Production of charmonium $\chi_{cJ}(2P)$ plus one $\omega$ meson by $e^+e^-$ annihilation

Inspired by the recent observation of $e^+e^-\to \omega X(3872)$ by the BESIII Collaboration, in this work we study the production of the charmonium $\chi_{cJ}(2P)$ by $e^+e^-$ annihilation. We find that the $e^+e^-\to\omega\chi_{c0}(2P)$ and $e^+e^-\to \omega\chi_{c2}(2P)$ have sizable production rates, when taking the cross section data from $e^+e^-\to \omega X(3872)$ as the scaling point and treating the $X(3872)$ as the charmonium $\chi_{c1}(2P)$. Considering that the dominant decay modes of $\chi_{c0}$ and $\chi_{c2}(2P)$ involve $D\bar{D}$ final states, we propose that $e^+e^-\to \omega D\bar{D}$ is an ideal process to identify $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$, which is similar to the situation that happens in the $D\bar{D}$ invariant mass spectrum of the $\gamma\gamma\to D\bar{D}$ and $B^+\to D^+{D}^- K^+$ processes. With continuous accumulation of experimental data, these proposed production processes offer a promising avenue for exploration by the BESIII and Belle II collaborations.