The pollution to the $Kπ$-puzzle from the isospin-breaking $π^0-η-η'$ mixing effect

The influence of the isospin-breaking $π^0\!\!-\!η\!-\!η'$ mixing effect on the $CP$-asymmetries of $B\to Kπ$ 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 $B\to Kπ$ processes and the $CP$-asymmetry difference of the $B^+\to K^+π^0$ and $B^0\to K^+π^-$, obscuring the significance of the $Kπ$-puzzle. This uncertainty can be so large that it is even possible to explain the $Kπ$-puzzle by the $π^0\!\!-\!η\!-\!η'$ mixing effect {\it alone}.

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