Dilaton stabilization in KKLT revisited

Min-Seok Seoa

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

We study the condition for the dilaton stabilization in Type IIB flux
compactifications consistent with the KKLT scenario. Since the Gukov-Vafa-
Witten superpotential depends linearly on the dilaton, the dilaton mass
squared is given by a sum of the gravitino mass squared and additional terms
determined by the complex structure moduli stabilization. If the dilaton mass
is not much enhanced from the gravitino mass, the mass mixing with the Kähler
modulus in the presence of the non-perturbative effect generates the saddle
point at the supersymmetric field values, hence the potential becomes
unstable. When the complex structure moduli other than the conifiold modulus
are neglected, the saddle point problem arises over the controllable parameter
space. We also point out that the dilaton stabilization condition is
equivalent to the condition on the NS 3-form fluxes,
$|H_{(1,2)}|>|H_{(0,3)}|$.

##  1 Introduction

The construction of convincing cosmological models from string theory has been
a challenging issue for more than two decades. This essentially requires
compactifications to four dimensions in which all the moduli are fixed at the
metastable de Sitter (dS) vacuum. The stabilization of the dilaton and the
complex structure moduli was shown to be possible by turning on 3-form fluxes
in Type-IIB Calabi-Yau orientifold compactifications with O3/O7-planes, which
we will call the Gidding-Kachru-Polchinski (GKP) solution [1]. On the other
hand, the Kähler modulus associated with the internal compactification volume
remains unfixed at tree level, as reflected in the no-scale structure. The
Kachru-Kallosh-Linde-Trivedi (KKLT) scenario [2] suggested that when the tree
level superpotential is comparable in size to the exponentially small non-
perturbative quantum effect, the Kähler modulus can be stabilized at the
supersymmetric anti-de Sitter (AdS) minimum. Supersymmetry is broken by
putting anti-D3 ($\overline{\rm D3}$)-branes at the tip of the Klebanov-
Strassler (KS)-like throat [3], provided the number of $\overline{\rm
D3}$-branes is tuned to be much smaller than the flux quanta [4]. Then the
minimum of the potential is uplifted to a metastable dS vacuum.

The description above implies that the KKLT scenario does not work without
fine-tuning of parameters, which indeed seems to be a common feature of string
models for the metastable dS vacuum including the Large Volume Scenario [5].
Presumably, there may be some as yet unknown reason in string theory (or in
quantum gravity) which forbids the metastable dS vacuum over a wide range of
(or even whole) parameter space as argued in the “dS swampland conjecture” [6]
(see [7, 8, 9] for a refined version, especially [9] and also [10, 11] for
attempts to find out the thermodynamic argument to support the conjecture).
Such a concern has motivated consistency checks for the KKLT scenario in
various directions, investigating whether the scenario works over the
controllable parameter space of the effective supergravity (e.g., [12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24] and especially [23] for a summary on
main issues and relevant references).

What makes the problem more complicated is that while the KKLT scenario is a
combination of several mechanisms, each of them is not necessarily completely
decoupled from others [25]. The dilaton or some of the complex structure
moduli can be so light that they are not decoupled from the stabilization
mechanism of the Kähler modulus. For example, in [19], the strongly warped
regime was explored, in which not only the Kähler modulus but also the
conifold modulus, the complex structure modulus associated with the size of
the 3-cycle of the deformed conifold [26] have exponentially suppressed
masses. On the other hand, the Gukov-Vafa-Witten (GVW) superpotential [27]
induced by the 3-form fluxes depends linearly on the dilaton. As a result, the
dilaton mass squared is written as a sum of the gravitino mass squared and
other terms determined by the stabilization of other moduli. In other words,
whether the dilaton is heavier than the gravitino is closely connected to how
the complex structure moduli are stabilized by fluxes. If the dilaton mass is
not considerably larger than the gravitino mass, the mass mixing between the
Kähler modulus and the dilaton needs to be carefully addressed since the
Kähler modulus mass is enhanced from the gravitino mass. As can be inferred
from [28], the mass mixing in this case may generate an unstable direction
such that a set of field values obtained from the supersymmetric condition is
in fact on the saddle point. The purpose of this article is to explore this
possibility in detail. Since the dilaton linearly couples to any complex
structure moduli in the GVW superpotential, we expect that our investigations
provide more stringent constraints on the stabilization of the complex
structure moduli with fluxes which are not evident when we just focus on the
conifold modulus.

This article is organized as follows. In section 2, we review main features of
the moduli stabilization in the original KKLT and GKP which assumed that the
Kähler modulus is decoupled from the dilaton and the complex structure moduli.
From this, we summarize the ingredients needed for our discussion and settle
the notations. In section 3, we illustrate the saddle point problem by
considering the extreme case in which the conifold modulus is the only complex
structure modulus or other complex structure moduli effects are negligibly
small, as frequently employed in the literatures for simplicity. For this
purpose we examine the low energy effective superpotential induced by
integrating out the conifold modulus. Then we apply the results of [28] to
show that the mass mixing between the dilaton and the Kähler modulus in the
presence of the non-perturbative effect easily destabilizes the potential. In
section 4 we interpret the condition for the absence of the saddle point in
light of the GVW superpotential in two equivalent ways : 1) the dilaton must
be considerably heavier than the gravitino mass and 2) the $(1,2)$ component
of the NS 3-form fluxes dominates over the $(3,0)$ component which is
consistent with the low energy supersymmetry breaking. This provides a more
stringent constraint on the flux vacua. Finally we conclude. Throughout this
work, we take the (reduced) Planck unit $m_{\rm Pl}=1$ unless otherwise
stated. Notations and conventions follow [29].

##  2 Moduli stabilization in KKLT and GKP

In this section we first briefly review the Kähler modulus stabilization in
the original KKLT scenario which assumed that the dilaton and the complex
structure moduli are already fixed through the GKP solution. We also write
down the Kähler potential and the GVW superpotential used in the GKP solution,
focusing on the dilaton and the conifold modulus. In both cases, the moduli
and the dilaton are stabilized through the F-term potential

$\begin{split}V=e^{K}\Big{(}G^{I\overline{J}}D_{I}WD_{\overline{J}}\overline{W}-3|W|^{2}\Big{)},\end{split}$
(1)

where $G^{I\overline{J}}$ is the inverse of the Kähler metric
$G_{I\overline{J}}=\partial_{I}\partial_{\overline{J}}K$ and
$D_{I}W=\partial_{I}W+\partial_{I}KW$.

###  2.1 KKLT scenario

In the KKLT scenario [2], the Kähler potential and the superpotential for the
Kähler modulus $\rho$ are given by

$\begin{split}K_{\rho}=-3\log[-i(\rho-\overline{\rho})],\quad\quad
W_{\rho}=W_{0}+Ce^{ia\rho},\end{split}$ (2)

before the uplift. If $W_{0}$ and $C$ have the same phase and $a$ is real, the
F-term potential has a supersymmetric AdS minimum with $\rho=i\sigma$ (i.e., a
scalar and a pseudoscalar parts are fixed at $\sigma$ and $0$, respectively),
satisfying $D_{\rho}W=0$ or

$\begin{split}W_{0}=-e^{-a\sigma}C\Big{(}1+\frac{2}{3}a\sigma\Big{)}.\end{split}$
(3)

For higher-order non-perturbative effects to be negligible, $a\sigma>1$ is
required, which means that the scenario works provided $W_{0}$ is
exponentially suppressed. The mass for $\sigma$ is given by

$\begin{split}m_{\sigma}^{2}=G^{\rho\overline{\rho}}\partial_{\sigma}^{2}V=\frac{2a^{3}W_{0}^{2}}{a\sigma}\Big{(}\frac{2+5(a\sigma)+2(a\sigma)^{2}}{(3+2a\sigma)^{2}}\Big{)}.\end{split}$
(4)

The pseudoscalar part of $\rho$ has a mass $m_{\rm
Re\rho}^{2}=2a^{3}W_{0}^{2}/(3+2(a\sigma))$, which is similar in size to
$m_{\sigma}^{2}$ for $a\sigma\gg 1$. After the uplift, the supersymmetry is
broken to give the gravitino mass

$\begin{split}m_{3/2}^{2}=e^{K}|W|^{2}=\frac{a^{2}C^{2}}{18\sigma}e^{-2a\sigma},\end{split}$
(5)

which is roughly $m_{3/2}^{2}\sim W_{0}^{2}/\sigma^{3}$ while $m_{\sigma}^{2}$
is not much affected, hence

$\begin{split}m_{\sigma}\sim(a\sigma)m_{3/2}>m_{3/2}.\end{split}$ (6)

###  2.2 GKP solution

The premise of the KKLT scenario is that the dilaton $\tau=C_{0}+ie^{-\phi}$
as well as the complex structure moduli $t^{a}$ are stabilized by the flux-
induced superpotential as suggested in GKP [1]. Here we must require that at
the stabilized field values, the string coupling constant is perturbatively
small $g_{s}=e^{\langle\phi\rangle}<1$ and $W_{0}$ is tuned to be
exponentially suppressed. In the original KKLT scenario, it was assumed that
$\tau$ and $t^{a}$ are heavier than $\rho$ such that their stabilization is
not much affected by the non-perturbative effect. However, This is not always
guaranteed and as we will see, if $\tau$ is as light as $m_{3/2}$ the vacuum
is typically destabilized by the mass mixing between $\rho$ and $\tau$.

To see this, we first review the structure of the Kähler potential and the
superpotential in the GKP solution. The Kähler potential for
$(\tau,\rho,t^{a})$ is written as

$\begin{split}K_{\tau}+K_{\rho}+K_{\rm
cs}=-\log[-i(\tau-\overline{\tau})]-3\log[-i(\rho-\overline{\rho})]-\log\Big{(}i\int_{X}\Omega\wedge\overline{\Omega}\Big{)},\end{split}$
(7)

whereas the flux-induced superpotential is given by the GVW superpotential,

$\begin{split}W_{\rm GVW}=c\ell_{s}^{-2}\int_{X}\Omega\wedge
G_{3}.\end{split}$ (8)

Here $\Omega$ is the holomorphic 3-form and $G_{3}=F_{3}-\tau H_{3}$. Since
$W_{\rm GVW}$ depends on $\tau$ through $G_{3}$, it is linear in $\tau$. A
coefficient $c$ is determined by matching the F-term potential induced by
$W_{\rm GVW}$ with the dimensional reduction of the Type-IIB supergravity
action for $G_{3}$, which turns out to be $c=g_{s}^{3/2}(4\pi)^{-1/2}$ [30].
The value of $W_{\rm GVW}$ at the stabilized field values will be identified
with $W_{0}$ in the KKLT scenario.

In order to see the explicit form of $K_{\rm cs}$ and $W_{\rm GVW}$, we
introduce a basis of the de Rham cohomology group $H^{3}(X,\mathbb{Z})$ given
by generators $\alpha_{I}$ and $\beta^{I}$ ($I=0,\cdots h^{2,1}(X)$), which
are Poincaré dual to the homology basis $(A^{I},B_{I})$ of
$H_{3}(X,\mathbb{Z})$. They satisfy

$\begin{split}&A^{I}\cdot A^{J}=B_{I}\cdot B_{J},\quad A^{I}\cdot
B_{J}=\delta^{I}_{J},\\\
&\int_{A^{J}}\alpha_{I}=-\int_{B_{I}}\beta^{J}=\int_{X}\alpha_{I}\wedge\beta^{J}=\delta_{I}^{J}.\end{split}$
(9)

In terms of $A$\- and $B$-periods of $\Omega$

$\begin{split}Z^{I}=\int_{A^{I}}\Omega,\quad\quad{\cal
F}_{I}=\int_{B_{I}}\Omega,\end{split}$ (10)

$\Omega$ is written as

$\begin{split}\Omega=A^{I}\alpha_{I}-{\cal F}_{I}\beta^{I}.\end{split}$ (11)

Then ${\cal F}_{I}$ corresponds to $\partial_{I}{\cal F}$, in which ${\cal
F}(Z)$ is connected to the prepotential $F(t^{a})$ through ${\cal
F}(Z)=(Z^{0})^{2}F(t^{a})$ with $t^{a}=Z^{a}/Z^{0}$ ($a=1,\cdots h^{2,1}(X)$).
From this, the Kähler potential for the complex structure moduli is written as

$\begin{split}K_{\rm
cs}=-\log\Big{(}i\int_{X}\Omega\wedge\overline{\Omega}\Big{)}=-\log\Big{(}i|Z^{0}|^{2}\Big{[}2(F-\overline{F})-(t^{a}-\overline{t}^{a})(F_{a}+\overline{F}_{a})\Big{]}\Big{)}.\end{split}$
(12)

Meanwhile, $F_{3}$ and $H_{3}$ fluxes are quantized with respect to
$H^{3}(X,\mathbb{Z})$ such that

$\begin{split}F_{3}=\ell_{s}^{2}(M^{I}\alpha_{I}+M_{I}\beta^{I})\quad\quad
H_{3}=\ell_{s}^{2}(K^{I}\alpha_{I}+K_{I}\beta^{I}).\end{split}$ (13)

The quanta $(M^{I},M_{I})$ and $(K^{I},K_{I})$ are restricted by the tadpole
cancellation condition,

$\begin{split}\frac{1}{\ell_{s}^{4}}\int_{X}F_{3}\wedge
H_{3}=M^{I}K_{I}-M_{I}K^{I}=-N_{D_{3}}+\frac{1}{2}N_{O_{3}}.\end{split}$ (14)

In the presence of the wrapped D7-brane, the D3 charge $\chi(X)/24$ can be
induced, where $\chi(X)$ is the Euler characteristic of the Calabi-Yau 4-fold
in the F-theory compactifications [31]. This value is added to the r.h.s. of
(14).

In the GKP solution, one of the complex structure moduli corresponds to the
conifold modulus $z$, a period of a three-cycle $A$ which vanishes as $S^{3}$
of a cone $S^{3}\times S^{2}$ shrinks to zero size. 111More precisely, the
deformed conifold is described by four complex coordinates $w_{i}$
($i=1,\cdots 4$) satisfying $\sum_{i}w_{i}^{2}=\epsilon^{2}$. The conifold
modulus $z$ determines the deformation parameter $\epsilon$ through the
relation
$\epsilon^{8/3}={\alpha^{\prime}}^{2}[-i(\rho-\overline{\rho})]|z|^{4/3}$. The
$\rho$ dependence here will be discussed later.  Setting $Z^{0}=1$, one finds
that

$\begin{split}\int_{A}\Omega=z\quad\quad\int_{B}\Omega={\cal
F}_{z}=\frac{z}{2\pi i}\log z+({\rm holomorphic}).\end{split}$ (15)

Let us assign the flux charges relevant to $(A,B)$ as

$\begin{split}\int_{A}F_{3}=\ell_{s}^{2}M,\quad\quad\int_{B}H_{3}=-\ell_{s}^{2}K,\end{split}$
(16)

i.e., $(M^{z},M_{z})=(M,0)$ and $(K^{z},K_{z})=(0,K)$. Moreover, we assume
that the prepotential has the form of $F(z,t^{i})=F_{1}(z)+F_{2}(t^{i})$ with
$i=2,\cdots h^{2,1}(X)$, such that

$\begin{split}F_{1}(z)=\frac{z^{2}}{4\pi i}\Big{(}\log
z-\frac{1}{2}\Big{)}+f_{0}+f_{1}z+\cdots.\end{split}$ (17)

Then the Kähler potential is written as

$\begin{split}K_{\rm
cs}=-\log\Big{(}&\frac{|z|^{2}}{2\pi}\log|z|^{2}+2i(f_{0}-f_{0}^{*})+i(f_{1}-f_{1}^{*})(z+{\overline{z}})+\cdots\\\
&+i[2(F_{2}-\overline{F}_{2})-(t^{i}-\overline{t}^{i})(F_{2,i}+\overline{F}_{2,i})]\Big{)},\end{split}$
(18)

where $\cdots$ includes higher order terms (${\cal O}(z^{2})$) that vanish as
$z\to 0$. Meanwhile, since $F_{0}=Z^{0}(2F-zF_{z}-t^{i}F_{i})$, the
superpotential with $Z^{0}=1$ is written in the form of

$\begin{split}W_{\rm GVW}=A(z,t^{i})-\tau B(z,t^{i}),\end{split}$ (19)

where

$\begin{split}A(z,t^{i})&=c(Z^{I}M_{I}+F_{I}M^{I})\\\
&=c\Big{(}M_{0}+(2f_{0}+f_{1}z)M^{0}+\big{(}\frac{z}{2\pi i}\log
z+f_{1}\big{)}M+(2F_{2}-t^{i}F_{i})M^{0}+t^{i}M_{i}+F_{i}M^{i}\Big{)},\\\
B(z,t^{i})&=c(Z^{I}K_{I}+F_{I}K^{I})\\\
&=c\Big{(}K_{0}+(2f_{0}+f_{1}z)K^{0}+zK+(2F_{2}-t^{i}F_{i})K^{0}+t^{i}K_{i}+F_{i}K^{i}\Big{)}.\end{split}$
(20)

Here terms higher order in $z$ are omitted as well. We note that even if
$t^{i}$ are integrated out, the effective superpotential should not contain
$z$ dependent terms enhanced from $z\log z$ and $\tau z$ since otherwise $z$
cannot be stabilized at the exponentially small value. The separation of $z$
and $t^{i}$ in both the Kähler potential and the superpotential is helpful for
this purpose. While discussion so far is generic, a concrete example, e.g.,
the mirror dual of the quintic $\mathbb{P}_{4}[5]^{(101,1)}$ gives the Kähler
potential and the superpotential of the same structures as those we have shown
[32] (see also [33, 34, 35]).

If $\tau$ as well as $t^{a}=(z,t^{i})$ are heavy enough to be decoupled from
the KKLT scenario, the non-perturbative effect can be ignored in their
stabilization. In this limit, since the Kähler potential for $\rho$ is given
by $K_{\rho}$ in (7) and $W_{\rm GVW}$ is independent of $\rho$, the F-term
potential has a no-scale structure,
$V=e^{K}G^{I\overline{J}}D_{I}WD_{\overline{J}}\overline{W}$, where the
indices $K,~{}L$ run over fields other than $\rho$. Then the second derivative
matrix of the potential at the minimum satisfying $D_{I}W=0$ consists of

$\begin{split}\partial_{I}\partial_{J}V=e^{K}G^{K\overline{L}}[\partial_{I}D_{K}W\partial_{J}D_{\overline{L}}\overline{W}+\partial_{J}D_{K}W\partial_{I}D_{\overline{L}}\overline{W}].\end{split}$
(21)

Meanwhile, the fluxes and the branes backreact on the geometry, resulting in
the warped geometry in which the metric around the throat is given by

$\begin{split}ds^{2}=e^{2A_{0}}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+e^{-2A_{0}}\widetilde{g}_{mn}dy^{m}dy^{n}.\end{split}$
(22)

In the dilute flux limit ($e^{-4A_{0}}\simeq 1$) the internal geometry can be
treated as if the Calabi-Yau manifold. In contrast, when the throat is highly
warped ($e^{-4A_{0}}\gg 1$), the warp factor near the throat is given by [3]

$\begin{split}&e^{-4A_{0}}\simeq
2^{2/3}\frac{(\alpha^{\prime}g_{s}M)^{2}}{\epsilon^{8/3}}{\cal I}(y),\\\
&{\cal
I}(y)=\int_{y}^{\infty}dx\frac{x\coth(x)-1}{\sinh^{2}x}(\sinh(2x)-2x)^{1/3},\end{split}$
(23)

where $\alpha^{\prime}=\ell_{s}^{2}/(2\pi)^{2}$ and $y$ is a coordinate along
the throat : $y=0$ corresponds to the tip of the throat and ${\cal I}(y=0)$
converges to a constant. We infer from the metric of the KS throat [3] that
the deformation parameter $\epsilon^{2}$ (see footnote 1) is proportional to
$|z|$. Moreover, from the equation of motion

$\begin{split}\tilde{\nabla}^{2}e^{4A}=e^{2A}\frac{G_{mnp}\overline{G}^{mnp}}{12{\rm
Im}\tau}+\cdots,\end{split}$ (24)

where $\tilde{\nabla}^{2}$ is the Laplacian with respect to the unwarped
internal (Calabi-Yau) metric $\widetilde{g}_{mn}$, one finds that under
$g_{mn}\to\lambda g_{mn}$, the warp factor for the throat part also scales as
$e^{-4A_{0}}\to\lambda^{-2}e^{-4A_{0}}$ while the overall volume
$V_{X}=\ell_{s}^{6}[-i(\rho-\overline{\rho})]^{3/2}$ scales as
$V_{X}\to\lambda^{3}V_{X}$. 222The scaling behavior of $V_{X}$ comes from
$\begin{split}V_{X}=\int d^{6}y\sqrt{\tilde{g}}e^{-4A_{\rm tot}}.\end{split}$
(25) Here $e^{-4A_{\rm tot}}$ is the warp factor for the overall internal
volume, which should not be confused with the warp factor for the throat
$e^{-4A_{0}}$.  This suggests the relation
$\epsilon^{8/3}={\alpha^{\prime}}^{2}[-i(\rho-\overline{\rho})]|z|^{4/3}$
[36]. In this strongly warped regime $W_{\rm GVW}$ is not changed but $K_{\rm
cs}$ in (12) is modified to be [37]

$\begin{split}K_{\rm
cs}=-\log\Big{(}i\int_{X}e^{-4A_{0}}\Omega\wedge\overline{\Omega}\Big{)},\end{split}$
(26)

such that the Kähler metric for $z$ has the form of [38]

$\begin{split}G_{z\bar{z}}=c_{0}+c_{1}\log|z|+c_{2}\frac{(g_{s}M)^{2}}{[-i(\rho-\overline{\rho})]|z|^{4/3}},\end{split}$
(27)

where the last term dominates provided $\sigma z^{4/3}\ll 1$. We also note
that in the strongly warped regime, the Kaluza-Klein modes can be light so the
four dimensional effective theory is spoiled (for more discussion see [19]).

We finally note that far away from the throat, the warp factor is given by
$e^{-4A_{0}}\sim 4\pi{\alpha^{\prime}}^{2}g_{s}N/r^{4}$ such that the geometry
is approximated by AdS${}_{5}\times$S5. Here $r$ is the radial distance from
the tip of the throat and $N$ is the total D-brane charges stored in the
fluxes and the branes, thus $N\simeq MK$. This shows that the throat size in
the string length unit is $(r_{\rm throat}/\ell_{s})^{4}\sim g_{s}MK$ in the
modified Einstein frame (or $\sim MK$ in the Einstein frame).

##  3 Saddle point Problem in dilaton stabilization

###  3.1 Constraints on the dilaton-conifold modulus system

In the GKP solution, the dilaton $\tau$ is stabilized at the minimum
satisfying $D_{\tau}W=0$ and $D_{a}W=0$. As the GVW superpotential is given in
the form of $W_{\rm GVW}=A(z,t^{i})-\tau B(z,t^{i})$, i.e., linearly dependent
on $\tau$, and the Kähler potential is given by (7), the condition
$D_{\tau}W_{\rm GVW}=0$ is written as

$\begin{split}D_{\tau}W_{\rm
GVW}=-\frac{1}{\tau-\overline{\tau}}c\int_{X}\Omega\wedge\overline{G}_{3}=-\frac{1}{\tau-\overline{\tau}}(A-\overline{\tau}B)=0,\end{split}$
(28)

or $\tau=\overline{A}/\overline{B}$ in which the complex structure moduli take
the stabilized values. Meanwhile, by the condition $D_{z}W_{\rm GVW}=0$ the
conifold modulus $z$ is stabilized at the exponentially small value

$\begin{split}z\sim{\rm
exp}\Big{[}-\frac{2\pi}{g_{s}M}\Big{(}K+f_{1}K^{0}-\frac{1}{2}\frac{f_{1}-f_{1}^{*}}{f_{0}-f_{0}^{*}}(K_{0}+2f_{0}K^{0})\Big{)}\Big{]}.\end{split}$
(29)

We note that the last term comes from $K_{z}W$ in the dilute flux limit, where
the Kähler potential is given by (18). This term vanishes for
$|f_{0}-f_{0}^{*}|\sim{\cal O}(1)$ and $f_{1}-f_{1}^{*}=0$, as supported by a
concrete example considered in [32]. The well known value of $z$ corresponds
to this case so hereafter we assume that $f_{1}$ is real such that
$D_{z}W_{\rm GVW}=\partial_{z}W_{\rm GVW}+{\cal O}(z)$. In the strongly warped
regime, the dimensional reduction of the Type IIB supergravity action just
replaces $G_{z\overline{z}}$ by (27), so the minimum condition
$\partial_{z}W_{\rm GVW}\simeq 0$ is still valid [38]. For the string higher
loops to be ignored in the effective theory, $g_{s}<1$, or equivalently
$-i(\tau-\overline{\tau})>1$ is required. However, $g_{s}$ cannot be
arbitrarily small by another effective theory validity condition that the size
of $S^{3}$ at the tip of the deformed conifold is larger than the string
length scale [19],

$\begin{split}R^{2}_{S^{3}}\sim
e^{-2A(0)}|\epsilon|^{4/3}\sim\alpha^{\prime}g_{s}|M|\gg\alpha^{\prime},\end{split}$
(30)

or $g_{s}|M|\gg 1$.

Now let us naively assume that the conifold modulus $z$ is the only complex
structure modulus. While it is sensible only if the effects of other complex
structure moduli are negligible, this illustrates the potential issues caused
by the linear dependence of $W_{\rm GVW}$ on $\tau$ and the parametric
constraints discussed above. Suppose $K^{0}=K_{0}=0$ such that $W_{\rm GVW}$
does not contain a (constant)$\times\tau$ term, i.e., $B$ does not have a
constant term,

$\begin{split}&A=c\Big{(}w_{0}+w_{1}z+M\frac{z}{2\pi i}\log
z\Big{)},\quad\quad B=cKz,\end{split}$ (31)

where $w_{0}=M_{0}+2f_{0}M^{0}+f_{1}M$ and $w_{1}=f_{1}M^{0}$. At the minimum,
conditions $D_{z}W_{\rm GVW}=0\simeq\partial_{z}W_{\rm GVW}$ and $D_{\tau}W=0$
give

$\begin{split}&\frac{M}{2\pi i}\log z=\tau K-w_{1}-\frac{M}{2\pi i},\\\
&zK\overline{\tau}=w_{0}+w_{1}z+M\frac{z}{2\pi i}\log z=w_{0}-\frac{M}{2\pi
i}z+zK\tau.\end{split}$ (32)

When Im$(\tau)$ is stabilized at $1/g_{s}$, they give the relation

$\begin{split}\frac{g_{s}}{2\pi}=\frac{2Kz}{2\pi iw_{0}-Mz}.\end{split}$ (33)

We note that $w_{0}$ should be nonzero and indeed not be suppressed compared
to $Mz$ since otherwise (33) gives $|z|\sim{\rm exp}[-(2\pi/g_{s})(K/M)]\simeq
e^{1/2}$, which is contradict to the exponentially small $|z|$. Thus, we find
that the absence of a (constant)$\times\tau$ term results in the exponentially
small $g_{s}\sim|z|$. Such an exponentially small $g_{s}$ calls for the
exponentially large $M$ and modestly large $K$ to satisfy $g_{s}|M|\gg 1$ and
$(2\pi/g_{s})(K/M)>1$ (for $|z|\ll 1$), respectively. In this case, a
coefficient $c=g_{s}^{3/2}(4\pi)^{-1/2}$ [30] becomes exponentially small
hence $W_{\rm GVW}\simeq cw_{0}$ can be as small as $e^{-a\sigma}(a\sigma)$
(or $\sigma\sim(ag_{s})^{-1}K/M>1$) as required by the KKLT scenario.
Meanwhile, flux charges $M$ and $K$ are constrained by the tadpole
cancellation condition (14), or $MK=-N_{D_{3}}+(1/2)N_{O_{3}}+\chi(X)/24$ then
the exponentially large positive number $MK$ requires the exponentially large
positive local charges in r.h.s.. Since $\chi(X)$ ranges between
$-240\leq\chi(X)\leq 1820448$ [39] and $\log(1820448/24)\simeq 11$, the value
of $\log(M)\sim-\log(g_{s})\sim(2\pi/g_{s})(K/M)$ smaller than 10 may be
allowed. However, as pointed out in [18], the throat size in the Einstein
frame $(r_{\rm throat}/\ell_{s})^{4}\sim MK$ is required to be smaller than
the overall internal volume size
$(V_{X}/\ell_{s})^{4/6}\sim-i(\rho-\overline{\rho})=2\sigma$. Then the
exponentially large $MK$ can be excluded by the modest value of $\sigma$. 333
Even if $M$ is not exponentially enhanced, the parametric constraints
generically call for large $M$, which gives rise to the similar issue. Away
from the condition $g_{s}|M|\gg 1$, for example, if the supersymmetry is
broken by the mechanism descirbed in [4], the metastability of the uplifting
term imposes (the number of $\overline{\rm D3}$ branes)/$|M|<0.08$. Then given
an integral number of $\overline{\rm D3}$ branes $M$ is required to be large.
Since $K$ is also not small as imposed by $(2\pi/g_{s})(K/M)>1$, the Kähler
modulus $\sigma$ needs to be fixed at a large value to satisfy $MK<\sigma$.
For example, we may consider (the number of $\overline{\rm D3}$ branes)=$2$,
$g_{s}=0.1$, $M=30$, and $K=2$, which leads to $MK=60$. Then
$e^{-a\sigma}<e^{-60a}$ so for $a\sim{\cal O}(1)$, we have a very tiny ratio
$m_{3/2}/m_{\rm Pl}\lesssim e^{-60}\simeq 10^{-26}$. In this way, the
consistency conditions may require the very large internal volume and the
extremely light gravitino mass. Then the low energy effective theory may be
altered by the Large Volume Scenario [5] (see also [40]) and we need to
introduce additional supersymmetry breaking mechanism to explain the
undiscovery of the supersymmetry at LHC, which may affect the uplift.

Alternatively, we can introduce a (constant)$\times\tau$ term through nonzero
$K_{0},~{}K^{0}$ or the stabilization of other complex structure moduli
$t^{i}$ [1] to fix $\tau$ at the modestly large value. Then $g_{s}$ is
perturbatively small but not exponentially suppressed. Especially, the
existence of other complex structure moduli is appealing. While the
requirement $W_{\rm GVW}\sim e^{-a\sigma}(a\sigma)$ is fulfilled through the
conspiracy of various stabilized field values, the low energy effective
superpotential for $\tau$ after integrating out the complex structure moduli
is no longer linear in $\tau$. As pointed out in [28], such a nontrivial
$\tau$ dependence of the effective superpotential is crucial to prevent the
destabilization of the vacuum. Even though the effective superpotential
constructed by integrating out $z$ in the absence of other complex structure
moduli is not linear in $\tau$ as well, it may not be enough to stabilize the
potential.

###  3.2 Saddle point problem

To see the instability issue discussed at the end of previous section in
detail, we first review the saddle point condition studied in [28], which
observed the mass matrix of $\rho$ and $\tau$ with respect to the Kähler
potential and the effective superpotential

$\begin{split}&K=-3\log[-i(\rho-\overline{\rho})]-\log[-i(\tau-\overline{\tau})],\\\
&W_{\rm eff}=W_{0}(\tau)+Ce^{ia\rho}.\end{split}$ (34)

Here the superpotential term $W_{0}$ as a function of $\tau$ is constructed
from $W_{\rm GVW}$ by integrating out $z$ as well as $t^{i}$. This assumes
that the complex structure moduli including $z$ are heavier than $\tau$, and
in fact, $m_{z}\gg m_{\tau}$ is generic in the dilute flux limit. This can be
easily seen from the potential generated by the Kähler potential (18) and the
superpotential (31). From (21), the $z$ mass is given by

$\begin{split}m_{z}^{2}\simeq e^{K}G^{z{\overline{z}}}\partial_{z}^{2}W_{\rm
GVW}\partial_{\overline{z}}^{2}W_{\rm
GVW}\simeq\frac{c^{2}g_{s}M^{2}}{32\pi\sigma^{3}|z|^{2}\log|z|^{-2}},\end{split}$
(35)

so for $|z|\ll 1$, $z$ is very heavy. Including a (constant)$\times\tau$ term
in $W_{\rm GVW}$ does not affect this result. In the strongly warped regime
the modification of the Kähler metric (27) makes $z$ light for $|z|\ll 1$, but
in this case the Kaluza-Klein modes can be even lighter than $z$ so they also
need to be taken into account in the effective theory for $(\rho,\tau)$ [19].

To make our discussion simple, we consider the special form of $W_{\rm eff}$
where every terms in $W_{0}$ has the same phase as $C$ and $a$ is real. By
doing this, $\rho$ and $\tau$ have purely imaginary values such that the
pseudoscalars are fixed to zero at the supersymmetric extremum. From (34), the
second derivative matrix of the potential for the scalars $\sigma={\rm
Im}\rho$ and $s={\rm Im}\tau$ is given by [28]

$\begin{split}V_{s}^{\prime\prime}=\frac{|dW_{\rm
0}/ds|^{2}}{8s\sigma^{5}}\left(\begin{array}[]{cc}3s^{2}(7+10a\sigma+4(a\sigma)^{2})&3s\sigma(3+2a\sigma-2\gamma)\\\
3s\sigma(3+2a\sigma-2\gamma)&\sigma^{2}(1-2\gamma+4\gamma^{2})\\\
\end{array}\right),\end{split}$ (36)

where

$\begin{split}\gamma=\frac{is(d^{2}W_{0}/d\tau^{2})}{(dW_{0}/d\tau)}=\frac{s(d^{2}W_{0}/ds^{2})}{(dW_{0}/ds)}.\end{split}$
(37)

Note that $\gamma$ is real in our phase assignment. The corresponding matrix
for the pseudoscalars has the same structure [28]

$\begin{split}V_{a}^{\prime\prime}=\frac{|dW_{\rm
0}/ds|^{2}}{8s\sigma^{5}}\left(\begin{array}[]{cc}3s^{2}(3+6a\sigma+4(a\sigma)^{2})&3s\sigma(1+2a\sigma-2\gamma)\\\
3s\sigma(1+2a\sigma-2\gamma)&\sigma^{2}(1-2\gamma+4\gamma^{2})\\\
\end{array}\right),\end{split}$ (38)

which is identical to $V_{s}^{\prime\prime}$ for $a\sigma\gg 1$, so we focus
on the scalar part $V^{\prime\prime}_{s}$ only. Then the matrix
$V^{\prime\prime}_{s}$ does not have a saddle point provided
det$(V^{\prime\prime}_{s})>0$, or

$\begin{split}\gamma>\frac{5+4a\sigma}{4(2+a\sigma)},\quad\quad\gamma<-\frac{2+a\sigma}{1+2a\sigma}.\end{split}$
(39)

Thus, we conclude that for the potential to be stabilized, $|\gamma|>{\cal
O}(1)$ should be imposed [28]. If $W_{0}$ is linear in $\tau$, i.e.,
$W_{0}=A-\tau B$ with $A,~{}B$ constants, $\gamma=0$ so the potential has an
unstable direction at the supersymmetric field values.

Now we go back to the case of the single complex structure modulus $z$, with
the superpotential given in the form of $W_{\rm GVW}=A(z)-\tau B(z)$, where

$\begin{split}A&=c\Big{(}w_{0}+w_{1}z+M\frac{z}{2\pi i}\log
z\Big{)},\quad\quad B=c\Big{(}(K+f_{1}K^{0})z+w_{2}\Big{)}.\end{split}$ (40)

While $w_{0}=M_{0}+2f_{0}M^{0}+f_{1}M$, $w_{1}=f_{1}M^{0}$ and
$w_{2}=K_{0}+2f_{0}K^{0}$, we will not attach to the explicit forms of these
coefficients in order to accommodate the superpotential as a function of
$(z,\tau)$ induced by integrating out $t^{i}$ but still linear in $\tau$.
444One may include the non-perturbative term $Ce^{ia\rho}$ in $cw_{0}$. Then
$\tau$ is stabilized at the same value as that obtained from $W_{\rm eff}$
after the $\rho$ stabilization. This does not change the stabilized value of
$g_{s}$ much provided $cw_{0}\gtrsim Ce^{-a\sigma}$. Moreover, to make our
discussion simple, we restrict our discussion to the case where $w_{0}$ is
pure imaginary, $w_{1}$ is pure imaginary or zero, and $w_{2}$ is real, which
is realized by taking $M_{0}=K^{0}=0$ and $f_{1}=0$. By doing this, $z$ is
stabilized at the real value while $\tau$ is fixed at $i/g_{s}$, i.e., pure
imaginary value. Remaining quanta $M^{0},~{}K_{0}$ are tuned such that
$|w_{0}-(i/g_{s})w_{2}|$ is exponentially suppressed. From the minimum
conditions $D_{z}W_{\rm GVW}=D_{\tau}W_{\rm GVW}=0$ one finds that

$\begin{split}\frac{g_{s}}{2\pi}=\frac{2Kz+w_{2}}{2\pi iw_{0}-Mz},\end{split}$
(41)

in which $g_{s}$ is, in general, not exponentially tiny. Since the condition
$D_{z}W=0\simeq\partial_{z}W_{\rm GVW}$ reads

$\begin{split}w_{1}+\frac{1}{2\pi i}(1+\log
z)M-\tau(f_{1}K^{0}+K)=0,\end{split}$ (42)

$z$ is expressed as a function of $\tau$,

$\begin{split}z={\rm exp}\Big{[}\frac{2\pi
i}{M}\Big{(}\tau(f_{1}K^{0}+K)-w_{1}\Big{)}-1\Big{]}.\end{split}$ (43)

Putting this back to $W_{\rm GVW}$ the effective superpotential in the absence
of the non-perturbative effect is written as

$\begin{split}W_{0}(\tau)\simeq c\Big{(}w_{0}-\tau w_{2}-\frac{M}{2\pi i}{\rm
exp}\Big{[}\frac{2\pi
i}{M}\big{(}\tau(f_{1}K^{0}+K)-w_{1}\big{)}-1\Big{]}\Big{)}.\end{split}$ (44)

It is remarkable that $W_{0}(\tau)$ here has an exponential term, which looks
like the non-perturbative effect under $s={\rm Im}(\tau)=1/g_{s}$. As pointed
out in [32], this shows that the shift symmetry $\tau\to\tau+\theta$ of the
Kähler potential is broken to the discrete one through the coupling between
$\tau$ and $z$ induced by the $H_{3}$ flux (the term $-\tau(f_{1}K^{0}+K)z$ in
(40)). More explicitly, if $w_{2}=0$ ($K_{0}=K^{0}=0$), the discrete symmetry
$\tau\to\tau+(M/K)n$ with $n\in\mathbb{Z}$ remains as a remnant of the shift
symmetry. This indeed is originated from the monodromy property of the
conifold modulus $z$, ${\cal F}_{z}\to{\cal F}_{z}+1$ while $z\to e^{2\pi i}z$
(see (15)), which is a part of Sp$(2(h^{2,1}+1))$ symplectic structure of the
holomorphic 3-form $\Omega$ as given by (11) [41]. In (40), we immediately
find that when $K_{0}=K^{0}=0$, the shift $\log z\to\log z+2\pi i$ from the
monodromy $z\to e^{2\pi i}z$ is compensated by the discrete shift
$\tau\to\tau+(M/K)$.

After the stabilization of $\tau$ ($s=g_{s}^{-1}$), we obtain

$\begin{split}\gamma=-\frac{2\pi}{g_{s}M}\frac{(f_{1}K^{0}+K)^{2}z}{w_{2}+(f_{1}K^{0}+K)z},\end{split}$
(45)

where $z$ is given by (43), which is an exponentially suppressed value. Thus,
when $w_{2}$, or a (constant)$\times\tau$ term in $W_{\rm GVW}$ is non-
vanishing and even not suppressed compared to $(f_{1}K^{0}+K)z$, $|\gamma|$ is
exponentially small. Then the potential has a saddle point by the mass mixing
with $\rho$ so becomes unstable. For $w_{2}=0$, $\gamma=-(2\pi K/g_{s}M)$ is
noting more than the exponent of $|z|$ then $|\gamma|>1$ is satisfied. As we
have seen in the discussion below (33), however, in this case $g_{s}$ becomes
exponentially small, which can be excluded by the condition $MK<\sigma$.

##  4 Interpretation of the stability condition

###  4.1 Comparison with gravitino mass

The physical interpretation of the stability condition $|\gamma|>1$ is clear
by comparing the dilaton mass with the gravitino mass $m_{3/2}$. Since the
Lagrangian is written as

$\begin{split}-G_{\tau{\overline{\tau}}}\partial_{\mu}\tau\partial^{\mu}\overline{\tau}-V(\tau)+\cdots=-\frac{1}{4s^{2}}(\partial
s)^{2}-\frac{1}{2}\frac{\partial^{2}V}{\partial s^{2}}\Big{|}_{\rm
min}s^{2}+\cdots,\end{split}$ (46)

the $(22)$ component of $V^{\prime\prime}_{s}$ is related to the $(22)$
component of the mass matrix by

$\begin{split}M^{2}_{ss}=2s^{2}\frac{\partial^{2}V}{\partial
s^{2}}\Big{|}_{\rm
min}=\frac{s}{4\sigma^{3}}\Big{(}\frac{dW_{0}}{ds}\Big{)}^{2}(1-2\gamma+4\gamma^{2}).\end{split}$
(47)

Meanwhile, the supersymmetric minimum conditions $D_{\tau}W_{\rm
eff}=0=D_{\rho}W_{\rm eff}$ read

$\begin{split}Ce^{-a\sigma}=-\frac{3}{3+2a\sigma}W_{0},\quad\quad
s\frac{dW_{0}}{ds}=\frac{a\sigma}{3+2a\sigma}W_{0},\end{split}$ (48)

from which the gravitino mass is given by

$\begin{split}m_{3/2}^{2}=e^{K}|W|^{2}=\frac{1}{16\sigma^{3}s}|W_{0}+Ce^{-a\sigma}|^{2}=\frac{s}{4\sigma^{3}}\Big{(}\frac{dW_{0}}{ds}\Big{)}^{2}.\end{split}$
(49)

Thus, in the limit $\gamma\to 0$, $M^{2}_{ss}$ becomes $m_{3/2}^{2}$. When the
stability condition $|\gamma|>1$ is satisfied, $M^{2}_{ss}$ gets considerably
larger than $m_{3/2}^{2}$. Since $m_{\sigma}\sim(a\sigma)m_{3/2}>m_{3/2}$,
this indicates that unless $|\gamma|\gg 1$, $\tau$ cannot be decoupled from
$\rho$ and the mass mixing caused by the non-perturbatuve effect should not be
neglected. The original KKLT scenario assumed $|\gamma|\gg 1$ for the
decoupling of $\tau$ from $\rho$. In order to see how the non-perturbative
effect changes $M_{ss}^{2}$, let us for a moment turn off the non-perturbative
effect by taking $a\to\infty$, and observe the property of $M_{ss}^{2}$
generated by $W_{0}$ only. In this case, the potential has a no-scale
structure
$V=e^{K}G^{\tau\overline{\tau}}D_{\tau}WD_{\overline{\tau}}\overline{W}$ and
we have a simple relation $2s(dW_{0}/ds)=W_{0}$ at the minimum. From this we
obtain

$\begin{split}\frac{d^{2}V}{ds^{2}}\Big{|}_{\rm
min}=\frac{|dW_{0}/ds|^{2}}{8\sigma^{3}s}|1+2\gamma|^{2},\end{split}$ (50)

which is positive definite for any value of $\gamma$ while this is not the
case in the presence of the non-perturbative effect, as can be seen from the
$(22)$ component of $V^{\prime\prime}_{s}$.

Since $W_{0}$ is obtained by integrating out the complex structure moduli, the
properties of $W_{0}$ including $\gamma$ encode how the complex structure
moduli are stabilized by $W_{\rm GVW}$. From $W_{\rm GVW}=A(t^{a})-\tau
B(t^{a})$ where $t^{a}=(z,t^{i})$, the minimum conditions read

$\begin{split}&D_{a}W_{\rm
GVW}=(\partial_{a}A-\tau\partial_{a}B)+\partial_{a}K(A-\tau B)=0,\\\
&D_{\tau}W_{\rm
GVW}=-\frac{1}{\tau-\overline{\tau}}(A-\overline{\tau}B)=0,\end{split}$ (51)

or equivalently,

$\begin{split}A=\overline{\tau}B,\quad\quad\partial_{a}A-\tau\partial_{a}B=-\partial_{a}K(\tau-\overline{\tau})B.\end{split}$
(52)

Then using (21) we obtain

$\begin{split}m_{\tau}^{2}&=G^{\tau{\overline{\tau}}}\partial_{\tau}\partial_{\overline{\tau}}V\\\
&=-(\tau-\overline{\tau})^{2}e^{K}\Big{(}G^{a{\overline{b}}}(\partial_{a}B+\partial_{a}KB)(\partial_{\overline{b}}\overline{B}+\partial_{\overline{b}}K{\overline{B}})+|B|^{2}\Big{)}.\end{split}$
(53)

This can be compared to 555While $m_{3/2}$ here corresponds to the
$a\to\infty$ limit of (49) and its numerical value is not much changed under
this limit as $W_{0}\sim e^{a\sigma}(a\sigma)\gg Ce^{-a\sigma}$, the
supersymmetry breaking occurs in a different way. In the GKP solution which
employs $W_{\rm GVW}$, the potential has a no-scale structure. Then it is
stabilized at the Minkowski vacuum through the cancellation between
$G^{\rho{\overline{\rho}}}|D_{\rho}W_{\rm GVW}|^{2}$ and $3|W_{\rm GVW}|^{2}$,
which implies that the supersymmetry is broken by nonzero $D_{\rho}W_{\rm
GVW}$. On the other hand, in the KKLT scenario all fields are stabilized at
the supersymmetric AdS minimum satisfying $DW=0$ and the supersymmetry is
broken by the uplift. Since the original KKLT assumed $\tau$ as well as
$t^{a}$ are heavy enough to be decoupled from the KKLT scenario, their
stabilized values are not much affected by the small non-perturbative effect.

$\begin{split}m_{3/2}^{2}=e^{K}|A-\tau
B|^{2}=-(\tau-\overline{\tau})^{2}e^{K}|B|^{2},\end{split}$ (54)

which is exactly the second term of (53). That is, in the absence of the
coupling to the complex structure moduli $m_{\tau}^{2}$ which is equivalent to
$M_{ss}^{2}$ is given by $m_{3/2}^{2}$. This can be found in (47) as well :
$\gamma$ is generated by the interaction between $\tau$ and the complex
structure moduli so when the interaction is truned off, $\gamma$ becomes zero,
and we obtain $M_{ss}^{2}=m_{3/2}^{2}$. Moreover, the first term of (53) is
positive definite, which is consistent with the positive definiteness of
$d^{2}V/ds^{2}$ obtained by turning off the non-perturbative effect.

In terms of the discussion above, the problem in considering the conifold
modulus $z$ as the only complex structure modulus can be understood in the
following way. Since

$\begin{split}&B=c\Big{(}(K+f_{1}K^{0})z+w_{2}\Big{)},\quad\quad\partial_{z}B=c(f_{1}K^{0}+K),\end{split}$
(55)

and the Kähler potential is given by (18) with $t^{i}$ turned off, the first
term of (53) becomes

$\begin{split}G^{z{\overline{z}}}|\partial_{z}B+\partial_{z}KB|^{2}\simeq\frac{2\pi
i(f_{0}-f_{0}^{*})}{\log|z|}|\partial_{z}B|^{2}\simeq\frac{2\pi
i(f_{0}-f_{0}^{*})}{\log|z|}c^{2}|f_{1}K^{0}+K|^{2}.\end{split}$ (56)

Since $-\log|z|=(2\pi/g_{s})(f_{1}K^{0}+K)/M>1$ and $B\simeq cw_{2}$ unless
$w_{2}<(f_{1}K^{0}+K)z$, this term is typically suppressed by $-\log|z|$
compared to $m_{3/2}^{2}$. It corresponds to the case $|\gamma|<1$ in which
the potential is destabilized when the non-perturbative effect and the mass
mixing are taken into account additionally. If $w_{2}<(f_{1}K^{0}+K)z$, e.g.,
$K_{0}=K^{0}=0$, $B\simeq(f_{1}K^{0}+K)z$ hence (56) becomes larger than
$m_{3/2}^{2}$. While this allows $|\gamma|>1$ and the potential is stable
against the mass mixing, as we have seen, this requires the exponentially
small $g_{s}$ and exponentially large $M$, easily violating the condition
$MK<\sigma$. In the highly warped regime, the Kähler metric is replaced by
(27), to give

$\begin{split}G^{z{\overline{z}}}|\partial_{z}B+\partial_{z}KB|^{2}&\simeq
c^{2}\Big{(}c_{0}+c_{1}\log|z|+c_{2}\frac{(g_{s}M)^{2}}{2\sigma|z|^{4/3}}\Big{)}^{-1}|f_{1}K^{0}+K|^{2}\\\
&\simeq\frac{2c^{2}|z|^{4/3}\sigma}{c_{2}(g_{s}M)^{2}}|f_{1}K^{0}+K|^{2},\end{split}$
(57)

which is typically smaller than $|B|^{2}$ unless $w_{2}<(f_{1}K^{0}+K)z$ thus
the potential is easily destabilized by the mass mixing.

###  4.2 Flux condition

We have seen that $\tau$ is stabilized without the saddle point provided
$\tau$ is considerably heavier than $m_{3/2}$, or equivalently

$\begin{split}G^{a{\overline{b}}}(\partial_{a}B+\partial_{a}KB)(\partial_{\overline{b}}\overline{B}+\partial_{\overline{b}}K{\overline{B}})>|B|^{2}.\end{split}$
(58)

Since $\tau$ as well as $t^{a}=(t^{i},~{}z)$ are stabilized by the fluxes,
such stabilization condition can be translated into the condition on the
fluxes. To see this, We first note that $B$ in $W_{\rm GVW}$ corresponds to

$\begin{split}B=c\int_{X}\Omega\wedge H_{3}.\end{split}$ (59)

Since the complex structure moduli $t^{a}$ are contained in $\Omega$ and
$\partial_{a}\Omega=k_{a}\Omega+\chi_{a}$, i.e. a combination of the $(3,0)$\-
and the primitive $(2,1)$ forms, we have 666For the strongly warped regime, a
factor $e^{-4A_{0}}$ is multiplied to each integrand, giving, e.g., (27) [37].

$\begin{split}&\partial_{a}K=\frac{\int_{X}\partial_{a}\Omega\wedge\overline{\Omega}}{\int_{X}\Omega\wedge\overline{\Omega}}=-k_{a},\quad\quad
G_{a\overline{b}}=-\frac{\int_{X}\chi_{a}\wedge\overline{\chi}_{\overline{b}}}{\int_{X}\Omega\wedge\overline{\Omega}},\end{split}$
(60)

from which we obtain

$\begin{split}\partial_{a}B+K_{a}B=c\int_{X}\chi_{a}\wedge H_{3}.\end{split}$
(61)

We note that for

$\begin{split}D_{a}W=c\int_{X}\chi_{a}\wedge G_{3}=0\end{split}$ (62)

to be satisfied at the minimum, $G_{3}$ is required to be imaginary self dual,
so $(3,0)$ and $(1,2)$ components do not exist [1]. This however does not mean
that $H_{3}$ does not have a $(1,2)$ component, but constrained by the
condition $G_{(1,2)}=F_{(1,2)}-\tau H_{(1,2)}=0$ at the minimum. Otherwise,
(61) becomes zero such that (58) is violated, which means that the potential
is destabilized. Then $H_{(1,2)}$ and $H_{(0,3)}$ are constrained by (58),
which reads

$\begin{split}\frac{\int_{X}\chi_{a}\wedge
H_{(1,2)}\int_{X}\chi_{\overline{b}}\wedge\overline{H}_{(2,1)}}{\int_{X}\chi_{a}\wedge\chi_{\overline{b}}}>\frac{\Big{|}\int_{X}\Omega\wedge
H_{(0,3)}\Big{|}^{2}}{\int_{X}\Omega\wedge\overline{\Omega}}\end{split}$ (63)

with the help of (60) and (61). One obvious way to satisfy this condition is
demanding $|H_{(1,2)}|>|H_{(0,3)}|$. Thus, the decoupling of the dilaton from
the low energy physics of the typical mass scale $m_{3/2}$ or
$m_{\sigma}\sim(a\sigma)m_{3/2}$ is achieved when the $(1,2)$ component of
$H_{3}$ dominates over the $(0,3)$ component. Since $W_{\rm
GVW}=c\int\Omega\wedge G_{3}=c\int\Omega\wedge G_{(0,3)}$, the low energy
supersymmetry breaking is realized provided the $G_{(0,3)}$ component is
small, say, $G_{(0,3)}\ll G_{(2,1)}$ [42]. The smallness of $H_{(0,3)}$ is
consistent with such requirement while there still remains the tuning to make
$W_{\rm GVW}=A-\tau B=-(2i/g_{s})B$ as small as $e^{-a\sigma}(a\sigma)$. In
[42], it was also pointed out that the flux vacua satisfying
$G_{(0,3)}/G_{(2,1)}<\epsilon$ take up about $\epsilon^{2}$ of the total
number of vacua [43, 44, 45, 46]. We expect that (63), or
$|H_{(1,2)}|>|H_{(0,3)}|$ provides a more stringent constraint on the search
for the flux vacua consistent with our universe.

##  5 Conclusions

We have discussed the condition for the dilaton stabilization consistent with
the KKLT scenario. The dilaton potential is generated by the flux-induced GVW
superpotential, as it breaks the shift symmetry of the Kähler potential for
the dilaton $\tau$. Moreover, the fact that the GVW superpotential is linearly
dependent on $\tau$ may give rise to the saddle point problem. That is, if
$\tau$ does not couple to any complex structure moduli, the $\tau$ mass is
simply given by the gravitino mass $m_{3/2}$. Then when we take the non-
perturbative effect of the Kähler modulus $\rho$ into account the
supersymmetric solution is on the saddle point hence the potential becomes
unstable through the mass mixing between $\tau$ and $\rho$. While $\tau$
couples to the conifold modulus $z$, another essential ingredient of both the
GKP solution and the KKLT scenario, this does not help to resolve the problem
in the controllable parameter space. Alternatively, we can introduce the
complex structure moduli $t^{i}$ other than $z$ in which the exponentially
small $W_{0}\sim e^{-a\sigma}(a\sigma)$ as required by the KKLT scenario is a
result of the conspiracy of stabilized field values or fluxes.

In summary, the saddle point problem is circumvented when $\tau$ is
considerably heavier than $m_{3/2}$ through the coupling to the complex
structure moduli $t^{i}$. In the low energy effective theory point of view,
the effective superpotential $W_{0}(\tau)$ generated by integrating out
$t^{i}$ as well as $z$ does no longer linearly depend on $\tau$, then an
appropriate functional form of $W_{0}(\tau)$ makes $m_{\tau}\gg m_{3/2}$. In
terms of the GVW superpotential which still linearly depends on $\tau$, this
can be understood as a result of the complex structure moduli stabilization
under the constraint $|H_{(1,2)}|>|H_{(0,3)}|$ on the flux. This is consistent
with the flux condition $G_{(0,3)}\ll G_{(2,1)}$ for the low energy
supersymmetry breaking, but more restrictive. It is remarkable that since
$\tau$ linearly couples to any complex structure moduli, its stabilization
provides the restriction from the unknown part of the internal manifold. Our
study which connects the vacuum stability to the flux condition can be an
example of such restriction.

We also point out that as a by-product, our study can shed light on the
cosmology of the multi-field axion inflation [47, 48]. The effective
superpotential after integrating out $z$ contains the non-perturbative effect-
like term, so after the stabilization the potential for $C_{0}$, the
pseudoscalar component of $\tau$ is given by the sum of the axion-like term
and the polynomial term as well as their mixing. The fixed scalar field value
${\rm Im}(\tau)=g_{s}^{-1}$ provides the axion decay constant. This implies
that the features of the $(\rho,\tau)$ system also appear in the two-field
axion inflation model with the perturbative correction [32]. In this regard,
we expect that our discussion on the stability provides some conditions on the
multi-field axion inflation model, which also can be strongly constrained by
the weak gravity conjecture [49] (see also e.g., [50, 51]). Moreover, the flux
condition as well as the parametric restrictions on the internal manifold
consistent with the dilaton and the moduli stabilization may be used to
revisit the inflation models using the throat, such as the DBI inflation model
[52] (see also, e.g., [53, 54, 55, 56, 57]).

### Acknowledgements

MS is grateful to Kang-Sin Choi and Bumseok Kyae for discussion and comments
while this work was under progress.

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