Observation of the Decay $B^{\pm }\to {\pi }^{\pm }{\pi }^0$, Study of $B^{\pm }\to K^{\pm }{\pi }^0$, and Search for $B^0\to {\pi }^0{\pi }^0$

We present results for the branching fractions and charge asymmetries in $B^{\pm }\to h^{\pm }{\pi }^0$ (where $h^{\pm }={\pi }^{\pm },K^{\pm }$) and a search for the decay $B^0\to {\pi }^0{\pi }^0$ using a sample of approximately $88\times {10}^6$ $B\overline{B}$ pairs collected by the BABAR detector at the PEP-II asymmetric-energy $B$ Factory at SLAC. We measure $\mathcal{B}(B^{\pm }\to {\pi }^{\pm }{\pi }^0)=({5.5}_{-0.9}^{+1.0}\pm 0.6)\times {10}^{-6}$, where the first error is statistical and the second is systematic. The $B^{\pm }\to {\pi }^{\pm }{\pi }^0$ signal has a significance of $7.7\sigma$ including systematic uncertainties. We simultaneously measure the $K^{\pm }{\pi }^0$ branching fraction to be $\mathcal{B}(B^{\pm }\to K^{\pm }{\pi }^0)=({12.8}_{-1.1}^{+1.2}\pm 1.0)\times {10}^{-6}$. The charge asymmetries are ${\mathcal{A}}_{{\pi }^{\pm }{\pi }^0}=-{0.03}_{-0.17}^{+0.18}\pm 0.02$ and ${\mathcal{A}}_{K^{\pm }{\pi }^0}=-0.09\pm 0.09\pm 0.01$. We place a 90% confidence-level upper limit on the branching fraction $\mathcal{B}(B^0\to {\pi }^0{\pi }^0)$ of $3.6\times {10}^{-6}$.