Hadronic decay of exotic mesons consisting of four charm quarks

In 2020, a di-$J/\psi$ resonance centered at around 6.9 $\mathrm{GeV}$ was discovered by the LHCb Collaboration, and was later confirmed by the ATLAS and CMS Collaborations. In addition to the 6.9-$\mathrm{GeV}$ resonance, its low energy counterpart centered at around 6552 $\mathrm{MeV}$ was also observed by the CMS Collaboration. In this paper, we calculate the inclusive decay width of such exotic mesons into light hadrons and into open charms, respectively, assuming them as either genuine tetraquarks or molecules composed of two $J/\psi$'s. Since the spins of these resonances have not yet been experimentally measured, regarding them as S-wave bound states, we consider all possible spin configurations, namely spin-0 and spin-2, that decay exclusively into double $J/\psi$, and find that the ratio ($\mathcal{R}$) of the decay width for the light-hadron final states to that for the charmed final states differs significantly among different spin configurations, which can help to explore the nature of the observed di-$J/\psi$ resonances and fix the mixing angle for the $J^{PC}=0^{++}$ tetraquark.