Investigating hybrid mesons with $0^{+-}$ and $2^{+-}$ exotic quantum numbers

Hybrid mesons occupy a unique position in the hadronic spectrum, offering a valuable insight into the non-perturbative nature of the glue field. Hybrid states with exotic $J^{PC}$ quantum numbers, such as the $1^{-+}$, $0^{+-}$ and $2^{+-}$, cannot mix with conventional $q\bar{q}$ mesons, making them critical for establishing the full spectrum of hybrid mesons. In this work, we investigate the masses and the strong decays of $0^{+-}$ and $2^{+-}$ hybrid mesons, which could be regarded as the first orbitally excited states of the $\pi_1(1600)$ and $\eta_1(1855)$ mesons. We develop a constituent gluon model and apply it to study the mass spectra and decay modes of these exotic states. Our results suggest that experiment may searched for these $0^{+-}$ and $2^{+-}$ hybrid mesons in the mass region between 2.0 GeV and 2.4 GeV. Additionally, we identify some key decay channels which should be targeted in future experiments.