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.