We make a detailed study on the $D_s$ meson leading-twist LCDA $phi_{2;D_s}$ by using the QCD sum rules within the framework of the background field theory. To improve the precision, its moments $langle xi^nrangle _{2;D_s}$ are calculated up to dimension-six condensates. At the scale $mu = 2{rm GeV}$, we obtain: $langle xi^1rangle _{2;D_s}= -0.261^{+0.020}_{-0.020}$, $langle xi^2rangle _{2;D_s} = 0.184^{+0.012}_{-0.012}$, $langle xi^3rangle _{2;D_s} = -0.111 ^{+0.007}_{-0.012}$ and $langle xi^4rangle _{2;D_s} = 0.075^{+0.005}_{-0.005}$. Using those moments, the $phi_{2;D_s}$ is then constructed by using the light-cone harmonic oscillator model. As an application, we calculate the transition form factor $f^{B_sto D_s}_+(q^2)$ within the light-cone sum rules (LCSR) approach by using a right-handed chiral current, in which the terms involving $phi_{2;D_s}$ dominates the LCSR. It is noted that the extrapolated $f^{B_sto D_s}_+(q^2)$ agrees with the Lattice QCD prediction. After extrapolating the transition form factor to the physically allowable $q^2$-region, we calculate the branching ratio and the CKM matrix element, which give $mathcal{B}(bar B_s^0 to D_s^+ ell u_ell) = (2.03^{+0.35}_{-0.49}) times 10^{-2}$ and $|V_{cb}|=(40.00_{-4.08}^{+4.93})times 10^{-3}$.
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