Direct $CP$ violation in multi-body $B$ decays with $a_0^0(980)-f_0(980)$ mixing

We predict that $a_0^0(980)-f_0(980)$ mixing would lead to large $CP$ violation. We calculate the localized direct $CP$ asymmetry in the decays $B^{\pm }\to f_0(980)[a_0^0(980)]{\pi }^{\pm }\to {\pi }^{+}{\pi }^{-}{\pi }^{\pm }$ via the $a_0^0(980)-f_0(980)$ mixing mechanism based on the hypothetical $q\overline{q}$ structures of $a_0^0(980)$ and $f_0(980)$ in the QCD factorization approach. It is shown that there is a peak for $CP$ violation, which could be as large as 58%, when the invariant mass of $\pi \pi$ is near the masses of $a_0^0(980)$ and $f_0(980)$. Since the $CP$ asymmetry is sensitive to $a_0^0(980)-f_0(980)$ mixing, measuring the $CP$-violating parameter in the aforementioned decays could provide a new way to verify the existence of $a_0^0(980)-f_0(980)$ mixing and be helpful in clarifying the configuration of the light scalar mesons.

Figure 1

The Feynman diagram for $a_0^0(980)-f_0(980)$ mixing.

Figure 2

The differential $CP$-violating parameter as a function of $m$ in the decays $B^{\pm }\to f_0(980)[a_0^0(980)]{\pi }^{\pm }\to {\pi }^{+}{\pi }^{-}{\pi }^{\pm }$. (a) For ${\rho }_{H(A)}=0.25$: the dashed, dotted, dot-dashed, and solid lines correspond to the cases when ${\varphi }_{H(A)}=0,\pi /2,\pi ,3\pi /2$, respectively. (b) The same as (a) but for ${\rho }_{H(A)}=0.5$.