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

Within the framework of nonrelativistic quantum chromodynamics, we study the production of $C=+$ charmonium states $H$ in $e^+e^-\to γ~+~H$ at BESIII with $H=η_c(nS)$ (n=1, 2, 3, and 4), $χ_{cJ}(nP)$ (n=1, 2, and 3), and $^1D_2(nD)$ (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 $e^+e^-\to γ~+~H$ at BESIII may help clarify the nature of these states. BESIII can search $XYZ$ states through two body process $e^+e^-\to γH$, where $H$ decay to $J/ψπ^+π^-$, $J/ψϕ$, 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 $e^+e^-\to γ+~H$ at B factories is also discussed.