Higgs Vacuum Decay from Particle Collisions?

We examine the effect of large extra dimensions on black hole seeded vacuum decay using the Randall-Sundrum model as a prototype for warped extra dimensions. We model the braneworld black hole by a tidal solution, and solve the Higgs equations of motion for the instanton on the brane. Remarkably, the action of the static instanton can be shown to be the difference in the bulk areas of the seed and remnant black holes, and we estimate these areas assuming the black holes are small compared to the bulk AdS radius. Comparing to the Hawking evaporation rate shows that small black hole seeds preferentially catalyse vacuum decay, thus extending our previous results to higher dimensional braneworld scenarios. The parameter ranges do not allow for Standard Model Higgs decay from collider black holes, but they can be relevant for cosmic ray collisions.

developed a description of vacuum decay catalysed by black holes, with the result that the strong local spacetime curvature of small black holes catalyses vacuum decay and dramatically changes the prediction for the lifetime of the universe 1 . Tunnelling is initiated by a black hole seed in the the false vacuum that decays into a remnant black hole surrounded by Higgs fields which have overcome the potential barrier and lie in a lower energy state.
The tunnelling rate is determined by the difference in action between the remnant black hole-instanton combination and the seed black hole false vacuum configuration that turns out to be proportional to the difference in horizon area of the seed and remnant black holes.
Because of this dependence on black hole area, enhancement occurs only for very small black holes, the obvious candidates being primordial black holes in our universe, indeed, there is an interesting thermal interpretation of our result, see for example [23][24][25].
There is however another possible scenario in which small black holes could occur, and that is in particle collisions. If we have a situation where our four dimensional Planck scale is derived from a higher dimensional Planck mass close to the standard model scale [26][27][28][29], then it is easier to form black holes in particle collisions [30][31][32][33]. Such higher dimensional theories are dubbed Large Extra Dimension scenarios, and the premise is that we live on a four dimensional "brane" in a higher dimensional spacetime. Our relatively high Planck scale, M p = 1/ √ 8πG N , is the result of a geometric hierarchy coming from an integration over the extra dimensions. Since the true Planck scale is the higher dimensional one, it is easier to form black holes in high energy processes, leading to the possibility of black holes being produced at the LHC (for a review see [34]). Given this exciting possibility for producing small black holes, we should revisit our four dimensional black hole instanton calculations and explore the impact of large extra dimensions.
As a first step in looking at vacuum decay with extra dimensions, we considered the impact of dimensionality on our toy model thin wall calculations in [19], finding that extra dimensions seemed to impede vacuum decay, however, these estimates were predicated on a rather crude higher dimensional generalization that did not take the braneworld aspect of the Large Extra Dimension models into account. In this paper, we revisit the role of large extra dimensions in vacuum decay, explicitly modelling the brane black hole and finding exact solutions for the instanton on the brane. We also make a more careful estimation of the black hole Hawking radiation rate on the brane. We find that, while for a given seed mass the higher dimensional tunnelling rate is indeed lower than the four dimensional one, what we gain from higher dimensions is that lower seed masses are allowed due to the lower value of the fundamental Planck scale, M D .
The layout of the paper is as follows: in the next section, we review the status of constructing instantons both in four dimensions with black holes, and for braneworlds in five dimensions without black holes, and discuss the problems involved in introducing a black hole to the higher dimensional calculation. In section III we discuss the calculation of the action of an approximate black hole instanton, showing that, as in four dimensions, the static instanton action is the difference in black hole horizon areas. In section IV we solve for the brane scalar field and find the instantons and their actions numerically. In section V we conclude.

II. BRANEWORLDS AND BLACK HOLES
It is perhaps worth recalling the various challenges in finding an instanton for vacuum decay in a braneworld setting. The braneworld paradigm describes our universe as an effective submanifold of a higher dimensional manifold, with standard model fields living only on the four-dimensional braneworld, but with gravity propagating throughout all of the dimensions, leading to the renormalization of Newton's constant. For one extra dimension we can consistently solve for the spacetime geometry using the Israel approach [35], giving the standard Randall-Sundrum (RS) braneworld [29], a paradigm for warped compactifications.
For higher codimension, there is no unique "delta-function" limit for a thin braneworld [36], and typically one resorts to approximate hybrid Kaluza-Klein/warped descriptions for gravity on a lower-dimensional brane. Thus, for a concrete gravitational description in this paper we will remain within the RS model.
The RS model supposes that we have one extra dimension, and that the higher dimensional spacetime, or bulk, has a negative cosmological constant. The braneworld has a positive tension, and the vacuum brane has an energy-momentum tensor that is parallel to the brane with energy and tension equal. The original solution presented by Randall and Sundrum had the tension tuned to give a flat brane: where the cusp in the warp factor at z = 0 corresponds to the brane. The local negative curvature of the bulk supports the brane tension that is easily calculated from the Israel junction conditions: and is tuned to fit with the cosmological constant Λ 5 = −6/ 2 . De-tuned branes, with tension greater or less than this critical value may also be embedded within the bulk AdS spacetime, although the natural embeddings now become either space-or time-like [37][38][39][40][41][42], but as long as the brane energy-momentum is approximately homogeneous (i.e. having a spatially isotropic pressure term only) the bulk solution can be fully integrated, and the brane trajectory found [41].
For a brane black hole solution, we must break this spatial homogeneity, but even with the added benefit of having only one codimension, the exact solution for a brane black hole has been extremely elusive [43,44]. The natural geometry of a Schwarzschild black hole that extends off the brane into a black string, found by Chamblin, Hawking and Reall [45], has the problem that it is neither representative of matter localised on the brane, nor is it stable, suffering from a Gregory-Laflamme type of instability [46,47]. A lower dimensional analogue of the brane black hole was found by Emparan et al. [48,49] by taking a 2 + 1 dimensional brane through the equatorial plane of a 4-dimensional AdS C-metric [50,51]. The black hole would be expected to be accelerating from the perspective of the bulk, since an observer hovering at fixed distance from the brane is in fact undergoing uniform acceleration towards it. Unfortunately, there is no known exact solution for a C-metric in more than 4 dimensions, thus no template for constructing a braneworld black hole plus bulk analytically.
To maintain an analytic approach one can explore the effective brane gravitational equations using the approach of Shiromizu et al. [52], leading to the tidal solution that we will use in this paper [53]. (One can also explore braneworlds with additional matter, either on the brane or in the bulk, to support analyticity of the brane embedding, see e.g. [54][55][56][57]) Alternately, one can take a numerical approach; the equations of motion to be solved are an elliptic system [58], with the brane junction conditions and asymptotic Poincare horizon providing the boundary conditions. The solutions for small black holes were found in [59], although the large black hole solutions have been far more tricky to determine due to the nonlinearity of the Einstein equations and the impact of the bulk warping of the horizon, however there has been some interesting recent work in this direction [60,61]. Coleman and collaborators in a series of papers [13][14][15], in which a Euclidean approach was used to find an instanton solution interpolating between the true and false vacua. A convenient approximation, extremely useful for visualisation, is to take the region over which the vacuum interpolates to be very narrow in comparison with the interior of the bubble. This "thin wall" then has a straightforward generalisation to gravity, as described in the paper with de Luccia [15] (CDL).
While this thin wall description is not appropriate for the Higgs vacuum decay [20], where the vacuum interpolation is very wide and relatively gentle, it nonetheless provides an excellent shorthand for visualising the process of decay.
The CDL picture however, is very symmetric, and assumes that both the initial and final states are completely devoid of features and are homogeneous. If instead one relaxes this assumption, minimally, by allowing for an inhomogeneity in the form of a black hole, the analytic approach of CDL can be preserved, and the equations of motion for the instanton are only minimally altered [17][18][19][20], however, the impact on the action of the instanton can be quite significant, and particularly for the thick scalar domain walls appropriate to the Higgs potential [20], tunnelling turns out to be significantly enhanced to the extent that if there are primordial black holes, false vacuum decay will happen.
Let us now consider how these arguments might lift to higher dimensions. In [62], the equivalent of the CDL instantons on a Randall-Sundrum braneworld were constructed, the 5D instanton being geometrically akin to the 4D representations of the CDL instantons. Suband super-critical branes follow spherical trajectories in the AdS bulk, so the tunneling of a Minkowski false vacuum to an AdS true vacuum is represented by a flat brane with a bubble sticking out, as shown in figure 1. As is usual with the RS model, two copies of the picture are identified, and the "bubble wall" is the sharp edge between the spherical and flat parts of the braneworld, appearing roughly as a codimension two object. Thus as a direct approach to finding the instanton seems problematic, we follow a more pragmatic approach, and rather than seeking an exact analytic solution, instead consider what a black hole instanton might approximately look like. From the intuition gleaned in the 4D black hole instantons, we expect that small black holes are the most dangerous, and that the dominant instanton will be the static instanton [20]. Then, analogous to the modelling of collider black hole phenomenology [64], we use the higher dimensional Schwarzschild-AdS solution as an approximation to the local bulk black hole: this allows us to construct a method of calculating the instanton action formally.
Finally, in order to correctly identify the asymptotics of our instanton, we need a way of interpolating between the near horizon and far-field brane solution, which we expect to have a 4D Schwarzschild G N M/r behaviour. This final step requires a choice for the braneworld solution, and we use the tidal brane solution of Dadhich et al. [53], found by considering vacuum solutions with a non-vanishing bulk Weyl tensor in the formalism of Shiromizu et al. [52]. The tidal solution has the attractive feature that it has the correct asymptotic form at large brane radius, but looks like the five dimensional Schwarzschild potential for small radius, indeed, it is similar to the Reissner-Nordstrom black hole, although the "tidal charge" term −r 2 Q /r 2 is negative. This tidal charge was not related to the mass in [53], but left as an arbitrary degree of freedom, therefore part of our task in section IV will be to relate the tidal charge to the mass of the black hole.
Our strategy is then as follows: we first take our brane black hole, approximately modelled by the 5D Sch-AdS solution, and continue to Euclidean time. We then compute the action of this solution in a rather general way, using the approach of Hawking and Horowitz [65]; as per usual, the direct way of computing the action leads to an apparent divergence that we cannot in this case regulate directly by introducing a cut-off as we will explain. Nonetheless, however we choose to regulate the action, the same method will apply for the false vacuum black hole and the instanton bubble solution, thus we simply subtract the seed and bubble actions to get the final amplitude for vacuum decay. Crucially, this turns out to be simply the difference in areas of the seed and remnant black hole horizon geometries. Finally, we integrate the scalar equations of motion on the brane to obtain the brane bubble solution, and use the tidal metric to relate the near horizon and asymptotic geometries. The nett result is an amplitude for brane black hole seeded vacuum decay that we can compare to the higher dimensional brane black hole evaporation rate to explore whether brane vacuum metastability is an issue.

III. THE EUCLIDEAN BRANE BLACK HOLE ACTION
In this section we will show that, just like in four dimensions, the Euclidean action of any static black hole solution can be expressed entirely by surface terms. This is a remarkable result, because it not only applies to the vacuum black hole, it also applies with a cosmological constant, with matter and even with a conical singularity at the horizon.
We begin by recalling the properties of the Euclidean Schwarzschild black hole in four dimensions where In order to explore the geometry near the 'horizon' r h = 2G N M , we expand using a new coordinate , defined by where κ is the surface gravity, κ = f (r h )/2. To leading order f (r) = κ 2 2 + O( 4 ), and close to the horizon, For small ≥ 0, the metric is geometrically the product of a disc with a sphere, provided that κτ is taken to be an angular coordinate with the usual range 2π. If κτ has a different range, then the manifold has a conical singularity at r h . Note that the Euclidean section is perfectly regular other than this, but only covers the exterior region of the original black hole. The event horizon of the original Lorentzian black hole is encoded in the topology of the Euclidean solution: the surface = 0 is a 2-sphere of radius r h .
For the brane black hole in five dimensions, the metric is extended into an additional direction, parametrised by χ in Kudoh et al. [59], who numerically constructed small brane black holes with horizon size less than the AdS radius . In [59], the metric was written in the form where the brane sits at χ = π/2, and χ ≤ π/2 is kept as the bulk. Clearly, in the small black hole limit, → ∞, we have the five dimensional Schwarzschild black hole: written here in homogeneous co-ordinates, rather than the area gauge. The local Euclidean horizon coordinate is = 2(ρ − ρ h ), and the horizon has area A = 4ρ 2 h , and surface gravity The black hole is corrected at order ρ/ by the conformal factor, and at order ρ h / in the other metric functions close to the horizon. Kudoh and collaborators integrated the functions T, B and C numerically, and found that the T function to a very good approximation extends hyperspherically off the brane. Although B and C are not precisely the same, their difference is roughly of order ρ h / as expected. At large ρ, T, B, C → 1, and the metric is asymptotically AdS in the Poincaré patch.
We do not use the explicit form of the metric, however, the features we require from the solutions of [59] are that the event horizon is topologically hyperspherical with constant surface gravity, and that the braneworld black hole asymptotes the Poincaré patch of AdS.
The coordinate transformation between the local black hole coordinates and the Poincaré RS coordinates is and we expect that the 'trajectory' of the brane in the black hole metric will bend slightly in response to the black hole at ρ h , giving rise to a four dimensional Newtonian potential as described in [66]. From the perspective of the {ρ, χ} coordinates, in which the brane sits at χ = π/2, this will show up as a 1/ρ correction to T, B, C. We therefore take our asymptotic metric to be of the form where . We can think of M (z) as coming from the brane bending term of M/ρ in the original coordinates.

A. Computing the Action
The action of the black hole instanton combination diverges and has to be regulated in some way. We do this by truncating the five dimensional manifold at large distances from the black hole, taking a surface at large radius R on the brane, and extending this along geodesics in the ±z directions orthogonal to the brane to produce the outer boundary surface ∂M R as indicated in the cartoon in figure 3. The interior is denoted by M R and the intersection of M R with the brane world is denoted by B. coordinates are suppressed, and the cut-off surface is indicated relative to the brane and bulk black hole horizon. Only one half of the Z 2 symmetric solution is shown. On the right, the Euclidean τ coordinate is shown but the bulk and angular coordinates are suppressed, and the "black hole cigar" geometry is indicated. Two circles denote the boundary ∂H of the region just outside the horizon and the boundary ∂M r at large radius.
The Euclidean action for this truncated instanton or black hole solution is where K denotes the extrinsic curvature of the boundary surface ∂M R defined with an inward pointing normal to the bulk manifold M R . The matter Lagrangian L m includes the contribution from any nontrivial Higgs field profile, as well as the brane stress-energy tensor.
The bulk integral is understood to range across all z, and includes the δ−function curvature at the brane source in the spirit of the Israel approach. Note that the gravitational constant in five dimensions is related to Newton's constant in four dimensions by G 5 = G N .
We now show that the tunnelling exponent, given by the difference between the actions of the instanton geometry with a remnant black hole, and the false vacuum geometry with the seed black hole: B = I inst − I FV , is finite in the limit R → ∞. The first step is to introduce a small ball, H, extending a proper distance of order O(ε) out from the black hole event horizon, to formally deal with any conical deficits arising from a generic periodicity in Euclidean time. This splits the action calculation into two terms, where 2 and B H = B ∩ H is the intersection of the event horizon cap with the brane.
In order to deal with the near-horizon contribution, we transform (7) to local horizon coordinates, analogous to the Euclidean Schwarzschild transformation, (5), so that where < ε inside H. Comparing to (7), we see The brane sits at ξ = π/2, and on the horizon, ξ ∈ [0, π].
As with the four dimensional Euclidean Schwarzschild, there is a natural periodicity of τ for which the Euclidean metric is nonsingular; this periodicity is β 0 = 2π/κ, where κ is the surface gravity of the black hole given in the original coordinates by (9), and in the horizon coordinates by ∂A/∂ . From nonsingularity of the geometry, we deduce N ∼ N 0 (ξ) + O( 2 ), . Now let us consider a general periodicity β for the Euclidean time τ , then we will have a conical singularity at = 0. In order to compute the action, we smooth this out by modifying the A function so that A (ε, ξ) = κ, but A (0, ξ) = κβ 0 /β. Computing the curvature for this smoothed metric gives which gives the bulk contribution to I hor R as 2 Note, the extrinsic curvature in the Gibbons-Hawking term is computed with an inward pointing normal, hence the same sign for that term in each expression.
where A 5 = 4π N 0 D 2 0 dξ is the area of the braneworld black hole horizon extending into the bulk (on both sides of the brane).
To compute the Gibbons-Hawking boundary term we note that the normal to ∂H is n = −d , hence the extrinsic curvature is and Thus the contribution to the action from the horizon region is In appendix A, we show that the external part I ext R can be simplified by taking a canonical decomposition based on a foliation of the manifold by surfaces of constant τ , Σ τ , and the part of the action outside the horizon cylinder reduces to simple surface terms, where 3 K are the extrinsic curvatures of codimension two surfaces of constant r, regarded as submanifolds of surfaces of constant τ , Σ τ as described in appendix A.
Close to the horizon, we use the metric (16) and find at the horizon = 0 for the behaviour of the metric coefficients D( , ξ) and N ( , ξ) given earlier. There is no contribution to the action from this boundary term.
At large distances, the metric approaches the perturbed Poincaré form (11), and we find hence Ideally, we would like to regularise this action either by background subtraction, or adding in boundary counterterms along the lines of [67,68], however, the counterterms of [68] do not regulate this action, and one cannot replace the interior of M R with a pure RS braneworld, due to the variation of M (z) along ∂M R . Instead, we note that the Higgs fields on the brane in any instanton solution will die off exponentially for large r, so from the intuition that M (z)/r ∼ M ∞ /ρ = M ∞ / r 2 + 2 (e |z|/ − 1) 2 , we then deduce that the mass function M (z) will be the same at leading order for both the false vacuum with the seed brane black hole, and the instanton solution, therefore the exterior terms will cancel when we take the difference between the instanton action and the false vacuum action: where A seed ∇ µ E µν = 0. Following [69], one uses the symmetry of the physical set up to write the Weyl tensor as This is manifestly tracefree, and the 'Bianchi' identity implies a conservation equation for U, Π. For the spherically symmetric static brane metric the conservation equation implies Even for the vacuum brane this is not a closed system, but if one assumes an equation of state, one can find an induced brane solution [70]. The tidal black hole corresponds to the choice Π = −2U, for which (29) is easily solved by U ∝ 1/r 4 .
The tidal black hole of Dadhich et al. [53], has δ(r) ≡ 0, and where r Q is a constant parameter related to the tidal charge Q of [53] by r 2 Q = −Q. The motivation for this solution is clear: at large distances, the Newtonian potential of a mass source has the conventional G N M/r behaviour due to a "brane-bending" term identified by Garriga and Tanaka [66]; the interpretation being that the brane shifts relative to the bulk in response to matter on the brane. At small distances on the other hand we would expect the higher dimensional Schwarzschild potential to be more appropriate, hence the −r 2 Q /r 2 term. The event horizon is distorted by the Weyl tensor, hence the name. Other choices for the Weyl tensor lead to different brane solutions [70], however these tend to have either wormholes or singularities (or both), therefore we do not consider these here.
For our bubble solution, we will need to find the fully coupled Higgs plus brane SMSgravitational equations of motion in the spherically symmetric gauge (28), and we will use the same tidal Ansatz for the equation of state of the Weyl tensor: Π = −2U. The beauty of the tidal Ansatz is that even with the Higgs fields taking a nontrivial bubble profile, the conservation equation for the Weyl tensor (29) is still solved by U = −r 2 Q /r 4 .
We also have some limited information about the form of the tidal black hole solution away from the brane from an expansion in the fifth coordinate. According to Maartens and Koyama, [71] the metric parallel to the brane at proper distance z from the brane is where S µν = T µν − 1 3 T g µν is composed of the energy momentum tensor of brane matter. In the false vacuum state, we have T µν = 0 and the metric expansion away from the brane reduces to which shows clearly how the horizon area decreases in the z direction. The horizon forms into a true bulk black hole when the area vanishes for some value of z of order r 2 h /r Q . Although this tidal black hole has many attractive features, the main difficulty that has to be overcome when finding the bubble solutions is that the tidal constant r Q is undetermined.
Clearly a nonsingular brane black hole, if approximately tidal, should have a relation between the asymptotic mass measured on the brane, M , and the tidal charge r 2 Q . For very large black holes, we expect the horizon radius to be predominantly determined by M , and this ambiguity is not relevant, however for the small black holes we are interested in, the horizon radius is primarily dependent on r Q , and we must confront this ambiguity.
We start by noting that the tidal black hole solution should be identical to the five dimensional Schwarzschild black hole in the limit that the AdS radius → ∞, as the brane stress-energy tensor, which is tuned to the cosmological constant, vanishes in this limit, and full SO(4) rotational symmetry is restored. Since G N = G 5 / , (30) implies that r 2 Q → r 2 h in this limit. Intuitively, we also expect that for small black holes, the bulk AdS scale should also be subdominant, and the black hole should look (near the horizon at least) mainly like a five dimensional black hole, i.e. r 2 Q → r 2 h as r h → 0. We will therefore assume analyticity in r h / and write for small r h / , where b is some constant independent of r h and , expected to be roughly of order unity. For the tidal black hole, a trivial rewriting of (30) gives the relation in other words, we have expressed the ambiguity in the tidal parameter for small black holes by the parameter b, and the relationship between the asymptotic mass of the black hole as measured on the brane and the horizon radius explicitly factors in this ambiguity. As we now see, this uncertainty can be absorbed into a redefinition of the low energy Planck scale in the tunnelling rate.
The tunnelling process starts with the uniform false vacuum φ v and a seed black hole with mass M S . This false vacuum configuration resembles the tidal black hole on the brane, and a slightly perturbed 5D Schwarzschild solution in the bulk [59]. The bubble solution represents the decay process to another state with the field asymptoting the same false vacuum at large distances but with the field approaching its true vacuum near the horizon of a remnant black hole with mass M R , which remains after tunnelling.
In the previous section we showed that the tunnelling exponent is given by where S represents the seed black hole area and R that of the remnant black hole (recall, this area is the full five dimensional area of the horizon extending into the bulk). To leading order in r h / , the small black hole horizon has an approximately hyperspherical shape, therefore the area will be well approximated by 2π 2 r 3 , hence using (35). In the limit that the difference in seed and remnant black hole masses is small, where M 5 = (8πG N ) −1/2 is the low energy Planck scale. Fortuitously, the uncertainty in the value of the tidal charge parameter b can be absorbed into our uncertainty in the low energy Planck scale, and so we let bM 5 → M 5 .

A. Higgs bubbles on the brane
The Higgs bubble will correspond to a solution of the brane SMS equations with an energy momentum tensor derived from the (Euclidean) scalar field Lagrangian 3 where V (φ) has a metastable false vacuum. The SMS equations for the bubble, assuming the general form (28) are derived in appendix B, and are where, for comparison with the vacuum case (30), we have defined a "mass" function µ(r) by These are integrated numerically from the black hole horizon r h to r → ∞ where φ is in the false vacuum.
The numerical results contained in this section are based on a Higgs-like potential, assuming that the standard model holds for energy scales up to the low energy Planck mass M 5 . The detailed form of the potential is determined by renormalisation group methods and depends on low-energy particle masses, with strong dependence on the Higgs and top quark masses. Of these, the top quark mass is less well known, and for masses in the range 171 − 174 GeV, Higgs instability sets in at scales from 10 10 − 10 18 GeV.
The Higgs potential is usually expressed in the form with a running coupling constant λ eff (φ) that becomes negative at some crossover scale Λ φ . Vacuum decay depends on the shape of the potential barrier in the Higgs potential around this instability scale, and in order to explore the likelihood of decay it is useful to  use an analytic fit to λ eff . In [20], we used a two parameter fit to λ eff , where one of the parameters was closely related to the crossover scale. We found that the dependence of the instanton action on the potential was strongly dependent on this parameter, but very weakly dependent on the second parameter, which was more related to the shape of the potential at low energy. For clarity therefore, here we take a one parameter analytic fit to λ eff , where the single parameter is the crossover scale Λ φ : and g(Λ φ ), chosen to fit the high energy asymptote of λ eff , varies very little across the range of Λ φ of relevance to the Standard Model λ eff . Figure 4 shows a sample of our analytic fit for the Higgs potential to the actual λ eff computed for M t = 172GeV. In four dimensions,  The change in the mass term is given by ∆µ(r) = µ(r) − µ(r h ). Near the horizon, ∆µ(r) is negative due to the negative potential V in equation 41. µ(r) becomes positive at large r where there is a positive contribution from the kinetic term and hence ∆M is positive.

B. Branching ratios
The calculation of the vacuum decay rate assumes a stationary background which only makes sense when the decay rate exceeds the Hawking evaporation rate. The brane black hole can radiate in the brane or into the extra dimension, but if we consider a scenario as close as possible to the standard model then most of the radiation will be in the form of quarks and leptons radiated into the brane, simply because these are the most numerous particles. (For a review of Hawking evaporation rates in higher dimensions see [72].) Black hole radiation is similar to the radiation from a black body with the same area as the black hole horizon and at the Hawking temperature, but with additional 'grey body' factors representing the effects of back-scattering of the radiation from the space-time curvature around the black hole. Following [72], we can express the energy loss rate due to evaporation asĖ, where on dimensional grounds (since r h is the only relevant dimensionful parameter) for some constant γ. The Hawking decay rate of the black hole Γ H , using (35) to eliminate the radius, is The vacuum decay rate is given by The pre-factor A contains a factor (B/2π) 1/2 from a zero mode and a vacuum polarisation term from the other modes, whose characteristic length scale is the bubble radius r b . We The branching ratio of the two is Vacuum decay is important when this ratio is larger than one.
In the case of small r h / , the five-dimensional black hole has a temperature which is double the temperature of a black hole solely in four dimensions. We would therefore expect to have energy flux on the brane roughly ∝ T 4 ∼ 16 times the flux solely in four dimensions. Numerical results actually give a factor of 14.2 for fermion fields, which give the largest contribution to the decay [73]. The energy loss due to a fermion in four dimensions contributes a factor of 7.88 × 10 −4 for each degree of freedom to γ, giving a total for 90 standard model fermion degrees of freedom of The branching ratio is plotted in figure 6 for M 5 = 10 15 GeV and Higgs instability scale around 10 12 GeV (corresponding to a top quark mass of 172 GeV). Note that the decay rates in this parameter range are larger than M 3 5 /M 2 S , i.e. they are extremely fast. The figure shows an example where black holes with masses between 10 17 GeV and 10 20 GeV, or 10 −7 g to 10 −4 g, would seed rapid Higgs vacuum decay.

V. CONCLUSIONS
In this paper we have explored the impact of large extra dimensions on black hole seeded vacuum decay. We used the Randall-Sundrum set-up as a concrete example for warped extra dimensions, and numerically computed the Higgs profile on the brane for vacuum decay assuming a tidal Ansatz for the Weyl tensor on the brane. Although the solution for a brane black hole is not known analytically, we were nonetheless able to construct an argument that the action for tunnelling would still be the difference in areas of the black hole horizons. In order to estimate these areas, we focussed on small brane black holes (expected to be the most relevant for vacuum decay), and used qualitative features of the numerical solutions to argue the black hole area would be very well approximated by the hyperspherical result 2π 2 r 3 h . We then used the tidal model for a brane black hole (in keeping with the tidal Ansatz for the Weyl tensor), expanded for small masses, to relate the 4D brane mass of the black hole, the 1/r fall-off of the Newtonian potential, to the horizon radius. This then allowed us to compute the amplitude for tunnelling.
Since a black hole can also radiate, we then have to consider whether the evaporation rate is so fast that the tunnelling amplitude is irrelevant, or whether the tunnelling probability becomes so high for small black holes (as was the case for purely four dimensional black holes [20]) that the black hole always initiates decay. We therefore estimated the nett evaporation rate by taking the integrated flux from [73], which is dominated by the fermion radiation, and summing up the effect from the standard model particles. The branching ratio plot of figure 6 demonstrates that, just as in 4D, small black holes in higher dimensions are overwhelmingly likely to initiate vacuum decay once they have radiated away sufficient mass to enter this danger range. As with pure 4D, any small black hole, formed either in the early universe, or in a high energy cosmic ray collision, will radiate, lose mass, then become sufficiently light that it seeds decay with a rate of order 10 3−5 T 5 . 4 What is interesting here is that what we mean by small is now very different to the pure 4D case.
With large extra dimensional scenarios, we generate a high 4D Planck scale geometrically, having a renormalization of the Newton constant coming from the 'volume' of the internal dimensions. Thus, in 4D, where the typical black hole seeding vacuum decay for the Higgs was in the range 10 5 − 10 9 M p 1g−10 tonnes, these black holes could only be primordial in origin, having far too high a mass to be produced in a particle collision. Here however, our Planck mass can be much lower, so 10 5 M 5 can potentially be sufficiently low that the black hole could be produced in cosmic ray collision. For example, the highest energy cosmic ray collisions [76][77][78] observed have an energy in excess of 10 11 GeV. Hut and Rees [79] have shown that there are at least 10 5 collisions with centre of mass energy exceeding 10 11 GeV in our past light cone. Thus, provided the higher dimensional Planck scale were below M 5 10 9 GeV, black holes could be formed in a cosmic ray collision that would be sufficiently light to catalyse vacuum decay.
In in flat space [75] when r h , The area depends on two rotation parameters a 1 and a 2 , but for a rotation axis aligned to the brane we can take a 2 = 0. In this case where r 0 is the horizon radius of the non-rotating black hole solution, The area is smaller than the non-rotating case. Furthermore, the Hawking temperature is reduced, since The numerical results for vacuum decay are shown in figure 7. The vacuum decay rate Ae −B with rotating seeds is larger than than with non-rotating seeds due to the reduced area.
While this is a rather rough argument, the basic intuition that the branching ratio will be enhanced both by the larger decay rate and the reduced Hawking evaporation rate is likely to be correct. In other words, if the existence of large extra dimensions does not destroy the vacuum metastability of the standard model Higgs, then ultra high energy particle collisions risk producing black hole seeds that will catalyse the decay of the vacuum.

ACKNOWLEDGMENTS
We are grateful for the hospitality of the In this appendix we review and extend the ideas given in [65] that provide a canonical decomposition of a manifold (in our case a Euclidean one) by a foliation of hypersurfaces Σ τ to recast the gravitational action in its Hamiltonian version.
The gravitational equations on a manifold M with boundary ∂M are obtained by the extremisation of the usual Einstein-Hilbert action plus a Gibbons-Hawking surface term: here L m is the matter Lagrangian, h ab = g ab − n a n b is the induced metric and K = g ab K ab = g ab h a c h b d ∇ c n d is the trace of the extrinsic curvature of the boundary ∂M with normal vector n a pointing in to M.
To simplify this action we make a foliation of the spacetime M by codimension one timeslices Σ τ , labelled by a periodic Euclidean time function τ which runs from τ = 0 to τ = β. The induced metric on the time-slices is written as where u a is a unit normal vector to the slice Σ τ . In general, ∂/∂τ and u a will not be aligned, but we can decompose ∂/∂τ into components along the normal and tangential directions, The lapse function, N , measures the rate of flow of proper time with respect to the coordinate time τ as one moves through the family of hypersurfaces. We construct the time-slices Σ τ to meet the boundary ∂M orthogonally for convenience. In the case of the region outside the horizon for I ext R (15), the boundary ∂M is composed of two surfaces of constant radius, Σ H near the horizon, and Σ R at large radius.
We use the Gauss identity to relate the Riemann tensor of g ab in five dimensions to the Riemann tensor of h ab in four, and the extrinsic curvatures of the constant time slices Notice this K is distinct from the extrinsic curvature of Σ R in (A1). Contracting (A4) gives and we obtain a relation between the second term of this expression and the extrinsic curvature by commuting covariant derivatives of the normal vector Combining these two expressions leads to the identity, which forms the basis of all canonical decompositions of the Einstein-Hilbert action.
When substituted in (A1), the last two terms of (A7) are reduced to boundary contributions on ∂M. The first of these vanishes due to orthogonality of ∂M R and Σ τ . The second combines with ∂M K from the original action, and gives on ∂M R (with a similar expression for ∂H) but this four dimensional integral can be viewed as an integral over τ of a three dimensional integrand that is precisely the three dimensional extrinsic curvature 3 K of a family of surfaces C R (τ ) = ∂M R ∩ Σ τ living in the boundary ∂M R . A similar term is obtained for the ∂H surface near the horizon however, for the black hole metrics, it turns out that 3 K → 0 as r → r h , and so this term does not contribute to the action.
Noticing that √ g = N √ h, and introducing a metric 3 h on C R , we can divide the spacetime integral into space and time, to express the action (A1) as Furthermore, we can see how the extrinsic curvature is related to the Lie derivative of the intrinsic metric with respect to τ via (A3): whereḣ ab = h c a h d b £ τ h cd and D a is the derivative associated with h ab . To obtain the Hamiltonian form of I we define the canonical momentum π ab conjugate to the intrinsic metric as This allows us to recast (A9) in terms of the canonical momentum Now we are ready to perform a Legendre transformation of the Lagrangian and using (A10) and (A11) to obtain the Hamiltonian formulation.
with the Hamiltonian constraint function H and the momentum constraint function H a given by Finally, for a static spacetime we haveḣ ab = 0 and in the non-rotating case N a = 0.
The metric is a solution to the field equations, so that in particular we have the constraint equations H = H a = 0. The only non-vanishing part of the action are the two boundary terms 3 K, For our black hole solutions, this diverges in the limit R → 0. However, the matter contributions to the black hole instanton solutions die off exponentially at large radii, so that the boundary terms cancel when we calculate the difference in actions between the instanton solutions and the false vacuum solutions with the same mass and periodicity β.
The Einstein equations for a five dimensional RS braneworld can be written as (5) G ab = −Λ 5 g ab + 8πG 5 δ(z)(−σh ab + T ab ), where z is a coordinate defined by taking the proper distance from the brane into the bulk, G 5 = G N and the cosmological constant of the bulk Λ 5 = −6/ 2 is given in terms of the AdS 5 radius . Notice that we use latin indices for the bulk spacetime whereas greek indices will be reserved for objects living on the brane. The brane is located at z = 0 and has an induced metric h ab , defined by where n a is a unit vector in the z−direction. The energy momentum tensor of the brane carries the effect of the tension σ and has a contribution T ab , coming from the fields living in the brane.
The Israel junction conditions for the brane allow us to write down a set of four dimensional Einstein equations (see [52]), where Λ eff is an effective four dimensional cosmological constant on the brane, and E µν is the projection of the five dimensional Weyl tensor onto the brane carrying information about the extra dimensional geometry to the brane. Due to the properties of the Riemann tensor, E µν is traceless and divergence free. In the critical RS brane that will be our false vacuum, the tension of the brane is tuned so as to set Λ eff to zero, i.e. σ = 3 4πG 5 .
Finally, the effective energy momentum tensor,T µν = T µν + π µν consists of the standard energy momentum tensor, together with second order terms