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Scalar-particle self-energy amplitudes and confinement in Minkowski space

We analyze the analytic structure of the Covariant Spectator Theory (CST) contribution to the self-energy amplitude for a scalar particle in a ϕ^2 χ-theory. To this end we derive dispersion relations in 1+1 and in 3+1 dimensional Minkowski space. The divergent loop integrals in 3+1 dimensions are regularized using dimensional regularization. We find that the CST dispersion relations exhibit, in addition to the usual right-hand branch cut, also a left-hand cut. The origin of this "spectator" left-hand cut can be understood in the context of scattering for a scalar ϕ^2 χ^2-type theory. If the interaction kernel contains a linear confining component, its contribution to the self-energy vanishes exactly.