Study on the structure of the $Z_c(3900)$ state

In this work, we studied the $Z_c(3900)$ state within the framework of effective field theory. We firstly show the construction of the Lagrangian describing meson-meson-meson and meson-diquark-diquark interactions. By using the Feynman rule, we calculate the effective potentials corresponding to the coupled channels of $D{\bar{D}}^{\ast }\mathrm{/}{D}^{\ast }\bar{D}$ and $S_{cq}{\bar{A}}_{cq}\mathrm{/}{A}_{cq}{\bar{S}}_{cq}$ with $S_{cq}$ ($A_{cq}$) the scalar (axial vector) diquark composed of $c$ and $q$ quarks. After solving the Bethe-Salpeter equation of the on-shell parametrized form and compare our numerical results with the experimental mass and width of $Z_c(3900)$, we find that the $Z_c(3900)$ state can be explained as the mixture of $D{\bar{D}}^{\ast }\mathrm{/}{D}^{\ast }\bar{D}$ and $S_{cq}{\bar{A}}_{cq}\mathrm{/}{A}_{cq}{\bar{S}}_{cq}$ components.