Uncertainties of the $B\to D$ transition form factor from the $D$-Meson Leading-Twist Distribution Amplitude

Abstract

The $B\to D$ transition form factor (TFF) $f_{+}^{B\to D}(q^2)$ is determined mainly by the $D$-meson leading-twist distribution amplitude (DA) , ${\varphi }_{2;D}$, 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 ${\varphi }_{2;D}$ and its impact on $f_{+}^{B\to D}(q^2)$. In this paper, we calculate the moments of ${\varphi }_{2;D}$ with the QCD sum rules under the framework of the background field theory. New sum rules for the leading-twist DA moments ${⟨{\xi }^n⟩}_D$ up to fourth order and up to dimension-six condensates are presented. At the scale $\mu =2\mathrm{GeV}$, the values of the first four moments are: ${⟨{\xi }^1⟩}_D=-0.418_{-0.022}^{+0.021}$, ${⟨{\xi }^2⟩}_D=0.289_{-0.022}^{+0.023}$, ${⟨{\xi }^3⟩}_D=-0.178\pm 0.010$ and ${⟨{\xi }^4⟩}_D=0.142_{-0.012}^{+0.013}$. Basing on the values of ${⟨{\xi }^n⟩}_D(n=1,2,3,4)$, a better model of ${\varphi }_{2;D}$ is constructed. Applying this model for the TFF $f_{+}^{B\to D}(q^2)$ under the light cone sum rules, we obtain $f_{+}^{B\to D}(0)=0.673_{-0.041}^{+0.038}$ and $f_{+}^{B\to D}(q_{\max }^2)=1.124_{-0.058}^{+0.053}$. The uncertainty of $f_{+}^{B\to D}(q^2)$ from ${\varphi }_{2;D}$ is estimated and we find its impact should be taken into account, especially in low and central energy region. The branching ratio $\mathcal{B}(B\to Dl{\bar{\nu }}_l)$ is calculated, which is consistent with experimental data.

12 pages, 8 figures

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