to the mechanism of final state interaction. We should note that in both cases, the intermediate production of $K\bar{K}$ states, and further rescattering to give $\pi^+\pi^-$ or $\pi^0 \eta$ in the final states, is a novelty of our approach compared to other approaches and an essential ingredient in the results due to the strong coupling of the $f_0(980)$ and $a_0(980)$ resonances to $K\bar{K}$.

The dominance of the internal emissions in this kind of processes is also supported in other works [@Stone:2013eaa; @Aaij:2014siy; @Li:2011pg; @Aaltonen:2011nk; @Abazov:2011hv]. In Ref. [@kaminski] a detailed discussion is made of results in different works. The $W$-exchange mechanism in Ref. [@kaminski] depends on two unknown form factors which are fitted to the data and a phase which is unknown. From a fit to the data, a minimal strength of about $20\%$ is obtained for the $W$-exchange mechanism, suggesting that the contribution could be bigger. It is clear that this issue is still open but the relative smallness of the $W$-exchange mechanism has many arguments in favor, and our study, producing a big ratio of $a_0(980)$ versus $f_0(980)$ production due to final state interaction in coupled channels, neglecting the $W$-exchange mechanism, provides extra support for its smallness. Note 