Measurement of the $D^0 to pi^- e^+ u_e$ differential decay branching fraction as a function of $q^2$ and study of form factor parameterizations

Based on a sample of 500 million $e^+e^- to cbar c$ events recorded by the BaBar detector at c.m. energies of close to 10.6 GeV, we report on a study of the decay $D^0 to pi^- e^+ u_e$. We measure the ratio of branching fractions $R_D = {cal B}(D^0 to pi^- e^+ u_e)/{cal B}(D^0 to K^- pi^+) = 0.0713 pm 0.0017_{stat.} pm 0.0024_{syst.}$, and use the present world average for ${cal B}(D^0 to K^- pi^+)$ to obtain ${cal B}(D^0 to pi^- e^+ u_e) = (2.770 pm 0.068_{{rm stat.}} pm 0.092_{{rm syst.}} pm 0.037_{{rm ext.}})times 10^{-3}$ where the third error accounts for the uncertainty on the branching fraction for the reference channel. The measured dependence of the differential branching fraction on $q^2$, the four-momentum transfer squared between the $D$ and the $pi$ meson, is compared to various theoretical predictions for the hadronic form factor, $f_{+,D}^{pi}(q^2)$, and the normalization $|V_{cd}| times f_{+,D}^{pi}(q^2=0)=0.1374pm0.0038_{{rm stat.}} pm0.0022_{{rm syst.}} pm0.0009_{{rm ext.}}$ is extracted from a fit to data. Using the most recent LQCD prediction of $f_{+,D}^{pi}(q^2=0)=0.666 pm 0.029$, we obtain $|V_{cd}|=0.206 pm 0.007_{rm exp.} pm 0.009_{rm LQCD}$. Assuming instead, $|V_{cd}|=|V_{us}|=0.2252 pm 0.0009$, we obtain $f_{+,D}^{pi}(q^2=0)=0.610 pm 0.020_{rm exp.} pm 0.005_{rm ext.}$. The $q^2$ dependence of $f_{+,D}^{pi}(q^2)$ is compared to a variety of multi-pole parameterizations. This information is applied to $B^0 to pi^- e^+ u_e$ decays and, combined with an earlier $B^0 to pi^- e^+ u_e$ measurement by BaBar, is used to derive estimates of $|V_{ub}|$.