The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects

TL;DR

This work develops a controlled semiclassical framework for 4d gauge theories with adjoint fermions on ${\mathbb R}^3\times S^1$, showing how center-stabilized holonomies can Higgs and partially abelianize the gauge group. It identifies monopole-instantons and two classes of topological molecules—magnetic and neutral bions—and explains how the Bogomolny–Zinn-Justin prescription renders neutral bions well-defined, uncovering a close link between their Borel-plane singularities and 4-d IR renormalons. By combining 1-loop perturbation theory with nonperturbative semiclassical effects, the paper derives mass gaps and confinement mechanisms in the abelianized regime and discusses the fate of nonabelian factors for various gauge groups. The resurgence framework is invoked to relate perturbative expansions to nonperturbative saddles via transseries, offering a path toward a continuum, nonperturbative definition of these theories and illuminating the role of neutral topological objects in the infrared structure of gauge theories.

Abstract

We study the dynamics of four dimensional gauge theories with adjoint fermions for all gauge groups, both in perturbation theory and non-perturbatively, by using circle compactification with periodic boundary conditions for the fermions. There are new gauge phenomena. We show that, to all orders in perturbation theory, many gauge groups are Higgsed by the gauge holonomy around the circle to a product of both abelian and nonabelian gauge group factors. Non-perturbatively there are monopole-instantons with fermion zero modes and two types of monopole-anti-monopole molecules, called bions. One type are "magnetic bions" which carry net magnetic charge and induce a mass gap for gauge fluctuations. Another type are "neutral bions" which are magnetically neutral, and their understanding requires a generalization of multi-instanton techniques in quantum mechanics - which we refer to as the Bogomolny-Zinn-Justin (BZJ) prescription - to compactified field theory. The BZJ prescription applied to bion-anti-bion topological molecules predicts a singularity on the positive real axis of the Borel plane (i.e., a divergence from summing large orders in peturbation theory) which is of order N times closer to the origin than the leading 4-d BPST instanton-anti-instanton singularity, where N is the rank of the gauge group. The position of the bion--anti-bion singularity is thus qualitatively similar to that of the 4-d IR renormalon singularity, and we conjecture that they are continuously related as the compactification radius is changed. By making use of transseries and Ecalle's resurgence theory we argue that a non-perturbative continuum definition of a class of field theories which admit semi-classical expansions may be possible.

Figures (7)

• Figure 1: A cartoon of the leading topological defects and molecules in the small-$S^1$ domain. Gray and white circles represent monopole-instantons and anti--monopole-instantons. Unpaired arrows represent fermion zero modes, paired monopole-instanton events are magnetic and neutral bions. See text for explanations.
• Figure 2: Gauge holonomy eigenvalues $\exp\{2\pi i {\varphi}_j\}$ for the classical Lie algebras at rank $N=9$. The red circles are the ${\varphi}^*$ predicted minima and the black "+"'s mark the values found numerically. The predicted minima are exact for $A_N$ and $D_N$, and thought to be correct only in the large-$N$ limit for $B_N$ and $C_N$.
• Figure 3: Gauge cells for the rank-2 Lie algebras in the coordinates of appendix \ref{['secB']}, shaded according to the values of the 1-loop potential. Green and red lines enclose fundamental domains for the action of the center $Z({\widetilde{G}})$ on ${\widehat{T}}$, red lines or dots are points of unbroken center symmetry, and blue dots are the minima of the 1-loop potential. The $B_2$ and $C_2$ cases are equivalent, but are expressed in different coordinate systems.
• Figure 4: Typical eigen-spectrum of the small-fluctuation operator (a) for a monopole-instanton, and (b) for a topological molecule, e.g., a magnetic bion. To get the correct prefactor for the magnetic bions, the quasi-zero modes integrals need to be done exactly.
• Figure 5: The integral over the quasi-zero mode---the separation between ${\mathcal{M}}_i$ and $\overline{\mathcal{M}}_j$---is dominated by separations $r \sim L/g^2$. In Euclidean space for $i\neq j$ the interaction between the two monopole-instantons is repulsive at short distances due to Coulomb repulsion and attractive at long distances due to fermion zero-mode exchange, leading to the stable saddle.