Light double-gluon hybrid states with the exotic quantum numbers $J^{PC}=1^{-+}$ and $3^{-+}$

We apply the QCD sum rule method to study the double-gluon hybrid states with the quark-gluon contents $\bar{q}qgg$ ($q=u\mathrm{/}d$) and $\bar{s}sgg$. We construct twenty-eight double-gluon hybrid currents, eleven of which are found to be zero due to some internal symmetries between the two gluons fields. We concentrate on the non-vanishing currents with the exotic quantum numbers $J^{PC}=1^{-+}$ and $3^{-+}$. Their masses are calculated to be $M_{\mathrm{\mid }\bar{q}qgg;1^{-+}⟩}=4.35_{-0.30}^{+0.26}$ GeV, $M_{\mathrm{\mid }\bar{s}sgg;1^{-+}⟩}=4.49_{-0.30}^{+0.25}$ GeV, $M_{\mathrm{\mid }\bar{q}qgg;3^{-+}⟩}=3.02_{-0.31}^{+0.24}$ GeV, and $M_{\mathrm{\mid }\bar{s}sgg;3^{-+}⟩}=3.16_{-0.28}^{+0.22}$ GeV. The decay behaviors of the $J^{PC}=3^{-+}$ states are studied, and we propose to search for them in the $\pi a_1(1260)\mathrm{/}\rho \omega \mathrm{/}\phi \phi$ channels in future particle experiments.