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We present the angular distribution of the four-fold $B\to\rho (\to\pi\pi)\mu^{+}\mu^{-}$ and $B\to a_{1}(\to\rho_{\parallel, \perp}\pi)\mu^{+}\mu^{-}$ decays both in the Standard Model and the family non-universal $Z^{\prime}$ model. At the quark level, these decays are governed by the $b\to d\mu^{+}\mu^{-}$ transition. Along with different angular observables, we also give predictions of differential branching ratios, forward-backward asymmetry, longitudinal polarization fraction of $\rho$, and $a_{1}$ mesons. Our analysis shows that the signatures of family non-universal $Z^{\prime}$ model are more distinct in the observables associated with the $B\to\rho(\to\pi\pi)\mu^{+}\mu^{-}$ decay, compared to that of the $B\to a_{1}(\to\rho_{\parallel, \perp}\pi)\mu^{+}\mu^{-}$ decay. Future measurements of the predicted angular observables, both at current and future high energy colliders, will add to the useful complementary data required to clarify the structure of the family non-universal $Z^{\prime}$ model in $|\Delta b|$=$|\Delta d|=1$ processes., Comment: 39 pages, 6 figures, 38 tables; version to be published in Chinese Physics C

We investigate the angular decay distribution of the four-fold $B_{c}\to D^{\ast}_{s}(\to D_{s}\gamma)\mu^{+}\mu^{-}$, and $B_{c}\to D^{\ast}_{s}(\to D_{s}\pi)\mu^{+}\mu^{-}$ decays that proceed through $b\to s\mu^{+}\mu^{-}$ quark level transition. We use the model independent effective Hamiltonian with vector and axial vector new physics operators to formulate the angular observables and study the implications of different latest new physics scenarios, taken from the global fits to all the $b\to s$ data, on these observables. We also give Standard Model and new physics predictions of several observables such as differential branching ratios, forward backward asymmetry, longitudinal polarization fraction of $D_s^{\ast}$, and the unpolarized and polarized lepton flavor universality violating ratios. Future measurements of the predicted angular observables, both at current and future high energy colliders, will add to the useful complementary data required to clarify the structure of new physics in $b\to s\ell\ell$ neutral current decays., Comment: 24 pages, 7 figures, 18 tables; ; version matching publication in NPB