Comments on a 4-derivative scalar theory in 4 dimensions

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 Poincaré-invariant scattering amplitudes by assuming that only massless oscillating (nongrowing) 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 nonunitarity of this theory manifests itself at the level of the one-loop massless scattering amplitude.