CP asymmetry Relations Between $\bar B^0\rightarrow ππ$ and $\bar B^0\rightarrow πK$ Rates

Abstract

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