Sum rule for rate and CP asymmetry in $B^+\to K^+π^0$

Abstract

A sum rule relating the ratio $R_c = 2 Γ(B^+ \to K^+ π^0)/Γ(B^+ \to K^0 π^+)$ and the CP asymmetry $A_{CP}(B^+ \to K^+π^0)$ is proved to first order in the ratio of tree to penguin amplitudes. The sum rule explains why it is possible to have $R_c$ consistent with 1 together with a small CP asymmetry in $B^+ \to K^+ π^0$. The measured ratio $A_{CP}(B^+\to K^+π^0)/A_{CP}(B^0\to K^+π^-)$ rules out a small strong phase difference between a color-suppressed and a color-favored tree amplitude contributing to $B^+\to K^+π^0$ as favored by QCD factorization.