Pionic transitions from $Z_c(4020)$ to $D$ wave charmonia

In the present work, we investigate the charmed meson loops contributions to the pionic transitions from $Z_c(4020)^+$ to the $D$ wave triplets charmonia by using an effective Lagrangian approach. Our estimations indicate that the predicted branching fraction of $Z_c(4020)^+ \to \pi^+ \psi(1^3D_J) , \ J=(1,2,3)$ are much smaller than the one of $Z_c(4020)^+ \to \pi^+ h_c$. Thus, searching $Z_c(4020)^\pm$ in the $\pi^\pm \psi(1^3D_J)$ invariant mass distributions is impossible. Thus, the observed peak structures at 4.04 and 4.13 GeV in the $\pi^\pm \psi(3770)$ invariant mass distributions should not come from the contributions of $Z_c(4020)^\pm$, and further precise experimental measurements of the $e^+ e^- \to \pi^+ \pi^- \psi(3770)$ process are needed to decode the nature of these two peak structures.