Search for $C=+$ charmonium and XYZ states in $e^{+}{e}^{-}\to \gamma +H$ at BESIII

Within the framework of nonrelativistic quantum chromodynamics, we study the production of $C=+$ charmonium states $H$ in $e^{+}{e}^{-}\to \gamma +H$ at BESIII with $H={\eta }_c(nS)$ (n=1, 2, 3, and 4), ${\chi }_{cJ}(nP)$ (n=1, 2, and 3), and ${}^1{D}_2(nD)$ (n=1 and 2). The radiative and relativistic corrections are calculated to next-to-leading order for $S$ and $P$ wave states. We then argue that the search for $C=+$ $XYZ$ states such as $X(3872)$, $X(3940)$, $X(4160)$, and $X(4350)$ in $e^{+}{e}^{-}\to \gamma +H$ at BESIII may help clarify the nature of these states. BESIII can search $XYZ$ states through two body process $e^{+}{e}^{-}\to \gamma H$, where $H$ decay to $J\mathrm{/}\psi {\pi }^{+}{\pi }^{-}$, $J\mathrm{/}\psi \varphi$, or $D\bar{D}$. This result may be useful in identifying the nature of $C=+$ $XYZ$ states. For completeness, the production of $C=+$ charmonium in $e^+e^-\to \gamma +~H$ at B factories is also discussed.