Search for $X(3872)\toπ^0π^0χ_{c1,2}$
Using 10.1 fb$^{-1}$ of $e^+e^-$ collision data collected by the BESIII detector with center-of-mass energies between 4.15 GeV and 4.30 GeV, we search for the decays $X(3872)\toπ^0π^0χ_{c1,2}$, where the $X(3872)$ is produced in $e^+e^-\toγX(3872)$. No evidence above $3σ$ is found for either decay. Upper limits at the $90\%$ C.L. on the branching fractions of $X(3872)\toπ^0π^0χ_{c1,2}$ normalized to the branching fraction of $X(3872)\toπ^+π^-J/ψ$ are set to be $\mathcal{B}(X(3872)\toπ^0π^0χ_{c1})/\mathcal{B}(X(3872)\toπ^+π^-J/ψ) < 1.1$ and $\mathcal{B}(X(3872)\toπ^0π^0χ_{c2})/\mathcal{B}(X(3872)\toπ^+π^-J/ψ) < 0.5$, taking into account both statistical and systematic uncertainties.