Triviality" and the Perturbative Expansion in $\lambda {\Phi }^4$ Theory

Abstract

The "triviality" of $(\lambda {\Phi }^4{)}_4$ quantum field theory means that the renormalized coupling ${\lambda }_R$ vanishes for infinite cutoff. That result inherently conflicts with the usual perturbative approach, which begins by postulating a non-zero, cutoff-independent ${\lambda }_R$. We show how a "trivial" solution ${\lambda }_R=0$ can be compatible with the known structure of perturbation theory to arbitrarily high orders, by a simple re-arrangement of the expansion. The "trivial" solution reproduces the result obtained by non-perturbative renormalization of the effective potential. The physical mass is finite, while the renormalized coupling strength vanishes: the two are NOT proportional. The classically scale-invariant $\lambda {\Phi }^4$ theory coupled to the Standard Model predicts a 2.2 TeV Higgs, but does NOT imply strong interactions in the scalar sector.

Comments: 10 pages, LaTeX, no figures, DE-FG05-92ER40717-13, MAD/TH/94-4