TREE AND NONTREE MULTIPARTICLE AMPLITUDES

We examine the behavior of the connected n-point function in a zero-dimensional scalar theory (L = - 1/2 m2-phi-2 - 1/4-lambda-phi-4). The problematic n!(square-root-lambda)n behavior for large n, previously found for tree graphs in scalar field theory, is again obtained in saddle-point approximation in this model. However, a similar behavior, n![f(lambda)]n, persists in the full theory, where f approximately square-root-lambda for lambda --> 0, f approximately lambda-1/4 for lambda >> 1. This behavior can be traced to the existence of zeros in the complex J plane of the generating functional Z(J), which in turn is controlled by the asymptotic behavior in (J) of Z(J) calculated in saddle-point approximation. The application of these results to field theory is discussed.