  The improved QCD light-cone sum rule (LCSR) provides an effective way to deal
with the heavy-to-light transition form factors (TFFs). Firstly, we adopt the
improved LCSR approach to deal with the $B\to D$ TFF $f^{+}(q^2)$ up to twist-4
accuracy. Due to the elimination of the most uncertain twist-3 contribution and
the large suppression of the twist-4 contribution, the obtained LCSR shall
provide us a good platform for testing the $D$-meson leading-twist DA. For the
purpose, we suggest a new model for the $D$-meson leading-twist DA
($\phi_{3D}$), whose longitudinal behavior is dominantly determined by a
parameter $B$. Moreover, we find its second Gegenbauer moment $a^D_2\sim B$.
Varying $B$ within certain region, one can conveniently mimic the $D$-meson DA
behavior suggested in the literature. Inversely, by comparing the estimations
with the experimental data on the $D$-meson involved processes, one can get a
possible range for the parameter $B$ and a determined behavior for the
$D$-meson DA. Secondly, we discuss the $B\to D$ TFF at the maximum recoil
region and present a detailed comparison of it with the pQCD estimation and the
experimental measurements. Thirdly, by applying the LCSR on $f^{+}(q^2)$, we
study the CKM matrix element $\Vcb$ together with its uncertainties by adopting
two types of processes, i.e. the $B^0/\bar{B}^0$-type and the $B^{\pm}$-type.
It is noted that a smaller $B \precsim 0.20$ shows a better agreement with the
experimental value on $\Vcb$. For example, for the case of $B=0.00$, we obtain
$|V_{cb}|(B^0/\bar{B}^0-{\rm type})=(41.28 {^{+5.68}_{-4.82}}
{^{+1.13}_{-1.16}}) \times 10^{-3}$ and $|V_{cb}|(B^{\pm}-{\rm type})=(40.44
{^{+5.56}_{-4.72}} {^{+0.98}_{-1.00}}) \times 10^{-3}$, whose first (second)
uncertainty comes from the squared average of the mentioned theoretical
(experimental) uncertainties.
