Observation of $\psi (3686)\to \gamma \eta (1405)$ via $\eta (1405)\to f_0(980){\pi }^0$

The decay $\psi (3686)\to \gamma {\pi }^{+}{\pi }^{-}{\pi }^0$ is studied using a sample of $(2712.4\pm 14.3)\times {10}^6$ $\psi (3686)$ events collected with the BESIII detector. The decay $\eta (1405)\to {\pi }^{+}{\pi }^{-}{\pi }^0$ is observed for the first time in $\psi (3686)$ decays via the intermediate state $f_0(980)$ and the product branching fraction $\mathcal{B}(\psi (3686)\to \gamma \eta (1405))\times \mathcal{B}(\eta (1405)\to f_0(980){\pi }^0)\times \mathcal{B}(f_0(980)\to {\pi }^{+}{\pi }^{-})$ is determined to be $(3.77\pm 0.43\pm 0.29)\times {10}^{-7}$, where the first uncertainty is statistical and the second is systematic. The isospin-violating decay of $\psi (3686)\to \gamma f_1(1285)\to \gamma f_0(980){\pi }^0\to \gamma {\pi }^{+}{\pi }^{-}{\pi }^0$ has been observed with signal significance of $2.9\sigma$. And the branching fraction $\mathcal{B}(\psi (3686)\to \gamma f_1(1285)\to \gamma f_0(980){\pi }^0\to \gamma {\pi }^{+}{\pi }^{-}{\pi }^0)$ is determined to be $(7.36\pm 2.25\pm 2.26)\times {10}^{-8}$. Since no ${\eta }_c$ signal is evident in either the ${\pi }^{+}{\pi }^{-}{\pi }^0$ or $f_0(980){\pi }^0$ mass spectrum, upper limits are set to be $\mathcal{B}(\psi (3686)\to \gamma {\eta }_c)\times \mathcal{B}({\eta }_c\to {\pi }^{+}{\pi }^{-}{\pi }^0)<3.09\times {10}^{-7}$ and $\mathcal{B}(\psi (3686)\to \gamma {\eta }_c)\times \mathcal{B}({\eta }_c\to f_0(980){\pi }^0)\times \mathcal{B}(f_0(980)\to {\pi }^{+}{\pi }^{-})<7.97\times {10}^{-8}$ at 90% confidence level, respectively.