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In this paper, we study the $B^{0} \to J/\psi h_{1}$ decays for the first time by using perturbative QCD approach up to the presently known next-to-leading order accuracy. The vertex corrections present significant contribution to the amplitude. In the calculation, the mixing between two light axial-vector mesons $h_{1}(1170)$ and $h_{1}(1415)$ are also studied in detail. The observables including the branching ratios, polarization fractions and $CP$ asymmetries are predicted and discussed explicitly. It is found that the $B^{0} \to J/\psi h_{1}$ decays have relatively large branching fractions, which are generally at the order of ${\cal O}(10^{-6}\sim10^{-3})$, and thus are possible to be observed by the LHCb and Belle-II experiments in the near future. Moreover, they are very sensitive to the mixing angle $\theta$ and can be used to test the values of $\theta$. In addition, some ratios between the branching fractions of $B^{0} \to J/\psi h_{1}$ decays can provide much stronger constraints on $\theta$ due to their relatively small theoretical errors. The $B^{0} \to J/\psi h_{1}$ decays are generally dominated by the longitudinal polarization contributions, specifically, $f_{L}(B^{0} \to J/\psi h_{1})>80\%$, except for the case that $\theta\sim 35^\circ$ and $-55^\circ$. Unfortunately, the direct $CP$ asymmetries of $B^{0} \to J/\psi h_{1}$ decays are too small to be observed soon even if the effect of $\theta$ is considered. The future precise measurements on $B^{0} \to J/\psi h_{1}$ decays are expected for testing these theoretical findings and exploring the interesting nature of $h_{1}(1170)$ and $h_{1}(1415)$., Comment: 23 pages, 6 figures, 8 tables