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 $(2712.4\pm14.3)\times10^6$ $\psi(3686)$ 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 $(3.77\pm0.43\pm0.29)\times10^{-7}$, 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\pm2.25\pm2.26)\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 $\mathcal{B}(\psi(3686)\to\gamma\eta_c)\times\mathcal{B}(\eta_c\to\pi^+\pi^-\pi^0)<3.09\times10^{-7}$ and $\mathcal{B}(\psi(3686)\to\gamma\eta_c)\times\mathcal{B}(\eta_c\to f_0(980)\pi^0)\times\mathcal{B}(f_0(980)\to\pi^+\pi^-)<7.97\times10^{-8}$ at 90\% confidence level, respectively.