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) $\phi^\parallel_{2;V}(x,\mu)$ 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 $\langle\xi^n\rangle_{2;V}^\parallel$ up to tenth order are calculated. In which, $\langle\xi^2\rangle^\parallel_{2;\rho}=0.225^{+0.013}_{-0.012}$, $\langle\xi^1\rangle^\parallel_{2;K^\ast}=-0.0228^{+0.0042}_{-0.0040}$, $\langle\xi^2\rangle^\parallel_{2;K^\ast}=0.217^{+0.007}_{-0.007}$, $\langle\xi^2\rangle^\parallel_{2;\phi}=0.209^{+0.020}_{-0.020}$, and the corresponding Gegenbauer moments $a^{2;\parallel}_{2;\rho}=0.074^{+0.039}_{-0.036}$, $a^{1;\parallel}_{2;K^\ast}=-0.038^{+0.007}_{-0.007}$, $a^{2;\parallel}_{2;K^\ast}=0.050^{+0.020}_{-0.019}$, $a^{2;\parallel}_{2;\phi}=0.027^{+0.058}_{-0.058}$ at the scale $\mu = 1~{\rm GeV}$, respectively. By fitting those $\langle\xi^n\rangle^\parallel_{2;V}(n = 1,2,\cdots,10)$ 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.