Relativistic effects in heavy-quarkonium spectroscopy

Abstract

It is shown that the discrepancy between the measured rate for ${\psi }^{\prime }\to {\chi }_0\gamma$ and that predicted by nonrelativistic models can be accounted for by ${\left(\frac{v}{c}\right)}^2$ relativistic corrections. A Breit-Fermi Hamiltonian is used to predict the energy level structure and $E1\mathrm{}$ transition rates in the charmonium and $\Upsilon$ 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 $E1\mathrm{}$ transition rates. It is used to predict the level structure and $E1\mathrm{}$ rates in the $\Upsilon$ system. It is shown that an extension of Siegert's theorem is valid in the relativistic regime. This result is useful in analyzing $E1\mathrm{}$ transition-matrix elements.