Black holes and Higgs stability

We study the effect of primordial black holes on the classical rate of nucleation of AdS regions within the standard electroweak vacuum. We find that the energy barrier for transitions to the new vacuum, which characterizes the exponential suppression of the nucleation rate, can be reduced significantly in the black-hole background. A precise analysis is required in order to determine whether the the existence of primordial black holes is compatible with the form of the Higgs potential at high temperature or density in the Standard Model or its extensions.


Introduction
The electroweak vacuum in the Standard Model is metastable.

Introduction
Matching the geometries The critical bubbles The Higgs profile Conclusions The tunnelling rate to the new vacuum is exponentially suppressed by the action of the instanton, which is the configuration that minimizes the Euclidean action where R is the Ricci scalar, κ = 1/M 2 Pl = 8πG with M Pl = M Pl / √ 8π, M Pl ≈ 1.22 × 10 19 GeV.
The instanton is an O(4)-symmetric configuration on a space with metric ds 2 = dr 2 + ρ(r ) 2 dΩ 2 , where dΩ is the volume element of the unit 3-sphere.

Introduction
Matching the geometries  There are also constraints on the scale of inflation.
In the absence of a large Higgs mass term, the evolution of the long wavelength modes of the Higgs field h is controlled by the Langevin equation where η is a Gaussian random noise with

Introduction
Matching the geometries The critical bubbles The Higgs profile Conclusions

Matching the geometries
Spherical AdS-Schwarzschild bubble within asymptotically flat or dS space, separated by a thin wall with surface tension σ.
The space inside the bubble has a metric The space outside the bubble is described by the metric

Introduction
Matching the geometries The critical bubbles The Higgs profile Conclusions The Israel junction conditions give where ϵ 1 = ±1, ϵ 2 = ±1 are possible sign choices. The square of this equation can be put in the form For ϵ 1 = 1,Ṙ ≪ 1 and G → 0, the above expression has a Newtonian interpretation.

Introduction
Matching the geometries The critical bubbles The Higgs profile Conclusions

The AdS crunch
The evolution after the wall reaches the timelike boundary of AdS, cannot be determined without additional boundary conditions. There is a Cauchy horizon. A spacelike singularity (the AdS 'crunch' of Coleman and De Luccia) must develop in the bubble interior. The coordinate change puts the AdS metric in the form ) .
This metric describes an homogeneous FRW universe that is born with a big 'bang' and collapses in a big 'crunch'. The coordinate singularity becomes a true physical singularity in the presence of a fluctuating Higgs field.

Introduction
Matching the geometries The critical bubbles The Higgs profile Conclusions

The critical bubbles
The mass of the bubble configuration is We concentrate on the bubble evolution for κ 2 = (Gσ) 2 ≪ 1/ℓ 2 . It is reasonable to assume that at the time of production of an AdS bubble the wall has small velocity (Ṙ ≃ 0). We set ϵ 1 = 1. Configurations with ϵ 1 = −1 correspond to shrinking bubbles. There is a critical value m cr , above which δM(R, m) is negative for all R. This value and the corresponding bubble radius R cr are The bubble radius R cr is always larger than the horizon radius of the black hole.
There are no bubbles with radii below a certain value. ForṘ = 0, the minimal radius R h satisfies If the bubble is located within the horizon, with vanishing wall velocity, it cannot grow. The mass parameter m is not a properly defined physical quantity.
A geometrical quantity that can be used to characterize the energy content of the central region is the horizon radius R h .
provides an alternative estimate for the energy barrier associated with the bubble. R h (m)/(2G) coincides with m only for 2Gm/ℓ ≪ 1, while it is much smaller than m for 2Gm/ℓ ≫ 1. As a result, δM ′ may provide an overestimate of the energy barrier for large m.

Introduction
Matching the geometries The critical bubbles The Higgs profile Conclusions For small κℓ and 1/ℓ ′ → 0, the minimal critical value of δM ′ is obtained for The points (R cr , m cr ), (R ′ cr , m ′ cr ) are close: can be compared with the barrier in the absence of the black hole, estimated by M(R 0 , 0), with ∂M(R 0 , 0)/∂R = 0. For small κℓ and 1/ℓ ′ → 0 For 1/ℓ ′ ≫ 1/ℓ, the quantity δM ′ (R, m) turns negative for Complete instability is expected for a strong dS background.

Introduction
Matching the geometries The critical bubbles The Higgs profile Conclusions The critical bubble mass can be estimated as for all values of 1/ℓ, 1/ℓ ′ .

Introduction
Matching the geometries The critical bubbles The Higgs profile Conclusions

Standard Model Higgs
The Higgs potential has the approximate form V ∼ λ(h) h 4 /4 for values of the Higgs field h above 10 6 GeV. The quartic coupling λ varies from 0.02 to −0.02 for Higgs values between 10 6 GeV and 10 20 GeV, respectively. The maximum of the potential is located at a value h max ∼ 5 × 10 10 GeV. Near the maximum the potential can be approximated as

Introduction
Matching the geometries The critical bubbles The Higgs profile Conclusions In the presence of a black hole, the energy barrier to be overcome in order to produce an AdS bubble is reduced significantly.
A primordial black hole can form when the density fluctuations are sufficiently large for an overdense region of horizon size to collapse.
Its maximal mass is of order the total mass within the particle horizon m bh ∼ M 2 Pl /H, while its maximal radius is R bh ∼ 1/H. These estimates are also valid for black holes that are pair-produced during inflation. We can estimate m cr m bh , V ∼ −λh 4 , and V ′ equal to the inflaton vacuum energy V inf during inflation, or to zero after its end. This ratio is always smaller than 1.

Introduction
Matching the geometries The critical bubbles The Higgs profile Conclusions

Beyond the thin-wall approximation
For the metric with N(r ) = 1 − 2GM(r )/r , the equations of motion become On the horizon: 2GM(R h ) = R h . For a finite ADM mass:

Introduction
Matching the geometries The critical bubbles The Higgs profile Conclusions The ratio δM ′ /M 0 has a minimum δM ′ max ≃ 0.473 at R h ≃ 11. There is a reduction of the energy barrier by approximately a factor of 2, instead of the factor of 3 estimated through the thin-wall approximation.
This reduction still has a profound effect on the nucleation rate. The energy barrier drops from approximately 300 to 150 in units of h max . The characteristic scale of the solutions is set by h max . This means that gravitational corrections are not relevant, as they are suppressed by powers of h 2 max /M 2 Pl . For our solutions κℓ ∼ √ Gh max ∼ h max /M Pl ≪ 1. The energy δM ′ associated with the bubble is much smaller than the mass of the central black hole, as estimated by R h /(2G). The gravitational background is induced mainly by the black hole, with the bubble being only a small perturbation.

Introduction
Matching the geometries The critical bubbles The Higgs profile Conclusions

Higgs potential and primordial black holes
Is the presence of primordial black holes consistent with the Standard Model Higgs?
The barrier for classical transitions to the AdS vacuum is reduced by roughly a factor of 2 in the presence of a black hole for an asymptotically flat false vacuum.
For an asympotically dS false vacuum, the barrier is eliminated by a sufficiently big black hole, indicating complete instability. Consider a high-temperature environment with T ∼ h max in asymptotically flat space. The bubble nucleation probability per unit time is dP/dt = T exp(−δM ′ /T ).
The smallest time interval that can be associated with the scale T is the Hubble time ∼ M Pl /T 2 . The number of causally independent regions, which are currently within our horizon, is roughly N ∼ 10 34 (T /GeV) 3 . Assume that a black hole can be produced within each of these regions with probability p.
The reduction of the energy barrier δM ′ from ∼ 300 to ∼ 150 in units of h max means that the exponental suppression is eliminated.
It must be emphasized, however, that the probability p for the creation of a primordial black hole may be very small, resulting in the suppression of the rate.