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Nature of the $X(6900)$ in partial wave decomposition of $J/ψJ/ψ$ scattering

In this letter, we perform partial wave decomposition on coupled-channel scattering amplitudes, $J/ψJ/ψ$-$J/ψψ(2S)$-$J/ψψ(3770)$, 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 $J/ψJ/ψ$ invariant mass spectra measured by LHCb and determine the unknown couplings. The pole parameters of the $X(6900)$ are extracted as $M=6861.0^{+6.3}_{-8.8}$~MeV and $Γ=129.0^{+5.6}_{-3.4}$~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.