Renormalons as Saddle Points

TL;DR

This paper proposes a non-perturbative path integral perspective on renormalons, linking them to saddle points of the effective action, and illustrates this with simple models to advance understanding in quantum field theory.

Contribution

It introduces a novel path integral explanation for renormalons as saddle points, connecting them to instantons and quantum anomalies in a unified framework.

Findings

Renormalons can be interpreted as saddle points of the 1-loop effective action.

Quantum scale anomaly plays a crucial role in enabling this saddle point interpretation.

Toy models demonstrate the potential for studying renormalons in realistic field theories.

Abstract

Instantons and renormalons play important roles at the interface between perturbative and non-perturbative quantum field theory. They are both associated with branch points in the Borel transform of asymptotic series, and as such can be detected in perturbation theory. However, while instantons are associated with non-perturbative saddle points of the path integral, renormalons have mostly been understood in terms of Feynman diagrams and operator product expansions. We suggest a non-perturbative path integral explanation of how both instantons and renormalons produce singularities in the Borel plane using representative finite-dimensional integrals. In particular, we build evidence that renormalons can be understood as saddle points of the 1-loop effective action, enabled by a crucial contribution from the quantum scale anomaly. These results are illustrated in simple toy models and indicate a possible route toward studying renormalons within realistic asymptotically-free field theories.