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Thermal false vacuum decay near black holes is aspherical

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Pith's one-line read Thermal false-vacuum decay near a Schwarzschild black hole is aspherical for all but the smallest holes, with the largest holes tunneling from a point on the horizon.

desk verdict Spherical-only BH vacuum decay is shown to fail at large rs with a solid negative-mode count, and the new aspherical saddles are genuinely interesting, but the central phase diagram is conditional on an unverified index-1 property the authors concede. read the letter →

arxiv 2608.12469 v1 pith:HJSG7OFG submitted 2026-08-12 gr-qc astro-ph.COhep-phhep-th

D. S. Gorbunov , D. G. Levkov , V. E. Maslov This is my paper · ORCID
classification gr-qcastro-ph.COhep-phhep-th
keywords falsevacuumdecaySchwarzschildblackholethermalactivationcriticalbubbleFubini-LipatovinstantonasphericalnegativeEuclideanmodesHiggsmetastability
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

A false vacuum is a metastable field configuration trapped behind an energy barrier; it can decay by nucleating a bubble of true vacuum. Near a Schwarzschild black hole in thermal equilibrium at the Hawking temperature, this paper shows, the decay is generically aspherical: the bubble that nucleates is not a sphere centered on the hole except for the smallest cases. In the scalar model $V(\varphi)=m^2\varphi^2/2-\lambda\varphi^4/4$, which mimics the Higgs sector at large field, the paper identifies three regimes: spherical activation for $r_s\lesssim0.194\,m^{-1}$, where the critical bubble (the static saddle configuration perched on the barrier top) is spherical; aspherical activation for $0.194\,m^{-1}\lesssim r_s\lesssim0.211\,m^{-1}$; and horizon-localized tunneling for $r_s\gtrsim0.211\,m^{-1}$. The largest-hole regime is governed by an infinitesimally thin Fubini-Lipatov bounce sitting at a point of the event horizon, with suppression $S_b=8\pi^2/(3\lambda)$. If this is right, earlier spherical calculations of black-hole-catalyzed vacuum decay understate the decay rate, and the mechanism matters for whether primordial black holes or hot plasma could have triggered electroweak vacuum decay.

What carries the argument

The machinery has four parts. The Euclidean action $S_E[\varphi]=\int d\tau\,d^3x\,\sqrt{g_E}\,[\frac{1}{2}g_E^{\mu\nu}\partial_\mu\varphi\,\partial_\nu\varphi+V(\varphi)]$ defines the semiclassical exponent $\Gamma\sim e^{-S_E}$. The critical bubble—a static field configuration sitting on top of the potential barrier—governs thermal activation, with one-period action $\beta E_{cb}$. The Fubini-Lipatov instanton, the conformally symmetric flat-space bounce $\varphi_b=\sqrt{8/\lambda}\,a/(a^2+\tau^2+x^2)$, has action $S_b=8\pi^2/(3\lambda)$ and in the massive model shrinks to $a\to0$; the same solution, written in locally flat coordinates near the horizon $\tilde\tau=\tau/8R_s$, $\varrho=4(R-R_s)$, $\varrho_\theta=4R_s\theta$, becomes a static, axially symmetric but angularly localized saddle at a point of the horizon. The decisive mechanism that forces asphericity is the negative-mode spectrum: the second variation of $S_E$ about a spherical bubble decomposes into radial eigenmodes with angular momentum $\ell$, and a physical saddle must have exactly one negative mode; counting these modes via the oscillation theorem and the large-$r_s$ reduction to a two-dimensional near-horizon problem with eigenvalue $\mu_{2d}\simeq-86.6\,m^2$ shows that the spherical bubble acquires $\ell=1$ negative modes for $r_s>r_s^{(a)}$, so the true saddle must break spherical symmetry. A constrained functional $F[\varphi]=S_E-\mu_Z\int(z-Z_c)\varphi^4$ is used to follow saddle branches versus asphericity and thereby discover the horizon bounce.

What would settle it

At $r_s=0.2\,m^{-1}$, solve the full eigenvalue problem for the second variation of $S_E$ around the numerically found aspherical bubble, and do the same around the regularized horizon bounce at, say, $r_s=0.216\,m^{-1}$. Count the negative eigenvalues. If either saddle has zero or more than one negative mode, the reported suppression exponents are not the decay rates; if both have exactly one, the paper's mechanism is confirmed.

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Extended reading notes

Core claim

The central claim is that in the model $V(\varphi)=m^2\varphi^2/2-\lambda\varphi^4/4$ near a Schwarzschild hole of radius $r_s$ equilibrated at Hawking temperature $T=(4\pi r_s)^{-1}$, the dominant semiclassical saddle describing false-vacuum decay is rotationally asymmetric about the hole for every $r_s>r_s^{(a)}\approx0.194\,m^{-1}$. For $r_s^{(a)}<r_s<r_s^{(b)}\approx0.211\,m^{-1}$ the decay is thermal activation: fluctuations create an aspherical critical bubble—a static saddle point on the barrier top—that hugs one side of the horizon and then expands. For $r_s\ge r_s^{(b)}$ the mechanism changes to vacuum tunneling, and the governing solution is an infinitesimally thin Fubini-Lipatov bounce placed at a point of the event horizon, with Euclidean action $S_b=8\pi^2/(3\lambda)$ independent of $r_s$; in the unregularized model this solution is singular (zero size, infinite field), and the paper identifies it by adding a small $\varphi^6$ regulator and taking the regulator to zero. The proof that spherical bubbles fail is that above $r_s^{(a)}$ they acquire dipole ($\ell=1$) negative modes in addition to the required single one, and the number of such modes grows with hole size; the suppression exponent $\lambda S_E(r_s)$ is therefore piecewise smooth, switching from spherical activation to aspherical activation to horizon tunneling.

Load-bearing premise

The result rests on assuming that the off-center bubble and the point-like bounce are each unstable in exactly one direction in the space of field configurations—the direction that leads to true vacuum. The paper itself notes in Sec. VI that this was not checked by computing the eigenspectra directly. If either solution were unstable in additional directions, the reported suppression exponents would not describe the decay.

Editorial extensions

If this is right

  • For holes with $r_s>0.194\,m^{-1}$, the dominant decay bubble is off-center, so spherical-bubble calculations overestimate the suppression and underestimate the catalyzed decay rate.
  • For $r_s>0.211\,m^{-1}$, vacuum tunneling through the infinitesimal bounce has the universal suppression $S_b=8\pi^2/(3\lambda)$, independent of hole size, and the nucleated bubble starts at a point on the horizon.
  • The suppression exponent is piecewise smooth in $r_s$, so a single formula for black-hole-catalyzed vacuum decay cannot capture the actual mechanism.
  • The first regime boundary corresponds to Hawking temperature $T^{(a)}_{cr}\approx0.41\,m$ and the second to $T^{(b)}_{cr}\approx0.377\,m$; for holes colder than $T^{(b)}_{cr}$, tunneling dominates.
  • Because the scalar model resembles the Higgs sector at large fields, the results call for reexamining finite-temperature Higgs decay near black holes and the resulting primordial-black-hole constraints.

Reading between the lines

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

  • We infer that the instability mechanism is generic: any static false-vacuum bubble localized at a horizon should acquire dipole negative modes once the hole radius exceeds the inverse field mass, so a similar $r_s^{(a)}\sim m^{-1}$ threshold should appear in other potentials beyond the quartic model.
  • We infer that in a theory with a running coupling, such as the Standard Model Higgs, the scale degeneracy of the bounce is lifted; the horizon-riding bounce is the natural winner because the black hole lowers the barrier, so the aspherical tunneling regime should persist and its boundaries may shift.
  • We infer that the same near-horizon local-flatness argument applies to any static horizon geometry, so horizon-localized bounces and aspherical activation should also appear for charged or higher-dimensional black holes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This paper studies false-vacuum decay of a scalar field with potential V = (1/2)m^2 φ^2 - (λ/4)φ^4 in Euclidean Schwarzschild spacetime at the Hawking temperature. The authors compute Euclidean actions of static critical bubbles and of an infinitesimal Fubini–Lipatov bounce placed on the horizon, and from these construct a phase diagram with three regimes: spherically symmetric activation for r_s ≲ 0.194/m, aspherical activation for 0.194/m ≲ r_s ≲ 0.211/m, and vacuum tunneling via a singular near-horizon bounce for r_s ≳ 0.211/m. The central technical results are a negative-mode count showing that spherically symmetric bubbles acquire additional negative modes above r_s^(a), a numerical construction of aspherical bubble branches, and an identification of the near-horizon extremum with the regulated Fubini–Lipatov instanton.

Significance. If correct, the asphericity result overturns the spherically symmetric ansatz used in previous thermal black-hole catalysis studies and gives a concrete, testable prediction for the decay exponent in a Higgs-like model. The paper's strengths include an explicit negative-mode count for spherical bubbles with analytic asymptotics and numerics in agreement (Fig. 7), a detailed matched-asymptotic treatment of the Fubini–Lipatov bounce, and a regulator-limit check (Figs. 14–16). The main caveat is that the physical interpretation of the new aspherical saddles as decay bounces with exactly one negative Euclidean mode is not directly verified; this is acknowledged in Sec. VI and is the key issue for the phase diagram.

major comments (2)
  1. [Secs. IV, V, and VI] The central claim of the paper is the three-regime suppression exponent in Fig. 3, obtained by comparing S_E among spherical critical bubbles, aspherical critical bubbles, and the near-horizon bounce. This comparison is valid only if each selected saddle has exactly one negative Euclidean mode. For the aspherical critical bubbles (Sec. IV), the one-negative-mode property is not demonstrated: the two-mode action surface S_E(A_0,A_1) in Fig. 8 covers only the axisymmetric subspace; the argument excluding φ-dependent modes is sound for m≠0 sectors of an axisymmetric background, but it says nothing about other axisymmetric modes (e.g., higher-ℓ deformations) of the full nonlinear solution; and the Newton–Raphson non-divergence argument would only detect an eigenvalue crossing zero exactly at the steps used in the r_s continuation, not a mode that becomes negative between steps. For the near-horizon bounce (Sec. V), the index-1 property is transferred from the flat-space Fubini–Lipatov instanton by assumption, and no eigenvalue spectrum of the regularized solution in the curved background is presented. Since extra negative modes would make these saddles unusable for the decay rate, the phase diagram is conditional on this unverified property; the authors themselves state this in Sec. VI. Please compute the eigenspectra of the aspherical bubbles and of the near-horizon bounce, or explicitly weaken the claims.
  2. [Sec. II and Appendix B] The exclusion of periodic instantons is argued in flat space (Sec. II) and for spherically symmetric bubbles (Appendix B shows no negative time-dependent modes around φ_cb^(s)), but no equivalent analysis is given for the aspherical bubbles or the near-horizon bounce. The phase-diagram interpretation of the intermediate regime as activation and the statement that periodic instantons are irrelevant near black holes rely on this assumption. Please either extend the stability check to the aspherical saddles or clearly mark this as an assumption imported from Refs. [13,28,51].
minor comments (5)
  1. [Fig. 4 caption] The caption writes 'periodic instanons'; this should be 'periodic instantons'.
  2. [Fig. 12 caption] The caption says 'at two values of r_s' although three panels (a)–(c) are shown; it should say 'three values'.
  3. [Eq. (B8)] The final eigenfunction in Eq. (B8) has a subscript mismatch: the right-hand side should be μ_{kℓn} ξ_{kℓn}, not ξ_{kn}.
  4. [Sec. II, after Eq. (11)] The word 'irrelevent' should be 'irrelevant'.
  5. [Secs. IV and V] The critical radii in Eqs. (4a)–(4c) are quoted to three digits without an error estimate; please state the numerical uncertainty from the lattice resolution and the r_s stepping.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the phase diagram and suppression exponents are computed outputs, not inputs.

full rationale

Walking the derivation chain, I find no circular step. The central outputs—the critical radii r_s^(a) ≈ 0.194 m^-1 and r_s^(b) ≈ 0.211 m^-1, the three-regime suppression S_E(r_s), and the aspherical bubble/bounce configurations—are obtained by solving the Euclidean field equations (Eqs. (16), (21), (27), and the lattice Newton–Raphson system (C2)) and comparing the resulting actions. No parameter is fitted to these outputs and then renamed a prediction. The regulator gamma_6 in Eq. (30) is a technical device with an explicit gamma_6 -> 0 limit; Eq. (35) and Figs. 14–16 verify that the numerical bounce approaches the flat-space Fubini–Lipatov instanton, an independently known solution. The flat-space action S_b = 8*pi^2/(3*lambda) is an imported standard result, checked numerically and derived in Appendix A; it is not obtained from the black-hole problem. The subdominance of periodic instantons is imported from Refs. [13,28,51], which are external to the present authors and not self-citations. The paper's own self-citations (Refs. [14,62]) are contextual and not load-bearing. The Sec. VI admission that the one-negative-mode property of aspherical bubbles and near-horizon bounces was not verified by explicit eigen-spectra is an acknowledged correctness gap, but it is not circularity: no equation uses the target result as an input. Thus the derivation is self-contained with respect to the circularity criteria, and the appropriate score is 0.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central computation relies on standard semiclassical saddle-point technology plus model assumptions about the scalar potential and black hole background. The load-bearing nonstandard inputs are the subdominance of periodic instantons and the unverified one-negative-mode property, both noted in the text.

free parameters (1)
  • regulator strength gamma_6 = 4e-7 and 4e-10
    Introduced in Eq. (30) to give finite-size bounces in the otherwise singular model (2). It is not fitted to any target; physical results are taken in the gamma_6 -> 0 limit, and Figs. 15 and 16 check convergence.
assumptions (6)
  • standard math Semiclassical approximation: decay rate Gamma ~ exp(-S_E) to leading order in lambda << 1.
    Used throughout, see Eq. (3) and Sec. II; standard for weak coupling but not exact.
  • standard math A Euclidean saddle describes the decay only if it has exactly one negative mode.
    Standard Coleman-Langer criterion; invoked to reject spherical bubbles with multiple negative modes and to identify physical aspherical saddles, Secs. III and IV.
  • domain assumption The Schwarzschild background is fixed and backreaction of the scalar field is neglected.
    Justified by inequality (13) M_BH >> E_cb; used in metric (14).
  • domain assumption Finite-energy periodic instantons are subdominant in black-hole thermal decay.
    Assumed in Sec. II on the basis of refs. [13,28,51]; not computed for the aspherical branches; load-bearing for the phase diagram.
  • domain assumption The model V = m^2 phi^2/2 - lambda phi^4/4 with true vacuum at phi -> infinity is a valid stand-in for the Higgs large-field sector.
    Central to the whole calculation; differences such as running coupling and thermal corrections are acknowledged in Sec. VI.
  • ad hoc to paper The phi^6 regulator with gamma_6 -> 0 does not change the physical conclusions.
    Regularization is used for numerical stabilization; authors provide gamma_6-independence checks, but the singular unregularized limit is subtle.

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Cite this review

Pith. "Pith review of Thermal false vacuum decay near black holes is aspherical." pith.science (2026). https://pith.science/paper/HJSG7OFG

@misc{pith2026260812469,
  author       = {Pith},
  title        = {Pith review of: Thermal false vacuum decay near black holes is aspherical},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HJSG7OFG}},
  note         = {Machine review of arXiv:2608.12469}
}
read the original abstract

We study decay of a scalar field false vacuum near a (3+1)-dimensional Schwarzschild black hole equilibrated at Hawking temperature with the environment. Our scalar field model has negative quartic self-coupling and thereby resembles Higgs sector of the Standard Model in the large-field limit. We demonstrate that if the black hole is not too small, the false vacuum in this model decays aspherically with regard to the black hole center: via formation of expanding true vacuum bubbles emerging on the outer side of the event horizon. More specifically, we identify three regimes of the decay. For the largest and coldest black holes, the main mechanism is quantum tunneling described by an infinitesimally thin bounce sitting at some point of the horizon. In the intermediate-mass regime, the vacuum is destroyed by thermal fluctuations creating aspherical critical bubbles in the horizon vicinity. Finally, near the smallest black holes thermal fluctuations still guide the decay but the dominant critical bubble is spherically symmetric and covers the entire horizon.

Figures

Figures reproduced from arXiv: 2608.12469 by the authors.

Figure 1
Figure 1. FIG. 1. Diagrams of bubbles formed during false vacuum [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (Schematic) Configuration space and mechanisms for [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Thermal false vacuum decay in flat space: sup [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figures from the paper (14 more)
Figure 6
Figure 6. Figure 6: FIG. 6. Euclidean actions [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Spherically-symmetric critical bubble [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Number of negative modes [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Euclidean action [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Aspherical critical bubble at [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. (a) Aspherical critical bubble from the second [PITH_FULL_IMAGE:figures/full_fig_p008_11.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Asphericities [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]
Figure 12
Figure 12. Figure 12: shows implementation of Zc fixation proce￾dure. We launch it from every critical bubble we know: spherical S, aspherical A, and second aspherical A2. Changing Zc in small steps, we repeatedly solve saddle￾point equations δF/δφ = ∂F/∂µZ = 0 using previous solution as i…
Figure 13
Figure 13. Figure 13: FIG. 13. Euclidean Schwarzschild spacetime (not to scale). [PITH_FULL_IMAGE:figures/full_fig_p010_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. The field [PITH_FULL_IMAGE:figures/full_fig_p010_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. (a) Euclidean action [PITH_FULL_IMAGE:figures/full_fig_p011_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Change [PITH_FULL_IMAGE:figures/full_fig_p013_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Solutions [PITH_FULL_IMAGE:figures/full_fig_p015_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Transfer function [PITH_FULL_IMAGE:figures/full_fig_p016_18.png]

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Works this paper leans on

72 extracted references · 41 canonical work pages

  1. [1]

    The respec- tive bounceφ b isO(4) symmetric, i.e

    Flat-space Fubini-Lipatov instantons Start with the model (2) in flat space. The respec- tive bounceφ b isO(4) symmetric, i.e. depends only on Euclidean four-radiusr 4 = √ τ 2 +x 2. We introduce its overlapping core and tail atr 4≪m−1 andr 4≫a. In- side the core, the mass term in Eqs. (5), (2) is suppressed. Indeed, rescaling with the bounce sizea, φb = √...

    work page

  2. [2]

    Regularized bounces in flat space To stabilize bounce size, we added the regulator (30) withγ 6≪1 to the scalar potential. The respective change of the flat-space bounce action can be obtained by substituting the unregularized solution from Ap- pendix A 1 into the regulator action, ∆Sγ6 E Sb ≡ γ6λ2 6m2Sb Z d4xφ 6 b = 8γ6 5(ma)2 +O(γ 6),(A14) where we used...

    work page

  3. [3]

    with no analytic solution

    = (1 +r′2 4 )−1,(A4) andφ 1 satisfies equation ∂2 r′ 4 + 3 r′ 4 ∂r′ 4 + 24φ′ 0 2 φ′ 1 =φ′ 0. with no analytic solution. However, the asymptotics φ′ 1 = 1 2 lnr′ 4 + c′ 0 2 +O(r′ 4)−2 atr ′ 4≫1,(A5) is fixed by the equation up to an unknown constantc ′ 0. In the tail regionr 4≫athe field mass is important, but the amplitude is already small. Changing the v...

    work page

  4. [4]

    +O(ma) 4 .(A3) Hereφ 0 is the Fubini-Lipatov instanton (6), φ′ 0(r′

    work page

  5. [5]

    Regularized bounces near large black holes We proceed by placing the small-size bounce onto the black horizon in the regularized model (30), see Fig. 13. We will assume that the black hole is large,r s≫m−1, and discuss generalization to smallerr s afterwards. This approach is simple because well-localized bounces barely feel the spacetime curvature. Indee...

    work page

  6. [6]

    Isidori, G

    G. Isidori, G. Ridolfi, and A. Strumia, On the metasta- bility of the standard model vacuum, Nucl. Phys. B609, 387 (2001), arXiv:hep-ph/0104016

    arXiv 2001

  7. [7]

    Isidori, V

    G. Isidori, V. S. Rychkov, A. Strumia, and N. Tetradis, Gravitational corrections to standard model vacuum de- cay, Phys. Rev. D77, 025034 (2008), arXiv:0712.0242

    arXiv 2008

  8. [8]

    A. V. Bednyakov, B. A. Kniehl, A. F. Pikelner, and O. L. Veretin, Stability of the Electroweak Vacuum: Gauge In- dependence and Advanced Precision, Phys. Rev. Lett. 115, 201802 (2015), arXiv:1507.08833

    arXiv 2015

Show all 72 references
  1. [9]

    Di Luzio, G

    L. Di Luzio, G. Isidori, and G. Ridolfi, Stability of the electroweak ground state in the Standard Model and its extensions, Phys. Lett. B753, 150 (2016), arXiv:1509.05028

    2016 arXiv

  2. [10]

    A. V. Bednyakov, A. S. Fedoruk, and D. I. Kazakov, Renormalization-group analysis of the SM: Loops, uncer- tainties, and vacuum stability, Phys. Rev. D113, 036018 (2026), arXiv:2509.03369

    2026

  3. [11]

    W. A. Hiscock, Can black holes nucleate vacuum phase transitions?, Phys. Rev. D35, 1161 (1987)

    1987

  4. [12]

    V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, Black holes initiate false vacuum decay, Phys. Rev. D43, 3112 (1991)

    1991

  5. [13]

    Gregory, I

    R. Gregory, I. G. Moss, and B. Withers, Black holes as bubble nucleation sites, JHEP2014(03), 081, arXiv:1401.0017

    arXiv

  6. [14]

    Burda, R

    P. Burda, R. Gregory, and I. Moss, Gravity and the sta- bility of the Higgs vacuum, Phys. Rev. Lett.115, 071303 (2015), arXiv:1501.04937

    2015 arXiv

  7. [15]

    Burda, R

    P. Burda, R. Gregory, and I. Moss, Vacuum metasta- bility with black holes, JHEP2015(08), 114, arXiv:1503.07331

    arXiv

  8. [16]

    Burda, R

    P. Burda, R. Gregory, and I. Moss, The fate of the Higgs vacuum, JHEP2016(06), 025, arXiv:1601.02152

    arXiv

  9. [17]

    Tetradis, Black holes and higgs stability, JCAP2016 (09), 036, arXiv:1606.04018

    N. Tetradis, Black holes and higgs stability, JCAP2016 (09), 036, arXiv:1606.04018

    arXiv

  10. [18]

    Briaud, A

    V. Briaud, A. Shkerin, and S. Sibiryakov, Thermal false vacuum decay around black holes, Phys. Rev. D106, 125001 (2022), arXiv:2210.08028

    2022 arXiv

  11. [19]

    Gorbunov, D

    D. Gorbunov, D. Levkov, and A. Panin, Fatal youth of the Universe: black hole threat for the electroweak vacuum during preheating, JCAP2017(10), 016, arXiv:1704.05399

    arXiv

  12. [20]

    Mukaida and M

    K. Mukaida and M. Yamada, False Vacuum Decay Cat- alyzed by Black Holes, Phys. Rev. D96, 103514 (2017), arXiv:1706.04523

    2017 arXiv

  13. [21]

    Hayashi, K

    T. Hayashi, K. Kamada, N. Oshita, and J. Yokoyama, On catalyzed vacuum decay around a radiating black hole and the crisis of the electroweak vacuum, JHEP2020 (08), 088, arXiv:2005.12808

    2005 arXiv

  14. [22]

    Shkerin and S

    A. Shkerin and S. Sibiryakov, Black hole induced false vacuum decay from first principles, JHEP2021(11), 197, arXiv:2105.09331

    arXiv

  15. [23]

    Shkerin and S

    A. Shkerin and S. Sibiryakov, Black hole induced false vacuum decay: the role of greybody factors, JHEP2022 (08), 161, arXiv:2111.08017

    arXiv

  16. [24]

    Strumia, Black holes don’t source fast Higgs vacuum decay, JHEP2023(03), 039, arXiv:2209.05504

    A. Strumia, Black holes don’t source fast Higgs vacuum decay, JHEP2023(03), 039, arXiv:2209.05504

    arXiv

  17. [25]

    Kohri and H

    K. Kohri and H. Matsui, Electroweak Vacuum Col- lapse induced by Vacuum Fluctuations of the Higgs Field around Evaporating Black Holes, Phys. Rev. D98, 123509 (2018), arXiv:1708.02138

    2018 arXiv

  18. [26]

    Hamaide, L

    L. Hamaide, L. Heurtier, S.-Q. Hu, and A. Cheek, Pri- mordial black holes are true vacuum nurseries, Phys. Lett. B856, 138895 (2024), arXiv:2311.01869

    2024 arXiv

  19. [27]

    D.-C. Dai, R. Gregory, and D. Stojkovic, Connecting the Higgs Potential and Primordial Black Holes, Phys. Rev. D101, 125012 (2020), arXiv:1909.00773

    2020 arXiv

  20. [28]

    Cuspinera, R

    L. Cuspinera, R. Gregory, K. M. Marshall, and I. G. Moss, Higgs Vacuum Decay in a Braneworld, Int. J. Mod. Phys. D29, 2050005 (2020), arXiv:1907.11046

    2020 arXiv

  21. [29]

    J. B. Hartle and S. W. Hawking, Path Integral Derivation of Black Hole Radiance, Phys. Rev. D13, 2188 (1976)

    1976

  22. [30]

    G. W. Gibbons and S. W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15, 2752 (1977)

    1977

  23. [31]

    Miyachi and J

    T. Miyachi and J. Soda, False vacuum decay in a two- dimensional black hole spacetime, Phys. Rev. D103, 085009 (2021), arXiv:2102.02462

    2021 arXiv

  24. [32]

    V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, O(3) Invariant Tunneling in General Relativity, Phys. Lett. B 207, 397 (1988)

    1988

  25. [33]

    P. B. Arnold, Gravity and false vacuum decay rates: O(3) solutions, Nucl. Phys. B346, 160 (1990)

    1990

  26. [34]

    A. N. Kuznetsov and P. G. Tinyakov, Periodic instanton bifurcations and thermal transition rate, Phys. Lett. B 406, 76 (1997), arXiv:hep-ph/9704242

    1997 arXiv

  27. [35]

    I. Y. Kobzarev, L. B. Okun, and M. B. Voloshin, Bubbles in Metastable Vacuum, Yad. Fiz.20, 1229 (1974)

    1974

  28. [36]

    S. R. Coleman, The Fate of the False Vacuum. 1. Semi- classical Theory, Phys. Rev. D15, 2929 (1977), [Erratum: Phys.Rev.D 16, 1248 (1977)]

    1977

  29. [37]

    S. R. Coleman, The Uses of Instantons, Subnucl. Ser.15, 805 (1979)

    1979

  30. [38]

    A. D. Linde, Fate of the False Vacuum at Finite Temper- ature: Theory and Applications, Phys. Lett. B100, 37 (1981)

    1981

  31. [39]

    A. D. Linde, Decay of the False Vacuum at Finite Tem- perature, Nucl. Phys. B216, 421 (1983), [Erratum: Nucl.Phys.B 223, 544 (1983)]

    1983

  32. [40]

    S. Y. Khlebnikov, V. A. Rubakov, and P. G. Tinyakov, Periodic instantons and scattering amplitudes, Nucl. Phys. B367, 334 (1991)

    1991

  33. [41]

    J. S. Langer, Statistical theory of the decay of metastable states, Annals Phys.54, 258 (1969)

    1969

  34. [42]

    D. Y. Grigoriev, V. A. Rubakov, and M. E. Shaposh- nikov, Sphaleron Transitions at Finite Temperatures: Numerical Study in (1+1)-dimensions, Phys. Lett. B 216, 172 (1989)

    1989

  35. [43]

    Pˆ ırvu, A

    D. Pˆ ırvu, A. Shkerin, and S. Sibiryakov, Thermal false vacuum decay in (1+1) dimensions: Evidence for nonequilibrium dynamics, Int. J. Mod. Phys. A39, 2445007 (2024), arXiv:2408.06411

    2024 arXiv

  36. [44]

    V. A. Rubakov,Classical theory of gauge fields(Prince- 18 ton University Press, Princeton, New Jersey, 2002)

    2002

  37. [45]

    N. S. Manton, Topology in the Weinberg-Salam Theory, Phys. Rev. D28, 2019 (1983)

    1983

  38. [46]

    F. R. Klinkhamer and N. S. Manton, A Saddle Point Solution in the Weinberg-Salam Theory, Phys. Rev. D 30, 2212 (1984)

    1984

  39. [47]

    C. G. Callan, Jr. and S. R. Coleman, The Fate of the False Vacuum. 2. First Quantum Corrections, Phys. Rev. D16, 1762 (1977)

    1977

  40. [48]

    Fubini, A New Approach to Conformal Invariant Field Theories, Nuovo Cim

    S. Fubini, A New Approach to Conformal Invariant Field Theories, Nuovo Cim. A34, 521 (1976)

    1976

  41. [49]

    L. N. Lipatov, Divergence of the perturbation-theory se- ries and the quasi-classical theory, Sov. Phys. JETP45, 216 (1977)

    1977

  42. [50]

    Affleck, On Constrained Instantons, Nucl

    I. Affleck, On Constrained Instantons, Nucl. Phys. B191, 429 (1981)

    1981

  43. [51]

    A. N. Kuznetsov and P. G. Tinyakov, False vacuum de- cay induced by particle collisions, Phys. Rev. D56, 1156 (1997), arXiv:hep-ph/9703256

    1997 arXiv

  44. [52]

    J. I. Kapusta and C. Gale,Finite-temperature field theory: Principles and applications, Cambridge Mono- graphs on Mathematical Physics (Cambridge University Press, 2011)

    2011

  45. [53]

    A. N. Kuznetsov and P. G. Tinyakov, Numerical study of induced false vacuum decay at high-energies, Mod. Phys. Lett. A11, 479 (1996), arXiv:hep-ph/9510310

    1996 arXiv

  46. [54]

    S. R. Coleman, V. Glaser, and A. Martin, Action Minima Among Solutions to a Class of Euclidean Scalar Field Equations, Commun. Math. Phys.58, 211 (1978)

    1978

  47. [55]

    S. D. Hampton, K. Lee, and S. Lee, Resurgence of the Thermal Transition between Bounce and Sphaleron (2026), arXiv:2606.13778

    2026

  48. [56]

    Shkerin, On thermal false vacuum decay around black holes, PoSICPPCRubakov2023, 018 (2024)

    A. Shkerin, On thermal false vacuum decay around black holes, PoSICPPCRubakov2023, 018 (2024)

    2024

  49. [57]

    Weyl, The theory of gravitation, Annalen Phys.54, 117 (1917)

    H. Weyl, The theory of gravitation, Annalen Phys.54, 117 (1917)

    1917

  50. [58]

    W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,Numerical Recipes: The Art of Scien- tific Computing (Third Edition)(Cambridge University Press, 2007)

    2007

  51. [59]

    Levkov, E

    D. Levkov, E. Nugaev, and A. Popescu, The fate of small classically stable Q-balls, JHEP12, 131 (2017), arXiv:1711.05279

    2017 arXiv

  52. [60]

    S. R. Coleman, Quantum Tunneling and Negative Eigen- values, Nucl. Phys. B298, 178 (1988)

    1988

  53. [61]

    S. V. Demidov and D. G. Levkov, Soliton-antisoliton pair production in particle collisions, Phys. Rev. Lett.107, 071601 (2011), arXiv:1103.0013

    2011 arXiv

  54. [62]

    Demidov and D

    S. Demidov and D. Levkov, High-energy limit of collision- induced false vacuum decay, JHEP06, 123 (2015), arXiv:1503.06339

    2015 arXiv

  55. [63]

    S. V. Demidov and D. G. Levkov, Semiclassical descrip- tion of soliton-antisoliton pair production in particle col- lisions, JHEP2015(11), 066, arXiv:1509.07125

    arXiv

  56. [64]

    S. V. Demidov, B. R. Farkhtdinov, and D. G. Levkov, Suppression exponent for multiparticle production inλϕ4 theory, JHEP02, 205 (2023), arXiv:2212.03268

    2023 arXiv

  57. [65]

    Avraham, K

    G. Avraham, K. Blum, O. Rosner, and I. G. Smith, The O(4)-breaking bubble (2026), arXiv:2606.16024

    2026

  58. [66]

    B. Hu, K. Kamada, and A. Shkerin, False vacuum de- cay catalyzed by black hole in a heat bath (2026), arXiv:2603.17008

    2026

  59. [67]

    Gazizov, D

    R. Gazizov, D. Gorbunov, and D. Levkov, (2026), to be published

    2026

  60. [68]

    G. W. Anderson, New Cosmological Constraints on the Higgs Boson and Top Quark Masses, Phys. Lett. B243, 265 (1990)

    1990

  61. [69]

    P. B. Arnold and S. Vokos, Instability of hot electroweak theory: bounds on m(H) and M(t), Phys. Rev. D44, 3620 (1991)

    1991

  62. [70]

    Delle Rose, C

    L. Delle Rose, C. Marzo, and A. Urbano, On the fate of the Standard Model at finite temperature, JHEP2016 (05), 050, arXiv:1507.06912

    arXiv

  63. [71]

    C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravi- tation(W. H. Freeman, San Francisco, 1973)

    1973

  64. [72]

    J. R. Shewchuk,An introduction to the conjugate gradient method without the agonizing pain, Tech. Rep. (Carnegie Mellon University, 1994)

    1994

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