Observation of the doubly Cabibbo-Suppressed decays $D^+\to K^+π^0π^0$ and $D^+\to K^+π^0η$

By analyzing $e^+e^-$ annihilation data corresponding to an integrated luminosity of $2.93\,\rm fb^{-1}$ collected at the center-of-mass energy of 3.773\,GeV with the BESIII detector, we report the first observations of the doubly Cabibbo-suppressed decays $D^+\to K^+π^0π^0$ and $D^+\to K^+π^0η$. The branching fractions of $D^+\to K^+π^0π^0$ and $D^+\to K^+π^0η$ are measured to be $(2.1 \pm 0.4_{\rm stat} \pm 0.1_{\rm syst})\times 10^{-4}$ and $(2.1 \pm 0.5_{\rm stat} \pm 0.1_{\rm syst})\times 10^{-4}$ with statistical significances of 8.8$σ$ and 5.5$σ$, respectively. In addition, we search for the subprocesses $D^+\to K^{*}(892)^{+}π^0$ and $D^+\to K^{*}(892)^{+}η$ with $K^{*}(892)^+\to K^+π^0$. The branching fraction of $D^+\to K^{*}(892)^{+}η$ is determined to be $({4.4^{+1.8}_{-1.5}}_{\rm stat}\pm0.2_{\rm syst})\times10^{-4}$, with a statistical significance of 3.2$σ$. No significant signal for $D^+\to K^{*}(892)^{+}π^0$ is found and we set an upper limit on the branching fraction of this decay at the 90\% confidence level to be $5.4\times10^{-4}$.