  We formulate supersymmetric non-Hermitian quantum field theories with PT
symmetry, starting with free chiral boson/fermion models and then including
trilinear superpotential interactions. We consider models with both Dirac and
Majorana fermions, analyzing them in terms of superfields and at the component
level. We also discuss the relation between the equations of motion, the
(non-)invariance of the Lagrangian and the (non-)conservation of the
supercurrents in the two models. We exhibit a similarity transformation that
maps the free-field supersymmetric PT-symmetric Dirac model to a supersymmetric
Hermitian theory, but there is, in general, no corresponding similarity
transformation for the Majorana model. In this model, we find generically a
mass splitting between bosons and fermions, even though its construction is
explicitly supersymmetric, offering a novel non-Hermitian mechanism for soft
supersymmetry breaking.
