  Calculations of high-energy processes involving the production of a large
number of particles in weakly-coupled quantum field theories have previously
signaled the need for novel non-perturbative behavior or even new physical
phenomena. In some scenarios, already tree-level computations may enter the
regime of large-order perturbation theory and therefore require a careful
investigation. We demonstrate that in scalar quantum field theories with a
unique global minimum, where suitably resummed perturbative expansions are
expected to capture all relevant physical effects, perturbation theory may
still suffer from severe shortcomings in the high-energy regime. As an example,
we consider the computation of multiparticle threshold amplitudes of the form
$1 \to n$ in $\varphi^6$ theory with a positive mass term, and show that they
may violate unitarity of the quantum theory for large $n$, even after the
resummation of all leading-$n$ quantum corrections. We further argue that this
is a generic feature of scalar field theories with higher-order
self-interactions beyond $\varphi^4$, thereby rendering the latter unique with
respect to its high-energy behavior.
