Measurement of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>D</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:msup><mml:mi>e</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msub><mml:mi>ν</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math>differential decay branching fraction as a function of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>q</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math>and study of form factor parametrizations

Based on a sample of 500 million ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}c\overline{c}$ events recorded by the BABAR detector at c.m. energies of close to 10.6 GeV, we report on a study of the decay ${D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{e}^{+}{\ensuremath{\nu}}_{e}$. We measure the ratio of branching fractions, ${R}_{D}=\mathcal{B}({D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{e}^{+}{\ensuremath{\nu}}_{e})/\mathcal{B}({D}^{0}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+})=0.0713\ifmmode\pm\else\textpm\fi{}0.001{7}_{\text{stat}}\ifmmode\pm\else\textpm\fi{}0.002{4}_{\text{syst}}$, and use the present world average for $\mathcal{B}({D}^{0}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+})$ to obtain $\mathcal{B}({D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{e}^{+}{\ensuremath{\nu}}_{e})=(2.770\ifmmode\pm\else\textpm\fi{}0.06{8}_{\text{stat}}\ifmmode\pm\else\textpm\fi{}0.09{2}_{\text{syst}}\ifmmode\pm\else\textpm\fi{}0.03{7}_{\text{ext}})\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}3}$ 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 $\ensuremath{\pi}$ meson, is compared to various theoretical predictions for the hadronic form factor, ${f}_{+,D}^{\ensuremath{\pi}}({q}^{2})$, and the normalization $|{V}_{cd}|\ifmmode\times\else\texttimes\fi{}{f}_{+,D}^{\ensuremath{\pi}}({q}^{2}=0)=0.1374\ifmmode\pm\else\textpm\fi{}\phantom{\rule{0ex}{0ex}}0.003{8}_{\text{stat}}\ifmmode\pm\else\textpm\fi{}0.002{2}_{\text{syst}}\ifmmode\pm\else\textpm\fi{}0.000{9}_{\text{ext}}$. is extracted from a fit to data. Using the most recent LQCD prediction of ${f}_{+,D}^{\ensuremath{\pi}}({q}^{2}=0)=0.666\ifmmode\pm\else\textpm\fi{}0.029$, we obtain $|{V}_{cd}|=0.206\ifmmode\pm\else\textpm\fi{}0.00{7}_{\text{exp}}\ifmmode\pm\else\textpm\fi{}0.00{9}_{\mathrm{LQCD}}$. Assuming, instead, $|{V}_{cd}|=|{V}_{us}|=0.2252\ifmmode\pm\else\textpm\fi{}0.0009$, we obtain ${f}_{+,D}^{\ensuremath{\pi}}({q}^{2}=0)=0.610\ifmmode\pm\else\textpm\fi{}0.02{0}_{\text{exp}}\ifmmode\pm\else\textpm\fi{}0.00{5}_{\text{ext}}$. The ${q}^{2}$ dependence of ${f}_{+,D}^{\ensuremath{\pi}}({q}^{2})$ is compared to a variety of multipole parametrizations. This information is applied to ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{e}^{+}{\ensuremath{\nu}}_{e}$ decays and, combined with an earlier ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{e}^{+}{\ensuremath{\nu}}_{e}$ measurement by BABAR, is used to derive estimates of $|{V}_{ub}|$.