  Theories with fourth-order derivatives, including the Lee-Wick finite QED
model and Quadratic Gravity, have a better UV behaviour, but the presence of
negative metric ghost modes endanger unitarity. Noticing that the ghost
acquires a complex mass by radiative corrections, Lee and Wick, in particular,
claimed that such complex ghosts would never be created by collisions of
physical particles because of energy conservation, so that the physical
S-matrix unitarity must hold.
  We investigate the unitarity problem faithfully working in the operator
formalism of quantum field theory. When complex ghosts participate, a complex
delta function (generalization of Dirac delta function) appears at each
interaction vertex, which enforces a specific conservation law of complex
energy. Its particular property implies that the naive Feynman rule is wrong if
the four-momenta are assigned to the internal lines after taking account of the
conservation law in advance. We show that the complex ghosts are actually
created and unitarity is violated in such fourth-order derivative theories. We
also find a definite energy threshold below which the ghosts cannot be created:
The theories are unitary and renormalizable below the threshold.
