Complete analytical solution to the Cornell potential and heavy quarkonium structure

Abstract

We use the recently proposed supersymmetric expansion algorithm (SEA) to obtain a complete analytical solution to the Schr\"{o}dinger equation with the Cornell potential. We find that the energy levels $E_{nl}(\lambda)$ depend on $n^{2}$ and $L^{2}=l(l+1)$. For a given $n$, the energy {\emph {decreases}} with $l$ and the radial probabilities have the Coulomb shape but their peaks are shifted toward smaller radius. We study the heavy quarkonium structure on the light of these results, showing that the measured $\bar{b}b$ and $\bar{c}c$ meson masses follow the inverted spectrum pattern predicted by the Cornell potential. Details of the structure of heavy quarkonium like the mean inverse radius and mean squared velocity for the different quarkonium configurations can be obtained from our solution. These details point to significant relativistic corrections for all the configurations of real heavy quarkonium. We calculate relativistic corrections using perturbation theory finding an expansion in $\alpha^{2}_{s}$ for the heavy quarkonium masses. The mass hierarchies in the fine splittings can be qualitatively understood from this expansion. The quantitative analysis of the Bohr-like levels and of the fine splittings in the $l=0$ sector allow us to make well defined predictions for the masses of some of the missing heavy quarkonium states, to identify the $\psi(4040)$ as the $3^{3}S_{1}$ $\bar{c}c$ state and the $\psi(3842)$, $\psi(3823)$ and $\psi(3770)$ as the $3^{3}D_{3}$, $3^{3}D_{2}$ and $3^{3}D_{1}$ $\bar{c}c$ states respectively.