Color-hyperfine splitting in heavy-quarkoniumPstates from perturbative QCD with a running coupling strength

We obtain an expression for the color-hyperfine interaction in heavy quarkonium in perturbative QCD including terms up to one loop, but going beyond one loop by letting the strong-interaction coupling constant ${\alpha }_s$ run. The resulting expression cannot be written analytically in position space, but we make approximations which enable us to obtain approximate upper and lower bounds on the $P$-wave color-hyperfine splittings in charmonium and bottomonium.