Bound States in Lee's Complex Ghost Model

TL;DR

The paper investigates whether a ghost-ghost bound state can form in Lee's complex ghost model using the canonical operator formalism. By computing the correlation function of the ghost composite operator ${\cal O}_{|\varphi|^2}$ and analyzing the bound-state pole equation on the Lee-Wick contour, it demonstrates that a term involving the complex delta function $\delta_c$ obstructs the emergence of a bound-state pole. The analysis, including a Pauli-Villars regularized loop integral and a Wick rotation to Euclidean space, shows that no nontrivial bound-state solution exists for ghost composites, and this remains true for other ghost-based composites. The result reinforces that unitarity violation in such higher-derivative theories may not be cured by simple bound-state formation and points to potential confinement mechanisms, such as BRST-based structures, as avenues for addressing the massive ghost problem in theories like quadratic gravity.

Abstract

Quantum field theories (QFTs) including fourth-derivative terms such as the Lee-Wick finite QED and quadratic gravity have a better ultra-violet behavior compared to standard theories with second-derivative ones, but the existence of ghost with negative norm endangers unitarity. Such a ghost in general acquires a pair of complex conjugate masses from radiative corrections whose features are concisely described by the so-called Lee model. Working with the canonical operator formalism of QFTs, we investigate the issue of bound states in the Lee model. We find that the bound states cannot be created from ghosts by contributions of a complex delta function, which is a complex generalization of the well-known Dirac delta function. Since the cause of unitarity violation in the Lee-Wick model is the existence of the complex delta function instead of the Dirac delta function, it is of interest to notice that the violation of the unitarity is also connected to the non-existence of bound states. Finally, the problem of amelioration of the unitarity in quadratic gravity is briefly discussed.