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arxiv: 2501.03105 · v1 · submitted 2025-01-06 · ✦ hep-th · hep-ph· math-ph· math.MP

Resurgence of the Tilted Cusp Anomalous Dimension

Gerald V. Dunne This is my paper

Pith reviewed 2026-05-23 06:04 UTC · model grok-4.3

classification ✦ hep-th hep-phmath-phmath.MP
keywords resurgencecusp anomalous dimensionBES equationsweak-strong coupling interpolationnon-perturbative methodsintegrable systemsanomalous dimensions
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The pith

Resurgent extrapolation from BES perturbative series recovers the full non-perturbative tilted cusp anomalous dimension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that resurgent extrapolation and continuation methods applied to the weak-coupling and strong-coupling perturbative expansions of the tilted cusp anomalous dimension, both generated from the BES equations, produce accurate analytic information across all coupling values. This yields a smooth interpolation between the two limits while locating the singularities that control the radius of convergence of the weak-coupling series and the asymptotic character of the strong-coupling series. A sympathetic reader would care because the input consists solely of perturbative data, yet the output matches the expected physical non-perturbative structure without additional assumptions.

Core claim

Applying resurgent extrapolation and continuation to the perturbative series obtained from the BES equations extracts detailed analytic information about the tilted cusp anomalous dimension, enabling accurate interpolation between weak and strong coupling while identifying the governing singularities.

What carries the argument

resurgent extrapolation and continuation methods applied to the weak-coupling and strong-coupling perturbative series generated by the BES equations

If this is right

  • A smooth, accurate interpolation function between the weak and strong coupling regimes is obtained.
  • The singularities that set the finite radius of convergence of the weak-coupling expansion are located.
  • The asymptotic character of the strong-coupling expansion is clarified by the same singularity structure.
  • The extracted non-perturbative values agree with the underlying physical structure of the anomalous dimension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same resurgent pipeline could be applied to other observables whose perturbative series are known from integrability but whose non-perturbative completions are not.
  • Intermediate-coupling benchmarks from lattice or other numerical methods would provide an independent test of the extracted singularity locations.

Load-bearing premise

The perturbative expansions generated from the BES equations contain sufficient information for resurgent extrapolation and continuation to recover the correct non-perturbative structure of the tilted cusp anomalous dimension.

What would settle it

A high-precision numerical evaluation of the tilted cusp anomalous dimension at an intermediate coupling strength that deviates from the resurgent interpolation result would falsify the claim.

Figures

Figures reproduced from arXiv: 2501.03105 by Gerald V. Dunne.

Figure 1
Figure 1. Figure 1: FIG. 1. Pad´e poles for the weak coupling expansions for the cusp [left] and hex [right], in the complex [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Extrapolation of the weak coupling expansion of the cusp ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Extrapolation of the weak coupling expansion of the hex ( [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The Darboux test ratios for the cusp [left] and hex [right], showing the ratio of the weak-coupling coefficients to the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The ratio on the left-hand-side of ( [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The ratio on the left-hand-side of ( [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The combination on the left-hand-side of ( [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Ratio tests using expression ( [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The Pad´e poles of the Borel transform, for [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The Pad´e poles (blue dots) in terms of the conformally mapped variable [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Rescaled Pad´e poles in the [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The poles in the new [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
read the original abstract

We use resurgent extrapolation and continuation methods to extract detailed analytic information about the tilted cusp anomalous dimension solely from its weak coupling and strong coupling expansions. This enables accurate and smooth interpolation between the weak and strong coupling limits, and identifies the relevant singularities governing the finite radius of convergence of the weak coupling expansion and the asymptotic nature of the strong coupling expansion. The input data is purely perturbative, generated from the BES equations, and these resurgent methods extract accurate non-perturbative information which matches the underlying physical structure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript applies resurgent extrapolation and continuation methods to the weak-coupling and strong-coupling perturbative series of the tilted cusp anomalous dimension generated from the BES equations. It claims these methods recover accurate non-perturbative information matching the underlying physical structure, identify governing singularities, and enable smooth interpolation between the two regimes using purely perturbative input.

Significance. If the central claim holds, the work illustrates how resurgence can extract non-perturbative data from perturbative expansions in an integrable AdS/CFT observable, providing a concrete example of analytic continuation across coupling regimes without direct non-perturbative input. This could strengthen the case for resurgence as a practical tool in gauge-theory anomalous dimensions.

minor comments (2)
  1. [Abstract] The abstract asserts 'accurate' extraction and matching to physical structure but supplies no explicit error estimates, convergence checks, or side-by-side comparison tables; adding these in a dedicated results section would strengthen verifiability.
  2. Clarify the precise resurgent procedure (e.g., which Borel singularities are retained, how the transseries is truncated) so that the interpolation can be reproduced from the quoted BES series alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of our manuscript. The report recommends minor revision but lists no specific major comments, so we have no point-by-point responses to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper generates input series from the established BES equations and applies resurgent extrapolation/continuation to recover non-perturbative information. The abstract states the input is 'purely perturbative, generated from the BES equations' and presents the match to physical structure as an output validation. No quoted equations or steps reduce the claimed result to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of resurgence theory to the BES-generated series and on the assumption that the resulting continuation reproduces the correct physical non-perturbative structure.

axioms (1)
  • domain assumption Resurgence theory applies to the perturbative expansions of the tilted cusp anomalous dimension
    Invoked when the paper states that resurgent extrapolation extracts non-perturbative information from the weak and strong coupling series.

pith-pipeline@v0.9.0 · 5602 in / 1159 out tokens · 35732 ms · 2026-05-23T06:04:41.363813+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Strong coupling structure of $\mathcal{N}=4$ SYM observables with matrix Bessel kernel

    hep-th 2026-02 unverdicted novelty 6.0

    Reorganizing the transseries of matrix Bessel kernel determinants at strong coupling yields a simple structure where non-perturbative corrections are directly determined by the perturbative series for N=4 SYM observables.

  2. Introductory Lectures on Resurgence: CERN Summer School 2024

    hep-th 2025-11 unverdicted novelty 2.0

    Introductory lectures cover resurgent asymptotics using examples like the Airy function, nonlinear Stokes phenomenon, Heisenberg-Euler action, and resurgent continuation.

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages · cited by 2 Pith papers · 8 internal anchors

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    This is straightforwardly identified numerically by Pad´ e approximants: see Figure

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    And since there are no singularities in the direction of g2 positive, the Pad´ e approximants also provide a simple -1.0 -0.5 0.5 Re[g2 ] -1.0 -0.5 0.5 1.0 Im[g2 ] -1.0 -0.5 0.5 Re[g2 ] -1.0 -0.5 0.5 1.0 Im[g2 ] FIG. 1. Pad´ e poles for the weak coupling expansions for the cusp [left] and hex [right], in the complex g2 plane. In each case the poles accumu...

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    The left-hand plot shows the extrapolation out to g2 = 1, and the right-hand plot extends out to g2 = 10

    based on 24 terms of the weak coupling expansion. The left-hand plot shows the extrapolation out to g2 = 1, and the right-hand plot extends out to g2 = 10. The blue curve is the 24-term weak coupling series expansion, whose breakdown at the radius of convergence, g2 = 1 16, can be clearly seen. The red curves plot the first 3 terms of the (divergent) stro...

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    The left-hand plot shows the extrapolation out to g2 = 1, and the right-hand plot extends out to g2 = 10

    based on 24 terms of the weak coupling expansion. The left-hand plot shows the extrapolation out to g2 = 1, and the right-hand plot extends out to g2 = 10. The blue curve is the 24-term weak coupling series expansion, whose breakdown at the radius of convergence, g2 = 1 16, can be clearly seen. The red curves plot the first 3 terms of the (divergent) stro...

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    Make a conformal map of the cut complex g2 plane to the interior of the unit disk in the z plane 16g2 = 4z (1 − z)2 ← → z = p 1 + 16g2 − 1p 1 + 16g2 + 1 (13) and re-expand about z = 0 to the same number of terms as the original expansion (this is provably optimal [9])

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    Make a Pad´ e approximant inside the unit disk in thez plane

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    The black dots in Figures 2 and 3 illustrate the improvement out to large g2

    Finally, map this Pad´ e approximant back to theg2 plane via the inverse conformal map. The black dots in Figures 2 and 3 illustrate the improvement out to large g2. The coefficients of the weak coupling expansion also encode information about the nature of the singularity at g2 = − 1

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    This follows from Darboux’s theorem [13], which states that the large order behavior of the coefficients of 4 the expansion about the origin encodes information about the expansion near the singularities. For example, numerical analysis from 24 terms of the weak coupling expansions suggests the leading large order behavior (for 0 < a < 1 2): bn(a) ∼ C (−1...

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    Note that expression (15) is also consistent with the fact that the limiting values for a → 0 and a → 1 2 are special

    The constant C depends on a in a non-trivial way. Note that expression (15) is also consistent with the fact that the limiting values for a → 0 and a → 1 2 are special. The singularities for a = 0 are logarithmic bn(0) = 8 π2 (−16)n (n + 1)(1 − 2−2n−2)ζ2n+2 ⇒ Γ0 = − 2 π2 log Γ 1 2 − 2ig Γ 1 2 + 2ig /π (16) And for a = 1 2 the tilted cusp becomes regular: ...

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    Note that sk(a) develops an order k pole at a = 1 2, so the limit a → 1 2 is most naturally studied in a double-scaling limit [7, 20]

    Here the sk(a) are functions of the tilt parameter a [1]: sk+1(a) = ψk(1) − ψk 1 2 + a + (−1)k ψk(1) − ψk 1 2 − a (19) where ψk(z) := d dz k+1 ln Γ(z). Note that sk(a) develops an order k pole at a = 1 2, so the limit a → 1 2 is most naturally studied in a double-scaling limit [7, 20]. The first few strong-coupling expansion coefficients are [1]: Γa = 2a ...

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    (21) Γhex ∼ 4ξ 3 √ 3 π 1 − ψ1 1 6 − ψ1 5 6 12ξ2 − 5ζ3 ξ3 +

    and octagon ( a = 0) cases we have (here K is the Catalan constant): Γcusp ∼ ξ 2π 1 − K ξ2 − 27ζ3 32ξ3 + . . . (21) Γhex ∼ 4ξ 3 √ 3 π 1 − ψ1 1 6 − ψ1 5 6 12ξ2 − 5ζ3 ξ3 + . . . ! (22) Γoct = ξ π2 + 2 π2 ∞X k=1 (−1)k+1 k e−k ξ 4k (23) The strong coupling expansion (17) is generically an asymptotic expansion, with factorially growing coefficients, as discuss...

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    The blue dots denote the raw combination, and the black dots show the 5th order Richardson acceleration, converging to (28). Given A, the next term in (25) identifies β as n 1 − A cn+1 n cn ∼ β + O 1 n (27) We find that β is the same linear function of the tilt parameter a (see Figure 6): β = (1 − 2a) (28) Given A and β, the next term in (25) determines t...

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    Pad´ e-Borel Analysis The numerical goal is to learn as much precise information as possible about the singularities of the Borel transform, as these correspond to the non-perturbative physics. But since the strong coupling expansion is truncated at a finite order, this means we are trying to probe the singularities of the Borel function given only a poly...

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    Pad´ e-Conformal-Borel Analysis The procedure of the Pad´ e-Conformal-Borel analysis is a simple generalization of the Pad´ e-Borel method described in the previous Section, but it has two distinct advantages [8]. First, it reveals the existence of higher singularities that might be hidden among the Pad´ e poles that appear to be representing a branch cut...

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    It is significantly more accurate to do the Pad´ e analysis in the Borel plane than in the original physical variable [8]

    First, go to the Borel plane. It is significantly more accurate to do the Pad´ e analysis in the Borel plane than in the original physical variable [8]

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    Make a conformal map from the Borel variable ζ to a new variable, z, which maps the cut Borel plane to the interior of the unit disk. In practice, one does not know the entire singularity structure in the Borel plane, but it is already an excellent approximation to use a conformal map based on the knownleading Borel singularities. For example, given the i...

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    Re-expand the truncated Borel transform in z to the same order (this is provably optimal [9])

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    Remarks: • The chosen conformal map only depends on the location of the branch points, not on their associated exponents

    Map back to the Borel ζ plane using the inverse conformal map. Remarks: • The chosen conformal map only depends on the location of the branch points, not on their associated exponents. • If the conformal map happens to be based on all the actual ζ singularities of the Borel transform, then by construction there cannot be any singularities inside the unit ...

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    Recall Figure 9

    Results of Pad´ e-Conformal-Borel Analysis We use the conformal map (44) based on two cuts along the real axis of the Borel plane, emanating from −2, and from (1 − 2a). Recall Figure 9. The resulting Pad´ e poles in the conformally mapped z plane are shown in Figure 10 for a chosen tilt parameter a = 1

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    This analysis can be done for any a, but this value is convenient for a reason that will become clear below. The first thing we learn from this Pad´ e-Conformal-Borel analysis is that there is a new singularity on the positive real Borel axis at ζ = (1 + 2a), which corresponds to a pair of complex conjugate points on the unit circle in the z plane, and wh...

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    Singularity Elimination Singularity elimination is a powerful method to probe higher Borel singularities [9]. The application of a linear operator followed by a suitable conformal map completely removes a chosen singularity, thereby enabling access to the fluctuations near the location of the removed singularity, and also access to higher Riemann sheets. ...

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    Poles appear outside the unit disk, accumulating to points on the unit circle, which are the z-plane images of true Borel singularities lying on the cuts along the real ζ axis in the Borel plane. The singularities for Re(ζ) > 0 are shown in red, and those for Re(ζ) < 0 are shown in green. The leading singularity at ζ = (1 − 2a) = 3 4 maps to z = +1, while...

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    This singularity has exponent β = 1 − 2 × 1 8 = 3 4

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    See [9], equation (30)

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    See Figure 1 in [11], and Figure 2 in [12]

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