A bstract Within the framework of nonrelativistic quantum chromodynamics, we study the production of C ?=?+ charmonium states H in e _(+) e _(?)?→?γ?+? H at BESIII with H ?=?η~(c)( nS ) (n?=?1, 2, 3, and 4), χ~( 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 _(?)?→?γ?+? H at BESIII may help clarify the nature of these states. BESIII can search for XY Z states through two body process e _(+) e _(?)?→?γ H , where H decay to J/ ψπ_(+)π_(?), J/ ψ ? , or D $ overline{D} $ . This result may be useful in identifying the nature of C ?=?+ XYZ states. For completeness, the production of C ?=?+ charmonium in e _(+) e _(?)?→?γ?+? H at B factories is also discussed.
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