Investigation of the $D^0\to K_S^0{\pi }^0\eta ,K_S^0{\pi }^0{\pi }^0$ decays

Inspired by the invariant mass distributions for the decays $D^0\to K_S^0{\pi }^0\eta$ and $D^0\to K_S^0{\pi }^0{\pi }^0$ reported by the BESIII Collaboration, we investigate these processes with an unified final state interaction formalism by incorporating both the $S$-wave pseudoscalar meson-pseudoscalar meson interactions within a chiral unitary approach and the $P$-wave contribution from the intermediate resonance ${\bar{K}}^{\ast }(892)$. By performing a combined fit to the invariant mass spectra and taking into account the coherence between the $S$- and $P$-waves, our results are in agreement with the experimental data. For the decay $D^0\to K_S^0{\pi }^0\eta$, the structure near 1.0 GeV in the ${\pi }^0\eta$ invariant mass distribution corresponds to the signal of the $a_0(980)$, which is dynamically generated from the $S$-wave interactions. In the case of $D^0\to K_S^0{\pi }^0{\pi }^0$, the near-threshold enhancement in the ${\pi }^0{\pi }^0$ mass distribution arises from the combined contributions of the $f_0(500)$ and the intermediate ${\bar{K}}^{\ast }(892)$, while the cusp-like structure around 1 GeV${}^2$ is associated with the $f_0(980)$.