\begin{document}$ B_c\to D^*h\to D\pi h $\end{document} with \begin{document}$ h = (K^0,\pi^0,\eta,\eta^{\prime}) $\end{document} using the perturbative QCD (PQCD) approach. The description of final state interactions of the \begin{document}$ D\pi $\end{document} pair is achieved through the two-meson distribution amplitudes (DAs), which are normalized to the time-like form factor. The PQCD predictions on the branching ratios of the quasi-two-body decays \begin{document}$ B_c\to D^*h\to D\pi h $\end{document} show an obvious hierarchy: \begin{document}${\rm Br}(B_{c}^+ \to D^{*+} K^{0}\to D^0\pi^+K^{0})=({5.22}_{-0.74}^{+0.86})\times{10}^{-6},\;\; {\rm Br}(B_{c}^+ \to D^{*+} \pi^{0}\to D^0\pi^+\pi^{0})=(0.93\pm0.26)\times{10}^{-7},$\end{document} \begin{document}${\rm Br}(B_{c}^+ \to $\end{document}\begin{document}$ D^{*+} \eta\to D^0\pi^+\eta) = ({2.83}_{-0.52}^{+0.59})\times{10}^{-8}$\end{document} and \begin{document}${\rm Br}(B_{c}^+ \to D^{*+} \eta^\prime\to D^0\pi^+\eta^\prime)=({1.89}_{-0.36}^{+0.40})\times{10}^{-8}$\end{document}. From the invariant mass \begin{document}$ m_{D\pi} $\end{document}-dependence of the decay spectrum for each channel, one can find that the branching fraction is concentrated in a narrow region around the \begin{document}$ D^{*} $\end{document} pole mass. Thus, one can obtain the branching ratios for the corresponding two-body decays \begin{document}$ B_c\to D^{*+}h $\end{document} under the narrow-width approximation. We find that the branching ratios of the decays \begin{document}$ B_c\to D^{*+}h $\end{document} are consistent with the previous PQCD calculations within errors. These predictions will be tested in future experiments."> Quasi-two-body decays <inline-formula><tex-math id="M1">\begin{document}${ \boldsymbol B_{\boldsymbol c}{\bf\to} {\boldsymbol D}^*{\boldsymbol h}{\bf\to}{\boldsymbol D}{\boldsymbol\pi}{\boldsymbol h} }$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/fc44e91c-d9cc-43dc-93c7-e9d0a3c15aba/CPC-2023-0060_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/fc44e91c-d9cc-43dc-93c7-e9d0a3c15aba/CPC-2023-0060_M1.png"/></alternatives></inline-formula> in perturbative QCD -
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