The impact of $a_0^0(980)-f_0(980)$ mixing on the localized $CP$ violations of the $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ decay

In the framework of the QCD factorization approach, we study the localized $CP$ violations of the $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ decay with and without $a_0^0(980)-f_0(980)$ mixing mechanism, respectively, and find that the localized $CP$ violation can be enhanced by this mixing effect when the mass of the ${\pi }^{+}{\pi }^{-}$ pair is in the vicinity of the $f_0(980)$ resonance. The corresponding theoretical prediction results are ${\mathcal{A}}_{CP}(B^{-}\to K{f}_0\to K^{-}{\pi }^{+}{\pi }^{-})=[0.24,0.36]$ and $$\mathcal{A}$_{CP}(B^-\rightarrow K^- f_0(a_0) \rightarrow K^-\pi^+\pi^-)=[0.33, 0.52]$, respectively. Meanwhile, we also calculate the branching fraction of the $B^{-}\to K^{-}{f}_0(980)\to K^{-}{\pi }^{+}{\pi }^{-}$ decay, which is consistent with the experimental results. We suggest that $a_0^0(980)-f_0(980)$ mixing mechanism should be considered when studying the $CP$ violation of the $B$ or $D$ mesons decays theoretically and experimentally.