The $D\to \rho$ semileptonic and radiative decays within the light-cone sum rules

The measured branching ratio of the $D$ meson semileptonic decay $D \to \rho e^+ \nu_e$, which is based on the $0.82~{\rm fb^{-1}}$ CLEO data taken at the peak of $\psi(3770)$ resonance, disagrees with the traditional SVZ sum rules analysis by about three times. In the paper, we show that this discrepancy can be eliminated by applying the QCD light-cone sum rules (LCSR) approach to calculate the $D\to \rho$ transition form factors $A_{1,2}(q^2)$ and $V(q^2)$. After extrapolating the LCSR predictions of these TFFs to whole $q^2$-region, we obtain $1/|V_{\rm cd}|^2 \times \Gamma(D \to \rho e \nu_e) =(55.45^{+13.34}_{-9.41})\times 10^{-15}~{\rm GeV}$. Using the CKM matrix element and the $D^0(D^+)$ lifetime from the Particle Data Group, we obtain ${\cal B} (D^0\to \rho^- e^+ \nu_e) = (1.749^{+0.421}_{-0.297}\pm 0.006)\times 10^{-3}$ and ${\cal B} (D^+ \to \rho^0 e^+ \nu_e) = (2.217^{+0.534}_{-0.376}\pm 0.015)\times 10^{-3}$, which agree with the CLEO measurements within errors. We also calculate the branching ratios of the two $D$ meson radiative processes and obtain ${\cal B}(D^0\to \rho^0 \gamma)= (1.744^{+0.598}_{-0.704})\times 10^{-5}$ and ${\cal B}(D^+ \to \rho^+ \gamma) = (5.034^{+0.939}_{-0.958})\times 10^{-5}$, which also agree with the Belle measurements within errors. Thus we think the LCSR approach is applicable for dealing with the $D$ meson decays.