Investigating the heavy quarkonium radiative transitions with the effective Lagrangian method

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.