Study of $B^0\to {\pi }^0{\pi }^0$, $B^{\pm }\to {\pi }^{\pm }{\pi }^0$, and $B^{\pm }\to K^{\pm }{\pi }^0$ decays, and isospin analysis of $B\to \pi \pi$ decays

We present updated measurements of the branching fractions and $CP$ asymmetries for $B^0\to {\pi }^0{\pi }^0$, $B^{\pm }\to {\pi }^{\pm }{\pi }^0$, and $B^{\pm }\to K^{\pm }{\pi }^0$. Based on a sample of $383\times {10}^6$ $\mathit{{\rm Y}}(4S)\to B\overline{B}$ decays collected by the BABAR detector at the PEP-II asymmetric-energy $B$ factory at SLAC, we measure $\mathcal{B}(B^0\to {\pi }^0{\pi }^0)=(1.47\pm 0.25\pm 0.12)\times {10}^{-6}$, $\mathcal{B}(B^{\pm }\to {\pi }^{\pm }{\pi }^0)=(5.02\pm 0.46\pm 0.29)\times {10}^{-6}$, and $\mathcal{B}(B^{\pm }\to K^{\pm }{\pi }^0)=(13.6\pm 0.6\pm 0.7)\times {10}^{-6}$. We also measure the $CP$ asymmetries ${\mathcal{C}}_{{\pi }^0{\pi }^0}=-0.49\pm 0.35\pm 0.05$, ${\mathcal{A}}_{{\pi }^{\pm }{\pi }^0}=0.03\pm 0.08\pm 0.01$, and ${\mathcal{A}}_{K^{\pm }{\pi }^0}=0.030\pm 0.039\pm 0.010$. Finally, we present bounds on the Cabibbo-Kobayashi-Maskawa angle $\alpha$ using isospin relations.

Figure 1

Distributions made with the event weighting and background subtraction method described in Ref. [15] and PDF projections for the likelihood fit variables in the $B^0\to {\pi }^0{\pi }^0$ fit. Shown are $m_{\mathrm{ES}}$ [(a),(b)], $\Delta E$ [(c),(d)], and $\mathcal{F}$ [(e),(f)] for signal [(a),(c),(e)] and continuum background [(b),(d),(f)].

Figure 2

Distributions made with the event weighting and background subtraction method described in Ref. [15] and PDF projections of the likelihood fit variables from the $B^{\pm }\to h^{\pm }{\pi }^0$ fit. Shown are $m_{\mathrm{ES}}$ [(a),(b)], $\Delta E$ [(c),(d)], and $\mathcal{F}$ [(e),(f)] distributions for signal [(a),(c),(e)] and continuum background [(b),(d),(f)]. PDF projections for the $B^{\pm }\to K^{\pm }{\pi }^0$ signal and background are shown as solid lines, while the PDF projections for the $B^{\pm }\to {\pi }^{\pm }{\pi }^0$ signal and background are shown as dashed lines. $B^{\pm }\to {\pi }^{\pm }{\pi }^0$ signal and background are shown as open circles while the $B^{\pm }\to K^{\pm }{\pi }^0$ is shown as solid circles.

Figure 3

Constraints on (a) the angle $\Delta \alpha =|\alpha -{\alpha }_{\mathrm{eff}}|$ and (b) $\alpha$ expressed as one minus the confidence level as a function of angle. We find an upper bound on $\Delta \alpha$ of $39°$ at the 90% confidence level. In (b) the curve shows the bounds on $\alpha$ using the isospin method alone, while the shaded region shows the result with the $SU(3)$ requirement as discussed in the text.