Predictability of charmonium levels from a range of good fits.

Abstract

A QCD-motivated potential with five parameters and a cutoff on high virtual momentum is used in a relativistic bound-state equation to examine the predictive power of valence quark potential models by computing fits to the seven well-established charmonium levels. The predictions of potential models are really ranges for new levels given by the ranges of parameters that give good fits to established levels, rather than the single best fit. The range of parameters that give good fits is presented in three-dimensional plots by isosurfaces of fits with fixed theoretical standard deviations in energy from the known levels. Next, for each value of a predicted splitting we show the minimum deviation found over all the fits that lead to that value of splitting. We also show correlations between predictions by finding for each pair of values for two predictions the minimum deviation among the fits that give those predicted values, and making contour plots. For the $^{1}P_{1}\ensuremath{-}^{3}P_{\mathrm{c}.\mathrm{g}.}$ splitting we find the predictions to cover a range of several MeV both positive and negative. A correlation of the spin-spin $^{1}P_{1}\ensuremath{-}^{3}P_{\mathrm{c}.\mathrm{g}.}$ splitting with the similar ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\eta}}_{c}^{\ensuremath{'}}$ splitting prediction is presented, and values for ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\eta}}_{c}^{\ensuremath{'}}$ have a range of order 20-50 MeV centered about 35 MeV. To range as high as the single experimental observation of 92 MeV for ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\eta}}_{c}^{\ensuremath{'}}$, such a poor fit to the seven levels is encountered that the model lacks any significant predictive power. Predictions for the $^{3}D_{2}$ and $^{1}D_{2}$ levels are also given.