Isolating the penguin-diagram contribution toCP violation in B ⁰_d vs.$B_d^0 {\mathbf{ }}vs.{\mathbf{ }}\bar B_d^0 \to \pi ^ + \pi ^ -$

Summary

A reliable prediction forCP violation in B ⁰_d vs.$B_d^0 {\mathbf{ }}vs.{\mathbf{ }}\bar B_d^0 \to \pi ^ + \pi ^ -$ suffers from the penguin-diagram induced uncertainty. With the help ofSU(3) relations, we show that not only the magnitude but also the strong phase shift of the penguin amplitude can be approximately determined only from the branching ratios of$B_d^0 {\mathbf{ }}vs.{\mathbf{ }}\bar B_d^0 \to \pi ^ + \pi ^ - ,B_d^0 \to K^ + \pi ^ -$, and$B_u^0 \to K^0 \pi ^ +$.