Termination of the Radial Ladder in Mixed-Flavor Heavy Mesons

Abstract

A fundamental open question in hadron spectroscopy is whether the radial excitation ladder of heavy mesons terminates at a finite level once confinement is screened by light-quark pair creation. We address this question for the mixed-flavor mesons $D_s$, $B_s$, and $B_c$ by solving a screened Godfrey--Isgur Hamiltonian with the Gaussian expansion method. The screening length is fixed by threshold-dependent string breaking analysis, and is matched to the charmonium and bottomonium anchors. The calculated spectra saturate at $3.36$, $6.78$, and $8.19~{\rm GeV}$ in $D_s$, $B_s$, and $B_c$ families, while the RMS radii grow by several fm and mass gaps fall below $10~{\rm MeV}$. Mass-radius decoupling places the end of the safe radial ladder at $n_{\mathrm{safe}}\approx 4$--$6$ for $D_s$, $5$--$7$ for $B_s$, and $9$--$11$ for $B_c$, so the ladder terminates first in $D_s$, then in $B_s$, and last in $B_c$. Above the limiting-mass windows, at $3.49$, $6.86$, and $8.24~{\rm GeV}$, states of the radial ladder cannot be discovered. This termination is directly testable with forthcoming high-statistics data from BESIII, Belle II, and LHCb.