the tetraquark scenario predicts an $SU(3)$ flavor 81-plet comprising three isoscalar states: $qq\bar{q}\bar{q}$, $qs\bar{q}\bar{s}$, and $ss\bar{s}\bar{s}$. Our analysis yields a mass of $2.22^{+0.18}_{-0.26}$ GeV for the $ss\bar{s}\bar{s}$ tetraquark state, which is expected to decay predominantly into the $\phi\phi$ and $\eta f_1(1420)$ final states. Therefore, experimental scrutiny of their invariant mass spectra is pivotal for distinguishing between hybrid and tetraquark interpretations.