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

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 $(D,D^{\ast })$ meson and a $P$-wave $(D_0^{\ast }(2300),D_1(2430),D_1(2420),D_2^{\ast }(2460))$ 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 $D^{\ast }{\bar{D}}_1(2420)$ and $D^{\ast }{\bar{D}}_2^{\ast }(2460)$ systems as candidates for the $Z_c(4430)$, while the significantly broader resonances in the $D{\bar{D}}_0^{\ast }(2300)$ and $D{\bar{D}}_1(2430)$ sectors are suggested as candidates for the $Z_c(4200)$. Focusing on the $D^{\ast }{\bar{D}}_2^{\ast }(2460)$ 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.

Comments: 23 pages, 4 figures, 7 tables