Measurement of branching fractions and charge asymmetries in B ± → ρ±π0 and B ± → ρ0π± decays, and search for B0 → ρ0π0

We present measurements of branching fractions and charge asymmetries in $B$-meson decays to ${\ensuremath{\rho}}^{+}{\ensuremath{\pi}}^{0}$, ${\ensuremath{\rho}}^{0}{\ensuremath{\pi}}^{+}$, and ${\ensuremath{\rho}}^{0}{\ensuremath{\pi}}^{0}$. The data sample comprises $89\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $\ensuremath{\Upsilon}(4S)\ensuremath{\rightarrow}B\overline{B}$ decays collected with the BABAR detector at the PEP-II asymmetric-energy $B$ Factory at SLAC. We find the charge-averaged branching fractions $\mathcal{B}({B}^{+}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{+}{\ensuremath{\pi}}^{0})=[10.9\ifmmode\pm\else\textpm\fi{}1.9\mathrm{(}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{)}\ifmmode\pm\else\textpm\fi{}1.9\mathrm{(}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{)}]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ and $\mathcal{B}({B}^{+}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}{\ensuremath{\pi}}^{+})=(9.5\ifmmode\pm\else\textpm\fi{}1.1\ifmmode\pm\else\textpm\fi{}0.9)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, and we set a $90%$ confidence-level upper limit $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}{\ensuremath{\pi}}^{0})l2.9\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$. We measure the charge asymmetries ${A}_{CP}^{{\ensuremath{\rho}}^{+}{\ensuremath{\pi}}^{0}}=0.24\ifmmode\pm\else\textpm\fi{}0.16\ifmmode\pm\else\textpm\fi{}0.06$ and ${A}_{CP}^{{\ensuremath{\rho}}^{0}{\ensuremath{\pi}}^{+}}=\ensuremath{-}0.19\ifmmode\pm\else\textpm\fi{}0.11\ifmmode\pm\else\textpm\fi{}0.02$.