Further studies on the exclusive productions of $J/ψ+χ_{cJ}$ ($J=0,1,2$) via $e^+e^-$ annihilation at the $B$ factories
By including the interference effect between the QCD and the QED diagrams, we carry out a complete analysis on the exclusive productions of $e^+e^- \to J/ψ+χ_{cJ}$ ($J=0,1,2$) at the $B$ factories with $\sqrt{s}=10.6$ GeV at the next-to-leading-order (NLO) level in $α_s$, within the nonrelativistic QCD framework. It is found that the $\mathcal O (α^3α_s)$-order terms that represent the tree-level interference are comparable with the usual NLO QCD corrections, especially for the $χ_{c1}$ and $χ_{c2}$ cases. To explore the effect of the higher-order terms, namely $\mathcal O (α^3α_s^2)$, we perform the QCD corrections to these $\mathcal O (α^3α_s)$-order terms for the first time, which are found to be able to significantly influence the $\mathcal O (α^3α_s)$-order results. In particular, in the case of $χ_{c1}$ and $χ_{c2}$, the newly calculated $\mathcal O (α^3α_s^2)$-order terms can to a large extent counteract the $\mathcal O (α^3α_s)$ contributions, evidently indicating the indispensability of the corrections. In addition, we find that, as the collision energy rises, the percentage of the interference effect in the total cross section will increase rapidly, especially for the $χ_{c1}$ case.