Study of lepton flavor universality and angular distributions in $D\to K^*_{J} (\to K\pi)\ell\nu_{\ell}$

The decay of the D meson into multibody final states is a complex process that provides valuable insights into the fundamental interactions within the Standard Model of particle physics. This study focuses on the decay cascade $D^{+}\to K^{*}_{J} \ell^{+}\nu \to K^{\pm}\pi^{\mp} \ell^{+}\nu$ where the $K^*_J$ resonance encompasses the $K^*(892),K^{*}(1410),K^{*}_0(1430)$ states. We employ the helicity amplitude technique to derive the angular distributions for the decay chain, enabling the extraction of one-dimensional and two-dimensional distributions. Utilizing form factors for the $D\to K^*$ transition derived from the quark model, we calculate the differential and integrated partial decay widths, explicitly considering the electron and muon masses. Decay branching fractions are calculated, the ratios of the branching fractions are found to be $\frac{\mathcal{B}r(D^{+}\to K^{*}(892)(\to K ^{-}\pi^{+}) \, \mu^{+}\nu_{\mu})}{\mathcal{B}r(D^{+}\to K^{*}(892)(\to K ^{-}\pi^{+}) \, e^{+}\nu_{e})} = 0.975$, $\frac{\mathcal{B}r(D^{+}\to K^{*}(1410)(\to K ^{-}\pi^{+}) \, \mu^{+}\nu_{\mu})}{\mathcal{B}r(D^{+}\to K^{*}(1410)(\to K ^{-}\pi^{+}) \, e^{+}\nu_{e})} = 0.714$ and $\frac{\mathcal{B}r(D^{+}\to K^{*}_{0}(1430)(\to K ^{+}\pi^{-}) \, \mu^{+}\nu_{\mu})}{\mathcal{B}r(D^{+}\to K^{*}_{0}(1430)(\to K ^{+}\pi^{-}) \, e^{+}\nu_{e})} = 0.774$. Results in this work will serve a calibration for the study of $c \to s $ in $D$ meson decays in future and provide useful information towards the understanding of the properties of the $K^{*}$ meson, as well as $K \pi$ system.