Light single-gluon hybrid states with various (exotic) quantum numbers

We apply the QCD sum rule method to study the light single-gluon hybrid states with various (exotic) quantum numbers. We construct twenty-four single-gluon hybrid currents, and use eighteen of them to calculate the masses of forty-four single-gluon hybrid states with the quark-gluon contents $\bar q q g$ ($q=u/d$) and $\bar s s g$. We concentrate on the hybrid states with the exotic quantum number $J^{PC} = 1^{-+}$, whose masses and widths are calculated to be $M_{|\bar q q g;1^-1^{-+}\rangle} =1.67^{+0.15}_{-0.17}$ GeV, $\Gamma_{|\bar q q g;1^-1^{-+}\rangle} = 530^{+540}_{-330}$ MeV, $M_{|\bar q q g;0^+1^{-+}\rangle} = 1.67^{+0.15}_{-0.17}$ GeV, $\Gamma_{|\bar q q g;0^+1^{-+}\rangle} = 120^{+160}_{-110}$ MeV, $M_{|\bar s s g;0^+1^{-+}\rangle} = 1.84^{+0.14}_{-0.15}$ GeV, and $\Gamma_{|\bar s s g;0^+1^{-+}\rangle} = 100^{+110}_{-~80}$ MeV. Our results support the interpretations of the $\pi_1(1600)$ and $\eta_1(1855)$ as the hybrid states $|\bar q q g;1^-1^{-+}\rangle$ and $|\bar s s g;0^+1^{-+}\rangle$, respectively. Considering the uncertainties, our results suggest that the $\pi_1(1600)$ and $\eta_1(1855)$ may also be interpreted as the hybrid states $|\bar q q g;1^-1^{-+}\rangle$ and $|\bar q q g;0^+1^{-+}\rangle$, respectively. To differentiate these two assignments and to verify whether they are hybrid states or not, we propose to examine the $a_1(1260) \pi$ decay channel in future experiments.