The $Z_b$ states as the mixture of the molecular and diquark-anti-diquark components within the effective field theory

In this study, we reconsider the states $Z_b(10610)$ and $Z_b(10650)$ by investigating the presence of diquark-anti-diquark components as well as the hadronic molecule components in the framework of effective field theory. The different masses of pseudoscalar mesons such as $\pi^{0}$, $\eta_{8}$, and $\eta_{0}$, as well as vector mesons like $\rho^{0}$ and $\omega$ violate the OZI rule that is well depicted under the $[U(3)_L\otimes U(3)_R]_{global}\otimes [U(3)_V]_{local}$ symmetry. To account for the contribution of intermediate bosons of heavy masses within the OBE model, we introduce an exponential form factor instead of the commonly used monopole form factor in the past. By solving the coupled-channel Schr\"{o}dinger equation with the Gaussian expansion method, our numerical results indicate that the $Z_b(10610)$ and $Z_b(10650)$ states can be explained as hadronic molecules slightly mixing with diquark-anti-diquark states.