Exploring high multiplicity amplitudes in quantum mechanics

Abstract

Calculations of 1→N amplitudes in scalar field theories at very high multiplicities exhibit an extremely rapid growth with the number N of final state particles. This either indicates an end of perturbative behavior, or possibly even a breakdown of the theory itself. It has recently been proposed that in the Standard Model this could even lead to a solution of the hierarchy problem in the form of a "Higgsplosion" [1]. To shed light on this question we consider the quantum mechanical analogue of the scattering amplitude for N particle production in φ4 scalar quantum field theory, which corresponds to transitions N|x^|0 in the anharmonic oscillator with quartic coupling λ. We use recursion relations to calculate the N|x^|0 amplitudes to high order in perturbation theory. Using this we provide evidence that the amplitude can be written as N|x^|0 ∼exp(F(λN)/λ) in the limit of large N and λN fixed. We go beyond the leading order and provide a systematic expansion in powers of 1/N. We then resum the perturbative results and investigate the behavior of the amplitude in the region where tree-level perturbation theory violates unitarity constraints. The resummed amplitudes are in line with unitarity as well as stronger constraints derived by Bachas [2]. We generalize our result to arbitrary states and powers of local operators N|x^q|M and confirm that, to exponential accuracy, amplitudes in the large N limit are independent of the explicit form of the local operator, i.e., in our case q.