$O({\alpha }_s)$ corrections to $J\mathrm{/}\psi +{\chi }_{cJ}$ production at $B$ factories

We investigate the $\mathcal{O}({\alpha }_s)$ corrections to $e^+e^-\to J/\psi(\psi’)+\chi_{cJ}$($J=0,1,2$) in the NRQCD factorization approach. These next-to-leading order (NLO) corrections are calculated at the level of helicity amplitude. We have made a detailed analysis for both polarized and unpolarized cross sections, and compared our predictions with the measurements at the $B$ factories. We also derive the asymptotic expressions for each of the NLO helicity amplitudes, and confirm the earlier speculation that at NLO in ${\alpha }_s$, the double logarithm of type ${\ln }^2(s\mathrm{/}{m}_c^2)$ appearing in the NRQCD short-distance coefficient is always associated with the helicity-suppressed channels.