\begin{document}$ B\to\rho (\to\pi\pi)\mu^{+}\mu^{-} $\end{document} and \begin{document}$ B\to a_{1}(\to\rho_{\parallel, \perp}\pi)\mu^{+}\mu^{-} $\end{document} decays in the Standard Model and family non-universal \begin{document}$ Z^{\prime} $\end{document} model. At the quark level, these decays are governed by the \begin{document}$ b\to d\mu^{+}\mu^{-} $\end{document} transition. Along with different angular observables, we provide predictions of differential branching ratios, forward-backward asymmetry, and longitudinal polarization fractions of \begin{document}$ \rho $\end{document} and \begin{document}$ a_{1} $\end{document} mesons. Our analysis shows that the signatures of the family non-universal \begin{document}$ Z^{\prime} $\end{document} model are more distinct in the observables associated with the \begin{document}$ B\to\rho(\to\pi\pi)\mu^{+}\mu^{-} $\end{document} decay than in those associated with the \begin{document}$ B\to a_{1}(\to\rho_{\parallel, \perp}\pi)\mu^{+}\mu^{-} $\end{document} decay. Future measurements of the predicted angular observables, both at current and future high energy colliders, will provide useful complementary data required to clarify the structure of the family non-universal \begin{document}$ Z^{\prime} $\end{document} model in \begin{document}$ |\Delta b| $\end{document}=\begin{document}$ |\Delta d|=1 $\end{document} processes."> Effects of family non-universal <i>Z</i> ′ model in angular observables of <inline-formula><tex-math id="Z-20240902105647">\begin{document}${\boldsymbol B {\bf\to} (\boldsymbol\rho, {\boldsymbol a}_{\bf 1})\boldsymbol\mu^+\boldsymbol\mu^- }$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/a8a88523-a2ac-4180-bd91-80b300a83236/CPC-2024-0148_Z-20240902105647.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/a8a88523-a2ac-4180-bd91-80b300a83236/CPC-2024-0148_Z-20240902105647.png"/></alternatives></inline-formula> decays -
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