We present a global analysis of the observed $Z_c$, $Z_{cs}$, and future $Z_{css}$-like spectra using (inverse) Laplace sum rule within stability criteria. Integrated compact QCD expressions of the leading order spectral functions up to dimension-six condensates are given. Next-to-leading order factorized perturbative contributions are included. We reemphasize the importance to include pertubative radiative corrections (though numerically small) for heavy quark sum rules in order to justify the (ad hoc) definition and value of the heavy quark mass used frequently at leading order in the literature. We also demonstrate that, contrary to a qualitative large $N_c$ counting, the two-meson scattering contributions to the four-quark spectral functions are numerically negligible confirming the reliability of the Laplace sum rule predictions. Our results are summarized in Tables III to VI. The $Z_c(3900)$ and $Z_{cs}(3983)$ spectra are well reproduced by the ${\mathcal{T}}_c(3900)$ and ${\mathcal{T}}_{cs}(3973)$ tetramoles (superposition of quasidegenerated molecules and tetraquark states having the same quantum numbers and with almost equal couplings to the currents). The $Z_c(4025)$ or $Z_c(4040)$ state can be fitted with the $D_0^{*}{D}_1$ molecule having a mass 4023(130) MeV while the $Z_{cs}$ bump around 4.1 GeV can be likely due to the $D_{s0}^{*}{D}_1\oplus D_0^{*}{D}_{s1}$ molecules. The $Z_c(4430)$ could be a radial excitation of the $Z_c(3900)$ weakly coupled to the current, while all strongly coupled ones are in the region $(5634\sim 6527)\mathrm{MeV}$. The double strange tetramole state ${\mathcal{T}}_{css}$, which one may identify with the future $Z_{css}$, is predicted to be at 4064(46) MeV. It is remarkable to notice the regular mass splittings of the tetramoles due to $SU(3)$ breakings: $M_{{\mathcal{T}}_{cs}}-M_{{\mathcal{T}}_c}\approx M_{{\mathcal{T}}_{css}}-M_{{\mathcal{T}}_{cs}}\simeq (73\sim 91)\mathrm{MeV}$.

• Received 17 January 2021
• Accepted 16 March 2021

DOI:https://doi.org/10.1103/PhysRevD.103.074015

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Published by the American Physical Society