Exact Nonperturbative Unitary Amplitudes for 1->N Transitions

Abstract

I present an extension to arbitrary N of a previously proposed field theoretic model, in which unitary amplitudes for $1->8$ processes were obtained. The Born amplitude in this extension has the behavior $A(1->N{)}^{tree}=g^{N-1}N!$ expected in a bosonic field theory. Unitarity is violated when $|A(1->N)|>1$, or when $N>{\mathbb{N}}_{c}rit\simeq e/g.$ Numerical solutions of the coupled Schrödinger equations shows that for weak coupling and a large range of $N>\text{ncrit},$ the exact unitary amplitude is reasonably fit by a factorized expression $|A(1->N)|\sim (0.73/N)\cdot \exp (-0.025/\text{g}2)$. The very small size of the coefficient $1/\text{g}2$ , indicative of a very weak exponential suppression, is not in accord with standard discussions based on saddle point analysis, which give a coefficient $\sim 1.$ The weak dependence on $N$ could have experimental implications in theories where the exponential suppression is weak (as in this model). Non-perturbative contributions to few-point correlation functions in this theory would arise at order $K\simeq \left((0.05/\text{g}2)+2lnN\right)/ln(1/\text{g}2)$ in an expansion in powers of $\text{g}2.$

Comments: 11 pages, 3 figures (not included)