$W$-loop and vertical $W$-loop. As shown in Ref. [@Cheng:2010vk], only the internal emission graph (Fig. \[Fig:feyn\] of the present work) and $W$-exchange [^2] contribute to the $D^0 \to \bar{K}^0 f_0(980)$ and $D^0 \to \bar{K}^0a_0(980)$ decays. In Ref. [@kaminski] the $D^0 \to \bar{K}^0 \pi^+\pi^-$ decay is studied. Hence, only the $D^0 \to K^0_s f_0(980)$ decay can be addressed, which is accounted for by proper form factors and taken into account by means of the $M_2$ ($K^0_s[\pi^+\pi^-]_s$) amplitude, which contains the tree level internal emission, and $W$-exchange (also called annihilation mechanism). In order to establish connection with the work of Ref. [@kaminski], let us draw the external emission and $W$-exchange diagrams pertinent to the $D^0 \to \bar{K}^0\pi^+\pi^-$ decay, as shown in Fig. \[wexchange\].

![External emission diagram \[(A)\] and the $W$-exchange diagram \[(B)\] for $D^0 \to \bar{K}^0 \pi^+ \pi^-$ decay[]{data-label="wexchange"}](wexchange.eps)

It is also instructive to recall the basic non-leptonic Hamiltonian at the quark level responsible for this transition [@Buchalla:1995vs; @ElBennich:2009da; @Leitner:2010ai] $$\begin{aligned}
H_{\rm W} &=& \frac{G_F}{\sqrt{2}} V_{cs} V_{ud} \bar{c}\gamma_{\mu}
(1   -   \gamma_5)s \bar{d}\gamma^{\mu} (1  - \gamma_5)u \nonumber
\\ &&  +   h.c.  \label{eq:hweak}\end{aligned}$$ This Hamiltonian transforms as an isospin $I=1$ operator. Consequently the decay amplitude of $D^0 \to K\pi\pi$ is $$\begin{aligned}
T(D^0 \! \to 