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In nonrelativistic QCD (NRQCD), we study $e^+ e^- \to J/\psi(\psi(2S))+\chi_{c0}$ at $B$ factories, where the P-wave state $\chi_{c0}$ is associatedwith an S-wave state $J/\psi$ or $\psi(2S)$. In contrast to the failure offactorization in most cases involving P-wave states, e.g. in B decays, we findthat factorization holds in this process at next to leading order (NLO) in$\alpha_s$ and leading order (LO) in $v$, where the associated S-wave stateplays a crucial rule in canceling the infrared (IR) divergences. We also givesome general analyses for factorization in various double charmoniumproduction. The NLO corrections in $e^+ e^- \to J/\psi(\psi(2S)) +\chi_{c0}$ at$\sqrt{s}=10.6$ GeV are found to substantially enhance the cross sections by afactor of about 2.8; hence crucially reduce the large discrepancy betweentheory and experiment. With $m_c=1.5{\rm GeV}$ and $\mu=2m_c$, the NLO crosssections are estimated to be $17.9(11.3)$ fb for $e^+ e^- \to J/\psi(\psi(2S))+\chi_{c0}$, which reach the lower bounds of experiment.

Factorization and NLO QCD correction in e^+ e^- \to J/\psi (\psi(2S))+ \chi_{c0} at B Factories}