Analysis of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>D</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>K</mml:mi><mml:mo>−</mml:mo></mml:msup><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>decay channel

Using $347.5\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of data recorded by the BABAR detector at the PEP-II electron-positron collider, $244\ifmmode\times\else\texttimes\fi{}{10}^{3}$ signal events for the ${D}^{+}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{e}^{+}{\ensuremath{\nu}}_{e}$ decay channel are analyzed. This decay mode is dominated by the ${\overline{K}}^{*}(892{)}^{0}$ contribution. We determine the ${\overline{K}}^{*}(892{)}^{0}$ parameters: ${m}_{{K}^{*}(892{)}^{0}}=(895.4\ifmmode\pm\else\textpm\fi{}0.2\ifmmode\pm\else\textpm\fi{}0.2)\text{ }\text{ }\mathrm{MeV}/{c}^{2}$, ${\ensuremath{\Gamma}}_{{K}^{*}(892{)}^{0}}^{0}=(46.5\ifmmode\pm\else\textpm\fi{}0.3\ifmmode\pm\else\textpm\fi{}0.2)\text{ }\text{ }\mathrm{MeV}/{c}^{2}$, and the Blatt-Weisskopf parameter ${r}_{\mathrm{BW}}=2.1\ifmmode\pm\else\textpm\fi{}0.5\ifmmode\pm\else\textpm\fi{}0.5\text{ }\text{ }(\mathrm{GeV}/c{)}^{\ensuremath{-}1}$, where the first uncertainty comes from statistics and the second from systematic uncertainties. We also measure the parameters defining the corresponding hadronic form factors at ${q}^{2}=0$ (${r}_{V}=\frac{V(0)}{{A}_{1}(0)}=1.463\ifmmode\pm\else\textpm\fi{}0.017\ifmmode\pm\else\textpm\fi{}0.031$, ${r}_{2}=\frac{{A}_{2}(0)}{{A}_{1}(0)}=0.801\ifmmode\pm\else\textpm\fi{}0.020\ifmmode\pm\else\textpm\fi{}0.020$) and the value of the axial-vector pole mass parametrizing the ${q}^{2}$ variation of ${A}_{1}$ and ${A}_{2}$: ${m}_{A}=(2.63\ifmmode\pm\else\textpm\fi{}0.10\ifmmode\pm\else\textpm\fi{}0.13)\text{ }\text{ }\mathrm{GeV}/{c}^{2}$. The $S$-wave fraction is equal to $(5.79\ifmmode\pm\else\textpm\fi{}0.16\ifmmode\pm\else\textpm\fi{}0.15)%$. Other signal components correspond to fractions below 1%. Using the ${D}^{+}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}$ channel as a normalization, we measure the ${D}^{+}$ semileptonic branching fraction: $\mathcal{B}({D}^{+}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{e}^{+}{\ensuremath{\nu}}_{e})=(4.00\ifmmode\pm\else\textpm\fi{}0.03\ifmmode\pm\else\textpm\fi{}0.04\ifmmode\pm\else\textpm\fi{}0.09)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$, where the third uncertainty comes from external inputs. We then obtain the value of the hadronic form factor ${A}_{1}$ at ${q}^{2}=0$: ${A}_{1}(0)=0.6200\ifmmode\pm\else\textpm\fi{}0.0056\ifmmode\pm\else\textpm\fi{}0.0065\ifmmode\pm\else\textpm\fi{}0.0071$. Fixing the $P$-wave parameters, we measure the phase of the $S$ wave for several values of the $K\ensuremath{\pi}$ mass. These results confirm those obtained with $K\ensuremath{\pi}$ production at small momentum transfer in fixed target experiments.

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