The $K\bar{K}^*$ interaction in the unitary coupled-channel approximation

The $K\bar{K}^*$ interaction Lagrangian is constructed when the $SU(3)$ hidden gauge symmetry is taken into account, and then the $K\bar{K}^*$ potential is obtained. In the low energy region, the $K\bar{K}^*$ potential mainly comes from the contribution of the $t-$channel interaction by exchanging $\rho$,$\omega$ and $\varphi$ mesons, respectively. The $K\bar{K}^*$ amplitude is investigated by solving the Bethe-Salpeter equation in the unitary coupled-channel approximation, where the loop function of the vector and pseudoscalar mesons are evaluated in the dimensional regularization scheme, and the contribution of the longitudinal part of the intermediate vector meson propagator is included in the calculation. Finally, it is found that a resonance state of $K\bar{K}^*$ is generated in the isospin $I=0$ sector, which might correspond to the $f_1(1420)$ particle in the review of the particle data group(PDG). Moreover, in the isospin $I=1$ sector, a pole of the $K\bar{K}^*$ amplitude is detected at $1425-i316$MeV on the complex plane of the total energy in the center of mass system, which is higher than the $K\bar{K}^*$ threshold. Thus this pole might be a resonance state of $K\bar{K}^*$ although no counterpart has been found in the PDG review.