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Using 106~million $\psi(3686)$ events collected with the BESIII detector, wemeasure multipole amplitudes for the decay$\psi(3686)\rightarrow\gamma\chi_{c1,2}\to\gamma\gamma J/\psi$ beyond thedominant electric-dipole amplitudes. The normalized magnetic-quadrupoleamplitude for $\psi(3686)\rightarrow\gamma\chi_{c1,2}\rightarrow\gamma\gammaJ/\psi$ and the normalized electric-octupole amplitudes for$\psi(3686)rightarrow\gamma\chi_{c2}$,~$\chi_{c2}\rightarrow\gamma J/\psi$ aredetermined. The M2 amplitudes for $\psi(3686)\rightarrow\gamma\chi_{c1}$ and$\chi_{c1,2}\rightarrow\gamma J/\psi$ are found to differ significantly fromzero and are consistent with theoretical predictions. We also obtain the ratiosof M2 contributions of $\psi(3686)$ and $J/\psi$ decays to $\chi_{c1,2}$,$b_{2}^{1}/b_{2}^{2} = 1.35\pm0.72$ and $a_{2}^{1}/a_{2}^{2} = 0.617\pm0.083$,which agree well with theoretical expectations. By considering the multipolecontributions of $\chi_{c1,2}$, we measure the product branching fractions forthe cascade decays $\psi(3686)\rightarrow\gamma\chi_{c0,1,2}\to\gamma\gammaJ/\psi$ and search for the process $\eta_{c}(2S)\to\gamma J/\psi$ through$\psi(3686)\rightarrow\gamma\eta_{c}(2S)$. The product branching fraction for$\psi(3686)\rightarrow\gamma\chi_{c0}\to\gamma\gamma J/\psi$ is 3$\sigma$larger than published measurements, while those of$\psi(3686)rightarrow\gamma\chi_{c1,2}\to\gamma\gamma J/\psi$ are consistent.No significant signal for the decay $\psi(3686)\to\gamma \eta_c(2S)\to\gamma\gamma J/\psi$ is observed, and the upper limit of the product branchingfraction at the 90\% confidence level is determined.

physical review. d/physical review. d.

Measurement of higher-order multipole amplitudes in  $\psi(3686)\rightarrow\gamma\chi_{c1,2}$ with $\chi_{c1,2}\to\gamma J/\psi$  and search for the transition $\eta_{c}(2S)\to\gamma J/\psi$