Status of the $D_s^{+}\to \phi {\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ decay with a chiral-odd $\phi$-meson light-cone distribution amplitude

The twist-2 distribution amplitude of the $\phi$-meson has attracted considerable interest due to its unique properties. In this work, we construct the transverse leading-twist light-cone distribution amplitude ${\phi }_{2;\phi }^{\mathrm{\perp }}(x,{\mu }_0)$ of the $\phi$-meson using the light-cone harmonic oscillator model, in which a parameter $B_{2;\phi }^{\mathrm{\perp }}$ dominantly control its longitudinal distribution. To explicitly isolate different twist contributions, we employ the right-handed chiral correlator for the QCD light-cone sum rules calculation of $D_s^{+}\to \phi$ decays, and further, we get the branching fraction, $\mathcal{B}(D_s^{+}\to \phi e^{+}{\nu }_e)=(2.271_{-0.243}^{+0.291})\times {10}^{-2}$ and $\mathcal{B}(D_s^{+}\to \phi {\mu }^{+}{\nu }_{\mu })=(2.250_{-0.240}^{+0.287})\times {10}^{-2}$, where errors are squared average of the mentioned error sources. Furthermore, we have extracted the Cabbibo-Kobayashi-Maskawa (CKM) matrix element $\mathrm{\mid }{V}_{cs}\mathrm{\mid }=0.975_{-0.066}^{+0.067}$ with improved precision through the analysis. Finally, we calculated the polarization parameter and asymmetry parameter for the $D_s^{+}\to \phi$ decays.