Fully-heavy structures in the invariant mass spectrum of $J/\psi \psi(3686)$, $J/\psi \psi(3770)$, $\psi(3686) \psi(3686)$, and $J/\psi \Upsilon(1S)$ at hadron colliders

Motivated by a recent successful dynamical explanation for the newly observed fully-charm structure $X(6900)$ in the mass spectrum of di-$J/\psi$ by LHCb [J.~Z.~Wang \textit{et al.} arXiv:2008.07430], in this work, we extend the same dynamical rescattering mechanism to predict the line shape of more potential fully-heavy structures in the invariant mass spectrum of $J/\psi \psi(3686)$, $J/\psi \psi(3770)$, $\psi(3686) \psi(3686)$, and $J/\psi \Upsilon(1S)$ at high energy proton-proton collisions, whose verification in experiments should be helpful to further clarify the nature of $X(6900)$. The above final states of vector heavy quarkonia can be experimentally reconstructed more effectively by a $\mu^+\mu^-$ pair in the muon detector compared with $Q\bar{Q}$ meson with other quantum numbers. Furthermore, the corresponding peak mass positions of each of predicted fully-heavy structures are also given. Our theoretical studies here could provide some valuable information for the future measurement proposals of LHCb and CMS, especially based on the accumulated data after completing Run III of LHC in the near future.