\! K\pi\pi) \! =  <K^0_s M_1 M_2|H_{\rm W}|D^0>,\end{aligned}$$ where the two meson system $M_1M_2$ ($\pi^+\pi^-$ here) can have $I=0,~1,~2$. This is the case in the diagram of Fig. \[wexchange\] (A) where the $c \bar{u}$, $\pi^+$ intermediate state can have $I=1/2,~3/2$, which allows the $\pi^+\pi^-$ system to have $I=0,~1,~2$ in the final $\bar{K}^0 \pi^+\pi^-$ state. However, the diagram of Fig. \[wexchange\] (A) will not contribute to our resonance production which requires the $\pi^+\pi^-$ $S$ wave loop, as seen in Fig. \[Fig:mesonmesonFSI\], due to the vector structure of Eq. (\[eq:hweak\]) in the $csW^+$ vertex of Fig. \[wexchange\] (A). This is also the case in the phenomenological analysis of Ref. [@Cheng:2010vk]. Then, in the remaining mechanisms of Fig. \[Fig:mesonmesonFSI\] and Fig. \[wexchange\] (B) the $\pi^+\pi^-$ can only be in $I=0$ or $1$.

In our study we have isolated the $S$ wave of the pions in order to get the $f_0(500)$, $f_0(980)$ resonances, and the $a_0(980)$ in the case of $\pi^0 \eta$. Certainly, the operator of Eq. (\[eq:hweak\]) allows other angular momenta, and indeed experimentally $\rho$ meson and other mesons can be obtained, but the experimental analysis of Refs. [@Muramatsu:2002jp; @Rubin:2004cq] with partial wave analysis separate the contributions of $f_0(980)$ and $a_0(980)$ production, which 