  We study perturbative unitarity in the scalar sector of the Myers-Pospelov
model. The model introduces a preferred four-vector $n$ which breaks Lorentz
symmetry and couples to a five-dimension operator. When the preferred
four-vector is chosen in the pure timelike or lightlike direction, the model
becomes a higher time derivative theory, leading to a cubic dispersion
relation. Two of the poles are shown to be perturbatively connected to the
standard ones, while a third pole, which we call the Lee-Wick-like pole, is
associated to a negative metric, in Hilbert space, threatening the preservation
of unitarity. The pure spacelike case is a normal theory in the sense that it
has only two solutions both being small perturbations over the standard ones.
We analyze perturbative unitarity for purely spacelike and timelike cases using
the optical theorem and considering a quartic self-interaction term. By
computing discontinuities in the loop diagram, we arrive at a pinching
condition which determines the propagation of particles and Lee-Wick-like
particles through the cut. We find that the contribution for Lee-Wick-like
particles vanishes for any external momenta, leaving only the contribution of
particles, thus preserving one-loop unitarity in both cases.
