Resurgence in Quantum Field Theory: Nonperturbative Effects in the Principal Chiral Model

Abstract

We explain the physical role of nonperturbative saddle points of path integrals in theories without instantons, using the example of the asymptotically free two-dimensional principal chiral model (PCM). Standard topological arguments based on homotopy considerations suggest no role for nonperturbative saddles in such theories. However, the resurgence theory, which unifies perturbative and nonperturbative physics, predicts the existence of several types of nonperturbative saddles associated with features of the large-order structure of the perturbation theory. These points are illustrated in the PCM, where we find new nonperturbative "fracton" saddle point field configurations, and suggest a quantum interpretation of previously discovered "uniton" unstable classical solutions. The fractons lead to a semiclassical realization of IR renormalons in the circle-compactified theory and yield the microscopic mechanism of the mass gap of the PCM.

Resurgence of the cusp anomalous dimension

“…Since then it has been applied at first in quantum mechanical systems [28,29] and only very recently it has been applied to quantum field theory [30][31][32] to obtain a weak coupling interpretation of the IR renormalons .…”

Hidden Topological Angles in Path Integrals

“…Providing a non-perturbative continuum definition of the path integral in quantum field theory is a challenging but important problem [1]. There is growing evidence that, if an ordinary integral or a path integral admits a Lefschetz-thimble decomposition [2,3] or resurgent transseries expansion [4][5][6][7][8][9][10][11][12] then either of these methods gives this long-sought non-perturbative definition. If this is indeed the case, then we expect that these new methods will provide new and deep insight into quantum field theory and quantum mechanics formulated in terms of path integrals.…”

Complex Path Integrals and Saddles in Two-Dimensional Gauge Theory

“…Such relations are central to the concept of resurgence, whereby different saddles are intertwined by monodromy properties that connect them and account for Stokes phases. The theory of resurgence has recently provided new insights into matrix models and string theories [1][2][3][4][5][6], and has been applied to asymptotically free QFTs and sigma models [7][8][9][10], and localizable SUSY QFTs [11]. One motivation for such QFT studies is to find a practical numerical implementation of a semiclassical expansion that could provide a Picard-Lefschetz thimble decomposition of gauge theory, either in the continuum or on the lattice, especially for theories with a sign problem [12,13].…”