Penguin Zoology in $B\to \pi \pi$ and the Extraction of the CKM Angle $\alpha$

Abstract

We reanalyze the decay $B_d\to {\pi }^{+}{\pi }^{-}$ 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 $\beta$. Nevertheless it is still possible to extract the CKM angle $\alpha$ in a transparent and almost model-independent way from the time-dependent CP-violating asymmetry of the transition $B_d\to {\pi }^{+}{\pi }^{-}$. 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.

8 pages, LaTeX, no figures. The complete paper is also available via anonymous ftp at ftp://www-ttp.physik.uni-karlsruhe.de/ , or via www at http://www-ttp.physik.uni-karlsruhe.de/cgi-bin/preprints/

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