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

Abstract

In this work, we focus on the light vector meson leading-twist longitudinal distribution amplitudes (DAs) $ϕ^\parallel_{2;V}(x,μ)$ with $V = ρ, K^\ast, ϕ$. 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 $ξ$-moments $\langleξ^n\rangle_{2;V}^\parallel$ up to tenth order are calculated. In which, $\langleξ^2\rangle^\parallel_{2;ρ}=0.225^{+0.013}_{-0.012}$, $\langleξ^1\rangle^\parallel_{2;K^\ast}=-0.0228^{+0.0042}_{-0.0040}$, $\langleξ^2\rangle^\parallel_{2;K^\ast}=0.217^{+0.007}_{-0.007}$, $\langleξ^2\rangle^\parallel_{2;ϕ}=0.209^{+0.020}_{-0.020}$, and the corresponding Gegenbauer moments $a^{2;\parallel}_{2;ρ}=0.074^{+0.039}_{-0.036}$, $a^{1;\parallel}_{2;K^\ast}=-0.038^{+0.007}_{-0.007}$, $a^{2;\parallel}_{2;K^\ast}=0.050^{+0.020}_{-0.019}$, $a^{2;\parallel}_{2;ϕ}=0.027^{+0.058}_{-0.058}$ at the scale $μ= 1~{\rm GeV}$, respectively. By fitting those $\langleξ^n\rangle^\parallel_{2;V}(n = 1,2,\cdots,10)$ with the least squares method, the behaviors of leading-twist longitudinal DAs for $ρ, K^\ast, ϕ$ are determined. Further, we recalculate the transition form factors and branching ratio of the $D\to(ρ,K^\ast)$, $D_s\toϕ$ semi-leptonic decay processes.