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arxiv: 2604.21736 · v1 · submitted 2026-04-23 · ✦ hep-th · gr-qc

Recognition: unknown

IR behaviour of one-loop complex mathbb{R}times S³ saddles

Shubhashis Mallik , Gaurav Narain
Authors on Pith no claims yet

Pith reviewed 2026-05-09 21:31 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords Hartle-Hawking wavefunctioncomplex saddlesinfrared divergencesno-boundary proposalde Sitter spaceone-loop gravityEuclidean quantum gravityPicard-Lefschetz
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The pith

Metric fluctuations around complex saddles produce secularly growing infrared divergences in the Hartle-Hawking wavefunction.

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

The paper studies the infrared behavior of one-loop metric fluctuations in the gravitational path integral over complex metrics with R times S cubed topology in four-dimensional Einstein-Hilbert gravity. It imposes a no-boundary condition at the initial surface and considers either Dirichlet or fixed-extrinsic-curvature conditions at the final surface, leading to a Euclidean-to-Lorentzian transition whose dominant saddles are complex in the Lorentzian phase. After regularizing the one-loop lapse action with the Hurwitz zeta function and subtracting ultraviolet divergences, the renormalized wavefunction is shown to acquire infrared divergences that grow secularly as the universe expands. This leading infrared divergence matches the one obtained for the pure Lorentzian de Sitter wavefunction, which has real saddles. The Picard-Lefschetz contour must be supplemented by an i epsilon prescription realized through a slight complexification of the cosmological constant.

Core claim

The UV-renormalized one-loop Hartle-Hawking wavefunction receives secularly growing infrared divergences from metric fluctuations as the Universe expands, and this leading IR divergence is identical to that of the one-loop dS wavefunction corresponding to a Lorentzian transition with fixed extrinsic curvature.

What carries the argument

The one-loop lapse action computed from metric fluctuations around the complex saddles in the ADM decomposition, regularized by the Hurwitz zeta function and renormalized to remove UV poles.

If this is right

  • The metric-fluctuation contributions produce secularly growing infrared divergences after UV renormalization in both the no-boundary and Lorentzian setups.
  • Picard-Lefschetz methods alone are insufficient and must be supplemented by an i epsilon prescription from complexifying Lambda.
  • The dominant saddles in the Lorentzian phase are a pair of complex metrics that superimpose for the chosen boundary conditions.
  • All boundary choices examined leave the saddles KSW-allowed.

Where Pith is reading between the lines

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

  • The matching infrared divergence suggests that the leading late-time behavior may be insensitive to the choice of initial boundary condition.
  • The result raises the possibility that resummation of fluctuations or non-perturbative effects are needed to obtain a finite late-time wavefunction in de Sitter quantum gravity.
  • Extending the same one-loop analysis to include matter fields or different spatial topologies could test whether the secular growth is universal.

Load-bearing premise

The one-loop approximation around the identified complex saddles, combined with Hurwitz-Zeta regularization and an i epsilon prescription obtained by complexifying the cosmological constant, correctly captures the infrared behavior without requiring resummation or non-perturbative effects.

What would settle it

An explicit higher-order or non-perturbative computation of the wavefunction that finds the coefficient of the secular infrared growth to be zero or to differ between the no-boundary and Lorentzian cases would falsify the one-loop claim.

Figures

Figures reproduced from arXiv: 2604.21736 by Gaurav Narain, Shubhashis Mallik.

Figure 1
Figure 1. Figure 1: FIG. 1. The H-H no boundary wave function of the universe (Ψ [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Showing the background No-boundary universe where, [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The plot showing the KSW allowability of no-boundary saddles for different values of [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The PL plot shows the relevant saddles (Blue dots) and irrelevant saddle (Blue square) [PITH_FULL_IMAGE:figures/full_fig_p030_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The plot showing Ψ [PITH_FULL_IMAGE:figures/full_fig_p032_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Saddle geometry describing the expanding phase in real de-Sitter ( [PITH_FULL_IMAGE:figures/full_fig_p034_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Picard-Lefschetz analysis for the classical boundary condition describing the real de-Sitter. [PITH_FULL_IMAGE:figures/full_fig_p036_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The Divergence shows up as the final hypersurface (Bd [PITH_FULL_IMAGE:figures/full_fig_p037_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. PL plot with Λ complex ¯ϵ [PITH_FULL_IMAGE:figures/full_fig_p040_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Plots showing the KSW allowability of the complex de-Sitter saddles for various param [PITH_FULL_IMAGE:figures/full_fig_p044_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The plot showing Ψ [PITH_FULL_IMAGE:figures/full_fig_p053_11.png] view at source ↗
read the original abstract

Gravitational path-integral over $\mathbb{R}\times S^3$ complex metrics with fluctuations is studied in 4D for Einstein-Hilbert gravity in Lorentzian signature, with the aim to investigate the IR properties of complex saddles for various boundary choices. General covariance doesn't allow arbitrary boundary choices for the background and fluctuations. In the ADM-decomposition, while imposing ``no-boundary'' condition at the initial boundary, two scenarios are considered for the final boundary: Dirichlet and fixed extrinsic curvature. Universe undergoes transition from a Euclidean to Lorentzian phase in either scenario, where the dominant saddle in Euclidean phase correspond to a Euclidean metric (imaginary time), while the Lorentzian phase has two complex metrics as dominant saddles which superimpose. One-loop corrected lapse action is computed using Hurwitz-Zeta regularization. UV-divergences canceled by suitable counter terms lead to a renormalized lapse action. One-loop renormalized Hartle-Hawking wave-function is computed using the Picard-Lefschetz and WKB methods, where the contributions coming from the metric-fluctuations show secularly growing infrared divergences as the Universe expands. This is compared with the situation in pure Lorentzian dS, corresponding to a Universe transitioning from an initial state of vanishing conjugate momenta to final state of fixed extrinsic curvature, thereby giving real saddles. Picard-Lefschetz methods alone are not sufficient to overcome the technical hurdles in the one-loop computation, which needs to be supplemented by an $i\epsilon$-prescription, achieved via slight complexification of the cosmological constant $\Lambda$. The UV renormalized one-loop dS wavefunction has the same leading IR divergence as for the Hartle-Hawking no-boundary Universe. Interestingly for all boundary choices considered, the saddles remain KSW-allowed.

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

3 major / 2 minor

Summary. The manuscript examines the IR behavior of one-loop corrections in the gravitational path integral for complex metrics on R×S³ in 4D Einstein-Hilbert gravity. It considers no-boundary initial conditions and two types of final boundary conditions (Dirichlet and fixed extrinsic curvature), finding a transition from Euclidean to Lorentzian phases with complex saddles in the latter. Using Hurwitz-Zeta regularization for the one-loop lapse action, counterterms to cancel UV divergences, and an iε prescription via complexified Λ, the authors compute the renormalized Hartle-Hawking wavefunction via Picard-Lefschetz and WKB methods, reporting secularly growing IR divergences from metric fluctuations as the universe expands, matching those in pure Lorentzian dS, with all saddles remaining KSW-allowed.

Significance. If the one-loop approximation and regularization procedure are robust, the result would highlight potential IR issues in the no-boundary proposal and the role of complex saddles in quantum cosmology, providing a comparison between different boundary conditions and Lorentzian dS. The explicit use of Picard-Lefschetz theorem combined with Hurwitz-Zeta regularization and counterterms adds technical value to analyses of gravitational path integrals in expanding backgrounds.

major comments (3)
  1. [One-loop lapse action and Hurwitz-Zeta regularization] The application of Hurwitz-Zeta regularization to the time-dependent fluctuation modes in the expanding Lorentzian phase (in the section computing the one-loop corrected lapse action) evaluates the zeta function on instantaneous spectra before integrating over the lapse. This procedure cancels UV poles but does not automatically isolate or resum secular IR logarithms/powers from the explicitly time-dependent background; a concrete verification against known results for time-dependent operators or an explicit resummation step is needed to support the claim of secular growth in the wavefunction.
  2. [iε prescription and complexification of Λ] The justification that complexifying the cosmological constant Λ yields a valid iε prescription that leaves the physical IR behavior unchanged (in the section on Picard-Lefschetz and WKB methods for the renormalized wavefunction) requires more detail. The time-dependent background for the mode sums remains Lorentzian, so it is unclear whether the reported leading IR divergence is robust or regulator-dependent; an explicit comparison of the divergence terms before and after the complexification would address this.
  3. [Comparison with pure Lorentzian dS] The claim that the UV-renormalized one-loop dS wavefunction has the same leading IR divergence as the Hartle-Hawking no-boundary case (in the comparison with pure Lorentzian dS) is load-bearing for the central IR result but relies on identical regularization choices in both scenarios. Explicit equations or tables showing the matching divergence coefficients after counterterms would confirm it is not an artifact.
minor comments (2)
  1. [Boundary conditions and ADM decomposition] The description of how general covariance constrains boundary choices for both background and fluctuations in the ADM decomposition could be expanded with explicit boundary terms for the metric fluctuations to improve clarity.
  2. [One-loop computation] A brief appendix or table listing the explicit mode sums and the form of the Hurwitz zeta function used for the lapse integral would aid reproducibility of the one-loop determinant evaluation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify several technical aspects of our analysis. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [One-loop lapse action and Hurwitz-Zeta regularization] The application of Hurwitz-Zeta regularization to the time-dependent fluctuation modes in the expanding Lorentzian phase (in the section computing the one-loop corrected lapse action) evaluates the zeta function on instantaneous spectra before integrating over the lapse. This procedure cancels UV poles but does not automatically isolate or resum secular IR logarithms/powers from the explicitly time-dependent background; a concrete verification against known results for time-dependent operators or an explicit resummation step is needed to support the claim of secular growth in the wavefunction.

    Authors: We agree that additional verification would strengthen the presentation. The Hurwitz-Zeta function is evaluated on the instantaneous spectra of the fluctuation operator for each fixed lapse, after which the lapse integral is performed; this yields the secular IR growth from the accumulating time-dependent modes as the scale factor increases. In the revised manuscript we will add a short subsection comparing our leading IR terms with known results for one-loop effective actions in time-dependent de Sitter backgrounds (e.g., the secular logarithms reported in the literature on expanding cosmologies). This comparison confirms that the IR growth is physical and not an artifact of the regularization order. revision: partial

  2. Referee: [iε prescription and complexification of Λ] The justification that complexifying the cosmological constant Λ yields a valid iε prescription that leaves the physical IR behavior unchanged (in the section on Picard-Lefschetz and WKB methods for the renormalized wavefunction) requires more detail. The time-dependent background for the mode sums remains Lorentzian, so it is unclear whether the reported leading IR divergence is robust or regulator-dependent; an explicit comparison of the divergence terms before and after the complexification would address this.

    Authors: We thank the referee for highlighting the need for explicit checks. The small imaginary part introduced via complex Λ shifts the integration contour in the complex lapse plane while leaving the physical Lorentzian background intact for the mode sums. In the revision we will insert a new paragraph (or appendix) that tabulates the UV and IR divergence coefficients computed with real Λ versus the complexified value, demonstrating that the leading secular IR term is unchanged to the order we work. This explicit comparison will be placed in the section discussing the iε prescription. revision: yes

  3. Referee: [Comparison with pure Lorentzian dS] The claim that the UV-renormalized one-loop dS wavefunction has the same leading IR divergence as the Hartle-Hawking no-boundary case (in the comparison with pure Lorentzian dS) is load-bearing for the central IR result but relies on identical regularization choices in both scenarios. Explicit equations or tables showing the matching divergence coefficients after counterterms would confirm it is not an artifact.

    Authors: We agree that an explicit side-by-side display is desirable. The matching of the leading IR coefficient follows directly from the identical form of the renormalized one-loop lapse action once the same counterterms are subtracted in both the no-boundary and pure Lorentzian dS setups. In the revised manuscript we will add a short table (or set of equations) in the comparison section that lists the coefficients of the secular terms for both cases after renormalization, making the equality manifest and confirming it is independent of the specific regularization details. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external regularization and contour methods on explicit mode sums

full rationale

The paper computes the one-loop lapse action via Hurwitz-Zeta regularization of the fluctuation spectrum, subtracts UV poles with counterterms, and obtains the wavefunction via Picard-Lefschetz plus WKB. The reported secular IR growth follows from integrating the time-dependent eigenvalues over the lapse in the expanding phase; this is not obtained by fitting parameters to the target IR result, nor by defining the output in terms of itself. The iε prescription (complex Λ) is introduced to define the contour and is independent of the final IR claim. No load-bearing self-citations or uniqueness theorems from the authors' prior work are invoked to force the result. The central claim therefore remains a genuine output of the calculation rather than a renaming or tautology.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on the Einstein-Hilbert action in Lorentzian signature, the validity of the one-loop saddle-point approximation, the applicability of Hurwitz-Zeta regularization to gravitational fluctuations, and the legitimacy of deforming the integration contour via Picard-Lefschetz while supplementing with a complexified cosmological constant.

free parameters (1)
  • cosmological constant Lambda
    Slightly complexified to implement the iε prescription that renders the contour integral well-defined.
axioms (2)
  • domain assumption General covariance restricts admissible boundary conditions for both background and fluctuations in the ADM decomposition.
    Invoked to justify the choice of no-boundary initial condition together with Dirichlet or fixed-extrinsic-curvature final conditions.
  • domain assumption The one-loop determinant around complex saddles can be evaluated with Hurwitz-Zeta regularization after UV counterterms are subtracted.
    Required for obtaining a finite renormalized lapse action.
invented entities (1)
  • complex metrics as dominant saddles no independent evidence
    purpose: To provide a Lorentzian continuation of the Euclidean Hartle-Hawking geometry while satisfying the boundary conditions.
    Postulated to allow the path integral to capture both Euclidean and Lorentzian phases; no independent falsifiable signature is given beyond the wavefunction itself.

pith-pipeline@v0.9.0 · 5634 in / 1805 out tokens · 51455 ms · 2026-05-09T21:31:36.261901+00:00 · methodology

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Reference graph

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    Lorentzian Regime:H 1(=H 1/H)<1 In this section, we compute the wave function when the extrinsic curvature (K) is fixed at the final hypersurface. This boundary condition corresponds to fixing the Hubble radius 30 (H1 =K/3), which is more physically sensible compared to the fixed size [29, 34]. We con- sider the no-boundary condition on the initial hypers...

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