$D \bar{D}^*$ and $\pi \psi$ interactions in a unitary coupled-channel approximation

The $D \bar{D}^*$ interaction via a $\pi \psi$ intermediate state is studied carefully in the isospin $I=1$ sector. By solving the Bethe-Salpeter equation in the unitary coupled-channel approximation, we obtain the S-wave amplitude as a function of the total energy of the system in the center of mass frame. A resonance state is generated dynamically in the 3900MeV region, which might correspond to the $Zc(3900)$ particle. Moreover, the loop function of a vector meson and a pseudoscalar meson is deduced explicitly in the dimensional regularization scheme and the contribution of the longitudinal part of the vector meson propagator is taken into account. The initial and final polarization vectors in the vertex of the vector meson and the pseudoscalar meson are eliminated when the Bethe-Salpeter equation is solved, and it is certified that the amplitude is still unitary in the calculation.