QCD sum rule predictions on gluonic tetraquark states with $J^{PC}=0^{+-},0^{--}$ and $1^{\pm \pm}$

Abstract: In this work, we present a systematic calculation of the mass spectrum for tetraquark hybrid states, focusing on the $$8_{[c\bar{c}]}\otimes 8_{[G]}\otimes 8_{[c\bar{c}]}$$ color configuration, within the framework of QCD sum rules. As an extension of our previous work on $0<sup>{++}$ and $0<sup>{-+}$ states, we now construct 18 distinct interpolating currents with $J^{PC} = 0<sup>{+-}$, $0<sup>{--}$, and $1<sup>{\pm\pm}$. Using operator product expansion (OPE) techniques and including nonperturbative contributions up to dimension six, we obtain key results: for the $0<sup>{+-}$, $1<sup>{--}$, and $1<sup>{-+}$ states, the predicted masses lie in the range of $7.2-7.3$ GeV, while the $1<sup>{+-}$ and $1<sup>{++}$ states have slightly lower masses, between 6.9 and 7.1 GeV. These predictions provide strong support for the possible existence of an $$8_{[c\bar{c}]}\otimes 8_{[G]}\otimes 8_{[c\bar{c}]}$$ component within the di-$J/ψ$ structure reported by LHCb. Moreover, our analogous calculations for tetrabottom hybrid states yield mass ranges of $19.4-19.5$ GeV (for $0<sup>{+-}$, $1<sup>{--}$, and $1<sup>{-+}$) and $19.2-19.3$ GeV (for $1<sup>{+-}$ and $1<sup>{++}$), offering crucial references for future searches.