Relativistic Particle-Antiparticle Stationary States in a Model Quantum Field Theory: Mesons from Quarks

Relativistic two-quantum particle-antiparticle bound states and scattering states are derived in a renormalized complex scalar field theory model which features a Heisenberg-type self-interaction term ${\lambda }_0{\left({\psi }^{*}\psi \right)}^2$ in the Lagrangian. The analysis is based on the Rayleigh-Ritz procedure for functionalities—a nonperturbative solutional method requiring the assumption that physically reasonable stationary states exist in the model. It is shown that the mass of a particle-antiparticle bound state can be one or more orders of magnitude less than the mass associated with a one-particle state. This property of the model suggests that a Heisenberg-type theory for a self-interacting quark spinor field may yield a physically realistic quark-antiquark meson with a mass one or two orders of magnitude less than the mass of a constituent quark.