the strength of the $f_0(500)$ with respect to the $f_0(980)$ is relatively bigger in this latter decay than in the present case (almost $50\%$ bigger). The $\bar{B}^0 \to J/\psi
f_0(500),~f_0(500) \to \pi^+ \pi^-$ decay mode, together with the $f_0(980)$ one have been identified in Ref. [@Aaij:2014siy] through a partial wave analysis, and the rates obtained are comparable with the findings of Ref. [@weihong]. Such a partial wave analysis is not available from the work of Ref. [@Muramatsu:2002jp], where the analysis was done assuming a resonant state and a stable meson, including many contributions, but not the $K^0_s f_0(500)$. Yet, a discussion is done at the end of the paper [@Muramatsu:2002jp] in which the background seen is attributed to the $f_0(500)$. With this assumption they get a mass and width of the $f_0(500)$ compatible with other experiments. Further analyses in the line of [@Aaij:2014siy] would be most welcome to separate this important contributions to the $D^0 \to
K^0_s \pi^+ \pi^-$ decay.

Further considerations
======================

Our results are based on the dominance of the quark diagrams of Fig. \[Fig:feyn\]. In the weak decay of mesons the diagrams are classified in six different topologies [@Chau:1982da; @Chau:1987tk]: external emission, internal emission, $W$-exchange, $W$-annihilation, horizontal 