Branching fraction and form-factor shape measurements of exclusive charmless semileptonic<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi></mml:math>decays, and determination of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>u</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo></mml:math>

We report the results of a study of the exclusive charmless semileptonic decays, ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$, ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$, ${B}^{+}\ensuremath{\rightarrow}\ensuremath{\omega}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$, ${B}^{+}\ensuremath{\rightarrow}\ensuremath{\eta}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$, and ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$ ($\ensuremath{\ell}=e$ or $\ensuremath{\mu}$) undertaken with approximately $462\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $B\overline{B}$ pairs collected at the $\ensuremath{\Upsilon}(4S)$ resonance with the BABAR detector. The analysis uses events in which the signal $B$ decays are reconstructed with a loose neutrino reconstruction technique. We obtain partial branching fractions in several bins of ${q}^{2}$, the square of the momentum transferred to the lepton-neutrino pair, for ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$, ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$, ${B}^{+}\ensuremath{\rightarrow}\ensuremath{\omega}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$, and ${B}^{+}\ensuremath{\rightarrow}\ensuremath{\eta}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$. From these distributions, we extract the form-factor shapes ${f}_{+}({q}^{2})$ and the total branching fractions $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\ell}}^{+}\ensuremath{\nu})=(1.45\ifmmode\pm\else\textpm\fi{}{0.04}_{\mathrm{stat}}\ifmmode\pm\else\textpm\fi{}{0.06}_{\mathrm{syst}})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ (combined ${\ensuremath{\pi}}^{\ensuremath{-}}$ and ${\ensuremath{\pi}}^{0}$ decay channels assuming isospin symmetry), $\mathcal{B}({B}^{+}\ensuremath{\rightarrow}\ensuremath{\omega}{\ensuremath{\ell}}^{+}\ensuremath{\nu})=(1.19\ifmmode\pm\else\textpm\fi{}{0.16}_{\mathrm{stat}}\ifmmode\pm\else\textpm\fi{}{0.09}_{\mathrm{syst}})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ and $\mathcal{B}({B}^{+}\ensuremath{\rightarrow}\ensuremath{\eta}{\ensuremath{\ell}}^{+}\ensuremath{\nu})=(0.38\ifmmode\pm\else\textpm\fi{}{0.05}_{\mathrm{stat}}\ifmmode\pm\else\textpm\fi{}{0.05}_{\mathrm{syst}})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. We also measure $\mathcal{B}({B}^{+}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}{\ensuremath{\ell}}^{+}\ensuremath{\nu})=(0.24\ifmmode\pm\else\textpm\fi{}{0.08}_{\mathrm{stat}}\ifmmode\pm\else\textpm\fi{}{0.03}_{\mathrm{syst}})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. We obtain values for the magnitude of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|{V}_{ub}|$ by direct comparison with three different QCD calculations in restricted ${q}^{2}$ ranges of $B\ensuremath{\rightarrow}\ensuremath{\pi}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$ decays. From a simultaneous fit to the experimental data over the full ${q}^{2}$ range and the FNAL/MILC lattice QCD predictions, we obtain $|{V}_{ub}|=(3.25\ifmmode\pm\else\textpm\fi{}0.31)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, where the error is the combined experimental and theoretical uncertainty.

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