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Search for $X(3872)\toπ^0χ_{c0}$ and $X(3872)\toππχ_{c0}$ at BESIII

Using 9.9 fb$^{-1}$ of $e^+e^-$ collision data collected by the BESIII detector at center-of-mass energies between 4.15 and 4.30 GeV, we search for the processes $e^+e^-\toγX(3872)$ with $X(3872)\rightarrowπ^0χ_{c0}$ and $X(3872)\rightarrowππχ_{c0}$. Depending on the fitting model, the statistical significance for $X(3872)\toπ^0χ_{c0}$ ranges from 1.3$σ$ to 2.8$σ$. We set upper limits (at 90\% C.L.) of $\frac{\mathcal{B}(X(3872)\rightarrowπ^0χ_{c0})}{\mathcal{B}(X(3872)\toπ^+π^-J/ψ)}<3.6$, $\frac{\mathcal{B}(X(3872)\rightarrowπ^+π^-χ_{c0})}{\mathcal{B}(X(3872)\toπ^+π^-J/ψ)}<0.68$, and $\frac{\mathcal{B}(X(3872)\rightarrowπ^0π^0χ_{c0})}{\mathcal{B}(X(3872)\toπ^+π^-J/ψ)}<1.7$. Combined with the BESIII measurement of $X(3872)\toπ^0χ_{c1}$, we also set an upper limit of $\frac{\mathcal{B}(X(3872)\rightarrowπ^0χ_{c0})}{\mathcal{B}(X(3872)\toπ^0χ_{c1})}<4.4$.