Search for $\psi (3686)\to \gamma {\eta }_c(\eta (1405))\to \gamma {\pi }^{+}{\pi }^{-}{\pi }^0$

Using a sample of $448.1\times {10}^6$ $\psi (3686)$ events collected with the BESIII detector, a search for the isospin violating decay ${\eta }_c\to {\pi }^{+}{\pi }^{-}{\pi }^0$ via $\psi (3686)\to \gamma {\eta }_c$ is presented. No signal is observed, and the upper limit on $\mathcal{B}(\psi (3686)\to \gamma {\eta }_c\to \gamma {\pi }^{+}{\pi }^{-}{\pi }^0)$ is determined to be $1.6\times {10}^{-6}$ at the $90\mathrm{\%}$ confidence level. In addition, a search for $\eta (1405)\to f_0(980){\pi }^0$ in $\psi (3686)$ radiative decays is performed. No signal is observed, and the branching fraction $$\mathcal{B}$(\psi(3686)\to\gamma\eta(1405)\to \gamma f_{0}(980)\pi^{0}\to\gamma \pi^+\pi^-\pi^0 )$iscalculatedtobelessthan$ 5.0\times10^{-7}$atthe$90%$ confidence level.