Asymptotic Behavior of Diagram Classes

Abstract

The asymptotic nature of perturbative expansions in quantum field theory can arise from the factorial growth in the number of Feynman diagrams with loop order, as with instantons, or from a series of individual diagrams whose values grow factorially, as with renormalon chains in QED. Other classes of diagrams are known also to grow factorially, such as the Hopf series of graphs in $\phi^3$ theory. This Hopf series was studied using Schwinger-Dyson equations and the Connes-Kreimer Hopf algebra of decorated rooted trees. We review the Hopf algebra approach and show that the same results can be obtained using analytic QFT techniques as with Hopf-algebraic ones. We present an efficient method to extract the asymptotic behavior and thereby generalize the analysis of the Hopf series to other classes of diagrams in other theories. We confront the question of whether these classes correspond to new types of asymptotic growth beyond instantons and renormalons, and find that they appear to be incomplete calculations of what would be renormalons in these theories if all diagrams were included. Although the Hopf algebra approach is not essential to deriving asymptotic behavior from Schwinger-Dyson equations, it does provide some other insights into quantum field theory. We therefore attempt also to provide a map between some relevant aspects of the Hopf algebra and quantum field theory.