The $\rho$-meson longitudinal leading-twist distribution amplitude within QCD background field theory

We revisit the $\rho$-meson longitudinal leading-twist distribution amplitude (DA) $\phi_{2;\rho}^\|$ by using the QCD sum rules approach within the background field theory. To improve the accuracy of the sum rules for its moments $\langle\xi_{n;\rho}^\|\rangle$, we include the next-to-leading order QCD correction to the perturbative part and keep all non-perturbative condensates up to dimension-six consistently within the background field theory. The first two moments read $\langle \xi_{2;\rho}^\| \rangle|_{1{\rm GeV}} = 0.241(28)$ and $\langle \xi_{4;\rho}^\| \rangle|_{1{\rm GeV}} = 0.109(10)$, indicating a double humped behavior for $\phi_{2;\rho}^\|$ at small energy scale. As an application, we apply them to the $B\to \rho $ transition form factors within the QCD light-cone sum rules, which are key components for the decay width $\Gamma(B\to \rho \ell \nu_\ell)$. To compare with the world-average of $\Gamma(B\to \rho \ell \nu_\ell)$ issued by Particle Data Group, we predict $|V_{\rm ub}| = 3.19^{+0.65}_{-0.62}$, which agrees with the BABAR and Omn\`{e}s parameterization prediction within errors.