Semiclassics, Resurgence, and Transseries

Semiclassics expands around saddles; resurgence studies how perturbative large order encodes other sectors; a transseries supplies the exponentials, powers, logarithms, and Stokes parameters needed to combine them. These methods can turn a divergent expansion into precise nonperturbative information, but only after the integration cycle, relevant saddles, Borel singularities, and observable are specified. Evidence from quantum mechanics and selected QFT regimes does not establish universal resurgent completeness of four-dimensional QFT.

Evidence cutoff. 11 August 2026.

Required background. Borel singularities, lateral sums, and Stokes data supplies the analytic ambiguity; transseries sectors and parameters supplies the completion; complex saddles and Lefschetz thimbles supplies contour geometry. Helpful background. Ambiguity cancellation supplies the main consistency test, renormalized saddle validity supplies QFT error checks, and evidence and limits for resurgence in QFT fixes the claim ceiling.

Saddles, large order, and nonperturbative sectors

Target observables include energy levels, partition functions, vacuum energies, condensates, mass gaps, Wilson loops, and large-order coefficients. A typical completion has the form

$$\mathcal O(g)\sim\sum_{k\ge0}\sigma^k e^{-kA/g}g^{\beta_k} \sum_{n\ge0}a_{k,n}g^n,$$

where the integration cycle and boundary conditions fix the transseries parameter $\sigma$. A formal list of sectors is not a result until lateral Borel sums and Stokes jumps combine into an unambiguous observable.

Method	Required inputs and best use	Dominant limitation
Real-saddle semiclassics	Small coupling or $\hbar$, isolated finite-action saddles, fluctuation determinants	Misses complex saddles; zero/negative modes and collective coordinates
Picard–Lefschetz analysis	Complexified fields, original integration cycle, thimble intersections	Infinite-dimensional contour definition and Stokes jumps in QFT
Borel–Padé or conformal resummation	Many perturbative coefficients and Borel-plane information	Finite-order pole artifacts; non-Borel-summable directions require lateral prescriptions
Full transseries/resurgence	Actions, characteristic exponents, Stokes constants, bridge equations	Completeness of sectors and determination of parameters are rarely proved in QFT
Small-circle/bion program	Center-stabilized compactification, weak coupling, continuity to target regime	Phase transitions or loss of adiabatic continuity block extrapolation

Bogomolny and Zinn-Justin showed in quantum-mechanical tunneling problems that ambiguities of non-Borel-summable perturbation theory cancel ambiguities of instanton–anti-instanton sectors (Bogomolny 1980; Zinn-Justin 1981). This is a benchmark for the mechanism, not a proof that the saddle catalog of an arbitrary QFT is complete.

Picard–Lefschetz theory makes the dependence on the original integration cycle explicit; Witten 2011 develops this structure for analytically continued Chern–Simons theory. In suitable finite-dimensional and quantum-mechanical problems, perturbative data can also recover instanton information once the correct analytic continuation is fixed (Serone, Spada, and Villadoro 2017). Neither result supplies a universal QFT saddle-completeness theorem.

In compactified two-dimensional $CP^{N-1}$, Dunne and Ünsal relate neutral bions, renormalon-scale ambiguities, and mass-gap data under an adiabatic-continuity hypothesis (Dunne and Ünsal 2012). The construction is a controlled QFT laboratory. Extending the microscopic renormalon–saddle correspondence to uncompactified four-dimensional Yang–Mills remains conjectural.

Error budgets, benchmarks, and independence

Errors separate into perturbative truncation, numerical Borel reconstruction, missing singularities, fluctuation-loop truncation, multi-saddle interactions, finite volume, and continuation from a deformed regime. The smallest term is not automatically the error: an omitted saddle with smaller action than the retained one dominates. A Stokes line is not a breakdown of the observable; it is where the decomposition into sectors changes and ambiguity cancellation must be demonstrated.

Strong benchmarks include zero-dimensional integrals with known contours, the anharmonic oscillator, double-well and periodic quantum mechanics, supersymmetric/localized observables with exact answers, two-dimensional sigma models, and selected matrix models. Large-order coefficients predicted from a saddle and a numerical match of the same coefficients share analytic inputs; more independent checks use exact quantization, direct Hamiltonian diagonalization, lattice data, or localization not used to fit the Stokes constants.

Known failure modes include choosing a thimble not homologous to the original contour, omitting complex or complex-bion saddles, treating Padé poles as physical singularities, fitting too many transseries parameters, ignoring renormalization and quasi-zero modes, and extrapolating across a phase transition. In gauge theory, gauge fixing and collective-coordinate measures add further obligations.

Claim ceiling. Resurgence can prove or strongly support an exact completion in finite-dimensional integrals, quantum mechanics, and special QFT observables. In a general interacting QFT it is a systematically testable framework, not yet a nonperturbative definition or proof of saddle completeness.

Choosing a semiclassical or resurgent route

Situation	Preferred start	Required cross-check
One dominant real saddle, small parameter	Conventional semiclassics	Negative/zero modes, next saddles, and direct numerics
Factorial perturbative growth with known coefficients	Borel analysis and conformal/Padé reconstruction	Stability under order and map; locate singularities independently
Non-Borel-summable physical direction	Lateral sums plus explicit nonperturbative sectors	Cancellation of imaginary ambiguities and reality of the result
Multiple complex saddles or parameter continuation	Picard–Lefschetz and transseries	Original-cycle intersection numbers across Stokes walls
Strongly coupled target reached from a small circle	Bion/compactification program	Center realization and absence of a phase transition along the path

Related routes are nonperturbative gauge dynamics, functional equations and functional RG, the Yang–Mills mass-gap dossier, and proposed confinement mechanisms.

Evidence cutoff and change criteria

The finite selection covers ambiguity cancellation, thimbles, transseries in quantum mechanics and QFT, compactified sigma models, and constructive Borel boundaries. Targeted arXiv, journal, mathematical-index, and citation-chain searches covered public evidence through 11 August 2026. Reassessment is triggered by a proved saddle-completeness theorem, a verified missing sector, failure of ambiguity cancellation, a phase transition invalidating continuity, or an independent exact/lattice benchmark.

References

• Bogomolny, E. B. “Calculation of Instanton–Anti-Instanton Contributions in Quantum Mechanics.” Physics Letters B 91 (1980): 431–435. DOI.
• Dunne, Gerald V., and Mithat Ünsal. “Resurgence and Trans-Series in Quantum Field Theory: The $CP^{N-1}$ Model.” Journal of High Energy Physics 2012, no. 11 (2012): 170. arXiv.
• Serone, Marco, Gabriele Spada, and Giovanni Villadoro. “Instantons from Perturbation Theory.” Physical Review D 96 (2017): 021701. DOI.
• Witten, Edward. “Analytic Continuation of Chern–Simons Theory.” AMS/IP Studies in Advanced Mathematics 50 (2011): 347–446. arXiv.
• Zinn-Justin, Jean. “Multi-Instanton Contributions in Quantum Mechanics.” Nuclear Physics B 192 (1981): 125–140. DOI.