Discovery potential of charmonium $2P$ states through the $e^+e^- \to \gamma D\bar{D}$ processes

In this work, we investigate the production of charmonium $2P$ states via the $e^+e^-\to \gamma D\bar{D}$ process at $\sqrt{s} = 4.23$ GeV. Using the measured cross-section data for $e^+e^-\to \gamma X(3872)$ as a reference, we calculate the cross sections for $e^+e^-\to \gamma \chi_{c0}(2P)$ and $e^+e^-\to \gamma \chi_{c2}(2P)$. Since the $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$ states predominantly decay into $D\bar{D}$ final states, we also predict the corresponding $D\bar{D}$ invariant mass spectrum for the $e^+e^-\to \gamma D\bar{D}$ process. Our results indicate that $e^+e^-\to \gamma D\bar{D}$ is an ideal process for identifying the $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$ states, analogous to the $\gamma\gamma\to D\bar{D}$ and $B^+\to D^+D^-K^+$ processes. This study highlights the discovery potential of charmonium $2P$ states at BESIII and Belle II.