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Scalar resonances in the final state interactions of the decays $D^{0} \rightarrow π^{0}π^{0}π^{0}, π^{0}π^{0}η,π^{0}ηη$

We investigate the scalar resonances in the processes of $D^{0} \rightarrow π^{0}π^{0}π^{0}, π^{0}π^{0}η, π^{0}ηη$ decays based on the chiral unitary approach for the final state interaction. We start from singly Cabibbo-suppressed production diagrams which provide a primary quark pair to hadronize two pseudoscalar mesons in $D^{0}$ decays. The resonances $f_{0}(500)$, $f_{0}(980)$ and $a_{0}(980)$ are dynamically produced from the final state interactions of the meson pairs. In our results, the experimental data for the $π^{0}η$ invariant mass spectrum of the $D^{0} \rightarrow π^{0}ηη$ decay can be described well. We also make the predictions for the $π^{0}π^{0}$ invariant mass spectrum of the $D^{0} \rightarrow π^{0}π^{0}π^{0}$, where the $f_{0}(980)$ can be found, and for the $π^{0}π^{0}$, $π^{0}η$ invariant mass spectrum of the $D^{0} \rightarrow π^{0}π^{0}η$, where the $f_{0}(500)$, $f_{0}(980)$ and $a_{0}(980)$ appear. Furthermore, the branching ratios of each decay channel are predicted. We expect more accurate measurements of these decays to better understand the nature of the states $f_{0}(500)$, $f_{0}(980)$ and $a_{0}(980)$.