Extracting $\alpha$ from the CP Asymmetry in $B^0 \to \pi^+ \pi^-$ Decays

The extraction of the CKM angle $\alpha$ from the asymmetry in $B^0 \to \pi^+\pi^-$ vs ${\bar B^0} \to \pi^+\pi^-$ suffers from a currently unknown penguin contribution. Experimentally, one can determine the magnitude and phase of the CP asymmetry from a time-dependent analysis of tagged events, and the average rate for $B^0$ and $\bar{B}^0$ decays to $\pi^+\pi^-$ from untagged events. These measurements, together with the magnitudes and relative phase of the tree and penguin diagrams, can in principle completely determine $\alpha$, free of discrete ambiguities. We perform an error analysis on $\alpha$ given assumptions on the values and uncertainties of both the measurements and theoretical inputs.