Unitarity Violation in Field Theories of Lee-Wick's Complex Ghost

TL;DR

This paper demonstrates that in higher-derivative field theories like Lee-Wick models, complex ghost modes can be created, leading to unitarity violation above a certain energy threshold, challenging previous assumptions of unitarity preservation.

Contribution

It provides a detailed operator formalism analysis showing how complex ghosts are generated and violate unitarity, contradicting earlier claims that ghosts are never produced in physical processes.

Findings

Complex ghosts are created during interactions, violating unitarity.

A specific energy threshold exists below which ghosts cannot be produced.

Theories are unitary and renormalizable only below this energy threshold.

Abstract

Theories with fourth-order derivatives, including the Lee-Wick finite QED model and Quadratic Gravity, have a better UV behaviour, but the presence of negative metric ghost modes endanger unitarity. Noticing that the ghost acquires a complex mass by radiative corrections, Lee and Wick, in particular, claimed that such complex ghosts would never be created by collisions of physical particles because of energy conservation, so that the physical S-matrix unitarity must hold. We investigate the unitarity problem faithfully working in the operator formalism of quantum field theory. When complex ghosts participate, a complex delta function (generalization of Dirac delta function) appears at each interaction vertex, which enforces a specific conservation law of complex energy. Its particular property implies that the naive Feynman rule is wrong if the four-momenta are assigned to the internal lines after taking account of the conservation law in advance. We show that the complex ghosts are actually created and unitarity is violated in such fourth-order derivative theories. We also find a definite energy threshold below which the ghosts cannot be created: The theories are unitary and renormalizable below the threshold.