Nature of the $X(6900)$ in partial wave decomposition of $J\mathrm{/}\psi J\mathrm{/}\psi$ scattering

In this letter, we perform partial wave decomposition on coupled-channel scattering amplitudes, $J\mathrm{/}\psi J\mathrm{/}\psi$-$J\mathrm{/}\psi \psi (2S)$-$J\mathrm{/}\psi \psi (3770)$, to study the resonance appears in these processes. Effective Lagrangians are used to describe the interactions of four charmed vector mesons, and the scattering amplitudes are calculated up to the next-to-leading order. Partial wave projections are performed, and unitarization is implemented by Padé approximation. Then we fit the amplitudes to the $J\mathrm{/}\psi J\mathrm{/}\psi$ invariant mass spectra measured by LHCb and determine the unknown couplings. The pole parameters of the $X(6900)$ are extracted as $M=6861.0_{-8.8}^{+6.3}$~MeV and $\mathrm{\Gamma }=129.0_{-3.4}^{+5.6}$~MeV. Our analysis implies that its quantum number prefers to be $0^{++}$. The pole counting rule and phase shifts show that it is a normal Breit-Wigner resonance and, hence, should be a compact tetraquark.