Unitarity of loop diagrams for the ghost-like $1/(k^2-M_1^2)-1/(k^2-M_2^2)$ propagator

Abstract

With fourth-order derivative theories leading to propagators of the generic ghost-like $1/(k^2-M_1^2)-1/(k^2-M_2^2)$ form, it would appear that such theories have negative norm ghost states and are not unitary. However on constructing the associated quantum Hilbert space for the free theory that would produce such a propagator, Bender and Mannheim found that the Hamiltonian of the free theory is not Hermitian but is instead $PT$ symmetric, and that there are in fact no negative norm ghost states, with all Hilbert space norms being both positive and preserved in time. Even though perturbative radiative corrections cannot change the signature of a Hilbert space inner product, nonetheless it is not immediately apparent how such a ghost-like propagator would not then lead to negative probability contributions in loop diagrams. Here we obtain the relevant Feynman rules and show that all states obtained in cutting intermediate lines in loop diagrams have positive norm. Also we show that due to the specific way that unitarity (conservation of probability) is implemented in the theory, negative signatured discontinuities across cuts in loop diagrams are cancelled by a novel and unanticipated contribution of the states in which tree approximation (no loop) graphs are calculated, an effect that is foreign to standard Hermitian theories. Perturbatively, the fourth-order derivative theory is thus viable. The theory associated with the pure massless $1/k^4$ propagator is equally shown to be perturbatively viable.