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

Abstract

A sum rule relating the ratio $R_c=2\Gamma (B^{+}\to K^{+}{\pi }^0)/\Gamma (B^{+}\to K^0{\pi }^{+})$ and the CP asymmetry $A_{CP}(B^{+}\to K^{+}{\pi }^0)$ 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 $B^{+}\to K^{+}{\pi }^0$. The measured ratio $A_{CP}(B^{+}\to K^{+}{\pi }^0)/A_{CP}(B^0\to K^{+}{\pi }^{-})$ rules out a small strong phase difference between a color-suppressed and a color-favored tree amplitude contributing to $B^{+}\to K^{+}{\pi }^0$ as favored by QCD factorization.

Comments: small correction