Sign of the color-hyperfine splitting in charmoniumPstates as a test of perturbative QCD

The sign of the difference in energy between the center of gravity of the ${}_{}{}^3{}_{}{}^{}P$ levels and the ${}_{}{}^1{}_{}{}^{}P$ level in heavy quarkonium is a test of the ${\alpha }_s^2$ terms in perturbative QCD with a fixed coupling constant. Whereas this energy difference is zero in the Fermi-Breit approximation and greater than zero in a generalized Fermi-Breit approximation with asymptotically free potentials, the energy difference is negative in perturbative QCD calculations to one-loop order with a fixed ${\alpha }_s$. For potentials which reasonably agree with the spin-averaged data, the absolute magnitude of the splitting is calculated to be about 4 MeV or less in charmonium and 2 MeV or less in bottomonium. Since the energy of the ${}_{}{}^3{}_{}{}^{}P$ center of gravity in charmonium is known, a measurement of the energy of the ${}_{}{}^1{}_{}{}^{}P$ level is needed to discriminate between these different schemes. Although the ${}_{}{}^1{}_{}{}^{}P$ levels in bottomonium will probably be harder to measure, we also present results for bottomonium.