First Measurement of the Form Factors in the Decays $D^0\to {\rho }^{\mathbf{-}}{e}^{\mathbf{+}}{\nu }_e$ and $D^{\mathbf{+}}\to {\rho }^0{e}^{\mathbf{+}}{\nu }_e$

The beauty to up quark coupling constant $|V_{ub}|$ can be extracted from $B\to \rho e^{+}{\nu }_e$ combined with the form factors for $D\to K^{*}{e}^{+}{\nu }_e$ and $B\to V{\ell }^{+}{\ell }^{-}$ and $D\to \rho e^{+}{\nu }_e$. Using the entire CLEO-c $\psi (3770)\to D\overline{D}$ event sample, corresponding to an integrated luminosity of $818{\mathrm{pb}}^{-1}$ and approximately $5.4\times {10}^6$ $D\overline{D}$ events, we measure the form factors for the decays $D^0\to {\rho }^{-}{e}^{+}{\nu }_e$ and $D^{+}\to {\rho }^0{e}^{+}{\nu }_e$ for the first time and the branching fractions with improved precision. A four-dimensional unbinned maximum likelihood fit determines the form factor ratios to be $V(0)/A_1(0)=1.48\pm 0.15\pm 0.05$ and $A_2(0)/A_1(0)=0.83\pm 0.11\pm 0.04$. Assuming Cabibbo-Kobayashi-Maskawa unitarity, the known $D$ meson lifetimes, and our measured branching fractions we obtain the form factor normalizations $A_1(0)$, $A_2(0)$, and $V(0)$. We also present a measurement of the branching fraction for $D^{+}\to \omega e^{+}{\nu }_e$ with improved precision.

Figure 1

Fits to the $U$ and hadron invariant mass distributions in data (filled circles with error bars) for (a) and (d) $D^0\to {\rho }^{-}{e}^{+}{\nu }_e$, ${\rho }^{-}\to {\pi }^{-}{\pi }^0$; (b) and (e) $D^{+}\to {\rho }^0{e}^{+}{\nu }_e$, ${\rho }^0\to {\pi }^{+}{\pi }^{-}$; and (c) and (f) $D^{+}\to \omega e^{+}{\nu }_e$, $\omega \to {\pi }^{+}{\pi }^{-}{\pi }^0$. The solid line represents the fit of the sum of the signal function and background function to the data. The dashed line indicates the background contribution. The arrows indicate a $\pm 48\mathrm{MeV}$ region around the $K_S^0$ mass, which has been removed for display.

Figure 2

Projections of the combined ${\rho }^{-}$ and ${\rho }^0$ data (points with statistical error bars) and the fit (solid histogram) onto $q^2$, $\cos {\theta }_e$, $\cos {\theta }_{\pi }$, and $\chi$. The dashed lines show the sum of the background distributions.