Predictions of the hybrid mesons with exotic quantum numbers $J^{PC}=2^{+-}$

We study the non-strange and strangeonium light hybrid mesons with $J^{PC}=2^{+-}$ by using the method of QCD sum rules. The local hybrid interpolating currents with three Lorentz indices are constructed to couple to such exotic quantum numbers. We calculate the correlation functions up to dimension eight condensates at the leading order of $\alpha_{s}$. In our results, the masses of the non-strange $b_2$ and $h_2$ hybrids are about $2.65~\mathrm{GeV}$, while that of the strangeonium $h_2^\prime$ hybrid is about $2.72~\mathrm{GeV}$. Such exotic $2^{+-}$ hybrids can be generated through both the two-gluon and three-gluon emission processes in the radiative decays of $\chi_{cJ}$. Moreover, these hybrid mesons may be detectable due to their peculiar decay behaviors and small decay widths. Using the high-statistics data samples of $\psi(3686)$ in BESIII and BelleII, it is possible to hunt for such hybrid states through the partial wave analyses in the $b_2\to\omega/a_1/h_1/a_2\pi\to4\pi$, $h_2\to\rho\pi\to3\pi$ and $h_2\to b_1\pi\to5\pi$ processes.