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

High-energy behavior of scattering amplitudes in theories with purely virtual particles