Toponia at the HL-LHC and FCC-ee

The hint of a pseudoscalar toponium state at the Large Hadron Collider (LHC) opens a new avenue for studying a novel class of QCD (quasi-)bound states with comparable formation and decay times. Compared with charmonium and bottomonium, toponium is a quasi-bound state, resembling a hydrogen atom of the strong interaction, although it appears as a broader resonance. We compute the masses and annihilation decay widths of the lowest $S$-wave ($\eta_t$, $\psi_t$) and $P$-wave ($\chi_{t0}$, $\chi_{t1}$) toponium states, and assess their discovery prospects at the High-Luminosity LHC (HL-LHC) and future lepton colliders, such as the $e^+e^-$ stage of the Future Circular Collider (FCC-ee). Detecting the vector $\psi_t$ state at the HL-LHC is hindered by the Landau-Yang theorem and the gluon-dominated production environment of the collider, whereas lepton colliders offer promising sensitivity through both constituent and two-body decays. A more precise measurement of the $\eta_t$ mass, approximately equal to that of $\psi_t$, at the LHC could help determine the optimal $t\bar{t}$ threshold center-of-mass energy for FCC-ee. The $P$-wave states remain challenging to observe at both the HL-LHC and future lepton colliders. We also discuss how toponium measurements can be used to probe top-quark properties and to conduct indirect searches for new physics, including light scalars that couple to the top quark.