Vibrational trajectories for heavy quarkonia.

Abstract

We show that vibrational states $\mathrm{vS}(v\ensuremath{\ge}0)$ of the heavy mesons provide a special window to the nature of the confining quark potential, extending here the studies of Kang, Schnitzer, and others of the case of linear confinement in the 3+1 dimensions. Linear confinement potentials yield an initial bending, followed by an approximate saturation of the spectroscopic interval ${\ensuremath{\Delta}}_{v}={M}_{v}^{2}\ensuremath{-}{M}_{v\ensuremath{-}1}^{2}$ as a function of $v$, in contrast with many other forms of confinement potentials with their characteristically different forms of ${\ensuremath{\Delta}}_{v}$. This feature of the linear potentials is roughly in accord with the available experimental data, and has some similarities with characteristics of the 't Hooft trajectories occurring in his (1+1)-dimensional theory of mesons.