Resurgence and Transseries

Choose a route by naming the observable, the asymptotic parameter, and the strength of conclusion you need. Begin with the Borel transform when the question starts from coefficients; use lateral sums when a Borel singularity lies on the integration ray; introduce transseries sectors only after their exponential weights and free data have a mathematical or physical origin; and finish with the evidence page before extending a result from an integral, quantum mechanics, or a compactified model to a generic quantum field theory.

This chapter connects factorial growth, Borel singularities, Stokes jumps, nonperturbative sectors, instantons, renormalons, complex saddles, quasi-zero modes, and constructive summability. Its central discipline is conditional reasoning: each implication has hypotheses, and ambiguity cancellation is a stringent consistency condition rather than a proof that every sector has been found.

Helpful background. Asymptotic scales, remainders, uniformity, and optimal truncation supplies the meaning of an asymptotic expansion and its least term. Stokes jumps, saddle dominance, and contour dependence supplies the integration-cycle interpretation of changing saddle decompositions.

Enter this chapter

You are ready to choose a short route if you can answer four questions.

• What is the series for? Specify the observable, state or boundary condition, regulator and renormalization scheme, and any volume or continuum limit. If only a string of coefficients is known, begin with large-order growth.
• What convention is being used? State whether the coefficient of $g^n$ is divided by $n!$ or $\Gamma(n+\beta)$, and include the normalization of the Laplace integral. If this is missing, repair it before comparing singularity locations or residues.
• What fixes the nonperturbative sectors? A differential equation, boundary condition, path-integral cycle, or independently calculated saddle must determine their allowed weights and parameters. If the answer is “the perturbative series alone,” continue through transseries sectors and complex saddles.
• What kind of support is claimed? Distinguish an exact integral, a theorem with stated hypotheses, a controlled semiclassical expansion, regulated numerical evidence, and a conjectural correspondence. Use the evidence limits whenever the claim crosses models, observables, dimensions, or orders of limits.

No score is needed. Each uncertain answer identifies a missing input and the page that supplies it.

Choose a route

Goal	Suggested route	Result
Learn the analytic core	Large order → lateral sums → transseries parameters	Convert factorial growth into Borel-plane data and track a Stokes jump without treating it as a physical prescription.
Test an instanton relation	Large order → instantons and late terms → ambiguity cancellation	Match action, power, phase, and normalization in a controlled quantum-mechanical example.
Diagnose a field-theory singularity	Lateral sums → renormalons and the OPE	Derive the bubble-chain factorial moment and match the ambiguity power to an allowed operator dimension.
Follow correlated events	Ambiguity cancellation → complex saddles and quasi-zero modes	Declare the continuation or contour that produces a phase and trace it into the full sector sum.
Ask what is proved	Constructive summability → evidence and limits	Translate theorem hypotheses and keep controlled examples separate from open four-dimensional claims.

The arrows give efficient reading orders. Every leaf states its own hard prerequisites; a suggested route never overrides them.

The logical structure

For the convention used throughout the chapter,

$$\Phi(g)\sim\sum_{n\ge0}a_n g^n, \qquad \widehat\Phi(\zeta)=\sum_{n\ge0}\frac{a_n}{n!}\zeta^n, \qquad \mathcal S_\theta\Phi(g) =\frac1g\int_0^{e^{i\theta}\infty} e^{-\zeta/g}\widehat\Phi(\zeta)\,\mathrm d\zeta.$$

The chapter develops five distinct links.

1. A Gevrey-one coefficient bound gives a Borel transform analytic near the origin. It does not give global continuation.
2. A continued singularity at action $A$ can produce $a_n\sim K\Gamma(n+\beta)/A^{n+\beta}$, with phases retained when several singularities have equal modulus.
3. A singularity on the Laplace ray produces two lateral sums. Their jump is a property of the asymptotic representation; boundary data or an integration cycle chooses the physical combination.
4. An instanton, renormalon, complex saddle, or quasi-zero-mode interpretation requires independent model-specific information. Matching only the exponential scale is insufficient.
5. Cancellation of lateral ambiguities checks the mutual normalization of included sectors. It does not establish existence, uniqueness, completeness, or numerical accuracy of the full theory.

A transseries parameter, conventionally denoted $\sigma$, is therefore not an adjustable fit coefficient. Its allowed value or Stokes transformation is fixed by the equation, boundary condition, or integration cycle of the problem.

Chapter guide

1. Large-Order Growth and the Borel Transform fixes the coefficient and Laplace conventions, derives the local analytic consequence of factorial growth, and solves an exactly Borel-summable integral. Continue here whenever a comparison quotes a Borel singularity without stating the transform convention.
2. Borel Singularities, Lateral Sums, and Stokes Data computes the upper-minus-lower discontinuity of a one-pole model and separates a Stokes jump from a physical discontinuity. Continue to transseries parameters when another sector must absorb the jump.
3. Transseries Sectors and Parameters derives a one-parameter solution in a differential-equation example and shows how the parameter changes across a Stokes ray. Continue to saddle pages only after boundary data have fixed the allowed solution.
4. Instantons and Large-Order Relations derives factorial-over-action growth from a neighboring singularity and tests all normalizations in the symmetric double well. It hands instanton measures and zero modes back to the instanton chapter.
5. Renormalons, OPE Ambiguities, and Transseries derives factorial growth from running-coupling momentum moments, matches the ambiguity power to the OPE, and explains why a bubble chain is a diagnostic rather than the complete Borel transform.
6. Ambiguity Cancellation and Transseries Consistency pairs perturbative and correlated-event ambiguities, including their signs and residues. It identifies exactly what cancellation tests and what it leaves open.
7. Complex Saddles, Quasi-Zero Modes, and Hidden Phases evaluates a subtracted separation integral with an explicit continuation prescription. It rejects any phase claim that omits the saddle, contour, observable, or regime.
8. Constructive Borel Summability and Its Boundaries states a Nevanlinna–Sokal reconstruction criterion and translates constructive scalar-field results into model, dimension, volume, cutoff, stability, and observable hypotheses.
9. Evidence and Limits for Resurgence in QFT compares exact integrals, ODEs, quantum mechanics, constructive models, localized theories, compactified gauge theories, and generic four-dimensional QFT without silently transferring conclusions between them.

Conventions and recurring checks

The chapter inherits the site’s global conventions. Each page declares the observable, coupling convention, analytic sector, lateral direction, and order of limits needed for its claim. The following quantities recur:

• $g$ is the small expansion parameter used in the displayed Borel–Laplace convention; it need not equal a canonically normalized gauge coupling.
• $A$ is a Borel-singularity location or an action difference in the same normalization as $g$. A relation between those meanings must be demonstrated in the model.
• $\beta$ controls the algebraic prefactor and hence the shift in $\Gamma(n+\beta)$.
• $\sigma$ is a transseries parameter fixed by global data, not the coefficient of a freely appended exponential.

Before accepting a claimed relation, check dimensions and normalization, preserve complex phases, identify the nearest singularity or all equal-modulus singularities, and state the sector in $\arg g$. For a QFT calculation, also state the regulator, scheme, observable, volume and continuum limits, and whether the argument controls those limits uniformly.

Review the chapter

Convention check. Suppose $a_n\sim C\,n!A^{-n}$ in the chapter convention. Where is the nearest candidate Borel singularity, and what extra information is needed before calling it an instanton?

Check: the radius points to $|\zeta|=|A|$, while signs or phases locate candidate points on that circle. A saddle solution, action normalization, integration cycle, fluctuation power, and residue test are still required. Repair with the Borel map and instanton normalization test.

Lateral calculation. For $\widehat\Phi(\zeta)=(1-\zeta/A)^{-1}$ with $A,g>0$, calculate $\mathcal S_{0^+}-\mathcal S_{0^-}$ using the chapter Laplace convention.

Check: the pole residue and contour orientation give $2\pi i(A/g)e^{-A/g}$. Repair with the one-pole derivation.

Parameter check. Explain why adding $C e^{-A/g}$ with arbitrary $C$ does not by itself construct a transseries completion.

Check: the equation must permit that sector, and boundary data or the integration cycle must fix $C$ or its Stokes transformation. Repair with the differential-equation example.

Field-theory transfer. A bubble-chain calculation has a positive-axis pole whose ambiguity scales as $Q^{-4}$. What can be concluded?

Check: the scale is compatible with a dimension-four OPE contribution in the declared scheme and convention. The diagnostic alone does not prove the full singularity, a unique saddle origin, or a universal transseries completion. Repair with renormalons and operator dimensions.

Status check. Compare a constructive scalar-field Borel theorem with a semiclassical small-circle gauge-theory calculation. Name at least four hypotheses that cannot be transferred by analogy.

Check: identify the exact model and observable, dimension, stability domain, cutoff and volume limits, analytic coupling sector, circle holonomy, and semiclassical scale hierarchy. Repair with constructive boundaries and QFT evidence limits.

Where to continue

• Return to semiclassical expansions for saddle scaling, thimbles, determinant control, and dilute ensembles.
• Use instantons and tunneling when the next task is the collective-coordinate measure, zero modes, or a Euclidean saddle calculation.
• Use controlled compactification for the monopole and bion mechanisms whose resurgent interpretation is discussed here only within its scale hierarchy.
• Use evidence validation to compare systematic errors and dependence across nonperturbative methods.
• Return to the Nonperturbative Dynamics overview to choose lattice, functional, integrable, strong-coupling, or rigorous-status routes.