Search for $D^0togammagamma$ and improved measurement of the branching fraction for $D^0 to pi^0pi^0$

Using $2.92$ fb$^{-1}$ of electron-positron annihilation data collected at $sqrt{s} = 3.773$~GeV with the BESIII detector, we report the results of a search for the flavor-changing neutral current process $D^0togammagamma$ using a double-tag technique. We find no signal and set an upper limit at $90%$ confidence level for the branching fraction of $B(D^0togammagamma) < 3.8times10^{-6}$. We also investigate $D^0$-meson decay into two neutral pions, obtaining a branching fraction of $B(D^0topi^0pi^0) = (8.24pm0.21(text{stat.})pm0.30(text{syst.}))times10^{-4}$, the most precise measurement to date and consistent with the current world average