Existence of the $DD^*\bar{K}^*$ and $BB^*K^*$ three-body molecular states

TL;DR

The paper addresses the existence of a $DD^*\bar{K}^*$ three-body molecular state within an OBE framework by recalibrating scalar and vector couplings: the $Z_c(3900)$ pole position is treated as the sole free parameter for the $\sigma$ exchange, while $X(3872)$ and $T_{cc}(3875)$ fix the $\rho$ and $\omega$ exchanges, and heavy-quark symmetry extends the same parameters to the $BB^*K^*$ system. The three-body problem is solved with the Gaussian Expansion Method and complemented by the Complex Scaling Method to search for resonances; results show a charm-sector bound state in $I(J^P)=\frac{1}{2}(0^-)$ when the $Z_c(3900)$ virtual pole is within about $-10$ MeV of the $D\bar{D}^*$ threshold, while the bottom sector allows bound states for a pole within roughly $-25$ to $-35$ MeV. No three-body resonances are found in any considered channel. The work connects three-body hadronic molecular states to the 위치 of the $Z_c(3900)$ pole and suggests experimental measurements of the pole or searches for the three-body bound state in relevant decay channels to shed light on the underlying dynamics.

Abstract

We investigate the existence of the three-body molecular state composed of $DD^*\bar{K}^*$ within the one-boson-exchange (OBE) model. A major challenge is that while the pseudoscalar-meson couplings are well-determined, the couplings for scalar- and vector-meson exchanges render significant model dependence. To ensure the reliability of our predictions and reduce model dependence, we recalibrate the coupling constants of the OBE model. We treat the pole position of $Z_c(3900)$, or equivalently the scalar $σ$-exchange coupling constant, as the only unknown parameter. The coupling constants for the vector $ρ$- and $ω$-exchanges are determined by the pole positions of the well established states $X(3872)$ and $T_{cc}(3875)$. We demonstrate that these parameter sets also successfully describe the $T_{cs0}(2870)$ without further tuning. For the three-body system, our results indicate that an $I\left(J^P\right)=1 / 2\left(0^{-}\right)$ three-body molecular bound state exists when $Z_c(3900)$ is a virtual state located within approximately $-10~\text{MeV}$ of the $D\bar{D}^*$ threshold. Furthermore, we extend our analysis to the complex energy plane using the complex scaling method to search for molecular resonances, though no evidence of resonances is found in considered channels. We also apply this formalism to the bottom analog $BB^*K^*$ system. In this sector, the conditions for the existence of a three-body bound state are more relaxed, as a $Z_c(3900)$ virtual state located within $-25~\text{MeV}$ below the threshold suffices, although three-body molecular resonances remain absent. We suggest that future experiments precisely measure the pole position of $Z_c(3900)$ or search for the three-body bound state in $DD\bar{K}ππ$ and $DD\bar{K}$ channels, as these efforts would mutually illuminate the nature of the associated states.

Figures (11)

• Figure 1: Feynman diagrams to derive OBE potentials: (a1) and (a2) for $DD^*$, (b) for $D^*\bar{K}^*$, and (c) for $D\bar{K}^*$.
• Figure 2: Effective potentials of the scalar and vector meson exchange in the $Z_c(3900)~[D\bar{D}^*]^{C=-1}_{I=1}$ system. The blue dashed, yellow dotted, and green dot-dashed lines represent the contributions from the $\rho$, $\omega$, and $\sigma$ exchanges, respectively. The red solid line denotes the total of them. The cutoff $\Lambda$ is set to $1.10~\text{GeV}$, and the parameters $(R_\beta, R_\lambda, R_s)$ are set to $(1.0, 1.0, 2.0)$ as an example.
• Figure 3: The ratio factors for the scalar- and vector-meson-exchange interactions vary with the $Z_c(3900)$ virtual state pole position across different cutoffs. The blue circles, yellow squares, and green diamonds represent $R_\beta$, $R_\lambda$, and $R_s$, respectively. These ratios are defined relative to the baseline values as detailed in Eq. \ref{['eq:coupling-ratio']} and Table \ref{['tab:lag_para']}.
• Figure 4: The three sets of Jacobi coordinates for the $DD^*\bar{K}^*$ three-body system.
• Figure 5: A typical solution of the complex-scaled Schrödinger equation. The scattering states align along $\operatorname{Arg}(E)=-2 \theta$, whereas bound states and resonances do not shift as $\theta$ changes.The resonance with energy $E_R=m_R-i \frac{\Gamma_R}{2}$ can be obtained when $2 \theta>\left|\operatorname{Arg}\left(E_R\right)\right|$.