Evidence and Limits for Resurgence in QFT

Resurgent structure in QFT has several sharply different levels of support. It is proved for selected constructive models and localized observables under exact hypotheses, calculationally established in some quantum-mechanical and lower-dimensional systems, controlled semiclassically in specified compactifications, and conjectural as a universal completion of generic four-dimensional QFT. A responsible claim names the observable, regulator, regime, boundary data, and independent check before assigning it any of those descriptions.

Required background. Large-order growth and the Borel transform supplies the coefficient convention; Borel singularities, lateral sums, and Stokes data supplies directional resummation; transseries sectors and parameters supplies the global data; instantons and large-order relations supplies the saddle test; renormalons, OPE ambiguities, and transseries supplies the momentum-region boundary; ambiguity cancellation and transseries consistency supplies the cancellation test; and complex saddles, quasi-zero modes, and hidden phases supplies the contour-dependent correlated sectors.

Helpful background. Correlated evidence, independence, and triangulation explains why shared perturbative input cannot be counted as an independent confirmation.

Evidence classes attach to a named claim

The comparison below uses sources checked through 2026-08-09. It is not an undated assertion about “resurgence in a theory.” Each row is limited to the object stated.

Setting	Named object	Support justified here	Essential boundary
Finite-dimensional Laplace integrals	Integral on a declared cycle	Exact steepest-descent and Borel reconstruction in analytic examples	Other cycles and singular actions can change the sector set
Rank-one analytic ODE systems	Sectorial solutions under theorem hypotheses	Rigorous transseries and Borel-summation results	Nonlinear system, rank, nonresonance, and sector hypotheses are essential
Symmetric double-well quantum mechanics	Energy levels with a global quantization condition	Multi-instanton transseries, large-order matching, and ambiguity cancellation established model by model	Not a theorem for arbitrary potentials or field theory
Massive weak-coupling $P(\phi)_2$ and $\phi^4_3$	Specified Euclidean Schwinger functions or free energy	Constructive Borel theorems	Dimension, stability, mass, renormalization, observables, and uniform limits are fixed
Localized supersymmetric gauge theories	Sphere partition functions and selected localized observables, sector by sector	Borel summability proved for the classes and contours in the localization analysis	Requires the stated Lagrangian theory, supersymmetry, manifold, localization formula, and observable
Two-dimensional integrable asymptotically free models	Ground-state energy in an external field and related thermodynamic quantities	Exact integral equations plus analytic/high-order renormalon evidence	Does not automatically cover arbitrary correlators
Weakly coupled $\mathbb R^3\times S^1$ gauge regimes	Monopole and bion amplitudes at specified holonomy	Controlled semiclassical sectors and ambiguity tests	Matter, global form, boundary conditions, circle size, and continuity assumptions remain explicit
Four-dimensional infinite-volume QCD or pure Yang–Mills	Generic observables or a universal transseries	Renormalon/OPE diagnostics and selected high-order evidence; universal completion unresolved	No general proof identifies all Borel singularities, sectors, or parameters

“Exact” in the table modifies the named object, not every interpretation built from it. “Controlled” means an expansion parameter and a breakdown test are explicit. “Evidence” means a quantitative pattern has survived checks but is not a theorem.

Results with theorem-level hypotheses

There are at least three distinct theorem types, none of which should be merged.

1. Asymptotic ODE theorems. Costin proves Borel summation and Stokes results for rank-one nonlinear systems under analytic and nonresonance conditions; see Costin 1998, §1 and Theorem 1. The conclusion is sectorial and equation-specific.
2. Constructive scalar QFT. The weakly coupled massive two-dimensional $P(\phi)$ and three-dimensional $\phi^4$ results identify named Euclidean functions with their Borel sums. Their hypotheses are detailed on constructive Borel summability and its boundaries.
3. Localization-based gauge-theory results. For classes of four-dimensional $\mathcal N=2$ and five-dimensional $\mathcal N=1$ Lagrangian theories, Honda analyzes localized matrix integrals and proves Borel summability of perturbative series for specified observables in fixed instanton sectors; see Honda 2016, §§2–4. Summing those sectorwise results uses the exact localization formula. It is not a theorem about nonsupersymmetric QCD.

These theorems differ in object, geometry, and proof. The shared word “Borel” does not make their scopes interchangeable.

Model-verified and controlled mechanisms

In the symmetric double well, the instanton action, global spectral condition, multi-instanton logarithms, large-order coefficients, and lateral cancellation can all be compared. The evidence is exceptionally strong for that Hamiltonian, and parts admit rigorous spectral analysis. The transferable lesson is the mechanism; the numerical Stokes constants and sector content remain model dependent.

Two-dimensional integrable field theories provide a different check. Exact Bethe-ansatz integral equations generate very high perturbative orders for a ground-state energy in an external field. The resulting growth agrees with predicted renormalon locations and subleading beta-function dependence in several models; see Mariño and Reis 2020, §§2–5. This is strong, observable-specific field-theory evidence. It is not a direct construction of every local correlator’s transseries.

On a small circle, a center-symmetric holonomy can Abelianize selected asymptotically free gauge theories. Monopole-instantons and magnetic or neutral bions then have computable actions, zero modes, and quasi-zero-mode contours. Ambiguity cancellation is controlled when the semiclassical hierarchy and diluteness tests pass. The proposed identification of neutral-bion scales with infinite-volume renormalons is a further continuity claim, explicitly conjectural in Argyres and Ünsal 2012.

Compactified control does not equal the four-dimensional limit

Compare two claims:

$$\begin{array}{ll} \text{A:}& \text{At }L\Lambda\ll1\text{ with specified holonomy, a neutral-bion}\\ &\text{ambiguity cancels a perturbative ambiguity in observable }\mathcal O;\\[2mm] \text{B:}& \text{The same sector is the complete leading renormalon mechanism}\\ &\text{for }\mathcal O\text{ on }\mathbb R^4. \end{array}$$

Claim A can be a controlled semiclassical calculation. Claim B additionally needs adiabatic continuity, control of the $L\to\infty$ limit, and exclusion or incorporation of sectors that become important before that limit. Matching an exponential scale is supporting evidence, not proof of continuity.

At large $N$, uniform circle control can require the stronger condition $NL\Lambda\ll1$, not merely fixed-$N$ control at $L\Lambda\ll1$. Matter representation, fermion boundary conditions, center symmetry, and gauge-group global form can change the allowed constituents. A claim that omits them cannot inherit the controlled status of the compactified calculation.

Unresolved generic-QFT statements

For ordinary four-dimensional QCD and pure Yang–Mills, perturbative coefficients, large-$\beta_0$ calculations, OPE power matching, lattice perturbation theory, and continuum models provide valuable information. They do not currently supply a general theorem that:

• analytically continues every observable’s Borel transform;
• classifies every singularity and its Stokes data;
• derives every exponential sector from a path-integral cycle;
• fixes all transseries parameters from first principles;
• proves cancellation to all orders and sectors; or
• constructs the continuum theory through that transseries.

Accordingly, “QCD is resurgent” is too broad to receive one status. A claim about an Adler-function renormalon, a compactified bion amplitude, a localized supersymmetric partition function, and a generic hadronic observable are different propositions.

The Borel and transseries map is the chapter’s common convention map. The evidence triangulation graph exposes shared inputs before methods are counted as independent, while the exact and rigorous status comparison provides the exact-model and theorem comparison used here.

A five-question claim test

Before accepting a resurgent QFT statement, ask:

1. Observable: Which correlator, energy, partition function, or Wilson coefficient is expanded?
2. Definition: What regulator, renormalization scheme, state, geometry, and order of limits define it?
3. Analytic relation: Are the Borel singularity, Stokes jump, and large-order coefficients all normalized and mutually checked?
4. Nonperturbative sector: Is it derived from an equation, boundary data, or contributing integration cycle, including zero and quasi-zero modes?
5. Independent check: Is there an exact equation, held-out coefficient range, numerical spectrum, theorem, or alternative regulator that could falsify the relation?

A quantum-mechanical analogy answers none of these questions for a QFT automatically. A high-order fit without a sector calculation and a sector calculation without late coefficients are complementary, not duplicate, evidence.

Common pitfalls

Using “proved in QFT” without the observable and model. A localized sphere partition function and an infinite-volume hadronic correlator have different definitions and theorems.

Counting internal consistency as independent confirmation. If the nonperturbative normalization was fitted from the same perturbative coefficients used to test it, the agreement is not held out.

Treating a successful compactification as a harmless regulator. The circle and its boundary conditions can change saddles, symmetry realization, and infrared dynamics; continuity is a scientific claim to test.

Exercises

1. Classify the claim “the $u=2$ singularity in an Adler-like observable has the right $(\Lambda/Q)^4$ scale” using the five questions above.

Solution

It is a normalized OPE-consistency statement once the Borel and beta-function conventions are fixed. It identifies the power expected for a dimension-four contribution. By itself it neither proves the singularity’s exact residue and branch type nor identifies a unique semiclassical sector, so it remains diagnostic evidence rather than a universal transseries construction.

2. A small-circle calculation finds exact cancellation through order $e^{-2S_0}$, and the mass gap remains nonzero as the circle is enlarged numerically over a finite range. What additional evidence is needed for an $\mathbb R^4$ mechanism claim?

Solution

One needs control or independent evidence through the region where the semiclassical expansion fails, preservation of the relevant symmetry realization and holonomy, a demonstrated connection to the same observable, and a limit study that excludes a phase transition or sector rearrangement. Finite-range smoothness supports continuity but does not prove it.

References

• Argyres, Philip C., and Mithat Ünsal. “The Semi-Classical Expansion and Resurgence in Gauge Theories: New Perturbative, Instanton, Bion, and Renormalon Effects.” Journal of High Energy Physics 08 (2012): 063. arXiv:1206.1890; doi:10.1007/JHEP08(2012)063.
• Costin, Ovidiu. “On Borel Summation and Stokes Phenomena for Rank-1 Nonlinear Systems of Ordinary Differential Equations.” Duke Mathematical Journal 93 (1998): 289–344. doi:10.1215/S0012-7094-98-09311-5.
• Honda, Masazumi. “Borel Summability of Perturbative Series in 4d $\mathcal N=2$ and 5d $\mathcal N=1$ Theories.” Physical Review Letters 116 (2016): 211601. arXiv:1603.06207; doi:10.1103/PhysRevLett.116.211601.
• Mariño, Marcos, and Tomás Reis. “Renormalons in Integrable Field Theories.” Journal of High Energy Physics 04 (2020): 160. arXiv:1909.12134; doi:10.1007/JHEP04(2020)160.