A toy model for hadronic bonds in exotic hadrons

Three hadronic-bond mechanisms were proposed in our previous studies to describe near-threshold exotic hadrons in terms of hadronic molecular configurations: the covalent bond arises from the sharing and exchange of valence light quarks between hadronic clusters, the creation bond originates from the activation of coherent sea $q\bar{q}$ degrees of freedom followed by the formation of effective covalent correlations, and the annihilation bond arises from valence $q\bar{q}$ annihilation and the associated mixing with a nearby compact state. A simple toy model was previously constructed only for the covalent bond, and in the present study we extend this model to incorporate the creation and annihilation bonds within a unified framework. The resulting framework organizes covalent-bond configurations as tending to lie below threshold and creation-bond configurations as tending to appear above threshold, while the annihilation bond can shift a pre-existing molecular configuration either downward or upward through mixing with a nearby compact state. An effective activation energy $E_{\mathrm{a}\mathrm{c}\mathrm{t}}$ is introduced for the coherent sea four-quark component involved in the creation bond, with the $Z_c(3900)$ providing the benchmark value $E_{\mathrm{a}\mathrm{c}\mathrm{t}}\sim 17\mathrm{M}\mathrm{e}\mathrm{V}$. Intriguingly, this value is numerically close to the reported mass of the $X_{17}$ particle, which we previously interpreted as a low-energy bare four-quark structure. This numerical proximity may therefore reflect common low-energy (four-quark) dynamics, although the microscopic connection remains to be established. For the isoscalar $D^{\ast }{\bar{D}}^{\ast }{B}^{\ast }{\bar{B}}^{\ast }$ configuration involving the $Z_c(4020)$ and $Z_b(10650)$ creation-bond subsystems linked by two additional covalent bonds, we estimate a binding energy of approximately $36\mathrm{M}\mathrm{e}\mathrm{V}$ below its four-meson threshold and propose searches in the $4\mu$ and $4\mu 2\pi$ final states.