Measurement of Branching Fractions and Charge Asymmetries in $B^{+-} \to \rho^{+-}\pi^0$ and $B^{+-} \to \rho^0\pi^{+-}$ Decays, and Search for $B^0 \to \rho^0\pi^0$

We present measurements of branching fractions and charge asymmetries in $B$-meson decays to $\rho^+ \pi^{0}$, $\rho^{0}\pi^+$ and $\rho^0\pi^0$. The data sample comprises $89 \times 10^6$ $\Upsilon(4S) \to B\bar{B}$ decays collected with the $BaBar$ detector at the PEP-II asymmetric-energy $B$ Factory at SLAC. We find the charge-averaged branching fractions ${\cal B}(B^{+}\to \rho^{+}\pi^0) = (10.9 \pm 1.9{\rm (stat)} \pm 1.9{\rm (syst)})\times 10^{-6}$ and ${\cal B}(B^{+} \to \rho^0 \pi^{+}) = (9.5 \pm 1.1{\rm (stat)} \pm 0.8{\rm (syst)}) \times 10^{-6}$, and we set a 90% confidence-level upper limit ${\cal B}(B^0 \to \rho^0\pi^0) < 2.9 \times 10^{-6}$. We measure the charge asymmetries $A_{CP}^{\rho^{+}\pi^0} = 0.24 \pm 0.16 \pm 0.06$ and $A_{CP}^{\rho^0\pi^{+}} = -0.19 \pm 0.11 \pm 0.02$.