Study of the $\omega$ meson family and newly observed $\omega$-like state $X(2240)$

Since the present $\omega$ meson family has not been established, in this work, we carry out an investigation of the mass spectrum and Okubo-Zweig-Iizuka a allowed two-body strong decay of the $S$-wave and the $D$-wave $\omega$ mesons, and make the comparison with the experimental data of these reported $\omega$ states and the $\omega$-like $X(2240)$ state observed by BESIII. By this study, we not only suggest the possible assignments to these observed $\omega$ states under the framework of the $\omega$ meson family, but also predict three $\omega$ mesons ($\omega(5S)$, $\omega(2D)$ , and $\omega(4D)$) which are still missing in experiment. The present study may provide valuable information to further construct the $\omega$ meson family. Considering the present running status of BESIII, we also suggest that BESIII should pay more attention to the issue of $\omega$ meson with accumulating more data.