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

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 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 $mathcal{A}_{CP}(B^-rightarrow K f_0 rightarrow K^-pi^+pi^-)=[0.24, 0.36]$ 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 $B^-rightarrow K^-f_0(980)rightarrow K^-pi^+pi^-$ 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 and experimentally.