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's equation). We use a static kernel consisting of a one-gluon-exchange component and a confining contribution. Salpeter's equation is known to be formally equivalent to a random-phase-approximation equation; as such, it can develop imaginary eigenvalues. Thus, our study can not be complete without first discussing the stability of Salpeter's equation. The stability analysis limits the form of the kernel and reveals that, contrary to the usual assumption, the confining component can not transform as a Lorentz scalar; it must transform as the timelike component of a vector. 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 ``semirelativistic'' 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.