  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.
