  We analyze the analytic structure of the Covariant Spectator Theory (CST)
contribution to the self-energy amplitude for a scalar particle in a \phi^2
\chi-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 \phi^2 \chi^2-type theory. If the
interaction kernel contains a linear confining component, its contribution to
the self-energy vanishes exactly.
