Penguin Zoology in $B\toππ$ and the Extraction of the CKM Angle $α$

Abstract

We reanalyze the decay $B_d\toπ^+π^-$ without assuming dominance of QCD penguins with internal top-quark exchanges. In that case the weak phase of the $\bar b \to\bar d$ QCD penguin amplitude is not related in a simple way to the CKM angle $β$. Nevertheless it is still possible to extract the CKM angle $α$ in a transparent and almost model-independent way from the time-dependent CP-violating asymmetry of the transition $B_d\toπ^+π^-$. Aside from that CP asymmetry this approach needs as an input only amplitudes of decays with branching ratios of order $10^{-5}$ and will thus be well within reach at future $B$-factories.