Comments on 4-derivative scalar theory in 4 dimensions

TL;DR

This paper reviews a classically scale-invariant 4-derivative scalar theory, discussing its properties, how to define scattering amplitudes, and the implications of its non-unitarity at one-loop level.

Contribution

It provides a detailed analysis of the 4-derivative scalar theory, including a method to define scattering amplitudes and insights into its non-unitarity.

Findings

No IR divergences in the scattering amplitudes despite $1/q^4$ propagators.

Non-unitarity manifests at the one-loop amplitude level.

Proposes a way to define Poincare-invariant amplitudes in this theory.

Abstract

We review and elaborate on some aspects of the classically scale-invariant renormalizable 4-derivative scalar theory $L= \phi\, \partial^4 \phi + g (\partial \phi)^4$. Similar models appear, e.g., in the context of conformal supergravity or in the description of the crystalline phase of membranes. Considering this theory in Minkowski signature we suggest how to define Poincare-invariant scattering amplitudes by assuming that only massless oscillating (non-growing) modes appear as external states. In such shift-symmetric interacting theory there are no IR divergences despite the presence of $1/q^4$ internal propagators. We discuss how non-unitarity of this theory manifests itself at the level of the one-loop massless scattering amplitude.