${\eta }_c{\eta }_c$ and $J\mathrm{/}\psi J\mathrm{/}\psi$ scatterings from lattice QCD

We investigate the $J^{PC}=(0,2{)}^{++}$ ${\eta }_c{\eta }_c$ and $J\mathrm{/}\psi J\mathrm{/}\psi$ scatterings in $N_f=2$ lattice QCD at two pion masses $m_{\pi }\approx 420$,MeV and 250,MeV. The quark field smearing scheme used in the distillation method strongly suppresses high-momentum states, thereby limiting the maximum accessible center-of-mass energy in this study to approximately 6.6,GeV. Given the observed near-decoupling of the ${\eta }_c{\eta }_c$ and $J\mathrm{/}\psi J\mathrm{/}\psi$ channels, we analyze their scattering properties using the single-channel Lüscher’s method. In $0^{++}$ channels, ${\eta }_c{\eta }_c$ and $J\mathrm{/}\psi J\mathrm{/}\psi$ have repulsive and attractive interactions, respectively, which are dominated by the quark exchange effects with the Fierz rearrangement determining the interaction characteristic. A likely $0^{++}$ virtual state is observed below the threshold by 20-40,MeV in the ${}^1{S}_0$ $J\mathrm{/}\psi J\mathrm{/}\psi$ scattering and may have connection with $X(6200)$. In $2^{++}$ channels, the near-threshold interaction for the ${}^5{S}_2$ $J\mathrm{/}\psi J\mathrm{/}\psi$ scattering is repulsive. The di-charmonium interactions are found to be insensitive to $m_{\pi }$ and can be consistently understood in terms of quark rearrangement effects combined with a one-gluon-exchange mechanism, both of which are supported by our lattice QCD results. Most importantly, we observe a $2^{++}$ resonance in the ${}^5{S}_2$ $J\mathrm{/}\psi J\mathrm{/}\psi$ scattering, whose properties are $(m_R,{\mathrm{\Gamma }}_R)=(6544(10),552(34))$,MeV for $m_{\pi }\approx 420$,MeV and $(6539(13),546(57))$,MeV for $m_{\pi }\approx 250$,MeV, which are compatible with the parameters of the fully-charmed broad structure $X(6600)$ (or $X(6400)$) reported by ATLAS and CMS.