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By including the interference effect between the QCD and the QED diagrams, wecarry out a complete analysis on the exclusive productions of $e^+e^- \toJ/\psi+\chi_{cJ}$ ($J=0,1,2$) at the $B$ factories with $\sqrt{s}=10.6$ GeV atthe next-to-leading-order (NLO) level in $\alpha_s$, within the nonrelativisticQCD framework. It is found that the $\mathcal O (\alpha^3\alpha_s)$-order termsthat represent the tree-level interference are comparable with the usual NLOQCD corrections, especially for the $\chi_{c1}$ and $\chi_{c2}$ cases. Toexplore the effect of the higher-order terms, namely $\mathcal O(\alpha^3\alpha_s^2)$, we perform the QCD corrections to these $\mathcal O(\alpha^3\alpha_s)$-order terms for the first time, which are found to be ableto significantly influence the $\mathcal O (\alpha^3\alpha_s)$-order results.In particular, in the case of $\chi_{c1}$ and $\chi_{c2}$, the newly calculated$\mathcal O (\alpha^3\alpha_s^2)$-order terms can to a large extent counteractthe $\mathcal O (\alpha^3\alpha_s)$ contributions, evidently indicating theindispensability of the corrections. In addition, we find that, as thecollision energy rises, the percentage of the interference effect in the totalcross section will increase rapidly, especially for the $\chi_{c1}$ case.

Further studies on the exclusive productions of $J/\psi+\chi_{cJ}$ ($J=0,1,2$) via $e^+e^-$ annihilation at the $B$ factories