In this paper, we fit the $B\to D$ vector transition form factor (TFF) by using the data measured by \textsl{ BABAR} and Belle Collaborations within Monte Carlo (MC) method. Meanwhile, the $B\to D$ TFF is also calculated by using the QCD light-cone sum rules approach (LCSRs) within right-handed chiral current correlation function. In the TFF, $D$-meson leading-twist light-cone distribution amplitude (LCDA) is the most important non-perturbative input parameter, the precise behavior of which is mainly determined by the parameter $B_{ 2;D}$ in the light-cone harmonic oscillator model. Through fitting the TFF, we determine the value $B_{ 2;D}=0.445$. Then, we present the curve of $D$-meson leading-twist LCDA in comparison with other theoretical approaches. Subsequently, the $B\to D$ TFF $f_{ +}^{ BD}(q^2)$ at the large recoil region is $f_{ +}^{ BD}(0)=0.648_{ -0.063}^{ +0.067}$, which is compared in detail with theoretical estimates and experimental measurements. Furthermore, we calculated the decay width and branching fractions of the Cabibbo-favored semileptonic decays $B\to D\ell^+\nu_{ \ell}$, which lead to the results $\mathcal{ B}(B^0\to D^-\ell ^+\nu _{ \ell}) =(2.10_{ -0.38}^{ +0.44})\times 10^{ -2}$ and $\mathcal{ B}(B^+\to \bar{ D}^0\ell ^+\nu _{ \ell}) =(2.26_{ -0.41}^{ +0.48})\times 10^{ -2}$. Finally, we predict the CKM matrix element with two scenarios $|V_{ cb}|_{ \rm SR}=41.55_{ -4.50}^{ +4.91}\times 10^{ -3}$ and $|V_{ cb} |_{ \rm MC}=41.47_{ -2.66}^{ +2.55 }\times 10^{ -3}$ from $B^0\to D^-\ell^+\nu_{ \ell}$, $|V_{ cb}|_{ \rm SR}=40.54_{ -3.80}^{ +4.33}\times 10^{ -3}$ and $|V_{ cb} |_{ \rm MC}=40.46_{ -1.38}^{ +1.40}\times 10^{ -3}$ from $B^+\to \bar{ D}^0\ell^+\nu_{ \ell}$ which are in good agreement with theoretical and experimental predictions.
Categories: hep-ph