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 $N>\N_crit\simeq e/g.$ Numerical solutions of the coupled Schr\"odinger equations shows that for weak coupling and a large range of $N>\ncrit,$ the exact unitary amplitude is reasonably fit by a factorized expression $|A(1->N)| \sim (0.73 /N) \cdot \exp{(-0.025/\g2)}$. The very small size of the coefficient $1/\g2$ , 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 $K\ \simeq\ \left((0.05/\g2)+ 2\ ln{N}\right)/ \ ln{(1/\g2)}$ in an expansion in powers of $\g2.$