Why is the $Z_c(3900)$ absent in the $h_c\pi$ final state?

Abstract

In this work, we perform a comprehensive phenomenological analysis of the exotic hadronic states $Z_c(3900)$, $Z_c(4020)$, $Z_b(10610)$ and $Z_b(10650)$ within the framework of Heavy Quark Spin Symmetry (HQSS) and its violation. By constructing S-wave contact interactions between elastic ($D\bar{D}^*/D^*\bar{D}^*$ or $B\bar{B}^*/B^*\bar{B}^*$) and inelastic ($J/\psi\pi, h_c\pi$ or $\Upsilon\pi$, $h_b\pi$) channels, we solve the Lippmann-Schwinger equation to obtain physical production amplitudes and perform a global fit to experimental invariant-mass spectra. Our results demonstrate a striking difference between the charm and bottom sectors: HQSS violation is negligible in the bottom system, leading to comparable peak structures for both $Z_b$ states in all hidden-bottom decay channels. In contrast, significant HQSS breaking is required to describe the $Z_c$ system, where the violation is predominantly concentrated in the elastic interactions. This explains the observed selectivity: $Z_c(3900)$ appears prominently only in $J/\psi\pi$, while $Z_c(4020)$ appears only in $h_c\pi$. Pole analysis confirms the molecular nature of the states, with the $Z_c(4020)$ likely arising from a threshold cusp effect. The model's robustness is verified against variations of the form factor and cutoff, showing stable results.