On the catalysis of the electroweak vacuum decay by black holes at high temperature
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Abstract
We study the effect of primordial black holes on the classical rate of nucleation of AdS regions within the standard electroweak vacuum at high temperature. We base our analysis on the assumption that, at temperatures much higher than the Hawking temperature, the main effect of the black hole is to distort the Higgs configuration dominating the transition to the new vacuum. We estimate the barrier for the transition by the ADM mass of this configuration, computed through the temperaturecorrected Higgs potential. We find that the exponential suppression of the nucleation rate can be reduced significantly, or even eliminated completely, in the blackhole background if the Standard Model Higgs is coupled to gravity through the renormalizable term \(\xi {\mathcal {R}} h^2\).
1 Introduction
It is wellknown that for the current measured values of the Higgs and top quark masses the Standard Model (SM) effective potential develops an instability. Due to the running of the quartic coupling, the effective potential of the Higgs reaches a maximum and then becomes unbounded from below at values of the Higgs field of about \(5\times 10^{10}\) GeV. Our electroweak vacuum has a lifetime which is many orders of magnitude larger than the age of the Universe and is therefore stable against decay through quantum tunnelling.
A pertinent question concerns the fate of the electroweak vacuum during the evolution of the Universe in situations in which gravity plays a pronounced role. The gravitational background can have a significant effect on the rate of vacuum decay, leading to its enhancement or suppression [1]. This issue is crucial for the decay of the electroweak vacuum, because of the extreme sensitivity of the Higgs potential to the Higgs and top masses [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. The fact that the gravitational effects are significant during inflation has led to an extensive investigation of the stability of the electroweak vacuum during this era, leading to bounds on the inflationary scale [14, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] in order to avoid a catastrophic singularity characterized as the AdS crunch [1, 31, 36].
Strong gravitational fields are also sourced by black holes, which raises the question of electroweak vacuum stability in their vicinity. It has been argued that black holes can act as impurities enhancing the quantum decay rate to a level incompatible with the age of the Universe [37, 38, 39]. It is natural to expect a similar enhancement for classical transitions to the true Higgs vacuum, induced by the hightemperature environment of the early Universe [40]. This second mechanism can be explored through a more intuitive approach, with fewer technical uncertainties than the calculation of the quantum tunnelling rate. In any case, the question whether the existence of primordial black holes [41, 42, 43, 44] is consistent with the stability of the electroweak vacuum has important implications for the compatibility of the Standard Model of particle physics and the cosmological model [45].
In this paper we explore further the scenario of temperatureinduced vacuum decay in the blackhole background [40]. Firstly, we repeat the analysis using the temperaturecorrected effective potential of the Higgs field, instead of the zerotemperature one employed in [40]. Secondly, we allow for a nonminimal coupling of the Higgs field to gravity. This additional coupling can have strong effects on the action of the bounce in the case of quantum tunnelling, or the free energy of the critical bubble in the case of thermal tunnelling. Such behavior is known to occur in the context of inflation [31] and similar features are expected within the strong gravitational field of a black hole.
The paper is organized as follows. In Sect. 2 we derive the equations of motion and the relevant expressions for the energy of the bubble configuration. In Sect. 3 we study the finitetemperature effects on the Higgs potential and estimate the size of the barrier to be overcome for vacuum decay in the presence of black holes. In Sect. 4 we discuss the stability of the electroweak vacuum and derive constraints for the allowed range of the nonminimal coupling of the Higgs field to gravity. In the final section we give our conclusions.
2 Equations of motion
We consider now the SM Higgs field in the presence of black holes that can trigger the decay of the electroweak vacuum by providing nucleation sites [37, 38, 39]. The decay takes place through the formation of bubbles of the new vacuum around the black holes. The critical bubble is a static configuration, whose profile can be obtained by solving the equation of motion of the Higgs field with appropriate boundary conditions. The field interpolates between values on either side of the maximum of the potential.
In a thermal environment, we expect that the probability for the transition to the true vacuum will be exponentially suppressed, with the exponent being proportional to 1 / T. When the effects of gravity and entropy are negligible, the proportionality factor is simply the energy of the dominant saddle point underlying the transition. If the entropy is large, the energy is replaced by the free energy. In the presence of significant gravitational effects, however, the generalization is not obvious. We shall assume here that the ADM mass of the critical Higgs bubble is a measure of the barrier that must be overcome for the field to fluctuate beyond the potential maximum. In order to take into account the entropy associated with the heat bath, we shall employ the temperaturecorrected Higgs potential, which accounts for the entropy density of the thermally excited Higgs particles.
The form of the above equations demonstrates that the Higgs configuration is not modified by the nonminimal coupling to gravity, and, therefore, is independent of \(\xi \). The bubble has a negligible effect on the gravitational background, which is determined by the blackhole mass to a very good approximation. Allowing for a nonminimal Higgs coupling to gravity through a term \(\sim h^2 {\mathcal {R}}\) in the action does not modify this conclusion, because the curvature \({\mathcal {R}}\) vanishes in a blackhole background. As a result, all modifications to the background due to the Higgs are suppressed by \(M_{\mathrm{Pl}}\).
3 Finitetemperature effects
The next step is to consider the immersion of the black holeHiggs system in the thermal environment of the early Universe. In a thermal environment, the quantity relevant for transitions between different states is the free energy, which accounts for the effect of entropy. In this respect, it is natural to employ the temperaturedependent effective potential, which can be identified with the freeenergy density. On the other hand, the gravitational field is sourced by the energymomentum tensor, which includes the energy density and is identified with the zerotemperature potential. The formal derivation of the appropriate expressions on a strong gravitational background is difficult. For example, the effects of temperature are usually taken into account by compactifying the Euclidean time direction. On a blackhole background, the compactification scale is set by the Hawking temperature, which does not coincide necessarily with the ambient temperature, leading to singularities in the underlying geometry. For these reasons our approach will be intuitive rather than formal.
As we saw in the previous section, the backreaction of the Higgs field is negligible, so that the background is described by the Schwarzschild metric to a very good approximation. The equations relevant for our problem, Eqs. (8) and (9), determine the shape and energy of the Higgs configuration that the fluctuating system on this background has to go through for the transition to occur. At nonzero temperature, energy is expected to be replaced by free energy as the relevant quantity in the calculation of the nucleation rate. It is then justifiable intuitively to replace the zerotemperature potential with the hightemperature one. The resulting equations have the correct limits for vanishing temperature or blackhole mass. We emphasize, however, that a formal derivation is lacking.
A very interesting feature is the effect of the nonminimal coupling on the bubble free energy. The dependence on \(\xi \) is linear, as can be seen from Eq. (22). Large positive values of \(\xi \) result in the growth of \(\delta M_\mathrm{tot}/T\) and the suppression of the nucleation rate. From Eq. (22) it is apparent that, for every blackhole mass at a given temperature, there is a critical negative value \(\xi _\mathrm{cr}=F_1(\tilde{R}_\mathrm{h})/F_2(\tilde{R}_\mathrm{h})\), for which the exponential suppression of the nucleation rate is eliminated. Clearly, the saddlepoint approximation breaks down before this point is reached. On the other hand, the probability of vacuum decay in the early Universe depends also on the total number of primordial black holes that can be generated. In the following section we discuss this point in detail.
4 Vacuum (in)stability in the presence of black holes
In Ref. [40] it was pointed out that \(A(R_{\mathrm{h}}T)\) can lead to the reduction of the effect of \((\delta M_\mathrm{tot}/T)_0\) by a factor of roughly 2. Our present analysis demonstrates that the nonminimal coupling \(\xi \) to gravity can have a more dramatic effect: For \(\xi =1/B(R_{\mathrm{h}}T)\), the exponential suppression of the nucleation rate can be eliminated completely. Clearly, the saddlepoint expansion around the bubble configuration breaks down before this point is reached. On the other hand, the presence of a large prefactor, coming from the huge number of causally independent regions in the early Universe, indicates that the electroweak vacuum is likely to become totally unstable.
As we have mentioned above, the largest uncertainty in the calculation is connected with the probability p to find a black hole within each causally independent region at the time when the ambient temperature is T. A primordial black hole can form when the density fluctuations are sufficiently large for an overdense region to collapse [41, 42, 43]. It is usually assumed that its typical mass is of the order of its maximal possible mass. The latter is given by the total mass within the particle horizon \(\sim M_{\mathrm{Pl}}^2/H\), while the maximal radius is \(\sim 1/H\). It is not clear, however, if these assumptions are consistent with the typical size of density fluctuations in the early Universe, or the constraints imposed by the observations of the microwave background.
As a concrete application, we consider the scenario of Ref. [45], which is consistent with the current bounds on the size of primordial density fluctuations. It is assumed that the inflationary era is followed by a period during which the inflaton oscillates and decays into particles. The effective equation of state is similar to that of a matterdominated Universe. It is possible, therefore, for perturbations that were generated during inflation to reenter the horizon, grow gravitationally and collapse into black holes. When the thermalization of the decay products takes place, the equation of state changes and the growth of perturbations is suppressed. In such a scenario, the black holes most relevant for our discussion are those produced just before thermalization, because they are the most massive ones. This must be contrasted with the scenario of the quantum decay of the electroweak vacuum in which the most relevant black holes are the light ones, because of their large Hawking temperature [37, 38, 39, 45].
It seems, therefore, that a high reheating temperature eliminates the danger posed by the presence of black holes. On the other hand, for Open image in new window , the typical mass of a black hole in this scenario is Open image in new window GeV \(\simeq 10^3\) g. Such small black holes evaporate very quickly and are of little phenomenological interest. In this respect, the possibility of a reheating temperature smaller than \(10^{7} M_{\mathrm{Pl}}\) seems more exciting. As an example, let us consider the possibility of a reheating temperature \(T\simeq 5\times 10^{11}\) GeV, for which Eq. (26) with \(\delta _i \simeq 10^{4}\) gives \(\tilde{R}_{\mathrm{h}}\simeq 0.5\), so that \(A(R_{\mathrm{h}}T)\simeq 0.5\) and \(B(R_{\mathrm{h}}T)\simeq 5.3\) in Eq. (24). Using Eq. (25), we obtain \(\ln (N p P)\simeq 1240.5 (1+5.3 \, \xi )(\delta M_\mathrm{tot}/T)_0\), with \((\delta M_\mathrm{tot}/T)_0\) computed in the absence of black holes. The probability becomes of order one for Open image in new window . The stability of the electroweak vacuum in the presence of primordial black holes imposes a strong constraint on the nonminimal coupling of the Higgs field to gravity, by forbidding this range.
Our discussion above neglects effects resulting from the accretion of the primordial plasma by the black hole. We expect that a stationary situation will develop with an accretion disc forming around the black hole. Even though the profile of the Higgs field will be distorted by this configuration, we still expect that the main effect will arise through the presence of the horizon and is captured by the boundary condition (10) that we assumed.
5 Conclusions
Given the absence of new physics in the LHC data, the stability of the SM electroweak vacuum has become a more pressing issue. Special conditions in the early Universe may not be left out when studying the rate of vacuum decay. As a particular example, black holes may trigger the decay, as first suggested in Refs. [37, 38, 39]. In this paper we have analyzed how the presence of primordial black holes may induce the SM electroweak vacuum decay in the early Universe at finite temperature. In particular, our findings indicate that a nonminimal, but renormalizable coupling between the SM Higgs field and gravity may alter considerably the decay rate.
Ultimately, the final fate of the electroweak vacuum is a modeldependent issue, which suffers from our ignorance of the precise early Universe dynamics and the exact black hole mass function as a function of time. Nevertheless, our results indicate that even moderate values of the coupling \(\xi \) between the Higgs and gravity can render more stable or unstable the electroweak vacuum, depending on the sign of the coupling. While this calls for a refinement of our analysis, e.g. by understanding the role of the blackhole entropy on the problem, or by determining more precisely the probability p for a black hole to exist in one of the many causally independent regions which are currently within our visible Universe, it demonstrates once more the importance of gravity for the issue of vacuum decay.
As we have mentioned before, we have not provided a formal treatment of the entropy of the Higgs configuration induced by the presence of the black hole. Instead, we have focused on the entropy related to the ambient temperature. This is not an unreasonable approximation in the scenario we discussed, because the leading effect is obtained at ambient temperatures \(T/T_\mathrm{H}={\mathcal {O}}(10^{2})\). The black hole is not in equilibrium with the thermal environment, but we expect that it has a small effect on it. On the other hand, the black hole distorts the Higgs configuration through gravity, thus changing its energy, or free energy in a thermal environment.

In principle, one should start from the Euclidean action and impose a periodicity \(\sim 1/T\) in the time direction. On the other hand, the presence of the black hole requires a periodicity \(\sim 1/T_{ \mathrm H}\), with \(T_\mathrm{H}\) the Hawking temperature, in order to avoid a conical singularity on the background. The origin of the difficulty is the absence of thermal equilibrium, which necessitates an outofequilibrium analysis.

The configuration we are interested in cannot be characterized as a black hole with hair. It is unstable by definition, as it must possess a negative mode. Therefore, it is not clear whether the notions of blackhole physics are applicable.

Blackhole thermodynamics is well defined in the Einstein frame, while the natural setup for our problem is provided by the Jordan frame. Through a redefinition of the metric, it is possible to consider the problem in the Einstein frame. However, this would introduce unphysical couplings between the Higgs field and the matter sector.
Footnotes
Notes
Acknowledgements
N.T. would like to thank A. Salvio, A. Strumia and A. Urbano for useful discussions. A.R. is supported by the Swiss National Science Foundation (SNSF), project Investigating the Nature of Dark Matter, Project number 200020159223.
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