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

TL;DR

This paper investigates the factorial divergence behavior of perturbation series in infrared-unstable superrenormalizable field theories, revealing non-commuting limits and identifying Borel singularities linked to soliton solutions.

Contribution

It provides a detailed analysis of large-order perturbation theory in $ield^3_5$, uncovering the non-trivial dependence on momentum and mass, and identifying the relevant soliton solutions affecting divergence structure.

Findings

Large N, p, and M limits do not commute.

Borel singularity associated with $g^2/M$, not $g^2/p$.

Existence of soliton solutions in massive theories, absence in massless theories.

Abstract

We study the factorial divergences of Euclidean $\phi^3_5$, a problem with connections both to high-energy multiparticle scattering in d=4 and to d=3 (or high-temperature) gauge theory, which like $\phi^3_5$ 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 $\phi^3_5$; 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.