Mode-dependent field renormalization and triviality in λφ4 theory

A continuum limit of the (λφ4)4 theory exists which is “trivial” (all scattering amplitudes vanish), but which does not correspond to the zero-renormalized-coupling limit of conventional perturbation theory. This allows Spontaneous Symmetry Breaking and a very heavy, but still weakly interacting, Higgs boson. We answer some criticisms of this work, emphasizing that results from renormalization-group-improved perturbation theory are not valid. The phase transition encountered as the bare mass-squared is made more and more negative is first order, contrary to the assumption of the conventional perturbative “RG-improvement” program.