Unitarity in Multi-Higgs Production

It has long been known that perturbative calculations in scalar multi-particle production could break down since fast growing amplitudes appear. A recent calculation in the regime $\lambda n \gg 1$, where $n$ is the multiplicity and $\lambda$ is the self-coupling, gives an amplitude which grows exponentially with the energy, resulting in a divergent propagator and leading to the violation of perturbative unitarity. In this paper the transition rate is calculated from the solution of the self-consistent Schwinger-Dyson equation in spectral representation. We get an amplitude growing quadratically with the energy which leads to an asymptotically decreasing propagator contrary to previous results. Hence, unitarity is not violated, as expected in the Standard Model.