  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.
