  We study a class of renormalizable quantum field theories with purely virtual
particles that exhibits nonrenormalizable behavior in the high-energy limit of
scattering cross sections, which grow as powers of the center-of-mass energy
squared and seems to violate unitarity bounds. We point out that the problem
should be viewed as a violation of perturbativity, instead of unitarity, and
show that the resummation of self energies fixes the issue. As an explicit
example, we consider a class of $O(N)$ theories at the leading order in the
large-$N$ expansion and show that the different quantization prescription of
purely virtual particles takes care of the nonrenormalizable behavior, making
the resummed cross sections to decrease at high energies and the amplitudes to
satisfy the unitarity bounds. We compare the results to the case of theories
with ghosts, where the resummation cannot change the behavior of cross sections
due to certain cancellations in the high-energy expansion of the self energies.
These results are particularly relevant for quantum gravity.
