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

Abstract

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

Comments: 9 pages, Revtex