Exploring High Multiplicity Amplitudes: The QM Analogue of the Spontaneously Broken Case

Abstract

Calculations of high multiplicity Higgs amplitudes exhibit a rapid growth that may signal an end of perturbative behavior or even the need for new physics phenomena. As a step towards this problem we consider the quantum mechanical equivalent of $1\to n$ scattering amplitudes in a spontaneously broken ${\varphi }^4$-theory by extending our previous results on the quartic oscillator with a single minimum to transitions $⟨n|\hat{x}|0⟩$ in the symmetric double-well potential with quartic coupling $\lambda$. Using recursive techniques to high order in perturbation theory, we argue that these transitions are of exponential form $⟨n|\hat{x}|0⟩\sim \exp \left(F(\lambda n)/\lambda \right)$ in the limit of large $n$ and $\lambda n$ fixed. We apply the methods of "exact perturbation theory" put forward by Serone et al. to obtain the exponent $F$ and investigate its structure in the regime where tree-level perturbation theory violates unitarity constraints. We find that the resummed exponent is in agreement with unitarity and rigorous bounds derived by Bachas.

Comments: 21 pages, 7 figures. v2: minor updates and corrections