Measurements of the branching fractions of 𝜒𝑐⁢𝐽→𝜙⁢𝜙⁢𝜂, 𝜙⁢𝜙⁢𝜂′ and 𝜙⁢𝐾+⁢𝐾−⁢𝜂

Using a sample of $(2712.4 \pm 14.3)\times10^{6}$ $\psi(3686)$ events collected by the BESIII detector at the BEPCII collider, we measure the branching fractions of the decays $\chi_{cJ}\to\phi\phi\eta$, $\chi_{cJ}\to\phi\phi\eta'$, and $\chi_{cJ}\to\phi K^+K^-\eta$ $(J=0,1,2)$. The obtained branching fractions are $$\mathcal{B}(\chi_{c0}\to\phi\phi\eta)=(7.40 \pm 0.23\,(\mathrm{stat}) \pm 0.55\,(\mathrm{syst}))\times10^{-4},$$ $$\mathcal{B}(\chi_{c1}\to\phi\phi\eta)=(3.33 \pm 0.14\,(\mathrm{stat}) \pm 0.25\,(\mathrm{syst}))\times10^{-4},$$ $$\mathcal{B}(\chi_{c2}\to\phi\phi\eta)=(5.46 \pm 0.17\,(\mathrm{stat}) \pm 0.40\,(\mathrm{syst}))\times10^{-4},$$ $$\mathcal{B}(\chi_{c0}\to\phi\phi\eta')=(2.96 \pm 0.23\,(\mathrm{stat}) \pm 0.29\,(\mathrm{syst}))\times10^{-4},$$ $$\mathcal{B}(\chi_{c1}\to\phi\phi\eta')=(0.69 \pm 0.10\,(\mathrm{stat}) \pm 0.08\,(\mathrm{syst}))\times10^{-4},$$ $$\mathcal{B}(\chi_{c2}\to\phi\phi\eta')=(0.65 \pm 0.09\,(\mathrm{stat}) \pm 0.07\,(\mathrm{syst}))\times10^{-4},$$ $$\mathcal{B}(\chi_{c0}\to\phi K^+K^-\eta)=(1.23 \pm 0.08\,(\mathrm{stat}) \pm 0.10\,(\mathrm{syst}))\times10^{-4},$$ $$\mathcal{B}(\chi_{c1}\to\phi K^+K^-\eta)=(1.00 \pm 0.07\,(\mathrm{stat}) \pm 0.07\,(\mathrm{syst}))\times10^{-4},$$ and $$\mathcal{B}(\chi_{c2}\to\phi K^+K^-\eta)=(1.82 \pm 0.09\,(\mathrm{stat}) \pm 0.14\,(\mathrm{syst}))\times10^{-4},$$ where the $K^+K^-$ pair does not originate from the decay of a $\phi$ meson. The first uncertainties are statistical and the second are systematic. The branching fractions of $\chi_{cJ}\to\phi\phi\eta$ are measured with a precision improved by factors of $1.5$--$1.9$, while those of $\chi_{cJ}\to\phi\phi\eta'$ and $\chi_{cJ}\to\phi K^+K^-\eta$ are measured for the first time.