The DOI matches its Crossref record; 130 of the 130 references it cites were found in a registry or an index; 20 of the 32 sources its claims cite were checked for support.
Vacuum metastability with black holes
Open accessAbstract
We consider the possibility that small black holes can act as nucleation seeds for the decay of a metastable vacuum, focussing particularly on the Higgs potential. Using a thin-wall bubble approximation for the nucleation process, which is possible when generic quantum gravity corrections are added to the Higgs potential, we show that primordial black holes can stimulate vacuum decay. We demonstrate that for suitable parameter ranges, the vacuum decay process dominates over the Hawking evaporation process. Finally, we comment on the application of these results to vacuum decay seeded by black holes produced in particle collisions.
Overview
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Written from the paper’s full text.
TL;DR
The paper argues that small black holes can seed vacuum decay of a metastable Higgs vacuum, and that in suitable parameter ranges this seeded decay can dominate over Hawking evaporation. Using a thin-wall instanton approximation, it finds greatly enhanced nucleation rates relative to Coleman–de Luccia, with implications for primordial black holes and possibly collider black holes.
Key findings
- Black holes can act as nucleation sites for false-vacuum decay, with the dominant process depending on seed mass: for \(M_+ < M_C\) the bubble removes the black hole, while for \(M_+ > M_C\) a static bubble leaves a remnant black hole.
- For modified Higgs-potential examples in Table 1, the unseeded CDL exponent is \(B_{\rm CDL}\sim 4\times 10^6\), while the critical seeded bounce is much smaller, with \(B_C/B_{\rm CDL}\) ranging from \(1.2\times 10^{-6}\) down to \(9.4\times 10^{-9}\), and \(M_C/M_p\) ranging from about \(3.5\) down to \(0.28\).
- The branching ratio of vacuum decay to Hawking evaporation is \(\Gamma_D/\Gamma_H \approx 44\,(M_+^2/M_p^2)\,B^{1/2}e^{-B}\), using \(\Gamma_H \approx 3.6\times 10^{-4}(G^2M_+^3)^{-1}\); the paper shows this can exceed unity for seed masses well above the Planck mass.
- Magnetically charged black holes are considered; their reduced Hawking flux does not particularly assist vacuum decay, and in higher-dimensional models the branching ratio decreases as the number of extra dimensions increases.
Methods
- Constructs thin-wall Euclidean instantons for a bubble wall separating false and true vacua on a black-hole background, using Israel junction conditions and a master bounce-action expression, Eq. (2.22).
- Applies the formalism to a modified Higgs potential \(V(\phi)=\frac14\lambda_{\rm eff}\phi^4+\frac14(\delta\lambda)_{\rm bsm}\phi^4+\frac16\lambda_6\phi^6/M_p^2+\cdots\), with thin-wall parameters chosen to allow the approximation.
- Numerically computes bounce actions and compares the seeded nucleation rate to the Hawking evaporation rate for uncharged and monopole-charged black holes, and outlines a higher-dimensional extension.
Limitations
- The paper states that the thin-wall approximation is valid only in part of the parameter space and requires a relatively large positive \(\phi^6\) term; a companion paper is planned to treat thick-wall bubbles and extend to \(\lambda_6=0\).
- The calculation assumes a static black hole and neglects its time dependence from Hawking evaporation, except by comparing timescales and requiring vacuum decay to be faster than evaporation.
- The higher-dimensional collider-black-hole discussion is explicitly described as crude and problematic because there is no exact braneworld black-hole solution and the Higgs should be confined to the brane.
Model-written summary. Check it against the paper before relying on it.
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Claims the paper makes
20 of 32 cited sources checked · 14 supported · 1 flagged · accuracy not yet measured
Every source the paper cites for this claim was checked and supports it.
In the paper
“As in our previous paper, we show that the nucleation probability is welldefined.”
§ Prepared for > 1 Introduction
What the sources say
- Supportedcites [21] R. Gregory, I.G. Moss and B. Withers, Black holes as bubble nucleation sites, JHEP 03 (2014) 081 [ arXiv:1401.0017 ] [ INSPIRE ]. source
The paper states that accounting for conical singularities makes the Euclidean action independent of the choice of period, removing a possible ambiguity in tunnelling rates for singular instantons. Its action derivation (section 3) likewise says that this source of ambiguity is annulled, supporting the claim that the nucleation probability is well-defined for the instantons considered. (LLM, conf=0.91)
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“The physical process of vacuum decay with an inhomogeneity can be represented gravitationally by a Euclidean solution with two 'Schwarzschild' bulks which have different cosmological constants separated by a thin wall with constant tension (for a general proof of this result in the context of braneworlds, see [45,46]).”
§ Prepared for > 2.1 Constructing the instanton
What the sources say
- Partly supportedcites [45] P. Bowcock, C. Charmousis and R. Gregory, General brane cosmologies and their global space-time structure, Class. Quant. Grav. 17 (2000) 4745 [ hep-th/0007177 ] [ INSPIRE ]. sourceExperimental — accuracy not yet measured
The Discussion describes cosmological brane solutions joining two Schwarzschild-AdS spacetimes, and Section IV discusses domain walls between bulk regions. But the cited paper does not establish the claimed Euclidean solution for vacuum decay with an inhomogeneity; its result concerns homogeneous and isotropic brane cosmologies. Searches for “Euclidean” and “vacuum decay” found no passage addressing that process. (LLM, conf=0.85)
- Partly supportedcites [46] R. Gregory and A. Padilla, Nested brane worlds and strong brane gravity, Phys. Rev. D 65 (2002) 084013 [ hep-th/0104262 ] [ INSPIRE ]. sourceExperimental — accuracy not yet measured
In the passage giving the general bulk metric (section labeled Abstract, page unavailable), the paper describes a brane boundary joining bulk regions and notes that one metric case is analogous to a Euclidean black hole. Its closing discussion says the approach could be modified to join spacetimes with different cosmological constants. It does not establish the claimed Euclidean representation of inhomogeneous vacuum decay; the solution it develops assumes reflection symmetry and a vanishing bulk integration constant. (LLM, conf=0.85)
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The bubble wall has constant tension.
- Partly supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“In recent work [21], following earlier work by Hiscock and Berezin [22,23], we looked at the effect of gravitational inhomogeneities acting as seeds of cosmological phase transitions in de Sitter space.We found that the decay rates were considerably enhanced by the presence of black holes.”
§ Prepared for > 1 Introduction
What the sources say
- Supportedcites [21] R. Gregory, I.G. Moss and B. Withers, Black holes as bubble nucleation sites, JHEP 03 (2014) 081 [ arXiv:1401.0017 ] [ INSPIRE ]. source
The introduction (section 1) identifies false-vacuum decay around a black hole in a Schwarzschild–de Sitter background as the study's setting and says the inhomogeneity enhances true-vacuum bubble production. The abstract and discussion (section 6) report substantially higher nucleation rates than the homogeneous Coleman–de Luccia case. The discussion notes an exception for seed black holes approaching the Nariai limit, which does not undermine the citing sentence's general summary. (LLM, conf=0.94)
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“Nevertheless, the recent discovery of the Higgs boson [1,2] raises the possibility that, even within the standard model of particle physics, the present vacuum state of the universe may not be stable, but only metastable, with another lower energy state at high expectation values of the Higgs field [3][4][5][6][7].”
§ Prepared for > 1 Introduction
What the sources say
- Supportedcites [3] G. Degrassi et al., Higgs mass and vacuum stability in the standard model at NNLO, JHEP 08 (2012) 098 [ arXiv:1205.6497 ] [ INSPIRE ]. source
Section 3.3 reports a significant preference for metastability of the Standard Model potential. The Conclusions state that the Higgs quartic coupling becomes negative at high energies while tunneling remains slow enough for the electroweak vacuum to be metastable, supporting the possibility of a lower-energy state at large Higgs-field values. Our copy does not provide page numbers for these passages. (LLM, conf=0.90)
- Supportedcites [4] A. Gorsky, A. Mironov, A. Morozov and T.N. Tomaras, Is the standard model saved asymptotically by conformal symmetry?, J. Exp. Theor. Phys. 120 (2015) 344 [ arXiv:1409.0492 ] [ INSPIRE ]. source
The Effective potential section says that, for central experimental Standard Model parameters, a second minimum is deeper than the Standard Model minimum and the present vacuum is metastable. The Introduction identifies that second minimum with a Higgs-field expectation value near the Planck scale. The claim's ‘may’ appropriately reflects the paper's qualification that the second minimum depends on the parameters. (LLM, conf=0.94)
- Supportedcites [5] F. Bezrukov and M. Shaposhnikov, Why should we care about the top quark Yukawa coupling?, J. Exp. Theor. Phys. 120 (2015) 335 [ arXiv:1411.1923 ] [ INSPIRE ]. source
Section III, “Vacuum Stability and Cosmology,” says that in the Standard Model an additional minimum can develop at large Higgs-field values. It states that when the top Yukawa coupling exceeds its critical value, this minimum is deeper than the electroweak vacuum, making our vacuum metastable. The citing sentence appropriately presents this as a possibility. (LLM, conf=0.94)
- Supportedcites [6] J. Ellis, Discrete glimpses of the physics landscape after the Higgs discovery, J. Phys. Conf. Ser. 631 (2015) 012001 [ arXiv:1501.05418 ] [ INSPIRE ]. source
Section 4.1, “The Instability of the Electroweak Vacuum,” says that Standard Model calculations turn the Higgs potential downward at large field values, making the present electroweak vacuum unstable in principle. It also says the vacuum's likely lifetime is much longer than the age of the Universe, consistent with the claim's possibility of metastability. (LLM, conf=0.94)
- Could not verifycites [7] K. Blum, R.T. D’Agnolo and J. Fan, Vacuum stability bounds on Higgs coupling deviations in the absence of new bosons, JHEP 03 (2015) 166 [ arXiv:1502.01045 ] [ INSPIRE ]. sourceExperimental — accuracy not yet measured
The Introduction discusses Standard Model vacuum stability and says the measured Higgs and top masses are consistent with stability up to very high scales. The paper’s instability analysis concerns models with new fermions beyond the Standard Model; the passages found do not establish the claimed possibility of a lower-energy state at high Higgs-field values within the Standard Model itself. (LLM, conf=0.73)
Load-bearingStrength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Supported
The sources with a verdict support this claim; the rest weren’t checked yet, have no full text we could read, or their verdicts aren’t rated yet.
In the paper
“Meanwhile the Hawking temperature of the higher dimensional black hole is [63,64] T H = (D -3) 4πr + (C.8)”
§ Prepared for > C Higher dimensional instantons
What the sources say
- Not checked — no full text of the sourcecites [63] R.C. Myers and M.J. Perry, Black holes in higher dimensional space-times, Annals Phys. 172 (1986) 304 [ INSPIRE ]. source
- Supportedcites [64] P. Kanti, Black holes in theories with large extra dimensions: a review, Int. J. Mod. Phys. A 19 (2004) 4899 [ hep-ph/0402168 ] [ INSPIRE ]. source
Section 3, Eq. (10) gives the Hawking temperature as T_H = (n+1)/(4πr_H). The same section, Eq. (7), defines D = 4+n, so n+1 = D−3; r_H is the horizon radius denoted r+ in the claim. Page numbers are not provided in our copy. (LLM, conf=0.94)
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesEvery source the paper cites for this claim was checked and supports it.
In the paper
“If there is a horizon at r = r h , then by properly treating the conical singularity as in [21], we get an additional area term contribution to the action”
§ Prepared for > B Alternative bounce action calculation
What the sources say
- Supportedcites [21] R. Gregory, I.G. Moss and B. Withers, Black holes as bubble nucleation sites, JHEP 03 (2014) 081 [ arXiv:1401.0017 ] [ INSPIRE ]. source
Section 3.1 describes smoothing and evaluating conical singularities, and the appendix (eq. A.5) finds that a defect contributes −A/4G to the action. This supports the cited method and the additional horizon-area term; the citation does not need to supply the citing paper’s separate βM term. (LLM, conf=0.93)
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“This result generalises a previous result of Hawking and Horowitz, who found the same formula for the Euclidean action of static Einstein-matter solutions with Λ = 0 and no conical singularities [67].”
§ Prepared for > B Alternative bounce action calculation
What the sources say
- Supportedcites [67] S.W. Hawking and G.T. Horowitz, The gravitational hamiltonian, action, entropy and surface terms, Class. Quant. Grav. 13 (1996) 1487 [ gr-qc/9501014 ] [ INSPIRE ]. source
Section 4, “Horizons and the Euclidean action” (page unavailable), gives the Euclidean-action formula ĨP = βHP − Abh/4 for nonextreme black holes and fixes the imaginary-time period by requiring a regular Euclidean solution at the horizon. The paper treats static solutions with matter; the claimed Λ = 0 case falls within its discussion. (LLM, conf=0.90)
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passes- Supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“To estimate the decay rate of the black hole, we will assume that the main channel is due to emission of particles on the brane [69,70], leading to”
§ Prepared for > C Higher dimensional instantons
What the sources say
- Supportedcites [69] R. Emparan, G.T. Horowitz and R.C. Myers, Black holes radiate mainly on the brane, Phys. Rev. Lett. 85 (2000) 499 [ hep-th/0003118 ] [ INSPIRE ]. source
The Introduction (page unavailable) concludes that most Hawking-radiation energy goes into fields on the brane. The citation is attached to the assumption that brane emission is the main channel; the decay-rate formula follows as the citing authors’ calculation, rather than a formula attributed to this source. (LLM, conf=0.91)
- Supportedcites [70] C.M. Harris and P. Kanti, Hawking radiation from a (4 + n)-dimensional black hole: exact results for the Schwarzschild phase, JHEP 10 (2003) 014 [ hep-ph/0309054 ] [ INSPIRE ]. source
The citation is attached to the assumption that brane emission is the main channel, not to the decay-rate formula that follows. Section 5.2 reports that brane-localized scalar emission dominates bulk scalar emission for the dimensions studied and says its results support the broader expectation that higher-dimensional black holes emit mainly brane modes. It notes that confirming this for gravitons requires further analysis. (LLM, conf=0.87)
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: In the higher dimensional model, the main channel for black hole decay is emission of particles on the brane.
- Supported
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“and the action can be computed analytically as B CDL = - 1 2G R( τ+ -τ-) = π G σγ 3 α(α + 1)(α + 1 -2σγ) Λ + =0 ---→ πℓ 2 G 16(σℓ) 4 (1 -4σ 2 ℓ 2 ) 2 (3.2)”
§ Prepared for > 3.1 Coleman de Luccia
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It is in fact even valid when the black hole is charged, as we will consider in the next section.”
§ Prepared for > 2.2 Computing the action
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes
- Probably not a substantial claim. 25
- It may state two separate claims as one. 51
- The quote only partly states this claim. 5
At least one cited source doesn’t support this claim or points to a different paper, and none confirmed it — see the verdicts below.
In the paper
“Note that unlike the figure in [21], where the lower limit for k 1 was dependent on σ for negative but not positive k 2 , in this case it is the lower limits for positive k 2 and not negative that depend on σ.”
§ Prepared for > A The limits on k 1 , k 2
What the sources say
- Disputedcites [21] R. Gregory, I.G. Moss and B. Withers, Black holes as bubble nucleation sites, JHEP 03 (2014) 081 [ arXiv:1401.0017 ] [ INSPIRE ]. sourceExperimental — accuracy not yet measured
Appendix B, “The limits on κ” (Eq. B.4; page unavailable), gives the lower bound on κ₁ as a function of both κ₂ and σ. It says a nontrivial lower bound occurs for negative κ₂ when σℓ < 1/2, but also treats positive κ₂ when σℓ > 1/2. Thus the claim that the source’s lower limits depend on σ for negative but not positive κ₂ is contrary to the source. (LLM, conf=0.86)
Checked by a reasoning model — may be wrong.
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
- Disputed
The sources with a verdict support this claim; the rest weren’t checked yet, have no full text we could read, or their verdicts aren’t rated yet.
In the paper
“Magnetically charged black holes may be produced in the early universe [27,28], and form the seeds for vacuum decay of an unstable standard model Higgs field.”
§ Prepared for > 3.3 Charged black hole instantons
What the sources say
- Not checked — no full text of the sourcecites [27] F. Mellor and I. Moss, Black holes and quantum wormholes, Phys. Lett. B 222 (1989) 361 [ INSPIRE ]. source
- Supportedcites [28] F. Mellor and I. Moss, Black holes and gravitational instantons, Class. Quant. Grav. 6 (1989) 1379 [ INSPIRE ]. source
The Introduction describes charged black-hole pair creation as a possible real process during a de Sitter stage of the very early universe. Section 2 identifies magnetic charge as one of the charges in the black-hole solutions. The citation is attached to the early-universe production clause, not the subsequent claim about Higgs-field vacuum decay. (LLM, conf=0.88)
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“The tidal black holes, for example, resemble black-hole monopoles, but with negative square monopole charge P 2 .”
§ Prepared for > 5 Conclusions
What the sources say
- Partly supportedcites [66] N. Dadhich, R. Maartens, P. Papadopoulos and V. Rezania, Black holes on the brane, Phys. Lett. B 487 (2000) 1 [ hep-th/0003061 ] [ INSPIRE ]. sourceExperimental — accuracy not yet measured
The Introduction describes a tidal black hole with the mathematical form of a Reissner–Nordström solution, and Section IV says negative tidal charge is the physically more natural case. Our copy does not compare it to a black-hole monopole or identify its charge as a negative square monopole charge P²; Section III instead calls the parameter q a tidal charge. (LLM, conf=0.85)
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
- Partly supported
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“Now consider 2σγ < 1/2, for which k 2 < 0 is allowed.A similar argument to [21] gives”
§ Prepared for > A The limits on k 1 , k 2
What the sources say
- Partly supportedcites [21] R. Gregory, I.G. Moss and B. Withers, Black holes as bubble nucleation sites, JHEP 03 (2014) 081 [ arXiv:1401.0017 ] [ INSPIRE ]. sourceExperimental — accuracy not yet measured
Appendix B derives a lower bound on κ₁ and allows κ₂ < 0 in part of the parameter range, supporting the claim that a similar bounding argument can be used. It discusses σℓ < 1/2 rather than the stated 2σγ < 1/2 condition, and it does not give the quoted α-dependent expression for positive κ₂. No page number is available in our copy. (LLM, conf=0.84)
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one. The quote only partly states this claim.
- Partly supported
Every source the paper cites for this claim was checked and supports it.
In the paper
“The equations of motion have the same schematic form as (2.6)”
§ Prepared for > C Higher dimensional instantons
What the sources say
- Supportedcites [46] R. Gregory and A. Padilla, Nested brane worlds and strong brane gravity, Phys. Rev. D 65 (2002) 084013 [ hep-th/0104262 ] [ INSPIRE ]. source
The passage labeled Abstract in our copy gives the bulk potential h(Z) in equation (5), then derives the brane equations of motion in equations (6a–c). It also defines σₙ = 8πGₙσ/[2(n−2)], equivalent to the stated 4πGσ/(D−2) notation. The displayed equations assume the integration constant c vanishes. (LLM, conf=0.86)
- Supportedcites [68] R. Gregory and A. Padilla, Brane world instantons, Class. Quant. Grav. 19 (2002) 279 [ hep-th/0107108 ] [ INSPIRE ]. source
Section II, equations (9a–c), gives the brane equations of motion in terms of the bulk function h and the normalized tension σₙ; equation (10) defines σₙ = 4πGₙσ/(n−2). The same section defines h in equation (7) and describes its c > 0 case as Schwarzschild–AdS. These passages support the claimed schematic form; the citing paper’s equation (2.6) is not part of the source paper. (LLM, conf=0.91)
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Supported
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Initially, the static bubble shrinks with increasing GM + , but remains well outside the Schwarzschild radius, however, as we increase GM + further, the bubble becomes constrained by the expanding black hole horizon, and becomes stretched just outside RSCH .”
§ Prepared for > 3.2 The general instanton
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Eventually, at GM C , the two roots of U meet, and the static branch begins.”
§ Prepared for > 3.2 The general instanton
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In this special case the symmetry of the bubble solution has been raised from O(3) to O(4), and the result for the tunnelling rate agrees with explicitly O(4) symmetric methods.”
§ Prepared for > 3.1 Coleman de Luccia
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This expression is the central result of this section, and is independent of any choice of periodicity of Euclidean time, and independent of the choices of cosmological constant on each side of the wall.”
§ Prepared for > 2.2 Computing the action
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Meanwhile, the remnant black hole mass GM -increases along the static branch and eventually becomes larger than GM + , however, because of the negative cosmological constant, the horizon radius, while increasing, does not increase as rapidly as RSCH .”
§ Prepared for > 3.2 The general instanton
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The values of RMAX / RMIN are well outside the black hole horizon radius, and move together as GM + is increased.”
§ Prepared for > 3.2 The general instanton
Strength:boosted· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This qualitative picture remains true irrespective of the values of Λ ± : for seed mass M + < M C , the dominant tunnelling process leaves behind a true vacuum region and removes the black hole.”
§ Prepared for > 3.2 The general instanton
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The static bubble action therefore increases as GM + increases, eventually becoming larger than B CDL (see figure 3).”
§ Prepared for > 3.2 The general instanton
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For fixed seed mass, M + , there will be a range of allowed k 1 and k 2 (see (A.5)), and a corresponding range of values for the bounce action.”
§ Prepared for > 3.2 The general instanton
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The static branch is the preferred instanton with nonzero GM -, i.e. with a black hole remnant, although non-static solutions exist with higher action and remnant mass.”
§ Prepared for > 3.2 The general instanton
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Putting all of these results together, we find that the action of the instanton solution is I E = - A - 4G + 1 2G dλ [(R -3GM + ) τ+ -(R -3GM -) τ-] + βM + Λ + ≤ 0 - A c 4G Λ + > 0 (2.21)”
§ Prepared for > 2.2 Computing the action
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The upper limit on k 2 is determined by M -= 0 for the static bounce, i.e. which is clearly inconsistent for Λ + ≤ 0. For Λ + > 0, the range closes off at”
§ Prepared for > A The limits on k 1 , k 2
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“thus GM -≥ 0 ⇒ k 1 ≥ 2(1 -α) k 2 α GM + ≥ 0 ⇒ k 1 ≥ 2(1 -α + 2σγ) k 2 α (A.6)”
§ Prepared for > A The limits on k 1 , k 2
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The plot of the dependence of the action on GM + /ℓ is surprisingly similar to the uncharged case at fixed ratio P/M + , with one small modification.”
§ Prepared for > 3.3 Charged black hole instantons
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The static branch meets this time-dependent branch at a critical mass M CP , where the static bubble now has an extremal black hole in its interior.”
§ Prepared for > 3.3 Charged black hole instantons
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In the paper
“Then setting R = αR/γ, λ = αλ/γ gives the equation of motion”
§ Prepared for > C Higher dimensional instantons
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In the paper
“The timescales for Hawking evaporation and vacuum decay will both be less than roughly a million Planck times.”
§ Prepared for > 4 Application to the Higgs vacuum.
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In the paper
“The overall picture is that for lower σ and higher ℓ (or more negative λ * / higher φ * ) the branching ratio is larger, and is consistently higher than unity over a larger range.”
§ Prepared for > 4 Application to the Higgs vacuum.
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In the paper
“The static bubble is the difference in areas of the seed and remnant black holes, which we can deduce from figure 4 to be roughly linear (there is actually a slightly stronger dependence on M + , however, what is important is that it is not quadratic)”
§ Prepared for > 4 Application to the Higgs vacuum.
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In the paper
“For a potential which allows decay from flat space to AdS, there are thin-wall bubble solutions with spherical symmetry and constant values of φ at the appropriate minima of the potential.”
§ Prepared for > 3.3 Charged black hole instantons
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A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In all cases the pure CDL tunneling action is extremely large (10 6-7 ), but the suppression of the critical tunneling action is also large, and increases as λ * becomes more negative.”
§ Prepared for > 4 Application to the Higgs vacuum.
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In the paper
“Secondly, the requirement of positivity of τ ± , of which the constraint on τ+ is the stronger:”
§ Prepared for > A The limits on k 1 , k 2
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In the paper
“The dip is however very minor, and the minimum action is well approximated by the value at M CP .”
§ Prepared for > 3.3 Charged black hole instantons
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In the paper
“The static action (which is what is needed for the branching ratio) is, as before, the difference in horizon areas:”
§ Prepared for > C Higher dimensional instantons
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In the paper
“Roughly speaking, as λ * becomes more negative, the values of λ 6 required for thin-wall become larger, similarly as φ * drops, λ 6 increases.”
§ Prepared for > 4 Application to the Higgs vacuum.
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In the paper
“From this, we see that Λ + > 0, 2σγ ≤ 1 + α/2, and k 1 is bounded below by k m 1 = 2(1 -α)k 2 /α from (A.6).”
§ Prepared for > A The limits on k 1 , k 2
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In the paper
“The time-dependent tunneling solutions prior to the switching on of the static bubbles now do not remove the black hole altogether as this would leave a naked singularity.Instead, the bubbles leave behind an extremal remnant, M -= M ext (P ), where”
§ Prepared for > 3.3 Charged black hole instantons
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In the paper
“which leads to a dimension dependent branching ratio”
§ Prepared for > 5 Conclusions
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In the paper
“The branching ratio Γ D /Γ H can now be re-computed using this evaporation rate and the vacuum decay rates from the previous section.The result is shown in figure 7.”
§ Prepared for > 4 Application to the Higgs vacuum.
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In the paper
“Therefore the tunnelling rate is determined by”
§ Prepared for > B Alternative bounce action calculation
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In the paper
“The range of Higgs model parameters λ * , φ * and λ 6 which allow thin-wall bubbles is set by ζ > > 0, and by the condition that the true vacuum lies at large φ.”
§ Prepared for > 4 Application to the Higgs vacuum.
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In the paper
“The field equations imply that H = H i = 0, and furthermore the symmetry implies ∂ τ φ = ∂ τ γ ij = 0. Therefore only the boundary term in (B.2) survives.”
§ Prepared for > B Alternative bounce action calculation
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In the paper
“From figure 5 we see the dip is most visible at large ratio P/M , however, perhaps surprisingly, it is also the case that at large P/M the catalytic effect of the black hole is much reduced.”
§ Prepared for > 3.3 Charged black hole instantons
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In the paper
“Since the bubble exterior is in the false vacuum, we have Λ + > Λ -.”
§ Prepared for > 2.1 Constructing the instanton
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In the paper
“In general, this would not conflict with observation because the lifetime of the present vacuum would be far longer than the age of the universe.”
§ Prepared for > 1 Introduction
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The nucleation process typically requires an instanton that has a conical singularity at the black hole horizon.”
§ Prepared for > 1 Introduction
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In the paper
“The fact that this has not happened therefore means that either the Higgs parameters are not the the relevant range (a small region of parameter space for this purely gravitational argument) or there are no primordial black holes in the observable universe.”
§ Prepared for > 1 Introduction
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- Rests on an assumption: A primordial black hole losing mass by the Hawking process decays down to a mass around 10-100 times the Planck mass and then seeds a vacuum transition.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In terms of the Higgs, this thin-wall bubble nucleation requires the potential to be relatively shallow at the true vacuum, and this requires a large positive φ 6 term.”
§ Prepared for > 1 Introduction
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- The authors state this as an assumption: The thin-wall bubble approximation is valid for the Higgs potential considered.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“what this expansion indicates is that the minimal bounce action for a particular σℓ can be extremely small, so that even if we are above the critical black hole mass, the decay rate can still be significant.”
§ Prepared for > 3.2 The general instanton
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In the paper
“Electrically charged black holes can discharge by the emission of charged particles [53], but magnetically charged black holes may be the lightest magnetically charged particles in the theory, in which case a large mass black hole evaporates towards the extremal limit, and the Hawking radiation flux falls to zero.”
§ Prepared for > 3.3 Charged black hole instantons
What the sources say
- Not checked — no full text of the sourcecites [53] G.W. Gibbons, Vacuum polarization and the spontaneous loss of charge by black holes, Commun. Math. Phys. 44 (1975) 245. source
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In the paper
“The present-day broken symmetry vacuum may therefore be a metastable state, but quantum tunnelling in the Higgs potential determined by the usual Coleman de Luccia expressions is very slow, and the lifetime of the false vacuum far exceeds the lifetime of the universe.”
§ Prepared for > 1 Introduction
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In the paper
“This latter solution is not stable, and small fluctuations will lead it to either expand as with the first situation completing the phase transition, or to collapse back inwards leaving the initial state unchanged.”
§ Prepared for > 1 Introduction
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In the paper
“The vacuum decay process is based on a static black hole, in which a bubble nucleates outside the black hole and either completely replaces the black hole with a bubble of true vacuum expanding outwards, or nucleates a static bubble leaving a remnant black hole surrounded by true vacuum.”
§ Prepared for > 1 Introduction
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In the paper
“There are no known exact solutions for these black hole plus brane systems, and instead the black hole is usually considered to be approximately a higher dimensional Schwarzschild or Myers-Perry [63] solution”
§ Prepared for > 5 Conclusions
What the sources say
- Not checked — no full text of the sourcecites [63] R.C. Myers and M.J. Perry, Black holes in higher dimensional space-times, Annals Phys. 172 (1986) 304 [ INSPIRE ]. source
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- Rests on an assumption: For brane black hole systems, the black hole can be approximated by a higher dimensional Schwarzschild or Myers-Perry solution.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Finally, for the case of a monopole charged black hole, we might expect the branching ratio to be larger due to their reduced Hawking radiation rate”
§ Prepared for > 4 Application to the Higgs vacuum.
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In the paper
“The expression (B.7) is of course the same as (2.16, 2.19), but we have not assumed anywhere for the static case that the bubble is in the thin-wall limit.”
§ Prepared for > B Alternative bounce action calculation
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In the paper
“A black hole produced in the early universe, for example, starts out with a mass well beyond the right-hand side of the plots, but at some point after evaporation, the vacuum decay rate becomes larger than the Hawking evaporation rate and the black hole seeds the transition to a new vacuum.”
§ Prepared for > 4 Application to the Higgs vacuum.
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In the paper
“Note this is only a crude calculation meant to give an indication of the effect of large extra dimensions, and is problematic as an approximation to an actual Higgs tunneling event due to the lack of an exact braneworld black hole solution, as well as the fact that the Higgs should be confined to the brane, hence the instanton should instead involve only a shift in the cosmological constant on the”
§ Prepared for > C Higher dimensional instantons
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- Rests on an assumption: The Higgs should be confined to the brane, so the instanton should involve only a shift in the cosmological constant on the brane.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“The actual value of the maximum will depend on the details of how B depends on M + , which requires a full calculation.”
§ Prepared for > 4 Application to the Higgs vacuum.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Uncharged black holes can easily evaporate before they seed a phase transition, but the charged black holes hang around for a longer time making them better candidates for vacuum decay nucleation sites.”
§ Prepared for > 3.3 Charged black hole instantons
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In the paper
“Note that for Λ + < 0, the range of k 1 , while initially narrowing as k 2 becomes negative, eventually opens out, as the linear term in (A.9) becomes dominant.Thus large AdS black holes can tunnel to an even larger AdS black hole with a more negative cosmological constant.”
§ Prepared for > A The limits on k 1 , k 2
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In the paper
“This has enabled us to simplify the analysis by employing a thin-wall approximation, which is valid in part of the parameter space.”
§ Prepared for > 5 Conclusions
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In the paper
“We have used a Higgs model which was based loosely on the two-loop effective potential, with an extra term motivated by high-energy physics, possibly quantum gravity.”
§ Prepared for > 5 Conclusions
Load-bearingStrength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
- The authors state this as an assumption: The Higgs model used is based loosely on the two-loop effective potential, with an extra term motivated by high-energy physics, possibly quantum gravity.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“however the exact normalisation of the plot will depend on the confidence of translating modifications of the Higgs potential to the new Planck scale and variables.”
§ Prepared for > 5 Conclusions
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In the paper
“From this expression we can see roughly how the branching ratio will depend on M + , even though B is, in principle, a complex function of M + .”
§ Prepared for > 4 Application to the Higgs vacuum.
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In the paper
“Although we would expect ζ for a thin wall bubble, numerical solutions show that the wall approximation is reasonably accurate even when ζ ∼ , therefore we use this lower bound for ζ.”
§ Prepared for > 4 Application to the Higgs vacuum.
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- The authors state this as an assumption: The thin-wall bubble approximation is valid for the Higgs potential considered.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“An SU (2) × U (1) Yang-Mills theory with Higgs field H in the fundamental SU (2) representation has no flat-space monopole solutions, but it does have Dirac and Yang-Mills black-hole monoples.”
§ Prepared for > 3.3 Charged black hole instantons
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In the paper
“The total decay rate for a subset of the standard model was evaluated by Page, [55].”
§ Prepared for > 4 Application to the Higgs vacuum.
What the sources say
- Not checked — no full text of the sourcecites [55] D.N. Page, Particle emission rates from a black hole: massless particles from an uncharged, nonrotating hole, Phys. Rev. D 13 (1976) 198 [ INSPIRE ]. source
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In the paper
“According to Callan and Coleman [18], this pre-factor is made up of a factor of (B/2π) 1/2 for each translational zero mode of the instanton and a determinant factor.”
§ Prepared for > 4 Application to the Higgs vacuum.
What the sources say
- Not checked — no full text of the sourcecites [18] C.G. Callan Jr. and S.R. Coleman, The fate of the false vacuum. 2. First quantum corrections, Phys. Rev. D 16 (1977) 1762 [ INSPIRE ]. source
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The seeded nucleation calculation presented here requires a black hole, and for large tunneling enhancement, this is expected to be a primordial black hole which is evaporating and nearing the end of its life.”
§ Prepared for > 5 Conclusions
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Clearly, if 2σγ > 1, we must have k 2 > 0, and U must be positive with positive gradient at R+ .”
§ Prepared for > A The limits on k 1 , k 2
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In the paper
“Solutions with a moving bubble wall (τ (λ), R(λ)) break the time-translation symmetry of the full space-time, but the canonical method can still be used if the wall is thin and the geometries on both sides of the bubble wall, M ± , individually possess the Killing vector, ∂ τ .”
§ Prepared for > B Alternative bounce action calculation
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The bubble solutions for tidal holes are simple generalisations of the ones we have already discussed.”
§ Prepared for > 5 Conclusions
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“we therefore expect the plot to be strongly exponentially suppressed at large M + , but rising as M + falls to a maximum around M + /M p = O(1), then dropping again below M p .”
§ Prepared for > 4 Application to the Higgs vacuum.
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The authors state this as an assumption.
The nucleation probability for a bubble of the new phase is given schematically by Γ = A e^{-B}, where B is the action of an imaginary-time solution to the Einstein-Higgs field equations.
“We follow the approach of Coleman and de Luccia [19], and assume that the nucleation probability for a bubble of the new phase is given schematically by Γ = Ae -B ,(1.1)”
The authors state this as an assumption.
The vacuum decay process can be treated using a static black hole background, ignoring time dependence due to Hawking evaporation, by considering only processes with timescales short compared to the evaporation rate.
“Of course, this description does not explicitly account for any time dependence of the black hole due to Hawking evaporation, however, we can apply the same argument as that employed for black hole particle production, namely, we consider only vacuum decay precesses which have timescales short compared to the evaporation rate.”
The authors state this as an assumption.
The observation of negative λeff assumes no corrections from new physics between the TeV scale and the Planck scale.
“The observation of negative λ eff of course assumes no corrections from new physics between the TeV scale and the Planck scale.”
The authors state this as an assumption.
Modifications to the Higgs potential from quantum gravity or BSM physics can be written as V = 1/4 λeff φ^4 + 1/4 (δλ)bsm φ^4 + 1/6 λ6 φ^6/Mp^2 + 1/8 λ8 φ^8/Mp^4 + ..., with polynomial terms representing unknown Planck-scale physics.
“On dimensional grounds, we can write modifications to the potential of the following form [37][38][39][40][41][42], where (δλ) bsm includes corrections from BSM physics, and the polynomial terms represent unknown physics from the Planck scale.”
The authors state this as an assumption.
The thin-wall bubble approximation is valid for the Higgs potential considered.
“Using a thin-wall bubble approximation for the nucleation process, which is possible when generic quantum gravity corrections are added to the Higgs potential, we show that primordial black holes can stimulate vacuum decay.”
The authors state this as an assumption.
The coefficients of Planck-suppressed operators are similar in magnitude.
“If these coefficients are similar in magnitude, then the small size of λ eff at the Planck scale has the consequence that there is an intermediate range of φ where the potential is determined predominantly by λ eff and λ 6 .”
The paper relies on this without support.
A primordial black hole losing mass by the Hawking process decays down to a mass around 10-100 times the Planck mass and then seeds a vacuum transition.
“A primordial black hole losing mass by the Hawking process would decay down to a mass around 10-100 times the Planck mass and then seed a vacuum transition.”
The authors state this as an assumption.
The bubble wall has constant tension.
“separated by a thin wall with constant tension”
The authors state this as an assumption.
In the higher dimensional model, the main channel for black hole decay is emission of particles on the brane.
“To estimate the decay rate of the black hole, we will assume that the main channel is due to emission of particles on the brane [69,70]”
The authors state this as an assumption.
The Higgs model used is based loosely on the two-loop effective potential, with an extra term motivated by high-energy physics, possibly quantum gravity.
“We have used a Higgs model which was based loosely on the two-loop effective potential, with an extra term motivated by high-energy physics, possibly quantum gravity.”
The paper relies on this without support.
For brane black hole systems, the black hole can be approximated by a higher dimensional Schwarzschild or Myers-Perry solution.
“There are no known exact solutions for these black hole plus brane systems, and instead the black hole is usually considered to be approximately a higher dimensional Schwarzschild or Myers-Perry [63] solution”
The paper relies on this without support.
The Higgs should be confined to the brane, so the instanton should involve only a shift in the cosmological constant on the brane.
“the Higgs should be confined to the brane, hence the instanton should instead involve only a shift in the cosmological constant on the brane.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“For very small seed masses, there are instanton solutions that remove the black hole, and the solution has the form of a perturbed CDL instanton.”
From The fate of the Higgs vacuum (2016) · cites this paper as [30]
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“It is also found that the seed black hole can catalyze the bubble nucleation process [15][16][17].”
From Vacuum decay and bubble nucleation in the anti-de Sitter black holes (2022) · cites this paper as [17]
The cited source says what the claim attributes to it.
In the citing paper
“In [29,30], we found that the tunnelling exponent did not depend on the periodicity of the Euclidean time, in other words, on the value of the temperature.”
From The fate of the Higgs vacuum (2016) · cites this paper as [30]
The cited source says what the claim attributes to it.
In the citing paper
“These Euclidean solutions exist in principle with all possible values of interior and exterior mass terms, however, for each seed (exterior) mass black hole, there exists a unique least action instanton with a unique remnant (interior) black hole mass.”
From The fate of the Higgs vacuum (2016) · cites this paper as [30]
In the citing paper
“(In ref. [30] we demonstrated that the condition of metric regularity could be loosened to allow conical singularities, but the resulting tunnelling rate was unaffected.)”
From The fate of the Higgs vacuum (2016) · cites this paper as [30]
In the citing paper
“For larger black hole seeds (beyond a critical mass M C depending on the vacuum energy and the surface tension of the wall) the remnant mass is non-zero and the minimum action solution becomes static in the complex time coordinate.”
From The fate of the Higgs vacuum (2016) · cites this paper as [30]
In the citing paper
“Applying these ideas to the Higgs vacuum, in [29,30] we recently provided a proof of principle that the lifetime of the vacuum could become precipitously short in the presence of primordial black holes, paralleling the intuition of impurities catalysing a phase transition.”
From The fate of the Higgs vacuum (2016) · cites this paper as [30]
In the citing paper
“The possibility of vacuum decay caused by black holes formed from collisions was considered JHEP06(2016)025 in [29,30].”
From The fate of the Higgs vacuum (2016) · cites this paper as [30]
In the citing paper
“The O(3) × U(1) symmetry results in a simple formula for the tunnelling exponent B, derived in ref. [30]:”
From The fate of the Higgs vacuum (2016) · cites this paper as [30]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Higher-dimensional sphere area fails the four-dimensional limit
MinorAt D = 4, the stated formula gives A2 = 8π2/3. But inserting the higher-dimensional metric (C.2) into the four-dimensional metric (2.1) requires A2 = 4π, so the normalization used for the higher-dimensional results is inconsistent.
“A D-2 = 2π D+1 2 /Γ[ D+1 2 ]”
“f (r) ≡ 1 - 2GM ± r - Λ ± r 2 3 ,(2.1)”
Positive exterior mass gives the opposite sign in the bound on k1
MinorSolving the stated expression for GM+ ≥ 0 gives k1 ≥ 2(1−α−2σγ)k2/α, not the printed bound with +2σγ.
“GM + ≥ 0 ⇒ k 1 ≥ 2(1 -α + 2σγ) k 2 α”
“GM + = γk 1 2α + k 2 γ α 2 (α -1 + 2σγ)”
Citations
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An Alternative to Compactification
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The hierarchy problem and new dimensions at a millimeter
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Flagged first, then by citations
Cited by
Sorted by citations
Items 101–144 of 144
Scalar lumps with two horizons
Lavrelashvili, G., & Lehners, J.-L. (2022) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.105.024051 · Cited by 3
Chang, Y., Tang, S., Deng, H., Wang, Y., Ding, R., & Huang, F. P. (2025) · arXiv (Cornell University) · doi:10.48550/arxiv.2511.11408 · Cited by 3
Chiral phase transition with primordial black holes: Distinct phase structure and catalysisPreprint
Tanaka, M., Wang, J.-C., & Zhang, J.-J. (2026) · arXiv (Cornell University) · doi:10.48550/arxiv.2602.06661 · Cited by 3
Vacuum decay and bubble nucleation in the anti-de Sitter black holes
Li, R., & Wang, J. (2022) · Journal of High Energy Physics · doi:10.1007/jhep09(2022)151 · Cited by 2
Bubble evolution around heterogeneities in ϕ4-field theories
Marín, J. F. (2021) · Journal of High Energy Physics · doi:10.1007/jhep02(2021)198 · Cited by 2
Gogberashvili, M., & Sakharov, A. (2025) · Galaxies · doi:10.3390/galaxies13010004 · Cited by 2
Zygmunt Lalak, A., Nakonieczna, A., & Nakonieczny, Ł. (2020) · Journal of High Energy Physics · doi:10.1007/jhep11(2020)132 · Cited by 2
Black Hole Entropy from Non-dirichlet Sectors, and a Bounce Solution
Park, I. Y. (2023) · Foundations of Physics · doi:10.1007/s10701-023-00719-5 · Cited by 2
Thermal false vacuum decay near black holes is asphericalPreprint
Gorbunov, D. S., Levkov, D. G., & Maslov, V. E. (2026) · arXiv (Cornell University) · doi:10.48550/arxiv.2608.12469 · Cited by 2
Testing Higher Derivative Gravity through Tunnelling
Gregory, R., & Hu, S.-Q. (2024) · Particles · doi:10.3390/particles7010008 · Cited by 1
Self-citationOn thermal false vacuum decay around black holes
Shkerin, A. (2024) · International Conference on Particle Physics and Cosmology (ICPPC) · doi:10.22323/1.455.0018 · Cited by 1
Deformed boson statistics and primordial black hole production under slow expansion conditions
Maleki, M., Sattari, F., & Mirershadi, S. (2025) · Chinese Journal of Physics · doi:10.1016/j.cjph.2025.06.046 · Cited by 1
Supersymmetric horizons at the edge of effective field theory
Çağan, S., Güleryüz, Ö., & Senisik, C. B. (2025) · Journal of High Energy Physics · doi:10.1007/jhep09(2025)151 · Cited by 1
False vacuum decay catalyzed by black hole in a heat bath
Hu, B., Kamada, K., & Shkerin, A. (2026) · Journal of High Energy Physics · doi:10.1007/jhep07(2026)049 · Cited by 1
Tokgöz, G., & Sakallı, İ. (2026) · The European Physical Journal C · doi:10.1140/epjc/s10052-026-16352-1 · Cited by 1
Backreaction of particle production on false vacuum decay
Lagger, C. (2016) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.93.025024 · Cited by 1
Two faces of Hawking radiation and thin-shell emission: pair-creation vs. tunneling
Yeom, D.-H. (2021) · SciPost Physics Proceedings · doi:10.21468/scipostphysproc.4.004 · Cited by 1
Stationary vacuum bubble in a Kerr–de Sitter spacetime
Saito, D., & Yoo, C. (2023) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.107.064043 · Cited by 1
Impacts of wormholes on false vacuum bubble tunneling
Wang, H., Wu, Y., Li, R., & Wang, J. (2025) · Physical review. D/Physical review. D. · doi:10.1103/7lvl-8vqb · Cited by 1
Creation of an inflationary universe out of black hole space–time
Oshita, N., & Jun’ichi Yokoyama, B. (2018) · International Journal of Modern Physics A · doi:10.1142/s0217751x18440050 · Cited by 1
Ai, W.-Y. (2019) · mediaTUM (Technical University of Munich) · Cited by 1
Nonperturbative aspects of gravity and field theory
Burda, P. (2015) · Durham e-Theses (Durham University)
Self-citationErratum: Backreaction of particle production on false vacuum decay [Phys. Rev. D93, 025024 (2016)]
Lagger, C. (2016) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.93.089902
Larsen, K. M. (2019) · Science and fiction · doi:10.1007/978-3-030-12206-5_7
Decay of false vacua via impurities in string theories
Ookouchi, Y., Nakai, Y., & Tanahashi, N. (2019) · Corfu Summer Institute School and Workshops on Elementary Particle Physics and Gravity (CORFU) · doi:10.22323/1.347.0078
Gravitational Waves and Fundamental Physics
Lagger, C. (2019) · The Sydney eScholarship Repository (The University of Sydney)
Classical and Quantum Thermodynamic Systems in Curved SpacetimePreprint
Quinta, G. M. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2005.13965
Cosmological phase transitions: extra dimensions and black holes
Manuel, J. P. (2020) · Sussex Research Online (University of Sussex)
Small-scale probes of the early universe
Cole, P. S. (2021) · Sussex Research Online (University of Sussex)
Report on scipost_202010_00007v1
(2020) · SciPost Physics Proceedings · doi:10.21468/scipost.report.2182
(2022) · SciPost Physics · doi:10.21468/scipost.report.5121
(2022) · SciPost Physics · doi:10.21468/scipost.report.5131
First-Order Cosmological Phase Transition
Gouttenoire, Y. (2022) · Springer theses · doi:10.1007/978-3-031-11862-3_6
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Morvan, E., van der Schaar, J. P., & Visser, M. R. (2022) · SciPost Physics · doi:10.21468/scipost.report.5058
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Symmetry restoration and vacuum decay from accretion around black holes
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Euclidean path integral, entanglement entropy, and quantum boundary conditions
Yeom, D.-H. (2024) · AIP conference proceedings · doi:10.1063/5.0216044
Planckian charged black holes and their cosmological ramifications
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The Gravitational Higgs Phenomena and the Vacuum Instability
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Vacuum bubbles from cosmic ripples
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Black Hole Singularity and the Fate of Black Holes
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Metadata
- Authors
- Burda, P., Gregory, R., & Moss, I. G.
- Affiliations
- Durham University (GB)
- Perimeter Institute (CA)
- Newcastle University (GB)
- Corresponding author
- Gregory, Ruth
- Venue
- Journal of High Energy Physics
- Published
- 2015-08
- Volume
- 2015
- Issue
- 8
- ISSN
- 1029-8479 (ISSN-L)
- OpenAlex
- W1896355628
- Cited by
- 144
- References
- 80
- Citations by year
- 2026: 9 · 2025: 14 · 2024: 9 · 2023: 16 · 2022: 15 · 2021: 11 · 2020: 15 · 2019: 17 · 2018: 12 · 2017: 9 · 2016: 11 · 2015: 6
- FWCI
- 6.06
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Licence
- cc-by
- Locations
- Journal of High Energy Physics (publishedVersion, cc-by)
- Durham Research Online (Durham University) (acceptedVersion, other-oa)
- arXiv (Cornell University) (submittedVersion)
- Durham Research Online (Durham University) (submittedVersion, cc-by)
- arXiv (Cornell University)
- Funders
- Government of Canada
- Wolfson Foundation
- Industry Canada
- Institut Périmètre de physique théorique
- Engineering and Physical Sciences Research Council
- Science and Technology Facilities Council (ST/L000407/1, ST/J000426/1)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Particle physics theoretical and experimental studies
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Keywords
- vacuum metastability, vacuum decay, primordial black holes, black hole nucleation, Higgs potential, quantum gravity corrections, Hawking evaporation
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-11-09, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-11-09 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Cited sources are checked with a language model’s help (gpt-6-sol).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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