An improved light-cone harmonic oscillator model for the $\phi$-meson longitudinal leading-twist light-cone distribution amplitude

In the present paper, we study the properties of $\phi$-meson longitudinal leading-twist light-cone distribution amplitude $\phi_{2;{\phi}}^{\|}(x,\mu)$ by starting from a light-cone harmonic oscillator model for its wavefunction. To fix the input parameters, we derive the first ten $\xi$-moments of $\phi_{2;{\phi}}^{\|}(x,\mu)$ by using the QCD sum rules approach under the background field theory. The shape of $\phi_{2;{\phi}}^{\|}(x,\mu=2~{\rm GeV})$ tends to be a single-peak behavior, which is consistent with the latest Lattice QCD result. As an application, we derive the $D^+_s \to \phi$ transition form factors (TFFs) by using the light-cone sum rules approach. At the large recoil point, we obtain $A_1(0) = 0.512_{-0.020}^{+0.030}$, $A_2(0) = 0.402_{-0.067}^{+0.078}$, $A_0(0) = 0.596_{-0.020}^{+0.025}$ and $V(0) = 0.882_{-0.036}^{+0.040}$. As for the two typical ratios $\gamma_V$ and $\gamma_2$, we obtain $\gamma_V = 1.723_{-0.021}^{+0.023}$ and $\gamma_2 = 0.785_{-0.104}^{+0.100}$. After extrapolating those TFFs to the physically allowable region, we then obtain the transverse, longitudinal and total decay widths for semi-leptonic decay $D^+_s\to\phi\ell^+\nu_{\ell}$. Then the branching fractions are ${\cal B}(D^+_s\to \phi e^+\nu_e) = (2.367_{-0.132}^{+0.256})\times 10^{-3}$ and ${\cal B}(D^+_s\to \phi \mu^+\nu_{\mu}) = (2.349_{-0.132}^{+0.255})\times 10^{-3}$, which show good agreement with the data issued by the BESIII, the CLEO, and the BABAR Collaborations. We finally calculate $D^+_s\to\phi\ell^+ \nu_\ell$ polarization and asymmetry parameters.