Footprint in fitting $B \to D$ vector form factor and determination for $D$-meson leading-twist LCDA

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.