Relativistic effects in heavy-quarkonium spectroscopy

It is shown that the discrepancy between the measured rate for ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{\rightarrow}{\ensuremath{\chi}}_{0}\ensuremath{\gamma}$ and that predicted by nonrelativistic models can be accounted for by ${(\frac{v}{c})}^{2}$ relativistic corrections. A Breit-Fermi Hamiltonian is used to predict the energy level structure and $E1$ transition rates in the charmonium and $\ensuremath{\Upsilon}$ systems. It is obtained from an instantaneous approximation to a Bethe-Salpeter equation whose kernel is composed of Coulomb-gauge gluon exchange and a scalar confining piece. The model accounts for the observed fine and hyperfine structure of the charmonium levels and for the $E1$ transition rates. It is used to predict the level structure and $E1$ rates in the $\ensuremath{\Upsilon}$ system. It is shown that an extension of Siegert's theorem is valid in the relativistic regime. This result is useful in analyzing $E1$ transition-matrix elements.