Spin forces in charmonium spectroscopy

Abstract

The observed ${\psi }^{\prime }\mathrm{}\left(3.68\right)-{\psi }^{\prime \prime }\mathrm{}\left(3.77\right)\mathrm{}$ structure in $e^{+}{e}^{-}$ annihilation implies the existence of a tensor-force effect in charmonium spectroscopy. We show that the magnitude of the effect is consistent with an analysis of ${}_{}{}^3{}_{}{}^{}P_J^{}{}_{}{}^{}$ levels using a phenomenological Hamiltonian linear in both $\overrightarrow{\mathrm{L}}·\overrightarrow{\mathrm{S}}$ and $S_{12}$, the tensor force. Adding a linear ${\overrightarrow{\sigma }}_1·{\overrightarrow{\sigma }}_2$ term one can account for all known charmonium levels below 4 GeV and predict the location of the rest of the spectrum, in particular the ${}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}$ state at 3.27 GeV.