Why is the $Z_c(3900)$ absent in the $h_cπ$ final state?

TL;DR

This work develops a coupled-channel, heavy-quark spin symmetry (HQSS) based framework to study exotic hadrons $Z_c(3900)$, $Z_c(4020)$, $Z_b(10610)$, and $Z_b(10650)$. By constructing S-wave contact interactions among elastic and inelastic channels and solving the Lippmann–Schwinger equation, the authors fit invariant-mass spectra and extract pole structures, quantifying HQSS violation in charm and bottom sectors. The results show negligible HQSS breaking in the bottom sector, yielding closely related poles and couplings for the two $Z_b$ states, while significant HQSS breaking in the charm sector concentrates in elastic interactions, naturally explaining why $Z_c(3900)$ appears mainly in $J/\psi\pi$ and $Z_c(4020)$ in $h_c\pi$, with the latter potentially arising from a threshold cusp. The model remains robust under regulator and cutoff variations, supporting a molecular interpretation for the $Z_b$ states and a more intricate, symmetry-broken picture for the charm-sector $Z_c$ states.

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 ($J/ψπ, h_cπ$ or $Υπ$, $h_bπ$) 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 $J/ψπ$, while $Z_c(4020)$ appears only in $h_cπ$. 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.

Figures (11)

• Figure 1: Graphical representation for the decay amplitudes of both elastic channel (a) and inelastic channel (b). The red filled squares and circles indicate the physical and bare decay amplitudes, respectively. The blue filled circle in panel (a) denotes the transitions between the two elastic channels, whereas that in panel (b) corresponds to the transitions between the elastic and inelastic channels.
• Figure 2: Fitted line shapes of both HQSV (red solid curves) and HQS (blue dashed curves) schemes for the charm system in comparison with the experimental data BESIII:2013mhiBESIII:2015pqwBESIII:2017bua from BESIII collaboration. Subfigures (a)–(c) correspond to the invariant-mass distributions at $\sqrt{s}=4.23$ GeV, while subfigures (d)-(h) show those at $\sqrt{s}=4.26$ GeV. The vertical two gray dashed lines indicate the $D\bar{D}^*$ and $D^*\bar{D}^*$ thresholds from left to right, in order. The lower panel in each subfigure presents the standardized residuals, where the red and blue points correspond to the HQSV and HQS cases, respectively.
• Figure 3: Fitted line shapes of both HQSV (red solid curves) and HQS (blue dashed curves) schemes for the bottom system in comparison with the experimental data Belle:2011aaBelle:2014vznBelle:2015upu from Belle collaboration. The two vertical gray dashed lines indicate the $B\bar{B}^*$ and $B^*\bar{B}^*$ thresholds from left to right, in order. The lower panels are depicted in the same manner as those in Fig. \ref{['line_shape_Zc']}.
• Figure 4: Pole positions in the $Z_c$ and $Z_b$ systems under the HQSV scheme for different cutoff values. The real parts of the poles in panels (a), (b), and (c) are shown relative to the thresholds of $D\bar{D}^*$, $B\bar{B}^*$ and $B^*\bar{B}^*$, respectively. (a) For the $Z_c$ system, the cutoff varies from 0.8 to 1.8 GeV in increments of $0.1~\mathrm{GeV}$, corresponding to eleven distinct values labeled from 1 to 11. (b)-(c) For the $Z_b$ system, the cutoff ranges are from 1.3 to 2.3 GeV with a step of 0.1 GeV, labeled by indices from 1 to 11.
• Figure 5: Pole positions in the $Z_b$ system under the HQS scheme for different cutoff values. The real parts of the poles in panels (a) and (c) are shown relative to the $B\bar{B}^*$ and $B^*\bar{B}^*$ thresholds, respectively. Figures (a) and (b) denote the real and imaginary parts of the $Z_b(10610)$ pole, while (c) and (d) depict those of the $Z_b(10650)$ pole. The cutoff varies from 1.2 to 3.0 GeV, with a step of 0.1 GeV (0.2 GeV) in the region $1.2-2.2$ GeV ($2.2-3.0$ GeV).