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Observation of the decays $χ_{cJ} \to ϕϕη$

Using a data sample of $(448.1\pm2.9)\times10^{6}$ $ψ(3686)$ decays collected by the BESIII detector at the Beijing Electron Positron Collider (BEPCII), we observe the decays $χ_{cJ}\to ϕϕη~(J=0,~1,~2)$, where the $χ_{cJ}$ are produced via the radiative processes $ψ(3686)\toγχ_{cJ}$. The branching fractions are measured to be $\mathcal B(χ_{c0}\toϕϕη)=(8.41\pm0.74\pm0.62)\times10^{-4}$, $\mathcal B(χ_{c1}\toϕϕη)=(2.96\pm0.43\pm0.22)\times 10^{-4}$, and $\mathcal B(χ_{c2} \to ϕϕη)=(5.33\pm0.52\pm0.39) \times 10^{-4}$, where the first uncertainties are statistical and the second are systematic. We also search for intermediate states in the $ϕϕ$ or $ηϕ$ combinations, but no significant structure is seen due to the limited statistics.