Determining the Penguin Effect on CP Violation in $B^0 \rightarrow \pi^+ \pi^-$

A major goal in B physics is measuring the CP-violating asymmetry in the decay $B^0 \rightarrow \pi^+ \pi^-$. In order to determine one of the phase angles in the CKM matrix from this decay it is necessary to determine the influence of the penguin amplitude. Here we show how, using $SU(3)$ symmetry, the penguin effect can be approximately determined from the ratio of the decay rates of $B^0 \rightarrow K^+ \pi^-$ and $B^0 \rightarrow \pi^+ \pi^-$.