${\mathcal A}_{\rm CP}$ Puzzle: Possible Evidence for Large Strong Phase in $B\to K\pi$ Color-Suppressed Tree Amplitude

In QCD Factorization(QCDF), the suppression of the color-suppressed tree amplitude relative to the color-allowed one in $B\to K\pi$ decay implies a direct CP asymmetry in $B^{-}\to K^{-}\pi^{0}$ to be of the same sign and comparable in magnitude to that in ${\bar B}^{0}\to K^{+}\pi^{-}$, in contradiction with experiment. This is the $A_{\rm CP}$ $B\to K\pi$ puzzle. One of the current proposal to solve this puzzle is the existence of a large color-suppressed amplitude with large strong phase which implies also a large negative ${\bar B}^{0}\to {\bar K}^{0}\pi^{0}$ CP asymmetry. In this paper, by an essentially model-independent calculation, we show clearly that the large negative direct CP asymmetry in ${\bar B}^{0}\to {\bar K}^{0}\pi^{0}$ implies a large $C/T$, the ratio of the color-suppressed to the color-allowed tree amplitude and a large negative strong phase for $C$. By adding to the QCDF amplitude an additional color-suppressed term to generate a large $C/T$ and a large strong phase for $C$ and an additional penguin-like contribution, we obtain branching ratios for all $B\to K\pi$ modes and CP asymmetry for ${\bar B}^{0}\to K^{+}\pi^{-}$ and $B^{-}\to K^{-}\pi^{0}$ in agreement with experiment, and a large and negative CP asymmetry in ${\bar B}^{0}\to {\bar K}^{0}\pi^{0}$ which could be checked with more precise measurements.