We investigate the direct $CP$ violation in the decay $D^\pm \to \pi^\pm \pi^+ \pi^-$ incorporating the $a_0^0(980)$-$f_0(980)$ mixing mechanism. The integrated mixing intensities $\overline \xi_{ fa}$ and $\overline \xi_{ af}$ are calculated using meson masses and coupling constants extracted from various theoretical models and experimental data, yielding values of appreciable magnitude. We find that when the invariant mass of the $\pi^+\pi^-$pair lies near the $f_0(980)$ resonance, this isospin breaking mechanism can enhance the $CP$ asymmetry. The enhancement is particularly pronounced when $f_0(980)$ carries a significant $n\bar{ n}$ quark component and the $f_0(980)$ and $\sigma(600)$ mixing angle is approximately $26^\circ$. It is emphasized that the $a_0^0(980)$-$f_0(980)$ mixing mechanism can be taken into account in both theoretical and experimental studies of $CP$ violation in $B$ or $D$ meson decays.
Categories: hep-ph