Large-Order Perturbation Theory in Infrared-Unstable Superrenormalizable Field Theories

Abstract

We study the factorial divergences of Euclidean $ϕ^3_5$, a problem with connections both to high-energy multiparticle scattering in d=4 and to d=3 (or high-temperature) gauge theory, which like $ϕ^3_5$ is infrared-unstable and superrenormalizable. At large external momentum p (or small mass M) and large order N one might expect perturbative bare skeleton graphs to behave roughly like $N!(ag^2/p)^N$ with a>0, so that no matter how large p is there is an $N\sim g^2/p$ giving rise to strong perturbative amplitudes. The semi- classical Lipatov technique (which works only in the presence of a mass) is blind to this momentum dependence, so we proceed by direct summation of bare skeleton graphs. We find that the various limits of large N, large p, and small M do not commute, and that when $N\gg p^2/M^2$ there is a Borel singularity associated with $g^2/M$, not $g^2/p$. This is described by the zero-momentum Lipatov technique, and we find the necessary soliton for $ϕ^3_5$; the corresponding sphaleron-like solution for unbroken Yang-Mills theory has long been known. We also show that the massless theories have no classical solitons. We discuss non-perturbative effects based partly on known physical arguments concerning the cancellation by solitons of imaginary parts due to the pert- urbative Borel singularity, and partly on the dressing of bare skeleton graphs by dressed propagators showing non-perturbative mass generation, as happens in d=3 gauge theory.