Models with quark confinement and linear trajectories without parity doubling

Abstract

We consider simple dynamic models where the quarks satisfy the Dirac equation with a confining potential. We show that if the confining potential is a Lorentz scalar, the solutions to the Dirac equation satisfy the MacDowell symmetry. We discuss the slope of Regge trajectories in these models and show that in order for the trajectory to be linear (in $s={E}^{2}$) the potential must have a specific $j$ dependence, which produces a cut in $j$ of the type first proposed by Carlitz and Kislinger. In other words, the same mechanism which produces linear trajectories in the first place also produces the cut which removes the parity partners. The specific $j$ dependence is expected in "bag" models, as well as relativistic solition solutions.