Bounding the effect of penguin diagrams inaCP(B0→π+π−)

A clean determination of the angle $\alpha$ of the unitary triangle from $\overrightarrow{B}\pi \pi$ decays requires an isospin analysis. If the $\overrightarrow{B}{\pi }^0{\pi }^0$ and $\overline{B}\to {\pi }^0{\pi }^0$ decay rates are small it may be hard to carry out this analysis. Here we show that an upper bound on the error on $\sin 2\alpha$ due to penguin diagram effects can be obtained using only the measured rate $\mathrm{B}\mathrm{}{(B}^{\pm }\to {\pi }^{\pm }{\pi }^0)$ and an upper bound on the combined rate $\mathrm{B}\left(\overrightarrow{B}{\pi }^0{\pi }^0\right)+\mathrm{B}(\overline{B}\to {\pi }^0{\pi }^0)$. Since no $b$ flavor tagging is needed to measure this combined rate, the bound that can be achieved may be significantly better than any approach which requires separate flavor-tagged neutral pion information.