Seeded vacuum decay with Gauss-Bonnet

We investigate false vacuum decay catalysed by black holes under the influence of the higher order Gauss-Bonnet term. We study both bubble nucleation and Hawking-Moss types of phase transition in arbitrary dimension. The equations of motion of “bounce” solutions in which bubbles nucleate around arbitrary dimensional black holes are found in the thin wall approximation, and the instanton action is computed. The headline result that the tunnelling action for static instantons is the difference in entropy of the seed and remnant black holes is shown to hold for arbitrary dimension. We also study the Hawking-Moss transition and find a picture similar to the Einstein case, with one curious five-dimensional exception (due to a mass gap). In four dimensions, we find as expected that the Gauss-Bonnet term only impacts topology changing transitions, i.e. when vacuum decay removes the seed black hole altogether, or in a (Hawking-Moss) transition where a black hole is created. In the former case, topology changing transitions are suppressed (for positive GB coupling α), whereas the latter case results in an enhanced transition.


Introduction
It is well known that while classically a particle in a local minimum of potential is stable to small perturbations, the same is not necessarily true quantum mechanically -it depends on whether the particle sits in a global or only a local minimum.If the latter, a process of quantum tunnelling occurs, where the particle "borrows" energy by quantum uncertainty to emerge on the other side of the barrier.While well-tested in non-relativistic systems, tunnelling from such a local minimum or false vacuum in Quantum Field Theory lacks experimental verification.
By far the most intuitive picture of vacuum decay is that of a first order phase transition: a bubble of true vacuum fluctuates into existence inside a false vacuum state, then expands to convert to true vacuum everywhere.This picture, best described in a sequence of papers by Coleman and collaborators [1,2] (see also [3]) originally focused on the field theory aspect; considering the problem more holistically however mandated including gravity due to the impact of the vacuum energy.As described in [4], this requires using the semi-classical partition function approach developed by Gibbons and Hawking [5] where the path integral is assumed to be dominated by its saddle points, or, solutions to the Euclidean Einstein/field theory equations; these are found, and the action of the bubble is computed with the amplitude for decay being, to leading order, P ∼ e −B = e −(I−I 0 ) , (1.1) where I denotes the Euclidean action of the bubble instanton solution, and I 0 the action of the false vacuum background.
Coleman and de Luccia (CDL) originally computed the gravitational action of a bubble of the maximally symmetric spacetime associated with the vacuum energyde Sitter for positive vacuum energy, Minkowski for zero vacuum energy, and anti-de Sitter for negative vacuum energy.This, however, is a very idealised picture of the vacuum, with no structure or inhomogeneity.In [6,7], an extension of the CDL approach was considered, where a bubble was assumed simply to be spherical, i.e. with SO(3) symmetry.A generalisation of the Birkhoff theorem in the presence of branes (the bubble wall) [8] shows that the general solution of a bubble with SO(3) symmetry is Schwarzschild (A/dS) -i.e. the general bubble will also surround a black hole, and in principle can have different 'masses' inside and outside the bubble 1 .The finding of [6,7] (see also [9][10][11][12][13][14][15][16] for applications to Higgs decay) was that the black hole could significantly reduce the instanton action, resulting in a strongly enhanced probability of false vacuum decay in the presence of primordial black holes [12,14,[17][18][19][20].The bounce formalism not only gives the transition rate but also the initial conditions for the real-time evolution of the bubble, which is obtained via analytic continuation of the Euclidean solution.A discussion of the correspondence between the Euclidean and Lorentzian sections can be found e.g. in [4,10,21].
The schematic plot of a one-dimensional potential that has two different minima, with the higher (lower) representing false (true) vacuum, labelled as ϕ F V and ϕ T V , respectively.The top of the potential barrier is labelled as ϕ T op .A particle initially at the local minimum can arrive at the global minimum by two mechanisms: (i) Bubble nucleation (red) -quantum tunnelling to true vacuum or (ii) HM transition (blue) -climbing to the top of the potential and then rolling down to the true vacuum.
Bubble nucleation is not the only decay channel for a false vacuum.The Hawking-Moss (HM) instanton describes an up-tunnelling of the false vacuum to a local maximum of the potential, allowing a roll-down to the true minimum [22].The two tunnelling processes can be best illustrated via the cartoon of a particle in a onedimensional potential barrier, as shown in Fig. 1.The first order, or bubble nucleation, process corresponds to a particle transitioning from the local minimum on one side of the barrier to the other side.In the HM transition on the other hand, the particle "jumps" to the top of the potential, then rolls down to the true minimum.This HM channel was shown to be relevant for potentials with a gently varying barrier (the bound being on the second derivative of V (ϕ) at the local maximum [22]).The impact of black holes on the HM transition was considered in [23,24], see also [25], with an outcome broadly similar to the bubble nucleation process, namely, that black holes enhance the probability of tunnelling.There is one crucial difference however that is relevant for the study here: For bubble nucleation, there can only be a remnant black hole if there is a black hole seed to nucleate the decay.For HM transitions however, it is possible to jump from a pure de Sitter false vacuum to a Schwarzschild de Sitter configuration at the top of the potential barrier.
It is worth noting that the classic results of Coleman and de Luccia are obtained within the thin wall approximation, in which the vacuum transitions completely and instantaneously across the wall.The thin wall approximation, while excellent for testing gravitational effects, is only a good approximation for vacuum transitions in scalar fields when the potential barrier is sharp.For vacuum energy sourced by a scalar field, there is a region around the "wall" where the scalar rolls to its vacuum value and for the important physical application of Higgs vacuum decay, the "wall" is a rather broad scalar distribution around the black hole.The early results of [7,9] stayed strictly within the semi-classical approximation, and did not include possible thermal corrections to the Higgs potential due to the black hole temperature.Later work [26][27][28][29][30][31] has explored these corrections, however there is still a lack of consensus as to the precise magnitude of any suppression.The problems lie around the choice of vacuum for the Higgs, as well as the validity of calculational methods.In [26], Hayashi et al. perform a Euclidean computation of the tunnelling action including the local thermal corrections to the Higgs potential and their backreaction, concluding that while there is some lowering of the instanton decay probability, it is not suppressed completely.While this is an explicit calculation, one can critique the Euclidean approach as in [27], where Strumia uses a sphaleron approximation for the instanton to derive expressions capturing the parametric dependancy in certain limits, resulting in an apparent suppression of the process altogether.While phenomenologically compelling, [27] focusses on the primary parametric dependence in key limits, and does not address important prefactors in these expressions, which can be key in the comparison of evaporation to tunnelling.In [29], Shkerin and Sibiryakov perform a first principles calculation for all of the quantum vacua (Boulware, Hartle-Hawking and Unruh) which indicates that tunnelling is still significant, however the argument is performed in two spacetime dimensions, although the authors argue that the result can be extrapolated to four dimensions.However, this would require care in taking boundary conditions on the tunnelling solutions corresponding to different initial vacua.In essence, the problem centres around how to correctly incorporate gravity in non-perturbative processes that do not even have experimental verification in the absence of gravity.While these concerns and arguments are important and merit continuing investigation, in this work we are interested in the effect of higher curvature gravitational terms on instantons in general, thus will focus on exact solutions which transition between different cosmological constants, and thus will not be considering a direct application to Higgs vacuum decay.Among the theories containing higher-order corrections the most natural are those that maintain second order equations of motion -the Lovelock terms [32] -and we focus here on the Gauss-Bonnet (GB) term [33][34][35] that is known in 4D not to alter the dynamics of the theory.It may seem strange to consider a correction that does not alter the dynamics of a system, however, when computing the probability of decay, it is the action of a process that is relevant, and the GB term will alter the action of a solution, potentially affecting the tunnelling amplitude.(Note, we are not considering the Glavan-Lin approach [36], as this has several problems [37][38][39].)Our main finding is that due to the topological nature of the GB term, the results for black hole seeded decay are not, in the main, compromised by the addition of this higher derivative term in 4D.However, we find that the presence of the GB term almost always suppresses topology-changing transitions.The main exception to this result is HM decay from pure de Sitter space, where the GB term can provide an enhancement to the decay for small black holes.
The paper is organized as follows: In Sec. 2 we set up the formalism of instanton and compute the instanton action in order to show the effects of GB term.In Sec. 3, we discuss the HM transitions and display applications in higher dimensions (d ≥ 5), and explain how GB terms contribute when topology changes.In Sec. 4, we restrict to D = 4 where bubble solutions are identical to Einstein gravity and examine the difference with that in Einstein's.Finally, we conclude our results in Sec. 5. We work throughout in the thin wall approach of Coleman and de Luccia [4], as this captures the gravitational aspects of tunnelling in a straightforward analytic fashion, thus should isolate effectively the impact of the GB term.We use natural units (c = ℏ = 1) throughout the paper.

Bubble-nucleated decay
We first briefly review the procedure for finding the instanton, noting how the GB term impacts the argument in general and deriving the action in the presence of the GB term.Gauss-Bonnet instantons with O(D)−symmetry were explored in [40] in the context of stability of the different branches of the Einstein-Gauss-Bonnet (EGB) vacua, here we are looking for solutions with a black hole, hence O(D − 1) symmetry.
Recall that EGB gravity is the simplest of the Lovelock extensions to General Relativity [32], with an action given by where α is the (dimensionful) coupling constant of the GB term, L GB , defined in (1.2).The general (Euclidean) spherically symmetric vacuum solution is derived in [41], where dΩ 2 D−2 gives the metric of the unit S D−2 and we take the 'Einstein' branch solution [42][43][44]: with α = (D − 3)(D − 4)α, and µ related to the mass through the standard Myers-Perry formula [45]: with A D−2 being the area of a unit (D − 2)−sphere.Note that as α → 0, we recover the Schwarzschild solution.Note also that if α < 0, there are régimes of parameter space where the argument of the root in (2.3) becomes negative, hence the solution is unphysical.While what follows is independent of the sign of α, we will focus on α > 0 for discussion.A bubble separating false vacuum from true vacuum is expected to have SO(3) symmetry, and a generalisation of the Birkhoff theorem in the presence of branes [8] shows that the general bulk solution is Schwarzschild (A/dS) with the bubble surrounding the black hole.This theorem was also demonstrated for GB gravity in [46][47][48], with the general brane junction conditions derived in [49].Although the thin wall approximation is an idealised description of bubble nucleated vacuum decay, it will suffice for the exploration of the impact of the GB term.
Following the steps of [6], we seek a Euclidean bubble solution R(λ) separating two black hole spacetimes with exterior mass M + (the seed) and false vacuum energy Λ + , and an interior mass M − , and true vacuum energy Λ − .The regions exterior and interior to the wall are given by where with the boundary R(λ) being given by the generalisation of the Israel prescription to EGB gravity as we now describe.
The full action of the composite system is I = I bulk + I brane , where represents the contribution from the spacetime away from the wall, and represents the net contribution from the wall, including both the wall tension σ, and the geometrical Gibbons-Hawking terms from the boundary submanifolds on each side: Here, K ± ab is the extrinsic curvature of the wall as determined from the local embedding into M ± , G ab is the local intrinsic Einstein tensor of the wall, and J is a cubic extrinsic tensor (see [49][50][51][52]) : (2.9)

The instanton solution
In order to find the instanton, we have to solve the equations of motion for the bubble wall, found by varying (2.7) and (2.8).As described in [49], the generalised Israel equations are: the divergence free part of the intrinsic Riemann tensor.
Because of the SO(3) symmetry of the bubble, these rather complex relations simplify a little; we write the bubble wall as R(λ), where λ is the proper time of an observer comoving with the wall: and choosing a normal always pointing towards increasing R for τ± > 0, gives, after some algebra, the junction condition as where σ is the surface tension of the wall.Substituting (2.12) leads to a rather lengthy cubic Friedmann-like equation for (2.15) where σ = 2πGσ/(D − 2) economises on notation, and represents the geometry on each side of the wall and is related to the standard Einstein potential (2.17) as required.(Here, fE ≡ (f E− + f E+ )/2 and ∆f ≡ f E+ − f E− .)Since α = 0 for D = 4, we see that the EGB equations of motion for the wall indeed reduce to the Einstein equations, as expected for a physical system.Therefore, in 4D, the motion of the wall is as the Einstein case described in [6,7]: for a given seed mass M + , there are a range of bubble solutions with remnant masses M − , however, we expect a unique lowest action solution with specific mass M − that we identify as the instanton with a remnant black hole of mass M − .

Computing the instanton action
In order to identify the lowest action solution (bearing in mind that even in 4D, the GB term could contribute to the action), we must now compute the bulk and wall contributions to the action, then subtract the background action of the initial state.

The background seed action
We start by computing the action for the background solution of mass M + , and bare vacuum energy ∝ Λ + ; this is simply a matter of computing the bulk term (2.7), bearing in mind two key points.The first is that if we are in vacuum or AdS spacetime (Λ + ≤ 0) then we must truncate our bulk integral at some finite radius that we choose to label r c .For a positive cosmological constant, r c will be the upper limit of integration at the cosmological horizon.The second point is that we must perform the integration at arbitrary Euclidean time periodicity β, hence at an event horizon, there will, in general, be a conical deficit2 , thus we must generalise the argument of [6] to include the GB term.
Computing the integrand of (2.1) for the metric (2.2) we see that, on-shell, the bulk integrand is a total derivative: Giving a contribution of I c − I h , where at the upper r c limit, and at the black hole horizon the contribution is proportional to the entropy.Note we have replaced the derivative of f at the horizon with f ′ (r h ) = 4π/β h , an expression involving the critical horizon periodicity β h for which there is no conical singularity at the horizon.We now complete the computation by taking into account possible conical defects at a horizon.Following the approach of [6] (see also [53]), we note that a conical deficit metric has the local form dρ 2 + λ 2 ρ 2 dθ 2 , where λ ̸ = 1 and the periodicity of θ is 2π.The deficit angle for this metric is δ = 2π(1 − λ).We write the black hole metric locally at the horizon as where ρ 2 = 4(r − r h )/f ′ (r h ), and A ′ (0) = 2π/β h .Comparing this with the canonical form above (θ = 2πτ /β), we see that λ = β/β h , hence the conical deficit is To compute the contribution of the conical deficit to the action, we split our bulk integral into an integral from a proper distance ε from the horizon, and smooth out the cone inside ε, by having A smoothly interpolate between being regular at the origin, A ′ (0) = 2π/β, and having the black hole geometry value A(ε) = 2π/β h at ρ = ε.We then take the limit ε → 0 to obtain the action with the conical singularity.
Computing the terms in the EGB Lagrangian for the metric (2.21): (2.23) and using C = r h + O(ρ 2 ), shows that near the horizon Thus the full contribution to the bulk action including the conical deficit is The large r contribution from r c is treated differently depending on whether Λ > 0 or not.For Λ > 0, we have a cosmological horizon, and must therefore take into account any conical deficit as described above.This procedure follows through in much the same way as for the black hole horizon, and (noting that I c < 0 for the cosmological horizon) for a de Sitter seed we obtain thus, just as for Einstein gravity, the action of a black hole in de Sitter space for EGB gravity is the sum of the entropies of the horizons.
If however we are in asymptotically flat or AdS spacetime (Λ < 0 or Λ = 0), we place a cut-off boundary at r c , and compute the action at that cut-off.Strictly, this means we have to also compute a Gibbons-Hawking boundary term for the artificial boundary r c , as well as perform a background subtraction to obtain a finite answer.However, in computing the action of the instanton, we calculate the action of the bubble geometry and subtract the action of the seed geometry; noting that the bubble and seed geometries are identical outside the bubble wall, both the contribution from the bulk integral I c (eq. (2.19)) as well as the Gibbons-Hawking term and any background subtractions will be the same.Thus we do not need to explicitly compute these large r terms and simply note that both seed and bubble geometries have an identical contribution at the large r cut-off, which we label I C :

The bubble geometry
For the bubble geometry, the main difference from the previous subsection is that we have the bubble wall at R(λ).If R varies with λ, then the periodicity of the solution is set by the periodicity of R. The previous subsection demonstrates how to compute the bulk action, which is now composed of two parts, the interior and exterior of the bubble.Using these results, we see that the bulk contribution to the bubble action, (2.7), reduces to the boundary terms at r h and r c , together with a contribution evaluated at the wall: (2.29) Turning to the wall integral (2.8), the trace of the Israel equations gives the wall integral as It then proves useful to define here, K 1 is the angular part of the extrinsic curvature, and p 0 , p 1 appear in the intrinsic Einstein tensor.The Israel conditions then give the following relation between the timelike and spacelike extrinsic curvature differentials: Using these relations, after some algebra we find the wall integral can be reduced to (2.34) we find that the combined bulk-term and wall integrals give (2.36) Putting together, we see that the action of the bubble geometry is hence the action for the tunnelling instanton is Therefore, we have shown that the amplitude for vacuum decay including the higher derivative GB term in arbitrary dimensions has the same form as the decay rate computed for vacuum decay in Einstein gravity in 4D.While we leave the numerical evaluation of this action for arbitrary M + and M − for future study, we note that the contribution from the wall integral vanishes for static instantons, which are the relevant instantons for Higgs vacuum decay [9,10].
The central result of this section is therefore that, as with Einstein gravity, the probability of thermal decay seeded by a black hole in EGB gravity is simply the difference in entropy of the seed and remnant black holes: (2.39)

The Hawking-Moss Instanton
In the previous section, we derived the bounce action as the difference in entropies between the cosmological and event horizons.However, bubble nucleation is not the only decay mechanism for false vacuum decay.Another process is the HM transition, that describes the field fluctuating from the false vacuum to the top of potential [22].
At the top of the potential, the field is at an unstable point and can roll down to either the false or true vacuum.The HM bounce is responsible for mediating the system's evolution into a new state, as illustrated in Fig. 1.The HM transition depends only on the values of the potential energy densities at the top of the barrier and at the false vacuum, rather than the details of the potential in between.In [23,24], the impact of black holes on the HM transition was studied, with the general finding that black holes enhanced the transition probability.In addition however, it was found that the action could potentially become negative for very large seed black holes, unless a thermodynamically motivation condition was applied -Cosmological Area Conjecture: that the cosmological horizon area can never increase in an up-tunnelling transition (see [24] for a discussion and motivation for this principle).
The general Black-Hole-Hawking-Moss (BHHM) transition will have a 'seed' black hole in the false vacuum and transition to a 'remnant' black hole at the top of the potential.We change the notation in (2.5) and (2.6) of "f ± " to "f F/T " to correspond to the labellings of the false vacuum (F ) and the top of the potential (T ).These will be characterised by mass parameters µ F/T , and vacuum parameters ℓ F/T , where ℓ 2 = (D − 1)(D − 2)/2Λ.The tunnelling rate is dominated by the Boltzmann factor: From (2.39), we see that the change in action is the difference in the entropies, justifying the interpretation of the HM probability as a Boltzmann suppression: Thus, the form of the BHHM action is the same as for Einstein gravity, however the entropies in the expression are now determined by the EGB formula: where the subscript i corresponds to the black hole or cosmological event horizon radius, which in general is also modified by the GB term.The likelihood of the BHHM instanton is obviously dependent on the details of the seed black hole and the dimension of the spacetime, but some general trends can be noted.For example, when a black hole is introduced, the total entropy drops (though note the exception we will discuss for 5D presently), thus for a given seed mass the BHHM transition will always prefer to jump to a pure de Sitter configuration, as this will maximise the negative contribution to (3.2).Similarly, introducing a seed black hole will also lower the action, since the overall entropy in the false vacuum will drop.Thus, just as with Einstein gravity, we expect that introducing seed black holes will also catalyse the HM transition.
There is a limit however to how far we can drop the action, and this is determined by the Cosmological Area Principle [24], that the area of the cosmological horizon must never increase: As we saturate this bound, the HM action becomes the difference in entropy of the seed and remnant black holes.Thus, we get broadly the same picture as for the Einstein case, where the preferred instanton (the one with lowest action) is a jump to a pure vacuum solution at the higher vacuum energy (i.e.no remnant black hole) for lower seed masses.As the seed mass increases, we hit the Cosmological Area Bound and the transition is to a black hole geometry with the "top" vacuum energy (see figures 2, 3).Therefore, broadly speaking the picture for the BHHM decay process looks qualitatively very similar for the EGB case as for the pure Einstein case across arbitrary dimensions with one interesting exception.Solving f (r h ) = 0 for the black hole horizon at r h yields the following expression for the mass: For D = 5, we see that while smaller black hole horizons correspond to lower black hole masses, it is not possible for M → 0 while maintaining a horizon.Indeed, inspecting the expression for f (r) at the origin for D = 5 yields f (0) = 1 − µ/α, which becomes positive as µ, hence M , becomes sufficiently small: GM < 3πα/4, indicating the loss of an horizon.This is consistent with numerical studies of collapse [54,55], which find that black holes will not form dynamically if the total massenergy content of the spacetime is below a critical value in both asymptotically flat and asymptotically AdS spacetime.This mass gap was also noted in [56] in an exploration of black hole formation during gravitational collapse, and differences in thermodynamic phenomenology and gravitational collapse with GB corrections between five and higher dimensions were noted in [57,58].Thus, the picture for the HM transition will be distinctive in 5 dimensions.We, therefore, present results for D = 5 separately and present the HM results for D = 6 as representative of higher dimensions.
• 5D HM Taking f (r h , r c ) = 0 and D = 5 in (2.3) (noting α = 2α in 5D) the black hole and cosmological event horizons can be obtained in a simple form: Recall that M C = 3πα/4, is the critical minimum mass that allows for an event horizon to exist, and the Nariai mass, M N , where the black hole and cosmological horizons merge [59,60] gives a maximum: Any solution with a mass parameter larger than M N will not have the correct signature, hence is considered unphysical, thus all EGB black hole solutions in 5D must satisfy M C ≤ M ≤ M N .Solutions with M < M C have no event horizon, but an examination of the geometry near r = 0 shows that there is a solid angular deficit at the origin, which can be regularised by a similar method to the conical deficit regularisation, which allows one to integrate the Euclidean action, resulting in no additional contribution.The space of massive particle solutions in 5D EGB can therefore be thought of as analogous to AdS solutions in 3D Einstein gravity, where there is a finite range of masses for which the particle is a conical deficit, above which the solution becomes a BTZ black hole [61,62].
We depict the constraints on the black hole mass parameter in Fig. 2 by plotting the ratio of the BHHM instanton action to the pure HM instanton action, B/B HM , as a function of the seed black hole mass, M F /M N .Here, B HM denotes the pure HM instanton without seed or remnant black holes, which only depends on the scale parameters of the initial and final states.The blue solid lines represent different remnant black hole masses, ranging from 0.2M N to M N in increments of 0.2M N from lower to higher, while the cyan line on the bottom represents a transition with no remnant black hole.The black line shows the lowest mass when a black hole horizon can exist.As with the Einstein case, the BHHM action increases with the remnant mass, indicating larger remnant black holes slow down the transition.Two vertical dashed lines denote the lower and upper bounds of the seed mass for a black hole, which are constrained by mass gap and Nariai limits respectively.The red curve shows the limiting constraint of the cosmological area conjecture.In all cases, the Nariai limit is an upper bound on the seed mass.
On the left hand side of Fig. 2, the dashed lines indicate the region where the seed mass is smaller than M C , hence there is no black hole horizon.There is still a mass at the origin however, and the cosmological horizon area is reduced as a result of the seed mass parameter, and linearly approaches the pure de Sitter area as M → 0. Note that as M increases from M C , there is a small uptick in the action, reflected in the detail of the inset.This is due to the dependence of the entropies on ∆M = M −M C .The black hole horizon entropy S BH ∝ √ ∆M , whereas S CH ∼ S DS − O(∆M ), thus the overall entropy of the seed actually (temporarily) increases as a result of adding mass to the system.Meanwhile, at the lower part of the plot along the red equal area line, the same phenomenon occurs, only this time causing a small downtick as the action of the remnant increases as the mass creeps above M C .
• 6D HM where the relative mass gap is M C ≈ 0.06M N .The blue solid lines represent remnant black hole masses ranging from lower to higher as "0.2MN , 0.4M N , 0.6M N , 0.8M N , M N ", respectively; while the black solid line represents a remnant critical mass M C .The vertical dashed lines indicate the lower and upper bounds of the seed black hole mass.The red curve corresponds to the cosmological equal area limit.The cyan line represents a transition to a final de Sitter state without a remnant black hole, while the dots on the left axis show the transitions from pure de Sitter to remnant masses.The dashed lines on the left part of the graph, and the inset, represent a seed mass below the mass gap.
Following the same procedure, Fig. 3 displays the ratio B/B HM as a function of seed black hole mass, M F , for different remnant black hole masses, M T , in 6D.Unlike in 5D there is no mass gap, and only the Nariai limit sets an upper bound on the size of black holes.As before, the cosmological area conjecture gives a cut-off on the allowed parameter space.
In Fig. 3, we see a plot qualitatively the same as for the Einstein BHHM transitions: the action decreases as the seed mass increases, with the jump to a pure de Sitter spacetime at the top of the potential shown in cyan.The cosmological area conjecture provides a cutoff at larger seed masses, shown in red, where the transition now leaves a remnant black hole.The generic BHHM transition with both seed and remnant is shown in blue, where the remnant mass along a blue line is constant.The solid dots at the left depict the transition from pure de Sitter spacetime to a spacetime with a remnant black hole, especially, the cyan dot is the original HM transition.

Four dimensions
Having derived results for vacuum decay with higher derivative gravity in general dimensions, we now restrict to D = 4 to explore whether there is an impact from the modification of the action -we have already confirmed that the bubble solutions are identical to Einstein gravity as expected in Sec. 2, therefore it only remains to compute the effect of the GB term on the action.

Bubble nucleated vacuum transitions
Recall that the general instanton is (Euclidean) time dependent and has both seed and remnant black holes with action given by (2.38) and the equation of motion: where and First, note that it is not possible to nucleate a bubble with a remnant black hole unless there is a seed.To see this, note that (4.1) implies that If we have no black hole seed, then f + = 1−Λ + r 2 /3, and ∆M = −M − .The equation of motion, Ṙ2 = F (R), must have two zeros of F (R) (unless the solution is static, F (R) ≡ 0).However, hence F (R) is a monotonically decreasing function of R and cannot have two zeros.We therefore conclude that vacuum decay by bubble nucleation either is a CDL bubble, or is seeded decay with an initial seed black hole, which will nucleate a bubble that may, or may not, have a remnant black hole.Turning to the computation of the action for the bubble, there are two parts: the difference in black hole entropies, and the contribution from the bubble wall, which is not obviously vanishing.Dealing with this latter term, the α dependent part of the contribution is I wall,α = I + W,α − I − W,α , where Here, K 0,1 were defined in (2.32), and for the wall trajectory (4.1) are we therefore see that the α contribution to the wall action is: But now we note that the middle term vanishes due to the periodicity of R, the final term is the same on each side of the wall which will therefore cancel.Turning to the first term, this can be rewritten using the equation of motion for the wall: thus the first part of the integral in (4.7) becomes: due to the periodicity of the solution R(λ), the arctan function is symmetric around the turning points of the solution, whereas Ṙ changes sign, thus this integral vanishes and there is no α dependent contribution from the wall.
Examining the contribution of the entropies to the action reveals the presence of an α−dependent, but constant, contribution from each black hole horizon in (2.38).The entropy term, S BH is From the perspective of black hole thermodynamics, this constant shift is irrelevant [63], but for our action, it is potentially important.If we have both a seed and remnant black hole, these constant terms will cancel, but if we have a seed black hole that is wiped out by bubble nucleation, then there will be a residual α term in action.
We therefore arrive at the result that the bubble action in 4D is identical to the Einstein action if there is either both a seed and remnant black hole or neither a seed nor remnant black hole.However, if we have a seed black hole and no remnant, then the GB term leaves an imprint on the action: Thus, for bubble nucleated decay, the GB term either has no impact or actually suppresses tunnelling (I B is increased), in the case that the decay removes the black hole altogether.This is perhaps not surprising, once one considers the fact that the GB term is a topological invariant in 4D, thus it should only impact on topology changing transitions.Perhaps more surprising is that the GB term inhibits topology changing decay.

Hawking-Moss Tunnelling
Armed with the results of the previous subsection, we can now easily deduce how the GB term impacts 4D HM transitions.If there is no topology change, there is no change in the action: i.e. for the pure HM instanton and for a transition with a seed black hole to a remnant black hole geometry, which corresponds to the (red) Cosmological Area Principle boundary.However, for the HM transitions that jump from a Schwarzschild de Sitter (SdS) spacetime to a pure de Sitter spacetime, there is an α−dependent contribution that suppresses this topology changing transition.We have the bizarre situation that the universe preferentially jumps to an SdS solution with a vanishingly small mass -this will have the same topology as the initial SdS state, but the tiny black hole will presumably instantaneously evaporate leaving effectively pure de Sitter at the top of the potential.Whether or not such a Planckian sized black hole should be included in a semi-classical description is a very good question!Finally, unlike the bubble nucleation, it is possible for a HM transition to occur from pure de Sitter false vacuum to an SdS spacetime at the top of the potential.For these (topology changing) configurations the action is now lowered by δI ∼ 4α/G − 2πℓM relative to the pure HM transition, thus again there is a preferred transition to a vanishingly small mass black hole universe.

Discussion
In this paper, we studied the impact of higher order terms in the gravitational action, focussing on the Gauss-Bonnet invariant to maintain well-posedness of the equations of motion.We considered both first order, tunnelling, vacuum decay transitions as well as Hawking-Moss jumps.For bubble nucleation, we derived the general equations of motion for a bubble, including the GB term in arbitrary dimension, and found that the general result of [6], that larger black holes catalysed a static instanton with the action determined by the entropy difference between the seed and remnant black hole remains true with the GB term, and in all dimensions.The HM transitions were algebraically easier to explore, with results qualitatively similar to the Einstein process.
Note that our results have been obtained within the thin wall approximation, in which the vacuum transitions completely and instantaneously across the wall.While we do not expect higher derivative terms to significantly change the discussion for scalar field decay -in particular thermal corrections -in 4D, the impact of the changed geometry in higher dimensions would be interesting to investigate.
In 4D, the fact that the GB term is a topological invariant means that it will not impact on instantons that have both seed and remnant black holes, however, when there is a transition from a seed black hole with no remnant, the GB term suppresses the transition (for positive α).For a situation with no seed black hole, only the HM transition can result in a geometry with a remnant black hole, and here the GB term enhances the decay, although the lack of continuity of the action as a function of black hole mass means that an arbitrarily small mass black hole would have the lowest action, likely taking the spacetime outside of the semi-classical regime.The special case of 5D, with its mass gap for α > 0, led to some interesting additional features in the transition amplitudes, and this case may merit further examination.
Interestingly, there are problems with a generic higher order curvature terms in the action.Without the well-posedness of these Lovelock terms, the higher order derivatives mean that singular instantons cannot be regularised in a well-defined and rigorous manner without a UV completion.It is also likely the case that the generalised Birkhoff theorems which specify the form of bubble transitions are no longer applicable in the presence of such terms [64,65].
Finally, it would be interesting to return to the analysis of pure EGB vacua of [40], and to consider more general black hole bubble solutions.The cubic Friedmann equation (2.15) would need to be solved (with appropriate modifications for including the GB branch), and without the link to the Einstein limit of the Einstein branch, all roots of the cubic might be equally valid.
What we have learned is that Lovelock terms appear to give a very similar picture to the black hole catalysed decay as the Einstein case, albeit with greatly more convoluted algebra!

Figure 2 .
Figure 2. The ratios of tunnelling components B/B HM change with the seed black hole mass in 5D spacetime, M F /M N , for fixed values of ℓ F = 5, ℓ T = 4.5, and α = 0.2where the relative mass gap is M C ≈ 0.06M N .The blue solid lines represent remnant black hole masses ranging from lower to higher as "0.2MN , 0.4M N , 0.6M N , 0.8M N , M N ", respectively; while the black solid line represents a remnant critical mass M C .The vertical dashed lines indicate the lower and upper bounds of the seed black hole mass.The red curve corresponds to the cosmological equal area limit.The cyan line represents a transition to a final de Sitter state without a remnant black hole, while the dots on the left axis show the transitions from pure de Sitter to remnant masses.The dashed lines on the left part of the graph, and the inset, represent a seed mass below the mass gap.

Figure 3 .
Figure 3.The ratios of tunnelling exponents B/B HM as a function of seed mass in 6D spacetime, M F /M N , shown for parameter values ℓ F = 5, ℓ T = 4.5 and α = 0.2.The dashed vertical line marks the Nariai limit for the seed mass.Blue solid lines represent different remnant black hole masses, ranging from 0.1M N to 0.7M N with steps of 0.1M N from lower to higher.The red curve gives the equal horizon area constraint.The cyan line represents a transition from a seed black hole to pure de Sitter spacetime.The dots represent a transition from pure de Sitter spacetime to a spacetime with a remnant black hole.