Uncertainties of the $$B\rightarrow D$$B→D transition form factor from the D-meson leading-twist distribution amplitude

The $$B\rightarrow D$$B→D transition form factor (TFF) $$f^{B\rightarrow D}_+(q^2)$$f+B→D(q2) is determined mainly by the D-meson leading-twist distribution amplitude (DA) , $$\phi _{2;D}$$ϕ2;D, if the proper chiral current correlation function is adopted within the light-cone QCD sum rules. It is therefore significant to make a comprehensive study of DA $$\phi _{2;D}$$ϕ2;D and its impact on $$f^{B\rightarrow D}_+(q^2)$$f+B→D(q2). In this paper, we calculate the moments of $$\phi _{2;D}$$ϕ2;D with the QCD sum rules under the framework of the background field theory. New sum rules for the leading-twist DA moments $$\left\langle \xi ^n\right\rangle _D$$ξnD up to fourth order and up to dimension-six condensates are presented. At the scale $$\mu = 2 \,\mathrm{GeV}$$μ=2GeV, the values of the first four moments are: $$\left\langle \xi ^1\right\rangle _D = -0.418^{+0.021}_{-0.022}$$ξ1D=-0.418-0.022+0.021, $$\left\langle \xi ^2\right\rangle _D = 0.289^{+0.023}_{-0.022}$$ξ2D=0.289-0.022+0.023, $$\left\langle \xi ^3\right\rangle _D = -0.178 \pm 0.010$$ξ3D=-0.178±0.010 and $$\left\langle \xi ^4\right\rangle _D = 0.142^{+0.013}_{-0.012}$$ξ4D=0.142-0.012+0.013. Basing on the values of $$\left\langle \xi ^n\right\rangle _D(n=1,2,3,4)$$ξnD(n=1,2,3,4), a better model of $$\phi _{2;D}$$ϕ2;D is constructed. Applying this model for the TFF $$f^{B\rightarrow D}_+(q^2)$$f+B→D(q2) under the light cone sum rules, we obtain $$f^{B\rightarrow D}_+(0) = 0.673^{+0.038}_{-0.041}$$f+B→D(0)=0.673-0.041+0.038 and $$f^{B\rightarrow D}_+(q^2_{\mathrm{max}}) = 1.117^{+0.051}_{-0.054}$$f+B→D(qmax2)=1.117-0.054+0.051. The uncertainty of $$f^{B\rightarrow D}_+(q^2)$$f+B→D(q2) from $$\phi _{2;D}$$ϕ2;D is estimated and we find its impact should be taken into account, especially in low and central energy region. The branching ratio $$\mathcal {B}(B\rightarrow Dl\bar{\nu }_l)$$B(B→Dlν¯l) is calculated, which is consistent with experimental data.