Decoding $Z_c(4430)$ and $Z_c(4200)$: The role of $P$-wave charmed mesons

TL;DR

This work develops a comprehensive OBE-based framework to study hidden-charm tetraquarks with $I^G(J^{PC})=1^+(1^{+-})$, focusing on molecular states formed by an $S$-wave $(D,D^*)$ pair and a $P$-wave $(D_0^*, D_1, D_1', D_2^*)$ partner. By solving the momentum-space Schrödinger equation with the Complex Scaling Method and incorporating three-body decay effects via cross-channel OBE contributions and unstable-meson self-energies, the authors show that these three-body dynamics critically shape pole positions and widths, yielding broad near-threshold resonances that can be identified with the observed $Z_c(4430)$ and $Z_c(4200)$. A detailed line-shape analysis for the $D^*ar{D}_2^*$ assignment uses a Flatté-like parametrization with energy-dependent self-energy to reproduce the experimental peak and predict open-charm decay patterns. Across 10 sectors (7 single-channel and 3 spin-mixed coupled channels), HQSS and chiral symmetry constrain the interactions, with $ ho$ and $ ext{ω}$ exchanges largely canceling and long-range dynamics dominated by $ ext{π}$ and $ ext{σ}$ exchange. The results provide a coherent, testable picture for broad near-threshold $Z_c$-type states and offer concrete predictions for future experimental investigations in three-body open-charm final states.

Abstract

In this work, we perform a systematic investigation of the hidden-charm tetraquark states with $I^G(J^{PC})=1^+(1^{+-})$ within the hadronic molecular picture, placing particular emphasis on systems composed of an $S$-wave $(D, D^*)$ meson and a $P$-wave $(D_0^*(2300), D_1(2430), D_1(2420), D_2^*(2460))$ meson. Adopting the One-Boson Exchange potential, we solve the Schrödinger equation in momentum space via the Complex Scaling Method. A crucial feature of our approach is the rigorous treatment of the unstable nature of the $P$-wave constituents by incorporating three-body decay effects arising from self-energy corrections and the static limit approximation. Our results demonstrate that these three-body dynamics play a crucial role in determining the pole positions, specifically in reproducing the large decay widths observed experimentally. We identify several broad resonances in the $D^*\bar{D}_1(2420)$ and $D^*\bar{D}_2^*(2460)$ systems as candidates for the $Z_c(4430)$, while the significantly broader resonances in the $D\bar{D}_0^*(2300)$ and $D\bar{D}_1(2430)$ sectors are suggested as candidates for the $Z_c(4200)$. Focusing on the $D^*\bar{D}_2^*(2460)$ assignment as a specific case study, we further analyze the line shape of the $Z_c(4430)$ candidate using a Flatté-like parametrization with energy-dependent self-energy terms, providing predictions for its open-charm decay modes to guide future experimental searches.

Figures (4)

• Figure 1: The eigenvalue distribution of the complex scaled Schrödinger equation for the two-body systems.
• Figure 2: The OBE potentials for the diagonal channels: $\{D\bar{D}^*\}(^3S_1)$, $D^*\bar{D}^*(^3S_1)$, $\{D\bar{D}_0^{*}\}(^1P_1)$, $\{D\bar{D}_1^\prime\}(^3P_1)$, $\{D\bar{D}_1\}(^3P_1)$, $[D\bar{D}_2^*](^5P_1)$, and $[D^*\bar{D}_0^*](^3P_1)$, with quantum numbers $I^G(J^{PC})=1^+(1^{+-})$. The results are obtained using the instantaneous approximation ($q_0=0$) and a non-local regulator with cutoff $\Lambda=0.6$ GeV.
• Figure 3: The OBE potentials for the diagonal sectors of $D^*\bar{D}_1^\prime$, $D^*\bar{D}_1$, and $D^*\bar{D}_2^*$ with quantum numbers $I^G(J^{PC})=1^+(1^{+-})$. The potentials for the non-vanishing spin-mixing transitions among the $(^{1,3,5}P_1)$ partial waves are also displayed. The parameters are the same as in Fig. \ref{['fig: potential']}.
• Figure 4: Production line shapes of resonance A (corresponding to the $Z_c(4430)$ candidate) decaying into (a) two-body final states $D^{*+}\bar{D}_2^{*0}$ and $\psi(2S) \pi^+$, and (b) three-body final states $D^{*+}D^-\pi^+$, $D^{*+}D^{*-}\pi^+$, and $D^{*+}\bar{D}^{*0}\pi^0$. The coupling constant $g_{\psi^\prime}$ is determined by scaling the $\psi(2S) \pi^+$ distribution to share the same peak height as the $D^{*+}\bar{D}_2^{*0}$ mode, serving solely for shape comparison.