MPP-2022-58

Mass Hierarchies and Quantum Gravity
Constraints in DKMM-refined KKLT

Ralph Blumenhagen1, Aleksandar Gligovic1,2, Seyf Kaddachi1,2,

_ 1 Max-Planck-Institut für Physik (Werner-Heisenberg-Institut),
Föhringer Ring 6, 80805 München, Germany _

_2 Ludwig-Maximilians-Universität München, Fakultät für Physik,
Theresienstr. 37, 80333 München, Germany_

###### Abstract

We carefully revisit the mass hierarchies for the KKLT scenario with an uplift
term from an anti D3-brane in a strongly warped throat. First, we derive the
bound resulting from what is usually termed “the throat fitting into the bulk”
directly from the Klebanov-Strassler geometry. Generating the small value of
the superpotential $W_{0}$ via the mechanism proposed by Demirtas, Kim,
McAllister and Moritz (DKMM), we identify two possible DKMM-refined KKLT
scenarios for stabilizing the light axio-dilaton modulus. Both of them require
unnaturally large flux numbers that are likely in conflict with tadpole
cancellation. The first scenario even spoils the expected hierarchy between
the bulk and the throat mass scales and implies that the cosmological constant
in the supposed de Sitter minimum is larger than the species scale of the
effective theory in the throat. We also analyze the bounds set by the trans-
Planckian censorship conjecture and show explicitly that both DKMM-refined
KKLT scenarios do not satisfy them. Hence, all evidence points to the
conclusion that the model is inconsistent with quantum gravity, placing it in
the swampland.

##  1 Introduction

The KKLT scenario [1] has been proposed to be a controlled set-up of string
moduli stabilization that eventually leads to a dS minimum. It is important to
see that it is not a fully fledged string construction but rather consists of
a number of ingredients that were argued to be generically present in
consistent string models, working together in an intricate manner to give dS
vacua. In view of the swampland program [2, 3] (see [4, 5, 6] for recent
reviews) this construction has been under intense scrutiny during the last
years, but we think it is fair to say that the KKLT construction has
nevertheless passed many non-trivial tests and that it is still not really
clear where it actually fails.

Let us briefly review, how KKLT works. One is working in an effective 4D
supergravity theory that results from a compactification of the type IIB
superstring with fluxes, branes and orientifold planes. Then one proceeds in
three steps: First, stabilize the complex structure and axio-dilaton moduli
via three-form fluxes in a no-scale non-supersymmetric Minkowski minimum with
$|W_{0}|\ll 1$. Second, one considers the effective theory of the light Kähler
modulus described by the leading order Kähler potential and a superpotential
of the form $W=W_{0}+A\exp(-aT)$. The non-perturbative effect conspires with
the tiny value of $W_{0}$ to stabilize $T$ in a supersymmetric AdS minimum.
Finally, one uplifts the AdS minimum to dS via $\overline{D3}$-branes at the
tip of a warped throat.

This set-up has been scrutinized from various sides in the past. It was
questioned whether an $\overline{D3}$-brane at the tip of a warped throat is
really a stable configuration (see [7] for a review). Moreover, it has been
questioned whether the 4D description of the KKLT AdS minimum does really
uplift to a full 10D solution of string theory [8, 9, 10, 11, 12, 13, 14, 15,
16, 17].

More recently, there were also growing concerns even about step 1. One needs
$|W_{0}|\ll 1$, so that this should better not be in the swampland. In the
large complex structure regime, Demirtas, Kim, McAllister and Moritz (DKMM)
provided a mechanism [18] that gives $W_{0}=0$ at leading order where
subleading (instanton-like) terms provide the stabilization of a final light
complex structure modulus. This leads to exponentially small values of
$W_{0}$. For the later uplift one actually needs a similar controllable
mechanism close to a point in the complex structure moduli space where one
modulus approaches a conifold point so that large warping can occur. The
generalization of the DKMM approach to this so-called coni-LCS regime has been
performed in [19, 20]. There, also techniques for the determination of the
periods of the Calabi-Yau (CY) threefold in this regime of the complex
structure moduli space have been developed (see also [21]). A more detailed
analysis using asymptotic Hodge theory has been performed in [22]. One of
their main results is that, depending on the asymptotic Hodge structure, an
exponentially small $W_{0}$ does not necessarily mean that the scalar
potential leads also to exponentially small mass eigenvalues. Other systematic
studies of these so-called perturbatively flat flux vacua for concrete Calabi-
Yau manifolds have been reported in [23, 24, 25, 26, 27].

Additionally, for the description of the uplift one has to invoke an effective
action that is valid in the strongly warped regime. Based on earlier work
[28], this question has been addressed recently [29, 30, 31, 32, 33]. One of
the main results of [29] is that the uplift term could destabilize111While
this work was completed, we became aware of [34], in which this
destabilization was questioned by computing more carefully the off-shell
effective action for the conifold modulus $Z$.  the complex structure modulus
$Z$, that controls the size of the 3-cycle that shrinks to zero size at the
conifold locus $Z=0$. This provides a lower bound on the parameter
$g_{s}M^{2}\gtrsim O(50)$ pointing into the direction that KKLT is maybe not
compatible with tadpole cancellation [35, 36, 37, 38], which in fact is
thought to be a genuine quantum gravity effect.

In [30] the effective action in the warped throat was analyzed in more detail,
where the parametric dependence of all quantities on the two moduli
$\\{Z,T\\}$ and on the parameters $\\{g_{s},M,N,y_{\rm UV}\\}$ was carefully
determined. Here $M$ is the flux at the bottom of the throat, $N$ its D3-brane
tadpole and $y_{\rm UV}$ the length of the throat (in Klebanov-Strassler (KS)
[39] coordinates) before it enters into the bulk Calabi-Yau. It became evident
that it is the largish parameter $g_{s}M^{2}$ that controls the appearing mass
hierarchies. In particular, the existence of a red-shifted tower of KK-modes
was established, whose mass scale turned out to be close to the mass of the
conifold modulus $Z$. Since this fitted nicely into the scheme of the
emergence proposal [40, 41, 42, 43], it could not conclusively be argued that
the appearance of these ultra-light KK-modes signal a complete breakdown of
the effective theory222In [34] these KK-modes were made responsible for the
change in the off-shell action..

It is the purpose of this work to combine the analysis of this latter approach
[30, 17] with the recent results [18, 19, 20] about generating an
exponentially small $W_{0}$. We call this the “DKMM-refined KKLT scenario”.
This is in the same spirit as the analysis for KKLT-like AdS minima carried
out recently in [23, 24], where here we are less concerned about actual string
model building on concrete CY manifolds than on revealing the parametric
dependence of all involved mass scales. Generalizing the approach of [30], now
we are careful about the three moduli $\\{Z,T,S\\}$ and the parameters
$\\{M,N,y_{\rm UV},a\\}$ where $S$ denotes the axio-dilaton and $a$ the
parameter in the non-perturbative KKLT superpotential.

The paper is organized as follows: In section 2 we review the local Klebanov-
Strassler geometry and derive conditions for this geometry to be glued into an
unwarped CY manifold. It turns out that one of these conditions is the one
that was termed “the throat fitting into the bulk” in [14]. The resulting
constraint was further promoted to the singular-bulk problem in [44]. In fact
in [14] as well as in [44] a more indirect argument based on the D3-brane
backreaction was used, so that it is satisfying that one can derive the same
condition using the explicit flux supported KS solution. In section 3 we
recall the effective theory in the warped throat and the moduli stabilization
scheme of [19, 20] giving $|W_{0}|\ll 1$. We also recall the appearance of
ultra-light KK-modes and the relation to the emergence picture. We distinguish
two possible scenarios for the stabilization of the light axio-dilaton
modulus, labelled as “Scenario 1” and “Scenario 2” throughout the paper.
Scenario 1 appears to be more constrained than Scenario 2, but possibly only
because for the latter, we treat certain parameters as essentially arbitrary,
even though in concrete models they will also be determined by three-form
fluxes.

Section 4 deals with the final stabilization of the Kähler modulus and the
extra constraints arising from a successful uplift to a dS minimum. We find a
number of inequalities the parameters need to satisfy. For our more
constrained Scenario 1, the constraints turn out to be not natural, in the
sense that exponentially small numbers are bounded from below by rational
expressions of the flux quanta. This has a number of consequences.

First, it turns out that the expected hierarchy between the bulk and the
throat mass scales is spoiled, invalidating the employed low-energy effective
action in the throat. In fact, the mass scale of the bulk complex structure
moduli turn out to be smaller than the red-shifted KK mass scale. This leads
to a lowering of the (throat) species scale, such that it becomes smaller than
the cosmological constant in the de Sitter minimum. We notice that in [45] it
was argued that holography imposes a bound on the value of the cosmological
constant already in the supersymmetric AdS vacuum that leads to an analogous
inconsistency for the (bulk) species scale.

Second, both in Scenario 1 and in a bit weaker form in Scenario 2, the above
mentioned constraints lead to much larger fluxes than previously envisioned
and are likely in conflict with tadpole cancellation respectively the recent
tadpole conjecture [35, 36, 37, 38]. We note that a recent analysis [46, 47,
48] of the regions of control in the context of the Large Volume Scenario
arrived at a very similar conclusion. As proof of prin-ciple, we provide in
section 5 two explicit examples showing the analytically derived behaviour for
the effective theory in the throat. These involve D3-brane tadpoles of the
order $O(10^{7})-O(10^{19})$ already from the two fluxes supported in the
throat.

To support further the suspected inconsistency of the DKMM-refined KKLT with
quantum gravity, we analyze in section 6 the bounds set by the trans-Planckian
censorship conjecture (TCC) [49]. Contrary to the very early expectation [1],
we show that in the DKMM-refined KKLT it is justified to use the thin-wall
approximation for the Coleman-de Luccia bubble decay. Plotting the two regions
where TCC is satisfied and where one has a controllable uplift shows that they
are disjoint, though barely touching in a region where the control over the
construction diminishes.

Thus, all evidence points to the conclusion that this DKMM-refined KKLT
construction is inconsistent with quantum gravity, hence in the swampland. For
Scenario 1 control over the effective action breaks down and both scenarios
require very large fluxes that are certainly in conflict with tadpole
cancellation [35, 36, 37, 38]. Moreover, the DKMM-refined KKLT is inconsistent
with TCC.

##  2 Geometric constraints in the warped throat

In this section, we derive a couple of geometric constraints arising from
designing a controlled picture of a Calabi-Yau compactification that develops
a strongly warped throat close to a conifold point in the complex structure
moduli space. These constraints will arise from gluing the local Klebanov-
Strassler solution to a bulk unwarped CY manifold. In particular, we directly
derive the bound for the Calabi-Yau volume modulus resulting from “the throat
fitting into the bulk” in [14].

Note that throughout this paper, we will only explicitly determine the
parametric dependence suppressing numerical prefactors. Moreover, though
certainly present we will not explicitly consider the compact axionic partners
of the three moduli, i.e. $\\{\arg(Z),{\rm Im}(T),{\rm Im}(S)\\}$, which are
stabilized by the same fluxes and non-perturbative effects that stabilize the
saxions. Moreover, the masses of these axions are expected to scale in the
same way as those of their saxionic partners. We are using natural units, i.e.
$M_{\rm pl}=1$ except when it is explicitly written.

###  2.1 The strongly warped regime

Our starting point is type IIB string theory compactified on a CY orientifold,
where three-form fluxes will stabilize some of the complex structure moduli
and the axio-dilaton. Strongly warped throats can develop close to a conifold
point $Z=0$ in the complex structure moduli space. There exists a proposal for
the effective action of the conifold modulus $Z$ and the warped volume modulus
${\cal V}_{w}$ in the strongly warped regime [28, 29]. This is supposed to be
valid at a point very close to a conifold singularity where warping effects
become relevant. At such points the geometry develops a long throat, at the
tip of which a three-cycle $A$ becomes small. The corresponding complex
structure modulus is defined as $Z=\int_{A}\Omega_{3}$.

Such a warped throat can be supported by turning on R-R three-form flux $M$ on
the A-cycle and an NS-NS three-form flux $K$ on the dual B-cycle. This induces
a contribution to the D3-brane tadpole

$\begin{split}N=M\cdot K\,\end{split}$ (2.1)

and self-consistently fixes the conifold modulus at the exponentially small
value

$\begin{split}|Z|\sim\exp\left(-{2\pi K\over g_{s}M}\right)\end{split}$ (2.2)

for $N\gg g_{s}M^{2}$. The full ten-dimensional metric can be parameterized as
[50]

$ds^{2}=e^{2A(y)}g_{\mu\nu}dx^{\mu}dx^{\nu}+e^{-2A(y)}\tilde{g}_{mn}dy^{m}dy^{n}\,$
(2.3)

where $g_{\mu\nu}$ is the four-dimensional spacetime metric and
$\tilde{g}_{mn}$ the Ricci-flat metric of the CY with coordinates $y^{m}$. The
warp factor $A(y)$ depends only on these internal coordinates. The regime of
strong warping is given by

$\begin{split}\mathcal{V}_{w}\absolutevalue{Z}^{2}\ll 1\end{split}$ (2.4)

where the warped volume $\mathcal{V}_{w}$ (measured in units of
$\alpha^{\prime}$) of the CY is given by

$\mathcal{V}_{w}=\frac{1}{g_{s}^{3/2}(\alpha^{\prime})^{3}}\int_{\rm
CY_{3}}d^{6}y\,e^{-4A(y)}\sqrt{\absolutevalue{\tilde{g}}}\sim\tau^{\frac{3}{2}}\,.$
(2.5)

The string coupling $g_{s}$ is related to the axio-dilaton
$S=e^{-\phi}+iC_{0}$ via $g_{s}=\langle s\rangle^{-1}$, with the saxion
$s=e^{-\phi}$. The saxion $\tau$ appears in the complexified Kähler modulus
$T=\tau+i\theta$.

A warped geometry on the deformed conifold is locally described by the
Klebanov-Strassler (KS) solution [39]. This is a cone over $T^{1,1}$, which is
cut off in the IR by a finite size $S^{3}$. The metric of the KS-throat is
explicitly known

$\begin{split}\widetilde{ds}^{2}_{\rm
KS}=\frac{1}{2}\absolutevalue{S}^{\frac{2}{3}}K(y)\Big{[}\frac{dy^{2}+(g^{5})^{2}}{3K^{3}(y)}+&\cosh^{2}{\Big{(}\frac{y}{2}\Big{)}}((g^{3})^{2}+(g^{4})^{2})\\\
+&\sinh^{2}{\Big{(}\frac{y}{2}\Big{)}}((g^{1})^{2}+(g^{2})^{2})\Big{]}\,\end{split}$
(2.6)

where the collection of einbeins $\\{g^{i}\\}$ provides a basis for the
$S^{2}\times S^{3}$ base of the cone. As explained in [30], the deformation
parameter $S$ is related to the conifold modulus $Z$ by a rescaling

$S=(\alpha^{\prime})^{\frac{3}{2}}\sqrt{g_{s}^{\frac{3}{2}}\mathcal{V}_{w}}Z\,$
(2.7)

and has units of $[\text{length}]^{3}$. The factor $K(y)$ is given by

$K(y)=\frac{\left(\sinh{(2y)}-2y\right)^{\frac{1}{3}}}{2^{\frac{1}{3}}\sinh{(y)}}\,$
(2.8)

and at leading order the warp factor takes the form [39, 51]

$e^{-4A(y)}\approx
1+2^{\frac{2}{3}}\frac{(\alpha^{\prime}g_{s}M)^{2}}{\absolutevalue{S}^{\frac{4}{3}}}\mathcal{I}(y)=1+2^{\frac{2}{3}}\frac{g_{s}M^{2}}{(\mathcal{V}_{w}\absolutevalue{Z}^{2})^{\frac{2}{3}}}\mathcal{I}(y)$
(2.9)

with

$\mathcal{I}(y)=\int_{y}^{\infty}dx\frac{x\coth(x)-1}{\sinh^{2}(x)}(\sinh(2x)-2x)^{\frac{1}{3}}\,.$
(2.10)

In the following, as usual we think of the total geometry as such a KS-throat
of length $y_{\rm UV}$ glued to the unwarped bulk Calabi-Yau manifold. A
sketch of the CY is shown in figure 1.

$S^{3}$$y$$S^{2}$$y_{{}_{\text{UV}}}$$\begin{aligned} \text{bulk
\,\,\,\,\,\,\,\,\,\,}\\\
e^{-4A}\sim\mathcal{O}(1)\end{aligned}$$\begin{aligned} \text{throat
\,\,\,}\\\ e^{-4A}\gg 1\end{aligned}$ Figure 1: A schematic figure of a
Calabi-Yau with a KS-throat. The KS-throat is glued to the bulk at the UV
scale $y_{\rm UV}$.

###  2.2 Geometric constraints

Generalizing the results from [30], let us now derive a couple of constraints
on the parameters $\\{g_{s},M,N,y_{\rm UV}\\}$ and the moduli $\\{\tau,Z\\}$
that guarantee a parametrically controlled geometry of this type. In
particular we will derive bounds on the allowed length $y_{\rm UV}$ of the KS
throat which upon combination will lead to a lower bound for the volume
modulus $\tau$ in terms of the D3 tadpole contribution $N=MK$.

First, in order for the supergravity, large radius description to be
consistent one demands that the size of the $S^{3}$ at the tip of the throat
stays larger than the string length. This can be read-off from the KS metric
(2.6) with the warp factor (2.9)

$\begin{split}R^{2}_{S^{3}}\sim e^{-2A(0)}\alpha^{\prime}\,g_{s}^{1/2}\,({\cal
V}_{w}|Z|^{2})^{1\over 3}\sim\alpha^{\prime}g_{s}|M|\end{split}$ (2.11)

leading to the constraint

$\begin{split}\boxed{g_{s}|M|>1}\,.\end{split}$ (2.12)

Second, in the region where the throat is glued to the bulk CY, warping should
be negligible meaning that the warp factor should be of $\mathcal{O}(1)$. By
using the leading order behaviour of $\mathcal{I}(y)$ for large $y$ we end up
with the lower bound

$\begin{split}e^{{4\over 3}y_{\rm UV}}>\left({g_{s}M^{2}\,y_{\rm
UV}\over\tau}\right){1\over|Z|^{4\over 3}}\,.\end{split}$ (2.13)

Third, we require that the throat fits into the bulk, which means that the
contribution of the throat to the warped volume (2.5) must be smaller than the
value of the total volume ${\cal V}_{w}$. To obtain the volume of the KS
throat, we proceed as in formula (2.5) , but now we integrate the radial
coordinate over the restricted domain $y\in[0,y_{\rm UV}]$. Making use of the
diagonal basis for the KS metric and the expression for the warp factor we
find the relevant scaling to be

$\mathcal{V}_{w}^{\rm KS}\sim
g_{s}M^{2}\,(\mathcal{V}_{w}\absolutevalue{Z}^{2})^{\frac{1}{3}}\int_{0}^{y_{\rm
UV}}dy\>\mathcal{I}(y)\sinh^{2}(y)\,.$ (2.14)

For large $y_{\rm UV}$ the integral $\mathcal{I}(y)$ found on the right-hand
side behaves like

$\mathcal{I}(y)\sim y\,e^{-4y/3}$ (2.15)

so that the integral in (2.14) can be approximated by

$\int_{0}^{y_{\rm UV}}dy\>\mathcal{I}(y)\sinh^{2}(y)\approx y_{\rm
UV}\,e^{2y_{\rm UV}/3}\,.$ (2.16)

Thus, imposing $\mathcal{V}_{w}^{\rm KS}<\mathcal{V}_{w}$ leads to the upper
bound

$\begin{split}e^{{4\over 3}y_{\rm UV}}<\left({\tau\over g_{s}M^{2}\,y_{\rm
UV}}\right)^{2}{1\over|Z|^{4\over 3}}\,.\end{split}$ (2.17)

Consistency of this with the lower bound (2.13) implies the condition

$\begin{split}\tau>g_{s}M^{2}\,y_{\rm UV}\,.\end{split}$ (2.18)

Noticing that the scaling of the lower (2.13) and the upper (2.17) bound with
$|Z|$ is the same and that according to (2.2) $|Z|^{-4/3}$ is the only
exponentially large quantity on the right hand side of the bounds, the
geometric cut-off $y_{\rm UV}$ must scale like

$\begin{split}y_{\rm UV}\sim-\log|Z|\sim{N\over g_{s}M^{2}}\,.\end{split}$
(2.19)

For $N\gg g_{s}M^{2}$ this is indeed a large number, i.e. our initial
assumption $y\gg 1$ was justified. Combining the two conditions (2.18) and
(2.19) yields the intriguing relation

$\begin{split}\boxed{\phantom{\Big{(}}\tau>N\ \,}\,,\end{split}$ (2.20)

i.e. the “size” of the CY is larger than the D3-brane tadpole. We notice that
this is the same condition as proposed in equation (5.7) of [14] via a more
indirect argument based on the D3-brane backreacted geometry. It is very
satisfying that the same condition also arises as a geometric constraint for
the explicit KS metric of the deformed conifold.

##  3 Mass hierarchies

Recall that the first step in the KKLT construction is to stabilize the
complex structure moduli such that the value of the superpotential in the
minimum is exponentially small, i.e. $|W_{0}|\ll 1$. How this can be achieved
in the large complex structure regime was described by Demirtas, Kim,
McAllister and Moritz in [18]. The generalization of this algorithm to the
conifold regime and the necessary computation of the periods in this regime
was done in [19, 20]. For more details we refer the reader to these papers.

Here we take these results and analyze what the implications for the various
mass scales in the KKLT scenario are. The algorithm allows us to find concrete
flux configurations which yield a perturbatively flat direction of the vacuum
satisfying

$W_{\rm pert}\rvert_{\vec{U}}=0\,,\qquad{\rm and}\qquad\vec{U}=\vec{p}\,S$
(3.1)

with $\vec{p}$ being related to the flux vectors. This means that all complex
structure moduli are fixed in terms of the axio-dilaton. The masses of the
massive complex structure moduli scale as

$\begin{split}M_{\rm cs}\sim{g_{s}^{1\over 2}\over{\cal V}_{w}}M_{\rm
pl}\,.\end{split}$ (3.2)

Since the string and the bulk Kaluza-Klein scales are actually $M_{\rm s}\sim
g_{s}^{1/4}M_{\rm pl}/{\cal V}^{1/2}_{w}$ and ${M_{\rm KK}\sim M_{\rm
pl}/{\cal V}^{2/3}_{w}}$, $M_{\rm cs}$ is the lightest mass scale in the
bulk333We will denote bulk masses by capital letters and throat masses by
lowercase letters..

After integrating out the $U_{i}$ one gets an effective superpotential for the
remaining two moduli, $Z$ and $S$

$\begin{split}W=-{M\over 2\pi i}Z\log
Z+i{KS}Z+M_{1}Z+O(Z^{2},e^{-S})\,,\end{split}$ (3.3)

where the non-perturbative $\exp(-S)$ terms arise from the world-sheet
instanton corrections $\exp(-U_{i})$ (in the mirror dual picture). The
coefficient $M_{1}$ is a complex parameter derived from the flux vectors [20].
Note that in the second step of the KKLT construction also non-perturbative
terms in the Kähler modulus $T$ will be taken into account. Since for KKLT we
need to work in the strongly warped throat, we employ the corresponding Kähler
potential from [28, 52]444The approximate numerical value of the order one
constant $c^{\prime}$ was determined in [28] to be $c^{\prime}\approx 1.18$.

$\begin{split}K=-3\log(T+\overline{T})-\log(S+\overline{S})+{2c^{\prime}M^{2}|Z|^{2\over
3}\over(S+\overline{S})(T+\overline{T})}+O(\xi^{2})\,\end{split}$ (3.4)

where we have promoted the string coupling $g_{s}$ to the full complex field
$S$. There will be higher order corrections in $\xi\sim M^{2}|Z|^{2\over
3}/(S+\overline{S})(T+\overline{T})$ which are under control if

$\begin{split}\left(|Z|\over{\cal V}_{w}\right)^{2\over 3}\lesssim{1\over
g_{s}M^{2}}\,.\end{split}$ (3.5)

This Kähler potential and the superpotential (3.3) are the defining data of
the effective strongly warped KKLT model whose mass scales we analyze in more
detail in the following. The resulting scalar potential takes the usual form

$\begin{split}V=e^{K}\left(G^{I\overline{J}}D_{I}WD_{\overline{J}}\overline{W}-3|W|^{2}\right)\,\end{split}$
(3.6)

with $I,J\in\\{Z,S,T\\}$.

###  3.1 Conifold modulus and axio-dilaton

Next one stabilizes the conifold modulus. As shown in [30], the leading order
kinetic term for the conifold modulus scales just in the right way to admit a
deformed no-scale structure for the Kähler modulus. Since there the explicit
dependence on the field $S$ was not taken into account, let us extend this
deformed no-scale structure to the three moduli case. Using the Kähler
potential (3.4), one still finds some leading order cancellations

$\begin{split}G^{S\overline{T}}&=0\\\
G^{Z\overline{A}}\partial_{\overline{A}}K&=0+O(\xi)\\\
G^{A\overline{B}}\partial_{A}K\partial_{\overline{B}}K&=3+O(\xi)\\\
G^{S\overline{Z}}\partial_{\overline{Z}}K&=0+O(\xi)\end{split}$ (3.7)

with the sums over the $A,B$ indices restricted to $A,B\in\\{Z,T\\}$. Using
these relations, it follows that for a superpotential $W(Z,S)$ the scalar
potential is at leading order given by

$\begin{split}V\approx e^{K}\left(G^{M\overline{N}}{\cal D}_{M}W{\cal
D}_{\overline{N}}\overline{W}\right)\,,\qquad M,N\in\\{Z,S\\}\end{split}$
(3.8)

with

$\begin{split}{\cal D}_{Z}W:=\partial_{Z}W\,,\quad{\cal
D}_{S}W:=D_{S}W=\partial_{S}W+\partial_{S}K\,W\,.\end{split}$ (3.9)

Hence, $\partial_{Z}W=0$ fixes the conifold modulus at the exponentially small
value

$\begin{split}Z\sim\zeta_{0}\exp\left(-{2\pi K\over
M}S\right)\,,\qquad\text{with }\ \zeta_{0}=e^{2\pi
i\frac{M_{1}}{M}-1}\,.\end{split}$ (3.10)

Plugging this back into the superpotential (3.3) leads to an effective
superpotential for the stabilization of $S$

$\begin{split}W_{\rm eff}={1\over 2\pi i}MZ+O(Z^{2},e^{-S})\sim\frac{M}{2\pi
i}\zeta_{0}\,e^{-{2\pi K\over
M}S}+a_{1}\,e^{-c_{1}S}+a_{2}\,e^{-c_{2}S}+\ldots\end{split}$ (3.11)

where we have indicated the first two leading (mirror dual) word-sheet
instanton corrections. Hence, the stabilization of the axio-dilaton can occur
via a racetrack scenario (see [53] for a KKLT application). Recall that for
large $s$ one can approximate the solution to the F-term equation for $S$ by
solving $\partial_{S}W=0$ instead of $D_{S}W=0$. This yields

$S_{0}=\frac{1}{c_{1}-c_{2}}\log\left(-\frac{c_{1}a_{1}}{c_{2}a_{2}}\right)$
(3.12)

so that $\absolutevalue{c_{1}-c_{2}}\ll c_{2}$ is necessary to get large
$s_{0}=g_{s}^{-1}$. The value of the superpotential at the minimum can be
approximated by

$\begin{split}W_{0}\sim{c_{2}-c_{1}\over
c_{2}}a_{1}e^{-c_{1}S_{0}}\,.\end{split}$ (3.13)

Thus, for a successful controllable stabilization of the axio-dilaton, the
involved coefficients $(M,K,M_{1},a_{1},a_{2},c_{1},c_{2})$ have to be tuned
to a certain degree. In practice these parameters are determined by the
underlying CY manifold and the fluxes turned on [18, 19, 20]. For our purpose,
let us distinguish two possible scenarios:

Scenario 1: the $MZ$-term contributes dominantly to the race-track potential
so that stabilization requires $c_{1}\approx 2\pi K/M$ and one finds
${|W_{0}|\approx|K||Z|}$.

Scenario 2: the $MZ$-term is sub-leading so that the two leading instanton
corrections stabilize $S$. In this case, we can at least say $|W_{0}|>|K||Z|$.

Next, let us estimate the masses of the two moduli $Z$ and $S$ which are given
by the eigenvalues of the mass matrix

$M^{i}_{\,\,\,k}=G^{ij}\partial_{j}\partial_{k}V\,$ (3.14)

where $G^{ij}$ denotes the (inverse) Kähler metric and $V$ is the full scalar
potential of our theory. Here the transformation to a canonically normalized
basis is already taken care of. The mass of the conifold modulus turns out to
be

$\begin{split}m_{Z}\simeq{1\over(g_{s}M^{2})^{1\over 2}}\left(|Z|\over{\cal
V}_{w}\right)^{1\over 3}M_{\rm pl}\,\end{split}$ (3.15)

and for the second and the first scenario, the mass of $S$ can be expressed as

$\begin{split}m^{(2)}_{S}\simeq{c_{1}\,c_{2}\,|W_{0}|\over g_{s}^{3\over
2}{\cal V}_{w}}M_{\rm pl}\,,\qquad\quad
m^{(1)}_{S}\simeq{K^{3}M\over(g_{s}M^{2})^{3\over 2}}\left(|Z|\over{\cal
V}_{w}\right)M_{\rm pl}\,.\end{split}$ (3.16)

Hence, generically the dilaton is parametrically lighter than the conifold
modulus555As pointed out in [22], at the coni-LCS boundary it can in principle
happen that despite the exponentially small value of $W_{0}$, the modulus $S$
receives a polynomial mass..

However, let us already mention that with respect to $|W_{0}|$ the mass of the
dilaton scales in the same way as the mass of the Kähler modulus, to be
discussed next. Therefore we also stabilized them simultaneously in our actual
computations. To leading order, we got the same masses for the moduli $S$ and
$T$ and the same minimum conditions as in the step-by-step procedure presented
for pedagogical reasons in this paper.

#### Destabilization of conifold modulus

For completeness, let us mention an issue that was first observed in [29] (and
questioned recently in [34]). Setting $S$ to its VEV and adding the
contribution of an $\overline{D3}$-brane at the tip of the throat to the
scalar potential one obtains

$\begin{split}V_{\rm tot}={9\over 8c^{\prime}M^{2}}{|Z|^{4\over 3}\over{\rm
Re}(T)^{2}}\bigg{[}\Big{|}{\textstyle-{M\over 2\pi i}\log Z+i{K\over
g_{s}}+M_{1}-{M\over 2\pi i}}\Big{|}^{2}+{c^{\prime}c^{\prime\prime}\over
g_{s}}\bigg{]}\,.\end{split}$ (3.17)

Thus, both the three-form flux and the $\overline{D3}$-brane contribution
scale in the same way with the moduli $Z$ and $T$. Plugging in concrete
numbers666The order one constant $c^{\prime\prime}=2^{1\over
3}/\mathcal{I}(0)\approx 1.75$ as mentioned in [29]. for the constants
$c^{\prime},c^{\prime\prime}$, it was shown that a minimum in the $Z$
coordinate only continues to exist for $\sqrt{g_{s}}|M|>6.8$.

###  3.2 Throat KK-modes and emergence

In the strongly warped throat geometry one direction (namely $y$ along the
throat) becomes much larger than the other bulk directions. Therefore, one is
dealing with a highly non-isotropic geometry that could support very light
Kaluza-Klein modes [54, 55, 56, 57]. It was explicitly shown in [30] that such
modes indeed exist and that, reminiscent of infinite distance limits their
mass scale is related to the distance of the conifold point in the complex
structure moduli space via emergence. This picture allowed one to derive the
cut-off scale $\Lambda$ of the effective theory in the throat. Since there are
KK-modes lighter than this cut-off, the actual cut-off is the species scale
[58, 59] $\tilde{\Lambda}_{\text{sp}}=\Lambda/\sqrt{N_{\rm sp}}$ , where
$N_{\text{sp}}$ denotes the number of light species below the species scale.

Since in [30] the computation was done in the regime $y_{\rm UV}<1$, closely
following this paper, we numerically determined the masses of the KK-modes in
a long warped throat, i.e. $y_{\rm UV}\gg 1$. For that purpose, we solved the
same six-dimensional warped Laplace equation for the radial part of the
eigenmodes. The equation reads

$3K^{2}(y)\partial_{y}^{2}\chi(y)+4\frac{\partial_{y}\chi(y)}{\sinh(y)K(y)}+k^{2}\mathcal{I}(y)\chi(y)=0$
(3.18)

where

$k^{2}=\frac{\alpha^{\prime}g_{s}^{\frac{3}{2}}M^{2}}{2^{\frac{1}{3}}(\mathcal{V}_{w}\absolutevalue{Z}^{2})^{\frac{1}{3}}}m_{\text{KK}}^{2}$
(3.19)

depends on the Kaluza-Klein masses. By respecting the Neumann boundary
conditions

$\partial_{y}\chi_{\rm
n}(y)\rvert_{y=0}=0\quad\text{and}\quad\partial_{y}\chi_{\rm
n}(y)\rvert_{y=y_{\rm UV}}=0$ (3.20)

and the normalization requirement

$\int_{0}^{y_{\rm UV}}\absolutevalue{\chi_{\rm n}(y)}^{2}=1$ (3.21)

for all mode numbers ${\rm n}$, we find eigenfunctions which are localized
near small values of $y$. The plots of the lowest two eigenmodes are shown in
figure 2.

Figure 2: Graphs of the numerically determined eigenfunctions associated to
the two lowest modes, assuming $y_{\rm UV}=10$.

The masses scale approximately linearly with the mode number. The right plot
in figure 3 shows this behaviour for the first few mode numbers for fixed
$y_{\rm UV}$. Remarkably, for large values of $y_{\rm UV}$ the masses become
independent of the UV-cutoff. Since the eigenfunctions are localized near
small $y$, the eigenvalues reach an asymptotic value for $y_{\rm UV}\gtrapprox
10$. In the left-hand side of figure 3 we display this behaviour for the first
eigenmode.

Figure 3: Left: KK mass squared of the first eigenmode as a function of
$y_{\rm UV}$. Right: KK mass squared of the nine lowest eigenmodes, assuming
$y_{\rm UV}=10$.

To summarize, we obtain the following scaling of the red-shifted KK masses

$m_{{\rm
KK,n}}^{2}\sim\frac{(\mathcal{V}_{w}\absolutevalue{Z}^{2})^{\frac{1}{3}}}{g_{s}^{\frac{3}{2}}M^{2}}{\rm
n}^{2}M_{\rm
s}^{2}\sim\frac{1}{g_{s}M^{2}}\left(\frac{\absolutevalue{Z}}{\mathcal{V}_{w}}\right)^{\frac{2}{3}}{\rm
n}^{2}M_{\rm pl}^{2}\,.$ (3.22)

Comparison with the mass of the $Z$-modulus from the previous subsection
yields

$\frac{m_{\rm KK,n}}{m_{Z}}=c\,{\rm n}$ (3.23)

with an order 1 constant $c$. The vast majority of KK-modes is therefore
expected to be heavier than the conifold modulus. Thus, the KK-modes are
parametrically of the same mass as the conifold modulus $Z$ and the
description in terms of the effective action is at its limit.

As shown in [30], the existence and mass scale of these red-shifted KK-modes
is consistent with the picture of emergence of the field space metric [40, 41,
42, 43], which means that integrating them out the field space metric should
get a one-loop correction proportional to the tree-level result. The one-loop
correction came out as

$\displaystyle g_{Z\overline{Z}}^{\rm 1-loop}\sim N_{\rm
sp}^{3}\frac{1}{g_{s}M^{2}}\frac{1}{\left(\mathcal{V}_{w}|Z|^{2}\right)^{2/3}}\,,$
(3.24)

so that matching this with the tree-level metric resulting from (3.4) fixed
the number $N_{\rm sp}$ of light species as

$N_{\rm sp}\sim\left(g_{s}M^{2}\right)^{2/3}\,.$ (3.25)

Using this scaling, we found that the one-loop corrections
$g_{I\overline{J}}^{\rm 1-loop}$ of the field space metric, where
$I,J\in\\{Z,S,T\\}$, are proportional to the tree-level expressions of the
field space metric coming from the Kähler potential $K\sim
M^{2}|Z|^{\frac{2}{3}}/(S+\overline{S})(T+\overline{T})$.

From the spacing and number of modes below the cutoff, one could derive for
the UV cut-off

$\begin{split}\Lambda\sim\sqrt{g_{s}M^{2}}\left({|Z|\over{\cal
V}_{w}}\right)^{\\!{1\over 3}}M_{\rm pl}\,\end{split}$ (3.26)

which intriguingly equals the mass of a D3-brane wrapping the A-cycle $S^{3}$
which vanishes at the conifold singularity. Then, for the generalized species
scale we get

$\begin{split}\tilde{\Lambda}_{\text{sp}}={\Lambda\over\sqrt{N_{\rm
sp}}}\sim{\Lambda\over\left(g_{s}M^{2}\right)^{1/3}}\,.\end{split}$ (3.27)

We can express $m_{Z}\sim\Lambda/(g_{s}M^{2})$ so that in the regime of
interest, $g_{s}M^{2}\gg 1$, one has the expected hierarchy
$m_{Z}<\tilde{\Lambda}_{\text{sp}}$.

In summary, up to this point we have the following hierarchy of mass scales in
the warped throat

$\begin{split}\Lambda>\tilde{\Lambda}_{\text{sp}}>m_{Z}\sim m_{\rm
KK}\,.\end{split}$ (3.28)

How the mass scales of the light $S$ modulus (3.16) and the Kähler modulus fit
into this hierarchy will be discussed in the next section. The bulk masses
$M_{\rm cs}<M_{\rm KK}<M_{\rm s}$ are assumed to be much heavier.

##  4 Kähler modulus

Now, let us consider the second step of KKLT and, after integrating out $Z$
and $S$, consider the effective superpotential

$\begin{split}W_{T}=W_{0}+A\exp(-aT)\end{split}$ (4.1)

where $W_{0}$ is the exponentially small value from the previous race-track
model for the $S$ modulus. The parameter $A$ is the so-called 1-loop Pfaffian,
considered to be independent of the complex structure moduli. The parameter
$a$ is defined as

$\begin{split}a={2\pi\over N_{c}}\gamma\end{split}$ (4.2)

where $N_{c}$ is the rank of the gauge theory featuring gaugino condensation
and $\gamma$ is related to the size of the bulk 4-cycle $\tau_{\rm
np}=\gamma\tau$ supporting this gauge theory. For an isotropic bulk Calabi-Yau
manifold one expects $\gamma=O(1)$.

One comment might be in order here. While this work was in preparation,
employing holography, the authors of [45] suggested that no supersymmetric AdS
minimum à la KKLT can possibly exist in a controlled manner. In particular,
they argue that the AdS energy scale is larger than the (bulk) species scale,
leading at best to a non-scale-separated AdS minimum. In addition, for the
DKMM construction of perturbatively flat flux vacua, the authors argue that no
supersymmetric AdS minimum can potentially exist after fixing the Kähler
moduli. They say that suitable corrections to the superpotential depending on
the Kähler moduli will not materialize. This means that string consistency
conditions, like absence of Freed-Witten anomalies, correct number of
instanton 0-modes etc., could generically forbid any such instanton
correction. Here we proceed under the usual assumption that such an instanton
exists and derive its consequences. This will eventually lead us to the
conclusion that the uplifted de Sitter minimum of the DKMM-refined KKLT
scenario is in the swampland. Thus, logically our result is consistent with
[45].

###  4.1 Mass scales

The position $\tau$ of the supersymmetric AdS minimum is determined by the
solution of the equation

$\begin{split}\absolutevalue{A}(2a\tau+3)=3|W_{0}|e^{a\tau}\end{split}$ (4.3)

yielding the value of the cosmological constant

$\begin{split}V_{\rm AdS}\sim-{g_{s}\over\tau^{3}}|W_{0}|^{2}\,M_{\rm
pl}^{4}\,.\end{split}$ (4.4)

For a successful uplift to a dS vacuum, this has to be of the same scale as
the $\overline{D3}$-brane tension

$\begin{split}V_{\rm up}\sim{1\over g_{s}M^{2}}\left({|Z|\over{\cal
V}_{w}}\right)^{\\!{4\over 3}}\,M_{\rm pl}^{4}\,.\end{split}$ (4.5)

This leads to

$\begin{split}|W_{0}|\sim{1\over g_{s}|M|}\tau^{1\over 2}|Z|^{2\over
3}\,.\end{split}$ (4.6)

We note that for an exponentially small value of $Z$, this seems to be
parametrically compatible with the race-track condition $|W_{0}|>|K||Z|$.
Indeed, combining the latter with (4.6) leads to

$\begin{split}{|Z|\over{\cal
V}_{w}}\lesssim{1\over(g_{s}N)^{3}}\,,\end{split}$ (4.7)

which, however, is parametrically saturated in Scenario 1. In this case, the
exponentially small $|Z|$ has to be parametrically equal to a polynomially
small parameter. This might appear fairly unnatural and certainly requires a
large value of $g_{s}N$. We will see that it implies severe problems with the
validity of the employed low energy effective action in the warped throat.
Moreover, we notice that the bound (4.7) is compatible with control over the
warped effective action, i.e. with the upper bound (3.5) if
$(g_{s}K^{2})\gtrsim 1$.

Let us proceed with the determination of the remaining parameters. For an
exponentially small value of $Z$, one can estimate the value of $\tau$ as

$\begin{split}a\tau\sim-\log|W_{0}|\lesssim-\log|Z|\sim{N\over
g_{s}M^{2}}\,.\end{split}$ (4.8)

Invoking the condition (2.20), namely $\tau>N$, one gets in both Scenario 1
and Scenario 2

$\begin{split}\boxed{a\lesssim{1\over g_{s}M^{2}}\ll 1}\end{split}$ (4.9)

which, as already observed in [14] (for a refinement of the argument see also
[44]), is violated for an isotropic CY and a D3-brane instanton, i.e.
$N_{c}=1$. We notice that the string frame volume of the 4-cycle is given by
$\hat{\tau}\gtrsim g_{s}N=(g_{s}|M|)|K|>1$ and therefore guaranteed to be in
the large volume regime.

The mass of the Kähler modulus scales as

$\begin{split}m_{T}\sim a{g_{s}^{1\over 2}\over{\cal V}_{w}^{1\over
3}}|W_{0}|\,M_{\rm pl}\end{split}$ (4.10)

which, as already mentioned, scales in the same way with $|W_{0}|$ as the mass
of the light $S$ modulus. Therefore, one might suspect that first integrating
out $S$ is not self-consistent. It was argued in [23, 24] that as long as the
parameter $A$ in the KKLT superpotential does not depend on any complex
structure modulus, the minimum prevails. We checked this numerically in
concrete examples.

###  4.2 Mass hierarchies extended

Now we would like to see how the mass scale of the uplift potential fits into
the hierarchies of scales in (3.28). Using (4.5) we can write

$\begin{split}V_{\rm up}^{1\over 4}\sim{\Lambda\over(g_{s}M^{2})^{3\over
4}}\,\end{split}$ (4.11)

which is smaller than the species scale and larger than the mass of $Z$.

Next we analyze the relation between the mass of the Kähler modulus and the
light KK scale. Using the relations (4.6) and (4.8) one gets

$\begin{split}{m_{T}}\lesssim{N\over(g_{s}M^{2})^{3\over
2}}\left({|Z|\over{\cal V}_{w}}\right)^{\\!{2\over 3}}M_{\rm
pl}\sim{N\over(g_{s}M^{2})^{2}}\left({|Z|\over{\cal V}_{w}}\right)^{\\!{1\over
3}}\,\Lambda\,.\end{split}$ (4.12)

Thus, one finds

$\begin{split}{m_{T}\over m_{\rm
KK}}\lesssim{N\over(g_{s}M^{2})}\left({|Z|\over{\cal
V}_{w}}\right)^{\\!{1\over 3}}\lesssim{1\over(g_{s}|M|)^{2}}<1\,,\end{split}$
(4.13)

where we have used the relation (4.7) in the second step and the constraint
(2.12) needed to be in the supergravity regime in the last step. Hence, we
have the desired hierarchy $m_{T}<m_{\rm KK}$.

Let us perform a similar analysis for the relation between the mass of the $S$
modulus and the light KK scale. Using the relation (4.6) one gets for Scenario
2

$\begin{split}{m^{(2)}_{S}\over m_{\rm KK}}\sim{c_{1}c_{2}\over
g_{s}^{2}}\left({|Z|\over{\cal V}_{w}}\right)^{\\!{1\over
3}}\lesssim{c_{1}c_{2}\over g_{s}^{3}N}\,,\end{split}$ (4.14)

where we have used the relation (4.7) in the last step. Hence a sufficient
condition for having the desired hierarchy $m^{(2)}_{S}<m_{\rm KK}$ is

$\begin{split}\boxed{c_{1}c_{2}<g_{s}^{3}N}\,.\end{split}$ (4.15)

In Scenario 1 the constraint (4.15) becomes

$\begin{split}\boxed{g_{s}M^{2}<N<g_{s}\left(g_{s}M^{2}\right)^{2}}\,,\end{split}$
(4.16)

where the first relation has to hold to guarantee $|Z|\ll 1$. Note that in
Scenario 1, for the ratio of the two lightest moduli we find
$m_{T}/m^{(1)}_{S}\sim g_{s}M/K$ which has no definite hierarchy.

Summarizing, our analytic analysis revealed the following order of all the
relevant red-shifted mass scales of the warped throat

$\Lambda>\tilde{\Lambda}_{\text{sp}}>V^{1\over 4}_{\text{up}}>m_{Z}\sim m_{\rm
KK}>m_{S/T}\,.$ (4.17)

#### Mass scales in Scenario 1

Let us first consider the mass scales in Scenario 1 in more detail. Upon
invoking the parametrically saturated relation (4.7), the cut-off can be
expressed as ${\Lambda\sim M_{\rm pl}/(g_{s}K^{2})^{1\over 2}}$. We have seen
that the masses in the throat are ordered in the expected manner. However,
using the saturated relation (4.7), one can now express the lightest bulk mass
scale (3.2) as

$\begin{split}M_{\rm cs}\sim{g_{s}|K|\over\tau^{3\over
2}}\Lambda\lesssim{g_{s}|K|\over N^{3\over 2}}\Lambda\,\end{split}$ (4.18)

which implies

$\begin{split}M_{\rm cs}\lesssim g_{s}^{3\over 2}\left({g_{s}M^{2}\over
N}\right)^{1\over 2}m_{\rm KK}\lesssim m_{\rm KK}\,.\end{split}$ (4.19)

This means that the bulk complex structure moduli are lighter than the throat
red-shifted KK-scale. Hence, in Scenario 1 the bulk and throat are not
energetically separated, spoiling completely the validity of the used
effective action in the throat.

Next we analyze the relation between the lightest bulk mass scale $M_{\rm cs}$
and the masses of the light moduli $S$ and $T$. Using the relation (4.6) along
with the saturated relations (4.7) and (4.8), we get the ratios

$\begin{split}{M_{\rm cs}\over
m_{T}}\sim{\left(g_{s}|M|\right)^{4}|K|\over\tau^{3\over 2}}\,,\qquad{M_{\rm
cs}\over m_{S}^{(1)}}\sim{\left(g_{s}|M|\right)^{5}\over\tau^{3\over
2}}\,.\end{split}$ (4.20)

To have the desired hierarchy $M_{\rm cs}>m_{S/T}$, we must have then that

$\begin{split}\left(g_{s}|M|\right)^{4}|K|\gtrsim\tau^{3\over 2}\qquad\text{
and }\qquad\left(g_{s}|M|\right)^{5}\gtrsim\tau^{3\over 2}\,.\end{split}$
(4.21)

To estimate the value of $g_{s}$ , let us recall that $S_{0}$ is obtained from
(3.12), where $a_{1}=M\zeta_{0}/(2\pi i)$ and $c_{1}=2\pi K/M$. Since
$c_{1}\approx c_{2}$ is required to obtain large $s_{0}=g_{s}^{-1}$, we can
set ${c_{2}=(c_{1}^{-1}-1)^{-1}}$. With $a_{2}\sim-a_{1}$, we can then
estimate the value of $g_{s}$ as

$\begin{split}g_{s}=s_{0}^{-1}\sim c_{1}^{2}\sim{K^{2}\over
M^{2}}\,.\end{split}$ (4.22)

Upon invoking this estimate of $g_{s}$ together with the condition (2.20),
namely $\tau>N$, the conditions in (4.21) imply the following constraint on
the fluxes

$\begin{split}|K|\gtrsim|M|^{13/17}\,.\end{split}$ (4.23)

Let us roughly estimate the number of bulk complex structure moduli to be
${h_{31}\sim\chi(Y)\gtrsim N}$, where in the final step we have assumed that
the large tadpole from the fluxes in the throat $N=MK$ is not cancelled by
other contributions. Since these extra fields and all their KK-modes are now
light, the species scale also becomes smaller and can be estimated as

$\begin{split}\tilde{\Lambda}_{\text{sp}}\lesssim{\Lambda\over N^{1\over
2}(g_{s}M^{2})^{1\over 3}}\,.\end{split}$ (4.24)

Apparently, this scale is smaller than $V_{\rm up}^{1\over 4}$ as

$\begin{split}\tilde{\Lambda}_{\text{sp}}\lesssim\left({g_{s}M^{2}\over
N}\right)^{1\over 2}V_{\rm up}^{1\over 4}\lesssim V_{\rm up}^{1\over
4}\,.\end{split}$ (4.25)

We notice that for this more specific model we have arrived at a very similar
conclusion as [45], though not using holography but the existence of an uplift
of the initial AdS-minimum. In table 1 we show the hierarchy of the bulk and
the throat mass scales. On the left we list the expected order, if there were
a separation between the bulk and the red-shifted throat. On the right, we
list the non-separated mass scales found for Scenario 1. The latter is indeed
confirmed by an explicit numerical computation for an example provided in the
next section.

bulk-throat separated scales
---
$M_{\rm cs}\sim{\sqrt{g_{s}}\over{\cal V}_{w}}M_{\rm pl}$
$\Lambda\sim{M_{\rm pl}/(g_{s}K^{2})^{1\over 2}}$
$\tilde{\Lambda}_{\text{sp}}\sim\eta^{\frac{1}{3}}\Lambda$
$V_{\rm up}^{1\over 4}\sim\eta^{\frac{3}{4}}\Lambda$
$m_{Z}\sim m_{\rm KK}\sim\eta\Lambda$
$m_{S}^{(1)}\sim N^{2}\eta^{7\over 2}\Lambda^{2}\,,\ \ m_{T}\sim
N\eta^{\frac{5}{2}}\Lambda^{2}$

bulk-throat mixed scales
---
$\Lambda\sim{M_{\rm pl}/(g_{s}K^{2})^{1\over 2}}$
$V_{\rm up}^{1\over 4}\sim\eta^{\frac{3}{4}}\Lambda$
$\tilde{\Lambda}_{\text{sp}}\lesssim{\eta\frac{1}{3}\over N^{1\over
2}}\Lambda$
$m_{Z}\sim m_{\rm KK}\sim\eta\Lambda$
$M_{\rm cs}\lesssim{g_{s}|K|\over N^{3\over 2}}\Lambda$
$m_{S}^{(1)}\sim N^{2}\eta^{7\over 2}\Lambda^{2}\,,\ \ m_{T}\sim
N\eta^{\frac{5}{2}}\Lambda^{2}$

Table 1: Expected separated and realistic mixed mass scales for Scenario 1. We
have defined the parameter $\eta=1/\left(g_{s}M^{2}\right)$.

In addition, we have a very large tadpole contribution from the fluxes. From
the now saturated relation (4.7), we can derive the lower bound

$\begin{split}|Z|\sim{\tau^{3\over 2}\over(g_{s}N)^{3}}\gtrsim{1\over
g_{s}^{3}N^{3\over 2}}\,,\end{split}$ (4.26)

where we also used (2.20). This bound is not parametrically natural in the
sense that an exponentially small number on the l.h.s. is larger than a
rational number on the r.h.s.. To get a better impression, let us assume that
for having control over the geometry and the effective action we need the
conifold modulus in a regime $|Z|<10^{-5}$. Then for $g_{s}\sim 0.1$ the bound
(4.26) tells us that $N\sim 10^{5}$, which implies $M\sim 10^{5/2}$ and
$a<10^{-4}$ (for $M\approx N^{1\over 2}$).

Thus, in addition to having the wrong hierarchy we face large integers, i.e.
large fluxes and numbers of branes, large tadpole contributions and either
large non-isotropies in the bulk or a high rank confining gauge group.

#### Mass scales in Scenario 2

This case is less constrained, as we treat the parameters $c_{1}$ and $c_{2}$
in the race-track potential as free parameters. Thus the constraint (4.7) is
mild now. Indeed, the lightest bulk mass scale $M_{\rm cs}$ is larger than the
cut-off $\Lambda$, if

$\begin{split}\left({\cal
V}_{w}^{2}|Z|\right)^{1/3}\lesssim\frac{1}{|M|}\,.\end{split}$ (4.27)

In this case the bulk and the throat are energetically decoupled and we have
the following hierarchies of scales

$M_{\rm cs}>\Lambda>\tilde{\Lambda}_{\text{sp}}>V^{1\over
4}_{\text{up}}>m_{Z}\sim m_{\rm KK}>m_{S/T}\,.$ (4.28)

As a proof of principle, in the next section we provide an example that
features the intended hierarchy (4.28) of all masses.

However, we can still derive a lower bound on the fluxes in the throat. From
the uplift condition (4.6) and the relations (3.13) and (3.10) it follows that

$\begin{split}{|c_{2}-c_{1}|\over c_{2}}\,|a_{1}|\,e^{-c_{1}s_{0}}\sim{1\over
g_{s}|M|}\tau^{1\over 2}|\zeta_{0}|^{2\over 3}e^{-{2\over 3}{2\pi K\over
M}s_{0}}\,.\end{split}$ (4.29)

Setting the exponentially small terms in (4.29) to be of the same scale, we
can estimate

$\begin{split}c_{1}\sim{2\pi K\over M}\,.\end{split}$ (4.30)

As usual, $S_{0}$ is given by (3.12) so that as in Scenario 1, we can set
${c_{2}=(c_{1}^{-1}-1)^{-1}}$. With $a_{2}\sim-a_{1}$, we can then estimate
the value of $g_{s}$ as

$\begin{split}g_{s}=s_{0}^{-1}\sim c_{1}^{2}\,.\end{split}$ (4.31)

Invoking now the (slightly concretized) bound $g_{s}|M|\gtrsim O(10^{n})$ from
(2.12), set by having a valid supergravity, large radius description, we get

$\begin{split}M\gtrsim{10^{n}\over c_{1}^{2}}\,\,,\qquad K\gtrsim{10^{n}\over
c_{1}}\,\,,\qquad N\gtrsim{10^{2n}\over c_{1}^{3}}\,.\end{split}$ (4.32)

Assuming $c_{1}\lesssim 0.1$ for a controlled race-track minimum and
$g_{s}|M|\gtrsim 10$ for a controlled supergravity description, one finds
$N\gtrsim 10^{5}$. This is still a large tadpole contribution that might be in
conflict with tadpole cancellation or the tadpole conjecture, respectively.
For fully fledged models we expect the bound to be much more stringent, as we
treated the race-track parameters as free parameters while for concrete
Calabi-Yau manifolds they are also determined by three-form fluxes
contributing to the tadpole themselves.

#### Comments on tadpole cancellation

Recall that the tadpole cancellation condition reads

$\begin{split}MK+N_{\rm flux}+N_{\rm brane}={\chi(Y)\over 24}\end{split}$
(4.33)

where $N_{\rm flux}$ denotes the contribution of the other present fluxes and
$N_{\rm brane}$ the contribution from D3-branes and magnetized D7-branes. For
such a large flux $N=MK$ one does not expect (almost fine tuned) cancellations
between the various flux and brane contributions to occur. Therefore, one
needs a Calabi-Yau fourfold with a very large Euler characteristic $\chi(Y)$.
This will have many complex structure moduli so that one very likely
encounters the problem of the tadpole conjecture [35, 36, 37, 38]. Namely,
that it is not possible to freeze all of these many moduli using three-form
fluxes, something that was silently assumed before we focused just on the
final two moduli $Z$ and $S$. Hence, we arrive at the conclusion that a
working DKMM-refined KKLT Scenario 1 or 2 would require very large fluxes that
are in conflict with the tadpole constraint. Note that a very similar
conclusion was drawn recently for a controlled Large Volume Scenario [46, 47,
48].

##  5 Numerical analysis

Since throughout our analysis we were invoking quite a number of
approximations, we need to provide a proof of principle. For this purpose, let
us now test the previous results by comparing them with concrete numerical
solutions of the problem.

We consider the full scalar potential after analytically stabilizing the
conifold modulus. We systematically search for a region in parameter space
with a dS minimum by making use of the uplift condition, which will be written
as

$\begin{split}\kappa{g_{s}|W_{0}|^{2}\over\tau_{0}^{3}}\sim{\xi\over
g_{s}M^{2}}{|Z|^{4\over 3}\over\tau_{0}^{2}}\end{split}$ (5.1)

where $\xi$ is the numerical prefactor777From (3.17) we could read off
$\xi={9\over 8}c^{\prime\prime}\simeq{63\over 32}$ but here we prefer to treat
it as a parameter. appearing in the uplift potential and we introduced an
order one “balance” parameter $\kappa$. Since order one factors do matter in
this analysis, we have at least taken factors of $(2\pi)$ into account. In
hindsight, utilizing results from moduli stabilization we derive a condition
on the fluxes $(M,K,M_{1})$ for a successful uplift.

It turns out that a working uplift requires quite a bit of tuning so that we
need a bit of preparation. Recall that after the stabilization of the conifold
modulus, the generic superpotential reads

$\begin{split}W=a_{1}e^{-c_{1}S}+a_{2}e^{-c_{2}S}\end{split}$ (5.2)

and its value at the minimum can be approximated by

$\begin{split}W_{0}\sim{c_{2}-c_{1}\over
c_{2}}a_{1}e^{-c_{1}S_{0}}\end{split}$ (5.3)

with $S_{0}$ given by (3.12) as usual. $W_{0}$ allows to estimate the value of
$\tau$ via (4.8). To improve the quality of our estimate the log-log
correction will be taken into account, meaning that

$\begin{split}a\tau\sim\log{\absolutevalue{W_{0}}^{-1}}+\log{\log{\absolutevalue{W_{0}}^{-1}}}.\end{split}$
(5.4)

This effectively increases our estimate for $a\tau$ in both scenarios.
Respecting the condition $\tau>N$, we can express it as

$\begin{split}a={\lambda\over
N}\left(\log{\absolutevalue{W_{0}}^{-1}}+\log{\log{\absolutevalue{W_{0}}^{-1}}}\right)\end{split}$
(5.5)

where we introduced a factor $0<\lambda<1$, which for each concrete numerical
example allows some tuning. As in section 4.2 we set
${c_{2}=(c_{1}^{-1}-1)^{-1}}$. By additionally setting
$a_{2}=-a_{1}(1-c_{1})\exp\left(\left(1-c_{1}\right)^{-1}\right)$ to reduce
the number of free parameters, one ends up with the compact expressions

$\begin{split}S_{0}\sim c_{1}^{-2}\,,\qquad W_{0}\sim
a_{1}c_{1}\exp\left(-{1\over c_{1}}\right)\,.\end{split}$ (5.6)

The Pfaffian $A$ is taken to be $-1$ and the coefficient $\xi$ is taken to be
$\frac{63}{32}$ in all concrete realizations presented below. For later
purposes, we define the two parameters

$\begin{split}X={M\over 2\pi K}\,,\qquad\quad Y=2\pi|K|\,.\end{split}$ (5.7)

###  5.1 Scenario 1

Recall that for Scenario 1 we have888We got rid of the ${1/i}$ factor in
$a_{1}$ (c.f. $W_{\rm eff}$ in (3.11)) by shifting the axion $C_{0}$.
$a_{1}=M\zeta_{0}/(2\pi)$ and $c_{1}=2\pi K/M$ so that one gets
$S_{0}\sim\left({M\over 2\pi K}\right)^{2}$. Consequently, the conifold
modulus and the superpotential are given by

$\begin{split}Z\sim\zeta_{0}\exp\left(-{M\over 2\pi K}\right)\,,\qquad
W_{0}\sim K\zeta_{0}\exp\left(-{M\over 2\pi K}\right)\end{split}$ (5.8)

so that the uplift condition (5.1) can be expressed as

$\begin{split}{\xi\over\kappa}{2\pi\over
Y^{2}}X^{3}\left|\zeta_{0}\right|^{-2/3}\exp\left({2X\over
3}\right)\simeq\lambda\,.\end{split}$ (5.9)

For really finding parameters allowing a controllable dS uplift, this
condition needs to be satisfied with a sufficient accuracy. Let us stress that
(5.5) leads to the following refinement of the condition (4.9)

$a\lesssim{2\pi\over g_{s}M^{2}}+\frac{1}{N}\log\left({2\pi\over
g_{s}|M|\left|\zeta_{0}\right|}\right)\ll 1\,.$ (5.10)

As a consequence, the search for a numerical realization becomes a lot easier.

Our concrete example comes with a flux number of order $M=O(10^{10})$. In
particular, the strong warping condition (2.4), the condition (4.23) needed to
have the hierarchy $M_{\rm cs}>m_{S/T}$ and the necessity of an exponentially
small $W_{0}$ enforce such a high flux in this highly constrained setup. More
concretely, we chose $\lambda=3/4$, ${\kappa=1/2}$, ${M=4\cdot 10^{10}}$ and
${K=158\,070\,553}$ and $M_{1}=\frac{M}{2\pi i}$. This fixes the coefficient
$a$ to be $a\approx 2.98\cdot 10^{-18}$.

The dS minimum was found at ${s\rvert_{\text{num}}\approx 1\,621.60}$ and
${\tau\rvert_{\text{num}}\approx 8.19\cdot 10^{18}}$ with the following value
of the potential

$\begin{split}V_{0}\rvert_{\text{num}}\approx 9.55\cdot
10^{-80}M_{\text{pl}}^{4}\,.\end{split}$ (5.11)

In table 2 we list the numerical values of the mass scales in the left-hand
column. The numerical values of the scales $\Lambda$, $V_{\rm up}^{1\over 4}$
and $\tilde{\Lambda}_{\text{sp}}$ were determined using the relations (3.26),
(3.17) and (4.24) respectively. The mass scales of the throat KK-modes and the
bulk complex structure moduli were obtained via (3.22) and (3.2) respectively.
Employing (3.14) we determined the masses of the lightest saxions $s$ and
$\tau$.

We cross-checked our numerical results with the theoretical predictions,
assuming $C_{0}=0$ and $\theta=0$. The minimal values for $s$ and $\tau$ were
obtained via (3.12) and (4.3), respectively. At $s\rvert_{\text{theo}}\approx
1\,622.03$ and $\tau\rvert_{\text{theo}}\approx 8.14\cdot 10^{18}$ the
potential has the value

$\begin{split}V_{0}\rvert_{\text{theo}}\approx 1.11\cdot
10^{-79}M_{\text{pl}}^{4}\,.\end{split}$ (5.12)

In the right-hand column of table 2, we list the theoretical predictions of
the mass scales, using the approximate formulas appearing on the right-hand
side of table 1 as well as the relation (4.18) for the scale $M_{\rm cs}$.

numerical values | theoretical predictions
---|---
$\Lambda\approx 5.15\cdot 10^{-7}{M_{\rm pl}}$ | $\Lambda\approx 2.55\cdot 10^{-7}{M_{\rm pl}}$
$V_{\rm up}^{1\over 4}\approx 1.95\cdot 10^{-20}M_{\text{pl}}$ | $V_{\rm up}^{1\over 4}\approx 8.14\cdot 10^{-21}M_{\text{pl}}$
$\tilde{\Lambda}_{\text{sp}}\lesssim 2.06\cdot 10^{-22}M_{\text{pl}}$ | $\tilde{\Lambda}_{\text{sp}}\lesssim 1.02\cdot 10^{-22}M_{\text{pl}}$
$m_{Z}\sim m_{\rm KK}\approx 5.22\cdot 10^{-25}M_{\text{pl}}$ | $m_{Z}\sim m_{\rm KK}\approx 2.58\cdot 10^{-25}M_{\text{pl}}$
$M_{\rm cs}\approx 1.06\cdot 10^{-30}M_{\text{pl}}$ | $M_{\rm cs}\approx 1.07\cdot 10^{-30}M_{\text{pl}}$
$m_{s}\approx 9.19\cdot 10^{-37}M_{\text{pl}}$ | $m_{s}\approx 2.72\cdot 10^{-39}M_{\text{pl}}$
$m_{\tau}\approx 5.35\cdot 10^{-39}M_{\text{pl}}$ | $m_{\tau}\approx 4.25\cdot 10^{-40}M_{\text{pl}}$
Table 2: The numerical values and the theoretical predictions of the mass
scales for Scenario 1.

We see that the theoretically expected values of the mass scales agree with
the numerical results. In figure 4 we show the form of the potential close to
the uplifted minimum and in figure 5 a 3D plot of the potential.

Figure 4: Plots of scalar potential showing a realization of Scenario 1. Left:
$V(s,\tau)$ for $\tau\approx 8.19\times 10^{18}$ and right: $V(s,\tau)$ for
$s\approx 1\,621.60$, both multiplied by $10^{78}$. Choice of parameters:
$\lambda=3/4$, $\kappa=1/2$, $M=4\cdot 10^{10}$, $K=158\,070\,553$,
$M_{1}=\frac{M}{2\pi i}$. Figure 5: Potential $V(s,\tau)$ for our example from
Scenario 1, multiplied by $10^{78}$. Choice of parameters: $\lambda=3/4$,
$\kappa=1/2$, $M=4\cdot 10^{10}$, $K=158\,070\,553$, $M_{1}=\frac{M}{2\pi i}$.

Note that the D3-brane tadpole is of order $N=O(10^{19})$ and hence extremely
large. This has certainly no chance to be compatible with any realistic
tadpole cancellation condition.

###  5.2 Scenario 2

In the second scenario, the racetrack coefficients are not determined by the
flux numbers $M,K,M_{1}$ and we dial them freely as long as we stay in the
controlled regime. However, keep in mind that in concrete realizations those
coefficients are not free parameters but also set by other flux numbers and
data of the underlying Calabi-Yau manifold. Therefore, the way we proceed is
very optimistic and eventually Scenario 2 will also be much more constrained.

Setting $a_{1}=1$, for convenience, we find

$\begin{split}Z\sim\zeta_{0}\exp\left(-\frac{1}{c_{1}^{2}X}\right)\,,\qquad
W_{0}\sim c_{1}\exp\left(-{1\over c_{1}}\right)\end{split}$ (5.13)

so that the uplift condition in Scenario 2 becomes

$\begin{split}{\xi\over\kappa}(2\pi
X)^{-1}c_{1}^{-6}|\zeta_{0}|^{\frac{4}{3}}\exp\left({2\over
c_{1}}-\frac{4}{3}\frac{1}{c_{1}^{2}X}\right)\simeq\lambda\,.\end{split}$
(5.14)

One numerical example is specified by $c_{1}=1/50$, $\lambda=1/2$,
$\kappa=1/2$, $M=70\,000$, ${K=402}$ and $M_{1}=\frac{M}{2\pi i}$. The
coefficient $a$ is therefore $a\approx 1.03\cdot 10^{-6}$. The D3-brane
tadpole contribution is of order ${N=O(10^{7})}$, which is smaller than in
Scenario 1 but still fairly large. For $s\rvert_{\text{num}}\approx 2\,499.58$
and $\tau\rvert_{\text{num}}\approx 5.61\cdot 10^{7}$ the dS minimum is found
at

$\begin{split}V_{0}\rvert_{\text{num}}\approx 1.29\cdot
10^{-74}M_{\text{pl}}^{4}\,.\end{split}$ (5.15)

In table 3 we show the numerical values of the mass scales in the left-hand
column. The numerical values of the mass scale of the bulk complex structure
moduli and the scales $\Lambda$ and $\tilde{\Lambda}_{\text{sp}}$ were
determined via the relations (3.2), (3.26) and (3.27) respectively. $V_{\rm
up}^{1\over 4}$ and the mass scale $m_{\rm KK}$ were obtained via (3.17) and
(3.22) respectively. The masses of the lightest saxions $s$ and $\tau$ were
obtained via (3.14).

This has to be compared with our theoretical predictions. Here we find the dS
minimum at $s\rvert_{\text{theo}}\approx 2\,500.00$ and
$\tau\rvert_{\text{theo}}\approx 5.60\cdot 10^{7}$ with the value of the
potential being

$\begin{split}V_{0}\rvert_{\text{theo}}\approx 1.46\cdot
10^{-74}M_{\text{pl}}^{4}\,.\end{split}$ (5.16)

In the right-hand column of table 3, we list the theoretical predictions of
the mass scales, using the same formulas as those used for the numerical
values, except for the masses of the lightest saxions, whose theoretical
values were obtained with the approximate formulas (3.16) and (4.10).

numerical values | theoretical predictions
---|---
$M_{\rm cs}\approx 4.76\cdot 10^{-14}M_{\text{pl}}$ | $M_{\rm cs}\approx 4.77\cdot 10^{-14}M_{\text{pl}}$
$\Lambda\approx 1.64\cdot 10^{-14}{M_{\rm pl}}$ | $\Lambda\approx 1.63\cdot 10^{-14}{M_{\rm pl}}$
$\tilde{\Lambda}_{\text{sp}}\approx 1.31\cdot 10^{-16}M_{\text{pl}}$ | $\tilde{\Lambda}_{\text{sp}}\approx 1.30\cdot 10^{-16}M_{\text{pl}}$
$V_{\rm up}^{1\over 4}\approx 3.71\cdot 10^{-19}M_{\text{pl}}$ | $V_{\rm up}^{1\over 4}\approx 3.69\cdot 10^{-19}M_{\text{pl}}$
$m_{Z}\sim m_{\rm KK}\approx 8.37\cdot 10^{-21}M_{\text{pl}}$ | $m_{Z}\sim m_{\rm KK}\approx 8.33\cdot 10^{-21}M_{\text{pl}}$
$m_{s}\approx 4.76\cdot 10^{-34}M_{\text{pl}}$ | $m_{s}\approx 4.69\cdot 10^{-34}M_{\text{pl}}$
$m_{\tau}\approx 4.84\cdot 10^{-36}M_{\text{pl}}$ | $m_{\tau}\approx 1.06\cdot 10^{-35}M_{\text{pl}}$
Table 3: The numerical values and the theoretical predictions of the mass
scales for Scenario 2.

Hence, the theoretical predictions for the mass scales are in agreement with
the numerical results. Again we show plots of the potential close to this
minimum in the figures 6 and 7.

Figure 6: Plots of scalar potential showing a realization of Scenario 2. Left:
$V(s,\tau)$ for $\tau\approx 5.61\cdot 10^{7}$ and right: $V(s,\tau)$ for
$s\approx 2\,499.58$, both multiplied by $10^{73}$. Choice of parameters:
$c_{1}=1/50$, $\lambda=1/2$, $\kappa=1/2$, $M=70\,000$, $K=402$,
$M_{1}=\frac{M}{2\pi i}$. Figure 7: Potential $V(s,\tau)$ for our example from
Scenario 2, multiplied by $10^{73}$. Choice of parameters: $c_{1}=1/50$,
$\lambda=1/2$, $\kappa=1/2$, $M=70\,000$, $K=402$, $M_{1}=\frac{M}{2\pi i}$.

This example for Scenario 2 indicates that the fluxes $M,K$ needed to
stabilize the conifold modulus and the axio-dilaton yield a very large
contribution $N=MK$ to the D3-brane tadpole on their own. It is hard to say
anything more concrete, as the tadpole constraint is in general quite
involved. Hence, it would be nice to have a simpler quantum gravity criterion
that could be applied to these KKLT models. The very constraining dS swampland
conjecture [60, 61, 62, 63] rules out any meta-stable dS minimum right from
the start so that in the next section we consider a subsequent milder version,
known as the trans-Planckian censorship conjecture [49]. Despite, the severe
non-decoupling of bulk and throat mass scale for Scenario 1, we also consider
the TCC bounds for both scenarios.

##  6 Transplanckian Censorship Conjecture

It is general wisdom that the KKLT scenario is not consistent with the TCC
bound [49], which says that the life-time of a meta-stable dS minimum with
cosmological constant $V_{0}=3H^{2}$ should be smaller than the
$\log$-corrected Hubble time

$\begin{split}T<{1\over H}\log\left({1\over H}\right)\,.\end{split}$ (6.1)

For a KKLT minimum the decay to the Minkowski minimum at large field value can
only occur via a non-perturbative process generically leading to an
exponentially large decay time that easily violates the TCC bound. In this
section, we will analyze this in more detail.

The decay of the KKLT dS minimum is expected to occur via the Coleman-de
Luccia bubble nucleation, which is a non-perturbative semiclassical
instability whose transition rate $\Gamma$ can be estimated in terms of an
Euclidean instanton solution as follows

$\begin{split}\Gamma=P\,e^{-S}\,.\end{split}$ (6.2)

Here $S$ is the Euclidean classical instanton action and $P$ is some prefactor
involving quantum fluctuations. Without entering a discussion on the
subtleties of this formula, we just employ a result derived in [64], namely
that in the thin-wall approximation these two quantities are given as

$\begin{split}S\simeq{2\pi^{2}T^{4}\over\Delta\Lambda^{3}}\,,\qquad\qquad
P\simeq{T^{4}\over\Delta\Lambda^{2}}\end{split}$ (6.3)

where $T$ denotes the tension of the domain wall and $\Delta\Lambda$ is the
difference of vacuum energies on the two sides of the bubble. In our case, we
just have $\Delta\Lambda=V_{0}$. These two relations hold in the thin-wall
approximation $T^{2}/V_{0}\ll 1$, where gravitational corrections are
negligible. The tension of the bubble is determined by the width and the
height of the energy barrier between the dS minimum and the runaway direction
towards Minkowski. It can be estimated via

$\begin{split}T\sim\Delta\phi\sqrt{\Delta V}\end{split}$ (6.4)

where $\Delta\phi\sim\phi_{1}-\phi_{0}$ is the width (for the canonically
normalized field) and the height of the barrier is given by ${\Delta
V=V_{1}-V_{0}}$.

For the KKLT scenario, one expects the fastest decay channel to be the
tunneling in the direction of the lightest modulus. As we have seen, this is
the Kähler modulus so that one needs to determine not only the location of the
local minimum $\tau_{0}$ but also of the local maximum $\tau_{1}$ of the
uplifted scalar potential. One then has to evaluate the potential at these
extrema and derive the barrier height from there. The details of this
calculation can be found in the appendix.

### Scenario 1

Defining for convenience $\tilde{\kappa}=\kappa-\frac{3}{16}$ (recall (5.1)
for the definition of the balance parameter $\kappa$), the TCC bound

$\begin{split}\Gamma>H^{4}\log^{-4}\left({1\over H}\right)\end{split}$ (6.5)

can be compactly expressed as

$\begin{split}\frac{1}{\sqrt{2}}\left(\frac{3\kappa}{4\xi}\right)^{\frac{3}{2}}\frac{1}{(2\pi)^{2}\tilde{\kappa}}\log(X)\left|\log\left(\left(\frac{\kappa}{\xi}\right)^{3}\frac{3}{\left(2\pi
X\right)^{4}\tilde{\kappa}}Y^{10}\right)\right|\frac{Y^{5}}{X^{2}(X+\log\frac{2\pi
X}{|\zeta_{0}|Y})}\\\
\exp\left(-\frac{3}{16}\left(\frac{3\kappa}{8\xi}\right)^{3}\frac{1}{(2\pi)^{2}\tilde{\kappa}^{3}}\left(\log
X\right)^{4}\frac{Y^{10}}{X^{4}\left(X+\log\frac{2\pi
X}{|\zeta_{0}|Y}\right)^{4}}\right)>1\,.\end{split}$ (6.6)

For a suitable choice of $|\zeta_{0}|$ and the balance parameter $\kappa$, one
can now perform a contour plot in the $(X,Y)$ plane to see which regions are
allowed by the KKLT uplift constraint (5.9) and the TCC bound (6.6). For
$\kappa=1/2$ and $|\zeta_{0}|=1$, the result is shown in figure 8. In general,
we don’t expect the uplift contour to sharply define the border where
parametric control is lost. This fact is graphically represented by a glow
around the blue line for both scenarios.

Uplift regionTCC fulfilled Figure 8: Scenario 1: within the blue area we have
a controlled uplift, inside the orange region the TCC bound is satisfied.
Parameter choice: $\kappa=1/2$, $|\zeta_{0}|=1$.

The two regions are disjoint, except in the region of low $X$ and $Y$, where
they barely touch. However, in this region, the control over the effective
action is diminishing and we can less trust the result.

### Scenario 2

Finally, let us briefly discuss the TCC bound in Scenario 2. Defining for
convenience the parameters

$\begin{split}\alpha=\exp\left[\frac{2-2c_{1}X}{c_{1}^{2}X}\right]\,,\qquad\beta=-\frac{\log{c_{1}}}{c_{1}^{-1}-2\log\left[{c_{1}}\right]}\,,\end{split}$
(6.7)

the TCC bound in Scenario 2 can be compactly expressed as

$\begin{split}\frac{1}{\sqrt{2}\tilde{\kappa}}\left(\frac{3\kappa}{4\xi}\right)^{\frac{3}{2}}\left(XY\right)^{3}c_{1}^{7}\frac{\alpha\beta}{|\zeta_{0}|^{2}}\left|\log\left(\left(\frac{\kappa}{\xi}\right)^{3}\frac{3}{\tilde{\kappa}}\left(XY\right)^{6}c_{1}^{14}\frac{\alpha^{2}}{|\zeta_{0}|^{4}}\right)\right|\\\
\exp\left(-\frac{3}{4}\left(\frac{3\kappa}{8\xi\tilde{\kappa}}\right)^{3}\pi^{2}\left(XY\right)^{6}c_{1}^{14}\frac{\alpha^{2}\beta^{4}}{|\zeta_{0}|^{4}}\right)>1\,.\end{split}$
(6.8)

As shown in figure 9, the TCC bound is also violated by the DKMM-refined KKLT
Scenario 2.

Uplift regionTCC fulfilled Figure 9: Scenario 2: within the blue area we have
a controlled uplift, inside the orange region the TCC bound is satisfied.
Parameter choice: $c_{1}=1/50$, $\kappa=1/2$, $|\zeta_{0}|=1$.

Therefore, we conclude that the DKMM-refined KKLT scenario is in clear
conflict with the TCC bound.

##  7 Conclusion

In this paper we extended the usual KKLT construction by the concrete
mechanism of DKMM to stabilize the complex structure moduli such that $W_{0}$
is guaranteed to be exponentially small. The objective was to derive what
additional constraints this imposes on the length and mass scales of the
geometry and the light fields involved.

First, using geometric consistency constraints we derived a lower and an upper
bound for the length of the throat. Mutual consistency then directly led us to
a bound for the four-cycle volume in terms of the D3-tadpole contribution
coming from the throat fluxes. Thus, this direct computation for the Klebanov-
Strassler throat is consistent with a former indirect argument using the
D3-brane backreaction [14, 44].

We then considered the explicit stabilization of the final light moduli
$(Z,S,T)$, working in a framework where all heavier complex structure moduli
have been assumed to be fixed via three-form fluxes. We employed the methods
and the analysis of [30], but generalized it such that we also included the
axio-dilaton and therefore the string coupling constant in the set of light
fields. Following the generalization of the DKMM mechanism to the conifold
regime we invoked a racetrack scenario for its stabilization. For the absence
of non-perturbative corrections in the Kähler moduli, the scalar potential was
found to be positive-semidefinite and to obey a modified no-scale structure.

Computing the appearing mass scales, we found that this DKMM-refined KKLT
scenario comes with a couple of strong constraints. In our more restricted
Scenario 1 we found that one is driven to a regime where the bulk mass scale
does not decouple from the throat mass scale. Clearly, this invalidates the
employed low energy effective action in the throat. Moreover, analogous to the
recent work [45] we found that as a consequence, the cosmological constant of
the dS minimum is parametrically larger than the (throat) species scale.
Recall that [45] claim that the whole DKMM construction ceases to be
controllable even before the uplift.

We also found that for both Scenario 1 and Scenario 2, the two flux quanta
$M,K$ need to be fairly large, in fact much larger than what was derived from
the non-destabilization of the conifold modulus in [29]. Moreover, also the
parameter $a$ that appears in the combination $aT$ in the non-perturbative
term needs to be extremely small, which requires either a high rank confining
gauge group or a highly non-isotropic 4-cycle. All this was confirmed by two
concrete numerical examples, whose throat mass scales were consistent with our
theoretical predictions. In view of the recent tadpole conjecture, this raises
some additional doubts that the DKMM-refined KKLT scenario is in the landscape
of string theory.

In order to provide more evidence, we estimated the life-time of the dS
minimum and compared it to the bound set by the trans-Planckian censorship
conjecture. Evaluating the Coleman-de Luccia bubble decay we realized that its
thin-wall approximation is justified. From the combined plot of the TCC
allowed region and the region of successful uplift, it was evident that they
are disjoint. If TCC is indeed a valid swampland conjecture, this again
supports the suspicion that the DKMM-refined KKLT construction belongs rather
to the swampland of quantum gravity theories.

#### Acknowledgments

We would like to thank Andriana Makridou and Erik Plauschinn for helpful
discussions.

##  Appendix A Computing the TCC region

Here we provide the derivation of all ingredients that are required to fully
express the TCC bound in terms of the flux numbers and some parameters that we
are free to choose. These expressions will be given as functions of $X$ and
$Y$ defined in (5.7), which are the directions of the contour plots appearing
in the body of this paper.

First we calculate the extrema of the total scalar potential. Setting its
derivative with respect to $\tau$ to zero leads to a quadratic equation for
$|W_{0}|\exp(a\tau)$ that is solved by

$\begin{split}|W_{0}|\,e^{a\tau}={a^{2}|A|\tau_{0}\tau\over
16\kappa}\left(1\pm\sqrt{1-{64\over 3}{\kappa\over
a\tau_{0}}}\right)\,.\end{split}$ (A.1)

The location of the minimum is then the solution to the transcendental
equation

$\begin{split}|W_{0}|\,e^{a\tau_{0}}\simeq{2\over 3}a|A|\tau_{0}\end{split}$
(A.2)

and the location of the maximum is at

$\begin{split}|W_{0}|\,e^{a\tau_{1}}\simeq{1\over
8\kappa}a^{2}|A|\tau_{0}\tau_{1}\,.\end{split}$ (A.3)

Combining these two relations, we can estimate

$\begin{split}a\Delta\tau\simeq\log(a\tau_{0})\,\end{split}$ (A.4)

so that for the canonically normalized field $\phi=\sqrt{3\over 2}\log\tau$ we
obtain

$\begin{split}\Delta\phi\simeq\sqrt{3\over 2}{a\Delta\tau\over
a\tau}\simeq\sqrt{3\over 2}{\log(a\tau_{0})\over a\tau_{0}}\,.\end{split}$
(A.5)

The value of the scalar potential at the minimum and the maximum can then be
estimated as

$\begin{split}V_{0}\simeq{g_{s}|W_{0}|^{2}\over\tau_{0}^{3}}\left(-{3\over
16}+\kappa\right)\,,\qquad\quad V_{1}\simeq{\kappa
g_{s}|W_{0}|^{2}\over\tau_{0}\tau^{2}_{1}}\,.\end{split}$ (A.6)

leading to the height of the barrier

$\begin{split}\Delta V\simeq{3\over
16}{g_{s}|W_{0}|^{2}\over\tau^{3}_{0}}\,.\end{split}$ (A.7)

###  A.1 Scenario 1

Using the uplift condition (5.1) and that $|W_{0}|\sim\frac{Y}{2\pi}|Z|$
(result from section 5.1) one gets for the VEV of the conifold modulus

$|Z|\simeq\left(\frac{\xi\tau_{0}}{\kappa}\right)^{\frac{3}{2}}\frac{(2\pi
X)^{3}}{Y^{6}}\,.$ (A.8)

The height of the barrier is therefore

$\Delta V\simeq\frac{3}{16}\left(\frac{\xi}{\kappa}\right)^{3}\frac{(2\pi
X)^{4}}{Y^{10}}\,.$ (A.9)

Thus, one can express the tension and the cosmological constant of the dS
minimum as

$\displaystyle T$ $\displaystyle\sim\Delta\phi\sqrt{\Delta
V}\simeq\frac{3}{4\sqrt{2}}\left(\frac{\xi}{\kappa}\right)^{\frac{3}{2}}\frac{\log(X)}{X+\log(\frac{2\pi
X}{|\zeta_{0}|Y})}\frac{(2\pi X)^{2}}{Y^{5}}$ (A.10) $\displaystyle V_{0}$
$\displaystyle\simeq\left(\frac{\xi}{\kappa}\right)^{3}\left(-\frac{3}{16}+\kappa\right)\frac{(2\pi
X)^{4}}{Y^{10}}\,.$ (A.11)

With these quantities one can compute the decay rate and finally formulate the
TCC condition (6.6). Note that one finds

$\begin{split}{T^{2}\over V_{0}}\sim\left(\frac{\log(X)}{X+\log(\frac{2\pi
X}{|\zeta_{0}|Y})}\right)^{2}\ll 1\end{split}$ (A.12)

(for $Y>1$ and $X\gg 1$ as needed to get large $s_{0}=g_{s}^{-1}$), so that
the thin-wall approximation is justified.

###  A.2 Scenario 2

We immediately insert (5.13) in (A.7). Using the uplift condition (5.14) to
eliminate $\lambda$ we end up with

$\Delta
V\simeq\frac{3}{16}\left(\frac{\xi}{\kappa}\right)^{3}(XY)^{-6}c_{1}^{-14}|\zeta_{0}|^{4}\exp\left[\frac{4}{c_{1}}-\frac{4}{Xc_{1}^{2}}\right]\,.$
(A.13)

The tension and the cosmological constant of the dS minimum are

$\displaystyle T$ $\displaystyle\sim\Delta\phi\sqrt{\Delta
V}\simeq\frac{3}{4\sqrt{2}}\left(\frac{\xi}{\kappa}\right)^{\frac{3}{2}}\frac{\log{c_{1}}}{2\log{c_{1}}-{c^{-1}_{1}}}(XY)^{-3}c_{1}^{-7}|\zeta_{0}|^{2}\exp\left[\frac{2}{c_{1}}-\frac{2}{Xc_{1}^{2}}\right]$
(A.14) $\displaystyle V_{0}$
$\displaystyle\simeq\left(\frac{\xi}{\kappa}\right)^{3}\left(-\frac{3}{16}+\kappa\right)(XY)^{-6}c_{1}^{-14}|\zeta_{0}|^{4}\exp\left[\frac{4}{c_{1}}-\frac{4}{Xc_{1}^{2}}\right]\,.$
(A.15)

Once again we verify

$\begin{split}{T^{2}\over
V_{0}}\sim\left(\frac{\log{c_{1}}}{2\log{c_{1}}-{c^{-1}_{1}}}\right)^{2}\ll
1\end{split}$ (A.16)

(for $c_{1}^{-1}\gg 1$ as needed to get large $s_{0}=g_{s}^{-1}$), so that the
thin-wall approximation is again justified.

## References

  * [1] S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, “De Sitter vacua in string theory,” Phys. Rev. D 68 (2003) 046005, hep-th/0301240.
  * [2] C. Vafa, “The String landscape and the swampland,” hep-th/0509212.
  * [3] H. Ooguri and C. Vafa, “On the Geometry of the String Landscape and the Swampland,” Nucl. Phys. B 766 (2007) 21–33, hep-th/0605264.
  * [4] T. D. Brennan, F. Carta, and C. Vafa, “The String Landscape, the Swampland, and the Missing Corner,” PoS TASI2017 (2017) 015, 1711.00864.
  * [5] E. Palti, “The Swampland: Introduction and Review,” Fortsch. Phys. 67 (2019), no. 6, 1900037, 1903.06239.
  * [6] M. van Beest, J. Calderón-Infante, D. Mirfendereski, and I. Valenzuela, “Lectures on the Swampland Program in String Compactifications,” 2102.01111.
  * [7] U. H. Danielsson and T. Van Riet, “What if string theory has no de Sitter vacua?,” Int. J. Mod. Phys. D 27 (2018), no. 12, 1830007, 1804.01120.
  * [8] J. Moritz, A. Retolaza, and A. Westphal, “Toward de Sitter space from ten dimensions,” Phys. Rev. D97 (2018), no. 4, 046010, 1707.08678.
  * [9] R. Kallosh, A. Linde, E. McDonough, and M. Scalisi, “de Sitter Vacua with a Nilpotent Superfield,” Fortsch. Phys. 2018 (2018) 1800068, 1808.09428.
  * [10] R. Kallosh, A. Linde, E. McDonough, and M. Scalisi, “4D models of de Sitter uplift,” Phys. Rev. D99 (2019), no. 4, 046006, 1809.09018.
  * [11] F. F. Gautason, V. Van Hemelryck, and T. Van Riet, “The tension between 10D supergravity and dS uplifts,” Fortsch. Phys. 67 (2019), no. 1-2, 1800091, 1810.08518.
  * [12] Y. Hamada, A. Hebecker, G. Shiu, and P. Soler, “On brane gaugino condensates in 10d,” JHEP 04 (2019) 008, 1812.06097.
  * [13] Y. Hamada, A. Hebecker, G. Shiu, and P. Soler, “Understanding KKLT from a 10d perspective,” JHEP 06 (2019) 019, 1902.01410.
  * [14] F. Carta, J. Moritz, and A. Westphal, “Gaugino condensation and small uplifts in KKLT,” JHEP 08 (2019) 141, 1902.01412.
  * [15] F. F. Gautason, V. Van Hemelryck, T. Van Riet, and G. Venken, “A 10d view on the KKLT AdS vacuum and uplifting,” JHEP 06 (2020) 074, 1902.01415.
  * [16] I. Bena, M. Graña, N. Kovensky, and A. Retolaza, “Kähler moduli stabilization from ten dimensions,” JHEP 10 (2019) 200, 1908.01785.
  * [17] R. Blumenhagen, M. Brinkmann, D. Kläwer, A. Makridou, and L. Schlechter, “KKLT and the Swampland Conjectures,” PoS CORFU2019 (2020) 158, 2004.09285.
  * [18] M. Demirtas, M. Kim, L. Mcallister, and J. Moritz, “Vacua with Small Flux Superpotential,” Phys. Rev. Lett. 124 (2020), no. 21, 211603, 1912.10047.
  * [19] M. Demirtas, M. Kim, L. McAllister, and J. Moritz, “Conifold Vacua with Small Flux Superpotential,” Fortsch. Phys. 68 (2020) 2000085, 2009.03312.
  * [20] R. Álvarez-García, R. Blumenhagen, M. Brinkmann, and L. Schlechter, “Small Flux Superpotentials for Type IIB Flux Vacua Close to a Conifold,” Fortsch. Phys. 68 (2020) 2000088, 2009.03325.
  * [21] R. Álvarez-García and L. Schlechter, “Analytic Periods via Twisted Symmetric Squares,” 2110.02962.
  * [22] B. Bastian, T. W. Grimm, and D. van de Heisteeg, “Engineering Small Flux Superpotentials and Mass Hierarchies,” 2108.11962.
  * [23] M. Demirtas, M. Kim, L. McAllister, J. Moritz, and A. Rios-Tascon, “Small cosmological constants in string theory,” JHEP 12 (2021) 136, 2107.09064.
  * [24] M. Demirtas, M. Kim, L. McAllister, J. Moritz, and A. Rios-Tascon, “Exponentially Small Cosmological Constant in String Theory,” Phys. Rev. Lett. 128 (2022), no. 1, 011602, 2107.09065.
  * [25] I. Broeckel, M. Cicoli, A. Maharana, K. Singh, and K. Sinha, “On the Search for Low $W_{0}$,” 2108.04266.
  * [26] F. Carta, A. Mininno, and P. Shukla, “Systematics of perturbatively flat flux vacua,” JHEP 02 (2022) 205, 2112.13863.
  * [27] F. Carta, A. Mininno, and P. Shukla, “Systematics of perturbatively flat flux vacua for CICYs,” 2201.10581.
  * [28] M. R. Douglas, J. Shelton, and G. Torroba, “Warping and supersymmetry breaking,” 0704.4001.
  * [29] I. Bena, E. Dudas, M. Graña, and S. Lüst, “Uplifting Runaways,” Fortsch. Phys. 67 (2019), no. 1-2, 1800100, 1809.06861.
  * [30] R. Blumenhagen, D. Kläwer, and L. Schlechter, “Swampland Variations on a Theme by KKLT,” JHEP 05 (2019) 152, 1902.07724.
  * [31] I. Bena, A. Buchel, and S. Lüst, “Throat destabilization (for profit and for fun),” 1910.08094.
  * [32] E. Dudas and S. Lüst, “An update on moduli stabilization with antibrane uplift,” JHEP 03 (2021) 107, 1912.09948.
  * [33] L. Randall, “The Boundaries of KKLT,” Fortsch. Phys. 68 (2020), no. 3-4, 1900105, 1912.06693.
  * [34] S. Lüst and L. Randall, “Effective Theory of Warped Compactifications and the Implications for KKLT,” 2206.04708.
  * [35] I. Bena, J. Blåbäck, M. Graña, and S. Lüst, “The tadpole problem,” JHEP 11 (2021) 223, 2010.10519.
  * [36] I. Bena, J. Blåbäck, M. Graña, and S. Lüst, “Algorithmically Solving the Tadpole Problem,” Adv. Appl. Clifford Algebras 32 (2022), no. 1, 7, 2103.03250.
  * [37] E. Plauschinn, “The tadpole conjecture at large complex-structure,” JHEP 02 (2022) 206, 2109.00029.
  * [38] M. Graña, T. W. Grimm, D. van de Heisteeg, A. Herraez, and E. Plauschinn, “The Tadpole Conjecture in Asymptotic Limits,” 2204.05331.
  * [39] I. R. Klebanov and M. J. Strassler, “Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities,” JHEP 08 (2000) 052, hep-th/0007191.
  * [40] B. Heidenreich, M. Reece, and T. Rudelius, “The Weak Gravity Conjecture and Emergence from an Ultraviolet Cutoff,” Eur. Phys. J. C78 (2018), no. 4, 337, 1712.01868.
  * [41] T. W. Grimm, E. Palti, and I. Valenzuela, “Infinite Distances in Field Space and Massless Towers of States,” JHEP 08 (2018) 143, 1802.08264.
  * [42] B. Heidenreich, M. Reece, and T. Rudelius, “Emergence of Weak Coupling at Large Distance in Quantum Gravity,” Phys. Rev. Lett. 121 (2018), no. 5, 051601, 1802.08698.
  * [43] P. Corvilain, T. W. Grimm, and I. Valenzuela, “The Swampland Distance Conjecture for Kähler moduli,” JHEP 08 (2019) 075, 1812.07548.
  * [44] X. Gao, A. Hebecker, and D. Junghans, “Control issues of KKLT,” Fortsch. Phys. 68 (2020) 2000089, 2009.03914.
  * [45] S. Lüst, C. Vafa, M. Wiesner, and K. Xu, “Holography and the KKLT Scenario,” 2204.07171.
  * [46] D. Junghans, “LVS de Sitter Vacua are probably in the Swampland,” 2201.03572.
  * [47] X. Gao, A. Hebecker, S. Schreyer, and G. Venken, “Loops, Local Corrections and Warping in the LVS and other Type IIB Models,” 2204.06009.
  * [48] D. Junghans, “Topological Constraints in the LARGE-Volume Scenario,” 2205.02856.
  * [49] A. Bedroya and C. Vafa, “Trans-Planckian Censorship and the Swampland,” JHEP 09 (2020) 123, 1909.11063.
  * [50] S. B. Giddings, S. Kachru, and J. Polchinski, “Hierarchies from fluxes in string compactifications,” Phys. Rev. D 66 (2002) 106006, hep-th/0105097.
  * [51] C. P. Herzog, I. R. Klebanov, and P. Ouyang, “Remarks on the warped deformed conifold,” in Modern Trends in String Theory: 2nd Lisbon School on g Theory Superstrings.  8, 2001.  hep-th/0108101.
  * [52] M. R. Douglas and G. Torroba, “Kinetic terms in warped compactifications,” JHEP 05 (2009) 013, 0805.3700.
  * [53] R. Kallosh and A. D. Linde, “Landscape, the scale of SUSY breaking, and inflation,” JHEP 12 (2004) 004, hep-th/0411011.
  * [54] A. R. Frey and A. Maharana, “Warped spectroscopy: Localization of frozen bulk modes,” JHEP 08 (2006) 021, hep-th/0603233.
  * [55] C. P. Burgess, P. G. Camara, S. P. de Alwis, S. B. Giddings, A. Maharana, F. Quevedo, and K. Suruliz, “Warped Supersymmetry Breaking,” JHEP 04 (2008) 053, hep-th/0610255.
  * [56] G. Shiu, G. Torroba, B. Underwood, and M. R. Douglas, “Dynamics of Warped Flux Compactifications,” JHEP 06 (2008) 024, 0803.3068.
  * [57] S. P. de Alwis, “Constraints on Dbar Uplifts,” JHEP 11 (2016) 045, 1605.06456.
  * [58] G. Dvali, “Black Holes and Large N Species Solution to the Hierarchy Problem,” Fortsch. Phys. 58 (2010) 528–536, 0706.2050.
  * [59] G. Dvali and M. Redi, “Black Hole Bound on the Number of Species and Quantum Gravity at LHC,” Phys. Rev. D 77 (2008) 045027, 0710.4344.
  * [60] G. Obied, H. Ooguri, L. Spodyneiko, and C. Vafa, “De Sitter Space and the Swampland,” 1806.08362.
  * [61] H. Ooguri, E. Palti, G. Shiu, and C. Vafa, “Distance and de Sitter Conjectures on the Swampland,” Phys. Lett. B 788 (2019) 180–184, 1810.05506.
  * [62] S. K. Garg and C. Krishnan, “Bounds on Slow Roll and the de Sitter Swampland,” JHEP 11 (2019) 075, 1807.05193.
  * [63] D. Andriot, “On the de Sitter swampland criterion,” Phys. Lett. B 785 (2018) 570–573, 1806.10999.
  * [64] A. Bedroya, M. Montero, C. Vafa, and I. Valenzuela, “de Sitter Bubbles and the Swampland,” 2008.07555.