Angular observables of the four-fold $B \to K_{1}(1270,1400)(\to V P) \ell^{+}\ell^{-}$ decays in and beyond the Standard Model

Recent measurements of the lepton flavor universality ratios $R^{\mu e}_{K}$ and $R^{\mu e}_{K^*}$ in $B\to \left(K, K^*\right)\mu^{+}\mu^{-}\left(e^+e^-\right)$ at LHCb align with the Standard Model predictions, necessitating search for the complementary decay modes. In this context, we derive the angular decay distributions of the four-fold $B \to K_{1}(1270,1400)(\to V P) \mu^{+}\mu^{-}$ decays, where $K_1 $ is an axial-vector meson, $V=\rho, K^{\ast}$, and $P=K, \pi$. Considering the weak effective Hamiltonian with vector and axial-vector new physics operators, and employing the helicity formalism, we obtain the four-dimensional differential decay distributions and extract various physical observables. These include differential branching ratios, lepton forward-backward asymmetry, forward-backward asymmetry for transversely polarized $K_1$ meson, longitudinal and transverse polarization fractions of the $K_1$ meson, and the normalized angular coefficients with longitudinally and transversely polarized final state vector meson $V$, in the cascade decay $K_1\to VP$. Based on the latest global fit analysis data for all $b\to s$ transitions, we predict the results of these observables in various bins of the square of the momentum transfer, $q^2$. Our findings indicate that these physical observables are not only sensitive to new physics but can also differentiate between various new physics scenarios in certain kinematical regions. The precise measurements of these observables in $B \to K_{1}(1270,1400)(\to V P) \mu^{+}\mu^{-}$ decays at current and future experiments will provide opportunities for the complementary searches for physics beyond the Standard Model in $b\to s\ell^+\ell^-$ decays.