# Perturbative unitarity of Lee-Wick quantum field theory

## TL;DR

This paper demonstrates that Lee-Wick quantum field theories maintain perturbative unitarity when formulated via nonanalytic Wick rotation, with one-loop diagrams satisfying unitarity cuts in each region of the complex energy plane.

## Contribution

It provides a detailed analysis of unitarity in Lee-Wick models using nonanalytic Wick rotation, including explicit calculations and addressing Minkowski formulation issues.

## Key findings

- One-loop diagrams satisfy unitarity cutting equations in each region.
- Unitarity is preserved in Lee-Wick models with nonanalytic Wick rotation.
- Minkowski formulation of higher-derivative models violates unitarity.

## Abstract

We study the perturbative unitarity of the Lee-Wick models, formulated as nonanalytically Wick rotated Euclidean theories. The complex energy plane is divided into disconnected regions and the values of a loop integral in the various regions are related to one another by a nonanalytic procedure. We show that the one-loop diagrams satisfy the expected, unitary cutting equations in each region: only the physical degrees of freedom propagate through the cuts. The goal can be achieved by working in suitable subsets of each region and proving that the cutting equations can be analytically continued as a whole. We make explicit calculations in the cases of the bubble and triangle diagrams and address the generality of our approach. We also show that the same higher-derivative models violate unitarity if they are formulated directly in Minkowski spacetime.