Production of $1^{-+}$ exotic charmonium-like states in electron-positron collisions

The absence of observed charmonium-like states with the exotic quantum numbers $J^{PC}=1^{-+}$ has prompted us to investigate the production rates of the $1^{-+}$ $D{\bar{D}}_1(2420)$ and $D^{\ast }{\bar{D}}_1(2420)$ hadronic molecules, which we refer to as ${\eta }_{c1}$ and ${\eta }_{c1}^{\mathrm{\prime }}$, respectively, in electron-positron collisions. Assuming a hadronic molecular nature for the vector charmonium-like states $\psi (4360)$ and $\psi (4415)$, we evaluate the radiative decay widths of $\psi (4360)\to \gamma {\eta }_{c1}$ and $\psi (4415)\to \gamma {\eta }_{c1}^{\mathrm{\prime }}$. Using these decay widths, we estimate the cross sections for producing ${\eta }_{c1}$ and ${\eta }_{c1}^{\mathrm{\prime }}$ in electron-positron annihilations, as well as the event numbers at the planned Super $\tau$-Charm Facility. Our results suggest that the ideal energy region for observing these states is around $4.44$ and $4.50$ GeV, just above the $D^{\ast }{\bar{D}}_1(2420)$ and $D^{\ast }{\bar{D}}_2^{\ast }(2460)$ thresholds, respectively.