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In this paper, we calculate the $B\to D$ transition form factors (TFFs)within the light-cone sum rules (LCSR) and predict the ratio $\mathcal{R}(D)$.More accurate $D$-meson distribution amplitudes (DAs) are essential to get amore accurate theoretical prediction. We construct a new model for the twist-3DAs $\phi^p_{3;D}$ and $\phi^\sigma_{3;D}$ based on the QCD sum rules under thebackground field theory for their moments as we have done for constructing theleading-twist DA $\phi_{2;D}$. As an application, we observe that the twist-3contributions are sizable in whole $q^2$-region. Taking the twist-2 and twist-3DAs into consideration, we obtain $f^{B\to D}_{+,0}(0) =0.659^{+0.029}_{-0.032}$. As a combination of the Lattice QCD and the QCD LCSRpredictions on the TFFs $f^{B\to D}_{+,0}(q^2)$, we predict $\mathcal{R}(D) =0.320^{+0.018}_{-0.021}$, which improves is about $1.5\sigma$ deviation fromthe HFAG average of the Belle and BABAR data. At present the data are still oflarge errors, and we need further accurate measurements of the experiment toconfirm whether there is signal of new physics from the ratio $\mathcal{R}(D)$.

The Ratio $\mathcal{R}(D)$ and the $D$-meson Distribution Amplitude