Measurement of the branching fractions of $χ_{cJ} \to π^{+}π^{-}π^{0}π^{0}$ via $ψ(3686) \to γχ_{cJ}$
Using $(2712.4\pm14.3)\times 10^6$ $ψ(3686)$ events collected with the BESIII detector operating at BEPCII, the branching fractions of $χ_{cJ}\toπ^+π^-π^0π^0$ ($J=0,~1,~2$) are measured via the radiative transition $ψ(3686)\toγχ_{cJ}$. The results are $\mathcal{B}(χ_{c0} \to π^{+}π^{-}π^{0}π^{0}) = (3.10 \pm 0.01 \pm 0.14) \times 10^{-2}$, $\mathcal{B}(χ_{c1} \to π^{+}π^{-}π^{0}π^{0}) = (1.16 \pm 0.01 \pm 0.05) \times 10^{-2}$, and $\mathcal{B}(χ_{c2} \to π^{+}π^{-}π^{0}π^{0}) = (1.92 \pm 0.01 \pm 0.08) \times 10^{-2}$, where the first uncertainties are statistical and the second systematic. The dominant intermediate states are found to be $χ_{cJ}\toρ^+ρ^-$. These results supersede the previous most precise measurements and provide significantly improved precision.