Confinement phenomenology in the Bethe-Salpeter equation

We consider the solution of the Bethe-Salpeter equation in the Euclidean metric for a q̅ q vector meson in the circumstance where the dressed quark propagators have timelike complex conjugate mass poles. This approximates features encountered in recent QCD modeling via the Dyson-Schwinger equations; the absence of real mass poles simulates quark confinement. The analytic continuation in the total momentum necessary to reach the mass shell for a meson sufficiently heavier than 1 GeV leads to the quark poles being within the integration domain for two variables in the standard approach. Through Feynman integral techniques, we show how the analytic continuation can be implemented in a way suitable for a practical numerical solution. We show that the would-be q̅ q width to the meson generated from one quark pole is exactly canceled by the effect of the conjugate partner pole; the meson mass remains real and there is no spurious q̅ q production threshold. The ladder kernel we employ is consistent with one-loop perturbative QCD and has a two-parameter infrared structure found to be successful in recent studies of the light SU(3) meson sector.