Euclidean wormholes stability analysis revisited

Bruno Missoni; Donald Marolf

REVIEW 2 major objections 4 minor 2 cited by

Euclidean wormholes stability analysis revisited

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Axion wormholes stay stable once the apparent divergences in even modes are resolved as artifacts of constraint solving, with finite positive actions after a full treatment.

desk verdict A mostly convincing resolution of the divergent even-mode puzzle, with the pure-axion case on solid ground and the dilaton case carrying a prescription-dependence caveat that a referee should probe. read the letter →

arxiv 2505.21118 v3 pith:CY6K7MWA submitted 2025-05-27 hep-th gr-qc

Donald Marolf , Bruno Missoni This is my paper · ORCID

classification hep-thgr-qc

keywords EuclideanwormholesaxiondilatonlinearizedstabilityquadraticfluctuationssingularintegralscomplexcontourSylvestercriterion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Previous stability analyses of Euclidean axion wormholes found that symmetric, even fluctuations produced divergent quadratic actions, and this was used to exclude them from the spectrum. The paper argues that those divergences were an artifact of how the Hamiltonian constraints were solved, not a physical pathology. A different choice of variables removes the singularities in the pure-axion case, while a small detour into the complex plane handles the axion-dilaton case and yields exact cancellation of the imaginary terms. The even modes therefore belong in the spectrum, and because their action is positive the earlier conclusion, that axion wormholes have no negative modes and remain stable saddles, is unchanged. A pseudo-analytic positivity check based on Sylvester's criterion supports this without heavy numerics.

What carries the argument

The load-bearing object is the quadratic fluctuation action written in gauge-invariant Hamiltonian variables, whose coefficients behave like $B\sim 1/r$ and $C\sim 1/r^2$ at the wormhole throat. Two devices carry the argument: in the pure-axion case, solving the constraints by integrating out $\psi$ and $\Pi_E$ instead of $\Pi_\psi$ removes the singular factor $1/H=1/\tanh(2r)$ entirely; in the axion-dilaton case, the singular integrals are defined through the complex-plane identities (4.9) and (4.13), replacing them by principal values plus imaginary terms whose residues cancel. Positive-definiteness is then shown by writing the action as $v^T M v$ and applying Sylvester's criterion to the resulting $4\times 4$ matrix $M$, whose principal minors are positive except for a small gap for $n=3$ near $b\simeq 1.45$ that is covered numerically.

What would settle it

Compute the $n=3$ even eigenfunction's quadratic action near $b\simeq 1.45$, where the analytic positivity proof has a gap, using an independent numerical Sturm-Liouville solver with a regularization that does not rely on principal values, and check whether the action stays finite and positive; a negative eigenvalue or a divergent action would refute the claim.

Watch

Extended reading notes

Core claim

The central claim is that the divergences found in earlier analyses of symmetric modes around asymptotically flat Euclidean axion wormholes are not physical. In the pure-axion model, solving the constraints by integrating out a different set of variables leads directly to a nonsingular quadratic action. In the axion-dilaton model, where gradient factors $1/H$ are unavoidable, the singular integrals are defined by a complex-plane contour that converts $1/r$ and $1/r^2$ terms into principal values plus imaginary residue terms; those imaginary terms cancel exactly in both the homogeneous and inhomogeneous sectors. With this definition the quadratic action is real and finite, even eigenfunctions must be included in the spectrum of the fluctuation operator, and their eigenvalues are positive. The wormhole therefore remains a stable Euclidean saddle.

Load-bearing premise

The axion-dilaton result depends on the complex-plane prescription of Section 4.1: if the singular integrals are defined by a different regularization, the even-mode action could differ, and with it the stability conclusion.

Editorial extensions

If this is right

Even eigenfunctions must be counted in the spectrum, reversing the exclusion in earlier work, and their inclusion does not change the no-negative-mode result.
The complex-plane bookkeeping provides a template for regularizing $1/H$-type singularities that will appear around any $\mathbb{Z}_2$-symmetric wormhole.
The Sylvester-criterion method offers a semi-analytic route to stability checks for wormholes without explicit conformal-gauge solutions, such as cosmological-constant or massive-dilaton cases.
The axion wormhole remains a viable saddle in the Euclidean path integral, so factorization-type puzzles are not resolved by a perturbative instability.
For AdS axion wormholes the question remains open because the boundary conditions differ; the present techniques do not settle that case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A generic $\mathbb{Z}_2$-symmetric wormhole whose radial gradient vanishes at the throat will produce the same $1/H$ singularity, and analogous cancellations should be expected only when the imaginary residue terms happen to vanish, which could serve as a quick stability filter.
If the same complex-plane prescription were applied to a wormhole with singular behavior at both asymptotic ends rather than a single throat, the imaginary terms would not obviously cancel, which would signal a genuinely unstable or ill-defined mode.
The $n=3$, $b\approx 1.45$ gap in the Sylvester proof could likely be closed by a symbolic inequality for the fourth principal minor, turning the numerical check into a fully analytic positivity proof.
The regularization choices amount to a choice of integration contour in field space, so a different contour could in principle give different even-mode actions; the physical content of the prescription deserves further clarification.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The headline: this paper likely solves the divergent even-mode problem in Euclidean axion wormhole stability analyses, and the pure-axion half of the argument is clean enough to stand on its own. The dilaton half is more conditional; the complex-plane prescription is a choice that is not uniquely fixed by the physics, and a referee should push on that.

What is actually new: the observation that the singularities in [24,25] come from integrating out Πψ, and that choosing different variables (ψ and ΠE) yields a regular action for pure axion wormholes. The cancellation of the added total derivative is checked explicitly. For the dilaton case, they keep the singular action from [25] and regulate it with a principal value plus imaginary delta-function terms; the imaginary parts cancel, and the remaining action is shown positive via a Sylvester-criterion analysis, with numerics filling a small gap near b=1.45 for n=3. The positivity technique is transferable.

What the paper does well: the algebra is explicit, the constraints are stated, and the boundary terms are tracked. The pure-axion result is a direct mathematical demonstration, not a numerical fit. The self-citation to [25] is not a problem here; the paper corrects a conclusion of that paper while using its setup.

Where it is soft: the axion-dilaton resolution depends on the choice to define singular integrals by a particular contour deformation. The imaginary parts cancel, which suggests the prescription is consistent, but nothing in the paper proves that a different regularization—say a hard cutoff with a counterterm—would give the same finite part for even modes. The stress-test note is right about that. The gap in the Sylvester proof for n=3 near b=1.45 is real but small; the numerical eigenvalues in Table 1 are the kind of evidence that can close it, and a reader can check the shooting method. I do not think this gap sinks the paper.

Who it is for: people working on Euclidean quantum gravity, wormhole stability, and the factorization puzzle. It deserves a serious referee. My own verdict would be: accept after revision, with the request that the authors either prove or explicitly flag the prescription-dependence of the dilaton regularization, and tighten the numerical evidence near b=1.45.

Recommendation: send it to peer review. The pure-axion result alone justifies the referee time.

Referee Report

2 major / 4 minor

Summary. The paper revisits the linearized stability analysis of asymptotically flat Euclidean axion wormholes, focusing on the even-mode divergences reported in prior work [24, 25]. The authors claim that those divergences are artifacts of a particular way of solving the Hamiltonian constraints and that a full treatment yields finite quadratic actions for these modes; the modes should therefore be included in the stability analysis, and because the resulting actions are positive, previous conclusions about wormhole stability are unaffected. For the pure axion case, the paper gives an explicit alternative constraint-solving procedure in Section 3.1 that integrates out ψ and Π_E rather than Π_ψ, leading directly to the regular action (3.22) with positive coefficients. For the axion-dilaton model, the paper abandons the constraint-solving route (Section 3.1, after Eq. (3.24)) and instead defines the singular 1/r and 1/r^2 integrands by the complex-plane prescription of Section 4.1, Eqs. (4.9) and (4.13), in which each singular integral is replaced by a Cauchy principal value plus an imaginary delta-function term; the imaginary parts cancel in the final action. The paper then uses a Sylvester-criterion argument on the 4x4 matrix M in (4.22) and numerical eigenvalues in Appendix B to argue that the even-mode action is positive for the dilaton wormhole as well. The conclusion is that even eigenfunctions belong to the spectrum and the wormhole remains perturbatively stable.

Significance. If the central claim holds, the paper resolves a controversy about whether even eigenfunctions must be excluded from the spectrum of the quadratic fluctuation operator in Euclidean axion wormholes, and it provides a technique—moving to the complex plane to handle 1/H-type singularities—that may be useful for other Z2-symmetric wormhole backgrounds. The pure-axion reduction in Section 3.1 is a clear and valuable result: it shows that a change of constraint-solving variables eliminates the singularities entirely, without any regularization prescription. The pseudo-analytic positivity proof based on the Sylvester criterion is an elegant way to reduce heavy numerics. However, the axion-dilaton resolution, which is the paper's main focus, depends on a distributional prescription that is not derived from the path-integral measure or matched against an independent gauge-invariant calculation. In addition, the positive-definiteness proof has a gap for the n=3 mode near b=1.45, where the determinant M4 is negative and only sparse numerics are offered. These issues affect the robustness of the headline claim for the dilaton wormhole, though they do not invalidate the pure-axion result.

major comments (2)

[Section 4.1, Eqs. (4.9) and (4.13)] The resolution of the axion-dilaton singularities rests entirely on the complex-plane replacement of 1/r and 1/r^2 integrands by principal values plus imaginary delta-function terms. This is a regularization prescription, not a derivation: deforming the contour in the opposite half-plane, or using a hard cutoff with a local counterterm, would generally change the finite part of the even-mode action. The pure-axion case provides an independent check because Section 3.1 eliminates the singularities by a change of constraint-solving variables, but for the axion-dilaton case the paper states (after Eq. (3.24)) that it cannot form a gauge-invariant variable and abandons that route. The central claim that the even-mode divergences are artifacts, and that the resulting action is finite and positive for the dilaton wormhole, therefore requires either a derivation of the principal-value prescription from the path-integral measure or an independent gauge-invariant calculation that reproduces the same finite part.
[Section 4.3, Fig. 4 and Table 1] The Sylvester-criterion proof that the matrix M in (4.22) is positive-definite fails for the n=3 mode in a range near b=1.45, where M4 is negative as shown in Fig. 4. The paper appeals to numerical eigenvalues in Appendix B, but Table 1 samples only b=0.5, 1.0, and 1.5, with no entry at b=1.45 and no systematic scan over the interval where M4 is negative. The positivity of the even-mode action is therefore not established exactly in the parameter region where the analytic proof breaks down; the claim should either be supported by a dedicated numerical computation in that region or explicitly qualified.

minor comments (4)

[Section 3.1 (final paragraph)] There is a typographical error: 'artifiact' should be 'artifact'.
[References, [26]] Reference [26] is listed as 'title TBD, to appear'; it should be updated with the full reference or removed before publication.
[Fig. 4 caption] The caption for Fig. 4 should explicitly state that the plot is for n=3 and should indicate the r-interval and the fixed parameters; currently it only mentions b=1.45, Q=0.5, c=-1, and n=3 in a terse manner.
[Appendix A and main text] The notation A_n, B_n, C_n, etc. in Appendix A partially conflicts with the coefficients A and B in Eq. (3.20) and with the Lagrange multiplier B in Eq. (3.9); a short remark clarifying the notational shift would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claim is a direct re-analysis of an explicitly cited quadratic action, not an assumption of it.

full rationale

The paper's conclusion that even eigenfunctions have finite positive action is obtained by direct computation, not by construction from its inputs. The quadratic actions in Eqs. (3.9) and (4.1) are taken from the authors' prior work [25], which is a self-citation; however, that action is the object being analyzed, not the conclusion being assumed. The paper's positive results are (i) an alternative constraint-solving procedure for the pure axion case that avoids 1/H singularities entirely (Sec. 3.1), (ii) a complex-plane principal-value bookkeeping for the axion-dilaton case (Sec. 4.1) in which the singular coefficients combine so that the imaginary parts cancel and the remaining integrands are regular, and (iii) a Sylvester-criterion check of the resulting explicit 4x4 matrix (4.22), supplemented by direct shooting-method numerics from [25] only in the narrow n=3, b~1.45 window. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported to forbid alternatives. The principal-value prescription is a regularization choice, but it is not used to force the positivity result: the positivity is established by evaluating the resulting regularized integrand. Any concern that another regularization could give a different finite part is a physical correctness issue, not a circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted; b, Q, c, n are theory inputs. The main external inputs are the background solutions and the starting quadratic action from [25]; the novel regularization prescription (4.9)-(4.13) is a paper-specific assumption. No new particles or fields are introduced.

assumptions (6)

domain assumption Background wormhole solutions (2.12), (2.13) and regularity condition b^2 < 8/3 (2.14) are taken from prior literature [5,6,32].
The stability analysis is linearized around these saddles; if these are not the relevant saddles, the fluctuation spectrum computed is moot.
domain assumption The quadratic action (3.9) equals Eq. (3.23) of [25]; the paper does not re-derive it.
All subsequent results depend on this starting expression; [25] is by coauthor Missoni et al.
domain assumption Euclidean Hamiltonian contour: momenta integrated along the imaginary axis, and A, B contours chosen to impose constraints (footnote 9).
The paper follows the literature but notes this is a 'necessarily complicated prescription' to avoid the conformal factor problem.
ad hoc to paper Distributional replacement rules (4.9) and (4.13) define the action of singular terms.
These rules are introduced as a bookkeeping device; the physical uniqueness of the finite part is not established.
domain assumption Boundary conditions: Dirichlet on dilaton and metric fluctuations, Neumann on axion; fluctuations vanish at infinity.
From [25]; the sign of the boundary terms and the self-adjointness of the Sturm-Liouville operator depend on them.
ad hoc to paper n=2 mode is pure gauge except for a boundary term and is set aside.
The paper states this and refers to [24,28]; the stability of this mode is asserted, not derived in this paper.

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Cite this review

Pith. "Pith review of Euclidean wormholes stability analysis revisited." pith.science (2026). https://pith.science/paper/CY6K7MWA

@misc{pith2026250521118,
  author       = {Pith},
  title        = {Pith review of: Euclidean wormholes stability analysis revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CY6K7MWA}},
  note         = {Machine review of arXiv:2505.21118}
}

read the original abstract

Previous studies of linearized stability of asymptotically flat Euclidean axion wormholes found that symmetric modes suffered from divergences. We show that such divergences were an artifact of a particular way of solving the constraints, and that a full treatment leads to finite actions for such modes. The modes must thus be included in a stability analysis. However, since the action for these modes turns out to be positive, this turns out not to affect previous statements about stability of axion wormholes. We also introduce a technique that allows us to show this positivity at a pseudo-analytic level that avoids heavy numerics. Our techniques should be useful to future studies of stabilities of other wormholes as well.

Discussion (0). Continue with ORCID to comment.

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [25]

Hertog, S

T. Hertog, S. Maenaut, B. Missoni, R. Tielemans and T. Van Riet,Stability of axion-saxion wormholes,JHEP11(2024) 151 [2405.02072]

arXiv 2024

[2] [1]

Coleman,Why There Is Nothing Rather Than Something: A Theory of the Cosmological Constant,Nucl

S.R. Coleman,Why There Is Nothing Rather Than Something: A Theory of the Cosmological Constant,Nucl. Phys. B310(1988) 643

work page 1988

[3] [2]

Giddings and A

S.B. Giddings and A. Strominger,Axion Induced Topology Change in Quantum Gravity and String Theory,Nucl. Phys. B306(1988) 890

1988

[4] [3]

Giddings and A

S.B. Giddings and A. Strominger,STRING WORMHOLES,Phys. Lett. B230(1989) 46

work page 1989

[5] [4]

Marolf and J.E

D. Marolf and J.E. Santos,AdS Euclidean wormholes,Class. Quant. Grav.38(2021) 224002 [2101.08875]

arXiv 2021

[6] [5]

Arkani-Hamed, J

N. Arkani-Hamed, J. Orgera and J. Polchinski,Euclidean wormholes in string theory,JHEP 12(2007) 018 [0705.2768]

arXiv 2007

[7] [6]

Bergshoeff, A

E. Bergshoeff, A. Collinucci, U. Gran, D. Roest and S. Vandoren,Non-extremal instantons and wormholes in string theory,Fortsch. Phys.53(2005) 990 [hep-th/0412183]

arXiv 2005

[8] [7]

Andriolo, G

S. Andriolo, G. Shiu, P. Soler and T. Van Riet,Axion wormholes with massive dilaton,Class. Quant. Grav.39(2022) 215014 [2205.01119]

arXiv 2022