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Diagrammatic computation of multi-Higgs processes at very high energies: scaling F holy grail with MadGraph

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.