How many Higgs bosons does it take. Consistency of scalar field theories at high energies

Computations of multiparticle scattering amplitudes in scalar field theories at high multiplicities hint at a rapid growth with the number of final state particles, rendering the theory in conflict with unitarity at high energies. This questions the validity of the perturbative approach or even the interpretation of the underlying quantum field theory. We study the quantum mechanical equivalent of high multiplicity amplitudes in λϕ${}^4$-theory, namely transition amplitudes from the vacuum to highly excited states in the anharmonic oscillator with a quartic potential. Using recursive relations, we compute these amplitudes to high order in perturbation theory and provide evidence that they can be written in exponential form. By resummation techniques, we then construct its exponent beyond leading order and investigate the behaviour of the amplitudes in the region where tree-level perturbation theory violates unitarity constraints. We find that for both the single- and the double-well potential the resummed amplitudes are in agreement with unitarity bounds. We then extend our results to anharmonic oscillators with general monomial potentials and point out possible problems of perturbative expansions even in potentials with a single minimum. Finally, we comment on the relevance of our results for the field theoretical problem.