the metric functions $C(r)$ and $B(r)$ satisfy the boundary conditions (8.2)
and (8.3), so that $\mathcal{R}$ is the BoN nucleation radius and $R_{KK}$ the
KK radius.

Integrating over the 5th dimension242424This leads to $1/\kappa=2\pi
R_{KK}/\kappa_{5}$. We set set $\kappa=1$ below., $y$, followed by a Weyl
rescaling of the $4d$ metric:

$g_{\mu\nu}\to g_{\mu\nu}e^{\sqrt{2/3}\phi}\ ,$ (10.4)

where the field $\phi$ is defined by the relation

$C(r)=\mathrm{e}^{-\sqrt{2/3}\phi}\ ,$ (10.5)

gives the $4d$ reduced action

$S=\int
d^{4}x\sqrt{-g}\left\\{\frac{R_{4}}{2}-\frac{(\partial\phi)^{2}}{2}-\frac{(\partial
f)^{2}}{2}-V(\phi,f)\right\\},$ (10.6)

where

$V(\phi,f)=e^{\sqrt{2/3}\phi}\left[\frac{n^{2}}{2R_{KK}^{2}}f^{2}e^{2\sqrt{2/3}\phi}+\frac{\lambda}{4}(f^{2}-\eta^{2})^{2}+\Lambda\right]\
.$ (10.7)

The relation between $4d$ and $5d$ quantities is

$f\equiv\sqrt{2\pi R_{KK}}\,f_{5}\ ,\quad\eta\equiv\sqrt{2\pi
R_{KK}}\,\eta_{5}\ ,\quad\Lambda\equiv 2\pi R_{KK}\,\Lambda_{5}\
,\quad\lambda\equiv\lambda_{5}/(2\pi R_{KK})\ .$ (10.8)

The $4d$ (Euclidean) metric can be written in CdL form
$ds^{2}=d\xi^{2}+\rho(\xi)^{2}d\Omega_{3}^{2}$ with

$\frac{dr}{d\xi}=e^{\sqrt{2/3}\phi/2}\
,\quad\quad\rho(\xi)=\mathcal{R}B(r)e^{-\sqrt{2/3}\phi/2}\ .$ (10.9)

In terms of these variables, and using relations like
$(\partial\phi)^{2}=g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi=e^{\sqrt{2/3}\phi}(\partial_{r}\phi)^{2}=(\partial_{\xi}\phi)^{2}$,
the Euclidean action for the $O(4)$ symmetric BoN takes the form

$S_{E}=2\pi^{2}\int
d\xi\rho^{3}\left\\{-\frac{1}{2}R_{4}+\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}\dot{f}^{2}+V(\phi,f)\right\\}\,$
(10.10)

where $\dot{x}\equiv dx/d\xi$ and

$R_{4}=\frac{6}{\rho}(1-\rho\ddot{\rho}-\dot{\rho}^{2})\ .$ (10.11)

The action (10.10) takes the form of the action for a multifield bounce. The
bounce equations of motion read

$\displaystyle\ddot{\phi}+3\frac{\dot{\rho}}{\rho}\dot{\phi}$ $\displaystyle=$
$\displaystyle\partial V/\partial\phi\ ,$ (10.12)
$\displaystyle\ddot{f}+3\frac{\dot{\rho}}{\rho}\dot{f}$ $\displaystyle=$
$\displaystyle\partial V/\partial f\ ,$ (10.13)
$\displaystyle\dot{\rho}^{2}-1$ $\displaystyle=$
$\displaystyle\frac{1}{3}\kappa\rho^{2}\left(\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}\dot{f}^{2}-V\right).$
(10.14)

It can be checked that the $5d$ BoN equations of motion derived in [12]
coincide with the bounce equations above, once $5d$ quantities are translated
to $4d$ quantities following (10.8), $r$-derivatives are expressed in terms of
$\xi$-derivatives and the relations (10.5) and (10.9) are used.

In terms of the four dimensional fields $\rho$ and $\phi$, the regularity
conditions (8.2) at the BoN surface, $\xi\to 0$, are given by the expressions
(8.10) which, in the case of a circle compactification reduce to

$\phi(\xi)\simeq-\sqrt{\frac{2}{3}}\log\left(\frac{3\xi}{2R_{KK}}\right)\to\infty\
,\qquad\rho(\xi)\simeq{\cal R}\left[\frac{3\xi}{2R_{KK}}\right]^{1/3}\to 0\ .$
(10.15)

As a consequence of these boundary conditions and (10.13), it is also possible
to show that $f(\xi)\sim\xi^{2n/3}\to 0$ in the same limit. The boundary
conditions (8.3) at infinity, $\xi\to\infty$, take the form

$\phi(\infty)=\phi_{m}=0,\quad\rho(\infty)\to\infty,\quad\rho^{\prime}(\infty)=1\
,$ (10.16)

with $f(\infty)=f_{m}$, a constant. The scalar potential has an AdS critical
point at $(\phi_{m},f_{m})$ provided we set

$f_{m}^{2}=\eta^{2}-\frac{n^{2}}{\lambda
R_{KK}^{2}}=\frac{2\eta^{2}}{5}(1+\frac{3}{2}\Delta)\leq\eta^{2},\quad\phi_{m}=0\
,$ (10.17)

and we impose the KK radius $R_{KK}$ to satisfy

$R_{KK}^{2}=-\frac{3n^{2}\eta^{2}}{4\Lambda}(1+\Delta)\leq-\frac{3n^{2}\eta^{2}}{2\Lambda}\
,$ (10.18)

where252525We follow the notation of [12] except for the KK radius (which we
denote by $R_{KK}$) and the size of the KK circle $R_{KK}C(r)$ (given by the
function $rC(r)$ in [12]).
$\Delta\equiv\sqrt{1+20\Lambda/(9\lambda\eta^{4})}\in[0,1]$. The value of the
potential at the critical point is

$V_{m}\equiv
V(\phi_{m},f_{m})=-\frac{9}{25}\left(\frac{2}{3}+\Delta\right)(1-\Delta)\lambda\eta^{4}\leq
0\ ,$ (10.19)

which agrees with the value of $H^{2}$ given in [12]. The following quantities
are the same in $4d$ or $5d$:
$\Lambda/(\lambda\eta^{4})=\Lambda_{5}/(\lambda_{5}\eta_{5}^{4})$,
$\kappa\Lambda=\kappa_{5}\Lambda_{5}$ and
$\eta^{2}/\Lambda=\eta_{5}^{2}/\Lambda_{5}$.

In the tunneling potential formalism, the multifield BoN can be described by a
$V_{t}$ that is related to the Euclidean quantities by

$V_{t}=V-\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\dot{f}^{2}\ ,$ (10.20)

and can be expressed as a function of a single field $\varphi$ which is
defined by

$d\varphi^{2}\equiv d\phi^{2}+df^{2}\ ,$ (10.21)

with $\varphi(\xi=0)=0$. Note that $\phi^{\prime 2}+f^{\prime 2}=1$ follows
from (10.21). We therefore have

$V_{t}(\varphi)=V-\frac{1}{2}\dot{\varphi}^{2}\ .$ (10.22)

The EoM for $V_{t}(\varphi)$ is of the same form as for the single-field case,
eq. (4.1), but with $x^{\prime}\equiv dx/d\varphi$. As now we have two fields,
in order to determine the trajectory in field space followed by the instanton
solution, there is an additional equation, which reads [30, 32]

$2(V-V_{t})\phi^{\prime\prime}=\nabla_{T}V\equiv\partial
V/\partial\phi-\phi^{\prime}V^{\prime}\ ,$ (10.23)

where $V^{\prime}=\partial V/\partial\varphi=(\partial
V/\partial\phi)\phi^{\prime}+(\partial V/\partial f)f^{\prime}$.

Figure 18: Upper plot: Potential $V(\phi,f)$ from (10.7), plotted
logarithmically, showing the minimum and the potential defile along $f=0$. The
orange line follows the trajectory for BoN vacuum decay. Lower plot: $V$ and
$V_{t}$ along the BoN, using $\phi$ as parameter. The dashed lines show the
asymptotic approximations from (10.25) and (10.26).

According to the boundary conditions (10.15), in the regime of small $\xi$ (or
large $\phi$) we have $\phi\to\infty$ and $f\to 0$ so that we can take $f$ as
a small correction and identify $\phi$ and $\varphi$. It is immediate to check
that this limiting behaviour is consistent with (10.23), since this equation
is trivially satisfied when we set $\varphi\simeq\phi$. The scalar potential
has the following asymptotic behaviour for $\varphi\simeq\phi\to\infty$

$V(\varphi\to\infty)=\left(\Lambda+\frac{\lambda\eta^{4}}{4}\right)e^{\sqrt{2/3}\varphi}+\frac{n^{2}}{2R_{KK}^{2}}f^{2}(\varphi)\,e^{3\sqrt{2/3}\varphi}+{\cal
O}\left(f^{2}e^{\sqrt{2/3}\varphi}\right)\ .$ (10.24)

Although $f$ is expected to be exponentially suppressed, it appears in the
potential enhanced by $e^{3\sqrt{2/3}\varphi}$ so that we should check if its
contribution is small or not compared with the first term in the potential
above. The limiting behaviour of $f$ when $\xi\to 0$ implies $f\simeq
f_{0}e^{-n\sqrt{2/3}\varphi}$ for some $f_{0}>0$ [see (10.15) and the
discussion below] and, therefore, for $n=1$ both terms in (10.24) are of the
same order, while for $n>1$ the flux contribution becomes subleading.
Substituting the $n=1$ form of the scalar potential

$V(\varphi)\simeq\left(\Lambda+\frac{\lambda\eta^{4}}{4}+\frac{f_{0}^{2}}{2R_{KK}^{2}}\right)\mathrm{e}^{\sqrt{2/3}\varphi}\
,$ (10.25)

in equation (4.1) we find that the tunneling potential should be of the
form262626We discard a second solution of type $-$ with $a=1/3$, outside the
consistency range $1/\sqrt{3}<a<1$, see the discussion in section 4.

$V_{t}(\varphi)\simeq-\frac{3}{4R_{KK}^{2}}e^{\sqrt{6}\varphi}\ ,$ (10.26)

which, according to our classification in section 4 corresponds to a type $0$
case (see table 1). This is as expected on general grounds from the general
discussion in section 8: from eq. (8.14), for a compactification from $5d$ to
$4d$ we get $\rho\sim\xi^{1/3}$, which is the behaviour corresponding to type
$0$ cases (see table 1). The equation of motion for the tunneling potential
(4.1) leaves the coefficient in front of the exponential as a free parameter,
which we have fixed using relation (10.20) and the regularity conditions
(10.15).

Figure 18, upper plot, shows the potential (10.7) for a choice of parameters
already considered in [12]: $n=1$, $\Lambda_{5}=-(0.347M_{P5})^{5}$,
$\eta_{5}=(0.630M_{P5})^{3/2}$ and $\lambda_{5}=1/(0.995M_{P5})$ where
$M_{P5}=1/\kappa_{5}^{1/3}$. The trajectory of the BoN, marked in orange, goes
out of the potential minimum and heads towards the narrow potential defile
along $f=0$ (with $f_{0}\simeq 1.2$). The values of $V$ and $V_{t}$ along that
BoN trajectory, as functions of $\phi$, are given in the lower plot of the
same figure. The dashed lines correspond to the analytical asymptotic
approximations in (10.25) and (10.26), which are excellent.

As anticipated at the end of section 8, this example shows how the presence of
an additional field, such as $f$, can modify the asymptotic scaling behaviour
of the BoN solution for $\phi\to\infty$: For a fixed value of $f$, one has
$V\sim e^{\sqrt{6}\phi}$ [see (10.7)], which corresponds to the expected
scaling of a flux-generated potential from 5 dimensions [see (8.19)]. However,
when one takes into account how $f$ responds to the BoN $\phi$ profile, the
effective potential along the BoN has a different scaling, $V\sim
e^{\sqrt{2/3}\phi}$, which is subleading and corresponds to a type 0 BoN. In
physical terms, this phenomenon can be related to the presence of the global
string wrapping the BoN surface, mentioned at the beginning of this section.
The flux and its induced potential, if not neutralised, are incompatible with
a smooth BoN geometry [see discussion in section 8], and thus they prevent the
collapse of the KK circle to zero size. The role of the global string soliton,
whose position is precisely the vanishing locus of $f$, is to absorb the flux,
and thus, to cancel its contribution to the scalar potential.

As we mentioned in the introduction, other smooth BoN solutions have been
studied in the context of field theory models of flux compactification, see
[15]. The situation in these other models seem to resemble the one presented
here. The presence of the p-form flux allows for the perturbatively stable
compactification to exist and their smooth solitonic source is the key
ingredient to be added to the bare BON to permit the quantum instability of
the type we are discussing here. Furthermore the solution is such that deep in
the core of the source the flux contribution is turned off leading to a
solution of the form given by Eqs. (8.10). In other words, the solitonic
nature of these solutions allow the solutions to be always of the type 0 BoN.

End of the world type of solutions have also been obtained before in field
theory models of flux compactification [15]. It would be interesting to
investigate these type of solutions from a $4d$ point of view to compare with
the solutions obtained in the potential tunneling formalism similarly to what
was done in previous sections. We leave this comparison for future work.

Finally, let us also note that in models of flux compactifications one can
think of the BoN as a limiting case of the flux transitions [12]. Therefore in
order to understand the most likely decay channel one should in principle
consider these type of tunneling processes as well [53]. The $4d$ description
of those transitions would also involve the presence of a brane coupled to the
moduli field [54, 42]. This should also be incorporated in the tunneling
potential formalism.

##  11 Summary and Outlook

Bubble of nothing (BoN) decays are of fundamental importance for the stability
of vacua in theories with compactified extra dimensions, like String Theory,
and they are closely connected to the cobordism conjecture of the Swampland
program. Their study is greatly facilitated by dimensional reduction to an
effective theory in 4 dimensions, with the size of the compact dimension(s)
described by a modulus scalar field, stabilized by some potential $V(\phi)$.
In this language, the BoN instanton reduces to a Coleman-De Luccia (CdL)
instanton with a singular behaviour at its core (where the field diverges,
$\phi\to\infty$) but having a finite action (that agrees with the action of
the BoN instanton in the high dimensional theory).

The $4d$ reduced setting is very convenient to study the properties of BoNs in
the presence of a modulus potential, $V(\phi)$, and has been used in the past
for this purpose, most recently in [26]. In this paper we have followed the
$4d$ approach but using the tunneling potential formalism rather than
traditional Euclidean methods. In this formalism, vacuum decay is described by
a tunneling potential function $V_{t}(\phi)$ that can be compared directly
with $V(\phi)$ without having to deal with the profiles of the modulus field
nor the Euclidean metric. The analysis is further facilitated by the fact that
all the possible decay channels of a given potential [BoNs, Coleman-de Luccia
or Hawking-Moss (HM) instantons as well as the so-called pseudo-bounces] can
be described in the same $V_{t}$ language, with different types of decay
connected with the others in a continuous way. Moreover, the tunneling action
for the different decay modes can be calculated using the simple universal
formula (2.1). The fact that this general formula reproduces the Euclidean BoN
action is an important result proven in Appendix B.

Using the $V_{t}$ formalism we have performed an analysis exploring which type
of modulus potentials $V(\phi)$ allow for BoN decays and which different types
of BoN exist. This study is similar in spirit to the one of [26]. We confirm
many of the findings of [26] and extend that work in several directions.

We identify four different types of BoN, with different asymptotic behaviour
in the compactification limit ($\phi\to\infty$, corresponding to the BoN core)
and different possible higher dimensional origin, as summarized in table 1.
Type 0 BoNs can appear when the compact manifold is a circle (these, for dS
vacua, were the main subject of [26]). Type $-$ BoNs can appear instead when
the compact manifold is an $S^{d}$ sphere, with $d>1$. The type $+$ or $-^{*}$
BoNs can be relevant vacuum decay channels for more complicated compact
geometries, that require the presence of some UV object or defect to allow the
shrinking to zero of the internal manifold in the BoN core.

For BoNs of types 0 or $-$, there is a simple link between the asymptotic
$\phi\to\infty$ behaviour of $V$, $V_{t}$ and important quantities derived
from them, like $D^{2}$ in (2.2), and the geometric properties of the BoN in
the higher dimensional theory (like the KK radius, $R_{KK}$, and the BoN
radius, ${\cal R}$). Assuming the compact manifold is $S^{d}$, we find for
$\phi\to\infty$,

$V\simeq-\frac{d(d-1)}{2\kappa
R_{KK}^{2}}e^{\sqrt{\frac{2(d+2)\kappa}{d}}\,\phi}+...\ ,\quad\quad
V_{t}\simeq-\frac{3d^{2}}{4\kappa
R_{KK}^{2}}e^{\sqrt{\frac{2(d+2)\kappa}{d}}\,\phi}+...\ ,$ (11.1)

and

$D^{2}\simeq\frac{9d(d+2)}{2\kappa^{3}R_{KK}^{2}{\cal
R}^{2}}e^{\sqrt{\frac{8\kappa}{d(d+2)}}\,(d+1)\phi}+...$ (11.2)

These relations, which capture information from the higher-dimensional theory,
can be confronted with the asymptotic behaviour of the $V_{t}$ solutions for
BoN decay that we find by solving the equation of motion for $V_{t}$ in (2.3)
from low field to high field values. Typically, for a given potential that
does not grow as fast as $e^{\sqrt{6\kappa}\phi}$, we find a continuous family
of possible type 0 BoN solutions $V_{t}(p;\phi)$, labeled by some parameter
$p$, with the $\phi\to\infty$ asymptotics

$V_{t}(p;\phi)\simeq V_{tA}(p)e^{\sqrt{6\kappa}\phi}+...\ ,\quad
D(p;\phi)\simeq D_{\infty}(p)e^{\sqrt{8\kappa/3}\phi}+...$ (11.3)

with $V_{tA}(p)<0$ and $D_{\infty}(p)>0$. For a fixed higher-dimensional
theory (thus fixed $R_{KK}$), matching (11.3) to (11.1) and (11.2) selects a
finite number of BoNs from the family [each with different radius ${\cal R}$,
determined by $D(p;\phi)$, and different tunneling action]. The number of such
selected BoN solutions is model dependent in the following way.

If the modulus has a single vacuum (or if gravity forbids its decay to other
minima), the selected BoN solution is unique (for a given $R_{KK}$). Moreover,
when the vacuum is a Minkowski or AdS one, there is an upper critical limit
$R_{KK}^{*}$ [corresponding to a lower limit on $-V_{tA}(p)$] for which the
BoN has infinite action and radius and corresponds to an end-of-the-world
brane. For theories with $R_{KK}>R_{KK}^{*}$, BoN decay is not allowed (this
is quenching by a CdL mechanism). Figures 2 and 3 show particular examples of
this and section 9 is fully devoted to discussing this mechanism.

However, when the modulus potential has additional vacua and admits standard
decay channels to them (via either Coleman-de Luccia or Hawking-Moss
instantons), there are at least two $V_{t}$ BoN solutions, the BoN solution
with lowest action being the relevant one. Also in this case there is a
critical value $R_{KK}^{*}$, which again corresponds to the minimum of
$-V_{tA}(p)$. However, this time criticality corresponds to the merging of the
two BoN solutions into one which is a saddle-point with finite tunneling
action. In any case, for $R_{KK}>R_{KK}^{*}$ again BoN decay is forbidden (a
dynamical quenching). Figures 5, 7 and 8 show some examples of this when the
false vacuum is Minkowski and dS. Interestingly, we found that BoN decay of dS
vacua is dynamically forbidden only when the EFT and KK energy scales are
comparable (without scale separation), that is, in regimes of parameter space
where the validity of the EFT is questionable.

When BoN solutions coexist with CdL or HM instantons (the second case just
described), the same parameter $p$ that labels the family of BoNs in some
range of values, naturally describes (outside the BoN range) the CdL/HM
instanton solutions as well as the so-called pseudo-bounces that connect CdL
and HM (an action valley in configuration space). The associated tunneling
actions can be easily calculated in the $V_{t}$ formalism and we find an
action $S(p)$ which is continuous across the boundaries between different
classes of $V_{t}$ solutions. End-of-the-world branes correspond to
$S(p^{*})\to\infty$, while the critical cases with dynamical quenching and
finite action correspond to a maximum of $S(p)$. Again, figures 5, 7 and 8
illustrate this behaviour.

The previous results for type 0 BoNs are in agreement with the findings of
[26] when there is overlap between both works. We believe that the powerful
$V_{t}$ approach sheds light from a new angle on the whole topic. Moreover, we
have extended the analysis to any kind of vacua (Minkowski, dS or AdS),
clarified the nature of the hybrid branches of solutions found in [26], and
studied in detail the two types of quenching of BoN decays. Concerning other
types of BoN, not studied in [26], we find that type $-$ BoN solutions for a
fixed potential (with the right asymptotic behaviour) appear singly, at the
boundary of type-0 BoN families. This is consistent with the fact that now a
given $R_{KK}$ fixes the asymptotic behaviour of both $V$ and $V_{t}$, see
(11.1). The BoN radius is as usual determined by the asymptotics of $D$.
Figures 10 and 13 show examples that illustrate the behaviour just described.

Using the $4d$ approach we have also found (both numerically and analytically)
examples of potentials that admit BoN solutions of types $+$ and $-^{*}$.
While the type $+$ can appear in a continuous family (like the type 0 BoNs,
see figure 14), the type $-^{*}$ BoNs appear as limiting cases of type 0 BoN
families (like the type $-$ Bons, see figures 15 and 16). Nevertheless, for
type $+$ and $-^{*}$ BoNs the picture is less complete as the higher
dimensional theory has a more complicated geometry for the compact space and
shrinking it to zero requires a defect or some other UV object, which we have
not considered in detail.

BoNs in flux compactifications are also of particular interest as their
existence is nontrivial: shrinking the compact space in the presence of flux
requires a charged object at the BoN core to absorb the flux. We have shown
that a $4d$ description of such a BoN (in a $5d$ flux compactification) is
also possible. Besides the modulus field, the $4d$ theory contains an
additional scalar field descended by dimensional reduction from the charged
object. In this case, a two-field $V_{t}$ can describe the BoN. Interestingly,
the $4d$ counterpart of the flux absorption is realized by a $V_{t}$ that
selects a direction in the two-field space with the right (type 0) asymptotic
behaviour rather than the generic type $+$ asymptotics expected in flux
compactifications, see figure 18 and the discussion in section 10.

There is a number of further directions for future work. The study of singular
BoNs (types $+$ and $-^{*}$) and how they are regulated by some UV object or
defect is of clear interest. Applying a $V_{t}$ approach to their description
seems possible and our first explorations look rather promising. Indeed, we
find that the $V_{t}$ description of Witten’s BoNs with defects [16, 43] is
rather straightforward via a simple rescaling of the prefactor of Witten’s BoN
in (3.22) to take into account the deficit angle associated to the singular
behaviour of the BoN. It would be interesting to see how this generalizes in
the presence of a nonzero potential for the modulus field.

Our results indicate that in some limits the existence of the BoN decay
channel is suppressed by some dynamical obstruction. It would be interesting
to understand this effect in the higher dimensional theory, in particular in
models of flux compactification. In some cases this quenching can be
understood as a CdL suppression where the bubble becomes static and infinite.
This type of objects have been found numerically in simple higher dimensional
theories in [15]. On the other hand, in other cases the suppression does not
have a limiting static wall and it would be interesting to explore such
solutions, and study if they can be consistently obtained within the regime of
validity of the $4d$ EFT. Finally, we would also like to investigate the
existence of the End of the World type of BoNs in connection with
supersymmetric vacua, similarly to what was done in [16].

## Acknowledgments

The work of J.R.E. and J.H. has been supported by the following grants: IFT
Centro de Excelencia Severo Ochoa CEX2020-001007-S and by PID2022-142545NB-C22
and PID2021-123017NB-I00 by MCIN/AEI/10.13039/501100011033 funded by “ERDF; A
way of making Europe”. The work by J.H. is supported by the FPU grant
FPU20/01495 from the Spanish Ministry of Education and Universities. J.J.B.-P.
has been supported in part by the PID2021-123703NB-C21 grant funded by MCIN/
AEI /10.13039/501100011033/ and by ERDF; ”A way of making Europe”; the Basque
Government grant (IT-1628-22) and the Basque Foundation for Science
(IKERBASQUE).

##  Appendix A Energy of the BoN

Consider the BoN decay of a false Minkowski vacuum. A nucleated BoN has zero
energy (so that energy is conserved in the decay process). In the case of
Witten’s BoN, for the decay of the $5d$ KK vacuum, this follows immediately
from the asymptotic behaviour of the metric (3.1) at $r\to\infty$ which does
not have $1/r$ terms (see e.g. [55]). This key property should be inherited by
the $4d$ description of the BoN.

Let us consider the $4d$ description of a general BoN mediating the decay of a
compactification with vanishing vacuum energy $V(\phi_{+})=0$ (i.e. the non-
compact component of space-time is Minkowski space). The usual energy integral

$E=4\pi\int_{0}^{\infty}d\rho\rho^{2}\left[\frac{1}{2}\dot{\phi}^{2}+V(\phi)\right]\
,$ (A.1)

which vanishes for regular CdL bubbles (see [56]) diverges for BoNs, even for
Witten’s one272727In that case $V(\phi)\equiv 0$ and then the integrand is
positive definite, making it impossible to have $E=0.$. To understand the
origin of this discrepancy, first note that the integrand in (A.1) is a total
derivative so that

$E=\left.\frac{4\pi}{3}\rho^{3}\left[V(\phi)-\frac{1}{2}\dot{\phi}^{2}\right]\right|_{\xi=0}^{\xi=\infty}\
,$ (A.2)

which diverges at $\xi=0$ (the $\xi=\infty$ term vanishes). One can connect
this expression with the asymptotic behaviour of the metric by using the
constraint equation (3.20) to get

$E=\left.\frac{4\pi}{\kappa}\left[\rho(1-\dot{\rho}^{2})\right]\right|_{\xi=0}^{\xi=\infty}\
.$ (A.3)

The $\xi=\infty$ contribution corresponds to the usual ADM mass, $M_{ADM}$,
which should vanish, while the $\xi=0$ contribution vanishes provided the
metric is smooth at the origin. However, since in the $4d$ description the BoN
space-time is singular at $\xi=0$, the integral (A.1) cannot be identified
with the ADM mass. A careful derivation of $M_{ADM}$ in $4d$ starting from
$5d$ (similar to the calculation of the Euclidean action in Subsection 3.2)
gives

$M_{ADM}=\frac{4\pi}{\kappa}\left[\rho(1-\dot{\rho}^{2})\right]_{\xi=0}+4\pi\int_{0}^{\infty}d\rho\rho^{2}\left[\frac{1}{2}\dot{\phi}^{2}+V(\phi)\right].$
(A.4)

Although both terms in the r.h.s. of the previous expression are separately
divergent, the divergences cancel exactly. Using (A.2), we get

$M_{ADM}=\frac{4\pi}{3}\rho^{3}\left[V(\phi)-\frac{1}{2}\dot{\phi}^{2}\right]_{\xi=\infty}\
,$ (A.5)

Plugging the asymptotic behaviour of $\rho$ and $\dot{\phi}$ for the $4d$
reduced Witten’s BoN it can be checked that indeed $M_{ADM}=0$.

In more general situations, but still imposing that $V(\phi_{+})=0$, we find
that the expression for the BoN energy, written in the $V_{t}$ formalism,
reads

$M_{ADM}=\left.36\pi\frac{\left[2(V-V_{t})\right]^{3/2}}{D^{3}}V_{t}\right|_{\phi\to\phi_{+}}$
(A.6)

For Witten’s BoN, it can be readily checked that using $V_{t}$ from (3.22)
leads to $M_{ADM}=0$. For other cases of Minkowski vacuum decay, we can get
the limit (A.6) by using the low-energy expansion
($\phi\to\phi_{\scriptscriptstyle+}=0$) of $V_{t}$ obtained in (5.3), which
gives $V\sim\phi^{2}$, $V_{t},D\sim\phi^{2}/\log\phi$ and leads to
$M_{ADM}=0$. This result can be extended to the decay of AdS vacua (for dS
vacua one cannot define energy).

##  Appendix B $S[V_{t}]=\Delta S_{E}$

The agreement between the action $S[V_{t}]$ of the $V_{t}$ formalism and the
Euclidean action difference $\Delta S_{E}$ of Coleman-De Luccia for regular
CdL transitions was proven in [28] (for the proof in general dimension, see
[36]). In this section we extend this proof to BoN decays.

Let us first review briefly the proof for regular CdL solutions. In [28, 36]
the Euclidean action is rewritten in the form

$S_{E}=-2\pi^{2}\int_{0}^{\xi_{\rm max}}\rho Vd\xi+S_{\rm GHY}\ ,$ (B.1)

where $\xi_{\rm max}=\infty$ for AdS or Minkowski false vacua and finite for
dS, while $S_{\rm GHY}$ is the Gibbons-Hawking-York boundary action [57] (the
value of this term is irrelevant to show the agreement between the actions in
both formalisms). Let us consider first the case of Minkowski or AdS vacuum
decays. The proof rewrites $\Delta
S_{E}=S_{E}[\phi_{B}]-S_{E}[\phi_{\scriptscriptstyle+}]$ for CdL decay as an
integral in field space, using the dictionary between formalisms, and then
establishes the identities

$\displaystyle s(\phi)-s_{E,B}(\phi)$ $\displaystyle=$
$\displaystyle\frac{dH_{B}}{d\phi}\ ,$ (B.2) $\displaystyle
s_{E,{\scriptscriptstyle+}}(\phi)$ $\displaystyle=$
$\displaystyle\frac{dH_{\scriptscriptstyle+}}{d\phi}\ ,$ (B.3)

where the action densities are defined by

$\displaystyle S[V_{t}]$ $\displaystyle=$
$\displaystyle\int_{\phi_{\scriptscriptstyle+}}^{\phi_{0}}s(\phi)d\phi=\frac{6\pi^{2}}{\kappa^{2}}\int_{\phi_{\scriptscriptstyle+}}^{\phi_{0}}\frac{(D+V_{t}^{\prime})^{2}}{DV_{t}^{2}}d\phi\
,$ (B.4) $\displaystyle\Delta S_{E}$ $\displaystyle=$
$\displaystyle\int_{\phi_{\scriptscriptstyle+}}^{\phi_{0}}\left[s_{E,B}(\phi)-s_{E,{\scriptscriptstyle+}}(\phi)\right]d\phi=-108\pi^{2}\int_{\phi_{\scriptscriptstyle+}}^{\phi_{0}}\frac{V-V_{t}}{D^{3}}\left(V+\frac{V_{\scriptscriptstyle+}V_{t}^{\prime}}{D_{\scriptscriptstyle+}}\right)d\phi\
,$ (B.5)

where $D_{\scriptscriptstyle+}^{2}\equiv
V_{t}^{\prime}{}^{2}+6\kappa(V-V_{t})(V_{t}-V_{\scriptscriptstyle+})$ and the
term proportional to $V_{\scriptscriptstyle+}$ comes from subtracting the
false vacuum action (see [28] for details). The functions $H_{B}$ and
$H_{\scriptscriptstyle+}$ are given by

$H_{B}=\frac{216\pi^{2}(V-V_{t})^{2}}{D^{3}}\left[\frac{V_{t}(V_{t}^{\prime}-2D)}{(D-V_{t}^{\prime})^{2}}\right]\
,\quad
H_{\scriptscriptstyle+}=\frac{216\pi^{2}(V-V_{t})^{2}}{D^{3}}\left[\frac{V_{\scriptscriptstyle+}(D_{\scriptscriptstyle+}+2D)}{(D+D_{\scriptscriptstyle+})^{2}}\right]\
.$ (B.6)

The identities (B.2) and (B.3) hold on-shell, that is, for any $V_{t}$
solution of the EoM (2.3) and thus holds both for CdL and BoN solutions, so
that we can still rely on them for the BoN proof below. Integration of the sum
of the identities (B.2) and (B.3) in the interval $(\phi_{+},\phi_{0})$ gives

$S[V_{t}]=\Delta
S_{E}+H_{B}(\phi_{0})+H_{\scriptscriptstyle+}(\phi_{0})-H_{B}(\phi_{+})-H_{\scriptscriptstyle+}(\phi_{+})\
,$ (B.7)

and the equality of the actions results from
$H_{B}(\phi_{0})=H_{\scriptscriptstyle+}(\phi_{0})=H(\phi_{+})=H_{\scriptscriptstyle+}(\phi_{+})=0$,
which follows from the behaviour of $V$ and $V_{t}$ at $\phi_{+}$ and
$\phi_{0}$ for CdL solutions describing $V_{\scriptscriptstyle+}\leq 0$
decays, see [28].

In the case of dS vacuum decay we have

$\Delta
S_{E}=\int_{\phi_{0{\scriptscriptstyle+}}}^{\phi_{0}}s_{E,B}(\phi)d\phi+\frac{24\pi^{2}}{\kappa^{2}V_{\scriptscriptstyle+}}\
,$ (B.8)

where the first term is the bounce action and the second (minus) the false
vacuum action, which is now finite. Integrating (B.2) in the interval
$(\phi_{0{\scriptscriptstyle+}},\phi_{0})$ one gets

$\int_{\phi_{0{\scriptscriptstyle+}}}^{\phi_{0}}s(\phi)d\phi=\Delta
S_{E}-\frac{24\pi^{2}}{\kappa^{2}V_{\scriptscriptstyle+}}+H_{B}(\phi_{0})-H_{B}(\phi_{0{\scriptscriptstyle+}})\
.$ (B.9)

It is straightforward to show that $H_{B}(\phi_{0})=0$ and
$H_{B}(\phi_{0{\scriptscriptstyle+}})=-24\pi^{2}/[\kappa^{2}V(\phi_{0{\scriptscriptstyle+}})]$.
This $H_{B}(\phi_{0{\scriptscriptstyle+}})$ combines with
$-24\pi^{2}/[\kappa^{2}V(\phi_{{\scriptscriptstyle+}})]$ to give the HM part
of the $S{V_{t}}$ action missing in the l.h.s. of (B.9) resulting in
$S[V_{t}]=\Delta S_{E}$.

To extend the previous proof to BoNs, we have to care about two differences
with respect to the standard case. First note that now $\Delta S_{E}$ has an
extra piece associated to the BoN core as given by (3.21),

$\delta_{BoN}\Delta
S_{E}=-\left.\pi^{2}\sqrt{\frac{2}{3\kappa}}\rho^{3}\dot{\phi}\,\right|_{\xi=0}\
,$ (B.10)

which will appear for any type of BoN vacuum decay, which has an additional
term from the extra-dimensional origin of the BoN. Second, for BoN solutions
the behaviour of $V$ and $V_{t}$ as $\phi\to\phi_{\scriptscriptstyle+}$ or
$\phi\to\phi_{0{\scriptscriptstyle+}}$ is similar to that for regular CdL
bounces so that $H_{B}$ and $H_{\scriptscriptstyle+}$ take the same previous
values as well, and would lead to the equality between $\Delta S_{E}$ and
$S[V_{t}]$ up to the extra BoN term (B.10). However, the asymptotic behaviour
of $V$ and $V_{t}$ as $\phi\to\phi_{0}$ is different than for regular CdL
solutions (as $\phi_{0}\to\infty$ for BoNs) and, it is remarkable that the
asymptotics of $H_{B,{\scriptscriptstyle+}}(\phi_{0}\to\infty)$ is such that
one gets precisely the term needed to reproduce (B.10), as we show below, thus
completing the proof of $\Delta S_{E}=S[V_{t}]$.

It can be shown that $H_{\scriptscriptstyle+}(\phi\to\infty)=0$ for any type
of BoN, simply using the different asymptotics summarized in Table 1. For
$H_{B}(\phi\to\infty)$, using the same table, we find the following.

For type 0 solutions, $D$ is subleading compared to $V_{t}^{\prime}$ and we
find

$H_{B}(\phi_{0}\to\infty)=\frac{216\pi^{2}(V-V_{t})^{2}V_{t}}{D^{3}V_{t}^{\prime}}=\frac{216\pi^{2}(V-V_{t})^{2}}{D^{3}\sqrt{6\kappa}}\
,$ (B.11)

where we have used $V_{t}\sim e^{\sqrt{6\kappa}\phi}$ to derive the last
expression. Using the $V_{t}$-to-Euclidean dictionary to rewrite this in terms
of Euclidean quantities, we immediately find

$H_{B}(\phi_{0}\to\infty)=-\sqrt{\frac{2}{3\kappa}}\pi^{2}\rho^{3}\dot{\phi},$
(B.12)

evaluated at $\xi=0$. This is precisely the piece $\delta_{BoN}\Delta S_{E}$
of (B.10) that should be added to the usual Euclidean action integral to get
the action for BoNs, and thus $S[V_{t}]$ agrees with it, as anticipated. In
this case it can also be checked that (B.12) gives a finite nonzero result.

For type $-$ and type $-^{*}$ solutions, it can be readily checked from the
asymptotic behaviours of $V$, $V_{t}$ and $D$ given in section 4 that
$H_{B}(\phi_{0}\to\infty)=0$. One also has $\rho^{3}\dot{\phi}\to 0$ for
$\xi=0$ and thus $\delta_{BoN}\Delta S_{E}=0$ and again we find that
$S[V_{t}]$ reproduces the Euclidean action.

For type $+$ solutions $D$ is also subleading compared to $V_{t}^{\prime}$ at
large field values, so that (B.12) also holds and this reproduces once again
$\delta_{BoN}\Delta S_{E}$ so that $S[V_{t}]$ agrees with the Euclidean
result. In this case, however, (B.12) diverges. Indeed we have

$H_{B}(\phi_{0}\to\infty)\sim\frac{e^{2a\sqrt{6\kappa}\phi}}{e^{3\sqrt{\kappa/6}(3a+1/a)\phi}}\to\infty\
,$ (B.13)

as $a>1$. In the Euclidean formulation this corresponds to

$\rho^{3}\dot{\phi}\sim-\xi^{1/a^{2}-1}\ ,$ (B.14)

which also diverges for $\xi\to 0$ and $a>1$. In spite of this divergence we
find cases of this type with a finite total action for $a>1/\sqrt{3}$. In the
Euclidean approach the divergence is cancelled by a contribution in $\Delta
S_{E}$.

In summary, for all types of BoN we find agreement between the Euclidean
action [supplemented by the needed extra term (B.10) from the higher
dimensional theory] and the usual action $S[V_{t}]$ (without any added terms).

##  Appendix C Large field expansion of $V_{t}(\phi)$ for
$V(\phi)=V_{A}e^{a\sqrt{6\kappa}\phi}$

In this appendix we derive the large field expansion of the tunneling
potential if we assume that $V(\phi)=V_{A}e^{a\sqrt{6\kappa}\phi}$ and
consider the four possible different types of asymptotic behaviour discussed
in the text. Although subleading terms in $V$ can have an influence on
$V_{t}$, it is still instructive to just study this simple $V$. Also, we
consider this $V$ not as being the whole potential, but just as valid in some
field range extending to infinity.

Before solving for $V_{t}$, it is convenient to discuss first the solutions of
the differential equation $D(\phi)^{2}=0$ [which we call
$\overline{V_{t}}(\phi)$]. In the context of regular CdL bounces for Minkowski
or AdS false vacua, $\overline{V_{t}}(\phi)$ with boundary conditions
$\overline{V_{t}}(0)=V_{\scriptscriptstyle+}$ and
$\overline{V}^{\prime}_{t}(0)=0$ plays a role in determining whether the false
vacuum decay is quenched by gravity or not [32]. Moreover, $\overline{V_{t}}$
solutions joining two vacua describe domain wall solutions. In the context of
BoN solutions, $\overline{V_{t}}(\phi)$ solutions correspond instead to
dynamical cobordisms/ end of the world branes [41].

Figure 19 shows the ranges of $a$ and ${\rm sign}(V_{A})$ for which the
different types of asymptotic behaviour discussed below apply.

Figure 19: For $V=V_{A}e^{a\sqrt{6\kappa}\phi}$ with varying $a$ and both
signs of $V_{A}$, the plot shows the ranges for the four different types of
asymptotic behavior of $V$ and $V_{t}$ discussed in the text.

###  C.1 $\overline{V_{t}}$ Solutions

The asymptotic behaviour of $\overline{V_{t}}(\phi\to\infty)$ in the four
different types considered in the text can be obtained directly from the
differential equation $D(\phi)=0$ with $\overline{V_{t}}^{\prime}<0$, which
gives

$\overline{V_{t}}^{\prime}=-\sqrt{-6\kappa(V-\overline{V_{t}})\overline{V_{t}}}\
.$ (C.1)

The solutions of this equation do not cross each other away from the points
with $\overline{V_{t}}=V$, and they foliate the region of the
$(V,\overline{V_{t}})$ plane with $\overline{V_{t}}<0$ and
$V-\overline{V_{t}}>0$. The asymptotic behaviour of $V_{t}$ must be at least
as strong as that of $\overline{V_{t}}$, as $D^{2}>0$ forces $V_{t}$ to have a
slope more negative than that of the $\overline{V_{t}}$’s.

Below, we also present some exact solutions, to obtain which it is convenient
to write

$\overline{V_{t}}(\phi)=-e^{a\sqrt{6\kappa}\phi}\frac{[F(\phi)-V_{A}]^{2}}{4F(\phi)}\
.$ (C.2)

Eq. (C.1) fixes $F(\phi)$ to be given implicitly by

$\frac{[(a-s)F(\phi)-(a+s)V_{A}]^{2a}}{[F(\phi)]^{a+s}}=M^{4(a+s)}e^{(1-a^{2})\sqrt{6\kappa}\phi}\
,$ (C.3)

where $s\equiv{\rm sign}(V_{A})$ and $M$ is an arbitrary integration constant
(with dimensions of mass). This relation can be solved for $F(\phi)$ for some
particular values of $a$, or used also to get the asymptotic behaviour of
$\overline{V_{t}}$ at $\phi\to\infty$. The results are as follows.

Type $0$ ($a<1$ and $V_{A}$ of either sign): The expansion of
$\overline{V_{t}}$ is

$\overline{V_{t}}=\overline{V}_{tA}e^{\sqrt{6\kappa}\phi}+\frac{V_{A}}{2(1-a)}e^{a\sqrt{6\kappa}\phi}+...$
(C.4)

Here, $\overline{V}_{tA}$ is the free parameter (which should be negative). An
explicit example of exact solution of this type can be obtained e.g. for
$a=1/2$, $V_{A}=s=\pm 1$ and $\kappa=1$, for which

$\overline{V_{t}}=-e^{\sqrt{3/2}\phi}\frac{(1-3sx+x^{2})^{2}}{4x(x-s)^{2}}\ ,$
(C.5)

with

$x\equiv\left[s+2P+2\sqrt{P(P+s)}\right]^{1/3}\ ,\quad P\equiv
M^{12}e^{(3/2)\sqrt{6}\phi}\ .$ (C.6)

The large $\phi$ expansion of this exact solution agrees with the general
expectation for this type.

Type $-$ ($V_{A}<0$ and $1/\sqrt{3}<a<1$): There are three qualitatively
different classes of $\overline{V_{t}}$. One is the exact solution

$\overline{V}_{t,c}=\frac{V_{A}}{1-a^{2}}e^{a\sqrt{6\kappa}\phi}\ .$ (C.7)

The second class is formed by $\overline{V_{t}}$’s lying above
$\overline{V}_{t,c}$, which do not extend to $\phi=\infty$ and intersect $V$
at some finite value of $\phi$. The third class is formed by
$\overline{V_{t}}$’s lying below $\overline{V}_{t,c}$, which do extend to
$\phi=\infty$ and have the expansion

$\overline{V_{t}}=V_{tA}e^{\sqrt{6\kappa}\phi}+\frac{V_{A}}{2(1-a)}e^{a\sqrt{6\kappa}\phi}+...$
(C.8)

where $V_{tA}<0$ is the free parameter. The asymptotic behaviour of
$\overline{V_{t}}$ in (C.8) is too steep for a type $-$ solution and it indeed
is exactly the same found above for type 0 solutions. For an exact example of
this class take $a=3/5$, $V_{A}=-1$ and $\kappa=1$. We get

$\overline{V_{t}}=-e^{\sqrt{3/5}\phi}\frac{(8+z+q)^{2}}{8(6+z+q)}\ ,$ (C.9)

with

$\displaystyle q$ $\displaystyle\equiv$ $\displaystyle\sqrt{12-z^{2}-16/z}\
,\quad z\equiv\sqrt{5-25P/x+5x/6}\ ,$ $\displaystyle P$ $\displaystyle\equiv$
$\displaystyle M^{16}e^{(8/5)\sqrt{6}\phi}\ ,\quad x\equiv
6\left[-P(1+\sqrt{1+P(5/6)^{3})})\right]^{1/3}\ ,$ (C.10)

and its large $\phi$ expansion gives (for $M\neq 0$)

$\overline{V_{t}}=\overline{V}_{tA}e^{\sqrt{6}\phi}-\frac{5}{4}e^{(3/5)\sqrt{6}\phi}+...$
(C.11)

which corresponds to the expansion in (C.8) for $a=3/5$. For $M\to 0$, one
gets instead

$\overline{V_{t}}=-\frac{25}{16}e^{3\sqrt{6}\phi/5}\ ,$ (C.12)

which is the exact case of (C.7) for $a=3/5$.

Type $+$ ($a>1$ and $V_{A}>0$): The expansion of $\overline{V_{t}}$ is

$\overline{V_{t}}=\frac{V_{A}}{1-a^{2}}e^{a\sqrt{6\kappa}\phi}+Be^{[(a+1/a)/2]\sqrt{6\kappa}\phi}-\frac{(1-a^{2})^{2}B^{2}}{4a^{2}V_{A}}e^{\sqrt{6\kappa}\phi/a}+...$
(C.13)

Here, $B$ is a free parameter [related to $M$ in (C.3)]. An explicit example
of exact solution of this type can be obtained e.g. for $a=3$, $V_{A}=1$ and
$\kappa=1$, for which

$\overline{V_{t}}=-e^{3\sqrt{6}\phi}\frac{[12x+6Px^{2}+P^{2}]^{2}}{48x(24x+6Px^{2}+P^{2})}\
,$ (C.14)

with

$P\equiv M^{8/3}e^{-(4/3)\sqrt{6}\phi}\ ,\quad
x\equiv\left[1+\sqrt{1-(P/6)^{3}}\right]^{1/3}\ .$ (C.15)

It can be checked that the expansion of this exact solution conforms to the
general expression in (C.13).

Special Case ($V_{A}>0$ and $a=1$): The expansion of type $+$ above is not
valid. However, this case can be solved exactly to get

$\overline{V_{t}}=-V_{A}e^{\sqrt{6\kappa}\phi}\frac{[W(e^{2\sqrt{6\kappa}(\phi-\phi_{C})})-1]^{2}}{4W(e^{2\sqrt{6\kappa}(\phi-\phi_{C})})}\
,$ (C.16)

where $\phi_{C}$ is arbitrary and $W(z)$ is Lambert’s (or product log)
function satisfying $We^{W}=z$. The large $\phi$ expansion of this solution
gives (setting $\kappa=1$)

$\overline{V_{t}}=-V_{A}e^{\sqrt{6}(\phi-\phi_{C})}\left\\{\sqrt{3/2}(\phi-\phi_{C})-\frac{1}{4}\left[2+\log(2\sqrt{6}(\phi-\phi_{C}))\right]+...\right\\}+...$
(C.17)

Type $-^{*}$ ($a>1$ and $V_{A}<0$): For this case there are no
$\overline{V_{t}}$ solutions extending to $\phi\to\infty$ (i.e. no cobordism
solutions). Indeed, from (C.1) and the fact that $V,\overline{V_{t}}<0$ at
large $\phi$, one has the inequality
$-\overline{V_{t}}^{\prime}<\sqrt{6\kappa}|\overline{V_{t}}|$ and this shows
that $\overline{V_{t}}$ cannot drop faster than
$\overline{V_{t}}\sim-e^{\sqrt{6\kappa}\phi}$. In other words, all
$\overline{V_{t}}$’s hit $V$ at some finite field value. In the context of
regular CdL bounces, this implies [32] that such potentials are necessarily
unstable against CdL decay. Nevertheless, the possible existence of BoNs is
still open, as is discussed below.

###  C.2 $V_{t}$ solutions

We can next derive the large $\phi$ expansion of the tunneling potentials
directly from their equation of motion. As this EoM is a second order
differential equation, we expect two arbitrary constants in the expansions of
$V_{t}$. In all types except type $-^{*}$, which is special, the $V_{t}$
expansions consist of the corresponding expansion for $\overline{V_{t}}$
discussed above (which already depends on one free parameter) plus additional
terms that depend on a second free parameter. This second free parameter gives
a positive $D^{2}$ (and setting it to zero one would recover
$\overline{V_{t}}$, which has $D^{2}=0$). We obtain the following results:

Type $0$ ($a<1$ and $V_{A}$ of either sign): The expansion of $V_{t}$ is

$V_{t}=V_{tA}e^{\sqrt{6\kappa}\phi}+\frac{V_{A}}{2(1-a)}e^{a\sqrt{6\kappa}\phi}+...+Be^{\sqrt{6\kappa}\phi/3}+...$
(C.18)

The two free parameters are $V_{tA}$ (which has to be negative) and $B$ (which
should be positive). Again we find that $V_{t}$ is given by $\overline{V_{t}}$
plus additional terms proportional to the free parameter B, which makes
$D(\phi)$ non zero.282828The intermediate ellipsis in (C.18), and similar
$V_{t}$ expansions below, stands for terms more relevant than the $B$ term
(the number of such terms depends on $a$). Calculating $D^{2}$ directly from
expression (2.2) requires to keep all these terms up to the $B$ one as they
are crucial for the cancellations that end up giving $D^{2}\propto B$. Indeed,
one has

$D(\phi)=\sqrt{-8V_{tA}B\kappa}\,e^{\sqrt{8\kappa/3}\phi}+...$ (C.19)

Special Case ($V_{A}>0$ and $a=1$): For this case, at the boundary between
types $+$ and 0, the expansion of $V_{t}$ takes the following form

$\displaystyle V_{t}$ $\displaystyle=$ $\displaystyle-
V_{A}e^{\sqrt{6\kappa}\phi}\left\\{\sqrt{\frac{3\kappa}{2}}\phi-\frac{1}{4}\left[2+\log(2\sqrt{6\kappa}(\phi-\phi_{c}))\right]+{\cal
O}\left(\frac{\log\phi}{\phi}\right)\right\\}$ (C.20) $\displaystyle+$
$\displaystyle\frac{A}{\phi^{1/3}}e^{\sqrt{6\kappa}\phi/3}\left[1+\frac{3+\log(2\sqrt{6\kappa}(\phi-\phi_{c}))}{6\sqrt{6\kappa}\phi}+{\cal
O}\left(\frac{\log^{2}\phi}{\phi^{2}}\right)\right]\ .$

The two free parameters are $\phi_{c}$ and $A$, with $V_{t}$ given by
$\overline{V_{t}}$ (first line) plus terms proportional to $A$ that make
$D(\phi)$ nonzero. One has

$D(\phi)=2\sqrt{\sqrt{6\kappa}\kappa
AV_{A}}\,\phi^{1/3}e^{\sqrt{8\kappa/3}\phi}+...$ (C.21)

Type $-$ ($V_{A}<0$ and $1/\sqrt{3}<a<1$): The expansion of $V_{t}$ is

$V_{t}=\frac{V_{A}}{1-a^{2}}e^{a\sqrt{6\kappa}\phi}+...+Ae^{\sqrt{6\kappa}\phi/(3a)}+...$
(C.22)

The only free parameter is $A$ (which has to be negative). In this case,
$V_{t}$ is given by the special $\overline{V}_{tc}$ of (C.7) plus terms
proportional to $A$ that make $D(\phi)$ nonzero

$D(\phi)=\sqrt{\frac{2(1+3a^{2})AV_{A}\kappa}{a^{2}-1}}\,e^{(3a+1/a)\sqrt{\kappa/6}\phi}+...$
(C.23)

Type $+$ ($a>1$ and $V_{A}>0$): The expansion of $V_{t}$ is

$V_{t}=\frac{V_{A}}{1-a^{2}}e^{a\sqrt{6\kappa}\phi}+Ae^{[(a+1/a)/2]\sqrt{6\kappa}\phi}+...+Be^{\sqrt{6\kappa}\phi/(3a)}+...$
(C.24)

The two free parameters are $A$ and $B$ (which should be positive). One
recognizes the expansion of type $+$ $\overline{V_{t}}$ solutions plus terms
proportional to the parameter $B$. One gets

$D(\phi)=\sqrt{\frac{2(1+3a^{2})BV_{A}\kappa}{a^{2}-1}}\,e^{(3a+1/a)\sqrt{\kappa/6}\phi}+...$
(C.25)

which indeed would give zero if $B=0$ (limit in which
$V_{t}\to\overline{V_{t}}$). We see that, for $a\to 1^{+}$, both $V_{t}$ and
$D$ above blow up.

Type $-^{*}$ ($a>1$ and $V_{A}<0$): The expansion of $V_{t}$ is

$V_{t}=\frac{3V_{A}}{2}e^{a\sqrt{6\kappa}\phi}+Ae^{[(1+a)/2]\sqrt{6\kappa}\phi}+Be^{\sqrt{6\kappa}\phi/3}+...$
(C.26)

The two free parameters are $A$ and $B$. In this case there was no
$\overline{V_{t}}$ extending to infinite field values and, therefore, $V_{t}$
does not take the standard form found in all the other types. One has

$D(\phi)=-V_{A}\sqrt{3\kappa/2}\,e^{a\sqrt{6\kappa}\phi}+...$ (C.27)

and indeed we see that $D(\phi)$ is nonzero even for zero values of the free
parameters $A$ and $B$.

##  Appendix D Potentials for $V_{t}(\phi)=-e^{a\sqrt{6\kappa}\phi}$

Instead of assuming that the potential follows a simple exponential, as done
in the previous appendix, it is also instructive to assume that $V_{t}$ is the
simple exponential

$V_{t}(\phi)=-e^{a\sqrt{6}\phi}\ ,$ (D.1)

(we take again $\kappa=1$ and assume $a>0$) and derive the corresponding
$V(\phi)$’s that would satisfy the $V_{t}$ equation of motion. This can be
done exactly following the technique explained in [32] and one gets

$V(\phi)=e^{\sqrt{6}\phi}\left[-1+\frac{a^{2}}{1+\frac{3a^{2}-1}{1+(3a^{2}-1)Ce^{(3a-1/a)\sqrt{2/3}\phi}}}\right]\
,$ (D.2)

where $C$ is an integration constant. (There is only one integration constant
as the EoM is a first order differential equation when solved for $V$). We
then get

$D(\phi)^{2}=\frac{6a^{2}(3a^{2}-1)e^{(6a+1/a)\sqrt{2/3}\phi}}{3a^{2}e^{\sqrt{2/3}\phi/a}+(3a^{2}-1)Ce^{\sqrt{6}a\phi}}\
.$ (D.3)

The asymptotics of $V_{t}(\phi)$ and $D(\phi)$ at $\phi\to\infty$ depends on
the value of $a$ as follows.

Case $a<1/\sqrt{3}$: In this case we get

$V(\phi)\simeq-\frac{2}{3}e^{\sqrt{6}\phi}+...$ (D.4)

which is a type $-^{*}$ example but leads to

$D(\phi)^{2}=-2(1-3a^{2})e^{2\sqrt{6}\phi}+...$ (D.5)

which is negative, so that this case is not acceptable.

Case $a>1/\sqrt{3}$: In this case we get

$V(\phi)\simeq(a^{2}-1)e^{\sqrt{6}\phi}-\frac{a^{2}}{C}e^{[1-a+1/(3a)]\sqrt{6}\phi}...$
(D.6)

which leads to

$D(\phi)^{2}=\frac{6a^{2}}{C}e^{2(3a+1/a)\phi/\sqrt{6}}+...$ (D.7)

So, for $a>1$ this is of type $+$ while for $1/\sqrt{3}<a<1$ it is of type
$-$.

Special Case $a=1$: In this case we get

$V(\phi)\simeq-\frac{1}{C}e^{\sqrt{6}\phi/3}...$ (D.8)

which leads to

$D(\phi)^{2}=\frac{6}{C}e^{2\sqrt{8/3}\phi}+...$ (D.9)

This corresponds to type $0$.

Special Case $a=1/\sqrt{3}$: In this case, the result (D.2) does not apply. We
get instead

$V(\phi)=(C-\sqrt{2}\phi)e^{-\sqrt{2}\phi}\ .$ (D.10)

which leads to

$D(\phi)^{2}=-4e^{2\sqrt{2}\phi}+6\sqrt{2}\phi-6C\ .$ (D.11)

As this is negative at $\phi\to\infty$, this case is not acceptable.

##  Appendix E Tunneling Potential for Constant $V(\phi)$

In this appendix we calculate the BoN tunneling potential in a region of the
potential that is constant, $V(\phi)=V_{\infty}$, in some region extending to
$\phi=\infty$, to be matched eventually to a different potential (e.g.
containing a minimum) valid in a different field range. That is, the $V_{t}$
solution below should be considered just as a part of a complete BoN solution.
We consider here the case with $V_{\infty}>0$, but $V_{\infty}<0$ can be
treated in the same way.

Solving the $V_{t}$ EoM (4.1) for constant $V$ directly is too hard and we
follow a different route. An alternative formulation of the EoM for $V_{t}$ is
possible rewriting it as a first order differential equation for $D$ [defined
in (2.2)] as in (2.4):

$\frac{D^{\prime}}{D}=\frac{3V^{\prime}-4V_{t}^{\prime}}{6(V-V_{t})}\ .$ (E.1)

For a constant $V=V_{\infty}$ this can be integrated to give

$D(\phi)=C\sqrt{6\kappa}V_{\infty}^{1/3}[V_{\infty}-V_{t}(\phi)]^{2/3}\ ,$
(E.2)

with $C$ a dimensionless integration constant (normalized with
$\sqrt{6\kappa}V_{\infty}^{1/3}$ for convenience later on).292929We know from
the general asymptotic behaviour discussed in subsection 4.1 that
$V_{t}(\phi\to\infty)\sim-e^{\sqrt{6\kappa}\phi}$. Plugging this in the
definition of $D$ shows there is a cancellation of the leading exponential
term. Equation (E.2) shows that $D(\phi\to\infty)\sim
e^{\sqrt{8\kappa/3}\phi}$, as expected for a type 0 BoN. Next we use the
definition of $D^{2}$ in (2.2) to get a differential equation for
$V_{t}^{\prime}$:

$(V_{t}^{\prime})^{2}=6\kappa\left[C^{2}V_{\infty}^{2/3}(V_{\infty}-V_{t})^{1/3}-V_{t}\right](V_{\infty}-V_{t})\
.$ (E.3)

Before showing how to deal with this equation, we can deduce some general
properties of the solutions $V_{t}(\phi)$. First, if $V_{t}(\phi)$ is a
solution of (E.3), other related solutions are $V_{t}(\phi+c)$ (given the
shift invariance of the potential) and $V_{t}(-\phi)$ (given that
$V_{t}^{\prime}$ appears quadratically). Also, the right hand side of (E.3),
restricted to positive values, is a monotonic function of $V_{t}$ for
$V_{t}\leq V_{\infty}$ that grows when $V_{t}\to-\infty$. These properties
allow us to take $V_{t}(\phi)$ symmetric around $\phi=0$, with
$V_{t}^{\prime}(\phi<0)>0$ and $V_{t}^{\prime}(\phi>0)<0$, with $\phi=0$
corresponding to a maximum with $V_{t}^{\prime}(0)=0$. Figure 20 shows how
such $V_{t}(\phi)$ looks. Setting $V_{t}^{\prime}=0$ in (E.3) we find that $C$
is related to $V_{t}(0)\equiv V_{T}$ by

$C^{2}=\frac{V_{T}/V_{\infty}}{(1-V_{T}/V_{\infty})^{1/3}}\ .$ (E.4)

This also shows that the maximum of $V_{t}$ occurs at $V_{T}\geq 0$.

We can then try to solve (E.3) for $\phi>0$ and this can be done analytically
for the inverse function $\phi(V_{t})$, which is as good as solving for
$V_{t}(\phi)$. Introducing the variable

$z\equiv(1-V_{t}/V_{\infty})^{1/3}\ ,$ (E.5)

we recast (E.3) as

$\kappa\left(\frac{d\phi(z)}{dz}\right)^{2}=\frac{3z}{2(z^{3}+C^{2}z-1)}\ ,$
(E.6)

which can be integrated in terms of elliptic functions. The polynomial in the
denominator has a real root, and two complex ones:

$z_{T}\equiv(1-V_{T}/V_{\infty})^{1/3}\ ,\quad\quad z_{\pm}\equiv\left(-1/2\pm
i\sqrt{1/z_{T}^{3}-1/4}\right)z_{T}\ ,$ (E.7)

with $0<z_{T}\leq 1$. In terms of $z_{T}$ we have $C^{2}=1/z_{T}-z_{T}^{2}$.

The solution of (E.6) can be written as

$\displaystyle\phi(V_{t})$ $\displaystyle=$
$\displaystyle\left.\pm\sqrt{\frac{6}{(2r_{+}+1)\kappa}}\right\\{F(\alpha(r)|z_{k})\,+\,(r_{+}-1)\,\Pi(r_{+};\alpha(r)|z_{k})\,+\,iK(1-z_{k})$
(E.8)
$\displaystyle\left.+i(r_{+}+2)\left[\Pi(1-z_{k}/r_{+}|1-z_{k})\,-\,K(1-z_{k})\right]\frac{}{}\right\\}\
,$

where we have adjusted the constant shift so that $\phi(V_{T})=0$ and $r\equiv
z/z_{T}$, $r_{\pm}\equiv z_{\pm}/z_{T}$, with

$z_{k}\equiv\frac{r_{+}(r_{+}+2)}{2r_{+}+1}\
,\quad\alpha(r)\equiv\arcsin\sqrt{\frac{1-r/r_{+}}{1-r}}\ ,$ (E.9)

and $F(a|m)$ and $\Pi(n;a|m)$ are the incomplete elliptic integrals of the
first and third kind, respectively, while $K(a)$ and $\Pi(a|m)$ are the
corresponding complete elliptic integrals.

Figure 20: Tunneling potentials $V_{t}(\phi)$ [the inverse of eq. (E.8)], for
a constant potential, $V(\phi)=V_{\infty}$, for different values of
$r_{0}\equiv V_{t}(0)/V_{\infty}$ as indicated by the labels.

Figure 20 shows $V_{t}$ for $V_{\infty}=1$ and several values of $r_{0}\equiv
V_{T}/V_{\infty}$. The limiting case $r_{0}=0$ admits a simple expression and
is given by

$V_{t}(\phi)=-V_{\infty}\sinh^{2}\sqrt{\frac{3\kappa\phi^{2}}{2}}\ .$ (E.10)

It can be checked that this case leads to $D=0$ so that it can at most be part
of an end-of-the-world brane rather than a bubble of nothing.

For $r_{0}=1-\delta^{3}$, with $0<\delta\ll 1$ (so that $z_{T}=\delta$), one
gets the approximation

$V_{t}(\phi)\simeq
V_{\infty}\left[1-\delta^{3}-\frac{1}{\delta^{3/2}}\sinh^{3/2}\left(\sqrt{\frac{2\kappa\phi^{2}}{3}}\phi\right)\right]\
.$ (E.11)

This corresponds to the small $R_{KK}$ limit of a Witten-like BoN.

Let us assume then that $V=V_{\infty}$ for $\phi\in(\phi_{m},\infty)$, while
$V$ has some non trivial structure (a dS minimum, a barrier, etc.) for
$\phi<\phi_{m}$ [call it $V_{L}(\phi)$]. Then the complete $V_{t}$ consist of
two parts, one for $\phi<\phi_{m}$ (call it $V_{tL}$) and the solution found
in this appendix for $\phi>\phi_{m}$ (call it $V_{tR}$), and both should be
matched at $\phi_{m}$ to get a continuous $V_{t}$ and $V^{\prime}_{t}$. If
$V_{L}$ and $V_{tL}$ are known for $\phi<\phi_{m}$, the solutions for
$\phi>\phi_{m}$ are matched as follows. First,
$V_{R}(\phi>\phi_{m})=V_{\infty}=V_{L}(\phi_{m})$. Calling $V_{tm}\equiv
V_{tL}(\phi_{m})$ and $V^{\prime}_{tm}\equiv V^{\prime}_{tL}(\phi_{m})$,
equation (E.3) fixes the constant $C^{2}$ as

$C^{2}=\frac{(V^{\prime}_{tm})^{2}+6\kappa(V_{\infty}-V_{tm})V_{tm}}{6\kappa
V_{\infty}^{2/3}(V_{\infty}-V_{tm})^{2/3}}\ ,$ (E.12)

to ensure $V^{\prime}_{tR}(\phi_{m})=V^{\prime}_{tm}$. Once $C^{2}$ is known,
$z_{T}$ can be determined, fixing the solution $\phi(V_{tR})$ in (E.8) up to
an overall shift. To also match $V_{tR}(\phi_{m})=V_{tm}$ we simply solve for
$\delta\phi_{m}$ in $\phi_{m}=\phi(V_{tm})+\delta\phi_{m}$. This procedure is
carried out for the example of subsection 7.2.

The tunneling action in the range $(\phi_{m},\infty)$, where the solution for
$V_{t}$ found in this appendix is assumed to hold, can be calculated
analytically. In order to do this we change the integration variable in (2.1)
from $\phi$ to $V_{t}$ (or, equivalently $z$). In this way one gets

$\Delta
S=\frac{6\pi^{2}}{\kappa^{2}}\int_{V_{tm}}^{-\infty}\frac{(D+V_{t}^{\prime})^{2}}{DV_{t}^{2}V_{t}^{\prime}}dV_{t}=\frac{-18\pi^{2}}{\kappa^{2}V_{\infty}}\int_{z_{m}}^{\infty}\frac{[D(z)+V_{t}^{\prime}(z)]^{2}z^{2}}{D(z)(1-z^{3})^{2}V^{\prime}_{t}(z)}dz\
,$ (E.13)

where $z_{m}\equiv(1-V_{tm}/V_{\infty})^{1/3}$, and

$D(z)\equiv C\sqrt{6\kappa}V_{\infty}z^{2}\ ,\quad V_{t}^{\prime}(z)\equiv-
V_{\infty}z^{3/2}\sqrt{6\kappa[C^{2}z-1+z^{3}]}\ .$ (E.14)

Performing the integral, we get

$\displaystyle\Delta S(z_{m})$ $\displaystyle=$
$\displaystyle\frac{12\pi^{2}}{V_{\infty}\kappa^{2}\sqrt{z_{T}^{-3}-1}}\left\\{1+\frac{\sqrt{z_{T}^{-3}-1}}{1-z_{m}^{3}}-\frac{(z_{T}^{-1}-z_{m}^{2})\sqrt{z_{m}(z_{m}^{3}+C^{2}z_{m}-1)}}{(z_{m}-z_{T})(1-z_{m}^{3})}\right.$
(E.15) $\displaystyle+$
$\displaystyle\left.\frac{(1-r_{+})\left[E(\gamma(\infty)|\psi)-E(\gamma(z_{m})|\psi)\right]+F(\gamma(z_{m})|\psi))-F(\gamma(\infty)|\psi)}{\sqrt{-1-2z_{T}^{2}z_{+}}}\right\\},\quad\quad$

where

$\gamma(z)\equiv\arcsin\sqrt{\frac{r(1-r_{+})}{r_{+}(1-r)}}\
,\quad\psi\equiv\frac{1-z_{+}z_{T}^{2}}{1-z_{-}z_{T}^{2}}\ ,$ (E.16)

and $E(a|m)$ is the elliptic integral of the second kind.

##  Appendix F More Families of Analytic Potentials

In this appendix we give the general potential solution for some simple
families of BoN tunneling potentials. For each simple $V_{t}$, following the
general technique presented in [29] we write

$V(\phi)=V_{t}(\phi)+\frac{[V_{t}^{\prime}(\phi)]^{2}}{6\kappa[1/F(\phi)-V_{t}(\phi)]}\
,$ (F.1)

and solve

$V_{t}^{\prime}F^{\prime}=2\kappa(1-FV_{t})\ ,$ (F.2)

for $F(\phi)$. We also have

$D(\phi)^{2}=\frac{V_{t}^{\prime}{}^{2}}{1-V_{t}F}\ .$ (F.3)

###  F.1 $V_{t}(\phi)=V_{T}-\cosh(a\sqrt{6}\phi)$

Inspired by the constant $V$ solution discussed in the previous appendix, we
take the first family of solutions to be generated by

$V_{t}(\phi)=V_{T}-\cosh(a\sqrt{6}\phi)\ .$ (F.4)

We take $\kappa=1$, $a>0$ and $V_{T}>1$. Solving (F.2) we get

$F(\phi)=\frac{1}{V_{T}+1}\,{}_{2}F_{1}(1,p_{+}-p_{-},1+p_{+};\cosh^{2}(a\sqrt{3/2}\phi))+C\frac{\left[\sinh(a\sqrt{3/2}\phi)\right]^{2p_{-}}}{\left[\cosh(a\sqrt{3/2}\phi)\right]^{2p_{+}}}\
,$ (F.5)

where $C$ is an integration constant and

$p_{\pm}\equiv\frac{V_{T}\pm 1}{6a^{2}}\ .$ (F.6)

The asymptotic behaviour of $F(\phi)$ at $\phi\to\infty$ is

$F(\phi)\simeq
4^{1/(3a^{2})}(C-C_{s})\exp\left(-\frac{\phi}{a}\sqrt{\frac{2}{3}}\right)+\frac{2}{(3a^{2}-1)}\exp\left(-a\sqrt{6}\phi\right)+...\
,$ (F.7)

where

$C_{s}=(-1)^{1-p_{+}+p_{-}}\frac{\Gamma(1+p_{-}-p_{+})\Gamma(p_{+})}{(V_{T}-1)\Gamma(p_{-})}\
,$ (F.8)

which is real and finite only if $1-p_{+}+p_{-}=1-1/(3a^{2})$ is a positive
integer, which cannot be accomplished.

To get a real $F(\phi)$ we therefore need to take $C=C_{s}$, and then
$1/F(\phi)\simeq[(3a^{2}-1)/2]\exp(a\sqrt{6}\phi)$. Using this in (F.1), we
get a type $-^{*}$ case with $V(\phi)\simeq-(1/3)\exp(a\sqrt{6}\phi)$. On the
other hand, (F.3) gives $D(\phi)^{2}\simeq[(3a^{2}-1)/2]\exp(2a\sqrt{6}\phi)$
and we see that $a>1/\sqrt{3}$ is needed to have $D(\phi)^{2}>0$.

The previous solution can be used in some field interval, as done in the text
with other analytic solutions. If that interval includes $\phi=\phi_{T}=0$,
point at which $V_{t}$ reaches its maximum, then to satisfy $V(0)>V_{t}(0)$ we
need to set $C$ to a particular value, $C_{T}$, which can be found by
inspecting the expansion of $F(\phi)$ around $\phi_{T}$ (see [29] for
details). We get

$C_{T}=(-1)^{p_{-}}\frac{\Gamma(p_{+})\Gamma(1-p_{-})}{(V_{T}-1)\Gamma(p_{+}-p_{-})}\
.$ (F.9)

In general $C_{T}\neq C_{s}$ but for the particular choice $V_{T}=6a^{2}n-1$
(so that $p_{+}=n$), with $n$ an integer, one has $C_{T}=C_{s}$ (and therefore
one gets a working type $-^{*}$ example).

###  F.2 $V_{t}(\phi)=-Ae^{\sqrt{6}\phi}+Be^{\sqrt{2/3}\phi}$

For this family of examples we take $\kappa=1$ and

$V_{t}(\phi)=-Ae^{\sqrt{6}\phi}+Be^{\sqrt{2/3}\phi}\ ,$ (F.10)

with $A>0,B>0$ so that $V_{t}$ has a maximum at
$\phi_{T}=\sqrt{3/2}\log\sqrt{B/(3A)}$. As in previous subsections, we
consider these examples as valid in some range $(\phi_{m},\infty)$, where
$V_{t}(\phi_{m})=V(\phi_{m})$. In order to construct the complete $V$ and
$V_{t}$, for some of the following examples, it might be necessary to match
these functions to other solutions valid for $\phi<\phi_{m}$.

From the tunneling potential $V_{t}$ in (F.10), following the procedure
mentioned at the beginning of the appendix, one gets

$V(\phi)=\frac{6A(4BC-1)e^{\sqrt{6}\phi}-8B(2BC-1)e^{\sqrt{2/3}\phi}}{9\left(2ACe^{\sqrt{8/3}\phi}+1-2BC\right)}\
,$ (F.11)

where $C$ is an integration constant. The large field expansion gives

$V(\phi\to\infty)=\frac{4BC-1}{3C}e^{\sqrt{2/3}\phi}+{\cal
O}(e^{-\sqrt{2/3}\phi})$ (F.12)

(which is valid for $C\neq 0$) so that $V$ is indeed subleading in the limit
$\phi\to\infty$, and of type $0$. As we show below, the case $C=0$ is special
and gives a type $-^{*}$ case.

Depending on the value of $C$ relative to $C_{*}\equiv 1/(4B)$ we can get

$V(\phi\to\infty)\to\left\\{\begin{array}[]{cl}+\infty&\mathrm{for}\quad
C>C_{*}\quad\mathrm{or}\quad C<0\ ,\\\ 0&\mathrm{for}\quad C=C_{*}\ ,\\\
-\infty&\mathrm{for}\quad 0\leq C<C_{*}\ .\end{array}\right.$ (F.13)

Figure 21: Potentials and tunneling potential, $V$ and $V_{t}$, for the
analytic examples of subsection F.2, for the indicated values of the
integration constant $C$.

We can illustrate the above cases with some numerical examples, taking
$A=B=1$, and the indicated values of $C$ to get the following simple
potentials

$\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle\frac{8}{9}e^{\sqrt{2/3}\phi}-\frac{2}{3}e^{\sqrt{6}\phi}\
,\quad(C=0)$ (F.14) $\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle\frac{4}{9}\mathrm{sech}(\sqrt{2/3}\phi)\ ,\quad(C=C_{*}=1/4)$
(F.15) $\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle\frac{2}{3}e^{\sqrt{2/3}\phi}\ ,\quad(C=1/2)$ (F.16)
$\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle\frac{8}{9}e^{\sqrt{2/3}\phi}\ ,\quad(C=3/4)\ .$ (F.17)

These solutions are plotted, together with $V_{t}$, in figure 21. We see that
the $C=0$ case corresponds to the asymptotic behaviour of a type $-^{*}$ case,
while the rest of cases are of type $0$. The last one has been already
presented in subsection 7.1.

For generic choices of $C$ outside the interval $(0,2C_{*})$ the solutions
above develop singularities at some $\phi_{s}$ and one should choose
$\phi_{m}>\phi_{s}$. The BoN action for the range $(\phi_{m},\infty)$ can be
obtained analytically but we do not give it here as it is not particularly
illuminating.

To end this appendix, we use input from the higher dimensional theory to gain
information on the range of parameters of the previous solutions which is of
physical interest. From the discussion in section 8, it follows that the
$C\neq 0$ instanton describes a BoN mediating the decay of a KK circle
compactification. In this simple model, the parameters $A$ and $C$ can be
related directly to the relevant scales of the BoN solution. First, the
regularity conditions on the BoN surface ($\phi\to\infty$), link $A$ to the KK
radius, $R_{KK}$, as (8.14) implies that
$V_{t}(\phi\to\infty)\simeq-3/(4R_{KK}^{2})e^{\sqrt{6}\phi}$, and therefore we
must set

$A=\frac{3}{4R_{KK}^{2}}\ .$ (F.18)

Moreover, the constant $C$ can be related to the BoN nucleation radius,
$\mathcal{R}$, as (8.10) implies that near the BoN core the metric function
$\rho$ behaves as

$\rho(\phi\to\infty)\simeq\mathcal{R}e^{-\sqrt{\kappa/6}\phi}\ .$ (F.19)

On the other hand, the asymptotic behaviour of the function $\rho$ can be
obtained from $V(\phi)$ and $V_{t}(\phi)$ taking the limit $\phi\to\infty$ in
(2.15). Comparing the resulting expression with (F.19) we find

$C=\frac{\mathcal{R}^{2}}{9}>0\ .$ (F.20)

###  F.3  $V_{t}=Ae^{\phi}-e^{b\phi}$

Consider now a more general family of examples with

$V_{t}=Ae^{\phi}-e^{b\phi}\ ,$ (F.21)

taking $A,b>0$ and $\kappa$ free.303030$\kappa$ can eventually be set to 1 by
rescaling the field. The $a$ parameter used in the general classification of
section 4.1, with $\kappa$ rescaled to 1, is then given by
$a=b/\sqrt{6\kappa}$. This family of tunneling potentials was already
discussed in [29] to find CdL solutions, but we use it now to find BoN
solutions instead. For this $V_{t}$, (F.2) is solved by

$F(\phi)=\frac{r}{q(b-1)}\left[\left(\frac{b}{A}\right)^{q}\left(e^{-\phi}-\frac{A}{b}e^{-b\phi}\right)^{r}C+e^{-b\phi}{}_{2}F_{1}\left(1,p;1+q;(A/b)e^{(1-b)\phi}\right)\right]\
,$ (F.22)

where $C$ is an integration constant and

$r\equiv 2\kappa/b\ ,\quad q\equiv 1+p/b\ ,\quad p\equiv(b-2\kappa)/(b-1)\ .$
(F.23)

The tunneling potential $V_{t}$ has a maximum at $\phi_{tT}=\log(A/b)/(b-1)$
and, imposing $V>V_{t}$ at $\phi_{tT}$ fixes the integration constant to be
[29]

$C=\frac{\pi\csc(\pi r)}{B(p,1+r)}\ ,$ (F.24)

where $B(x,y)$ is the Euler beta function.

The different possible asymptotic behaviours of $F$ and thus of $V$ when
$\phi\to\infty$ can be studied generically depending on the parameter ranges.
Instead of exploring that, we show how a judicious choice of the constants
appearing in this general solution leads to a more or less simple expression
for $V$, illustrating all the different types of solutions. In all the cases,
we choose the integration constant that appears in $V$ so that
$V_{t}(\phi_{tT})<V(\phi_{tT})$. This allows to give $V$ in a complete
interval from its intersection with $V_{t}$ all the way to $\phi=\infty$ so
that we can calculate the CdL part of the BoN tunneling action.

Many of the particular examples given in Subsection 7 belong to this family.
For instance, the example in subsection 7.5 is a case with $b=2$ and
$\kappa=3/2$. The expressions given in that subsection correspond to $\kappa$
being rescaled to 1, with $a=2/3$. Different choices and field rescalings
generate the examples of Subsections 7.1 and 7.6. Other examples are the
following.

A type $0$ example can be obtained for $b=3/2$ and $\kappa=3/8$ (or $a=1$,
$\kappa=1$) for which we get

$V(\phi)=Ae^{\phi}-e^{3\phi/2}+\frac{(2A/3-e^{\phi/2})^{2}e^{\phi}}{e^{\phi/2}-A+A^{2}/(9e^{\phi/2}-3A)}\
.$ (F.25)

It can be checked that $V(\phi\to\infty)\sim(2A/3)e^{\phi}$, which corresponds
to a type $0$ example.

A type $-$ example is obtained for $b=2$ and $\kappa=3/4$ (or $a=2\sqrt{2}/3$
and $\kappa=1$) for which

$V=Ae^{\phi}-e^{2\phi}+\frac{2(A-2e^{\phi})^{2}e^{\phi}/3}{3e^{\phi}-3A+\frac{5A^{5/4}e^{\phi}}{2(2e^{\phi}-A)^{3/4}e^{\phi/2}\sqrt{\pi}\Gamma(9/4)/\Gamma(7/4)+A^{5/4}{}_{2}F_{1}(1/2,1;9/4;Ae^{-\phi}/2)}}\
.$ (F.26)

It can be checked that $V(\phi\to\infty)\sim-e^{2\phi}/9$, which indeed
corresponds to a type $-$ example. In this case there is a Hawking-Moss
instanton that can mediate vacuum decay. A numerical comparison shows that the
BoN action is lower than the HM one, so that vacuum decay proceeds
preferentially via a BoN.

We end up giving one further example of type $-^{*}$. We take $b=3/2$ and
$\kappa=1$ (so that $a=\sqrt{3/8}<1$). The potential is quite simple:

$V=\frac{5A}{6}e^{\phi}-\frac{2}{3}e^{3\phi/2}\ .$ (F.27)

The HM and BoN actions313131As discussed before, this part of the action
corresponds to the field interval from the contact point
$\phi_{0{\scriptscriptstyle+}}$ of $V$ and $V_{t}$ to $\phi=\infty$. There is
an additional contribution from the false vacuum $\phi_{\scriptscriptstyle+}$
to $\phi_{0{\scriptscriptstyle+}}$ that is common to both decays. can be
computed analytically as

$S_{\rm BoN}=\frac{64\pi^{2}}{A^{3}}\ ,\quad S_{\rm HM}=\frac{132}{125}S_{\rm
BoN}\ ,$ (F.28)

so that $S_{\rm HM}>S_{\rm BoN}$ and BoN again dominates decay over HM.

##  Appendix G Parametric Dependences of the Tunneling Action

In the text we have often found families of tunneling potentials, parametrized
by some free parameter, $p$, related to the behaviour of $V_{t}(\phi)$ near
the false vacuum. This parameter was $p=A$ for AdS/Minkowski vacua, with $A$
appearing in the low-field expansion of $V_{t}$ as in (5.2) and (5.5), and
$p=\phi_{i}$ for dS vacua, with $\phi_{i}$ being the field value at which
$V_{t}$ starts to depart from $V$. In this appendix we compute how the
tunneling action depends on such parameters, that is, we calculate $dS/dA$ or
$dS/d\phi_{i}$, although the formulas (G.4) and (G.7) are of more general
validity and apply for any parameter appearing in $V_{t}$ (and not in $V$).

Let us write the tunneling action in general as

$S[V_{t}]=\frac{6\pi^{2}}{\kappa^{2}}\int_{\phi_{\scriptscriptstyle+}}^{\phi_{e}}\frac{(D+V_{t}^{\prime})^{2}}{DV_{t}^{2}}d\phi=\int_{\phi_{\scriptscriptstyle+}}^{\phi_{e}}s(V,V_{t},V_{t}^{\prime})\
d\phi.$ (G.1)

As we have seen, for dS decay $\phi_{i}\neq\phi_{\scriptscriptstyle+}$ and the
action integral has a HM-like contribution from the interval
$(\phi_{\scriptscriptstyle+},\phi_{i})$ (in which $V_{t}=V$) and a CdL-like
contribution from the interval $(\phi_{i},\phi_{e})$ (in which $V_{t}\neq V$).
Concerning $\phi_{e}$, for pseudo-bounces or CdL bounces, $\phi_{e}$ takes
some finite value while for BoNs we have $\phi_{e}=\infty$.

Consider a family of tunneling potential solutions, $V_{t}(p;\phi)$, that
depend on some parameter $p$ which does not appear in $V(\phi)$. Let us
discuss first AdS/Minkowski vacua. We find

$\frac{dS}{dp}=s\left.\frac{d\phi}{dp}\right|_{\phi_{\scriptscriptstyle+}}^{\phi_{e}}+\int_{\phi_{\scriptscriptstyle+}}^{\phi_{e}}\left(\frac{\partial
s}{\partial V_{t}}\frac{dV_{t}}{dp}+\frac{\partial s}{\partial
V^{\prime}_{t}}\frac{dV^{\prime}_{t}}{dp}\right)d\phi\ .$ (G.2)

Using $dV^{\prime}_{t}/dp=d(dV_{t}/dp)/d\phi$ and integrating by parts, we get

$\frac{dS}{dp}=\left(s\left.\frac{d\phi}{dp}+\frac{\partial s}{\partial
V^{\prime}_{t}}\frac{dV_{t}}{dp}\right)\right|_{\phi_{\scriptscriptstyle+}}^{\phi_{e}}+\int_{\phi_{\scriptscriptstyle+}}^{\phi_{e}}\left[\frac{\partial
s}{\partial V_{t}}-\frac{d}{d\phi}\left(\frac{\partial s}{\partial
V^{\prime}_{t}}\right)\right]\frac{dV_{t}}{dp}d\phi\ .$ (G.3)

The integral above vanishes due to the EoM for $V_{t}$ and, using the explicit
form of the action density $s$ from (G.1), we get

$\frac{dS}{dp}=B(p;\phi_{e})-B(p;\phi_{\scriptscriptstyle+})$ (G.4)

with

$B(p;\phi)\equiv\frac{6\pi^{2}}{\kappa^{2}}\frac{(D+V_{t}^{\prime})^{2}}{D^{3}V_{t}^{2}}\left[D^{2}\frac{d\phi}{dp}+(2D-V_{t}^{\prime})\frac{dV_{t}}{dp}\right]\
.$ (G.5)

Although this formula is more general, in the text we apply it to the
parameter $p=A$ describing families of $V_{t}$ solutions.

Let us discuss next dS vacua. We start from

$S[V_{t}]=\frac{6\pi^{2}}{\kappa^{2}}\int_{\phi_{\scriptscriptstyle+}}^{\phi_{e}}\frac{(D+V_{t}^{\prime})^{2}}{DV_{t}^{2}}d\phi=-\left.\frac{24\pi^{2}}{\kappa^{2}V}\right|_{\phi_{\scriptscriptstyle+}}^{\phi_{i}}+\int_{\phi_{i}}^{\phi_{e}}s(V,V_{t},V_{t}^{\prime})\
d\phi,$ (G.6)

where the first term comes from the HM-like part of the dS tunneling action.
Following the same procedure as before we get

$\frac{dS}{dp}=B(p;\phi_{e})-B(p;\phi_{i})+\frac{24\pi^{2}}{\kappa^{2}}\frac{V^{\prime}}{V^{2}}\left.\frac{d\phi}{dp}\right|_{\phi_{\scriptscriptstyle+}}^{\phi_{i}}\
.$ (G.7)

As before, this formula is general but in the text we apply it to the
parameter $p=\phi_{i}$ describing families of $V_{t}$ solutions. In the
expressions above, $d\phi_{\scriptscriptstyle+}/dp=0$ and, for
$\phi_{e}=\infty$, $d\phi_{e}/dp=0$.

The terms $B(A,\phi_{\scriptscriptstyle+})$ and $B(\phi_{i},\phi_{i})$ only
depend on the expansion of $V_{t}$ near $\phi_{\scriptscriptstyle+}$ and
$\phi_{i}$ respectively. Therefore, they only depend on the type of false
vacua considered and not on the type of instanton driving the decay (pseudo-
bounce, CdL or BoN). Before discussing these different types of decay let us
then examine first these “false vacuum” terms, using the low field expansions
derived in section 5 to get the following results.

For Minkowski false vacua, as $\phi\to 0$, $V\sim\phi^{2}$,
$V_{t}\sim\phi^{2}/\log\phi$, $V_{t}^{\prime}\sim\phi/\log\phi$, and
$dV_{t}/dA\sim\phi^{2}/\log^{2}\phi$ while $D\sim\phi/\log\phi$ and
$D+V_{t}^{\prime}\sim\phi^{3}$. Using these asymptotics we find
$B(A,\phi_{\scriptscriptstyle+})=0$ in this case.

For AdS false vacua, we have, for $\phi\to 0$, $V,V_{t}\sim
V_{\scriptscriptstyle+}$, $V_{t}^{\prime}\sim\phi$, and
$dV_{t}/dA\sim\phi^{2+\alpha}$, with $\alpha=2\kappa
V_{\scriptscriptstyle+}/m_{t}^{2}$, with $m_{t}^{2}=(3\kappa
V_{\scriptscriptstyle+}/2)\sqrt{1-4m^{2}/(3\kappa
V_{\scriptscriptstyle+})}<0$, so that $0<\alpha<4/3$. We also have
$D\sim\phi^{1+\alpha/2}$. Using these asymptotics we find that
$B(A,\phi)\sim\phi^{2-\alpha/2}$ and therefore
$B(A,\phi_{\scriptscriptstyle+})=0$ also in this case.

Finally, for dS decays, $p=\phi_{i}\neq\phi_{\scriptscriptstyle+}$, so that
$d\phi_{i}/dp=1$. We also have, for $\phi\to\phi_{i}$, $V=V_{t}=V_{i}\equiv
V(\phi_{i})$, $V_{t}^{\prime}=D=3V_{i}^{\prime}/4\equiv
3V^{\prime}(\phi_{i})/4$, and, taking a derivative of the boundary condition
$V_{t}(p=\phi_{i};\phi_{i})=V(\phi_{i})$ with respect to $\phi_{i}$ we find
$dV_{t}/dp|_{i}=V_{i}^{\prime}/4$. Using these results we find

$B(\phi_{i};\phi_{i})=\frac{24\pi^{2}V_{i}^{\prime}}{\kappa^{2}V_{i}^{2}}\ ,$
(G.8)

which cancels with the last term in (G.7).

We conclude that

$\frac{dS}{dp}=B(p;\phi_{e})\ ,$ (G.9)

for any type of vacua. As $B(p;\phi_{e})$ depends on the type of instanton
considered, we discuss separately the different cases below.

###  G.1 Pseudo-Bounces

For pseudo-bounces, the boundary conditions at $\phi_{e}$ are
$V_{t}(\phi_{e})=V(\phi_{e})\equiv V_{e}$, $V_{t}^{\prime}(\phi_{e})=0$, which
imply $D(\phi_{e})=0$. In particular, pseudo-bounces are characterized [34] by
$\rho(\phi\to\phi_{e})\equiv\rho_{e}\neq 0$ and, using
$\rho=3\sqrt{2(V-V_{t})}/D$, we deduce that $D$ and $V_{t}^{\prime}$ go to
zero as $\sqrt{V-V_{t}}$ when $\phi\to\phi_{e}$. More precisely,

$D\sim\frac{3\sqrt{2(V-V_{t})}}{\rho_{e}}\ ,\quad
V_{t}^{\prime}\sim-\frac{3}{\rho_{e}}\sqrt{2(V-V_{t})}\sqrt{1-\frac{\kappa
V_{e}\rho_{e}^{2}}{3}}\ .$ (G.10)

Using the results above in (G.9) we get

$\frac{dS_{PS}}{dp}=\frac{6\pi^{2}}{\kappa^{2}V_{e}^{2}}\left(1-\sqrt{1-\frac{\kappa
V_{e}\rho_{e}^{2}}{3}}\right)^{2}\left(2+\sqrt{1-\frac{\kappa
V_{e}\rho_{e}^{2}}{3}}\right)\left.\frac{dV_{t}}{dp}\right|_{\phi_{e}}\ .$
(G.11)

Consider first the Minkowski/AdS case, with $p=A$. As we have seen in the
text, as $A$ increases, $\phi_{e}$ gets larger and $V_{t}(\phi_{e})$ more
negative. We thus expect $dV_{t}/dA|_{\phi_{e}}<0$ and therefore $dS/dA<0$.
This corresponds to the decreases of the pseudo-bounce action towards the CdL
minimum as $A$ increases toward $A_{CdL}$.

For dS vacua ($p=\phi_{i}$), as $\phi_{i}$ increases towards $\phi_{B}$ (the
field value of the top of the potential barrier), $\phi_{e}$ decreases (also
toward $\phi_{B}$), with $V_{t}(\phi_{e})$ increasing. We thus expect
$dV_{t}/d\phi_{i}|_{\phi_{e}}>0$ and thus $dS/d\phi_{i}>0$. This corresponds
to the increase of the pseudo-bounce action towards the HM maximum as
$\phi_{i}\to\phi_{B}$.

Finally, we can also calculate the $\kappa\to 0$ limit of (G.11). As in the
$\kappa=0$ limit the value of $V_{\scriptscriptstyle+}$ is irrelevant, instead
of $A$ or $\phi_{i}$ we can use $\phi_{e}$ as parameter (as was done in [34]).
With this trivial modification, the $\kappa\to 0$ expansion of (G.11) gives

$\frac{dS}{d\phi_{e}}=\frac{\pi^{2}}{2}\rho_{e}^{4}\left.\frac{dV_{t}}{d\phi_{e}}\right|_{\phi_{e}}\
,$ (G.12)

which reproduces the result in [34].

###  G.2 Coleman-De Luccia and Hawking-Moss Instantons

We can also use (G.11) to show that both the CdL and the HM limits (of pseudo-
bounce solutions) are stationary. The CdL limit corresponds to $\rho_{e}\to
0$. Expanding (G.11) at small $\rho_{e}$ does give $dS/dp=0$. The HM limit
instead corresponds to $\phi_{i}\to\phi_{B}$ and $\phi_{e}\to\phi_{B}$. At the
field value $\phi_{T}$ where $V_{t}$ has its maximum (not to be confused with
$\phi_{B}$), we have $\rho^{2}=3/(\kappa V_{t})$ and therefore, in the HM
limit $\kappa V_{e}\rho_{e}^{2}/3\to 1$ and again we get $dS/dp=0$ from
(G.11).

###  G.3 Bubbles of Nothing

Consider first type-0 BoNs, for which $V_{t}\sim V_{tA}e^{\sqrt{6\kappa}\phi}$
and $D\sim D_{\infty}e^{\sqrt{8\kappa/3}\phi}$ for $\phi\to\infty$. Plugging
these asymptotics in (G.9), with $d\phi_{e}/dp=0$, we get

$\frac{dS_{BoN,0}}{dp}=-36\pi^{2}\sqrt{\frac{6}{\kappa}}\frac{V_{tA}}{D_{\infty}^{3}}\frac{dV_{tA}}{dp}\
.$ (G.13)

As $V_{tA}<0$, we find that the sign of $dS_{BoN,0}/dp$ is the same as the
sign of $dV_{tA}/dp$. We show this relation at work in subsection 6. As a
trivial but nice example, we can check that this formula reproduces the
correct result for Witten’s action, see section 3, as follows. Using
$V_{tA}=-3/(4\kappa R_{KK}^{2})$ and $D_{\infty}^{2}=27/(2\kappa R_{KK}^{4})$,
and taking $p=R_{KK}$ we get $dS/dR_{KK}=2\pi^{2}R_{KK}/\kappa$, integrating
which one reproduces Witten’s BoN action, (3.26).

For type $+$ BoNs, with $V\sim V_{A}e^{a\sqrt{6\kappa}\phi}$ we have

$V_{t}=\frac{V_{A}}{1-a^{2}}e^{a\sqrt{6\kappa}\phi}+...+Xe^{(a+1/a)\sqrt{6\kappa}\phi/2}+...,\quad
D=D_{\infty}e^{(3a+1/a)\sqrt{\kappa/6}\phi}+...,$ (G.14)

where $X$ and $D_{\infty}$ are free parameters, and $D$ is subleading with
respect to $V_{t}^{\prime}$. Using these expressions in the general formula
(G.9) we get the action slope

$\frac{dS_{BoN,{\scriptscriptstyle+}}}{dX}=\sqrt{\frac{6}{\kappa}}\frac{36\pi^{2}a^{3}V_{A}}{(a^{2}-1)D_{\infty}^{3}}\
.$ (G.15)

As shown in the text, BoNs of types $-$ and $-^{*}$ appear as limiting cases
of families of other BoNs (of types $0$ or $+$). Therefore, it does not make
sense to calculate the slope of the action of such BoNs except as the limit of
actions of type $0$ or $+$, for which one can use the formulas given above.

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