Radiative decays of the neutral $Z_c(3900)$ and $Z_c(4020)$

We study the radiative decays $Z_c(3900)/Z_c(4020) \to \gamma \chi_{cJ}(\gamma\chi_{cJ}^\prime)$ ($J=0, 1, 2$), with the assumption that the $Z_c(3900)$ and $Z_c(4020)$ couple strongly to $D\bar D^* +c.c$ and $D^*{\bar D}^*$ channel, respectively. By considering the contributions of intermediate charmed mesons triangle loops within an effective Lagrangian approach, it is shown that the calculated partial widths of $Z_c(3900) \to \gamma \chi_{cJ}$ are about a few hundreds keVs, while the obtained partial widths $Z_c(4020) \to \gamma \chi_{cJ}$ are about tens of keVs. The predicted partial widths of $Z_c(3900)\to\gamma\chi_{c0,1}^\prime$ are less than 1 keV, which mainly due to the very small phase space. For $Z_c(4020)\to\gamma\chi_{c0,2}^\prime$, the calculated partial widths are usually smaller than 1 keV. For the $Z_c(4020)\to\gamma\chi_{c1}^\prime$ process, the obtained partial widths can reach up to the order of 10 keV. Furthermore, the dependence of these ratios between different decay modes on the masses of $Z_c(3900)$ or $Z_c(4020)$ are also investigated, which may be a good quantity for the experiments. It is hoped that these calculations here could be tested by future experiments.