Roles of the scalar $f_0(500)$ and $f_0(980)$ in the process $D^0\to \pi^0\pi^0 \bar{K}^0$

Motivated by the near-threshold enhancement and the dip structure around 1~GeV in the $\pi^0\pi^0$ invariant mass distribution of the process $D^0\to \pi^0\pi^0\bar{K}^0$ observed by the CLEO Collaboration, we have investigated this process by taking into account the contribution from the $S$-wave pseudoscalar meson-pseudoscalar meson interactions within the chiral unitary approach, and also the one from the intermediate resonance $K^{*}(892)$. Our results are in good agreement with the CLEO measurements, which implies that, the near-threshold enhancement near the $\pi^0\pi^0$ threshold is mainly due to the contributions from the scalar meson $f_0(500)$ and the intermediate $K^*$, and the cusp structure around 1~GeV in the $\pi^0\pi^0$ invariant mass distribution should be associated with the scalar meson $f_0(980)$.