Confinement in a Bethe-Salpeter model with logarithmic potential

Abstract

We show that the covariant Bethe-Salpeter equation for two scalar quarks confined through a kernel of the form K~[(p-p')²]^-3/2 possesses a discrete spectrum: this kernel is a simple covariant generalization of the potential in configuration space proportional to lnr. To investigate the spectrum, we adopt a new approximation scheme to convert the covariant Bethe-Salpeter equation into a three-dimensional one, the consistency of the scheme being checked by applying it to the Wick-Cutkosky equation for the covariant Coulomb problem and obtaining a Balmer-type formula. The spectrum as revealed by investigating the S-wave part of the resulting three-dimensional equation by semiclassical methods is essentially of the type obtained from the Schrodinger equation with a lnr potential; relativistic effects depress the energy levels.