# Unitarity, stability and loops of unstable ghosts

## TL;DR

This paper demonstrates that theories with unstable ghost-like resonances, such as quadratic gravity and Lee-Wick models, are unitary at all orders and that quantum effects ensure their stability and proper treatment of unstable states.

## Contribution

It provides a new understanding of unstable ghosts, proving unitarity without including ghost states in the spectrum and clarifying the role of quantum corrections and contours.

## Key findings

- Unitarity holds to all orders in theories with unstable ghosts.
- Quantum corrections cause the ghost resonances to be unstable and not appear in the asymptotic spectrum.
- Proper contour choices are essential for correct one-loop calculations involving unstable ghosts.

## Abstract

We present a new understanding of the unstable ghost-like resonance which appears in theories such as quadratic gravity and Lee-Wick type theories. Quantum corrections make this resonance unstable, such that it does not appear in the asymptotic spectrum. We prove that these theories are unitary to all orders. Unitarity is satisfied by the inclusion of only cuts from stable states in the unitarity sum. This removes the need to consider this as a ghost state in the unitarity sum. However, we often use a narrow-width approximation where we do include cuts through unstable states, and ignore cuts through the stable decay products. If we do this with the unstable ghost resonance at one loop, we get the correct answer only by using a contour which was originally defined by Lee and Wick. The quantum effects also provide damping in both the Feynman and the retarded propagators, leading to stability under perturbations.