From Light to Heavy Quarkonia: An Integrated Bethe-Salpeter View

A modification in the (harmonic) kernel of a QCD-oriented Bethe-Salpeter equation known for good fits to $q\overline{q}$ and $\mathrm{qqq}$ spectroscopy gives a comprehensive description of $q\overline{q}$, $Q\overline{q}$, and $Q\overline{Q}$ systems. The new assumptions are (i) proportionality of the "reduced" spring constant ${\tilde{\omega }}^2$ to ${\alpha }_s$ (${\tilde{\omega }}^2={\omega }_0^2{\alpha }_s$) and (ii) the replacement $r^2\to \frac{r^2}{{(1+A_0{m}_1{m}_2{r}^2)}^{\frac{1}{2}}}-\frac{c_0}{{\omega }_0^2}$. These provide a more direct QCD motivation for confinement and effect a smooth transition from harmonic ($q\overline{q}$) to linear ($Q\overline{Q}$) confinement.