We study the factorial divergences of Euclidean ${\phi }_5^3$, a problem with connections both to high-energy multiparticle scattering in $d=4\mathrm{}$ and to $d=3\mathrm{}$ (or high-temperature) gauge theory, which like ${\phi }_5^3$ 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 $a>\mathrm{}0\mathrm{}$, so that no matter how large $p$ is there is an $N\sim {p/g}^2$ giving rise to strong perturbative amplitudes. The semiclassical Lipatov technique (which only works 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 mass $M$ 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 ${\phi }_5^3$; the corresponding sphaleronlike solution for unbroken Yang-Mills theory has long been known. We also show that the massless theories have no classical solitons. We discuss nonperturbative effects based partly on known physical arguments concerning the cancellation by solitons of imaginary parts due to the perturbative Borel singularity, and partly on the dressing of bare skeleton graphs by dressed propagators showing nonperturbative mass generation, as happens in $d=3\mathrm{}$ gauge theory.

• Received 3 January 1997

DOI:https://doi.org/10.1103/PhysRevD.55.6209