The $X(4500)$ state considered as the mixture of hadronic molecule and diquark-antidiquark within effective field theory

In the present work, we construct the Lagrangians including three-meson, meson-diquark-antidiquark vertices, such that the diquark-antidiquark component as well as the molecular component are introduced within the effective field theory. With the obtained effective potentials projecting to spin 0, 1 and 2, we solve the Bethe-Salpeter equation with the on-shell approximation, and find that $X(4500)$ can be explained as the mixture of components $D_{s}^{*+}D_{s}^{*-}$, ${A}_{cq}\bar{A}_{cq}$ and ${A}_{cs}\bar{A}_{cs}$ with $I^G(J^{PC})=0^+(0^{++})$. In addition, another two resonances with quantum numbers $I^G(J^{PC})=0^+(1^{++})$ and $I^G(J^{PC})=0^+(2^{++})$ are predicted.