Spin splitting of the $L\ge 1$ quarkonium levels for a class of static potentials

Implications of a spin-dependent interaction generated by a static potential consisting of a vector [$V\left(r\right)\mathrm{}$] and a scalar [$S\left(r\right)\mathrm{}$] part are discussed for the mass splitting of the $L\ge 1$ levels of quarkonia. It is shown that when $V\left(r\right)\mathrm{}$ is a pure power law ($\sim r^{\alpha }$, attractive for all $r$) and $S\left(r\right)\mathrm{}$ is arbitrary, a relation exists between the power $\alpha$ and the four masses for a given $L\ge 1$ for any quarkonium system. For various values of $\alpha$ chosen recently to predict the center of gravity of the four levels, we use this relation to predict the mass of the ${}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{} c\overline{c}$ and $b\overline{b}$ systems as well as the mass of the ${}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}$ level of the $u\overline{s}$ system. It is argued that an accurate measurement of the masses of the $1 {}_{}{}^1{}_{}{}^{}P_1^{}{}_{}{}^{}\left(c\overline{c}\right)$ and $1 {}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}\left(u\overline{s}\right)$ levels can already be decisive in ascertaining the viability of this class of static potentials for quarkonia.