Truncation of the Radial Ladder in Heavy Quarkonia

Abstract

A fundamental open question in hadron spectroscopy is whether the radial excitation ladder of quarkonia truncates at a finite level---a possibility that would challenge the conventional quark-antiquark bound-state picture and offer decisive clues to the nonperturbative strong interaction. Taking advantage of the newly observed high-mass hadronic states, we address this issue by solving a screened Godfrey--Isgur Hamiltonian for charmonium and bottomonium using the Gaussian expansion method. The calculated spectra saturate at $4.74\,\mathrm{GeV}$ ($c\bar{c}$) and $11.67\,\mathrm{GeV}$ ($b\bar{b}$), while root-mean-square radii grow to $\sim10\,\mathrm{fm}$---an order of magnitude above the confinement scale---where adjacent mass gaps drop below $10\,\mathrm{MeV}$. Combining a threshold-based mass-gap criterion, defined by the onset of mass--radius decoupling, with additional diagnostics, we locate the operational upper radial limits at $n\approx8$--$10$ for charmonium and $n\approx12$--$13$ for bottomonium. Beyond these limits, the conventional $q\bar{q}$ description ceases to apply. This work provides the first quantitative determination of these upper limits, and the proposed criterion is directly testable with forthcoming high-statistics data from BESIII, Belle II, and LHCb.