Constraining the $DDD^*$ three-body bound state via the $Z_c(3900)$ pole

In this study, we propose using the $Z_c(3900)$ pole position to constrain the existence of the $DDD^*$ three-body bound state within the one-boson-exchange (OBE) model. The existence of the $DDD^*$ bound state remains uncertain due to significant variations in the OBE interaction, particularly in the strength of scalar-meson-exchange interactions, which can differ by a factor about 20 between two commonly used OBE models. This discrepancy renders the $DDD^*$ system highly model-dependent. To address this issue, we constrain the scalar-meson-exchange interaction using the $Z_c(3900)$ pole position, where the pseudoscalar-meson coupling is well-determined, and the $\rho$- and $\omega$-exchange interactions nearly cancel each other out, leaving the coupling constant of the $\sigma$-exchange as the only unknown parameter. Our results indicate that the isospin-$\frac{1}{2}$ $DDD^*$ bound states exist when $Z_c(3900)$ is a virtual state of $D\bar{D}^*/\bar{D}D^*$ located within approximately $-15$ MeV of the threshold. However, the three-body bound state is gone when the $Z_c(3900)$ virtual state pole is more than $20$ MeV away from the threshold. Each experimental progress, either on the $DDD^*$ state or the $Z_c(3900)$, can shed light on the nature of the other state. Another significant outcome is a refined set of OBE model parameters calibrated using the pole positions of $X(3872)$, $T_{cc}(3875)$, and $Z_c(3900)$, rigorously addressing the cutoff dependence. These parameters provide a valuable resource for more accurate calculations of systems involving few-body $D$, $D^*$ and their antiparticles. Additionally, we find no evidence of the $DDD^*$ three-body resonances after extensive search using a combination of the Gaussian expansion method and the complex scaling method.