Triviality of the Higgs Field

We give a critical review of the "triviality" of the λϕ 4 theory, i.e., the vanishing of the renormalized self-coupling. Evidence from perturbation theory and Monte-Carlo simulations are cited. It is noted that (a) the theory is "trivial" but not entirely free, for there is spontaneous symmetry breaking; (b) perturbation theory is unreliable. Soluble examples with similar behavior are compared, in particular the Lee model and the 3D δ function potential. The latter case is especially important, for it shows that triviality is a symptom that the interaction is too singular, and suggests a cure. The import for the Higgs sector of the standard model is discussed. It is argued that, like the Fermi pseudopotential, the Higgs field is a long-wavelength approximation that should be used in lowest order perturbation theory only.