Comments on 4-derivative scalar theory in 4 dimensions

We review and elaborate on some aspects of classically scale-invariant renormalizable 4-derivative scalar theory $L= \p \del^4 \p + g (\del \p)^4$ in 4 dimensions. Similar models appear, e.g., in the context of conformal supergravity or in the description of crystalline phase of membranes. Considering this theory in Minkowski signature we discuss how to define consistent (Lorentz-invariant) scattering amplitudes by assuming that only oscillating (non-growing) modes appear as external states. In the shiftsymmetric interacting theory the corresponding S-matrix is IR-soft despite having $1/p^4$ internal propagators. We demonstrate how non-unitarity of this theory manifests itself at the level of the one-loop scattering amplitude.