  We tackle the issue of the factorial growth in the amplitudes of multi-Higgs
production at high energy by developing a phenomenological approach based on
the Higgs splitting functions and Sudakov factors. We utilize the method of
generating functionals to define several jet observables for the Higgs sector.
Our results suggest that pure Higgs splittings should retain a good UV behavior
in contrast to the common picture represented by the breakdown of perturbation
theory and violation of unitarity due to the high multiplicity of particles
produced at or near threshold, which is found in scalar theories. We thus argue
that the issue of the factorial growth in the amplitude of multi-Higgs
production is probably associated with applying perturbation theory in a regime
where it is no longer valid, as opposed to being a sign of new physics.
