Study of X(6900) with unitarized coupled channel scattering amplitudes

Abstract

In this paper, we study the resonant state X(6900). The scattering amplitudes of coupled channels, $J/\psi J/\psi$-$J/\psi \psi (2S)$-$J/\psi \psi (3770)$, are constructed with the interaction of four vector mesons described by effective Lagrangians. The amplitudes are calculated up to one loop, decomposed by partial wave projection, and unitarized by Pad$\acute{e}$ approximation. These amplitudes are fitted to the latest experimental data sets of di-$J/\psi$ and $J/\psi \psi (2S)$ invariant mass spectra of LHCb, CMS, and ATLAS. High-quality solutions are obtained. With these partial wave amplitudes, we extract the pole parameters of the X(6900). Its quantum number is likely to be $0^{++}$. According to the pole counting rule as well as analysis of the phase shifts of the partial waves, it supports our previous conclusion that the X(6900) prefers to be a compact tetra-quark.