Mass spectra of $Z_c$ and $Z_b$ exotic states as hadron molecules

We construct charmoniumlike and bottomoniumlike molecular interpolating currents with quantum numbers $J^{PC}=1^{+-}$ in a systematic way, including both color singlet-singlet and color octet-octet structures. Using these interpolating currents, we calculate two-point correlation functions and perform QCD sum rule analyses to obtain mass spectra of the charmoniumlike and bottomoniumlike molecular states. Masses of the charmoniumlike $\overline{q}c\overline{c}q$ molecular states for these various currents are extracted in the range 3.85–4.22 GeV, which are in good agreement with observed masses of the $Z_c$ resonances. Our numerical results suggest a possible landscape of hadronic molecule interpretations of the newly observed $Z_c$ states. Mass spectra of the bottomoniumlike $\overline{q}b\overline{b}q$ molecular states are similarly obtained in the range 9.92–10.48 GeV, which support the interpretation of the $Z_b(10610)$ meson as a molecular state within theoretical uncertainties. Possible decay channels of these molecular states are also discussed.

Figure 1

OPE convergence for the current $J_{1\mu }^{(\mathbf{8})}$ with $s_0\to \infty$.

Figure 2

Variations of the charmoniumlike molecule hadron mass $m_X$ with $s_0$ and $M_B^2$ for the current $J_{1\mu }^{(\mathbf{8})}$.

Figure 3

Variations of the bottomoniumlike molecule hadron mass $m_X$ with $s_0$ and $M_B^2$ for the current $J_{1\mu }^{(\mathbf{8})}$.