The pollution to the $K\pi$-puzzle from the isospin-breaking ${\pi }^0-\eta -{\eta }^{\prime }$ mixing effect

Abstract

The influence of the isospin-breaking ${\pi }^0-\eta -{\eta }^{\prime }$ mixing effect on the $CP$-asymmetries of $B\to K\pi$ processes is examined for the first time. It is found that this mixing effect brings a large uncertainty both to the $CP$-asymmetry sum rule of $B\to K\pi$ processes and the $CP$-asymmetry difference of the $B^{+}\to K^{+}{\pi }^0$ and $B^0\to K^{+}{\pi }^{-}$, obscuring the significance of the $K\pi$-puzzle. This uncertainty can be so large that it is even possible to explain the $K\pi$-puzzle by the ${\pi }^0-\eta -{\eta }^{\prime }$ mixing effect {\it alone}.

Comments: 7 pages, with no figure