D ec 1 99 4 OITS - 566 CP asymmetry Relations Between B̄ 0 → ππ And B̄ 0 → πK Rates ∗

We prove that CP violating rate difference ∆(B̄0 → π+π−) = Γ(B̄0 → π+π−)− Γ(B0 → π−π+) is related to ∆(B̄0 → π+K−) = Γ(B̄0 → π+K−)− Γ(B0 → π−K+) in the three generation Standard Model. Neglecting small annihilation diagrams, and in the SU(3) symmetry limit, we show ∆(B̄0 → π+π−) = −∆(B̄0 → π+K−). The SU(3) breaking effects are estimated using factorization approximation, and yield ∆(B̄0 → π+π−) ≈ −(fπ/fK)∆(B̄ → π+K−). Usefulness of this remarkable relation for determining phases in the CKM unitarity triangle is discussed. Typeset using REVTEX Work supported in part by the Department of Energy Grant No. DE-FG06-85ER40224. 1 Detection of CP violation and verification of the unitarity triangle of the CKM matrix is a major goal of B factories. Measurement of rate asymmetry in certain channels not only establishes direct CP violation, but can aid in determining some of the angles of the unitarity triangle. In this letter we shall prove a remakable relationship between rate difference ∆(B̄ → ππ(ππ)) = Γ(B̄ → ππ(ππ)) − Γ(B → ππ(ππ)) and ∆(B̄ → πK(πK̄)) = Γ(B̄ → π+K−(π0K̄0))−Γ(B0 → πK(πK)). This relationship follows purely from SU(3) symmetry and ignoring annihilation diagrams which can be shown to make negligible contributions. The usefulness of such a relationship lies in that difficult to measure rate difference like ∆(B̄ → ππ) can be related to easier measurement of ∆(B̄ → πK) which is a self-tagging mode. Similarly, it might prove easier to measure ∆(B̄ → πK̄) than the rate difference of the suppressed mode B̄ → ππ. In the Standard Model (SM) the effective Hamiltonian for B → ππ and B → πK decays can be written as follows: H eff = GF √ 2 [VubV ∗ uq(c1O q 1 + c2O q 2)− 10