$\mathcal{O}({\alpha }_s^2)$ corrections to $J\mathrm{/}\psi +{\chi }_{c0,1,2}$ production at $B$ factories

We compute the $\mathcal{O}({\alpha }_s^2)$ corrections to the exclusive channels $e^{+}{e}^{-}\to J\mathrm{/}\psi +{\chi }_{cJ}$ ($J=0,1,2$) at $\sqrt{s}=10.58$ GeV within the nonrelativistic QCD (NRQCD) factorization framework. The validity of NRQCD factorization at $\mathcal{O}({\alpha }_s^2)$ has been confirmed for these double-charmonium exclusive production processes. We analyze the impact of the $\mathcal{O}({\alpha }_s^2)$ corrections on the polarized and unpolarized cross sections, as well as the $J\mathrm{/}\psi$ angular distributions, which largely reduce the renormalization scale dependence but increase the $\mathcal{O}({\alpha }_s)$ NRQCD predictions to some extent for ${\chi }_{c0,1}$. With high numerical accuracy, our predictions for $\sigma (J\mathrm{/}\psi +{\chi }_{c1,2})$ through ${\mathcal O}(\alpha_s^2)$ are compatible with the upper limit of the $\mathtt{\text{Belle}}$ measurement. Although the theoretical prediction for $\sigma (J\mathrm{/}\psi +{\chi }_{c0})$ is consistent with both $\mathtt{\text{Belle}}$ and $\mathtt{\text{BaBar}}$ measurements within uncertainties, there still exists serious tension between the predicted and the measured profiles for the $J\mathrm{/}\psi$ angular distribution. Regarding the bright observation prospect of the $e^{+}{e}^{-}\to J\mathrm{/}\psi +{\chi }_{c1,2}$ channels in $\mathtt{\text{Belle}}\mathtt{\text{2}}$ experiment, it is interesting to compare the future measurements with our NRQCD predictions. The more accurate measurement of $e^{+}{e}^{-}\to J\mathrm{/}\psi +{\chi }_0$ at $\mathtt{\text{Belle}}\mathtt{\text{2}}$ will also help to clarify the long-standing puzzle of $J\mathrm{/}\psi$ angular distribution.