Kaluza-Klein tower and bubble nucleation in six dimensional Einstein-Maxwell
theory

Min-Seok Seoa

aDepartment of Physics Education, Korea National University of Education,
Cheongju 28173, Republic of Korea

We study the implication of the distance and the cobordism conjecture on the
6-dimensional Einstein-Maxwell theory compactified on $S^{2}$. In this toy
model, the radion potential is stabilized by the conspiracy of the curvature
of $S^{2}$ and the flux through $S^{2}$ parametrized by $f$, and uplifted by
the positive 6-dimensional cosmological constant parametrized by $\lambda$.
When $\lambda=0$, the radion is stabilized at the anti-de Sitter (AdS) vacuum,
which cannot be interpolated to the Minkowski vacuum since the Kaluza-Klein
(KK) tower descends from UV in the vanishing limit of the 4-dimensional
cosmological constant. For nonzero $\lambda$ which realizes the metastable de
Sitter (dS) vacuum, as well as the AdS and the Minkowski vacuum, such an
obstruction can be found provided the combination $f^{2}\lambda$ is fixed and
the limit $\lambda\to 0$ is taken. Moreover, the 6-dimensional Einstein-
Maxwell theory allows the transition between vacua through the nucleation of
the bubble. In this case, the values of the 4-dimensional cosmological
constant inside and outside the bubble are different as $f$ is changed at the
bubble wall, while $\lambda$ remains unchanged. Regarding the AdS vacuum with
the vanishing curvature radius as the ‘nothing’, we find that the transition
from the metastable dS vacuum to the nothing is not prevented by the descent
of the KK tower since $f^{2}\lambda$ is not fixed.

##  1 Introduction

Whereas the values of physical parameters in the low energy effective field
theory (EFT) can be measured experimentally, we expect that the underlying
fundamental theory provides a dynamical mechanism explaining how these values
are determined. In string theory, the moduli stabilization does the job :
under the compactification, parameters in the 4-dimensional EFT are controlled
by the size and the shape of the extra dimensions, which are fixed by the
stabilization of scalars called moduli. Typically, a large number of
possibilities for choosing the flux quanta in the six extra dimensions allow
the moduli potential to have a local minimum [1], the collection of which is
called the string landscape [2]. But still, in order to explain the tiny,
positive cosmological constant as we observe it, the moduli need to be
stabilized at the metastable de Sitter (dS) vacuum, rather than the anti-de
Sitter (AdS) vacuum. This is achieved by introducing the uplift generated by
anti-branes in addition [3]. Several models for the metastable dS vacuum built
in this way have been proposed [4, 5], which contributed to the active debate
on whether the metastable dS vacuum can be realized in the controllable
parameter regime [6].

The roles of the flux and the brane in the moduli stabilization can be simply
illustrated by the toy model, the 6-dimensional Einstein-Maxwell theory
compactified on $S^{2}$ with the magnetic flux through $S^{2}$ [8, 9] (see [7]
for a pedagogical review). In this model, the radion, the modulus determining
the radius of $S^{2}$, is stabilized through the conspiracy of two potential
terms, each of which is generated by the curvature of $S^{2}$ and the magnetic
flux through $S^{2}$, respectively. This indeed is a typical way that the
moduli are stabilized in string theory, in which the curvature term may be
replaced by another flux term or the non-perturbative effect, depending on the
model. Moreover, the uplift term for the metastable dS vacuum is provided by
the positive 6-dimensional cosmological constant.

On the other hand, recent swampland program [10] (for reviews, see [11, 12,
13, 14, 15, 16]) suggests to revisit the moduli stabilization and see if there
is a missing ingredient which makes the EFT description inconsistent with
quantum gravity. Many studies along this line rely on the distance conjecture
[17]. It states that the infinite distance limit of the moduli space
corresponds to a corner of the landscape, at which the EFT description becomes
invalid as a tower of states descends from UV. The 6-dimensional Einstein-
Maxwell theory naturally contains a tower of states, a set of Kaluza-Klein
(KK) modes, the mass scale of which becomes extremely tiny in the
decompactification limit, i.e., the limit of the infinitely large radius of
$S^{2}$. Therefore, it is instructive to investigate how the radius of
$S^{2}$, hence the KK mass scale, is determined by the flux and the uplift in
the theory in detail, focusing on the decompactification limit. Since the
4-dimensional cosmological constant $\Lambda_{4}$ is determined by the flux
and the uplift as well, we will eventually arrive at the scaling law obeyed by
$\Lambda_{4}$ and the KK mass scale. Regarding this, there have been
conjectures which try to argue that the tiny value of $\Lambda_{4}$ implies
the low tower mass scale [18, 19, 20, 21, 22, 23]. If it were the case, our
universe having $\Lambda_{4}\simeq 10^{-123}M_{4}^{4}$, where $M_{4}$ is the
4-dimensional Planck mass, is quite close to the corner of the landscape.
Presumably, a relevant tower of states may be the KK tower, which implies the
large size of the extra dimensions [24]. Moreover, $\Lambda_{4}$ provides the
IR mass scale of the EFT, thus from the scaling law we can see if there is the
scale separation between the IR and the KK mass scale [25, 26, 27]. The first
purpose of this article is to explore the scaling law and the scale separation
in the 6-dimensional Einstein-Maxwell theory and discuss the implications of
them.

It is also remarkable that the 6-dimensional Einstein-Maxwell theory admits
the instanton solution describing the tunnelling between different flux vacua,
which in particular includes the nucleation of the bubble of ‘nothing’. Here
the bubble of nothing indicates the bubble containing not only no matter, but
also no spacetime, which is known to exist in the presence of the extra
dimensions [28]. At the wall of the bubble of nothing, the radius of the extra
dimensions shrinks to zero size, and at the same time noncompact 4-dimensional
spacetime pinches off such that it cannot extend to the region inside the
bubble wall. In the 6-dimensional Einstein-Maxwell theory, the magnetic flux,
one of ingredients determining the value of $\Lambda_{4}$, can be discharged
by the black 2-brane forming the bubble wall [29]. Then two vacua separated by
the bubble wall have different values of $\Lambda_{4}$ due to the different
values of the flux, while they have the same value of the 6-dimensional
cosmological constant which generates the uplift. Moreover, a new bubble can
be nucleated inside the bubble, which leads to the sequential nucleation of
bubbles having different values of $\Lambda_{4}$. When the magnetic flux is
completely discharged by the brane, i.e., the flux inside the bubble vanishes,
the value of $\Lambda_{4}$ inside the bubble becomes negative infinity, which
corresponds to the AdS vacuum with the vanishing curvature radius. Since the
radius of $S^{2}$ vanishes as well, we can regard this bubble as the bubble of
nothing. It indicates that spacetime can reach the nothing through a (series
of) nucleation of the bubble(s), which is extensively studied in [30, 31, 32].

Recently, the bubble of nothing has drawn attention (see, for example, [33,
34, 35, 36, 37, 38, 39, 40]) in the context of the swampland program. For
instance, the cobordism conjecture [41] (see [42] for a review) argues that
the compact space consistent with quantum gravity can shrink to zero size
without any topological obstruction. This conjecture is closely connected to
an old idea that the global symmetry is broken by quantum gravity [43]. For an
illustration, we may consider the compact ball made of the black hole
interior, which can smoothly evolve into the nothing according to the
cobordism conjecture [42]. This is nothing more than the complete evaporation
of the black hole, in which the global symmetry is not conserved since
otherwise information is lost. If the conjecture is true, the transition to
the nothing in which the internal manifold ($S^{2}$ in our case) shrinks to
zero size must be the topology preserving process. This motivates us to
discuss the implication of the cobordism conjecture on the vacuum transition
in the 6-dimensional Einstein-Maxwell theory. The vacuum transition to the
nothing through the nucleation of the bubble is quite different process from
that in [33, 34, 35, 36, 37, 38] where the radion is not stabilized but rolls
along the potential, but this is another way that the change in the value of
the radion takes place along some particular direction of the coordinate space
(see [44] for a similar discussion).

This article is organized as follows. In Sec. 2, we review the features of the
6-dimensional Einstein-Maxwell theory compactified on $S^{2}$ relevant to our
discussion. Based on this, we consider in Sec. 3 the implication of the
distance conjecture on the radion stabilization, presenting the relation
between $\Lambda_{4}$ and the KK mass scale. Sec. 4 is devoted to the
discussion on the transition between different flux vacua through the
nucleation of the bubble, focusing on the transition to the nothing which is
relevant to the cobordism conjecture. Then we conclude. Some details of the
bubble are reviewed in App. A.

##  2 Review on the 6-dimensional Einstein-Maxwell theory

The action for the 6-dimensional Einstein-Maxwell theory is

$\begin{split}S_{\rm EM}=\int d^{6}x\sqrt{-G}\Big{[}\frac{M_{6}^{4}}{2}{\cal
R}^{(6)}-\frac{1}{4}F_{AB}F^{AB}-\Lambda_{6}\Big{]},\end{split}$ (1)

where $M_{6}$ is the 6-dimensional Planck mass, $F_{AB}$ is the
electromagnetic field strength ($A,B=0,1,\cdots 5$), and $\Lambda_{6}$ is the
6-dimensional cosmological constant of mass dimension six. Solutions to the
equations of motion describing the compactification on $S^{2}$ by the magnetic
flux are given by

$\begin{split}&ds^{2}=e^{-\frac{\phi-\langle\phi\rangle}{M_{4}}}g_{\mu\nu}dx^{\mu}dx^{\nu}+e^{\frac{\phi-\langle\phi\rangle}{M_{4}}}\big{(}e^{\frac{\langle\phi\rangle}{M_{4}}}R_{0}^{2}\big{)}(d\psi^{2}+\sin^{2}\psi
d\omega^{2}),\\\ &F_{2}=\frac{g_{6}N}{4\pi}\sin\psi d\psi\wedge
d\omega,\quad\quad N\in\mathbb{Z}.\end{split}$ (2)

While $R_{0}$, the fiducial radius of $S^{2}$, can be chosen arbitrarily, the
physically meaningful radius is fixed by the stabilization of the radion
$\phi$ to $\langle\phi\rangle$ such that $\langle
R\rangle=e^{\frac{\langle\phi\rangle}{2M_{4}}}R_{0}$, from which the squared
4-dimensional Planck mass is written as

$\begin{split}M_{4}^{2}=4\pi\langle R\rangle^{2}M_{6}^{4}=4\pi
e^{\frac{\langle\phi\rangle}{M_{4}}}R_{0}^{2}M_{6}^{4}.\end{split}$ (3)

Meanwhile, the magnetic coupling $g_{6}$ has the mass dimension one as the
dimensionless 4-dimensional electric coupling is obtained by the relation
$e_{4}^{2}=e_{6}^{2}/(4\pi\langle R\rangle^{2})$, where $e_{6}$ is the
6-dimensional electric coupling, the inverse of $g_{6}$. The dependence of the
6-dimensional metric on $\phi$ is determined by the requirement that the
kinetic term of $\phi$ is written in the canonical form in the Einstein frame
action after integrating out the extra dimensions 111When D-dimensional
spacetime is compactified on the internal manifold of dimension $n=D-4$, the
metric giving the canonical kinetic term of the volume modulus $\phi$ in the
Einstein frame action is written as
$\begin{split}ds^{2}=e^{-\sqrt{\frac{2n}{n+2}}\phi}g_{\mu\nu}dx^{\mu}dx^{\nu}+e^{2\sqrt{\frac{2}{n(n+2)}}\phi}g_{mn}dy^{m}dy^{n}.\end{split}$
(4) Our case corresponds to $n=2$.  :

$\begin{split}S=\int d^{4}x\sqrt{-g}\Big{[}\frac{M_{4}^{2}}{2}{\cal
R}_{(4)}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\Big{]},\end{split}$
(5)

where the radion potential $V(\phi)$ is given by

$\begin{split}V(\phi)=\frac{4\pi}{M_{4}^{4}}\Big{[}\frac{1}{32\pi^{2}}\frac{(g_{6}N)^{2}}{M_{6}^{2}}\frac{e^{-3\frac{\phi}{M_{4}}}}{R_{0}^{6}M_{6}^{6}}-\frac{e^{-2\frac{\phi}{M_{4}}}}{R_{0}^{4}M_{6}^{4}}+\frac{\Lambda_{6}}{M_{6}^{6}}\frac{e^{-\frac{\phi}{M_{4}}}}{R_{0}^{2}M_{6}^{2}}\Big{]}.\end{split}$
(6)

We note that the first and the second term are generated by the magnetic flux
and the curvature of the internal manifold ($S^{2}$ in our case),
respectively. While these two terms suffice to stabilize $\phi$, as we will
see, the vacuum energy density at $\langle\phi\rangle$, or equivalently, the
4-dimensional cosmological constant $\Lambda_{4}$, cannot be positive with
$\Lambda_{6}=0$. To obtain the metastable dS vacuum, we need the uplift term
induced by the positive $\Lambda_{6}$ in addition.

For the ease of discussion, we define the following dimensionless quantities,

$\begin{split}v(x)=\frac{4\pi}{M_{4}^{2}}V(x),\quad\quad
x=\frac{e^{-\frac{\phi}{M_{4}}}}{R_{0}^{2}M_{6}^{2}},\quad\quad
f^{2}=\frac{(g_{6}N)^{2}}{M_{6}^{2}},\quad\quad\lambda=\frac{\Lambda_{6}}{M_{6}^{6}},\end{split}$
(7)

with respect to which (6) is rewritten as

$\begin{split}v(x)=\frac{f^{2}}{32\pi^{2}}x^{3}-x^{2}+\lambda x.\end{split}$
(8)

Then one finds that the potential has a local minimum at

$\begin{split}\langle
x\rangle=\frac{e^{-\frac{\langle\phi\rangle}{M_{4}}}}{R_{0}^{2}M_{6}^{2}}=\frac{32\pi^{2}}{3f^{2}}\Big{[}1+\sqrt{1-\frac{3f^{2}}{32\pi^{2}}\lambda}\Big{]}.\end{split}$
(9)

Here $M_{6}^{2}\langle x\rangle$ is nothing more than $\langle R\rangle^{-2}$,
or equivalently, the squared KK mass scale $m_{\rm KK}^{2}$. Using (3), the KK
mass scale also can be expressed as $m_{\rm
KK}=\frac{M_{4}}{\sqrt{4\pi}}\langle x\rangle$. This is useful because it is
typical to take $M_{4}$, instead of $M_{6}$, to be a fixed value given by
$2.4\times 10^{18}$GeV and treat $M_{4}$ and $\langle R\rangle$ as independent
parameters. 222 Indeed, if we fix the value of $M_{6}$, $M_{4}$ becomes $0$ in
the limit $\langle R\rangle\to 0$ hence the EFT assuming weak gravity breaks
down even in the very low energy regime. On the other hand, the opposite limit
$\langle R\rangle\to\infty$ corresponds to the limit $M_{4}\to\infty$,
indicating the decoupling of gravity in the 4-dimensional EFT description.
Putting (9) into (8), one finds that the value of the potential at $\langle
x\rangle$ in the unit of $M_{4}$ is given by

$\begin{split}v(\langle
x\rangle)=\Big{(}\frac{32\pi^{2}}{3f^{2}}\lambda\Big{)}\Big{[}1-\frac{2}{3}\frac{32\pi^{2}}{3f^{2}\lambda}\Big{(}1+\Big{(}1-\frac{3f^{2}}{32\pi^{2}}\lambda\Big{)}^{3/2}\Big{)}\Big{]}.\end{split}$
(10)

We note from (9) that the potential has the local minimum provided
$\frac{3}{32\pi^{2}}f^{2}\lambda\leq 1$ : if $\frac{3}{32\pi^{2}}f^{2}\lambda$
becomes larger than $1$, the potential monotonically decreases with respect to
$\phi$ (or equivalently, $1/x$) without the local minimum. In other words, the
potential in this case shows the runaway behavior. On the other hand, (10)
tells us that $v(\langle x\rangle)$ is positive, i.e., the potential is
stabilized at the metastable dS vacuum, provided
$\frac{3}{32\pi^{2}}f^{2}\lambda>\frac{3}{4}$. If $\lambda=0$, $v(\langle
x\rangle)$ is always negative, then the potential is stabilized at the AdS
vacuum.

Figure 1: (Left): Behavior of the potential with respect to $RM_{6}=x^{-1/2}$
when $\lambda=0$. The flux for each plot is given by $f=0,5$, and $10$,
respectively. (Right): Relation between the stabilized values $\langle
R\rangle$ and $V(\langle R\rangle)$. Orange and green dot indicate the values
of $\langle R\rangle$ and $V(\langle R\rangle)$ for $f=5$ and $10$,
respectively, which correspond to plots in the left panel. This shows that as
$f$ increases, $\langle R\rangle$ gets larger while $V(\langle R\rangle)$ gets
closer to zero.

Figure 2: (Left): Behavior of the potential with respect to $RM_{6}=x^{-1/2}$
when $f=5$. The 6-dimensional cosmological constant in each plot is given by
$\lambda=0,1,3.2,3.5$ and $5$, respectively. (Right): Relation between
$\langle R\rangle$ and $V(\langle R\rangle)$, showing that as $\lambda$
increases, both $\langle R\rangle$ and $V(\langle R\rangle)$ increase. In
particular, as $\lambda$ becomes larger than $3.2$, $V(\langle R\rangle)$
changes the sign from negative (AdS) to positive (dS). Blue, orange, green,
and red dot indicate the values of $\langle R\rangle$ and $V(\langle
R\rangle)$ for $\lambda=0,1,3.2$ and $3.5$, respectively, which correspond to
plots in the left panel.

To investigate the stabilization of the radion in detail, we first consider
the case of $\lambda=0$, i.e., the vanishing $\Lambda_{6}$, where the
potential is minimized at $\langle x\rangle=\frac{64\pi^{2}}{3f^{2}}$. Since
$\langle x\rangle$ is identified with $1/(M_{6}\langle R\rangle)^{2}$, the
radius of $S^{2}$ is given by

$\begin{split}\langle
R\rangle=\frac{\sqrt{3}}{8\pi}\frac{f}{M_{6}}=\frac{3}{32\pi^{3/2}}\frac{f^{2}}{M_{4}},\end{split}$
(11)

where (3) is used for the last equality. Then as mentioned above, the
potential at the minimum (in the unit of $M_{4}$),

$\begin{split}v(\langle
x\rangle)=-\frac{4096\pi^{4}}{27}\frac{1}{f^{4}}=-\frac{1}{3M_{6}^{4}\langle
R\rangle^{4}}=-\frac{4\pi}{3}\frac{1}{M_{4}^{2}\langle
R\rangle^{2}},\end{split}$ (12)

is negative, thus the potential is stabilized at the AdS vacuum. We note that
in the absence of the flux $(f=0)$, $\langle R\rangle$ is given by $0$, at
which the potential becomes negative infinity. As evident from plots in Fig.
1, as $f$ increases, $\langle R\rangle$ monotonically increases such that
$S^{2}$ is eventually decompactified, and at the same time, the depth of the
potential becomes smaller, approaching the Minkowski vacuum indefinitely.
Moreover, we infer from (3) that the stabilized value of the potential
$V(\langle x\rangle)=\frac{M_{4}^{4}}{4\pi}v(\langle x\rangle)$ is just given
by $-\frac{4\pi}{3}M_{6}^{4}$. If we take $M_{6}$ to be a fixed value,
$V(\langle x\rangle)$ is independent of $\langle R\rangle$. However, it is
typical to treat $\langle R\rangle$ and the fixed value $M_{4}$ as independent
parameters, in which case the qualitative features of $V(\langle x\rangle)$
are the same as those of $v(\langle x\rangle)$ discussed so far. As we will
see, $M_{6}$ in this case is interpreted as the ‘species scale’, which gets
lower as $\langle R\rangle$ gets larger.

On the other hand, when $\lambda\neq 0$, the potential is stabilized at the
AdS (dS) vacuum for $\frac{3}{32\pi^{2}}f^{2}\lambda<\frac{3}{4}$
($\frac{3}{32\pi^{2}}f^{2}\lambda>\frac{3}{4}$). Suppose $f$ is fixed but
$\lambda$ is allowed to vary. Then one finds that the transition between the
AdS and the dS vacuum at $\frac{3}{32\pi^{2}}f^{2}\lambda=\frac{3}{4}$
(Minkowski vacuum) is smooth with respect to $\lambda$. As can be found in
(9), for a fixed value of $f$, $\langle R\rangle=1/(M_{6}\langle
x\rangle^{1/2})$ increases as $\lambda$ increases. At the same time, the value
of $v(\langle x\rangle)$ given by (10) increases from the negative to the
positive value, thus $|\Lambda_{4}|$ of the AdS vacuum
($\frac{3}{32\pi^{2}}f^{2}\lambda<\frac{3}{4}$) gets smaller until the
Minkowski vacuum ($\frac{3}{32\pi^{2}}f^{2}\lambda=\frac{3}{4}$ hence
$|\Lambda_{4}|=0$) is reached, after which $|\Lambda_{4}|$ of the dS vacuum
($\frac{3}{32\pi^{2}}f^{2}\lambda>\frac{3}{4}$) gets larger. Moreover, the
values of $\langle x\rangle$ and $v(\langle x\rangle)$ in the limit
$\lambda\to 0$ coincide with those in the case of $\lambda=0$ we previously
discussed. Such a behavior of the potential is shown in Fig. 2.

We may also fix $\lambda$ to some finite value, and investigate the behavior
of the potential by changing $f$, as depicted in Fig. 3. When $f=0$, $\langle
x\rangle$ becomes infinity, or equivalently, $\langle R\rangle=0$, at which
the value of the potential is given by negative infinity. This corresponds to
the AdS vacuum with the vanishing curvature radius. As $f$ increases, the
value of the potential increases and when
$\frac{3}{32\pi^{2}}f^{2}\lambda=\frac{3}{4}$ is satisfied, we obtain the
Minkowski vacuum where the transition from the AdS to the dS vacuum takes
place.

Figure 3: (Left): Behavior of the potential with respect to $RM_{6}=x^{-1/2}$
when $\lambda=3.5$. The flux for each plot is given by $f=0,4,4.75,5$ and $6$,
respectively. (Right): Relation between $\langle R\rangle$ and $V(\langle
R\rangle)$, showing that as $f$ increases, both $\langle R\rangle$ and
$V(\langle R\rangle)$ increase allowing the transition from the AdS to the dS
vacuum. Orange, green, and red dot indicate the values of $\langle R\rangle$
and $V(\langle R\rangle)$ for $f=4,4.5$, and $5$, respectively, which
correspond to plots in the left panel.

Even more interesting case is that both $f$ and $\lambda$ are allowed to vary
but their combination $\frac{3}{32\pi^{2}}f^{2}\lambda$ is fixed. We postpone
the discussion on this case to Sec. 3 since it is closely relevant to the
swampland conjectures concerning the distance conjecture. In any case, even if
both $f$ and $\lambda$ are nonzero, they cannot be arbitrarily large since as
$\frac{3}{32\pi^{2}}f^{2}\lambda$ gets larger than $1$, the potential shows
the runaway behavior, i.e., monotonically decreases with respect to
$R=M_{6}^{-1}x^{-1/2}$ without the local minimum. The values of the stabilized
radius and the potential at $\frac{3}{32\pi^{2}}f^{2}\lambda=1$ are given by

$\begin{split}&\langle x\rangle=\frac{1}{M_{6}^{2}\langle
R\rangle^{2}}=\frac{32\pi^{2}}{3f^{2}}=\lambda,\quad\quad v(\langle
x\rangle)=\frac{1024\pi^{4}}{27}\frac{1}{f^{4}}=\frac{\lambda^{2}}{3},\end{split}$
(13)

respectively, which correspond to the maximum values of $\langle R\rangle$ and
$v(\langle x\rangle)$ that the metastable dS vacuum can have.

We also note that putting the relation

$\begin{split}\frac{3f^{2}}{32\pi^{2}}=2(M_{6}\langle
R\rangle)^{2}-\lambda(M_{6}\langle R\rangle)^{4},\end{split}$ (14)

which is obtained from (9), into (10), we can express $v(\langle x\rangle)$ in
terms of $\langle R\rangle$ :

$\begin{split}v(\langle x\rangle)=\frac{-1+2\lambda(M_{6}\langle
R\rangle)^{2}}{3(M_{6}\langle R\rangle)^{4}}.\end{split}$ (15)

This is useful in the later discussion.

Figure 4: The ratio $m_{R}^{2}/m_{\rm KK}^{2}$ as a function of
$\frac{3}{32\pi^{2}}f^{2}\lambda$.

Meanwhile, the squared mass of $\phi$, the canonically normalized radion, is
given by

$\begin{split}m_{R}^{2}&=\frac{d^{2}V}{d\phi^{2}}\Big{|}_{\phi=\langle\phi\rangle}=\frac{1}{M_{4}^{2}}x\frac{d}{dx}\Big{(}x\frac{dV}{dx}\Big{)}\Big{|}_{x=\langle
x\rangle}\\\
&=\frac{512\pi^{3}}{9}\frac{M_{4}^{2}}{f^{4}}\sqrt{1-\frac{3f^{2}}{32\pi^{2}}\lambda}\Big{(}1+\sqrt{1-\frac{3f^{2}}{32\pi^{2}}\lambda}\Big{)}^{2},\end{split}$
(16)

where we use the relation between $\phi$ and $x$ as well as that between $V$
and $v$, both of which can be found in (7). If $\lambda=0$, it is given by
$m_{R}^{2}=\frac{2048\pi^{3}}{9f^{4}}M_{4}^{2}$, which also can be written as
$\frac{128\pi^{2}}{3f^{2}}M_{6}^{2}$. When
$\frac{3}{32\pi^{2}}f^{2}\lambda=\frac{3}{4}$ is satisfied, i.e., the radion
is stabilized at the Minkowski vacuum, the radion mass is given by
$m_{R}^{2}=\frac{64\pi^{3}}{f^{4}}M_{4}^{2}=\frac{16\pi^{2}}{f^{2}}M_{6}^{2}$,
or equivalently, $m_{R}^{2}=\frac{\lambda^{2}}{\pi}M_{4}^{2}=2\lambda
M_{6}^{2}$. Of course, the radion becomes massless when
$\frac{3}{32\pi^{2}}f^{2}\lambda=1$, as for
$\frac{3}{32\pi^{2}}f^{2}\lambda>1$ the potential is not stabilized but
exhibits the runaway behavior. It is instructive to compare this with the KK
mass scale, $m_{\rm KK}^{2}=\langle R\rangle^{-2}=M_{6}^{2}\langle x\rangle$,
where $\langle x\rangle$ is given by (9). The ratio

$\begin{split}\frac{m_{R}^{2}}{m_{\rm
KK}^{2}}=2\sqrt{1-\frac{3f^{2}}{32\pi^{2}}\lambda},\end{split}$ (17)

monotonically decreases from $2$ to $0$ as the combination
$\frac{3}{32\pi^{2}}f^{2}\lambda$ varies from $0$ to $1$, which is evident
from Fig. 4. In particular, the ratio becomes $1$ at the Minkowski vacuum,
where $\frac{3}{32\pi^{2}}f^{2}\lambda=\frac{3}{4}$ is satisfied. Therefore,
$m_{R}^{2}$ is always larger than but not exceeding twice of $m_{\rm KK}^{2}$
in the AdS vacuum while is smaller than $m_{\rm KK}^{2}$ in the dS vacuum.
When either $\lambda$ or $f$ is zero, the ratio is fixed to $2$.

This shows that in the AdS vacuum, the 4-dimensional EFT description for the
radion stabilization is more or less valid since $m_{R}$ is not so much
enhanced compared to $m_{\rm KK}$ hence only a few KK modes are found at the
energy scale $m_{R}$. Moreover, the ratio $m_{R}^{2}/m_{\rm KK}^{2}$ does not
diverge, implying that even if $m_{\rm KK}$ gets closer to $0$, it does not
decrease more rapidly compared to $m_{R}$, so the radion stabilization may not
be obstructed by the descent of the KK modes. However, even in this case, the
UV cutoff for any other 4-dimensional EFT description can decrease rapidly in
the limit $m_{\rm KK}\to 0$. Indeed, more realistic model can contain the
dynamics which is not directly relevant to the radion stabilization, but well
described in the 4-dimensional EFT framework. For example, the size of
$\Lambda_{4}$ resulting from the moduli stabilization and that of the
electroweak scale can be connected only in the indirect way. For this reason,
we will regard the vanishing limit of $m_{\rm KK}$ as a signal that the
generic 4-dimensional EFT description becomes invalid, as well known.

##  3 Distance conjecture : scale separation and scaling law

Since two extra dimensions are compactified on $S^{2}$, the 6-dimensional
Einstein-Maxwell theory we reviewed in Sec. 2 contains the KK tower as a
natural tower of states. When the KK mass scale $m_{\rm KK}$ becomes zero, the
KK tower descends from UV, invalidating the 4-dimensional EFT description.
Recent distance conjecture claims that such a descent of a tower of states is
what happens when the EFT is close to the corner of the landscape. Moreover,
it is also conjectured that a tower of states invalidating the EFT descends
from UV as the size of $\Lambda_{4}$ gets smaller, emphasizing that the (A)dS
vacuum in the vanishing limit of $\Lambda_{4}$ cannot be continuously
interpolated into the Minkowski vacuum in the EFT framework [18]. In this
section, we discuss the conjectures concerning a tower of states in the
context of the 6-dimensional Einstein-Maxwell theory, in which both $m_{\rm
KK}$ and $\Lambda_{4}$ are determined by the radion stabilization thus the
flux and the uplift.

We begin our discussion with the scale separation between the IR mass scale
obtained from $\Lambda_{4}$ and $m_{\rm KK}$. When the background geometry is
given by dS space, the natural IR mass scale is the Hubble parameter $H$
satisfying $\Lambda_{4}=3M_{4}^{2}H^{2}$, which indeed is the inverse of the
horizon radius. In the case of the AdS background, the inverse of the
curvature radius, which will be also denoted by $H$, is the natural IR mass
scale. This can be justified by considering the metric of AdS space [45] (see
also [46, 37])

$\begin{split}ds^{2}=\frac{1}{H^{2}}\Big{[}-\Big{(}\frac{1+r^{2}}{1-r^{2}}\Big{)}dt^{2}+\frac{4}{(1-r^{2})^{2}}(dr^{2}+r^{2}d\Omega^{2})\Big{]},\end{split}$
(18)

where $r\in[0,1)$ and $r=1$ corresponds to the boundary. When we imagine the
surface of constant $r$, the area of the surface and the volume of the region
surrounded by the surface are given by

$\begin{split}&A=4\pi\Big{(}\frac{2r}{H(1-r^{2})}\Big{)}^{2}\\\ &V=4\pi\int
drr^{2}\Big{(}\frac{2}{H(1-r^{2})}\Big{)}^{3}=\frac{4\pi}{2H^{3}}\Big{[}\frac{2r(1+r^{2})}{(1-r^{2})^{2}}+\log\Big{(}\frac{1-r}{1+r}\Big{)}\Big{]},\end{split}$
(19)

respectively. Then the ratio $A/V$ is a monotonically decreasing function of
$r$ and minimized at the boundary ($r=1$) where $A/V=2H$.

Figure 5: The ratio $H^{2}/m_{\rm KK}^{2}$ as a function of
$\frac{3f^{2}\lambda}{32\pi^{2}}$.

In terms of $H$, the value of $\Lambda_{4}$ in dS space and that in AdS space
are written as $3H^{2}M_{4}^{2}$ and $-3H^{2}M_{4}^{2}$, respectively : as
(A)dS space gets closer to Minkowski space, $|\Lambda_{4}|$ gets smaller, and
$H$ becomes $0$. In the 6-dimensional Einstein-Maxwell theory, the value of
$H$ is given by

$\begin{split}H^{2}&=\frac{|V(\langle\phi\rangle)|}{3M_{4}^{2}}=\frac{M_{4}^{2}}{12\pi}|v(\langle
x\rangle)|\\\
&=\frac{M_{4}^{2}}{12\pi}\Big{(}\frac{32\pi^{2}}{3f^{2}}\lambda\Big{)}\Big{|}1-\frac{2}{3}\frac{32\pi^{2}}{3f^{2}\lambda}\Big{(}1+\Big{(}1-\frac{3f^{2}}{32\pi^{2}}\lambda\Big{)}^{3/2}\Big{)}\Big{|}.\end{split}$
(20)

We note that the potential for $f=0$ (the vanishing magnetic flux) is
minimized at $\langle R\rangle=0$ where $V=-\infty$ hence $H=\infty$. As
mentioned in Sec. 2, this will be interpreted as the AdS vacuum in the limit
of the vanishing curvature radius. Using the expression for $\langle
x\rangle=m_{\rm KK}^{2}/M_{6}^{2}$ given by (9) and the fact that
$M_{4}^{2}=4\pi M_{6}^{4}/m_{\rm KK}^{2}$ (see (3)), we obtain the ratio

$\begin{split}\frac{H^{2}}{m_{\rm
KK}^{2}}&=\frac{1}{3}\Big{(}\frac{3f^{2}}{32\pi^{2}}\lambda\Big{)}\frac{\Big{|}1-\frac{2}{3}\frac{32\pi^{2}}{3f^{2}\lambda}\Big{(}1+\Big{(}1-\frac{3f^{2}}{32\pi^{2}}\lambda\Big{)}^{3/2}\Big{)}\Big{|}}{\Big{(}1+\sqrt{1-\frac{3f^{2}}{32\pi^{2}}\lambda}\Big{)}^{2}}\\\
&=\frac{1}{9}|-1+2\lambda(M_{6}\langle R\rangle)^{2}|\\\
&=\frac{1}{9}\Big{|}-1+\frac{\lambda}{\pi}\frac{M_{4}}{m_{\rm
KK}}\Big{|},\end{split}$ (21)

the behavior of which as a function of $\frac{3f^{2}\lambda}{32\pi^{2}}$ is
depicted in Fig. 5. From this, one finds that when
$\frac{3f^{2}\lambda}{32\pi^{2}}=1$, i.e., the dS background is about to be
destabilized by the runaway behavior of the potential, $H^{2}/m_{\rm
KK}^{2}=1/9$. The same ratio $H^{2}/m_{\rm KK}^{2}=1/9$ is also obtained in
the limit $\frac{3f^{2}\lambda}{32\pi^{2}}\to 0$. In these cases, The IR mass
scale $H$ is of the same order as $m_{\rm KK}$, so the scale separation does
not occur.

In particular, when $\lambda=0$, the value of $H^{2}/m_{\rm KK}^{2}$ is $1/9$
for any finite value of $f$ since $\frac{3f^{2}\lambda}{32\pi^{2}}=0$ is
satisfied, while the radion is always stabilized at the AdS vacuum. Then we
obtain the scaling law obeyed by $\Lambda_{4}$ and $m_{\rm KK}$ given by
$m_{\rm KK}=|3\Lambda_{4}|^{1/2}/M_{4}$. The exponent $1/2$ here is claimed to
be the upper bound on the exponent in the AdS/dS distance conjecture [18],
which can be found in the supersymmetric AdS vacuum. The scaling law shows
that as $|\Lambda_{4}|\to 0$, i.e., the AdS background gets closer to
Minkowski space, $m_{\rm KK}$ becomes zero, invalidating the 4-dimensional EFT
description. In terms of $\langle\phi\rangle$, the relation $m_{\rm
KK}=\frac{M_{4}}{\sqrt{4\pi}}\langle x\rangle$ with $x\sim e^{-\phi/M_{4}}$
indicates that $|\Lambda_{4}|/M_{4}^{4}\sim e^{-2\phi/M_{4}}$. We note that
since the size of both $\langle x\rangle\sim f^{-2}$ and $v(\langle
x\rangle)\sim f^{-4}$ are controlled by the flux $f$, the smallness of
$|\Lambda_{4}|$ indicating the low KK mass scale can be understood as a result
of the large flux, as also discussed in Sec. 2.

On the contrary, when $\lambda\neq 0$, as can be inferred from the last
expression in (21), the ratio between $H$ and $m_{\rm KK}$ is no longer a
constant unless $\lambda$ is close to $0$. Indeed, as discussed in Sec. 2 and
summarized in Fig. 2 and Fig. 3, given the fixed value of $f$ ($\lambda$), the
transition between the AdS vacuum and the dS vacuum through the Minkowski
vacuum is smooth with respect to $\lambda$ ($f$). The KK mass scale does not
vanish at the Minkowski vacuum ($\frac{3f^{2}\lambda}{32\pi^{2}}=\frac{3}{4}$)
in this case, and as is evident from Fig. 5, $H$ (hence $|\Lambda_{4}|$)
decreases even more rapidly compared to $m_{\rm KK}$, thus the KK tower can be
decoupled from the EFT at the IR mass scale. This is quite different from the
prediction of the AdS/dS distance conjecture and seems to realize the scenario
similar to the KKLT [4] or the large volume scenario [5], in which the radion
is mainly stabilized by the flux and then slightly uplifted, while the KK
tower does not descend from UV even in the Minkowski limit.

The situation can be changed if we allow both $f$ and $\lambda$ to vary but
fix their combination $\frac{3f^{2}\lambda}{32\pi^{2}}$ to some value smaller
than $1$ (to avoid the runaway potential). This is equivalent to fixing the
combination $\lambda(M_{6}\langle
R\rangle)^{2}=(\lambda/\sqrt{4\pi})M_{4}\langle R\rangle=\lambda\langle
x\rangle^{-1}$, which is evident from (21) or the fact that $\lambda\langle
x\rangle^{-1}$ depends on $f$ and $\lambda$ through the combination
$\frac{3f^{2}\lambda}{32\pi^{2}}$ only (see (9)). In this case, $H^{2}/m_{\rm
KK}^{2}$ becomes a constant smaller than $1/9$, resulting in the scaling law
$m_{\rm KK}\sim|\Lambda_{4}|^{1/2}$ as in the case of $\lambda=0$ or $1$. Of
course, depending on the value of $\frac{3f^{2}\lambda}{32\pi^{2}}$ , the
vacuum we consider can be not only AdS or dS, but also Minkowski. Moreover,
under the fixed value of $\frac{3f^{2}\lambda}{32\pi^{2}}$, $f^{2}$ gets
larger if we choose smaller $\lambda$. Equivalently, the value of
$\lambda(M_{6}\langle R\rangle)^{2}=\frac{M_{4}}{\sqrt{4\pi}}\lambda\langle
R\rangle$ can be fixed if we take $\langle R\rangle$ to be infinitely large
(hence $m_{\rm KK}\to 0$) in the limit $\lambda\to 0$.

Figure 6: Radion potentials giving the same value of the AdS cosmological
constant under the fixed value of $M_{6}$. While
$\frac{3f^{2}\lambda}{32\pi^{2}}$ is fixed to $0.2$, the value of $\lambda$
for each plot is chosen to be $0.5$, $2$ and $10$, respectively.

Now, from (10), the stabilized value of the potential which is identified with
$\Lambda_{4}$ can be written as

$\begin{split}\Lambda_{4}&=V(\langle x\rangle)=\frac{M_{4}^{4}}{4\pi}v(\langle
x\rangle)\\\
&=\frac{M_{4}^{4}}{4\pi}\lambda^{2}\Big{(}\frac{32\pi^{2}}{3f^{2}\lambda}\Big{)}\Big{[}1-\frac{2}{3}\frac{32\pi^{2}}{3f^{2}\lambda}\Big{(}1+\Big{(}1-\frac{3f^{2}}{32\pi^{2}}\lambda\Big{)}^{3/2}\Big{)}\Big{]}\\\
&=4\pi M_{6}^{8}(\lambda\langle
R\rangle^{2})^{2}\Big{(}\frac{32\pi^{2}}{3f^{2}\lambda}\Big{)}\Big{[}1-\frac{2}{3}\frac{32\pi^{2}}{3f^{2}\lambda}\Big{(}1+\Big{(}1-\frac{3f^{2}}{32\pi^{2}}\lambda\Big{)}^{3/2}\Big{)}\Big{]}.\end{split}$
(22)

If we take $M_{6}$ to be a fixed value, $\lambda\langle R\rangle^{2}$ in the
last expression of (22), hence $\Lambda_{4}$ is completely determined by the
value of $\frac{3f^{2}\lambda}{32\pi^{2}}$. We can compare cases of different
values of $\lambda$ giving the same value of $\frac{3f^{2}\lambda}{32\pi^{2}}$
(hence the same value of $\Lambda_{4}$) as follows (see also Fig. 6). When we
separate the radion stabilization into two steps, the stabilization of the
radion at the AdS vacuum by $f$ and the uplift by $\lambda$, the larger value
of $f$ stabilizes the radion at the AdS vacuum of the smaller depth, but since
$\lambda$ becomes smaller to give the same value of
$\frac{3f^{2}\lambda}{32\pi^{2}}$, the potential is uplifted by the smaller
amount, giving the same value of $\Lambda_{4}$. 333The final value of
$\Lambda_{4}$ is not necessarily positive : the sign of $\Lambda_{4}$ is
determined by the value of $\frac{3f^{2}\lambda}{32\pi^{2}}$. In particular,
we can take the limit $\lambda\to 0$ without changing $\Lambda_{4}$, in which
both $f$ and $\langle R\rangle$ becomes infinity, resulting in $m_{\rm KK}\to
0$. Since the KK tower descends from UV in this case, the 4-dimensional EFT
description is not reliable. We can find the similar situation in more
realistic model like the KKLT or the large volume scenario, in which the
uplift potential is generated by $\overline{\rm D3}$-branes at the tip of the
Klebanov-Strassler throat [22, 23]. In this case, the uplift potential and the
tower mass scale $m$ satisfy the scaling law $m\sim V_{\rm up}^{\alpha}$. The
value of $\alpha$ is given by either $1/3$ or $1/4$, depending on the type of
tower (the KK or the string tower) and how strongly the throat is warped.

Figure 7: Radion potentials for different choices of $\lambda$ under the fixed
value of $M_{4}$. While $\frac{3f^{2}\lambda}{32\pi^{2}}$ is fixed to $0.2$,
the value $\lambda$ for each plot is chosen to be $0.5$, $2$ and $5$,
respectively.

On the other hand, if we fix the value of $M_{4}$, the second expression of
(22) tells us that given the fixed value of $\frac{3f^{2}\lambda}{32\pi^{2}}$,
the smaller value of $\lambda$ leads to the smaller size of $\Lambda_{4}$. But
since the sign of $\Lambda_{4}$ cannot be changed by varying $\lambda$ in this
case, the transition between the AdS and the dS vacuum does not take place.
Moreover, while the limit $\lambda\to 0$ corresponds to the Minkowski limit
$\Lambda_{4}\to 0$ of (A)dS space (the sign of $\Lambda_{4}$ depends on the
value of $\frac{3f^{2}\lambda}{32\pi^{2}}$), $\langle R\rangle$ becomes
infinity as well, thus $m_{\rm KK}\to 0$. More concretely, from (9), the KK
mass scale can be written as

$\begin{split}m_{\rm KK}=\frac{M_{4}}{\sqrt{4\pi}}\langle
x\rangle=\frac{M_{4}}{\sqrt{4\pi}}\lambda\Big{(}\frac{32\pi^{2}}{3f^{2}\lambda}\Big{)}\Big{[}1+\sqrt{1-\frac{3f^{2}}{32\pi^{2}}\lambda}\Big{]},\end{split}$
(23)

so when $M_{4}$ and $\frac{3f^{2}\lambda}{32\pi^{2}}$ are fixed, $m_{\rm KK}$
becomes $0$ as $\lambda\to 0$. Therefore, when $\lambda$ is very close to $0$,
the 4-dimensional EFT is no longer reliable as the KK tower descends from UV,
which is consistent with the claim of the AdS/dS distance conjecture.
Meanwhile, when $\frac{3f^{2}\lambda}{32\pi^{2}}$ is fixed to $\frac{3}{4}$,
the radion is stabilized at the Minkowski vacuum ($\Lambda=0$), regardless of
the size of $\lambda$ : the transition between the Minkowski and the (A)dS
vacuum does not take place. Even in this case, the radion stabilization in the
limit of $\lambda\to 0$ is not reliable since $m_{\rm
KK}=\frac{M_{4}}{\sqrt{\pi}}\lambda$ becomes $0$ as well. In terms of the
moduli stabilization, this indicates that the realization of the Minkowski
vacuum through the tiny uplift from very shallow AdS potential generated by
the large flux is invalidated by the descent of the KK tower.

We close this section with the remark on the species scale $\Lambda_{\rm sp}$,
above which gravity is no longer weakly coupled : denoting the number of
states below $\Lambda_{\rm sp}$ by $N_{\rm sp}$, $\Lambda_{\rm sp}$ is defined
by the requirement that the gravity coupling $N_{\rm sp}\Lambda_{\rm
sp}/M_{4}^{2}$ becomes ${\cal O}(1)$, i.e., $\Lambda_{\rm
sp}=M_{4}/\sqrt{N_{\rm sp}}$ [47, 48]. Then $\Lambda_{\rm sp}$ is a natural
cutoff scale of the EFT in which the gravitational interaction is weak. When
$m_{\rm KK}$ decreases, the descent of the KK tower leads to the increase in
$N_{\rm sp}$, thus the decrease in $\Lambda_{\rm sp}$. In our case, two extra
dimensions are compactified on $S^{2}$, then the squared mass of the KK mode
is written as $m_{\mathbf{n}}^{2}=\langle
R\rangle^{-2}(n_{1}^{2}+n_{2}^{2})=m_{\rm KK}^{2}(n_{1}^{2}+n_{2}^{2})$, with
$n_{1}$ and $n_{2}$ integers (for more generic case in which the sizes of two
extra dimensions are different, see, e.g., [49] or Appendix A of [50]). Since
we expect that $m_{\rm KK}\ll\Lambda_{\rm sp}$, $N_{\rm sp}$ is approximated
by a quarter of the area of the disc with radius $\Lambda_{\rm sp}/m_{\rm
KK}$. Omitting ${\cal O}(1)$ coefficient, $N_{\rm sp}$ is estimated as
$\Lambda_{\rm sp}^{2}/m_{\rm KK}^{2}$. Combining this with the relation
$\Lambda_{\rm sp}=M_{4}/\sqrt{N_{\rm sp}}$, we obtain

$\begin{split}N_{\rm sp}=\frac{M_{4}}{m_{\rm KK}},\quad\quad\Lambda_{\rm
sp}=m_{\rm KK}^{1/2}M_{4}^{1/2}.\end{split}$ (24)

Moreover, using the relation $M_{4}=\sqrt{4\pi}M_{6}^{2}/m_{\rm KK}$ (see
(3)), one finds that $\Lambda_{\rm sp}=M_{6}$. That is, the strong
gravitational interaction becomes evident above the more fundamental gravity
scale $M_{6}$ (see also [51] for more discussion). This indeed is consistent
with the well known proposal that the very high mass scale $M_{4}$ may be the
result of the large extra dimensions, in which case the Planck mass of the
higher dimensional UV completion ($M_{6}$ in our case) is low [52]. It is also
remarkable that $N_{\rm sp}$ is nothing more than $\langle x\rangle^{-1}$,
since

$\begin{split}N_{\rm sp}\sim M_{4}\langle R\rangle=\sqrt{4\pi}(M_{6}\langle
R\rangle)^{2}=\frac{\sqrt{4\pi}}{\langle x\rangle}.\end{split}$ (25)

Therefore, using (9) we can parametrize $N_{\rm sp}$ in terms of $f$ and
$\lambda$. For example, when $\lambda=0$, the relation $N_{\rm sp}\sim\langle
x\rangle^{-1}=\frac{3f^{2}}{64\pi^{2}}$ is satisfied, showing that the number
of KK modes below $\Lambda_{\rm sp}=M_{6}$ increases as $f$ increases.
Meanwhile, when $\lambda\neq 0$ and the value of
$\frac{3f^{2}\lambda}{32\pi^{2}}$ is fixed, the combination
$\lambda(M_{6}\langle R\rangle)^{2}=\lambda\langle x\rangle^{-1}$ is also
fixed, which leads to $N_{\rm sp}\sim 1/\lambda$ up to ${\cal O}(1)$ constant.
This explicitly shows the descent of the KK tower in the limit $\lambda\to 0$.

##  4 Vacuum transition and the bubble of nothing

As we have seen in Sec. 2, the 6-dimensional Einstein-Maxwell theory contains
essential ingredients of the moduli stabilization, the flux and the uplift
parametrized by $f$ and $\lambda$, respectively. A large number of
(meta)stable vacua in the landscape can be realized by various choices of $f$
and $\lambda$, provided the EFT description is protected from the descent of
the KK tower. Moreover, the 6-dimensional Einstein-Maxwell theory admits the
instanton solution describing the nucleation of the bubble, which allows a
transition between vacua having different values of $\Lambda_{4}$ [29, 30, 31,
32]. Among possible solutions for the bubble, the ‘bubble of nothing’ is of
particular interest in light of the cobordism conjecture [41]. In this
solution, noncompact spacetime cannot extend inside the bubble as it pinches
off at the bubble wall, where the radius of the extra dimensions shrinks to
zero size [28].

To investigate various aspects of the bubble solution including the bubble of
nothing in detail, we first note that in the bubble solution of the
6-dimensional Einstein-Maxwell theory, the magnetic flux through $S^{2}$ is
‘discharged’ by the black 2-brane at the bubble wall. Here the black 2-brane
is extended over noncompact $(1+2)$-dimensions and described by the metric
[29]

$\begin{split}ds^{2}=\Big{(}1-\frac{\sqrt{3}g_{6}}{8\pi
M_{6}^{2}\rho}\Big{)}^{2/3}(-dt^{2}+dx^{2}+dy^{2})+\Big{(}1-\frac{\sqrt{3}g_{6}}{8\pi
M_{6}^{2}\rho}\Big{)}^{-2}d\rho^{2}+\rho^{2}(d\psi^{2}+\sin^{2}\psi
d\omega^{2}),\end{split}$ (26)

which is the solution to the Einstein equation in the presence of the magnetic
flux through $S^{2}$, i.e., the electromagnetic field strength given by
$F_{2}=\frac{g_{6}}{4\pi}\sin\psi d\psi\wedge d\omega$. Then the flux quanta
outside and inside the bubble are given by $N$ (see (2)) and $N-n$,
respectively, where $n$ is the number of branes. Even in this case, the values
of the 6-dimensional cosmological constant $\Lambda_{6}$ (hence $\lambda$)
inside and outside the bubble are the same, since $\Lambda_{6}$ is not changed
by the black 2-brane. As a result, the values of $\Lambda_{4}$ inside and
outside the bubble are different and the bubble solution well describes the
transition between two vacua through the tunnelling [53]. We also expect that
another bubble can be nucleated inside the bubble, which leads to the
sequential change in $\Lambda_{4}$ along some particular direction (normal to
the bubble wall) in the coordinate space.

When the flux inside the bubble vanishes ($f=0$), $\Lambda_{4}$ becomes
negative infinity, hence the geometry inside the bubble corresponds to AdS
space with the vanishing curvature radius ($H=\infty$). In this case, $\langle
R\rangle=0$ and as can be inferred from (19), the volume vanishes for any
finite value of $r$. Then the region inside the bubble may be regarded as
being empty, i.e., not filled with spacetime. From this, [32] claimed that the
bubble of vanishing flux can be identified with the bubble of nothing, which
was extensively studied in [30, 31, 32].

If $\Lambda_{6}=0$ ($\lambda=0$), as we discussed in Sec. 2 (and summarized in
Fig. 1), the radion is always stabilized at the AdS vacuum where $\Lambda_{4}$
is negative. The decrease in the flux by the black 2-brane leads to the
nucleation of the bubble in which the geometry is given by AdS space with the
smaller value of $\Lambda_{4}$ (or equivalently, the larger value of $H$).
Then through the sequential nucleation of the bubbles lowering $\Lambda_{4}$,
the AdS vacuum with $\Lambda_{4}=-\infty$ can be reached, which is interpreted
as a nucleation of the bubble of nothing. Transition to the bubble of nothing
from dS or Minkowski space can be realized when $\Lambda_{6}\neq 0$
($\lambda\neq 0$ : see also discussion in Sec. 2 which is summarized in Fig.
2). Since the nucleation of the bubble just changes $f$, but not $\lambda$,
the transitions between dS, Minkowski, and AdS space are not obstructed by the
descent of the KK tower.

These features are evident in the thin wall approximation, in which the
geometry inside (or outside) the bubble is well described by (Euclidean) (A)dS
space, except for the narrow region near the bubble wall. Moreover, we assume
that the 4-dimensional part of the metric (2) describing the geometry inside
the bubble respects the spatial isotropy thus can be written as

$\begin{split}g_{\mu\nu}dx^{\mu}dx^{\nu}=d\xi^{2}+a(\xi)^{2}d\Omega_{3}^{2},\end{split}$
(27)

where we use the Euclidean signature since the bubble is an instanton
solution. In addition, $\phi$ depends only on $\xi$. We also expect that the
metric describing the region outside the bubble can be written in the same
form, but $\Lambda_{4}$ becomes different as the flux outside and inside the
bubble are different. Restricting our attention to the region inside the
bubble, the action is given by

$\begin{split}S_{E}^{\rm in}&=\int
d^{4}x\sqrt{g}\Big{[}-\frac{M_{4}^{2}}{2}{\cal
R}+\frac{1}{2}g^{ij}\partial_{i}\phi\partial_{j}\phi+V(\phi)\Big{]}\\\
&=2\pi^{2}\int
d\xi\Big{[}3M_{4}^{2}(a^{2}a^{\prime\prime}+a{a^{\prime}}^{2}-a)+a^{3}\Big{(}\frac{1}{2}{\phi^{\prime}}^{2}+V(\phi)\Big{)}\Big{]},\end{split}$
(28)

where the prime indicates the derivative with respect to $\xi$. Thus, $a(\xi)$
and $\phi(\xi)$ satisfy

$\begin{split}&\frac{3({a^{\prime}(\xi)}^{2}-1)}{a(\xi)^{2}}=\frac{1}{M_{4}^{2}}\Big{(}\frac{\phi^{\prime}(\xi)^{2}}{2}-V(\phi)\Big{)}\\\
&\phi(\xi)^{\prime\prime}+3\frac{a^{\prime}(\xi)}{a(\xi)}\phi^{\prime}(\xi)=\frac{dV(\phi)}{d\phi}.\end{split}$
(29)

In the thin wall approximation, the geometry inside the bubble is more or less
well described by the solution $\phi=\langle R\rangle=$constant (hence
$V(\phi)=V(\langle R\rangle)$), i.e., the stabilized radion. Depending on the
sign of $V(\langle R\rangle)$, $a(\xi)$ describes the Euclidean version of AdS
($V(\langle R\rangle)<0)$, Minkowski ($V(\langle R\rangle)=0$), and dS
($V(\langle R\rangle)>0$) space, respectively. Since $|V(\langle
R\rangle)|=3M_{4}^{2}H^{2}$, $a(\xi)$ in the AdS vacuum ($V(\langle
R\rangle<0$) is given by

$\begin{split}a(\xi)=\frac{1}{H}\sinh[H(\xi-\xi_{0})],\end{split}$ (30)

whereas $a(\xi)$ in the dS vacuum ($V(\langle R\rangle>0$) is given by

$\begin{split}a(\xi)=\frac{1}{H}\cos[H(\xi-\xi_{0})].\end{split}$ (31)

From (21), $H$ can be written as

$\begin{split}H=\frac{1}{3\langle
R\rangle}\Big{|}\frac{\lambda}{\pi}M_{4}\langle
R\rangle-1\Big{|}^{1/2}\end{split}$ (32)

in both cases. For Euclidean Minkowski space ($V(\langle R\rangle)=0$), we
just have $a(\xi)=\xi$.

The action for the whole region, which includes the bubble wall as well as the
regions inside and outside the bubble, gives the profiles of $a(\xi)$ and
$\phi(\xi)$ describing the interpolation between two different vacua. Even
though both $a(\xi)$ and $\phi(\xi)$ are continuous over the whole range of
$\xi$, their derivatives $a^{\prime}(\xi)$ and $\phi^{\prime}(\xi)$ show
discontinuity at the bubble wall in the thin wall approximation. This is well
explored in [31], which is also reviewed in App. A to make the discussion
self-contained. Moreover, whereas we are mainly interested in so-called the
‘down tunnelling’, the nucleation of the bubble with the smaller value of
$\Lambda_{4}$, there can be the ‘up tunnelling’, the nucleation of the bubble
with the larger value of $\Lambda_{4}$. Regarding this, we note that Euclidean
dS space is compact, leading to the finite value of the Euclidean action
$S_{E}$ even if the integration is taken over the region outside the bubble.
In contrast, Euclidean AdS or Minkowski space is noncompact, hence $S_{E}$ for
the region outside the bubble diverges. As pointed out in [32] (and also
[53]), such a difference prevents the transition from Minkowski or AdS space
to the vacuum with larger $\Lambda_{4}$. This is also reviewed in App. A.

Figure 8: The qualitative profile of $V(\phi(\xi))$ in the sequential
nucleation of bubbles. The flux changing black 2-branes are located at
$\xi_{1}$, $\xi_{2}$, and $\xi_{3}$, respectively, and the value of
$\Lambda_{4}$ in each bubble, i.e., the region between branes, is almost
constant in the thin wall approximation.

When the sequential nucleation of the bubbles takes place, the change in
$\Lambda_{4}$ in the moduli space (determined by the value of
$\langle\phi\rangle$) also can be realized in the real coordinate space along
$\xi$ (see Fig. 8). In particular, consider the transition toward the nothing,
the AdS vacuum with $H=\infty$, along the direction of $\xi\to 0$ through the
sequential nucleation of the bubbles. To describe this, suppose the geometry
before the nucleation of the bubble is given by AdS space and the bubble walls
are located at $\xi_{1},\xi_{2},\cdots,$ respectively. Then $\Lambda_{4}$ can
get smaller through the sequential nucleation of the bubbles until it reaches
the negative infinity, i.e., the ‘nothing’. 444In fact, the production of the
bubble with the larger value of $\Lambda_{4}$ in AdS space is suppressed. In
particular, As mentioned above, the nucleation of the small bubble with the
larger value of $\Lambda_{4}$ in the noncompact Euclidean AdS space is
forbidden, as argued in [32] and also reviewed in App. A. Denoting the value
of $H$ in the region $\xi\in(\xi_{i},\xi_{i+1})$ by $H_{i}$, one can infer
from (30) that $a(\xi)$ in this region is written as

$\begin{split}a_{i}(\xi)=\frac{1}{H_{i}}\sinh[H_{i}(\xi-\xi^{i}_{0})].\end{split}$
(33)

We note that in the bubble of nothing, $|a(\xi)|$ diverges since $H\to\infty$.
Ignoring the profile of $a(\xi)$ at the bubble wall which is assumed to be
very narrow, it is a good approximation to require that $a(\xi)$ continuously
changes from $a_{i}$ to $a_{i+1}$ at $\xi_{i+1}$, which gives

$\begin{split}\xi_{0}^{i}=\xi_{i+1}-\frac{1}{H_{i}}\sinh^{-1}\Big{[}\frac{H_{i}}{H_{i+1}}\sinh[H_{i+1}(\xi_{i+1}-\xi_{0}^{i+1})]\Big{]}.\end{split}$
(34)

In this way, every value of $\xi_{0}^{i}$ is determined by the value of
$\xi_{0}^{i}$ in the region far outside the bubble. Example of the profile of
$a(\xi)$ is shown in Fig. 9.

Figure 9: Profile of $a(\xi)$ describing the transition between AdS spaces,
where $H$ increases (hence $\Lambda_{4}(<0)$ decreases) discontinuously along
the direction of $\xi\to 0$ : the values of $H$ are taken to be $5$ for
$1<\xi<2$, $3$ for $2<\xi<3$, $1$ for $3<\xi<4$, and $0.1$ for $4<\xi<5$,
respectively, and $\xi_{0}^{4}=0.2$.

When the spacetime geometry before the bubble nucleation is given by dS space,
$a_{i}(\xi)$ in the dS region (the region in which $\Lambda_{4}>0$) is

$\begin{split}a_{i}(\xi)=\frac{1}{H_{i}}\cos[H_{i}(\xi-\xi^{i}_{0})].\end{split}$
(35)

This shows that in the dS region, $a(\xi)$ strongly oscillates for large value
of $H$, but as we have seen in Sec. 2, $H$ is restricted to be smaller than
$M_{4}\lambda/(6\sqrt{\pi})$ at which $\frac{3f^{2}\lambda}{32\pi^{2}}=1$ is
satisfied (see (20)) since otherwise the radion is not stabilized by the
runaway behavior of the potential. This is remarkable because the obstruction
of the radion stabilization is naturally connected to that of the
semiclassical approximation : when $H>M_{4}$ the period of $a(\xi)$ is smaller
than $M_{4}^{-1}$ but in this case $H$ is not fixed by the radion
stabilization. For the transition between two dS spaces at $\xi_{i+1}$, the
continuity between $a_{i}$ and $a_{i+1}$ leads to

$\begin{split}\xi_{0}^{i}=\xi_{i+1}-\frac{1}{H_{i}}\cos^{-1}\Big{[}\frac{H_{i}}{H_{i+1}}\cos[H_{i+1}(\xi_{i+1}-\xi_{0}^{i+1})]\Big{]}.\end{split}$
(36)

On the other hand, for the transition from dS to AdS space at $\xi=\xi_{i+1}$,
the continuity condition becomes

$\begin{split}\xi_{0}^{i}=\xi_{i+1}-\frac{1}{H_{i}}\sinh^{-1}\Big{[}\frac{H_{i}}{H_{i+1}}\cos[H_{i+1}(\xi_{i+1}-\xi_{0}^{i+1})]\Big{]}.\end{split}$
(37)

An example of the transition from dS to AdS space through the nucleation of
the bubble is shown in Fig. 10.

Figure 10: Profile of $a(\xi)$ describing the transition from dS to AdS space
(at $\xi=3$), where $\Lambda_{4}$ decreases discontinuously along the
direction of $\xi\to 0$ : the values of $H$ are taken to be $3$ for $1<\xi<2$
(AdS), $1$ for $2<\xi<3$ (AdS), $3$ for $3<\xi<4$ (dS), and $10$ for $4<\xi<5$
(dS), respectively, and $\xi_{0}^{4}=0.2$.

As we have investigated, the transition between vacua through the bubble
nucleation connects spacetime with any allowed value of $\Lambda_{4}$ to the
nothing. Indeed, since the nothing is the AdS vacuum with
$\Lambda_{4}=-\infty$, the successful transition from the dS vacuum to the
nothing requires that the transition from the dS to the AdS or the Minkowski
vacuum takes place without obstruction by the descent of the KK tower.
Regarding this, we recall that the descent of the KK tower becomes problematic
when the combination $\frac{3f^{2}\lambda}{32\pi^{2}}$ is kept fixed and the
limit $\lambda\to 0$ is taken. In our case, in contrast, the transition takes
place by changing $f$, while $\lambda$ remains fixed so the transition from
the dS vacuum to the nothing is not obstructed by the descent of the KK tower.
If we can realize the UV completion which gives rise to the simultaneous
changes in $f$ and $\lambda$ with $f^{2}\lambda$ fixed, dS space may not be
connected to the nothing through the bubble nucleation since $\Lambda_{4}$
does not change the sign under varying $\lambda$ and the KK tower descends
from UV in the limit $\lambda\to 0$. Then we may say that the realization of
the metastable dS vacuum is a counterexample of the cobordism conjecture or
the metastable dS vacuum cannot exist according to the cobordism conjecture.

##  5 Conclusions

The 6-dimensional Einstein-Maxwell theory compactified on $S^{2}$ contains
essential ingredients of the moduli stabilization like the curvature of the
extra dimensions, the flux through the extra dimensions, and the uplift, which
also can be found in the string compactification. This motivates us to
investigate the swampland conjectures concerning the moduli stabilization in
the context of the 6-dimensional Einstein-Maxwell theory. In this article, we
first consider the scale separation and the scaling law, both of which are
based on the distance conjecture. We found that in two cases, the AdS vacuum
with the vanishing uplift ($\lambda=0$) and any type of the vacuum with the
fixed value of the combination $f^{2}\lambda$, the KK mass scale and the value
of $\Lambda_{4}$ obey the scaling law. This indicates that, as claimed by the
AdS/dS distance conjecture, (A)dS space with the tiny value of $\Lambda_{4}$
is not interpolated to the Minkowski space, but is close to the swampland in
which the KK tower descends from UV. Moreover, the 6-dimensional Einstein-
Maxwell theory allows the transition between vacua through the nucleation of
the bubble. Of the particular interest is the case of $\Lambda_{4}=-\infty$,
which was claimed to be identified with the ‘nothing’. According to the
cobordism conjecture, the background geometry consistent with quantum gravity
can smoothly evolve into the nothing at which the compact $S^{2}$ shrinks to
zero size. Since the combination $f^{2}\lambda$ is not fixed in this case,
that is, the flux is changed by the bubble wall but the uplift remains fixed,
the transition from the metastable dS vacuum to the nothing is not obstructed
by the descent of the KK tower. This implies that the metastable dS vacuum in
the 6-dimensional Einstein-Maxwell is not excluded by the cobordism
conjecture.

It is quite remarkable that whether the given vacuum belongs to the landscape
or the swampland crucially depends on the relation between parameters
representing different ingredients of the moduli (radion in this case)
potential. That is, if the flux $f$ and the uplift $\lambda$ are independent,
the metastable dS vacuum is not obstructed by the descent of the KK tower and
is cobordant to the nothing. Then the distance and the cobordism conjecture
cannot be used to claim that the metastable dS vacuum belongs to the
swampland. In contrast, the opposite conclusion can be drawn if $f$ and
$\lambda$ are no longer independent, but constrained by
$f^{2}\lambda$=(constant). Since such a relation can be explained in the UV
completion, this may suggest that the EFT alone is not sufficient to
distinguish the landscape from the swampland, but the appropriate UV
completion needs to be taken into account.

### Acknowledgements

##  Appendix A Details of the bubble solution

In the thin wall approximation, the value of $\Lambda_{4}$ inside (or outside)
the bubble can be approximated by a constant unless the narrow region near the
bubble wall is considered. Thus, the geometry of the region inside (or
outside) the bubble is well described by Euclidean (A)dS or Minkowski space,
depending on the sign of $\Lambda_{4}$. Meanwhile, at the bubble wall, the
black 2-brane which changes the flux, hence $\Lambda_{4}$, is located. Then
even though $a(\xi)$ and $\phi(\xi)$ are continuous at the bubble wall, their
derivatives are not, as studied in [31]. To see this in detail, we consider
the action describing the whole region, i.e., the bubble wall as well as the
regions inside and outside the bubble :

$\begin{split}S_{E}=&S_{E}^{\rm in}+S_{E}^{\rm brane}+S_{E}^{\rm out}\\\
=&2\pi^{2}\int_{0}^{\overline{\xi}}d\xi\Big{[}a^{3}\Big{(}\frac{1}{2}{\phi^{\prime}}^{2}+V\Big{)}+3M_{4}^{2}\Big{(}\frac{d}{d\xi}(a^{2}a^{\prime})-a{a^{\prime}}^{2}-a\Big{)}\Big{]}\\\
&+2\pi^{2}a^{3}Te^{-\frac{3}{2}\frac{\phi}{M_{4}}}\Big{|}_{\overline{\xi}}\\\
&+2\pi^{2}\int_{\overline{\xi}}^{\infty}d\xi\Big{[}a^{3}\Big{(}\frac{1}{2}{\phi^{\prime}}^{2}+V\Big{)}+3M_{4}^{2}\Big{(}\frac{d}{d\xi}(a^{2}a^{\prime})-a{a^{\prime}}^{2}-a\Big{)}\Big{]},\end{split}$
(A.1)

where $\overline{\xi}$ is the location of the bubble wall, and we included the
DBI action for the black 2-brane. The brane tension is given by
$T=\frac{2}{\sqrt{3}}(ng_{6})M_{6}^{2}$ ($n$: the number of branes), which is
identified with the ADM mass of the brane solution (26) (see Eq. (2.6) of
[55]). Under the variations $\delta a$ and $\delta\phi$, $\delta S_{E}^{\rm
in}$ and $\delta S_{E}^{\rm out}$ provide the surface term containing
$a^{\prime}$ and $\phi^{\prime}$. That is, under $\delta\phi$ the kinetic term
gives $2\pi^{2}a^{3}\phi^{\prime}\delta\phi$. Meanwhile, under $\delta a$ the
Einstein-Hilbert term gives $12\pi^{2}M_{4}^{2}aa^{\prime}\delta
a+12\pi^{2}M_{4}^{2}a^{2}\delta
a^{\prime}=\frac{d}{d\xi}(6\pi^{2}M_{4}^{2}a^{2}\delta a)$ from the first
surface term and $-12\pi^{2}M_{4}^{2}aa^{\prime}\delta a$ from the second
term, respectively. The former, the surface term of the surface term is
defined on the boundary of the boundary which is empty, so we may ignore it.
Identifying the sum of these variations with the variation of the brane term,
$(6\pi^{2}a^{2}\delta
a-3\pi^{2}a^{3}\frac{\delta\phi}{M_{4}})Te^{-\frac{3}{2}\frac{\phi}{M_{4}}}$,
we obtain

$\begin{split}&\Delta\phi^{\prime}\equiv\phi^{\prime}_{\rm
out}-\phi^{\prime}_{\rm
in}=-\frac{3}{2}\frac{T}{M_{4}}e^{-\frac{3}{2}\frac{\phi}{M_{4}}},\\\ &\Delta
a^{\prime}\equiv a^{\prime}_{\rm out}-a^{\prime}_{\rm
in}=-\frac{T}{2M^{2}}ae^{-\frac{3}{2}\frac{\phi}{M_{4}}}.\end{split}$ (A.2)

In particular, the second relation gives

$\begin{split}\Delta\Big{(}\frac{1}{2}{\phi^{\prime}}^{2}-V\Big{)}&=M_{4}^{2}\Delta\frac{3{a^{\prime}}^{2}-1}{a^{2}}=M_{4}^{2}\frac{6a^{\prime}}{a}\Delta
a\\\ &=-3T\frac{a^{\prime}}{a}e^{-\frac{3}{2}\frac{\phi}{M_{4}}},\end{split}$
(A.3)

where $a^{\prime}$ can be taken to be the mean value between
$a^{\prime}({\overline{\xi}}+\epsilon)$ and
$a^{\prime}({\overline{\xi}}-\epsilon)$ ($\epsilon\ll 1$).

On the other hand, the transition between vacua through the nucleation of the
bubble includes the ‘up tunnelling’, in which the value of $\Lambda_{4}$
inside the bubble is larger than that outside the bubble. The qualitative
feature of the up tunnelling rate crucially depends on the sign of
$\Lambda_{4}$ outside the bubble. In particular, if the geometry outside the
bubble is AdS or Minkowski space, the Euclidean version of which is
noncompact, the up tunnelling rate vanishes [32]. To see this, consider two
vacua, say, A vacuum and B vacuum where the value of $\Lambda_{4}$ in A vacuum
is larger than that in B vacuum. Then the tunnelling rate from A vacuum to B
vacuum satisfies [53]

$\begin{split}\Gamma_{A\to B}\sim e^{-(S_{E}({\rm
instanton})-S_{E}(A))}.\end{split}$ (A.4)

At the same time, there can be the ‘up tunnelling’, the transition from B
vacuum to A vacuum, the rate of which is given by

$\begin{split}\Gamma_{B\to A}\sim e^{-(S_{E}({\rm
instanton})-S_{E}(B))}.\end{split}$ (A.5)

Here the instanton action $S_{E}$(instanton) in $\Gamma_{A\to B}$ is the same
as that in $\Gamma_{B\to A}$ and finite. Now, $S_{E}(A)$ and $S_{E}(B)$ can be
obtained by putting the solutions to the equations of motion (29) into (28).
Replacing ${\phi^{\prime}}^{2}/2$ by
$V+3M_{4}^{2}[({a^{\prime}}^{2}-1)/a^{2}]$, we obtain [54]

$\begin{split}S_{E}^{\rm in}=4\pi^{2}\int
d\xi(a^{3}V-3M_{4}^{2}a)+6\pi^{2}M_{4}^{2}\int
d\xi\frac{d}{d\xi}(a^{2}a^{\prime}),\end{split}$ (A.6)

where the last surface term will not be considered since we are interested in
the behaviors of $a(\xi)$ and $\phi(\xi)$ in the bulk. The first equation of
motion in (29) gives the relation

$\begin{split}da=d\xi\sqrt{1-\frac{V}{3M_{4}^{2}}a^{2}},\end{split}$ (A.7)

from which the first integral in (A.6) becomes

$\begin{split}S_{E}^{\rm in}=-12\pi^{2}M_{4}^{2}\int
daa\sqrt{1-\frac{V}{3M_{4}^{2}}a^{2}}=12\pi^{2}\frac{M_{4}^{2}}{V}\Big{(}1-\frac{V}{3M_{4}^{2}}a^{2}\Big{)}^{3/2}\Big{|}_{a_{i}}^{a_{f}},\end{split}$
(A.8)

where the surface term is omitted. Moreover, $S_{E}^{\rm out}$, the action for
the region outside the bubble is also given by the same expression. Therefore,
if the geometry of the region outside the bubble is AdS space, $V<0$ and
$a_{f}$ extends to infinity, thus $S_{E}^{\rm out}({\rm AdS})=-\infty$. Then
the ratio of the transition rates $\Gamma_{B\to A}/\Gamma_{A\to
B}=e^{S_{E}(B)-S_{E}(A)}$ becomes zero if the geometry of B vacuum (outside
the bubble wall) is AdS space. The same conclusion can be drawn when the
geometry of B vacuum is Minkowski space, in which $V=0$ and $a=\xi$, thus
$S_{E}^{\rm out}=-6\pi^{2}M_{4}^{2}(a_{f}^{2}-a_{i}^{2})$ with $a_{f}=\infty$.
This shows that the nucleation of the bubble of the lager value of
$\Lambda_{4}$ in AdS or Minkowski space is forbidden.

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