  At very high energies scattering amplitudes in a spontaneously broken gauge
theory into multi-particle final states are known to grow factorially with the
number of particles produced. Using simple scalar field theory models with and
without the VEV, we compute total cross-sections with up to 7 particles in the
final state at the leading order in perturbation theory with MadGraph. By
exploring the known scaling properties of the multi-particle rates with the
number of particles, we determine from these the general $n$-point
cross-sections in the large-$n$ limit. In the high-multiplicity regime we are
considering, n>>1 and lambda n=fixed, the perturbation theory becomes strongly
coupled with the higher-order loop effects contributing increasing powers of
lambda n. In the approximation where only the leading loop effects are
included, we show that the corresponding perturbative cross-sections grow
exponentially and ultimately violate perturbative unitarity. This occurs at
surprisingly low energy scales ~50 TeV with multiplicities above ~130. It is
expected that a repair mechanism or an extension of the theory has to set-in
before these scales are reached, possibly involving a novel non-perturbative
dynamics in the a priori weakly coupled theory.

High Energy Physics - Phenomenology;High Energy Physics - Theory