Search for the decay $χ_{c1}(3872)\toπ^{+}π^{-}χ_{c1}$
Using a data sample corresponding to an integrated luminosity of 10.9 fb$^{-1}$ collected at center-of-mass energies from 4.16 to 4.34 GeV with the BESIII detector, we search for the decay $χ_{c1}(3872) \to π^{+}π^{-}χ_{c1}$ in the radiative production $e^{+}e^{-} \to γχ_{c1}(3872)$. No significant signal is observed, and the ratio for the branching fraction of $χ_{c1}(3872) \to π^{+}π^{-}χ_{c1}$ to $χ_{c1}(3872) \to π^{+}π^{-}J/ψ$ is measured as $\mathcal{R}\equiv\frac{\mathcal{B}[χ_{c1}(3872) \to π^{+}π^{-}χ_{c1}]}{\mathcal{B}[χ_{c1}(3872) \to π^{+}π^{-} J/ψ]}<0.18$ at 90$\%$ confidence level. The upper limit on the product of the cross section $σ[e^{+}e^{-}\toγχ_{c1}(3872)]$ and the branching fraction $\mathcal{B}[χ_{c1}(3872)\toπ^{+}π^{-}χ_{c1}]$ at each center-of-mass energy is also given. These measurements favor the non-conventional charmonium nature of the $χ_{c1}(3872)$ state.