Electroweak Vacuum Collapse induced by Vacuum Fluctuations of the Higgs Field around Evaporating Black Holes

In this paper, we discuss the Higgs vacuum stability around evaporating black holes. We provide a new approach to investigate the false vacuum decay around the black hole and clearly show how vacuum fluctuations of the Higgs induce a gravitational collapse of the vacuum. Furthermore, we point out that the backreaction of the Hawking radiation can not be ignored and the gravitational vacuum decay is exponentially suppressed. However, a large number of the evaporating (or evaporated) primordial black holes threaten the existence of the Universe and we obtain a new upper bound on the evaporating PBH abundance from the vacuum stability. Finally, we show that the high-order corrections of the BSM or QG would not destabilize the Higgs potential, otherwise even a single evaporating black hole triggers a collapse of the electroweak vacuum.

In this paper, we investigate how the gravitational vacuum polarization affects stabilities of the electroweak vacuum in the Schwarzschild black-hole background. By using the renormalized vacuum field fluctuation δφ 2 ren which is approximately expressed by the thermal Hawking temperature TH around the black-hole horizon (r ≈ 2MBH ), we discuss the vacuum stability around the Schwarzschild black hole. In particular, we newly investigate the stability of the electroweak vacuum around evaporating primordial black holes (PBHs) by taking into account the back-reaction effects on the effective Higgs potential V eff (φ) which was ignored in past works. By incorporating these effects and analyzing the stability of the vacuum, we show that one evaporating black hole does not cause serious problems in vacuum stability of the standard model Higgs and obtain an upper bound on the evaporating PBH abundance β O 10 −21 mPBH/10 9 g 3/2 not to induce any catastrophes.

I. INTRODUCTION
In the late 1970s, Hawking [1] showed that black holes emit thermal radiation at the Hawking temperature T H = 1/8πM BH due to the vacuum-polarization effects on the strong gravitational field [2] where M BH is the black-hole mass. The gravitational polarization around the black hole determines the fate of the evaporating black hole which is still unknown and closely related with the information loss puzzle [3], and furthermore, leads to the spontaneous symmetry restoration [4] or the false vacuum decay around the black hole [5][6][7], which bring cosmological singular possibility. Thus, the gravitational vacuum-polarization effects on the black hole background have a great impact on the stability of the vacuum.
Particularly recent discussion about the stability of the electroweak vacuum around the evaporating black * kohri@post.kek.jp † matshiro@post.kek.jp hole has been growing. There are no general mechanisms to prevent the formation of such small black holes which finally evaporate during the history of the Universe. Especially, the primordial black holes (PBH) are formulated by large density fluctuations in the early universe [47][48][49] which are incompatible with the above suggestion [50,51]. These phenomena could imposed serious constraints on cosmology or beyond Standard Model. However, there is some controversy about whether the evaporating black hole can be a trigger of the false vacuum decay on the electroweak vacuum. It is because the backreaction of the thermal Hawking radiation can not be ignored [45,46,50,52]. In the literature [40][41][42], the false vacuum decay in the Schwarzschild black hole has been investigated by the Coleman-De Luccia (CDL) instanton method [53] with the zero-temperature effective Higgs potential. However, it is reasonable intuitively to assume the high-temperature effective Higgs potential [54][55][56][57][58] instead of the zero-temperature potential in the environment of the thermal Hawking flux. The thermal corrections can generally stabilize the effective Higgs potential, and furthermore, the false vacuum decay from strong gravitational field can only happen around the black hole horizon. That is approximately we can obtain the vacuum decay ratio of the Minkowski spacetime only far from the black hole, which is extremely small. From this viewpoint, the probability of vacuum decay around the black hole can be expected to be lower than what was considered in the literature [40][41][42]. Nevertheless, there are no formal descriptions how the gravitational vacuum polarization like the Hawking radiation modify the effective Higgs potential and definitely affect the stability of the vacuum so far.
Formally, the gravitational vacuum polarization effects can be described by the vacuum expectation value of the energy momentum tensor T µν or the two-point correlation function δφ 2 in the quantum field theory (QFT) in curved spacetime. The former T µν provides a exact description of the quantum back-reaction on the geometry [59] and it is crucial in order to determine the backreaction of the conformal anomaly or the fate of the evaporating black hole. The two-point correlation function δφ 2 corresponds to the vacuum field fluctuation and plays the essential role in the vacuum stability. In the present paper, we provide a quantitative description of the vacuum stability on the Schwarzschild background by considering the vacuum field fluctuation δφ 2 , and investigate the electroweak vacuum stability near the black hole. In our analysis, we provide a new description of these phenomena and reach opposite conclusions in past works.
The organization of this paper is as follows. In Sec. II, we introduce renormalized vacuum field fluctuations for various vacua in the Schwarzschild spacetime. In Sec. III, stabilities of the electroweak vacuum around the block hole are discussed. Sec. IV is devoted to our conclusions and future outlooks.

II. THE RENORMALIZED VACUUM FIELD FLUCTUATION IN SCHWARZSCHILD SPACETIME
In this section, we consider the renormalized vacuum field fluctuation in Schwarzschild black-hole spacetime. The renormalized expression of the vacuum fluctuation for the massless scalar field has been well-know and analytical estimation is possible. However, the massive case requires complicated numerical calculations. For briefness we consider the renormalized vacuum field fluctuation for the massless scalar field in Boulware, Unruh and Hartle-Hawking vacuum following Candelas work [60]. In Section III, we will discuss the vacuum stability by using the renormalized vacuum field fluctuation.
The metric in the Schwarzschild coordinates where we ignore the quantum backreaction of the scalar field on the geometry can be written by which can cover the exterior region r > 2M BH of the spacetime where M BH is the black hoe mass. The above singularity at the horizon r = 2M BH can be removed by transforming to Kruskal coordinates. By taking the Kruskal coordinates, we can obtain the following metric where these coordinates U and V are formally given by The Penrose-Carter diagram of the maximally extended Schwarzschild manifold. Regions I or II are asymptotically flat, Region III is the black hole, and Region IV is the white hole. H + corresponds to the (future) black hole horizon, H − is the (past) black hole horizon, J + corresponds to the (future) null infinity and J − is the (past) null infinity.
The Schwarzschild coordinates of Eq. (1) cover only a part of the spacetime, whereas the Kruskal coordinates of Eq. (2) cover the extended spacetime and becomes regular at the black hole horizon. These features of the Schwarzschild geometry are summarized in the Penrose-Carter diagrams as Fig. 1. In the curved spacetime, there are no unique vacua and we must take an appropriate vacuum state. In the Schwarzschild spacetime, there are three well defined vacua, namely: the Boulware vacuum (vacuum polarization around a static star) [61,62], the Unruh vacuum (black hole evaporation) [63] and the Hartle-Hawking vacuum (black hole in thermal equilibrium) [64] which correspond to the definitions of the normal ordering on the respective coordinates.
The Klein-Gordon equation for the massless scalar field φ (t, x) can be given by where we drop the curvature term ξRφ 2 because the Ricci scalar becomes R = 0 in Schwarzschild spacetime [65] for simplicity, but this approximation may brake down when the quantum backreaction on the metric can not be neglected. In the exterior region of Schwarzschild spacetime, the scalar field φ (t, r, θ, ϕ) can be decomposed into the from where these mode functions u in ωlm and u out ωlm defines the vacuum state that a ωlm |0 = b ωlm |0 = 0 which corresponds to the boundary conditions. In the Schwarzschild spacetime, these mode functions u in ωlm and u out ωlm for the massless scalar field are given by where these radial functions R in l (r; ω) and R out l (r; ω) have the well-known asymptotic forms, (ω) and B l (ω) are the reflection and transmission coefficients [66]. The Boulware vacuum |0 B is defined by taking ingoing and outgoing modes to be positive frequency with respect to the Killing vector ∂ t of the Schwarzschild metric [61] and constructed by using the scattering theory interpretation. This state closely reproduces the Minkowski vacuum |0 M at infinity because 0 B | δφ 2 |0 B → 1/r 2 in the limit r → ∞. However, the Boulware vacuum is singular on the event horizons r = 2M BH and hence unacceptable near the black-hole horizon. Therefore, the Boulware vacuum is considered to be the appropriate vacuum state which describes the vacuum polarization around a static star.
The two-point correlation function δφ 2 related with the Boulware vacuum |0 B can be given by [60,67]: where the sum of these radial functions R in l (r; ω) and R out l (r; ω) have the asymptotic forms, Therefore, the two-point correlation function δφ 2 of Eq. (8) has clearly UV divergences and must be regularized. There are several regularization methods to eliminate the UV divergences in the quantum field theory (QFT), but the point-splitting regularization is the extremely powerful and standard method to obtain the renormalized expression in the curved spacetime. Let us consider temporarily δφ 2 (x) → δφ (x) δφ (x ) to remove the divergences and afterwards take the coincident limit x → x, where δφ (x)δφ (x ) div express the divergence part and is namely the DeWitt-Schwinger counter-term, which can be generally given by [68] δφ where σ is the biscalar associated with the short geodesic, R or R αβ are respectively the Ricci scalar or tensor and γ express the Euler-Mascheroni constant. The renormalization parameter µ corresponds to the mass m of the scalar field, and therefore, the massless case lead to the well-known ambiguity [69], but the renormalization procedure can eliminate this ambiguity for T µν by the cosmological experiment or observation. In the Schwarzschild metric for the massless scalar field where m = 0 and R = 0, we can simplify the DeWitt-Schwinger counter-term of δφ (x)δφ (x ) div to be For simplicity we take the time separation as x = (t, r, θ, ϕ) and x = (t + , r, θ, ϕ) and the renormalized expression of δφ 2 in the Boulware vacuum |0 B can be given by By taking a second-order geodesic expansion we can obtain the following expression [68] σ where −2 satisfy the following relation By using Eq. (12), Eq. (13) and Eq. (14), we can obtain the renormalized expression of the Boulware vacuum |0 B [60], For the Boulware vacuum |0 B we have the asymptotic expression of the renormalized vacuum field fluctuation δφ 2 ren [60], where the renormalized expression of δφ 2 ren is singular on the event horizons r = 2M BH and ill-defined near the black-hole horizon. In the case of the energy momentum tensor T µν , the renormalized expression of the energy momentum tensor T µν ren was given by Ref. [60,[69][70][71][72][73][74] and shows similar properties to the renormalized expression of δφ 2 ren [75]. Therefore, the usual interpretation of the above result is that the Boulware vacuum |0 B is considered to be the appropriate vacuum state around a static star and not a black hole.
Next, we consider the Unruh vacuum |0 U , which corresponds to the evaporating black hole in the empty space. The Unruh vacuum |0 U is formally defined by taking ingoing modes to be positive frequency with respect to ∂ t , but outgoing modes to be positive frequency with respect to the Kruskal coordinate ∂ U [63]. The Unruh vacuum corresponds to the state where the black hole radiates at the Hawking temperature T H = 1/8πM BH in the empty space, and therefore, the vacuum field fluctuation δφ 2 approaches the thermal fluctuation near the black-hole horizon as 0 U | δφ 2 |0 U → O T 2 H in the limit r → 2M BH . Therefore, the Unruh vacuum is considered to be appropriate vacua which describe the evaporating black hole formed by gravitational collapse [60].
For the Unruh vacuum we obtain the two-point correlation functions δφ 2 [60,67], where we introduce κ = (4M BH ) −1 which is the surface gravity of the black hole and the factor of coth πω κ originates from the thermal features of the outgoing modes. The renormalized vacuum field fluctuation in the Unruh vacuum |0 U can be give by For the Unruh vacuum |0 U , we can obtain asymptotic expression of the renormalized vacuum field fluctuation δφ 2 ren [60], The Hartle-Hawking Vacuum |0 HH is formally defined by taking ingoing modes to be positive frequency with respect to ∂ V , and outgoing modes to be positive frequency with respect to the Kruskal coordinate ∂ U [64]. In the Hartle-Hawking vacuum |0 HH , we can obtain the twopoint correlation functions, For the Hartle-Hawking vacuum we can get the asymptotic expression of the renormalized vacuum field fluctuation δφ 2 ren [60], where the renormalized vacuum field fluctuation δφ 2 becomes exactly the thermal fluctuation at infinity, i.e 0 HH | δφ 2 |0 HH → T 2 H /12 in the limit r → ∞. Therefore, the Hartle-Hawking vacuum corresponds to a black hole in thermal equilibrium at T H = 1/8πM BH .

III. THE STABILITY OF THE ELECTROWEAK VACUUM AROUND THE BLOCK HOLE
In this section, we describe how the vacuum field fluctuation affects the stability of the vacuum and then discuss the electroweak vacuum stability around the evaporating block hole where r ≈ 2M BH . Now, for briefness we restrict our attention to the scalar field theory and assume a simple scalar potential V (φ) where the scalar field φ couples the extra scalar field ϕ with the positive interaction coupling g, where we assume that the self-interaction coupling λ is positive (λ > 0). As previously discussed, the renormalized vacuum field fluctuation δφ 2 ren near the block hole can be approximately given by The vacuum field fluctuation δφ 2 ren from the gravitational vacuum polarization is completely classic and can modify the scalar potential of Eq. (19) as follows: where we shift the scalar field φ 2 → φ 2 + δφ 2 ren and ϕ 2 → ϕ 2 + δϕ 2 ren in order to take into account the gravitational vacuum polarization. For the relatively large vacuum fluctuation of φ to be λ δφ 2 ren g δϕ 2 ren and δφ 2 ren m 2 /λ, the scalar potential V (φ) is destabilized. On the other hand, the scalar potential of Eq. (21) can be stabilized and becomes metastable state when λ δφ 2 ren g δϕ 2 ren . However, the vacuum fluctuation can cause directly the false vacuum decay in the metastable vacuum. In fact, when the inhomogeneous and local scalar field described stochastically by δφ 2 ren exceeds the hill of V (φ), the localized scalar field can classically form the true vacuum domains or bubbles.
In the case of the electroweak vacuum around the black-hole, the Higgs field fluctuation work to push down the Higgs potential due to the negative running correction of the Higgs self-coupling, whereas the contributions of the gauge bosons or the fermions raise the effective Higgs potential. Therefore, it turns out that the Higgs vacuum stability around the black hole corresponds approximately to the local thermal situations although it is complicated task to investigate the stability of the electroweak vacuum around the black hole.
Let us consider the effective Higgs potential V eff (φ) in the standard model (SM) where φ is the Higgs field. The one-loop standard model Higgs potential without the curvature mass ξRφ 2 in the 't Hooft-Landau gauge can be given by [86][87][88] where λ eff (φ) is the effective self-coupling of φ written by where µ is the renormalization scale and ρ Λ is the cosmological constant. The coefficients n i and C i are given by n W = 6, n Z = 3, n t = −12, n G = 3, n H = 1, C W = C Z = 5/6, C t = C G = C H = 3/2, and the mass terms m 2 i (φ) of the W and Z bosons, the top quark, the Nambu-Goldstone bosons, and the Higgs boson can be give by where g, g , y t are the SU (2) L , U (1) Y , top Yukawa couplings and λ φ is the Higgs self-coupling. From the viewpoint of the renormalization group (RG), the effective self-coupling λ eff (φ) of Eq. (23) becomes negative when the classic Higgs field is larger than the instability scale to be φ Λ I [20]. Now let us improve the effective Higgs potential of Eq. (22) by including δφ 2 ren in the Schwarzschild background. In order to include the back-reaction of the vacuum fluctuation of the Higgs field, let us shift the Higgs field φ 2 → φ 2 + δφ 2 ren . Thus, the effective Higgs potential with the Higgs vacuum fluctuation can be given by where the effective self-coupling λ eff (φ, δφ 2 ren ) can be negative and the effective Higgs potential is destabilized when δφ 2 1/2 ren Λ I . In the Schwarzschild background, the renormalized expression of δφ 2 ren proportional to the inverse of the black-hole mass M BH or the Hawking temperature T H , and therefore, the effective self-coupling λ eff (φ, T H ) becomes negative with T H Λ I . However, the vacuum fluctuation of the W and Z bosons or the top quark can raise the effective Higgs potential, and therefore, the effective Higgs potential V eff (φ) including the vacuum fluctuation of the various fields around the block hole can be written by where κ is defined by g, g , y t and λ, and we assume that the vacuum fluctuation of the various field like the Higgs, It is evident that the effective Higgs potential around the evaporating black hole reproduce the thermal one [54][55][56][57][58]. Therefore, the effective Higgs potential can be stabilized even in the Schwarzschild black-hole background. However, there still remains the possibility of the false vacuum decay via the vacuum fluctuation on the blackhole background. These estimations are usually calculated by using the CDL instanton method [53] which requires a full numerical analysis for a range of parameter space. In the present paper, we discuss the false vacuum decay by using the stochastic method proposed by Linde [89][90][91] where it was applied to estimate the probability of the inflationary universe creation. This approach is exactly consistent with the instanton method and extremely simple even in the investigation of the false vacuum decay with taking account of the gravitational effects. Now, we consider the electroweak vacuum stability in the Schwarzschild black-hole background by using the stochastic method. The probability of the local Higgs fields where the vacuum fluctuation δφ 2 ren exists can be given by [32] P φ, δφ 2 ren = 1 2π δφ 2 ren exp − φ 2 2 δφ 2 ren .
By using Eq. (26), we can obtain the probability not to exceed the hill of the effective Higgs potential as follows: where we define φ max to be the maximal field value of the effective Higgs potential. The probability that the localized Higgs fields go into the negative Planck-energy true vacuum is estimated to be Then, the constraint from the vacuum decay for the Higgs field is represented by [32]: where V express the volume factor or the number of the correlated patches. By substituting Eq. (29) into Eq. (31), we can simplify obtain a constraint from the vacuum stability, in order not to induce a transition due to large vacuum fluctuations. In the inflationary Universe, the volume factor of V and the renormalized expression of δφ 2 ren can be given by [32] V e 3N hor , δφ 2 where N hor is the e-folding number which can be N hor N CMB 60 [93]. In the Unruh vacuum |0 U corresponding to the vacuum state around the evaporating black hole, the renormalized expression of δφ 2 ren approximately approach the value of the the Hawking temperature T H near the horizon. But at the infinity δφ 2 ren attenuates rapidly and becomes zero. Therefore, we can summarize the renormalized vacuum fluctuation δφ 2 ren around the evaporating black hole as follows: However, the most uncertain thing in the stochastic formalism is how to determine the volume factor of V. We took V to be the volume of the domains in which the vacuum fluctuation δφ 2 ren governs. It is obvious that we can not take the entire volume of the Universe as V because the vacuum fluctuation of δφ 2 ren approach zero far from the black hole and the large vacuum fluctuation exists only near the black-hole horizon. Therefore, let us assume that the volume factor V can be given by V = N PBH · O (1) where N PBH is the number of the evaporating or evaporated primordial black holes during the cosmological history of the Universe. By using Eq. (29), Eq. (30) and Eq. (31), we can estimate the constraint of the number on the evaporating primordial black holes as follows: The Eq. (35) on the left side shows the number of the evaporating black holes which cause the Higgs vacuum collapse. Therefore, we can obtain the constraint on the number of the evaporating primordial black holes as N PBH O 10 43 which is extremely huge in order to threaten the Higgs metastable vacuum. Thus, one evaporating black hole can not cause serious problems in vacuum stability of the standard model Higgs case. The total number of the evaporating black hole (or the PBHs) strongly depends on the cosmological models at the early Universe [50,51], and therefore, we provide the upper bound on the yield of the PBHs Y PBH / ≡ n PBH /s as follows.
where s 0 denotes the entropy density at present (≈ 3 × 10 −4 eV 3 ), and H 0 the current Hubble constant (≈ 10 −33 eV). Note that Y PBH is constant from the formation time to the evaporation time.
It is convenient to transform this bound into an upper bound on β, which is defined by taking values at the formation of the PBH to be where ρ PBH and ρ tot are the energy density of the PBHs and the total energy density of the Universe including the PBHs at the formation, respectively. It is remarkable that β means the number of the PBHs per the horizon volume at the formation (β ∼ n PBH /H 3 . This bound can be stronger than the known one for m PBH 10 9 g [94]. However, at the final stage of the evaporation of the black hole, the black-hole mass M BH becomes extremely small and the Hawking temperature T H approaches to the Planck scale M Pl = 2.4 × 10 18 GeV where M Pl is the reduced Planck mass. Therefore, the UV corrections of the beyond Standard Model (BSM) and the quantum gravity (QG) can not be ignored at the last stage of the evaporation and undoubtedly contribute to the Higgs vacuum metastability. In the rest of this section, we discuss how the UV or Planck scale physics affect the electroweak vacuum stability. When the Hawking temperature approaches to the Planck scale as T H → O (M Pl ), the contribution of the Planck physics determines the stability of the vacuum. Therefore, if the UV or Planck scale physics destabilize the effective Higgs potential at the high energy, the catastrophe vacuum decay can happen by even a single evaporating black hole. Now, we consider the effective Higgs potential with the corrections of the BSM and the QG. For convenience we add two higher dimension operators φ 6 and φ 8 via the Planck scale physics to the effective Higgs potential as follows: where δλ bsm express the running corrections from the BSM, λ 6 and λ 8 dimensionless coupling constants. Thees higher-dimension contributions of λ 6 and λ 8 are usually negligible except for the Planck scale excursion φ ≈ M Pl . On the other hand, these corrections can affect the false vacuum decay via the quantum tunneling given by the literature [13][14][15][16]. In the Schwarzschild black-hole background, these enhancements were discussed in the literature [40][41][42] although the author entirely neglected the back-reaction of the Hawking thermal radiation. However, at the final stage of evaporation of the black hole where T H → O (M Pl ), these higher-dimension contributions of λ 6 and λ 8 can not be neglected and have a strong impact on the vacuum stability around the black hole. As previously discussed, the effective Higgs potential of Eq. (41) is modified by the Hawking thermal fluctuation around the black hole: where the Planck scale contributions govern the effective Higgs potential at T H → O (M Pl ). If the effective Higgs potential is destabilized by these UV scale corrections via δφ 2 ren ≈ O T 2 H and the effective Higgs potential becomes negative to be ∂V eff (φ)/∂φ < 0, the local Higgs fields around the black-hole classically roll down into the negative Planck-energy true vacuum and Higgs Anti-de Sitter (AdS) domains whose sizes are about the blackhole horizon are formed. Not all Higgs AdS domains threaten the existence of the Universe, which highly depends on their evolutions (see Ref. [32,44] for the detail discussions). However, Higgs AdS domains generally expand eating other regions of the electroweak false vacuum and finally consume the entire Universe.
Therefore, even a single evaporating black hole can be completely catastrophic for the stability of the electroweak vacuum via the extremely high Hawking temperature T H → O (M Pl ) although this possibility strongly depends on the BSM or the Planck scale physics and the detail of the evaporation of the black hole.

IV. CONCLUSION AND OUTLOOK
In this paper, we have investigated how gravitational vacuum polarization around a Schwarzschild black hole affects stability of the (electroweak) vacuum. We have discussed the problems by using the renormalized vacuum field fluctuation δφ 2 ren describing approximately the Hawking thermal fluctuation near the black-hole horizon (r ≈ 2M BH ). In particular, we have studied the stability of the electroweak vacuum around an evaporating black hole by taking account of the back-reaction effects on the effective Higgs potential V eff (φ) which was ignored in previous works of the Higgs vacuum stability.
When we incorporate the back-reaction effects and reanalyze the stability of the vacuum around the black hole, the stability conditions of the vacuum approximately reproduce the ones in the local thermal situations. Therefore, one evaporating black hole does not cause catastrophic problems in vacuum stability of the standard model Higgs, but we have obtained an upper bound on the evaporating PBH abundance β O 10 −21 m PBH /10 9 g 3/2 not to induce any instabilities during the history of the Universe. However, at the final stage of the evaporation of the black hole, the black-hole mass M BH becomes extremely small and the Hawking temperature reaches the Planck scale where T H → O (M Pl ). Therefore, the Planck scale physics or the beyond Standard Model (BSM) directly may intervene and have a strong impact on the vacuum stability around the black hole. Our discussion will be changed if the Planck-scale physics or the BSM destabilize the Higgs field.
In the numerical analysis, we can approximate the maximal field value to be φmax 10 · m eff . [93] In the case of the reheating era where the entire Universe is thermalized after inflation, the physical volume of V and the thermal fluctuation of δφ 2 ren can be given as follows [35]: V e 3N hor , δφ 2 ren T 2 12 − m eff T 4π (43)