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 $$8_{[c\bar{c}]}\otimes 8_{[G]}\otimes 8_{[c\bar{c}]}$$ 8 [ c c ¯ ] ⊗ 8 [ G ] ⊗ 8 [ c c ¯ ] color configuration, within the framework of QCD sum rules. As an extension of our previous work on $$0^{++}$$ 0 + + and $$0^{-+}$$ 0 - + states, we now construct 18 distinct interpolating currents with $$J^{PC} = 0^{+-}$$ J PC = 0 + - , $$0^{--}$$ 0 - - , and $$1^{\pm \pm }$$ 1 ± ± . Using operator product expansion (OPE) techniques and including nonperturbative contributions up to dimension six, we obtain key results: for the $$0^{+-}$$ 0 + - , $$1^{--}$$ 1 - - , and $$1^{-+}$$ 1 - + states, the predicted masses lie in the range of $$7.2-7.3$$ 7.2 - 7.3 GeV, while the $$1^{+-}$$ 1 + -