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

Inspired by the significant effect of the QED single-photon-fragmentation diagrams on the process of $e^+e^- \to J/\psi+\eta_c$, by introducing the cross terms up to $\mathcal O (\alpha^3\alpha_s^2)$ order between the QCD and QED diagrams, we carry out a further study on the exclusive productions of $e^+e^- \to J/\psi+\chi_{cJ}$ ($J=0,1,2$) at the $B$ factories with $\sqrt{s}=10.6$ GeV, within the nonrelativistic QCD framework. We find that, for $J/\psi+\chi_{c0}$, the newly introduced interference terms are mild, much smaller than the well-known large QCD corrections. However, for the $\chi_{c1}$ and $\chi_{c2}$ cases, the $\mathcal O (\alpha^3\alpha_s)$-order terms representing the tree-level interference can reach up to $70-80\%$ of the usual QCD corrections. To explore 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 able to largely counteract the $\mathcal O (\alpha^3\alpha_s)$ contributions, indicating the indispensability of the corrections. Consequently, due to the large cancellation between the $\mathcal O (\alpha^3\alpha_s)-$ and $\mathcal O (\alpha^3\alpha_s^2)-$ order terms, the effect of the newly introduced interference terms on the total cross sections is moderate, however, more and more significant as the collision energy rises.