Hadronic molecular states from the $K\bar{K}^*$ interaction

In this work, the $K\bar{K}^*$ interaction is studied in a quasipotential Bethe-Salpeter equation approach combined with the one-boson-exchange model. With the help of the hidden-gauge Lagrangian, the exchanges of pseudoscalar mesons ($\pi$ and $\eta$) and vector mesons ($\rho$, $\omega$ and $\phi$) are considered to describe the $K\bar{K}^*$ interaction. Besides the direct vector-meson exchange which can be related to the Weinberg-Tomozawa term, pseudoscalar-meson exchanges also play important roles in the mechanism of the $K\bar{K}^*$ interaction. The poles of scattering amplitude are searched to find the molecular states produced from the $K\bar{K}^*$ interaction. In the case of quantum number $I^G(J^{PC})=0^+(1^{++})$, a pole is found with a reasonable cutoff, which can be related to the $f_1(1285)$ in experiment. Another bound state with $0^-(1^{+-})$ is also produced from the $K\bar{K}^*$ interaction, which can be related to the $h_1(1380)$. In the isovector sector, the interaction is much weaker and a bound state with $1^+(1^{+})$ relevant to the $b_1(1235)$ is produced but at a larger cutoff. Our results suggest that in the hadronic molecular state picture the $f_1(1285)$ and $b_1(1235)$ are the strange partners of the $X(3872)$ and $Z_c(3900)$, respectively.