Vector mesons leading-twist longitudinal distribution amplitudes and related semi-leptonic decays within QCD sum rules

In this work, we focus on the light vector meson leading-twist longitudinal distribution amplitudes (DAs) ${\phi }_{2;V}^{\parallel }(x,\mu )$ with $V=\rho ,K^{\ast },\phi$. In order to obtain their accurate behaviors, a new scheme of QCD sum rule research with respect to DA suggested in 2021 by us is adopted. With an improved sum rule formula, the $\xi$-moments $⟨{\xi }^n{⟩}_{2;V}^{\parallel }$ up to tenth order are calculated. In which, $⟨{\xi }^2{⟩}_{2;\rho }^{\parallel }=0.225_{-0.012}^{+0.013}$, $⟨{\xi }^1{⟩}_{2;K^{\ast }}^{\parallel }=-0.0228_{-0.0040}^{+0.0042}$, $⟨{\xi }^2{⟩}_{2;K^{\ast }}^{\parallel }=0.217_{-0.007}^{+0.007}$, $⟨{\xi }^2{⟩}_{2;\phi }^{\parallel }=0.209_{-0.020}^{+0.020}$, and the corresponding Gegenbauer moments $a_{2;\rho }^{2;\parallel }=0.074_{-0.036}^{+0.039}$, $a_{2;K^{\ast }}^{1;\parallel }=-0.038_{-0.007}^{+0.007}$, $a_{2;K^{\ast }}^{2;\parallel }=0.050_{-0.019}^{+0.020}$, $a_{2;\phi }^{2;\parallel }=0.027_{-0.058}^{+0.058}$ at the scale $\mu =1\mathrm{G}\mathrm{e}\mathrm{V}$, respectively. By fitting those $⟨{\xi }^n{⟩}_{2;V}^{\parallel }(n=1,2,\cdots ,10)$ with the least squares method, the behaviors of leading-twist longitudinal DAs for $\rho ,K^{\ast },\phi$ are determined. Further, we recalculate the transition form factors and branching ratio of the $D\to (\rho ,K^{\ast })$, $D_s\to \phi$ semi-leptonic decay processes.