allows us to compare directly with these data without the need to look into other channels. Also, although in principle the amplitudes depend on two independent Mandelstam variables as seen in Ref. [@kaminski], the fact that we do not consider the $\bar{K}^0 \pi^-$ interaction (leading to the $\kappa$), which would just provide a background in the $\pi^+\pi^-$ mass distribution for the reasons discussed at the end of section II, makes our amplitude dependent upon the invariant mass of $\pi^+\pi^-$ or $\pi^0 \eta$.

Concerning the $W$-exchange diagrams, which we have ignored in our approach, we would like to argue in favor of its relative smallness with two arguments: firstly, in Fig. \[Fig:feyn\] (A) we can see that the $\bar{u}$ quark of the $D^0$ is a spectator. We thus have a one body operator at the $D^0$ quark level. However, in the $W$-exchange one involves the two quarks of $D^0$ and the amplitude squared involves the probability to find two quarks, smaller than that of finding one quark. This situation is typical in nuclear reactions, where the $W$-exchange would have its equivalent in the exchange currents [@Gil:1997bm]. The second argument is that in the $W$-exchange diagram of Fig. \[wexchange\] (B) there is 