Search for $X\left(3872\right)\to {\pi }^{0}{\chi }_{c0}$ and $X\left(3872\right)\to \pi \pi {\chi }_{c0}$ at BESIII

Using $9.9\text{}\text{}{\mathrm{fb}}^{-1}$ 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 ${e}^{+}{e}^{-}\to \gamma X\left(3872\right)$ with $X\left(3872\right)\to {\pi }^{0}{\chi }_{c0}$ and $X\left(3872\right)\to \pi \pi {\chi }_{c0}$. We set upper limits (at 90% C.L.) of $\frac{B\left(X\left(3872\right)\to {\pi }^{0}{\chi }_{c0}\right)}{B\left(X\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi \right)}<3.6$, $\frac{B\left(X\left(3872\right)\to {\pi }^{+}{\pi }^{-}{\chi }_{c0}\right)}{B\left(X\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi \right)}<0.56$, and $\frac{B\left(X\left(3872\right)\to {\pi }^{0}{\pi }^{0}{\chi }_{c0}\right)}{B\left(X\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi \right)}<1.7$. Combined with the BESIII measurement of $X\left(3872\right)\to {\pi }^{0}{\chi }_{c1}$, we also set an upper limit of $\frac{B\left(X\left(3872\right)\to {\pi }^{0}{\chi }_{c0}\right)}{B\left(X\left(3872\right)\to {\pi }^{0}{\chi }_{c1}\right)}<4.5$.