Borel Resummation of Gevrey Series

• Borel resummation of Gevrey series is a method that reconstructs analytic functions from divergent asymptotic expansions, capturing essential nonperturbative effects.
• It employs Borel-Padé approximants and Darboux analysis to accurately identify singularity structures like poles and branch points from a limited number of series terms.
• The approach integrates resurgent transseries and Borel-Écalle medianization, yielding robust analytic approximants with direct applications in quantum field theory and beyond.

Borel resummation of Gevrey series is a central analytic method for extracting nonperturbative content from divergent asymptotic expansions arising as formal solutions to nonlinear ordinary differential equations (ODEs), particularly those of resurgent type relevant in quantum field theory. The core challenge is to reconstruct, from a truncated Gevrey-1 series (with coefficients growing as $n!$), an analytic function or transseries that captures the correct singularity structure in the Borel plane, and to do so robustly even when only a few terms are available. The methodology synthesizes Borel-Padé approximants, Darboux analysis, resurgent approximant construction, and Borel-Écalle medianization to obtain accurate, practical, and resurgent-consistent resummations from finite data.

1. Structure of Divergent Gevrey Series from Nonlinear ODEs

Nonlinear ODEs of the type

$$y'(x) = -\lambda y(x) - A \frac{y(x)}{x} + f(x) + \sum_{n\geq 2, m} k_{n,m} y^n x^{-m}$$

admit formal asymptotic solutions $y(x) \sim \sum_{n} a_n x^{-n}$, where the coefficients $a_n$ exhibit $n!$ growth, identifying the series as Gevrey-1. This divergence is rooted in non-linear effects, such as renormalons in QFT, and leads to singularities in the Borel transform $$B[y](z) = \sum_{n=1}^{\infty} \frac{a_n}{(n-1)!} z^n$$. The singularity locations—typically at $z = \lambda n$ ($n \in \mathbb{N}$)—and their nature are determined by the ODE parameters, with branching indices or pole orders set by the coefficient $A$.

2. Analytic Structure Determination: Borel-Padé Approximation and Darboux Analysis

Given limited terms of the asymptotic Gevrey series, the analytic structure in the Borel plane can be inferred using two principal methods:

• Borel-Padé Approximants: Rational (Padé) approximants are constructed for the truncated Borel transform. Diagonal Padé approximants efficiently locate simple poles, as in Euler-type ODEs. For algebraic branch points, such as $\sqrt{1-z}$, direct Padé approximation is ineffective, but Padé applied to the derivative of the logarithm of the Borel transform exposes branch points via residue analysis; e.g.,

$$\frac{d}{dz} \log \left( \frac{1}{\sqrt{1-z}} \right) = \frac{1}{2(1-z)}.$$

• Darboux Analysis: Large-order behavior of $n$th Borel coefficients $B_n$ provides direct access to the singularity index $b$ via the ratio test:

$$\frac{B_n}{B_{n-1}} = 1+\frac{b-1}{n} + \frac{(b-1)s}{n(b-n-2)} + \mathcal{O}\left(\frac{1}{n^3}\right).$$

Even with as few as 10 coefficients, this method refines both singularity position and branching exponent (or pole order) estimates.

3. Construction of Resurgent Approximants

Once singularity positions and types are diagnosed, an analytic resurgent approximant is built to match the expected Borel-plane structure:

• Pole-type Singularities: Approximant

$$P(z) = \frac{\sum_{n=0}^N c_n z^n}{\prod_{n=1}^{N'} (n-z)}$$

with numerator coefficients $c_n$ fitted to the truncated expansion, $N'$ being the number of expected poles.

• Branch-point Singularities: For algebraic singularities,

$$B(z) = \frac{\sum_{n=0}^N c_n z^n}{\prod_{n=1}^{N'} (n-z)^{b}}$$

and for square-root types ($b=1/2$),

$$B(z) = \frac{\sum_{n=0}^N c_n z^n}{\prod_{n=1}^{N'} \sqrt{(n-z)}}.$$

This construction ensures the approximant encodes not just the singularity positions but also the correct local monodromy, an essential ingredient for a faithful resurgent analytic structure.

4. Borel-Écalle Resummation: Medianization and Transseries Structure

For series whose Borel transform exhibits singularities along the Laplace integration contour (as is the case for ODEs arising from nontrivial physical problems), traditional Borel-Laplace resummation becomes ambiguous due to Stokes phenomena. The Borel-Écalle framework resolves this via:

• Medianization: A prescription for defining an unambiguous, real-valued sum via the balanced average of lateral Borel sums,

$$\mathcal{A}_{med} := G^{-1/2}\circ \mathcal{A}^- = G^{1/2}\circ \mathcal{A}^+,$$

where $G$ is the Stokes automorphism encoding the discontinuity.

• Transseries Resummation: The solution is not merely a Borel sum but a resurgent transseries,

$$y(x) = \sum_{n=0}^{\infty} C^n e^{-\lambda n x} y_n(x),$$

in which each sector $y_n(x)$ is recursively determined by the bridge equation (alien calculus), connecting discontinuity data to the emergent nonperturbative structure.

For practical truncated data, the paper gives an explicit medianized resurgent resummation (see formula 38) up to quartic in Stokes/discontinuity data, and demonstrates accurate transseries-level approximation for $\mathcal{O}(10)$ initial coefficients.

5. Implementation Workflow and Practical Performance

Algorithmic Pipeline:

Step	Method	Key Formula / Result
1	Truncate asymptotic Gevrey series	$a_n x^{-n}$, $n!$ growth
2	Compute Borel transform	$B[y](z) = \sum a_n/(n-1)! z^n$
3	Infer singularity structure (Padé, Darboux)	Ratio tests, residues
4	Build resurgent analytic approximant	$P(z)$, $B(z)$ with correct singularity
5	Perform Borel-Écalle resummation (medianization, transseries)	Medianized sum with transseries sectors
6	Evaluate Laplace integral	$y_{med}(x) = PV[\ldots] + \text{transseries}$

Empirically, the method permits accurate analytic reconstruction of the ODE solution, robust even with limited input. Numerical examples (see Figures 1–5 in the paper) show rapid convergence and preservation of the transseries structure fundamental to the underlying physical model.

6. Physical and Mathematical Implications

The procedure is immediately applicable to divergent series in quantum field theory, notably those with renormalon-type singularities (arising from renormalization group equations, etc.), where locating and characterizing Borel-plane singularities is essential for disentangling nonperturbative ambiguities. The resummation strategy presented here—centered on resurgent matching—enables practical extraction of controlled, physically meaningful nonperturbative predictions from truncated perturbative inputs, directly addressing limitations of standard operator-product expansions.

The method generalizes to any problem yielding divergent series from a normal form, nonresonant first-order ODE, with ramifications for contexts as diverse as quantum mechanics, Painlevé transcendents, and nonlinear difference equations.

7. Illustrative Example: Euler-Type ODE

As a benchmark, consider the prototypical Euler ODE: $$y'(x) = -y(x) + \frac{1}{x}, \qquad y(x) = \sum_{n} (n-1)! x^{-n}.$$ Its Borel transform is $B[y](z) = (1-z)^{-1}$, with a simple pole at $z=1$. The resurgent Borel sum (after medianization) yields

$y_{res}(x) = e^{-x} \mathrm{Ei}(x) + C e^{-x},$

where $C$ is a real transseries parameter capturing nonperturbative ambiguity.

Conclusion

The approach articulated in (Maiezza et al., 30 Sep 2024) establishes a concrete and generalizable workflow for Borel resummation of Gevrey series generated by nonlinear ODEs using Borel-Padé analysis, Darboux singularity extraction, and resurgent-consistent analytic approximants. The method achieves accurate nonperturbative analytic reconstruction from modest data and is tailored to the singularity structure of the underlying equations, ensuring compatibility with resurgent QFT frameworks and opening avenues for systematic application in various fields requiring extraction of analytic content from finite, divergent expansion data.