Pathologies of dimension-zero scalar fields

It has been claimed in a series of papers that scalar fields with a fourth-order Lagrangian $\sim(\Box\varphi)^2$ can solve the cosmological constant problem by canceling the loop contributions from standard model fields, and that their fluctuations can be the source of the primordial density perturbations of the Universe, without the need for inflation. We dispute these claims. The spectrum of the theory includes a ghost, which leads to classical instabilities and quantum violation of unitarity. We show that the new scalar particles cannot cancel the standard model contributions to the cosmological constant, unless they include a unitarity-violating ghost at the quantum level. Further, the coupling of such scalars to the particles of the standard model induces a confining fifth force which rules it out as a source of density perturbations in the early Universe.