Vacuum metastability with black holes

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Unlike the figure in [21], where the lower limit for k1 was dependent on σ for negative but not positive k2, in this case it is the lower limits for positive k2 and not negative that depend on σ.

“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.”

• cites [21] R. Gregory, I.G. Moss and B. Withers, Black holes as bubble nucleation sites, JHEP 03 (2014) 081 [ arXiv:1401.0017 ] [ INSPIRE ]. sourceDisputed

  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.

  Experimental — accuracy not yet measured

It may state two separate claims as one.

Magnetically charged black holes may be produced in the early universe and form seeds for vacuum decay of an unstable standard model Higgs field.

“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.”

• cites [27] F. Mellor and I. Moss, Black holes and quantum wormholes, Phys. Lett. B 222 (1989) 361 [ INSPIRE ]. sourceNot checked — no full text of the source
• cites [28] F. Mellor and I. Moss, Black holes and gravitational instantons, Class. Quant. Grav. 6 (1989) 1379 [ INSPIRE ]. sourceSupported

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The tidal black holes resemble black-hole monopoles, but with negative square monopole charge P^2.

“The tidal black holes, for example, resemble black-hole monopoles, but with negative square monopole charge P 2 .”

• cites [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 ]. sourcePartly supported

  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)

  Experimental — accuracy not yet measured

It may state two separate claims as one.

The equations of motion have the same schematic form as (2.6) but with σ = 4πGσ/(D-2) and f the higher dimensional Schwarzschild potential.

“The equations of motion have the same schematic form as (2.6)”

• cites [46] R. Gregory and A. Padilla, Nested brane worlds and strong brane gravity, Phys. Rev. D 65 (2002) 084013 [ hep-th/0104262 ] [ INSPIRE ]. sourceSupported
• cites [68] R. Gregory and A. Padilla, Brane world instantons, Class. Quant. Grav. 19 (2002) 279 [ hep-th/0107108 ] [ INSPIRE ]. sourceSupported

It may state two separate claims as one.

As GM+ increases, the static bubble shrinks initially but remains well outside the Schwarzschild radius; further increase constrains it by the expanding black hole horizon, stretching it just outside RSCH.

“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 .”

It may state two separate claims as one.

At GM_C, the two roots of U meet, and the static branch begins.

“Eventually, at GM C , the two roots of U meet, and the static branch begins.”

It may state two separate claims as one.

In the CDL special case, the symmetry of the bubble solution is raised from O(3) to O(4), and the result for the tunnelling rate agrees with explicitly O(4) symmetric methods.

“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.”

It may state two separate claims as one.

The expression for the bounce action (2.22) is independent of any choice of periodicity of Euclidean time and independent of the choices of cosmological constant on each side of the wall.

“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.”

It may state two separate claims as one.

The remnant black hole mass GM- increases along the static branch and eventually becomes larger than GM+, but because of the negative cosmological constant, the horizon radius does not increase as rapidly as RSCH.

“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 .”

It may state two separate claims as one.

The values of RMAX/RMIN are well outside the black hole horizon radius, and move together as GM+ is increased.

“The values of RMAX / RMIN are well outside the black hole horizon radius, and move together as GM + is increased.”

It may state two separate claims as one.

For seed mass M+ < M_C, the dominant tunnelling process leaves behind a true vacuum region and removes the black hole.

“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.”

It may state two separate claims as one.

The static bubble action increases as GM+ increases, eventually becoming larger than B_CDL.

“The static bubble action therefore increases as GM + increases, eventually becoming larger than B CDL (see figure 3).”

It may state two separate claims as one.

For fixed seed mass M+, there is a range of allowed k1 and k2 and a corresponding range of values for the bounce action.

“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.”

It may state two separate claims as one.

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.

“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.”

Probably not a substantial claim.

The action of the instanton solution is given by equation (2.21).

“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)”

It may state two separate claims as one.

The upper limit on k2 is determined by M-=0 for the static bounce, which is inconsistent for Λ+ ≤ 0; for Λ+ > 0, the range closes off at k2,min = α^2(2σγ-1)/(γ^3 Λ+^{3/2}).

“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”

It may state two separate claims as one.

GM- ≥ 0 implies k1 ≥ 2(1-α)k2/α; GM+ ≥ 0 implies k1 ≥ 2(1-α+2σγ)k2/α.

“thus GM -≥ 0 ⇒ k 1 ≥ 2(1 -α) k 2 α GM + ≥ 0 ⇒ k 1 ≥ 2(1 -α + 2σγ) k 2 α (A.6)”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

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.

“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.”

It may state two separate claims as one.

The static branch meets the time-dependent branch at a critical mass M_CP, where the static bubble has an extremal black hole in its interior.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The quote only partly states this claim.

Setting R=αR/γ and λ=αλ/γ gives an equation of motion of the same form as (2.10) albeit with different exponents of R.

“Then setting R = αR/γ, λ = αλ/γ gives the equation of motion”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.

The timescales for Hawking evaporation and vacuum decay will both be less than roughly a million Planck times.

“The timescales for Hawking evaporation and vacuum decay will both be less than roughly a million Planck times.”

It may state two separate claims as one.

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.

“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.”

It may state two separate claims as one.

The static bubble is the difference in areas of the seed and remnant black holes, which can be deduced from figure 4 to be roughly linear in M+ (not quadratic).

“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)”

Probably not a substantial claim. The paper may give no reason why it holds.

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.

“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.”

It may state two separate claims as one.

In all cases in table 1, 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.

“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.”

It may state two separate claims as one. The quote only partly states this claim.

For 2σγ < 1/2, k2 < 0 is allowed, and a similar argument to [21] gives k1 ≥ ... for k2 ≤ 0, and k1^m = 2(1-α)k2/α for k2 > 0.

“Now consider 2σγ < 1/2, for which k 2 < 0 is allowed.A similar argument to [21] gives”

• cites [21] R. Gregory, I.G. Moss and B. Withers, Black holes as bubble nucleation sites, JHEP 03 (2014) 081 [ arXiv:1401.0017 ] [ INSPIRE ]. sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

It may state two separate claims as one.

The requirement of positivity of τ±, of which the constraint on τ+ is stronger, is k2 R2 + R α(1-2σγ) ≥ 0.

“Secondly, the requirement of positivity of τ ± , of which the constraint on τ+ is the stronger:”

It may state two separate claims as one.

The dip in action near the critical point is very minor, and the minimum action is well approximated by the value at M_CP.

“The dip is however very minor, and the minimum action is well approximated by the value at M CP .”

The quote only partly states this claim.

The static action needed for the branching ratio is the difference in horizon areas: B_D = A_{D-2}/4G (r+^{D-2} - r-^{D-2}).

“The static action (which is what is needed for the branching ratio) is, as before, the difference in horizon areas:”

It may state two separate claims as one.

As λ* becomes more negative, the values of λ6 required for thin-wall become larger; similarly as φ* drops, λ6 increases.

“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.”

It may state two separate claims as one.

From U>0, Λ+ > 0, 2σγ ≤ 1 + α/2, and k1 is bounded below by k1^m = 2(1-α)k2/α from (A.6).

“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).”

It may state two separate claims as one.

The time-dependent tunneling solutions in the charged case 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).

“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”

The quote only partly states this claim.

Appendix C leads to a dimension dependent branching ratio Γ_D/Γ_H ≈ 550 M+ r+ (D-3)^4 B_D e^{-B_D} = 550 √B_D e^{-B_D} (D-3)^4 [16πG_D M+^{D-2}/((D-2)A_{D-2})]^{1/(D-3)}.

“which leads to a dimension dependent branching ratio”

Probably not a substantial claim. It may state two separate claims as one.

The branching ratio Γ_D/Γ_H for monopole charged black holes can be re-computed using this evaporation rate and the vacuum decay rates from the previous section, and the result is shown in figure 7.

“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.”

It may state two separate claims as one.

Therefore the tunnelling rate is determined by B = A+/(4G) - A-/(4G) - 1/2 ∫W σ√h - 1/16πG ∫W (f+ τ+ - f- τ-)√h, which is identical to (2.22) after using the relation f+ τ+ - f- τ- = -2σR for the wall integral.

“Therefore the tunnelling rate is determined by”

It may state two separate claims as one. The quote only partly states this claim.

The range of Higgs model parameters λ*, φ*, and λ6 which allow thin-wall bubbles is set by ζ > ε > 0, and by the condition that the true vacuum lies at large φ.

“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.”

It may state two separate claims as one.

The field equations imply H=Hi=0, and symmetry implies ∂τφ=∂τγij=0; therefore only the boundary term in (B.2) survives.

“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.”

It may state two separate claims as one.

At large P/M, the dip is most visible, but the catalytic effect of the black hole is much reduced.

“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.”

Probably not a substantial claim.

Since the bubble exterior is in the false vacuum, Λ+ > Λ-.

“Since the bubble exterior is in the false vacuum, we have Λ + > Λ -.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.

The lifetime of the present vacuum would be far longer than the age of the universe, so metastability would not conflict with observation.

“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.”

Probably not a substantial claim. The paper may give no reason why it holds.

The nucleation process typically requires an instanton that has a conical singularity at the black hole horizon.

“The nucleation process typically requires an instanton that has a conical singularity at the black hole horizon.”

It may state two separate claims as one.

Since a primordial black hole has not seeded a vacuum transition, either the Higgs parameters are not in the relevant range or there are no primordial black holes in the observable universe.

“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.”

• AssumptionRests 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.

Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.

Thin-wall bubble nucleation in the Higgs potential requires the potential to be relatively shallow at the true vacuum, and this requires a large positive φ6 term.

“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.”

• AssumptionThe authors state this as an assumption: The thin-wall bubble approximation is valid for the Higgs potential considered.

It may state two separate claims as one.

The minimal bounce action for a particular σl can be extremely small, so that even above the critical black hole mass, the decay rate can still be significant.

“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.”

It may state two separate claims as one.

Electrically charged black holes can discharge by emission of charged particles, but magnetically charged black holes may be the lightest magnetically charged particles, in which case a large mass black hole evaporates towards the extremal limit and the Hawking radiation flux falls to zero.

“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.”

• cites [53] G.W. Gibbons, Vacuum polarization and the spontaneous loss of charge by black holes, Commun. Math. Phys. 44 (1975) 245. sourceNot checked — no full text of the source

It may state two separate claims as one.

The present-day broken symmetry vacuum may be metastable, 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.

“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.”

It may state two separate claims as one.

The static bubble with a remnant black hole is not stable; small fluctuations will lead it either to expand completing the phase transition or to collapse back leaving the initial state unchanged.

“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.”

It may state two separate claims as one.

The vacuum decay process based on a static black hole can either completely replace the black hole with a bubble of true vacuum expanding outwards, or nucleate a static bubble leaving a remnant black hole surrounded by true vacuum.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

There are no known exact solutions for black hole plus brane systems, and instead the black hole is usually considered to be approximately 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”

• cites [63] R.C. Myers and M.J. Perry, Black holes in higher dimensional space-times, Annals Phys. 172 (1986) 304 [ INSPIRE ]. sourceNot checked — no full text of the source

• AssumptionRests on an assumption: For brane black hole systems, the black hole can be approximated by a higher dimensional Schwarzschild or Myers-Perry solution.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

For a monopole charged black hole, the branching ratio is expected to be larger due to their reduced Hawking radiation rate.

“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”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

The expression (B.7) is the same as (2.16, 2.19), but the authors have not assumed anywhere for the static case that the bubble is in the thin-wall limit.

“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.”

It may state two separate claims as one.

A black hole produced in the early universe starts with mass well beyond the right-hand side of the plots, but after evaporation the vacuum decay rate becomes larger than the Hawking evaporation rate and the black hole seeds the transition to a new vacuum.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

The higher dimensional instanton model 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 and the fact that the Higgs should be confined to the brane.

“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”

• AssumptionRests 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.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The actual value of the maximum branching ratio depends on details of how B depends on M+, which requires a full calculation.

“The actual value of the maximum will depend on the details of how B depends on M + , which requires a full calculation.”

It may state two separate claims as one.

Uncharged black holes can easily evaporate before they seed a phase transition, but charged black holes hang around longer making them better candidates for vacuum decay nucleation sites.

“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.”

It may state two separate claims as one.

For Λ+ < 0, the range of k1 initially narrows as k2 becomes negative but 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.

“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.”

Probably not a substantial claim.

The thin-wall approximation is valid in part of the parameter space.

“This has enabled us to simplify the analysis by employing a thin-wall approximation, which is valid in part of the parameter space.”

Probably not a substantial claim.

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.”

• AssumptionThe 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.

Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it.

The exact normalisation of the higher-dimensional branching ratio plot depends on the confidence of translating modifications of the Higgs potential to the new Planck scale and variables.

“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.”

It may state two separate claims as one.

The branching ratio depends on M+ even though B is in principle a complex function of M+.

“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 + .”

It may state two separate claims as one.

The wall approximation is reasonably accurate even when ζ ∼ ε, so the lower bound ζ > ε is used.

“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.”

• AssumptionThe authors state this as an assumption: The thin-wall bubble approximation is valid for the Higgs potential considered.

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.

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.

“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.”

Probably not a substantial claim.

The total decay rate for a subset of the standard model was evaluated by Page.

“The total decay rate for a subset of the standard model was evaluated by Page, [55].”

• cites [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 ]. sourceNot checked — no full text of the source

Probably not a substantial claim.

According to Callan and Coleman, the pre-factor is made up of a factor (B/2π)^{1/2} for each translational zero mode of the instanton and a determinant factor.

“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.”

• cites [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 ]. sourceNot checked — no full text of the source

Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.

The seeded nucleation calculation 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.

“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.”

It may state two separate claims as one.

If 2σγ > 1, then k2 > 0, and U must be positive with positive gradient at R+.

“Clearly, if 2σγ > 1, we must have k 2 > 0, and U must be positive with positive gradient at R+ .”

It may state two separate claims as one.

Solutions with a moving bubble wall 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 individually possess the Killing vector ∂τ.

“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 .”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

The bubble solutions for tidal holes are simple generalisations of the ones already discussed.

“The bubble solutions for tidal holes are simple generalisations of the ones we have already discussed.”

It may state two separate claims as one.

The branching ratio plot is expected to be strongly exponentially suppressed at large M+, rising as M+ falls to a maximum around M+/Mp = O(1), then dropping again below Mp.

“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 .”