Sum Rule for Rate and Cp Asymmetry in B + →

A sum rule relating the ratio Rc = 2Γ(B + → Kπ)/Γ(B → Kπ) and the CP asymmetry ACP (B + → Kπ) is proved to first order in the ratio of tree to penguin amplitudes. The sum rule explains why it is possible to have Rc consistent with 1 together with a small CP asymmetry in B → Kπ. The measured ratio ACP (B → Kπ)/ACP (B → Kπ) rules out a small strong phase difference between C and T as favored by QCD factorization. PACS Categories: Partial decay rates and CP asymmetries in B → Kπ decays have been mapped out with increasing accuracy thanks to high-statistics studies by the BaBar [1, 2, 3] and Belle [4] Collaborations. The experimental situation for branching ratios and CP asymmetries is summarized in Table I [5, 6, 7], defining asymmetries by ACP ≡ [Γ(B̄ → f̄) − Γ(B → f)]/[Γ(B̄ → f̄) + Γ(B → f)]. These decays are dominated by an effective b̄ → s̄ penguin amplitude, whose isospin-preserving property leads to the rate predictions for B and B̄ decays, Γ(B → Kπ) = 2Γ(B → Kπ) = Γ(B → Kπ) = 2Γ(B → Kπ) . (1) Tests for non-penguin amplitudes are provided by useful CP-averaged ratios [8, 9, 10, 11], R ≡ Γ̄(B 0 → Kπ) Γ̄(B → Kπ) = 0.92± 0.07 , Rc ≡ 2Γ̄(B → Kπ) Γ̄(B → Kπ) = 1.11± 0.07 , Rn ≡ Γ̄(B → Kπ) 2Γ̄(B → Kπ) = 0.99± 0.07 , (2) where Γ̄(B → f) ≡ [Γ(B → f) + Γ(B̄ → f̄)]/2. We have converted ratios of branching ratios to ratios of rates in the first relation using the ratio τ+/τ0 = 1.076 ± 0.008 of B