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In the framework of the QCD factorization approach, we study the localized$CP$ violations of the $B^-\rightarrow K^- \pi^+\pi^-$ decay with and without$a_0^0(980)-f_0(980)$ mixing mechanism, respectively, and find that thelocalized $CP$ violation can be enhanced by this mixing effect when the mass ofthe $\pi^+\pi^-$ pair is in the vicinity of the $f_0(980)$ resonance. Thecorresponding theoretical prediction results are$\mathcal{A}_{CP}(B^-\rightarrow K f_0 \rightarrow K^-\pi^+\pi^-)=[0.24, 0.36]$and $\mathcal{A}_{CP}(B^-\rightarrow K^- f_0(a_0) \rightarrowK^-\pi^+\pi^-)=[0.33, 0.52]$, respectively. Meanwhile, we also calculate thebranching fraction of the $B^-\rightarrow K^-f_0(980)\rightarrow K^-\pi^+\pi^-$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 andexperimentally.

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