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$J/ψ+χ_{cJ}$ Production at the $B$ Factories under the Principle of Maximum Conformality

Under the conventional scale setting, the renormalization scale uncertainty usually constitutes a systematic error for a fixed-order perturbative QCD estimation. The recently suggested principle of maximum conformality (PMC) provides a principle to eliminate such scale ambiguity in a step-by-step way. Using the PMC, all non-conformal terms in perturbative expansion series are summed into the running coupling, and one obtains a unique, scale-fixed, scheme-independent prediction at any finite order. In the paper, we make a detailed PMC analysis for both the polarized and the unpolarized cross sections for the double charmonium production process, $e^{+} + e^{-} \to J/ ψ(ψ') + χ_{cJ}$ with $(J=0,1,2)$. The running behavior for the coupling constant, governed by the PMC scales, are determined exactly for the specific processes. We compare our predictions with the measurements at the $B$ factories, BABAR and Belle, and the theoretical estimations obtained in the literature. Because the non-conformal terms are different for various polarized and unpolarized cross sections, the PMC scales of these cross sections are different in principle. It is found that all the PMC scales are almost independent of the initial choice of renormalization scale. Thus, the large renormalization scale uncertainty usually adopted in the literature up to $sim40%$ at the NLO level, obtained from the conventional scale setting, for both the polarized and the unpolarized cross sections are greatly suppressed. It is found that the charmonium production is dominated by J=0 channel. After PMC scale setting, we obtain $σ(J/ψ+χ_{c0}) = 12.25^{+3.70}_{-3.13}$ fb and $σ(ψ'+χ_{c0}) =5.23^{+1.56}_{-1.32}$ fb,......