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

Abstract

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

and

$$ \eta _1(1855) $$ η 1 ( 1855 )

as candidate tetraquark states with exotic quantum numbers

$$ J^{PC} = 1^{-+}$$ J PC = 1 - +

. Within the hybrid framework, an

$$ SU(3) $$ S U ( 3 )

flavor nonet is expected, featuring two isoscalar configurations,

$$ q{\bar{q}}g $$ q q ¯ g

and

$$s{\bar{s}}g$$ s s ¯ g

, where

$$q = u/d $$ q = u / d

. Moreover, the decay width of the

$$ s{\bar{s}}g $$ s s ¯ g

state into the

$$ K^* {\bar{K}}^* $$ K ∗ K ¯ ∗

channel ( VV ) is found to be two orders of magnitude smaller than that into the

$$ \eta \eta ^\prime $$ η η ′

channel ( PP ) [1]. In contrast, the tetraquark scenario predicts an

$$ SU(3) $$ S U ( 3 )

flavor 81-plet comprising three isoscalar states:

$$ qq{\bar{q}}{\bar{q}} $$ q q q ¯ q ¯

,

$$ qs{\bar{q}}{\bar{s}} $$ q s q ¯ s ¯

, and

$$ ss{\bar{s}}{\bar{s}} $$ s s s ¯ s ¯

. Our analysis yields a mass of

$$ 2.22^{+0.18}_{-0.26}~\textrm{GeV} $$ 2 . 22 - 0.26 + 0.18 GeV

for the

$$ ss{\bar{s}}{\bar{s}} $$ s s s ¯ s ¯

tetraquark state, which is expected to decay predominantly into the

$$ \phi \phi $$ ϕ ϕ

channel ( VV ) and the

$$ \eta f_1(1420) $$ η f 1 ( 1420 )

channel, while its decay into the the

$$ \eta \eta ^\prime $$ η η ′

channel ( PP ) is significantly suppressed. Therefore, experimental searches focusing on the VV decay channels are essential for discriminating between hybrid and tetraquark scenarios.