chiral Lagrangians because the $[\phi,\partial_{\mu}\phi]$ structure of the Lagrangians renders this term inoperative. In Ref. [@weihong] the ordinary $\phi$ matrix of chiral perturbation theory was also used. Here we consider the full $\phi$ matrix of Eq. (\[eq:phimatrix\]) since we are concerned with physical $\eta$ plus $\pi^0$ production.

Hence upon hadronization of the $u\bar{u}$ component we shall have $$\begin{aligned}
&& u\bar{u}(\bar{u}u + \bar{d}d + \bar{s}s)  \equiv  (\phi \cdot
\phi)_{11} = \frac{1}{2}\pi^0 \pi^0 \nonumber \\
&& + \frac{1}{3} \eta \eta + \frac{2}{\sqrt{6}} \pi^0 \eta  +
\pi^+\pi^- + K^+ K^- , \label{eq:phiphi11}\end{aligned}$$ where we have omitted the $\eta'$ term because of its large mass. This means that upon hadronization of the $u\bar{u}$ component we have $D^0 \to \bar{K}^0 PP$, where $PP$ are the different meson meson components of Eq. (\[eq:phiphi11\]). This is only the first step, because now these mesons will interact among themselves delivering the desired meson pair component at the end: $\pi^+
\pi^-$ for the case of the $f_0(500)$ and $f_0(980)$, and $\pi^0
\eta$ for the case of the $a_0(980)$.

The multiple scattering of the mesons is readily taken into account as shown diagrammatically in Fig. \[Fig:mesonmesonFSI\].

![image](mesonmesonfsi.eps)

Analytically we shall have $$\begin{aligned}
&& t(D^0 \to \bar{K}^0 \pi^+ \pi^-) = V_P 