CP asymmetry Relations Between ${\bar{B}}^0\to \pi \pi$ and ${\bar{B}}^0\to \pi K$ Rates

We prove that CP violating rate difference $\Delta (\bar B^0 \rightarrow \pi^+\pi^-) = \Gamma (\bar B^0 \rightarrow \pi^+\pi^-) - \Gamma ( B^0 \rightarrow \pi^-\pi^+)$isrelatedto$\Delta (\bar B^0 \rightarrow \pi^+ K^-) = \Gamma (\bar B^0 \rightarrow \pi^+K^-) - \Gamma (B^0 \rightarrow \pi^-K^+)$ in the three generation Standard Model. Neglecting small annihilation diagrams, and in the SU(3) symmetry limit, we show $\Delta (\bar B^0 \rightarrow \pi^+\pi^-) = - \Delta (\bar B^0 \rightarrow \pi^+ K^-)$. The SU(3) breaking effects are estimated using factorization approximation, and yield $\Delta (\bar B^0\rightarrow \pi^+\pi^-) \approx -(f_\pi/f_K)^2\Delta (\bar B^0 \rightarrow \pi^+K^-)$. Usefulness of this remarkable relation for determining phases in the CKM unitarity triangle is discussed.