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We analytically calculate the cross sections of double charmonium productionin $e^+ e^- \to J/\psi(\psi(2S))\chi_{cJ}$ (J=0,1,2) at next-to-leading order(NLO) in $\alpha_s$ in nonrelativistic QCD, and confirm factorization of theseprocesses. In contrast to $\chi_{c0}$ production, for which the NLO correctionis large and positive, the NLO corrections for $\chi_{c1,2}$ production can benegative, resulting in decreased $K$ factors of 0.91 and 0.78 for J=1 and 2respectively when $\mu = 2 m_{c}$. Consequently, the NLO QCD correctionsmarkedly enlarge the difference between cross sections of $\chi_{c0}$ and$\chi_{c1,2}$. This may explain why $e^+ e^- \to J/\psi(\psi(2S))\chi_{c0}$ butnot $e^+ e^- \to J/\psi(\psi(2S))\chi_{c1,2}$ is observed experimentally.Moreover, for $J/\psi(\psi(2S))\chi_{c1,2}$, the NLO QCD correctionssubstantially reduce the $\mu$ dependence and lead to predictions with smalltheoretical uncertainties.

QCD corrections to e^+ e^- to J/\psi(\psi(2S))+\chi_{cJ} (J=0,1,2) at B Factories