\begin{document}$ B_s\to D_s^*\ell \bar\nu_{\ell} $\end{document}. First, we derive the moments of the \begin{document}$ D_s^* $\end{document}-meson longitudinal leading-twist light-cone distribution amplitude (LCDA) based on QCD sum rules within the background field theory framework. Considering the contributions of the vacuum condensates up to dimension-six, its first ten non-zero \begin{document}$ \xi $\end{document}-moments at the initial scale \begin{document}$ \mu_0 = 1.3\; {\rm{GeV}} $\end{document} are \begin{document}$ \langle \xi^{\|, 1}_{2; D_s^*} \rangle|_{\mu_0} = -0.302_{-0.046}^{+0.038} $\end{document}, \begin{document}$ \langle\xi^{\|, 2}_{2;D_s^*}\rangle|_{\mu_0} = +0.229_{-0.034}^{+0.039} $\end{document}, \begin{document}$ \langle\xi^{\|, 3}_{2;D_s^*}\rangle|_{\mu_0} = -0.121_{-0.019}^{+0.015} $\end{document}, \begin{document}$ \langle\xi^{\|, 4}_{2;D_s^*}\rangle|_{\mu_0} = +0.101_{-0.014}^{+0.017} $\end{document}, \begin{document}$ \langle\xi^{\|, 5}_{2; D_s^*} \rangle|_{\mu_0} = -0.066_{-0.010}^{+0.009} $\end{document}, \begin{document}$ \langle\xi^{\|, 6}_{2;D_s^*}\rangle|_{\mu_0} = +0.053_{-0.007}^{+0.009} $\end{document}, \begin{document}$ \langle\xi^{\|, 7}_{2;D_s^*}\rangle|_{\mu_0} = -0.041_{-0.007}^{+0.006} $\end{document}, \begin{document}$ \langle\xi^{\|, 8}_{2;D_s^*}\rangle|_{\mu_0} = +0.037_{-0.005}^{+0.006} $\end{document}, \begin{document}$ \langle\xi^{\|, 9}_{2; D_s^*} \rangle|_{\mu_0} = -0.026_{-0.004}^{+0.003} $\end{document}, and \begin{document}$ \langle\xi^{\|, 10}_{2;D_s^*}\rangle|_{\mu_0} = +0.025_{-0.004}^{+0.004} $\end{document}. We also construct the \begin{document}$ D_s^* $\end{document}-meson longitudinal leading-twist LCDA by using the light-cone harmonic oscillator model. Then, using the above moments, we fix the model parameters \begin{document}$ \alpha_{2;D_s^*} $\end{document} and \begin{document}$ B_1^{2;D_s^*} $\end{document} using the least squares method and apply them to calculate \begin{document}$ B_s \to D_s^* $\end{document} transition form factors \begin{document}$ A_1(q^2), A_2(q^2) $\end{document} and \begin{document}$ V(q^2) $\end{document} that are derived using the QCD light-cone sum rules. In the large recoil region, we obtain \begin{document}$ A_1(0) = 0.632_{-0.135}^{+0.228}, A_2(0) = 0.706_{-0.092}^{+0.109} $\end{document}, and \begin{document}$ V(0) = 0.647_{-0.069}^{+0.076} $\end{document}. These form factors are then extrapolated to the allowed whole physical \begin{document}$ q^2 $\end{document}-region through the simplified series expansion. Finally, we obtain the branching fractions for the two decay channels of \begin{document}$ B_s\to D_s^*\ell\bar\nu_\ell $\end{document}, \begin{document}$ {\cal{B}}(B_s^0 \to D_s^{*+}e^-\bar\nu_e) = (5.45_{-1.57}^{+2.15})\times 10^{-2} $\end{document} and \begin{document}$ {\cal{B}}(B_s^0 \to D_s^{*+}\mu^-\bar\nu_\mu) = $\end{document}\begin{document}$ (5.43_{-1.57}^{+2.14})\times 10^{-2} $\end{document}. In addition, we present the CKM matrix element \begin{document}$ |V_{cb}| $\end{document} by matching the LHCb Collaboration branching fraction, yielding a value of \begin{document}$ |V_{cb}| = (40.11_{-7.49}^{+6.54})\times 10^{-3} $\end{document}."> Probing <i>D</i><sub><i>s</i></sub><sup>*</sup>-meson longitudinal twist-2 LCDA -
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