Stability analysis of the instantaneous Bethe-Salpeter equation and the consequences for meson spectroscopy

We investigate the light and heavy meson spectra in the context of the instantaneous approximation to the Bethe-Salpeter equation (Salpeter close-quote s equation). We use a static kernel consisting of a one-gluon-exchange component and a confining contribution. Salpeter close-quote s equation is known to be formally equivalent to a random-phase-approximation equation; as such, it can develop imaginary eigenvalues. Thus our study cannot be complete without first discussing the stability of Salpeter close-quote s equation. The stability analysis limits the form of the kernel and reveals that a Lorentz scalar confining interaction in the Salpeter equation leads to instabilities (imaginary eigenvalues), whereas one transforming as the time component of a vector does not. Moreover, the stability analysis sets an upper limit on the size of the one-gluon-exchange component; the value for the critical coupling is determined through a solution of the open-quote open-quote semirelativistic close-quote close-quote Coulomb problem. These limits place important constraints on the interaction and suggest that a more sophisticated model is needed to describe the light and heavy quarkonia. copyright 1996 The American Physical Society