Perturbative S-matrix unitarity and higher-order Lorentz violation

We investigate the preservation of unitarity in a Lorentz and CPT-violating QED model containing higher-order operators. In particular, we consider modifications in the fermion sector with dimension-five operators. The higher-order operators lead to an indefinite metric and a pseudo-unitarity relation for the $S$-matrix. However, we show that the pseudo-unitarity condition can be promoted to a genuine unitarity relation by i) restricting the energies to the effective region far below the Planck mass and ii) considering stable particles to have positive metric. In the context of the optical theorem, we focus on the one-loop Bhabha and Compton scattering processes. We show that no ghost states get propagated through the cuts, thus satisfying the unitarity condition. Further, we show that discontinuities of propagators are equivalent to replacing physical Dirac functionals in the cutting equation. The physical Dirac functionals are defined to select only mode solutions of stable particles. The provided extension of Cutkosky rule may be helpful for analyzing perturbative unitarity in higher-order diagrams.