111Corresponding authoraainstitutetext: Department of Physics, King’s College
London, The Strand, London WC2R 2LS, UKbbinstitutetext: Perimeter Institute,
31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada

# Seeded vacuum decay with Gauss-Bonnet

Ruth Gregory a,1 Shi-Qian Hu<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

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 $\alpha$), whereas the latter case results in an enhanced
transition.

##  1 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
Coleman:1977py ; Callan:1977pt  (see also Kobzarev:1974cp ) 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 Coleman:1980aw , this requires using the semi-classical partition
function approach developed by Gibbons and Hawking Gibbons:1979xm  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,

${\cal P}\sim e^{-{\cal B}}=e^{-(I-I_{0})}\;,$ (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 energy
– de 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 Gregory:2013hja
; Burda:2015yfa , 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) Bowcock:2000cq  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 bubble222Note that the mass here refers to the local tidal forces, or
Riemann curvature, which have the form $GM/r^{3}$ locally.. The finding of
Gregory:2013hja ; Burda:2015yfa  (see also Burda:2015isa ; Burda:2016mou ;
Cuspinera:2018woe ; Tetradis:2016vqb ; Mukaida:2017bgd ; Dai:2019eei ;
Oshita:2018ptr ; DeLuca:2022cus  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 Hawking:1971ei ; Carr:1974nx ; Khlopov:2008qy ;
Tetradis:2016vqb ; Canko:2017ebb ; Dai:2019eei . 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 Coleman:1980aw ; Brown:2007sd ;
Burda:2016mou .

Figure 1: The schematic plot of a one-dimensional potential that has two
different minima, with the higher (lower) representing false (true) vacuum,
labelled as $\phi_{FV}$ and $\phi_{TV}$, respectively. The top of the
potential barrier is labelled as $\phi_{Top}$. 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
Hawking:1981fz . The two tunnelling processes can be best illustrated via the
cartoon of a particle in a one-dimensional 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 $V(\phi)$ at the local maximum
Hawking:1981fz ). The impact of black holes on the HM transition was
considered in Gregory:2020cvy ; Gregory:2020hia , see also Gomberoff:2003zh ,
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
Burda:2015yfa ; Burda:2015isa  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 Hayashi:2020ocn ;
Strumia:2022jil ; Miyachi:2021bwd ; Shkerin:2021zbf ; Briaud:2022few ;
Shkerin:2021rhy  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 Hayashi:2020ocn , 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 Strumia:2022jil , 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, Strumia:2022jil
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 Shkerin:2021zbf , 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
Lovelock:1971yv  – and we focus here on the Gauss-Bonnet (GB) term
Lanczos:1938sf ; Zumino:1985dp ; Zwiebach:1985uq

${\cal L}_{GB}=R^{2}-4R_{ab}^{2}+R_{abcd}^{2}$ (2)

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 Glavan:2019inb , as this has several problems
Arrechea:2020gjw ; Gurses:2020rxb ; Ai:2020peo .)

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 ($d\geq 5$), 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 Coleman:1980aw , 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 ($c=\hbar=1$) throughout the paper.

##  2 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 Charmousis:2008ce  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 Lovelock:1971yv , with an action given by

$I=-\frac{1}{16\pi G}\int_{\cal M}d^{D}x\sqrt{g}\left[{\cal
R}-2\Lambda+\alpha{\cal L}_{GB}\right]$ (3)

where $\alpha$ is the (dimensionful) coupling constant of the GB term, ${\cal
L}_{GB}$, defined in (2). The general (Euclidean) spherically symmetric vacuum
solution is derived in Boulware:1985wk ,

$ds^{2}=f(r)d\tau^{2}+f^{-1}(r)dr^{2}+r^{2}d\Omega^{2}_{D-2}$ (4)

where $d\Omega^{2}_{D-2}$ gives the metric of the unit $S^{D-2}$ and we take
the ‘Einstein’ branch solution Wheeler:1985nh ; Wheeler:1985qd ; Myers:1988ze
:

$f(r)=1+\frac{r^{2}}{2\tilde{\alpha}}\left(1-\sqrt{1+\frac{8\tilde{\alpha}\Lambda}{(D-1)(D-2)}+\frac{4\tilde{\alpha}\mu}{r^{D-1}}}\right)$
(5)

with $\tilde{\alpha}=(D-3)(D-4)\alpha$, and $\mu$ related to the mass through
the standard Myers-Perry formula Myers:1986un :

$\mu=\frac{16\pi GM}{(D-2){\cal A}_{D-2}}$ (6)

with ${\cal A}_{D-2}$ being the area of a unit $(D-2)-$sphere. Note that as
$\tilde{\alpha}\to 0$, we recover the Schwarzschild solution. Note also that
if $\alpha<0$, there are régimes of parameter space where the argument of the
root in (5) becomes negative, hence the solution is unphysical. While what
follows is independent of the sign of $\alpha$, we will focus on $\alpha>0$
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 Bowcock:2000cq  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 Wiltshire:1985us ; Charmousis:2002rc ;
2010AIPC.1241..521B , with the general brane junction conditions derived in
Davis:2002gn . 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 Gregory:2013hja , we seek a Euclidean bubble solution
$R(\lambda)$ separating two black hole spacetimes with exterior mass $M_{+}$
(the seed) and false vacuum energy $\Lambda_{+}$, and an interior mass
$M_{-}$, and true vacuum energy $\Lambda_{-}$. The regions exterior and
interior to the wall are given by

$ds^{2}_{\pm}=f_{\pm}d\tau^{2}_{\pm}+f_{\pm}^{-1}dr^{2}+r^{2}d{\Omega}^{2}_{D-2},$
(7)

where

$f_{\pm}=1+\frac{r^{2}}{2\tilde{\alpha}}\left(1-\sqrt{1+\frac{8\tilde{\alpha}\Lambda_{\pm}}{(D-1)(D-2)}+\frac{4\tilde{\alpha}\mu_{\pm}}{r^{D-1}}}\right),$
(8)

with the boundary $R(\lambda)$ 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

$\displaystyle I_{bulk}=-\frac{1}{16\pi G}\Biggl{(}$ $\displaystyle\int_{\cal
M_{+}}d^{D}x\sqrt{g_{+}}\left[{\cal R}_{+}-2\Lambda_{+}+\alpha{\cal
L}^{+}_{GB}\right]$ (9) $\displaystyle+\int_{\cal
M_{-}}d^{D}x\sqrt{g_{-}}\left[{\cal R}_{-}-2\Lambda_{-}+\alpha{\cal
L}^{-}_{GB}\right]\Biggr{)}$

represents the contribution from the spacetime away from the wall, and

$\displaystyle
I_{brane}=\int_{\mathcal{W}}d^{D-1}x\sqrt{h}\left(\sigma+\frac{1}{8\pi
G}\left(\Delta K-2\alpha\left[2\mathcal{G}_{ab}\Delta
K^{ab}-\Delta\mathcal{J}\right]\right)\right)$ (10)

represents the net contribution from the wall, including both the wall tension
$\sigma$, and the geometrical Gibbons-Hawking terms from the boundary
submanifolds on each side: $\partial{\cal M}_{+}=\partial{\cal M}_{-}={\cal
W}$. Here, $K_{{}_{\pm}ab}$ is the extrinsic curvature of the wall as
determined from the local embedding into ${\cal M}_{\pm}$, ${\cal G}_{ab}$ is
the local intrinsic Einstein tensor of the wall, and ${\cal J}$ is a cubic
extrinsic tensor (see Davis:2002gn ; Brihaye:2008xu ; Brihaye:2008ns ;
Liu:2008zf ) :

${\cal
J}_{ab}=\frac{1}{3}\left(2KK_{ac}K^{c}_{\;b}+K_{cd}K^{cd}K_{ab}-2K_{ac}K^{cd}K_{db}-K^{2}K_{ab}\right).$
(11)

###  2.1 The instanton solution

In order to find the instanton, we have to solve the equations of motion for
the bubble wall, found by varying (9) and (10). As described in Davis:2002gn ,
the generalised Israel equations are:

$\Delta K_{ab}-\Delta Kh_{ab}+2\alpha\left[3\Delta{\cal J}_{ab}-\Delta{\cal
J}h_{ab}-2{\cal P}_{acbd}\Delta K^{cd}\right]=8\pi G\sigma h_{ab}$ (12)

with

${\cal P}_{abcd}={\cal R}_{abcd}+2{\cal R}_{b[c}h_{d]a}-2{\cal
R}_{a[c}h_{d]b}+{\cal R}h_{a[c}h_{d]b}$ (13)

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 $R(\lambda)$, where $\lambda$
is the proper time of an observer comoving with the wall:

$f_{\pm}(R)\dot{\tau}^{2}_{\pm}+\frac{\dot{R}^{2}}{f_{\pm}(R)}=1,$ (14)

and choosing a normal always pointing towards increasing $R$ for
$\dot{\tau}_{\pm}>0$,

$n_{\pm}=\left(-\dot{R}d\tau_{\pm}+\dot{\tau}_{\pm}dr_{\pm}\right),$ (15)

gives, after some algebra, the junction condition as

$\left(f_{+}\dot{\tau}_{+}-f_{-}\dot{\tau}_{-}\right)\left[1+\frac{2\tilde{\alpha}}{R^{2}}-\frac{4\tilde{\alpha}}{3}\frac{{\dot{R}}^{2}}{R^{2}}\right]-\frac{2\tilde{\alpha}}{3R^{2}}\left(f_{+}^{2}\dot{\tau}_{+}-f_{-}^{2}\dot{\tau}_{-}\right)=-\frac{8\pi
G\sigma R}{(D-2)},$ (16)

where $\sigma$ is the surface tension of the wall. Substituting
$f_{\pm}\dot{\tau}_{\pm}=\sqrt{f_{\pm}-\dot{R}^{2}}$ from (14) leads to a
rather lengthy cubic Friedmann-like equation for $X=(1-\dot{R}^{2})/R^{2}$,

$\displaystyle 0$
$\displaystyle=\tilde{\alpha}^{2}X^{3}+\tilde{\alpha}X^{2}\left(\frac{3}{2}-\frac{(S_{+}^{2}-S_{-}^{2})^{2}}{256\tilde{\alpha}\bar{\sigma}^{2}}\right)$
(17)
$\displaystyle+\frac{X}{4}\left(3-\frac{3(S_{+}^{2}+S_{-}^{2})}{8}-\frac{(S_{+}^{2}-S_{-}^{2})(3S_{+}^{2}-3S_{-}^{2}+S_{+}^{3}-S_{-}^{3})}{192\tilde{\alpha}\bar{\sigma}^{2}}\right)$
$\displaystyle+\frac{(8-3S_{+}^{2}-3S_{-}^{2}-S_{+}^{3}-S_{-}^{3})}{64\tilde{\alpha}}-\frac{(3S_{+}^{2}-3S_{-}^{2}+S_{+}^{3}-S_{-}^{3})^{2}}{9216\tilde{\alpha}^{2}\bar{\sigma}^{2}}-\frac{9\bar{\sigma}^{2}}{16}$

where $\bar{\sigma}=2\pi G\sigma/(D-2)$ economises on notation, and

$S_{\pm}=\sqrt{1+\frac{8\tilde{\alpha}\Lambda_{\pm}}{(D-1)(D-2)}+\frac{4\tilde{\alpha}\mu_{\pm}}{r^{D-1}}}=\sqrt{1+\frac{4\tilde{\alpha}}{r^{2}}(1-f_{{}_{\pm}E})}$
(18)

represents the geometry on each side of the wall and is related to the
standard Einstein potential $f_{E}=1-\mu/r^{D-3}-2\Lambda r^{2}/(D-1)(D-2)$.
Note that $S\simeq 1+{\cal O}(\tilde{\alpha})$, so as $\tilde{\alpha}\to 0$,
(17) reduces to

$\displaystyle\frac{9}{16}\left[X-\bar{\sigma}^{2}-\frac{(1-\bar{f}_{E})}{R^{2}}-\frac{(\Delta
f_{E})^{2}}{16\bar{\sigma}^{2}R^{4}}\right]=0$ (19)
$\displaystyle\Rightarrow\qquad\left(\frac{\dot{R}}{R}\right)^{2}=\frac{\bar{f}_{E}}{R^{2}}-\bar{\sigma}^{2}-\frac{(\Delta
f_{E})^{2}}{16\bar{\sigma}^{2}R^{4}}$

as required. (Here, $\bar{f}_{E}\equiv(f_{E-}+f_{E+})/2$ and $\Delta f\equiv
f_{E+}-f_{E-}$.) Since $\tilde{\alpha}=0$ 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 Gregory:2013hja ; Burda:2015yfa : 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
$M_{-}$.

###  2.2 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.

####  2.2.1 The background seed action

We start by computing the action for the background solution of mass $M_{+}$,
and bare vacuum energy $\propto\Lambda_{+}$; this is simply a matter of
computing the bulk term (9), bearing in mind two key points. The first is that
if we are in vacuum or AdS spacetime ($\Lambda_{+}\leq 0$) 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 $\beta$,
hence at an event horizon, there will, in general, be a conical
deficit333Although the presence of a conical deficit can be disconcerting, the
singularity is mild, and can be resolved by a process analogous to the
discussion of the Dirac delta-distribution (as described in the appendix of
Gregory:2013hja ). As discussed in Gibbons:1979xm , the horizon is a fixed
point of the isometry of the metric – a “bolt” – which has codimension 2,
hence is not a boundary of the spacetime in the strict sense., thus we must
generalise the argument of Gregory:2013hja  to include the GB term.

Computing the integrand of (3) for the metric (4) we see that, on-shell, the
bulk integrand is a total derivative:

$\sqrt{g}\left[{\cal R}-2\Lambda+\alpha{\cal
L}_{GB}\right]=-\left\\{r^{D-4}f^{\prime}\left[r^{2}+\frac{2(D-2)\tilde{\alpha}}{(D-4)}(1-f)\right]\right\\}^{\prime}.$
(20)

Giving a contribution of $I_{c}-I_{h}$, where

$I_{c}=\frac{\beta{\cal A}_{D-2}}{16\pi
G}r_{c}^{D-4}f^{\prime}(r_{c})\left(r_{c}^{2}+\frac{2(D-2)\tilde{\alpha}}{(D-4)}(1-f(r_{c}))\right)$
(21)

at the upper $r_{c}$ limit, and at the black hole horizon

$I_{h}=\frac{\beta{\cal A}_{D-2}}{16\pi
G}\times\frac{4\pi}{\beta_{h}}r_{h}^{D-2}\left(1+\frac{2(D-2)\tilde{\alpha}}{(D-4)r_{h}^{2}}\right)=\frac{\beta}{\beta_{h}}S_{BH}$
(22)

the contribution is proportional to the entropy. Note we have replaced the
derivative of $f$ at the horizon with $f^{\prime}(r_{h})=4\pi/\beta_{h}$, an
expression involving the critical horizon periodicity $\beta_{h}$ 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 Gregory:2013hja  (see also
Li:2023men ), we note that a conical deficit metric has the local form
$d\rho^{2}+\lambda^{2}\rho^{2}d\theta^{2}$, where $\lambda\neq 1$ and the
periodicity of $\theta$ is $2\pi$. The deficit angle for this metric is
$\delta=2\pi(1-\lambda)$. We write the black hole metric locally at the
horizon as

$ds^{2}=A^{2}(\rho)d\tau^{2}+d\rho^{2}+C^{2}(\rho)d\Omega_{D-2}^{2}$ (23)

where $\rho^{2}=4(r-r_{h})/f^{\prime}(r_{h})$, and
$A^{\prime}(0)=2\pi/\beta_{h}$. Comparing this with the canonical form above
($\theta=2\pi\tau/\beta$), we see that $\lambda=\beta/\beta_{h}$, hence the
conical deficit is

$\delta=2\pi\frac{\beta_{h}-\beta}{\beta_{h}}.$ (24)

To compute the contribution of the conical deficit to the action, we split our
bulk integral into an integral from a proper distance $\varepsilon$ from the
horizon, and smooth out the cone inside $\varepsilon$, by having $A$ smoothly
interpolate between being regular at the origin, $A^{\prime}(0)=2\pi/\beta$,
and having the black hole geometry value $A(\varepsilon)=2\pi/\beta_{h}$ at
$\rho=\varepsilon$. We then take the limit $\varepsilon\to 0$ to obtain the
action with the conical singularity.

Computing the terms in the EGB Lagrangian for the metric (23):

$\displaystyle{\cal R}$
$\displaystyle=-2\frac{A^{\prime\prime}}{A}-2(D-2)\left(\frac{A^{\prime}C^{\prime}}{AC}-\frac{C^{\prime\prime}}{C}\right)+(D-2)(D-3)\frac{(1-C^{\prime
2})}{C^{2}}$ (25) $\displaystyle\alpha{\cal L}_{GB}$
$\displaystyle=-4(D-2)\tilde{\alpha}\Biggl{[}\frac{(1-C^{\prime
2})}{(D-4)C^{2}}\frac{A^{\prime\prime}}{A}+\frac{(1-C^{\prime
2})}{C^{2}}\left(\frac{A^{\prime}C^{\prime}}{AC}+\frac{C^{\prime\prime}}{C}\right)$
$\displaystyle\hskip 85.35826pt-(D-5)\frac{(1-C^{\prime
2})^{2}}{4C^{4}}\Biggr{]}$

and using $C=r_{h}+{\cal O}(\rho^{2})$, shows that near the horizon

$AC^{D-2}\left[{\cal R}-2\Lambda+\alpha{\cal
L}_{GB}\right]\sim-2\left(A^{\prime}\left[C^{D-2}+\frac{2\tilde{\alpha}(D-2)}{(D-4)}C^{D-4}\right]\right)^{\prime}+{\cal
O}(\rho)$ (26)

hence

$\displaystyle\int_{0}^{\varepsilon}d\rho AC^{D-2}\left[{\cal
R}-2\Lambda+\alpha{\cal L}_{GB}\right]$ $\displaystyle=$
$\displaystyle-2r_{h}^{D-2}\left(1+\frac{2(D-2)\tilde{\alpha}}{(D-4)r_{h}^{2}}\right)\left[A^{\prime}(\varepsilon)-A^{\prime}(0)\right]+{\cal
O}(\varepsilon^{2})$ (27) $\displaystyle=$
$\displaystyle-2\left(\frac{4GS_{BH}}{{\cal
A}_{D-2}}\right)\frac{2\pi(\beta-\beta_{h})}{\beta\beta_{h}}$

Thus the full contribution to the bulk action including the conical deficit is

$\displaystyle-\int d^{D}x\sqrt{g}\frac{{\cal L}_{EGB}}{16\pi G}$
$\displaystyle=-\lim_{\varepsilon\to 0}\frac{\beta{\cal A}_{D-2}}{16\pi
G}\left[\int_{0}^{\varepsilon}AC^{D-2}{\cal
L}_{EGB}d\rho+\int^{r_{c}}_{r_{h}+{\cal
O}(\varepsilon^{2})}\hskip-28.45274ptr^{D-2}{\cal L}_{EGB}dr\right]$ (28)
$\displaystyle=\frac{(\beta-\beta_{h})}{\beta_{h}}S_{BH}-\frac{\beta}{\beta_{h}}S_{BH}+I_{c}=I_{c}-S_{BH}$

The large $r$ contribution from $r_{c}$ is treated differently depending on
whether $\Lambda>0$ or not. For $\Lambda>0$, 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

$I_{seed}=-S_{CH}-S_{BH}$ (29)

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 ($\Lambda<0$ or
$\Lambda=0$), 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. (21)) 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 $I_{C}$:

$I_{seed}=I_{C}-S_{BH}(M_{+},\Lambda_{+})$ (30)

####  2.2.2 The bubble geometry

For the bubble geometry, the main difference from the previous subsection is
that we have the bubble wall at $R(\lambda)$. If $R$ varies with $\lambda$,
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, (9), reduces
to the boundary terms at $r_{h}$ and $r_{c}$, together with a contribution
evaluated at the wall:

$\displaystyle I_{bulk,{\cal W}}=$ $\displaystyle-\frac{{\cal A}_{D-2}}{16\pi
G}\int d\lambda
R^{D-4}\Bigl{[}f^{\prime}_{+}\dot{\tau}_{+}\left(R^{2}+\frac{2(D-2)\tilde{\alpha}}{(D-4)}(1-f_{+})\right)$
(31)
$\displaystyle-f^{\prime}_{-}\dot{\tau}_{-}\left(R^{2}+(1-f_{-})\right)\Bigr{]}.$

Turning to the wall integral (10), the trace of the Israel equations

$(D-1)8\pi G\sigma=-(D-2)\Delta K+\frac{2\tilde{\alpha}}{(D-3)}\left[{\cal
G}_{ab}\Delta K^{ab}-\Delta{\cal J}\right]$ (32)

gives the wall integral as

$I_{\cal W}=\frac{{\cal A}_{D-2}}{8\pi G}\int
d\lambda\frac{R^{D-2}}{(D-1)}\left(\Delta
K-\frac{6\tilde{\alpha}}{(D-3)(D-4)}\left[2\mathcal{G}_{ab}\Delta
K^{ab}-\Delta\mathcal{J}\right]\right).$ (33)

It then proves useful to define

$K_{0}=K_{\lambda\lambda}=\frac{f^{\prime}-2\ddot{R}}{2f\dot{\tau}}\;,\qquad
K_{1}=\frac{f\dot{\tau}}{R}\;,\qquad
p_{0}=\frac{(1-\dot{R}^{2})}{2R^{2}}\;,\qquad p_{1}=\frac{\ddot{R}}{R}$ (34)

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:

$\Delta K_{1}=\Delta K_{0}+2\tilde{\alpha}\left[2p_{0}\Delta
K_{0}-2p_{1}\Delta K_{1}-\Delta(K_{0}K_{1}^{2})+\Delta K_{1}^{3}-6p_{0}\Delta
K_{0}\right].$ (35)

Using these relations, after some algebra we find the wall integral can be
reduced to

$\begin{aligned} I_{\cal W}&=\frac{{\cal A}_{D-2}}{8\pi G}\int d\lambda
R^{D-2}\left[\Delta K_{0}+\frac{2(D-2)\tilde{\alpha}}{(D-4)}\left(2p_{0}\Delta
K_{0}-2p_{1}\Delta K_{1}-\Delta(K_{0}K_{1}^{2})\right)\right]\\\ &=\frac{{\cal
A}_{D-2}}{8\pi G}\int d\lambda
R^{D-2}\Delta\left[K_{0}\left(1+\frac{2(D-2)\tilde{\alpha}(1-f)}{(D-4)R^{2}}\right)-\frac{4(D-2)\tilde{\alpha}}{(D-4)}\frac{\ddot{R}}{R}K_{1}\right].\end{aligned}$

(36)

Noting that

$\frac{f^{\prime}\dot{\tau}}{2}=\dot{\tau}\left(f\dot{\tau}K_{0}+\ddot{R}\right)=\left(1-\frac{\dot{R}^{2}}{f}\right)K_{0}+\dot{\tau}\ddot{R},$
(37)

we find that the combined bulk-term and wall integrals give

$\displaystyle I_{wall}=$ $\displaystyle\frac{{\cal A}_{D-2}}{8\pi G}\int
d\lambda
R^{D-2}\Delta\Biggl{[}-\frac{4(D-2)\tilde{\alpha}}{(D-4)}\frac{\ddot{R}}{R}K_{1}$
(38) $\displaystyle\hskip
28.45274pt+\left(\frac{\dot{R}^{2}}{f}K_{0}-\dot{\tau}\ddot{R}\right)\left(1+\frac{2(D-2)\tilde{\alpha}(1-f)}{(D-4)R^{2}}\right)\Biggr{]}.$

Putting together, we see that the action of the bubble geometry is

$I_{bubble}=I_{wall}+I_{C}-S_{BH}(M_{-})$ (39)

hence the action for the tunnelling instanton is

$I_{\cal B}=I_{bubble}-I_{seed}=S_{BH}(M_{+})-S_{BH}(M_{-})+I_{wall}.$ (40)

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 Burda:2015isa ; Burda:2016mou .

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:

${\cal P}\sim\text{exp}\left[-(S_{seed}-S_{remnant})/\hbar\right]$ (41)

##  3 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 Hawking:1981fz . 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 Gregory:2020cvy ; Gregory:2020hia , 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 Gregory:2020hia  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 (7) and (8) of “$f_{\pm}$” 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
$\mu_{F/T}$, and vacuum parameters $\ell_{F/T}$, where
$\ell^{2}=(D-1)(D-2)/2\Lambda$. The tunnelling rate is dominated by the
Boltzmann factor:

$\Gamma_{FV\rightarrow Top}\sim e^{-B}.$ (42)

From (41), we see that the change in action is the difference in the
entropies, justifying the interpretation of the HM probability as a Boltzmann
suppression:

$B_{FV\rightarrow Top}=I_{T}-I_{F}=[S_{CH}+S_{BH}]_{F}-[S_{CH}+S_{BH}]_{T}.$
(43)

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:

$S_{i}=\frac{{\cal
A}_{D-2}}{4G}r_{i}^{D-2}\left(1+\frac{2(D-2)\tilde{\alpha}}{(D-4)r_{i}^{2}}\right)$
(44)

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 (43). 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_ Gregory:2020hia , that the
area of the cosmological horizon must never increase:

$S_{c}|_{T}\leq S_{c}|_{F}$ (45)

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:

$M=\frac{(D-2)\mathcal{A}_{D-2}}{16\pi
G}\mu=\frac{(D-2)\mathcal{A}_{D-2}r_{h}^{D-5}}{16\pi
G}\left(\tilde{\alpha}+r_{h}^{2}-\frac{r_{h}^{4}}{\ell^{2}}\right).$ (46)

For $D=5$, we see that while smaller black hole horizons correspond to lower
black hole masses, it is not possible for $M\to 0$ while maintaining a
horizon. Indeed, inspecting the expression for $f(r)$ at the origin for $D=5$
yields $f(0)=1-\sqrt{\mu/\tilde{\alpha}}$, which becomes positive as $\mu$,
hence $M$, becomes sufficiently small: $GM<3\pi\alpha/4$, indicating the loss
of an horizon. This is consistent with numerical studies of collapse
Deppe:2012wk ; PhysRevLett.114.071102 , which find that black holes will not
form dynamically if the total mass-energy content of the spacetime is below a
critical value in both asymptotically flat and asymptotically AdS spacetime.
This _mass gap_ was also noted in Frolov:2015bta  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 Cai:2001dz ; Wu:2021zyl . 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.

$\bullet$ 5D HM

Taking $f(r_{h},r_{c})=0$ and $D=5$ in (5) (noting $\tilde{\alpha}=2\alpha$ in
5D) the black hole and cosmological event horizons can be obtained in a simple
form:

$r_{h}=\left(\frac{\ell^{2}}{2}-\sqrt{2\alpha\ell^{2}+\frac{\ell^{4}}{4}-\frac{8GM\ell^{2}}{3\pi}}\right)^{\frac{1}{2}},~{}~{}~{}r_{c}=\left(\frac{\ell^{2}}{2}+\sqrt{2\alpha\ell^{2}+\frac{\ell^{4}}{4}-\frac{8GM\ell^{2}}{3\pi}}\right)^{\frac{1}{2}}$

(47)

Recall that $M_{C}=3\pi\alpha/4$, 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 nariai1950some ; Nariai1951OnAN  gives a
maximum:

$GM_{N}=\frac{3\pi}{32}\left(\ell^{2}+8\alpha\right).$ (48)

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 $M_{C}\leq M\leq M_{N}$. 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 Witten:1988hc ;
Banados:1992wn .

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 $M\to 0$. 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 $\Delta M=M-M_{C}$. The
black hole horizon entropy $S_{BH}\propto\sqrt{\Delta M}$, whereas $S_{CH}\sim
S_{DS}-{\cal O}(\Delta M)$, 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 $M_{C}$.

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
$\ell_{F}=5$, $\ell_{T}=4.5$, and $\alpha=0.2$ where the relative mass gap is
$M_{C}\approx 0.06M_{N}$. The blue solid lines represent remnant black hole
masses ranging from lower to higher as “$0.2M_{N}$, $0.4M_{N}$, $0.6M_{N}$,
$0.8M_{N}$, $M_{N}$”, 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.

$\bullet$ 6D HM

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.

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
$\ell_{F}=5,\ell_{T}=4.5$ and $\alpha=0.2$. 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.

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.

##  4 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.

###  4.1 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 (40) and the equation of
motion:

$\displaystyle\dot{R}^{2}$
$\displaystyle=1-\left(\bar{\sigma}^{2}+\frac{\bar{\Lambda}}{3}+\frac{(\Delta\Lambda)^{2}}{144\bar{\sigma}^{2}}\right)R^{2}-\frac{2G}{R}\left(\bar{M}+\frac{\Delta
M\Delta\Lambda}{24\bar{\sigma}^{2}}\right)-\frac{(G\Delta
M)^{2}}{4\bar{\sigma}^{2}R^{4}}$ (49)
$\displaystyle=f_{\pm}-\left(\omega_{\pm}R+\frac{\Delta
M}{2\bar{\sigma}R^{2}}\right)^{2}=F(R)$

where

$\omega_{\pm}=\frac{\Delta\Lambda}{12\bar{\sigma}}\mp\bar{\sigma}$ (50)

and $f_{\pm}=1-2M_{\pm}/r-\Lambda_{\pm}r^{2}/3$ is the Schwarzschild
potential.

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 (49) implies that

$f_{+}\dot{\tau}_{+}=\omega_{+}R+\frac{\Delta M}{2\bar{\sigma}R^{2}}\geq 0$
(51)

If we have no black hole seed, then $f_{+}=1-\Lambda_{+}r^{2}/3$, and $\Delta
M=-M_{-}$. The equation of motion, $\dot{R}^{2}=F(R)$, must have two zeros of
$F(R)$ (unless the solution is static, $F(R)\equiv 0$). However,

$F^{\prime}(R)=f_{+}^{\prime}-2\left(\omega_{+}R+\frac{\Delta
M}{2\bar{\sigma}R^{2}}\right)\left(\omega_{+}-\frac{\Delta
M}{\bar{\sigma}R^{3}}\right)=-\frac{2\Lambda
R}{3}-2(f_{+}\dot{\tau}_{+})\left(\omega_{+}+\frac{M_{-}}{\bar{\sigma}R^{3}}\right)<0$

(52)

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
$\alpha$ dependent part of the contribution is $I_{wall,\alpha}=I^{+}_{{\cal
W},\alpha}-I^{-}_{{\cal W},\alpha}$, where

$I^{\pm}_{{\cal W},\alpha}=\frac{2\alpha}{G}\int
d\lambda\left[\left(K_{0}^{\pm}-\frac{f^{\prime}_{\pm}\dot{\tau}_{\pm}}{2}\right)\left(1-f\right)-2R\ddot{R}K_{1}^{\pm}\right]$
(53)

Here, $K_{0,1}$ were defined in (34), and for the wall trajectory (49) are

$K_{0}^{(\pm)}=\omega_{\pm}-\frac{\Delta M}{\bar{\sigma}R^{3}}\quad,\qquad
K_{1}^{(\pm)}=\omega_{\pm}+\frac{\Delta M}{2\bar{\sigma}R^{3}}$ (54)

we therefore see that the $\alpha$ contribution to the wall action is:

$\begin{aligned} I^{\pm}_{{\cal W},\alpha}&=\frac{2\alpha}{G}\int
d\lambda\left[\left(K_{0}^{\pm}-\frac{f^{\prime}_{\pm}\dot{\tau}_{\pm}}{2}\right)\left(1-f\right)-2R\ddot{R}K_{1}^{\pm}\right]\\\
&=\frac{2\alpha}{G}\int
d\lambda\left[\frac{K_{0}^{\pm}\dot{R}^{2}-R\ddot{R}K_{1}^{\pm}}{f_{\pm}}-K_{0}^{\pm}\dot{R}^{2}-R\ddot{R}K_{1}^{\pm}\right]\\\
&=\frac{2\alpha}{G}\int
d\lambda\left[\frac{\dot{R}^{2}}{f_{\pm}}\frac{d~{}}{d\lambda}\left(\frac{\omega_{\pm}R}{\dot{R}}+\frac{\Delta
M}{2\bar{\sigma}\dot{R}R^{2}}\right)-\omega_{\pm}\frac{d(R\dot{R})}{d\lambda}+\frac{\Delta
M}{2\bar{\sigma}R^{3}}(2\dot{R}^{2}-R\ddot{R})\right]\end{aligned}$

(55)

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:

$\frac{\omega_{\pm}R}{\dot{R}}+\frac{\Delta
M}{2\bar{\sigma}\dot{R}R^{2}}=\frac{\sqrt{f_{\pm}-\dot{R}^{2}}}{\dot{R}}=\text{sign($\dot{R}$)}\,\sqrt{\frac{f_{\pm}}{\dot{R}^{2}}-1}$
(56)

thus the first part of the integral in (55) becomes:

$\int\text{sign($\dot{R}$)}\,\frac{\dot{R}^{2}}{f_{\pm}}\frac{d~{}}{d\lambda}\left(\sqrt{\frac{f_{\pm}}{\dot{R}^{2}}-1}\right)\,d\lambda=\int\text{sign($\dot{R}$)}\,\text{arctan}\left(\sqrt{\frac{f_{\pm}}{\dot{R}^{2}}-1}\right)\,d\lambda$
(57)

due to the periodicity of the solution $R(\lambda)$, the arctan function is
symmetric around the turning points of the solution, whereas $\dot{R}$ changes
sign, thus this integral vanishes and there is no $\alpha$ dependent
contribution from the wall.

Examining the contribution of the entropies to the action reveals the presence
of an $\alpha-$dependent, but constant, contribution from each black hole
horizon in (40). The entropy term, $S_{BH}$ is

$S_{BH}=\frac{\pi r_{+}^{2}}{G}\left(1+\frac{4\alpha}{r_{+}^{2}}\right).$ (58)

From the perspective of black hole thermodynamics, this constant shift is
irrelevant Clunan:2004tb , 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 $\alpha$ 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:

$I_{{\cal B},NR}=\frac{\pi r_{+}^{2}}{G}+\frac{4\pi\alpha}{G}+I_{wall}$ (59)

Thus, for bubble nucleated decay, the GB term either has no impact or actually
suppresses tunnelling ($I_{{\cal B}}$ 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.

###  4.2 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 $\alpha-$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 $\delta I\sim 4\alpha/G-2\pi\ell
M$ relative to the pure HM transition, thus again there is a preferred
transition to a vanishingly small mass black hole universe.

##  5 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
Gregory:2013hja , 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 $\alpha$). 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 $\alpha>0$, 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
Oliva:2011xu ; Oliva:2012zs .

Finally, it would be interesting to return to the analysis of pure EGB vacua
of Charmousis:2008ce , and to consider more general black hole bubble
solutions. The cubic Friedmann equation (17) 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!

###### Acknowledgements.

We would like to thank Christos Charmousis and Sam Patrick for discussions and
useful suggestions. This work was supported in part by the STFC Consolidated
Grant ST/P000371/1 (RG), the Chinese Scholarship Council and King’s College
London (SH), the Aspen Center for Physics, supported by National Science
Foundation grant PHY-2210452 (RG), and the Perimeter Institute for Theoretical
Physics (RG). Research at Perimeter Institute is supported by the Government
of Canada through the Department of Innovation, Science and Economic
Development Canada and by the Province of Ontario through the Ministry of
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