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

## TL;DR

This paper investigates the $D ar{D}^*$ and $\pi \psi$ interactions using a coupled-channel approach, dynamically generating a resonance near 3900 MeV that could be the $Z_c(3900)$, with detailed regularization and unitarity considerations.

## Contribution

The study applies a unitary coupled-channel Bethe-Salpeter approach to meson interactions, explicitly deriving loop functions and ensuring unitarity, leading to the dynamical generation of a resonance near 3900 MeV.

## Key findings

- Resonance near 3900 MeV identified as potential $Z_c(3900)$
- Explicit loop function derivation in dimensional regularization
- Unitarity maintained in the coupled-channel calculation

## Abstract

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.