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In this paper, we study the non-leptonic $B_{c}\to J/\psi V$ and$B_{c}^{*}\to\eta_{c}V$~$(V=\rho\,,K^{*})$ weak decays in the framework of QCDfactorization. In the evaluation, the form factors are calculated by using theBauer-Stech-Wirbel model and the light-front quark model, respectively. Besidesthe longitudinal amplitude, the power-suppressed transverse contributions arealso evaluated at next-to-leading order. The predictions for the observables of$B_{c}\to J/\psi V$ and $B_{c}^{*}\to\eta_{c}V$ decays are presented. We findthat the NLO QCD contribution presents about $ 8\%$ correction to the branchingratios, and the longitudinal polarization fractions of these decays are at thelevel of $(80\sim90)\%$. In addition, we suggest direct measurements on someuseful ratios, $R_{K^{*}/\rho}^{(\lambda=0)}$ and$\widetilde{R}^{(\lambda=0)}_{K^{*}/\rho}$, which are very suitable for testthe consistence between theoretical prediction and data because theirtheoretical uncertainties can be well-controlled.

Study of $B_{c}$$\to$$ J/\psi V $ and $B_{c}^{*}$$\to$$\eta_{c}V$ decays within the QCD factorization