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 million $B\bar{B}$ pairs collected by the BABAR detector at the PEP-II asymmetric-energy $B$ Factory at SLAC. We measure ${\cal B} (B^{\pm} \to \pi^{\pm} \pi^{0}) = (5.5 ^{+1.0}_{-0.9} \pm 0.6) \times 10^{-6}$, 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 $ {\cal B} (B^{\pm} \to K^{\pm} \pi^{0}) = (12.8 ^{+1.2}_{-1.1} \pm 1.0) \times 10^{-6}$. The charge asymmetries are ${\cal A}_{\pi^{\pm}\pi^{0}} = -0.03_{-0.17}^{+0.18} \pm 0.02$ and ${\cal A}_{K^{\pm}\pi^{0}} = -0.09 \pm 0.09 \pm 0.01$. We place a 90% confidence-level upper limit on the branching fraction ${\cal B} (B^{0} \to \pi^{0} \pi^{0})$ of $3.6 \times 10^{-6} $.