Branching Fractions and $CP$ Asymmetries in $B^0\to {\pi }^0{\pi }^0$, $B^{+}\to {\pi }^{+}{\pi }^0$, and $B^{+}\to K^{+}{\pi }^0$ Decays and Isospin Analysis of the $B\to \pi \pi$ System

Based on a sample of $227\times {10}^6$ $B\overline{B}$ pairs collected by the BABAR detector at the PEP-II asymmetric-energy $B$ Factory at SLAC, we measure the branching fraction $\mathcal{B}(B^0\to {\pi }^0{\pi }^0)=(1.17\pm 0.32\pm 0.10)\times {10}^{-6}$, and the asymmetry $C_{{\pi }^0{\pi }^0}=-0.12\pm 0.56\pm 0.06$. The $B^0\to {\pi }^0{\pi }^0$ signal has a significance of $5.0\sigma$. We also measure $\mathcal{B}(B^{+}\to {\pi }^{+}{\pi }^0)=(5.8\pm 0.6\pm 0.4)\times {10}^{-6}$, $\mathcal{B}(B^{+}\to K^{+}{\pi }^0)=(12.0\pm 0.7\pm 0.6)\times {10}^{-6}$, and the charge asymmetries ${\mathcal{A}}_{{\pi }^{+}{\pi }^0}=-0.01\pm 0.10\pm 0.02$ and ${\mathcal{A}}_{K^{+}{\pi }^0}=0.06\pm 0.06\pm 0.01$. Using isospin relations, we find an upper bound on the angle difference $|\alpha -{\alpha }_{\mathrm{eff}}|$ of 35° at the 90% C.L.