Connecting the Higgs Potential and Primordial Black Holes
PHYSICAL REVIEW D 101, 125012 (2020)
Connecting the Higgs potential and primordial black holes
De-Chang Dai ,1,2 Ruth Gregory ,3,4 and Dejan Stojkovic5 1Center for Gravity and Cosmology, School of Physics Science and Technology, Yangzhou University, 180 Siwangting Road, Yangzhou City, Jiangsu Province 225002, People’s Republic of China 2CERCA/Department of Physics/ISO, Case Western Reserve University, Cleveland, Ohio 44106-7079, USA 3Centre for Particle Theory, Durham University, South Road, Durham DH1 3LE, United Kingdom 4Perimeter Institute, 31 Caroline Street, Waterloo, Ontario N2L 2Y5, Canada 5HEPCOS, Department of Physics, SUNY at Buffalo, Buffalo, New York 14260-1500, USA
(Received 6 September 2019; accepted 3 June 2020; published 15 June 2020)
It was recently demonstrated that small black holes can act as seeds for nucleating decay of the metastable Higgs vacuum, dramatically increasing the tunneling probability. Any primordial black hole lighter than 4.5 × 1014 g at formation would have evaporated by now, and in the absence of new physics beyond the standard model, would therefore have entered the mass range in which seeded decay occurs, however, such true vacuum bubbles must percolate in order to completely destroy the false vacuum; this depends on the bubble number density and the rate of expansion of the universe. Here, we compute the fraction of the universe that has decayed to the true vacuum as a function of the formation temperature (or equivalently, mass) of the primordial black holes, and the spectral index of the fluctuations responsible for their formation. This allows us to constrain the mass spectrum of primordial black holes given a particular Higgs potential and conversely, should we discover primordial black holes of definite mass, we can constrain the Higgs potential parameters.
DOI: 10.1103/PhysRevD.101.125012
I. INTRODUCTION electroweak phase transition might be followed by a first order phase transition was explored. One of the most fascinating implications of the meas- For a black hole to seed vacuum decay, we must be sure urement of the Higgs mass at the LHC [1,2] is that the that the half-life for decay is less than the evaporation rate standard model vacuum appears to be metastable [3–11]. of the black hole. This means that the branching ratio of Initially, this was not thought to be a problem for our tunnelling to decay must be greater than one. In [18,19] this universe, as standard techniques for computing vacuum 106−9 decay [12–15] indicated that the half-life was many order was found to occur for black holes of order Mp or so, 10−23 of magnitude greater the age of the universe. However, by which point the half-life for decay is of order s. vacuum decay represents a first order phase transition, and Clearly this process is not relevant for astrophysical black in nature these typically proceed via catalysis: a seed or holes, however, it has been hypothesised that there exist impurity acts as a nucleus for a bubble of the new phase to very light black holes, formed from extreme density form. In [16–20], the notion that a black hole could act as fluctuations in the early universe [30–32] dubbed primor- such a seed was explored, with the finding that black holes dial black holes. Such black holes have been proposed as a can dramatically shorten the lifetime of a metastable source for dark matter [33], and although this has now been vacuum (see also [21–28]). Interestingly, before the dis- ruled out [34], they could still constitute a component of the covery of the Higgs particle, the electroweak phase dark matter of the universe. Indeed, it has even been transition was usually described as a second order tran- proposed that the Higgs vacuum instability could generate sition, and in [29] the idea that the usual second order primordial black holes in the early universe [35]. Given that we are in a current metastable Higgs vacuum, we can be sure that there has been no primordial black hole that has evaporated in our past lightcone, however, how strong a constraint on primordial black holes can we place? Published by the American Physical Society under the terms of For the universe to have decayed, the black hole must not the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to only have evaporated sufficiently to reach the mass range in the author(s) and the published article’s title, journal citation, which catalysis spectacularly dominates, but the consequent and DOI. Funded by SCOAP3. bubble (or bubbles) of true vacuum must have percolated
2470-0010=2020=101(12)=125012(9) 125012-1 Published by the American Physical Society DAI, GREGORY, and STOJKOVIC PHYS. REV. D 101, 125012 (2020) to engulf the current Hubble volume. Thus, this is a state- The spacetime geometry is taken to have SOð3Þ × Uð1Þ ment about the relative volume in the percolated bubble, symmetry, in other words, it is spherically symmetric which is itself a statement on the primordial black hole “around” the black hole, and has time translation symmetry density and mass. In this paper, we draw together all these along the Euclidean time direction, τ: aspects of the problem, linking the primordial black hole spectral index and formation epoch to the standard model dr2 ds2 ¼ fðrÞe2δðrÞdτ2 þ þ r2ðdθ2 þ sin2 θdφ2Þ; ð4Þ parameters. fðrÞ The outline of the paper is as follows. In Sec. II we review the physics of the Higgs vacuum decay in the presence of with gravity. In Sec. III we relate the primordial black hole masses that can trigger vacuum decay with the parameters in the 2GμðrÞ fðrÞ¼1 − : ð5Þ effective Higgs potential. In Sec. IV we put this scenario in r the cosmological context: Every black hole that can trigger the vacuum decay will create a bubble of true vacuum. These We can think of μðrÞ as the local mass parameter, bubbles then expand with the speed of light, but their number however caution must be used in pushing this analogy. density decreases due to the expansion of the universe. For a For an asymptotic vacuum of Λ ¼ 0, then μð∞Þ truly successful phase transition, the bubbles have to percolate, so is the Arnowitt-Deser-Misner (ADM) mass of the black we define a quantity P, which represents the portion of the hole, however, locally, μ also includes the effect of any universe that has already transitioned to the new vacuum. For vacuum energy: for a pure Schwarzschild-(A)dS solution, 3 P ≥ 1, the universe would be destroyed, thus the associated μðrÞ¼M þ Λr =6G. Since we are interested in seeding the range of parameters is excluded. We summarize and discuss decay of our current Standard Model (SM) vacuum, we our findings in Sec. V. will take Λþ ¼ 0, so that the asymptotic value of μ is indeed the seed black hole mass, Mþ, responsible for triggering the phase transition. The remnant mass, which is a leftover from II. FALSE VACUUM DECAY WITH BLACK HOLES the seed black hole after some of its energy is invested into The high energy effective Higgs potential has been the bubble formation, may not be precisely μðrhÞ, however, determined by a two-loop calculation in the standard model since we will be interested only in the area of the remnant as [7] black hole horizon, it turns out that μðrhÞ is in fact the desired quantity. 1 ðϕÞ¼ λ ðϕÞϕ4 ð Þ The Higgs and gravitational field equations of motion are V 4 eff ; 1 2 ϕ00 þ 0ϕ0 þ ϕ0 þ δ0 ϕ0 − ¼ 0 where λ ðϕÞ is the effective coupling constant that runs f f f f V;ϕ eff r with scale. We now review the calculations in [19], 1 adopting the same conventions. The running of the cou- μ0 ¼ 4π 2 ϕ02 þ r 2 f V pling constant can be excellently modelled over a large range of scales by the three parameter fit: δ0 ¼ 4πGrϕ02; ð6Þ ϕ 2 ϕ 4 ≡∂ ∂ϕ ¼ λ ¼ λ þ þ ð Þ where V;ϕ V= . The black hole horizon is at r rh, eff b ln c ln ; 2 ð Þ¼0 Mp Mp at which f rh . We have to solve these equations of motion numerically in order to get the function ϕðrÞ, and to −2 where Mp ¼ 8πG. By fitting the two-loop calculation do this, we start from the horizon rh with a particular ¼ 2 μ with a simple analytic form, we can easily investigate not remnant parameter, rh G −, and some value for the ϕ only the standard model, but also beyond the standard Higgs field h. At the horizon therefore the fields satisfy model potentials, allowing us to explore possible future the boundary conditions corrections to the standard model results. μð Þ¼μ δð Þ¼0 The Higgs potential supports a first order phase transition rh −; rh mediated via nucleation of bubbles of new vacuum inside the r V ϕðϕ Þ ϕ0ð Þ¼ h ; h ð Þ old, false, vacuum. The nucleation rate in the presence of rh 2 ; 7 1 − 8πGr VðϕhÞ gravity is determined by a saddle point “bounce” solution of h the Euclidean (signature þ; þ; þ; þ) action: and as r → ∞, Z 1 1 pffiffiffiffiffiffi lim ϕðrÞ → 0;Mþ ¼ lim μðrÞ: ð8Þ ab 4 r→∞ r→∞ SE ¼ − R þ g ∂aϕ∂bϕ þ VðϕÞ −gd x: M 16πG 2 We use a shooting method starting at rh with ϕ ¼ ϕh and ð3Þ integrate out, altering ϕh until a solution is obtained with ϕ
125012-2 CONNECTING THE HIGGS POTENTIAL AND PRIMORDIAL … PHYS. REV. D 101, 125012 (2020) tending to 0 for very large values of r. In practice, rather than setting the asymptotic mass μð∞Þ¼Mþ, we set the initial (remnant) value of μ− and deduce the seed mass from (8), repeating the integration for a range of values of μ−.We then determine μ−ðMþÞ by inversion. The decay rate of the Higgs vacuum, ΓD, is then determined by computing the difference in entropy between the seed and remnant black holes: B 1=2 Γ ¼ ð Þ−1 −B ð Þ D 2π GMþ e 9 where
2 2 Mþ − μ− B ¼ : ð10Þ 2M2 p FIG. 1. The branching ratio between the tunneling and evapo- – ration rate as a function of the seed black hole mass, Mþ, for As pointed out in [17 19], a black hole can also radiate and some sample choices of the potential parameters. The unit lose mass, eventually disappearing in Hawking radiation, at branching ratio, ΓD=ΓH ¼ 1, is labeled by the dotted line. We a rate initially estimated by Page [36], see also [37–41]: −5 set b ¼ 1.5 × 10 , c ¼ 0. Mþ is given in units of Mp.
Γ ¼ 3 6 10−4ð 2 3 Þ−1 ð Þ 14 H . × G Mþ : 11 universe is approximately 4.5 × 10 grams, meaning that all black holes lighter than this value would have already Thus, we define the branching ratio between the tunneling evaporated. Along the way, they will inevitably end up in and evaporation rate as the range given by Eq. (13). This however does not 2 automatically imply that all the primordial black holes Γ M 14 D ¼ 43 8 þ 1=2 −B ð Þ lighter than 4.5 × 10 grams are excluded for this choice of Γ . 2 B e : 12 H Mp parameters. To destroy the universe the bubbles of the true vacuum have to percolate, which takes time. We will study This equation contains all the information we need. In the this in the next section. next two sections, we will study the consequences of the The same Fig. 1 indicates that if we set λ ¼ −0.00045, gravitationally induced false Higgs vacuum decay. and keep b ¼ 1.5 × 10−5 and c ¼ 0, then the branching ratio is always smaller than one (these values are not consistent with a pure standard model effective coupling, III. THE VACUUM DECAY RATE AND THE however, indicate the principle of model dependence of the HIGGS EFFECTIVE POTENTIAL branching ratio). In that case, the primordial black holes of any mass (i.e., Mp
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bubbles. In our scenario, every black hole that can initiate the false vacuum decay will create a bubble, so the number of the bubbles is equal to the number of such primordial black holes. Thus, whether the initiated vacuum decay can be completed crucially depends on the production mecha- nism and age of the universe when the primordial black holes were formed. It is usually assumed that primordial black holes are produced by density fluctuations caused by oscillations of some (scalar) field. If density fluctuations are large enough [42–47], the whole causally connected region (i.e., the horizon volume at some time) collapses and forms a black hole. The horizon mass in a radiation-dominated universe (in units of grams) is
ΓD FIG. 2. A plot of the line Γ ¼ 1 in ðb; λ Þ parameter space for 7 2 H 10 −8 18 GeV the values c ¼ 0, 6.3 × 10 as labeled. Above the curve, the MH ≅ 10 g: ð14Þ Γ T branching ratio D is always less than one for any value of the seed ΓH black hole mass. Below the curve, there is always a range of the ΓD where T is the temperature of radiation. Obviously, the black hole masses for which the branching ratio Γ is greater than H earlier the black holes are formed, the lighter they are, one. The dependence on c is minimal. hence their lifetime is shorter. Their lifetime is given as [36–41]. The boundary with c ¼ 6.3 × 10−8 is lower than that with 3 c ¼ 0 because of the contribution from the quartic terms in τ ¼ 4 99 10−44 M ð Þ evap . × s: 15 the Higgs potential. However, numerical experiments MP indicate that the curves do not change significantly as we vary the parameter c. Black holes of mass M ≳ 4.5 × 1014 g have a lifetime According to [19] the standard model parameter greater than 1.38 × 1010 years, or the age of the universe. space corresponding to the allowed range of top quark Therefore only lighter primordial black holes will have the mass (172–174 GeV) is 1.2 × 10−5 ≲ b ≲ 1.4 × 10−5 and potential to destroy the universe, based on the SM −0.02 ≲ λ ≲ −0.007. Unfortunately, this lies inside the parameter ranges. We focus mostly on these lighter black ΓD potentially dangerous Γ > 1 region. If some new physics holes which, according to Eq. (14), are created at temper- H ≳ 4 7 108 beyond the standard model modifies the Higgs potential, atures higher than TF . × GeV. then we can potentially move outside of these parameter After primordial black holes are formed at TF, their ranges. In the best case, the Higgs vacuum gets stabilized number density changes with temperature as and the first order phase transition disappears altogether β 3 from the discussion. However, there is also a potentially i T nbðTÞ ∼ ρr;i : ð16Þ dangerous region where beyond the standard model cor- Mi TF rections give a “safe” long false vacuum lifetime in the β absence of black holes, yet unacceptably short in their where i is the mass fraction of the universe in black holes ρ ¼ π2 4 presence. Thus, one can always tension new beyond the at formation, while r;i 30 gFTF is the radiation energy standard model physics with the existence of the primordial density at that time, with gF ≈ 100 being the number of black holes of a certain mass. degrees of freedom of radiation species at TF. MF is the mass of the primordial black holes at formation, and we IV. PRIMORDIAL BLACK HOLE MASSES take MF ¼ MHðTFÞ as usual. The mass fraction βi can be AND PERCOLATING BUBBLES found assuming a Gaussian perturbation spectrum of fluctuations that lead to black hole formation (see e.g., In the previous section, we saw that any primordial black [42,48,49]) hole that had enough time to evaporate sufficiently to fit into an appropriate mass range for the corresponding choice δ2 min σ ð Þ − 2 of the parameters λ and b, could initiate false vacuum H TF 2σ ðT Þ β ≈ pffiffiffiffiffiffi e H F ; ð17Þ i 2πδ decay. The bubbles of true vacuum then expand with the min speed of light, but the background universe expands as δ ≈ 0 3 well. Successful completion of the first order phase The parameter min . is the minimum density contrast transition depends on the number density of the created required for black hole creation, while σHðTÞ is the mass
125012-4 CONNECTING THE HIGGS POTENTIAL AND PRIMORDIAL … PHYS. REV. D 101, 125012 (2020) variance evaluated at horizon crossing at the temperature T moment (which depends on the initial black hole mass) defined as [48] they seed vacuum decay and form a bubble of true vacuum that then expands at the speed of light. For a successful n−1 n−1 ð Þ 6 M ðT Þ 4 phase transition, the bubbles have to percolate, so we σ ð Þ¼σ ð Þ MH T0 H eq ð Þ H TF H T0 ð Þ ð Þ : 18 compute the overall volume of true vacuum in the expand- MH Teq MH TF ing universe from the volume of an individual bubble and ≈ 0 79 the number density of black holes. Here, Teq . eV is the temperature at the matter/ −4 The present time number density of the bubbles, n ðT0Þ, radiation equilibrium, T0 ¼ 2.725K ¼ 2.35 × 10 eV is b the present temperature of the universe, while n is the is shown in Fig. 3. It is calculated from Eq. (16) following spectral index of the fluctuations that lead to black hole the procedure outlined above. The present time radius of formation, i.e., PðkÞ ∝ kn. Note, the cosmic microwave the bubble depends on the time it was created. If an object background data indicate that the value of the spectral index (in this case a bubble of true vacuum) is created at a of the inflaton field is n ≈ 1, however the cosmic micro- cosmological redshift Z, its present age, t,isgivenby wave background (CMB) data probe the scales between Z 45 60 1 Z dz 10 and 10 times larger than those probed by primordial tðZÞ¼ ð19Þ black holes. It is expected that primordial black holes are H0 0 Eð1 þ zÞ formed by fluctuations of fields other than the inflaton (e.g., during phase transitions), and the typical value of n used in where this context is between 1.23 and 1.31 [42,49,50].To 2 normalize Eq. (18) we use the mass variance evaluated 2 H 3 4 −5 E ¼ ¼ Ω ð1 þ ZÞ þ Ω ð1 þ ZÞ σ ðT0Þ¼9 5 10 2 m rad at the horizon crossing H . × . H0 We now have all the elements to calculate the black hole þ Ω ð1 þ Þ2 þ Ω ð Þ abundance for any set of desired parameters. After for- k Z Λ: 20 mation, primordial black holes evaporate, and at some stage Ω Ω Ω Ω of their life they will trigger false vacuum decay. When Here m, rad, k and Λ are the present values of the dark exactly this will happen depends on the specific parameters matter, radiation, curvature, and dark energy density of the Higgs potential; we must be above the threshold respectively. We take their numerical values from Planck Ω ¼ 0 315 Ω ¼ 9 23 10−5 Ω ¼ 0 value of the branching ratio, or in the range of parameters results [51], m . , rad . × , k and below the curve in Fig. 2, where it is guaranteed that the ΩΛ ¼ 0.684. H0 is the present time Hubble parameter, −1 −1 phase transition will be initiated for some black hole H0 ¼ 67.4 km s Mpc . The connection between the mass range. temperature, T, scale factor, a, and the redshift, Z,in To illustrate the procedure, we calculate the excluded an expanding Friedman Robertson Walker universe is primordial black hole parameter space for the example from T ∝ 1=a ∝ ð1 þ ZÞ. Sec. II, i.e., for the values of the potential parameters −5 λ ¼ −0.004, b ¼ 1.5 × 10 , c ¼ 0. As shown in Sec. II, the branching ratio is larger than one for the seed black 6 hole masses Mp ≲ Mþ ≲ 10 Mp, therefore all the black 6 holes that have evaporated down to 10 Mp or less by the present time will trigger false vacuum decay for this set of parameters. We note that this number is effectively the same as the number of the black holes that have evaporated completely by the present time, since it takes only a fraction 6 of the second for a black hole to evaporate from 10 Mp to zero. For comparison, we will also add black hole masses that correspond to the parameters beyond the standard model. The scenario is as follows. Suppose that primordial black holes are formed at a temperature TF with some initial mass Mi. They then evaporate until they reach a mass at which vacuum decay is catalyzed (106−9M for the standard p FIG. 3. The present number density of the true vacuum bubbles, model values, which is essentially equivalent to a complete plotted as a function of the temperature at which the primordial evaporation, given the scale of the lifetimes involved). black holes that seed nucleation are formed. The number density For the parameters of the Higgs potential outside of the is shown for two values of the spectral density index, n, of the standard model, these masses could be much higher. At that perturbations responsible for the primordial black hole creation.
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Since dr ¼ cdt=a ¼ cdtð1 þ ZÞ, the current physical In contrast, for the range of parameters in the lower part of radius of the true vacuum bubble formed at redshift ZB is the plot, the universe is safe, though the primordial black Z holes may initiate false vacuum decay. We distinguish c ZB dz between the initial black hole mass Mi (determined by the R ¼ ð21Þ H0 0 E formation temperature TF) which evaporates and triggers the vacuum decay at some lower value Mdec, determined by Γ the requirement D > 1. To show how the results vary with where E is given by Eq. (20). The redshift, ZB, is calculated ΓH at the moment when a black hole of a certain seed mass the mass of the black hole that triggers the vacuum decay, (formed at the temperature TF) evaporates enough to fit into we plot the curves corresponding to several values of Mdec. the appropriate mass window where it can trigger the false With the help of Eq. (14), we can convert the temperature vacuum decay. of the universe at the time of the primordial black hole Thus, the portion of the universe which is already in the formation to the primordial black hole mass. This is shown new vacuum at the present time is in Fig. 5. We can see that lighter black holes are more dangerous than the more massive ones because they 4π evaporate quickly, form the true vacuum bubbles earlier, P ¼ 3 ð Þ ð Þ 3 R nb T0 ; 22 and the bubbles have more time to grow. Black holes much heavier than M ≳ 1017 g are not constrained since they did Figure 4 shows the boundary of the P ¼ 1 region. For not have enough time to evaporate down to the dangerous the range of parameters in the upper part of the plot the range of masses. A potentially interesting range of P universe today is destroyed, since the bubbles percolate. the parameter space is where is nonzero but not too close to one. This would mean that there are a few bubbles in the universe here and there, but they do not yet dominate the universe. This situation is shown in Fig. 6. Since the bubbles expand with the speed of light, if they are far from us, this state could last for billions of years. It would be very interesting to study observational effects of the presence of such bubbles within our horizon.
FIG. 4. This plot shows the temperature of the universe at the time of the primordial black hole formation, TF, as a function of the spectral index, n for several values of the black hole masses that can trigger the vacuum decay. The lines represent the P ¼ 1 value for the portion of the universe which already transitioned to the new vacuum at present time. Above these lines, we have P > 1 and the whole universe today would be destroyed. Below P 1 these lines we have < , and the universe is safe. A black hole FIG. 5. This plot shows the primordial black hole masses (in of the initial mass Mi (determined by the formation temperature grams) at the time of their formation, Mi, as a function of the TF) evaporates and triggers the vacuum decay at some lower spectral index, n. The lines represent the P ¼ 1 value for the value Mdec.IfMi >Mdec, the onset of the phase transition is portion of the universe which already transitioned to the new delayed by the time needed for a black hole to enter the window P 1 Γ vacuum at present time. Below this line we have > and the where it can trigger the vacuum decay with D > 1. The solid line ΓH whole universe today would be destroyed. Above this line, we ≪ 5 1014 P 1 corresponds to the black hole masses Mdec × g, i.e., have < , and the universe is safe. The solid line corresponds ≈ 0 ≪ 5 1014 ≈ 0 Mdec . The dashed, dot, dashed dot, and dashed dot dot lines to the black hole masses Mdec × g, i.e., Mdec . The 14 15 16 17 correspond to Mdec values of 5 × 10 g, 10 g, 10 g, 10 g. dashed, dot, dashed dot, and dashed dot dot lines correspond to 14 15 16 17 The short dashed line corresponds to Mi 125012-6 CONNECTING THE HIGGS POTENTIAL AND PRIMORDIAL … PHYS. REV. D 101, 125012 (2020) scalar field bounce solution. Therefore, any potential that has a very thick bounce solution and correspondingly small δM will have a much higher threshold of black hole mass for vacuum decay catalysis. We explored the threshold for a range of masses including the primordial black holes that could contribute to dark matter. Such black holes are outside the current SM parameter ranges, but could potentially be relevant for beyond the standard model potentials. However, just triggering the decay is not enough to destroy the universe, and automatically exclude associated black hole range. For a successful completion of the first order phase transition the bubbles have to percolate, which in turn depends on the number density of the created bubbles. Since every black hole that can initiate the false vacuum decay will create a bubble, the number of the FIG. 6. This plot shows the primordial black hole masses (in bubbles is equal to the number of such primordial black grams) at the time of their formation, Mi, as a function of the holes. We then trace evolution of the bubbles. The bubbles P ¼ 1 spectral index, n. The solid lines represent the value for the of the true vacuum expand with the speed of light, but the portion of the universe which already transitioned to the new P 1 background universe expands as well, so their number vacuum at present time. Below this line we have > and the P whole universe today would be destroyed. Above this line, we density decreases. We define a quantity , which represents have P < 1, and the universe is safe. The dashed, dot, dashed dot a portion of the universe which already transitioned to the −1 P ≥ 1 and dashed dot dot lines represent the values P ¼ 10 , new vacuum at the current time. For , the universe is P ¼ 10−2, P ¼ 10−3 and P ¼ 10−4 respectively, where the phase destroyed, and the associated range of parameters is transition has not completed yet, but there are bubbles of true excluded. vacuum present in the universe. Our procedure can be used in two ways. If we use the Higgs potential parameters as an input, we can constrain the black hole masses and the physics of formation (e.g., V. CONCLUSIONS the spectral index of perturbations). In turn, if we ever We demonstrated here that it is possible to connect the discover primordial black holes of definite mass, we can parameters of the Higgs potential with the primordial black use it to constrain the Higgs potential parameters, or indeed hole masses and physics of their formation (in our case the the presence of extra dimensions [52,53]. spectral index of perturbations that leads to their forma- tion). We used the recent result that corrections due to black ACKNOWLEDGMENTS hole seeds can significantly increase the tunneling proba- bility from the false to true Higgs vacuum. Any primordial It is a pleasure to thank Ian Moss for helpful discussions. black hole that had enough time to lose its mass from its D. C. D. was supported by the National Science Foundation formation till today to fit into an appropriate mass range for of China (Grants No. 11433001 and No. 11775140), the corresponding choice of the Higgs potential parameters National Basic Research Program of China (973 could trigger the false Higgs vacuum decay. If there is no Program 2015 CB857001), and the Program of Shanghai new physics beyond the standard model, any black hole Academic/Technology Research Leader under Grant lighter than 4.5 × 1014 g could trigger the decay. If some No. 16XD1401600. R. G. is supported in part by the new physics beyond the standard model modifies the Higgs Leverhulme grant Challenging the Standard Model with potential, then the black hole masses change appropriately. Black Holes, in part by STFC consolidated grant No. ST/ The decay rate is proportional to the exponential of the P000371/1, and in part by the Perimeter Institute. D. S. difference in entropy of the seed and remnant black holes was partially supported by the U.S. National Science masses, roughly ðMþ þ μ−ÞðMþ − μ−Þ. Numerically, this Foundation, under Grant No. PHY-1820738. Research at exponent varies slowly with Mþ, and so can be regarded as Perimeter Institute is supported by the Government of MδM. 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