Tree and nontree multiparticle amplitudes

We examine the behavior of the connected n-point function in a zero-dimensional scalar theory (scrL=-1/2${\mathit{m}}^2$ ${\mathrm{\phi }}^2$-1/4λ${\mathrm{\phi }}^4$). The problematic n!(√λ ${)}^{\mathit{n}}$ 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(λ)${]}^{\mathit{n}}$, persists in the full theory, where f∼ √λ for λ→0, f∼${\lambda }^{\mathrm{-}1/4}$ for λ≫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.