  In this work, we study the radiative decay of heavy quarkonium states by using the effective Lagrangian approach. Firstly, we construct the spin-breaking terms in the effective Lagrangian for the $nP\leftrightarrow mS$ transitions and determine the some of the coupling constants by fitting the experimental data. Our results show that in $\chi_{cJ}$, $\psi(2S)$, $\Upsilon(2S)$, and $\Upsilon(3S)$ radiative decays, the spin-breaking effect is so small that can be ignored. Secondly, we investigate the radiative decay widths of the $c\bar{c}(1D)$ states and find the if $\psi(3770)$ is a pure $^3D_1$ state its radiative decay into $\chi_{cJ}+\gamma$ roughly preserve the heavy-quark spin symmetry, while if it is a $S-D$ mixing state with mixing angel $12^{\circ}$ the heavy quark-spin symmetry in its radiative decay and in the radiative decay of $\psi(3686)$ will be largely violated. In the end, we show that combining the radiative decay and the light hadron decay of $P$-wave $\chi_{bJ}(1,2P)$ can provide another way to extract the information of the color-octet matrix element in the context of non-relativistic QCD (NRQCD) effective theory, and our result is consistent with potential NRQCD hypothesis. 
Investigating the heavy quarkonium radiative transitions with the   effective Lagrangian method