aainstitutetext: Department of Mathematical Sciences, University of Liverpool,
Liverpool L69 7ZL, UKbbinstitutetext: Department of Physics, Ashoka
University, Plot 2, Rajiv Gandhi Education City, P.O. Rai, Sonipat 131029,
Haryana, India ccinstitutetext: Department of Physics, Swansea University,
Singleton Park, SA2 8PP, UK

# De Sitter vacua – when are ‘subleading corrections’ really subleading?

Bruno Valeixo Bento b Dibya Chakraborty a Susha Parameswaran c Ivonne Zavala
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider various string-loop, warping and curvature corrections that are
expected to appear in type IIB moduli stabilisation scenarios. It has recently
been argued, in the context of strongly-warped LVS de Sitter vacua, that it is
impossible to achieve parametric suppression in all of these corrections
simultaneously Junghans:2022exo . We investigate corrections in the context of
the recently discovered weakly-warped LVS de Sitter vacua, which represent a
distinct branch of solutions in type IIB flux compactifications, supported by
small conifold flux numbers $MK\lesssim 32$. Warping corrections become less
problematic in this regime, and some corrections even help to reach the
weakly-warped regime of parameter space. Other corrections continue to be
dangerous and would require numerical coefficients to be computed — and found
to be small — in order not to destroy the consistency of the weakly-warped LVS
de Sitter solution.

##  1 Introduction

The microscopic nature of the Dark Energy that dominates our Universe today
and drives its accelerated expansion remains arguably the biggest question in
fundamental physics today. So far, cosmological observations are consistent
with Dark Energy being a tiny, positive cosmological constant, or vacuum
energy, sourcing a de Sitter Universe. The string theory landscape could in
principle provide an explanation for this tiny, positive cosmological
constant, with its prediction of a multitude of four-dimensional Universes,
each one with a different total vacuum energy, including ones that match the
observed value. However, this landscape relies on de Sitter vacua obtained
with a delicate balance of tree-level and leading-order perturbative and/or
non-perturbative corrections in two expansions; using weak string coupling and
large volume. There is an ongoing fruitful debate on the validity of these
solutions, with constructions becoming ever-more explicit and tests on their
robustness ever-more rigorous. Any candidate construction must also face the
fundamental issues around de Sitter vacua, such as how to formulate an
S-matrix in spacetimes with cosmological horizons.

This paper is inspired by Junghans:2022exo , which made a comprehensive
analysis of the various subleading corrections to type IIB moduli
stabilisation scenarios that have been considered in the literature and
assessed how dangerous they are to strongly-warped Large Volume Scenario de
Sitter vacua, with light volume and conifold moduli.111This analysis was
generalized to Calabi-Yau 3-folds with arbitrary topologies in
Junghans:2022kxg , which concludes that controlling all corrections in the
context of strongly-warped LVS would require large tadpoles. The conclusion in
that paper is that there are several curvature, warping and $g_{s}$
corrections that — although parametrically suppressed in the off-shell
potential — appear enhanced in the moduli vevs and vacuum energy. This
behaviour was called non-perturbative no-scale. Moreover, it was found that
some corrections have no parametric suppression at all and — even assuming
these unsuppressed corrections to be vanishing for some reason — the remaining
corrections could not be simultaneously parametrically suppressed in small
$g_{s}$ and large $\mathcal{V}$. Therefore, in order for strongly-warped LVS
dS vacua to survive all corrections, the topological choices in the 10d
compactification must be such that some of their numerical coefficients happen
to be small. To establish whether the strongly-warped LVS dS vacua are in the
string landscape or in the ‘string theory swampland’, one would need
computational control over all the potentially dangerous subleading
corrections, which may well be developed in the coming years.

At the same time, it is interesting to ask if similar issues arise for other
proposed de Sitter scenarios. In this paper, we revisit the recently
discovered _weakly-warped_ de Sitter vacua Bento:2021nbb . This construction
is related to the Large Volume Scenario, but emerges when considering a
previously unexplored regime of parameter space, where the Klebanov-Strassler
solution is only very weakly warped and there is only a mild hierarchy between
UV and IR scales. It is thus associated with small flux numbers threading the
conifold, $MK\lesssim 32$, which can be well-described within perturbative
type IIB. It should be emphasised that the weakly-warped LVS solutions are
more than just a different regime of parameter space of the strongly-warped
LVS solutions; their form in terms of the moduli — in particular the conifold
deformation modulus — are distinct, as different terms dominate in the warped
Kähler potential that mixes the volume and conifold moduli.

Strongly-warped KKLT and LVS constructions are vulnerable to runaway
instabilities after uplifting anti-de Sitter vacua to de Sitter
upliftingrunaways2019 ; Crino:2022sey . Avoiding a runaway volume modulus
consistently with the supergravity expansion $g_{s}M\gg 1$ requires large
conifold flux numbers, $MK\gtrsim\mathcal{O}(1000)$ and $\mathcal{O}(100)$ for
KKLT and LVS respectively, leading to a tadpole problem upliftingrunaways2019
; Crino:2022sey . The same was argued to hold true for the conifold modulus in
upliftingrunaways2019 ; Crino:2022sey , although Lust:2022xoq  subsequently
suggested that higher dimensional constraint equations, which are important
when warped-down light KK modes are not included in the 4d low-energy
effective field theory, may help evade the conifold instability. The weakly-
warped LVS scenario instead evades both volume and conifold runaways via large
$AW_{0}$, where $W_{0}$ is the bulk superpotential arising from the stabilised
complex structure moduli and the dilaton that have been integrated out and $A$
is the coeffiecient of the non-perturbative contribution. As well as providing
a possible route to ameliorate the tadpole problem tadpoleProblem , the
weakly-warped LVS solution has no bulk-singularity problem Carta_2019 ;
Gao:2020xqh , and no problem with the conifold fitting into the bulk
Carta:2021lqg . The warped KK scale is greater than the mass of the conifold
and volume moduli, making the solution consistent with the Kaluza-Klein
truncation.

The purpose of this paper is to explore the robustness of this weakly-warped
LVS de Sitter solution against subleading curvature, warping and $g_{s}$
corrections. We will find that the weak warping renders some the corrections
less dangerous, compared to the strongly-warped LVS case. Some of the
corrections even help in bringing us into the weakly-warped regime of
parameter space. Other corrections, however, continue to be dangerous,
representing contributions to the off-shell scalar potential, moduli vevs and
vacuum energy that have no parametric suppression in small $g_{s}$ or large
volume. Our results are summarised in Table 3. In order for the weakly-warped
dS solution to be consistent, with the neglected corrections genuinely
suppressed, the higher dimension construction must be such that certain
numerical coefficients are small or even vanishing.

The paper is organised as follows. In Sections 2–4 we review the weakly-warped
de Sitter vacua, in a presentation that supersedes that in222We correct an
error in the original presentation Bento:2021nbb . Bento:2021nbb . We begin
with a description of warped compactifications and the warped deformed
conifold, and the 4d low energy effective field theory for the volume and
conifold moduli. Then we discuss the conifold modulus stabilisation and
finally we treat the full volume and conifold moduli stabilisation in the
context of a weakly-warped LVS scenario. The main part of the paper is Section
5, where we compute the subleading corrections to the off-shell scalar
potential, and then examine their consequences for the moduli vevs and vacuum
energy. We conclude with a summary and outlook in Section 6.

##  2 The weakly-warped deformed conifold

In this section we briefly summarise the main ingredients of type IIB flux
compactifications with warped throats and their 4d effective field theory
description. In particular, we are mostly interested in the volume modulus and
the deformation modulus of a Klebanov-Strassler (KS) warped deformed conifold.

###  2.1 Warped compactifications and the volume modulus

We start from the type IIB supergravity bosonic action in the Einstein-frame
given by333In our conventions, the string and Einstein frame 10d metrics are
related by $G_{MN}^{S}=e^{\frac{\phi-\phi_{0}}{2}}G_{MN}^{E}$, where
$e^{\phi_{0}}=g_{s}$ with $\phi_{0}$ being the background value of the dilaton
ValeixoBento:2023afn . This means that $\kappa=\kappa_{10}g_{s}$, hence
$2\kappa^{2}=(2\pi)^{7}g_{s}^{2}\alpha^{\prime 4}$, and volumes are frame
independent.

$\displaystyle S_{IIB}^{boson}$ $\displaystyle=\frac{1}{2\kappa^{2}}\int
d^{10}x\sqrt{-g_{10}}\left\\{R_{10}-\frac{\partial_{M}\tau\partial^{M}\bar{\tau}}{2(\text{Im}\tau)^{2}}-\frac{g_{s}|G_{3}|^{2}}{2(\text{Im}\tau)}-\frac{g_{s}^{2}|F_{5}|^{2}}{4}\right\\}$
$\displaystyle\quad-\frac{ig_{s}^{2}}{8\kappa^{2}}\int\frac{C_{4}\wedge
G_{3}\wedge\overline{G}_{3}}{\text{Im}\tau}\,,$ (2.1)

where $\tau=C_{0}+ie^{-\phi}$ is the axio-dilaton, $G_{3}=F_{3}-\tau H_{3}$ is
the 3-form flux, and $F_{5}=dC_{4}-\frac{1}{2}C_{2}\wedge
H_{3}+\frac{1}{2}B_{2}\wedge F_{3}$ with $C_{0},C_{2},C_{4}$ the RR
potentials.

We consider a flux compactification, whose internal compact space consists of
a finite portion of a warped throat described by the warped non-singular
deformed conifold KS2000supergravity , glued to a compact Calabi-Yau. The 10d
metric can be written as:

$\displaystyle ds^{2}$
$\displaystyle=\left[1+\frac{e^{-4A_{0}(y)}}{c(x)}\right]^{-1/2}e^{2\Omega(x)}g_{\mu\nu}dx^{\mu}dx^{\nu}+\left[1+\frac{e^{-4A_{0}(y)}}{c(x)}\right]^{1/2}c(x)^{1/2}g_{mn}dy^{m}dy^{n}\,,$
(2.2)

where $e^{2\Omega(x)}$ is the Weyl-rescaling needed to go to the Einstein
frame in 4d, and we define the warp factor as

$h\equiv 1+\frac{e^{-4A_{0}(y)}}{c(x)}\,.$ (2.3)

In the limit where $c(x)\rightarrow\infty$ we identify
$c(x)=\mathcal{V}^{2/3}$, with $\mathcal{V}\leavevmode\nobreak\ l_{s}^{6}$ the
unwarped volume of the compact space and we take coordinates such that
$V_{6}=\int d^{6}y\sqrt{g_{6}}=l_{s}^{6}$. We define a warped throat as the
region in which $h\gtrsim 2$, that is, $e^{-4A_{0}(y)}\gtrsim c(x)$. In what
follows, we will be interested in weakly warped regions where $1\lesssim
h\lesssim 2$.

After dimensionally reducing the 10d Einstein-Hilbert term in (2.1) down to 4d
we identify the Weyl-rescaling parameter as

$\displaystyle e^{2\Omega(x)}=\frac{V_{\rm w}^{0}}{c(x)^{3/2}\int
d^{6}y\sqrt{g_{6}}\leavevmode\nobreak\ h}=\frac{V_{\rm w}^{0}}{V_{\rm w}},$
$\displaystyle V_{\rm w}=\leavevmode\nobreak\ c(x)^{3/2}\int
d^{6}y\sqrt{g_{6}}\leavevmode\nobreak\ h\,.$ (2.4)

We further choose444This choice is arbitrary and all mass scales in Planck
units will be independent of the normalisation of $e^{2\Omega(x)}$.  $V_{\rm
w}^{0}\equiv\langle V_{\rm w}\rangle$, such that $\langle
e^{2\Omega(x)}\rangle=1$. That is, the two frames are the same at the vev and
the string and 4d Planck scales are related via the vev of the volume modulus.
In the limit of small warping (or for no warping at all) $c(x)\gg
e^{-4A_{0}(y)}$ and $h\approx 1$ for most of the internal space and thus
$V_{\rm w}^{0}\approx\langle\mathcal{V}\rangle l_{s}^{6}$. With our choice of
Weyl-rescaling (2.4), the definition of $M_{p}$ is

$\displaystyle\frac{M_{p}^{2}}{2}=\frac{1}{2\kappa_{4}^{2}}=\frac{V_{\rm
w}^{0}}{2\kappa^{2}}\implies\frac{m_{s}}{M_{p}}=\frac{g_{s}}{\sqrt{4\pi\mathcal{V}_{\rm
w}^{0}}}\,,$ (2.5)

where $V_{\rm w}^{0}=\mathcal{V}_{\rm w}^{0}\leavevmode\nobreak\ l_{s}^{6}$.

The mass associated with the Kaluza-Klein towers of massive states for modes
localised in the bulk, and thus experiencing $h\sim 1$, is given by555The
factor of $2\pi$ follows by analogy with a toroidal compact space, for which
$(2\pi R_{CY})^{6}=\langle\mathcal{V}\rangle l_{s}^{6}$.

$\displaystyle
m_{KK}=\frac{2\pi}{\langle\mathcal{V}\rangle^{1/6}}m_{s}\approx\frac{2\pi
g_{s}}{\sqrt{4\pi}\mathcal{V}^{2/3}}M_{p}\,.$ (2.6)

###  2.2 The warped deformed conifold

The region of the compact space which takes the form of a (warped) deformed
conifold is described by the metric candelas1990conifolds ; Minasian:1999tt ;
Aganagic:1999fe ; KS2000supergravity

$\displaystyle
ds_{con}^{2}=\frac{\epsilon^{4/3}}{2}\mathcal{K}(\eta)\Bigg{(}\frac{1}{3\mathcal{K}^{3}(\eta)}\left(d\eta^{2}+(g^{5})^{2}\right)$
$\displaystyle+\sinh^{2}(\eta/2)\left((g^{1})^{2}+(g^{2})^{2}\right)$
$\displaystyle+\cosh^{2}(\eta/2)\left((g^{3})^{2}+(g^{4})^{2}\right)\Bigg{)}\,,$
(2.7)

where $\epsilon$ is a non-zero complex parameter ($\epsilon=0$ for the
singular conifold),
$\mathcal{K}(\eta)=\frac{(\sinh(2\eta)-2\eta)^{1/3}}{2^{1/3}\sinh(\eta)}$ and
$g^{i}$ are a basis of one-forms found in candelas1990conifolds .

Near the tip of the conifold, the topology of the metric is $S^{2}\times
S^{3}$ with the $S^{3}$ of finite size and the full 10d metric becomes
KS2000supergravity

$\displaystyle ds_{10}^{2}$
$\displaystyle=e^{2A_{0}}c(x)^{-1/2}g_{\mu\nu}dx^{\mu}dx^{\nu}+e^{-2A_{0}}\left(dr_{0}^{2}+\frac{r_{0}^{2}}{8}d\Omega_{S^{2}}^{2}+R_{\epsilon}^{2}d\Omega_{S^{3}}^{2}\right),$
(2.8)

where
$r_{0}^{2}=\frac{\epsilon^{4/3}}{4}\left(\frac{2}{3}\right)^{1/3}{\eta}^{2}$,
$R_{\epsilon}^{2}=\frac{\epsilon^{4/3}}{2}\left(\frac{2}{3}\right)^{1/3}$, and
the warp factor is given by

$\displaystyle e^{-4A_{0}(\eta)}$
$\displaystyle=2^{2/3}\frac{(\alpha^{\prime}g_{s}M)^{2}}{\epsilon^{8/3}}I(\eta),\quad\quad
I(\eta)\equiv\int_{\eta}^{\infty}dx\leavevmode\nobreak\
\frac{x\coth(x)-1}{\sinh^{2}{x}}(\sinh(2x)-2x)^{1/3}\,.$ (2.9)

Here $M$ is the 3-form RR-flux through the $S^{3}$ at the tip of the throat,

$\displaystyle\frac{1}{(2\pi)^{2}\alpha^{\prime}}\int_{S^{3}}F_{3}=M\,.$
(2.10)

The physical size of the $S^{3}$ at the tip is $R_{S^{3}}\sim\sqrt{g_{s}M}$;
the $\alpha^{\prime}$-expansion is thus well under control when $g_{s}M\gg 1$.

For large $\eta$, the metric (2.7) approaches the singular conifold metric
candelas1990conifolds ; Minasian:1999tt ; KS2000supergravity ; KT

$\displaystyle ds_{con}^{2}$
$\displaystyle=dr_{\infty}^{2}+r_{\infty}^{2}\left(\frac{1}{9}(g^{5})^{2}+\frac{1}{6}\sum_{i=1}^{4}(g^{i})^{2}\right),\quad\text{
as }\eta\to\infty\,,$ (2.11)

where the coordinate $r_{\infty}$ is defined as
$r_{\infty}^{2}=\frac{3}{2^{5/3}}\epsilon^{4/3}e^{2\eta/3}$. The warp factor
becomes KT

$\displaystyle
e^{-4A_{0}}\approx\frac{L^{4}}{r_{\infty}^{4}}\left[1+\frac{3g_{s}M}{8\pi
K}+\frac{3g_{s}M}{2\pi
K}\log\Big(\frac{r_{\infty}}{r_{UV}}\Big{missing})\right]\,,$ (2.12)

where we define

$\displaystyle
L^{4}=\frac{27\pi}{4}\frac{g_{s}MK}{(2\pi)^{4}}\leavevmode\nobreak\
l_{s}^{4}\,,$ (2.13)

with $K$ being the NSNS-flux number threading the 3-cycle that goes along the
throat, which is dual to the $S^{3}$ at the tip,

$\displaystyle\frac{1}{(2\pi)^{2}\alpha^{\prime}}\int_{\eta\leq\eta_{\Lambda}}\int_{S^{2}}H_{3}=K\,.$
(2.14)

The cut-off $\eta_{\Lambda}$ in this integral will be defined in the next
subsection.

###  2.3 Gluing the conifold to a compact CY

Combining our discussion on the volume modulus and warped deformed conifold,
we use only a finite portion of the non-compact Klebanov-Strassler solution,
with the latter assumed to be glued smoothly onto a compact Calabi-Yau 3-fold.
We thus introduce a cutoff scale, $\Lambda_{UV}\sim r_{UV}$ where the conifold
connects to the bulk, which is related to $\eta_{\Lambda}$ via

$\displaystyle
r_{UV}^{2}=\frac{3}{2^{5/3}}\epsilon^{4/3}e^{2\eta_{\Lambda}/3}.$ (2.15)

A natural choice for this cut-off is where $e^{-4A_{0}(r)}\sim c(x)$ in the
warp factor (2.3), such that $r_{UV}\sim\frac{L}{c^{1/4}}$; note that this is
the minimum choice one can make for the cut-off so that the conifold is
unwarped in the gluing region. However, one may choose to extend the conifold
further, thereby including an unwarped piece, before gluing onto the compact
CY.

Note that $r_{UV}$ thus defined is also related to the radial size around the
top of the conifold as $R_{con}\equiv h^{1/4}(r_{UV})\leavevmode\nobreak\
c^{1/4}\leavevmode\nobreak\ r_{UV}\sim c^{1/4}\leavevmode\nobreak\ r_{UV}$. We
need $R_{con}\lesssim\langle\mathcal{V}\rangle^{1/6}l_{s}$, which implies
$\Lambda_{UV}\leq l_{s}$, in order for the conifold to ‘fit into’ the (roughly
isotropic) bulk (Fig. 1).

Figure 1: A cartoon of the conifold fitting into the bulk. The precise
numerical factors depend on the geometry of the compact CY onto which the
conifold is glued (see e.g. Appendix A of Gao:2020xqh ). If we consider an
isotropic toroidal compactification of size $2\pi R_{\rm CY}$, then the
conifold will fit into the bulk when $R_{con}<\pi R_{\rm CY}$. Note that one
may choose to extend the conifold beyond the point where the warp factor (2.3)
becomes $h\sim 2$, thereby including a weakly-warped and even unwarped piece,
before gluing onto the compact CY.

Upon dimensional reduction, the deformation parameter $\epsilon$ becomes a
complex structure modulus, which is part of a chiral superfield in the 4d
effective action, denoted by $|S|=\epsilon^{2}$, with units of $(length)^{3}$,
or the dimensionless $z=S/l_{s}^{3}$. The volume modulus
$c(x)=\mathcal{V}^{2/3}$ also falls into a chiral superfield with scalar
component $\rho$ and $c=\text{Im}\rho$. The radial cutoff in these coordinates
is then given by

$\displaystyle\Lambda_{0}^{2}=\frac{3}{2^{5/3}}|z|^{2/3}e^{2\eta_{\Lambda}/3}\,,$
(2.16)

where $\Lambda_{0}=\Lambda_{UV}/l_{s}$, with $\Lambda_{0}\leq 1$.

From (2.16) it follows that

$\displaystyle K$
$\displaystyle=\frac{1}{(2\pi)^{2}\alpha^{\prime}}\int_{\eta\leq\eta_{\Lambda}}\int_{S^{2}}H_{3}\approx\frac{g_{s}M}{2\pi}\eta_{\Lambda}=\frac{g_{s}M}{2\pi}\Big{(}\log\frac{\Lambda_{0}^{3}}{|z|}+\frac{3}{2}\log\frac{2^{5/3}}{3}\Big{)}\,,$
(2.17)

which therefore, neglecting contributions from the bulk, must be imposed as a
consistency condition for the solution $|z|$ together with the parameters
$g_{s},M,K,\Lambda_{0}$. This is a higher-dimensional, topological derivation
of the GKP solution $z\sim\Lambda_{0}\leavevmode\nobreak\ e^{\frac{2\pi
K}{g_{s}M}}$, which follows also from the supersymmetry condition $D_{z}W=0$
for the superpotential derived below. It shows that the flux number $K$ is not
an independent parameter, but is related to the parameters $g_{s},M$ and
$\Lambda_{0}$, which is nothing but a reflection of the fact that the warped
conifold solution KS2000supergravity  only has a free flux number $M$, since
it is a solution for constant dilaton and therefore satisfies the relation
$g_{s}^{2}|F_{3}|^{2}=|H_{3}|^{2}$.

It will be useful to bear in mind the warp factor at the tip of the throat,

$\displaystyle
e^{-4A_{0}^{tip}}=\frac{2^{2/3}I(0)}{(2\pi)^{4}}\frac{(g_{s}M)^{2}}{|z|^{4/3}}\,,$
(2.18)

and the hierarchy between the bulk (UV), where $e^{-4A_{UV}}\ll c(x)$, and the
tip of the throat (IR) is

$\displaystyle\frac{h_{IR}}{h_{UV}}=\frac{1+\frac{e^{-4A_{0}^{tip}}}{c(x)}}{1+\frac{e^{-4A_{UV}}}{c(x)}}\approx
1+\frac{e^{-4A_{0}^{tip}}}{c(x)}\,,$ (2.19)

which is large for $e^{-4A_{0}^{tip}}\gg c(x)=\mathcal{V}^{2/3}$.

###  2.4 The low-energy effective field theory

The Kähler potential for a type IIB flux compactification can be written as
giddings2003scales

$\displaystyle\mathcal{K}/M_{p}^{2}=$
$\displaystyle\underbrace{\parbox[c][28.45274pt][c]{0.0pt}{}-2\log\left(\mathcal{V}_{\rm
w}\right)}_{\text{K\"{a}hler
moduli}}\underbrace{\parbox[c][28.45274pt][c]{0.0pt}{}-\log\left(-i(\tau-\bar{\tau})\right)}_{\text{Dilaton}}\underbrace{\parbox[c][28.45274pt][c]{0.0pt}{}-\log\left(\frac{i}{l_{s}^{6}}\int
h\leavevmode\nobreak\ \Omega\wedge\bar{\Omega}\right)}_{\text{Complex
structure moduli}}\,.$ (2.20)

Here and below we keep $V_{6}=l_{s}^{6}$ explicit for clarity. The metric in
the complex structure moduli space is given by (for $|z|\ll\Lambda_{0}^{3}$)
douglas2007warping

$\displaystyle
G_{z\bar{z}}=\frac{l_{s}^{6}}{\pi||\Omega||^{2}V_{6}}\left(\log\frac{\Lambda_{0}^{3}}{|z|}+\frac{c^{\prime}}{(2\pi)^{4}}\frac{1}{\mathcal{V}^{2/3}}\frac{(g_{s}M)^{2}}{|z|^{4/3}}\right)\,,$
(2.21)

where we define the constant $c^{\prime}=1.18$, and
$||\Omega||^{2}=\frac{1}{3!}\Omega_{mnp}\Omega^{mnp}=8$. This metric
corresponds to a contribution to the Kähler potential of the form

$\displaystyle\mathcal{K}(z,\bar{z})=\frac{l_{s}^{6}}{\pi||\Omega||^{2}V_{6}}\left[|z|^{2}\left(\log\frac{\Lambda_{0}^{3}}{|z|}+1\right)+\frac{9c^{\prime}(g_{s}M)^{2}}{(2\pi)^{4}\mathcal{V}^{2/3}}|z|^{2/3}\right]\,.$
(2.22)

From here we see that the warping contribution to the Kähler potential mixes
the deformation modulus $z$ and the volume modulus $\mathcal{V}$, in such a
way that large volumes suppress the effect of the warping. Combining the
Kähler potentials for the two moduli we can write (see e.g. Dudas:2019pls )

$\displaystyle\mathcal{K}=-3\log\left(\mathcal{V}^{2/3}-\frac{1}{(2\pi)^{4}}\frac{3c^{\prime}(g_{s}M)^{2}}{\pi||\Omega||^{2}}\frac{l_{s}^{6}}{V_{6}}|z|^{2/3}\right)+\dots\quad.$
(2.23)

Notice that despite the mixing between $z$ and $\mathcal{V}$, the no-scale
structure of the Kähler potential is preserved giddings2006dynamics ;
Blumenhagen:2019qcg .

The superpotential takes the well-known Gukov-Vafa-Witten form666 The
normalisation of the superpotential $W$ changes depending on the way we write
the volume modulus and axio-dilaton terms in the Kähler potential. If we write
$-3\log(-i(\rho-\bar{\rho}))$, where $\text{Im}\rho=c(x)=\mathcal{V}^{2/3}$,
there is an extra factor of $2^{3/2}$ in $W$. If the Kähler potential for the
axio-dilaton is written as $-\log(\imaginary\tau)$, there is an extra factor
of $2^{-1/2}$. Similarly, if the transformation between the string and
Einstein frame metrics,
$G_{\mu\nu}^{S}=e^{\frac{\phi-\langle\phi\rangle}{2}}G_{\mu\nu}^{E}$, does not
include the vev $\langle\phi\rangle$, the factor of $g_{s}$ will not be
present.  Gukov:1999ya

$\displaystyle W/M_{p}^{3}=\frac{g_{s}^{3/2}}{\sqrt{4\pi}\cdot l_{s}^{5}}\int
G_{3}\wedge\Omega\,,$ (2.24)

which takes the final form GKP ; douglas2007warping

$\displaystyle
W/M_{p}^{3}=\frac{g_{s}^{3/2}}{\sqrt{4\pi}}\left[W_{0}\,e^{i\sigma}-\frac{M}{2\pi
i}z\left(\log\frac{\Lambda_{0}^{3}}{z}+1\right)-i\frac{K}{g_{s}}z\right]\,.$
(2.25)

Here, $W_{0}\,e^{\sigma}$ contains the contribution from bulk fluxes, with
$\sigma$ some constant phase. We will assume that the bulk complex structure
moduli and the dilaton are stabilised by the bulk fluxes, with
$\langle\tau\rangle=ig_{s}^{-1}$ and $W_{0}e^{i\sigma}$ evaluated at the
constant vevs.

In uplifting scenarios like KKLT and LVS, an $\overline{\text{D3}}$-brane is
placed at the tip of the warped throat in order to uplift the potential from
an AdS minimum for the volume and conifold moduli into de Sitter. The
potential is obtained from the brane action in the warped background

$\displaystyle S_{\overline{\text{D3}}}=S_{\rm DBI}+S_{\rm CS}=-2T_{3}\int
d^{4}x\sqrt{-\det G_{\mu\nu}}\,,$ (2.26)

where $G_{\mu\nu}=h^{-1/2}e^{2\Omega(x)}g_{\mu\nu}$ is the Einstein frame
metric, giving

$\displaystyle S_{\overline{\text{D3}}}$
$\displaystyle={\color[rgb]{0,0,1}-}2T_{3}\int d^{4}x\sqrt{-\det
g_{\mu\nu}}\left(\frac{V_{\rm w}^{0}}{V_{\rm w}}\right)^{2}h^{-1}\,.$ (2.27)

Using $T_{3}=\frac{1}{(2\pi)^{3}g_{s}\alpha^{\prime 2}}$, the relation between
$m_{s}$ and $M_{p}$ (2.5), and $\mathcal{V}_{\rm w}\approx\mathcal{V}$, we
obtain

$\displaystyle
V_{\overline{\text{D3}}}=\left(\frac{g_{s}^{3}}{8\pi}\right)\frac{2}{\mathcal{V}^{2}}h^{-1}M_{p}^{4}\,.$
(2.28)

Since the $\overline{\text{D3}}$-brane is placed at the tip of the throat, it
is common to take $e^{-4A_{0}(\eta_{brane})}\gtrsim c(x)$ and therefore
$h\approx\frac{e^{-4A_{0}(\eta_{brane})}}{c(x)}$. Given that the potential is
minimised for $\eta_{brane}=0$, $V_{\overline{\text{D3}}}$ becomes (cf. 2.18)

$\displaystyle
V_{\overline{\text{D3}}}=\left(\frac{g_{s}^{3}}{8\pi}\right)\frac{(2\pi)^{4}}{\mathcal{V}^{4/3}}c^{\prime\prime}\frac{|z|^{4/3}}{(g_{s}M)^{2}}M_{p}^{4}\,.$
(2.29)

where $c^{\prime\prime}=\frac{2^{1/3}}{I(0)}\approx 1.75$. However, as our
interest is in the weakly-warped case, where $h\lesssim 2$ throughout the
conifold, we will use the exact expression777This corrects an error in
Bento:2021nbb , where (2.29) was used inconsistently with the weakly-warped
regime. (2.28). Note that the $\overline{\text{D3}}$-brane can be described in
a supersymmetric way within the low energy effective supergravity theory using
constrained superfields Kallosh:2014wsa ; Bergshoeff:2015jxa ; Kallosh:2015nia
; Garcia-Etxebarria:2015lif ; Dasgupta:2016prs ; Vercnocke:2016fbt ;
Kallosh:2016aep ; Aalsma:2017ulu ; GarciadelMoral:2017vnz ; Cribiori:2019hod ,
but this will not be important for our purposes.

##  3 Deformation modulus stabilisation and de Sitter vacua

In this section we review the deformation modulus stabilisation in the weakly-
warped regime, before including the volume modulus stabilisation by turning to
the Large Volume Scenario LV1 ; originalLVS , where a new metastable de Sitter
minimum was found in Bento:2021nbb .

Combining the Kähler potential and superpotential above, eqs. (2.22) and
(2.25), the resulting scalar potential is

$\displaystyle V_{\rm KS}$
$\displaystyle=\left(\frac{g_{s}^{3}}{8\pi}\right)\frac{\pi
g_{s}}{\mathcal{V}^{2}}\left(\log\frac{\Lambda_{0}^{3}}{|z|}+\frac{1}{(2\pi)^{4}}\frac{c^{\prime}(g_{s}M)^{2}}{\mathcal{V}^{2/3}|z|^{4/3}}\right)^{-1}\left|\frac{M}{2\pi}\log\frac{\Lambda_{0}^{3}}{z}-\frac{K}{g_{s}}\right|^{2}M_{p}^{4}\,.$
(3.1)

In addition to this scalar potential originating from the fluxes, we add the
contribution from a probe $\overline{\text{D3}}$-brane (2.28) at the tip of
the conifold,

$\displaystyle V_{\overline{\text{D3}}}=c_{\rm
D3}\left(\frac{g_{s}^{3}}{8\pi}\right)\frac{2}{\mathcal{V}^{2}}\Bigg{\\{}1+\frac{1}{(2\pi)^{4}}\frac{2}{c^{\prime\prime}}\frac{(g_{s}M)^{2}}{\mathcal{V}^{2/3}|z|^{4/3}}\Bigg{\\}}^{-1}M_{p}^{4}\,,$
(3.2)

where $c_{\rm D3}=1$ in the presence of the $\overline{\text{D3}}$-brane, and
zero otherwise. The deformation modulus appears in the brane potential through
the warp factor of the metric, and we see that the suppression is provided by
the vev of this modulus, through $|z|^{4/3}$. In the classic KKLT scenario
KKLT , this energy suppression ensures that the positive energy density from
the probe D-brane uplifts an otherwise AdS minimum for the volume modulus to a
near Minkowki minimum, instead of dominating the potential and causing a
runaway. How much suppression is required, and hence how large the hierarchy
(2.19) needs to be, depends on the stabilisation mechanism of the volume
modulus and, in particular, on the depth of the AdS minimum prior to the
uplift.

In Bento:2021nbb  we introduced the useful parameter

$\displaystyle\beta\equiv\frac{\mathcal{V}^{2/3}\log\frac{\Lambda_{0}^{3}}{\zeta}}{\frac{c^{\prime}}{(2\pi)^{4}}\frac{(g_{s}M)^{2}}{\zeta^{4/3}}}=C\,\mathcal{V}^{2/3}\Lambda_{0}^{4}\leavevmode\nobreak\
x\leavevmode\nobreak\ e^{-\frac{4}{3}x},$ (3.3)

with $z=\zeta e^{i\theta}$, $C=\frac{(2\pi)^{4}}{c^{\prime}(g_{s}M)^{2}}$, and
$x\equiv\log\frac{\Lambda_{0}^{3}}{\zeta}$. The parameter $\beta$ is useful
because it determines the warping regime, with large values of $\beta\gg 1$
reflecting weak warping — this can be seen both in the deformation modulus
metric in (3.1) and in the warp factor in the brane potential (3.2). Indeed
from the definition (3.3),

$\beta\approx\frac{\mathcal{V}^{2/3}}{e^{-4A_{0}^{tip}}}\log\frac{\Lambda_{0}^{3}}{\zeta}=(h-1)^{-1}\log\frac{\Lambda_{0}^{3}}{\zeta}\,,$
(3.4)

so that leaving the $\beta\ll 1$ regime requires us to have weak warping.
Notice in particular that $\beta\propto\zeta^{4/3}\mathcal{V}^{2/3}$, so that
the warping regime relies on the competition between small $\zeta$ and large
$\mathcal{V}$. Defining also the constants

$\displaystyle\varepsilon=\frac{g_{s}M}{2\pi K}\,,$
$\displaystyle\delta_{1}=\frac{g_{s}^{3}}{8}\times\frac{K^{2}}{g_{s}}\,,$
$\displaystyle\delta_{2}=\frac{g_{s}^{3}}{8\pi}\times
c^{\prime\prime}\frac{c^{\prime}}{\delta_{1}}=\frac{1}{\pi}\times
c^{\prime\prime}c^{\prime}\frac{g_{s}}{K^{2}}\,,$
$\displaystyle\delta_{3}\equiv\frac{(2\pi)^{4}}{2}\frac{c^{\prime\prime}}{(g_{s}M)^{2}}\,,$
(3.5)

the potentials (3.1) and (3.2) become (in Planck units, $M_{p}=1$)

$\displaystyle V$
$\displaystyle=\delta_{1}C\Lambda_{0}^{4}\frac{e^{-\frac{4}{3}x}}{\mathcal{V}^{4/3}}\left[(1+\beta)^{-1}(1-\varepsilon
x)^{2}+c_{\rm
D3}\delta_{2}\Big{(}1+\delta_{3}\mathcal{V}^{2/3}\Lambda_{0}^{4}e^{-\frac{4}{3}x}\Big{)}^{-1}\right]\,,$
(3.6)

with the axion already fixed at $\langle\theta\rangle=0$ .

Since the weakly-warped regime where the warping term in (2.22) is
subdominant888To be precise, we consider a “weakly-but-still-warped scenario”,
by which we mean that the $c^{\prime}$ term in the deformation modulus
metric—the strong warping correction to GKP by DST douglas2007warping
—provides an important subleading correction to the scalar potential (3.7),
even though the interplay of all the ingredients is such that the warping is
small. Starting with a conifold provides us with an explicit background metric
onto which we can add a probe anti-D3 brane. However, to stabilise the
conifold modulus, we need fluxes, which then source a warped throat.
corresponds to $\beta\gg 1$, the potential is approximately

$\displaystyle V$
$\displaystyle\approx\frac{\delta_{1}}{\mathcal{V}^{2}}\frac{1}{x}\left[\Big{(}1-\frac{1}{\beta}\Big{)}\Big{(}1-\varepsilon
x\Big{)}^{2}+\beta\leavevmode\nobreak\ c_{\rm
D3}\delta_{2}\Big{(}1+\delta_{3}\mathcal{V}^{2/3}\Lambda_{0}^{4}e^{-\frac{4}{3}x}\Big{)}^{-1}\right]\,.$
(3.7)

The minimum can be found perturbatively in $1/\beta$,

$x_{min}=x_{0}+\frac{x_{1}}{\beta}+\mathcal{O}\Big{(}\frac{1}{\beta^{2}}\Big{)}\,,$
(3.8)

plugging $x_{min}$ into $V^{\prime}$ and solving at each order in $1/\beta$,
which gives

$x_{min}=\frac{1}{\varepsilon}+c_{\rm
D3}\frac{2\delta_{2}C^{2}}{3\delta_{3}^{\phantom{3}2}\varepsilon^{4}}\frac{1}{\beta}\Bigg{|}_{x=1/\varepsilon}+\mathcal{O}\Big{(}\frac{1}{\beta^{2}}\Big{)}\,.$
(3.9)

At leading order we obtain the GKP solution,
$x_{0}=1/\varepsilon\Rightarrow\zeta_{0}=\Lambda_{0}^{3}\exp\\{-\frac{2\pi
K}{g_{s}M}\\}\equiv\zeta_{\rm GKP}$, while the $1/\beta$ corrections depend
explicitly on $c_{\rm D3}$, reflecting the fact that it is a correction due to
the presence of the brane. This gives for $\zeta_{min}$

$\zeta_{min}\approx\zeta_{\rm GKP}\cdot\exp\left\\{-c_{\rm
D3}\frac{4K}{3\pi^{2}c^{\prime}M\zeta_{\rm
GKP}^{4/3}\mathcal{V}^{2/3}}\right\\}\,,$ (3.10)

corresponding to a small shift of the GKP solution towards smaller values of
$z$. Note the presence of the combination $\zeta_{\rm
GKP}^{4/3}\mathcal{V}^{2/3}$ in the correction, which reminds us of the
$1/\beta$ suppression and how it relies on the competition between small
$\zeta$ and large $\mathcal{V}$. We always find a solution provided $\beta$ is
large enough so that the expansion in $1/\beta$ is valid, showing that in the
weakly-warped limit there is no uplifting runaway bound on the fluxes and the
anti-brane does not destabilise the conifold modulus (cf.
upliftingrunaways2019 ). Moreover, since the solution is always only a small
perturbation away from $\zeta_{\rm GKP}$, we do not expect it to be affected
by the discussion of Lust:2022xoq , which argues that higher-dimensional
constraint equations become important when studying the 4d potential far away
from the GKP minimum.

Since $V\to 0$ as $\zeta\to 0$ 999Note that this is only true because the
warping correction to the deformation modulus metric is still present. The
$\mathcal{O}\left(1/\beta\right)$ correction itself relies on the warping
being _small but non-vanishing_. and there is a minimum, there must be a
maximum at some $0<\zeta_{max}<\zeta_{min}$. If $\zeta_{max}$ is sufficiently
smaller than $\zeta_{min}$, we will have $\varepsilon x_{max}\gg 1$. Taking
this limit and in terms of the variable $y\equiv
h-1=\frac{2}{(2\pi)^{4}}\frac{(g_{s}M)^{2}}{c^{\prime\prime}\mathcal{V}^{2/3}\zeta^{4/3}}\neq
0$, one can show that $V^{\prime}$ is proportional to

$V^{\prime}\propto-y^{3}+\left(\frac{3}{2c^{\prime}c^{\prime\prime}}-2\right)y^{2}+\left(\frac{3}{c^{\prime}c^{\prime\prime}}-\frac{16\pi
c_{\rm
D3}}{c^{\prime}c^{\prime\prime}(g_{s}M^{2})}-1\right)y+\frac{3}{2c^{\prime}c^{\prime\prime}}\,.$
(3.11)

The sign of the discriminant of this cubic equation is only a function of
$(g_{s}M^{2})$ and always negative — there can only be one real solution. On
the other hand, the variable $y=h-1<1$ and the coefficients are all
$\mathcal{O}(1)$ or more. Hence, neglecting the cubic term, we find the
approximate solution for the maximum,

$\displaystyle y=$ $\displaystyle\leavevmode\nobreak\
\frac{(3-c^{\prime}c^{\prime\prime})(g_{s}M^{2})-16\pi c_{\rm
D3}}{(g_{s}M^{2})(4cc^{\prime\prime}-3)}$
$\displaystyle+\frac{\sqrt{c^{\prime}c^{\prime\prime}(cc^{\prime\prime}+6)(g_{s}M^{2})^{2}-32\pi
c_{\rm D3}(g_{s}M^{2})(3-c^{\prime}c^{\prime\prime})+64(2\pi)^{2}c_{\rm
D3}^{2}}}{(g_{s}M^{2})(4c^{\prime}c^{\prime\prime}-3)}\,.$ (3.12)

In Fig.2 we plot the potential (3.6) in the regime $\beta\gg 1$ for a specific
set of parameters, which shows both the minimum near the GKP solution and a
maximum at $\zeta_{max}$. Note also that the value of $\zeta_{min}$ in the
weakly-warped case should not be too small in order for $\beta$ (3.3) to be
large.

Figure 2: Potential (3.6) in the regime $\beta\gg 1$ (with the $\zeta$-axis
presented in log scale), with the choice of parameters
$\Lambda_{0}=0.43,\leavevmode\nobreak\
g_{s}=0.17,M=16,K=2,\leavevmode\nobreak\ \mathcal{V}=10^{4}$ and given
$||\Omega||^{2}=8,\leavevmode\nobreak\ c^{\prime}=1.18,\leavevmode\nobreak\
c^{\prime\prime}=1.75$; for this parameter set $\zeta_{min}\approx 5.57\times
10^{-4}$ and $\zeta_{max}\approx 6.89\times 10^{-5}$. For comparison,
$\zeta_{\rm GKP}\approx 7.83\times 10^{-4}$.

Since $\beta\gg 1$ is also associated with large volumes, it is natural to use
the Large Volume Scenario LV1 ; originalLVS  to explicitly stabilise the
volume modulus. This relies on balancing leading perturbative and non-
perturbative contributions to the 4d low-energy effective field theory.
Several further corrections to the scalar potential may arise from warping,
curvature and string-loop effects, which may affect the stabilisation studied
in Bento:2021nbb . In the Section 4 we review the construction in
Bento:2021nbb  and in Section 5 we will analyse the effect of sub-leading
corrections on this leading solution.

##  4 Weakly-warped LVS de Sitter vacua

In this section we review the full moduli stabilisation and dS solution in the
weakly-warped regime $(\beta\gg 1)$, including the coupled system of Kähler
and complex structure moduli, by embedding the above system into the Large
Volume Scenario. The bulk Kähler moduli field content is therefore extended to
include the two moduli of a “Swiss cheese” Calabi-Yau.

In the Large Volume Scenario originalLVS , the Kähler moduli are stabilised by
balancing different quantum corrections against each other. It relies on both
perturbative $\alpha^{\prime}$ corrections to the Kähler potential101010Note
that the form of the correction term depends on the convention used for the
change between string frame and Einstein frame Becker:2002nn . In our
conventions, it reads (in terms of the Einstein frame volume $\mathcal{V}$)
$-2\log[\mathcal{V}+\frac{\xi}{2}e^{-\frac{3}{2}(\phi-\langle\phi\rangle)}\Big{]}=-2\log[\mathcal{V}+\frac{\xi}{2}\Big{]}\,,$
(4.1) once the dilaton $\phi$ is stabilised at $\langle
e^{\phi}\rangle=g_{s}$. This is in contrast with LVSdS:2010.15903 ;
Junghans:2022exo , in which the change of frames does not include the dilaton
vev. Notice that also the definition of $\xi$ differs by a factor of $2$. and
non-perturbative corrections to the superpotential,

$\displaystyle\mathcal{K}/M_{p}^{2}$
$\displaystyle=-2\log[\mathcal{V}+\frac{\xi}{2}\Big{]}\,,$ (4.2)
$\displaystyle W$
$\displaystyle=W^{cs}+\sum_{i}A_{i}e^{-\frac{a_{i}}{g_{s}}T_{i}}\,,$ (4.3)

where $\xi=-\frac{\chi(X_{6})\zeta(3)}{2(2\pi)^{3}}$, $\chi(X_{6})$ being the
Euler characteristic of the Calabi-Yau 3-fold $X_{6}$ that describes the
internal space with $\zeta(3)\approx 1.202$, $W^{cs}$ is the flux
superpotential responsible for stabilising the complex structure moduli and
the dilaton, $i$ runs over the Kähler moduli and the real part of
$T_{i}=\tau_{i}+i\theta_{i}$ are the volumes of internal four-cycles, being
therefore related to the volume of the internal space, while the axions
$\theta_{i}$ correspond to deformations of the RR-form $C_{4}$. The
superpotential for $T_{i}$ arises from non-perturbative effects — it can be
generated by gaugino condensation on a stack of D7-branes
($a_{i}=\frac{2\pi}{N}$, where $N$ is the rank of the world-volume gauge
group) or by Euclidean D3-brane instantons ($a_{i}=2\pi$); the coefficients
$A_{i}$ may generically depend on complex structure moduli, the dilaton, and
brane moduli that have been integrated out. As in LVSdS:2010.15903 ;
Bento:2021nbb , we consider a CY 3-fold with $h^{1,1}=2$ and a “Swiss cheese”
form for the volume

$\displaystyle\mathcal{V}=\tau_{b}^{3/2}-\kappa_{s}\tau_{s}^{3/2}\,,$ (4.4)

where only the leading non-perturbative effect is considered LV1 ; originalLVS
; LVSdS:2010.15903 ; Bento:2021nbb ,

$\displaystyle W=W^{cs}+Ae^{-\frac{a}{g_{s}}T_{s}}\,.$ (4.5)

Therefore, we consider the following Kähler and super potentials

$\displaystyle\mathcal{K}/M_{p}^{2}=$
$\displaystyle-2\log\left[\mathcal{V}+\frac{\xi}{2}\right]-\log\left(-i(\tau-\bar{\tau})\right)-\log\left(||\Omega||^{2}\right)$
$\displaystyle+\frac{1}{\pi||\Omega||^{2}}\left(|z|^{2}\left(\log\frac{\Lambda_{0}^{3}}{|z|}+1\right)+\frac{1}{(2\pi)^{4}}\frac{9c^{\prime}(g_{s}M)^{2}}{\mathcal{V}^{2/3}}|z|^{2/3}\right)\,,$
(4.6) $\displaystyle W/M_{p}^{3}=$ $\displaystyle\leavevmode\nobreak\
\frac{g_{s}^{3/2}}{\sqrt{4\pi}}\left(W_{0}\,e^{i\sigma}+\left[-\frac{M}{2\pi
i}z\left(\log\frac{\Lambda_{0}^{3}}{z}+1\right)-i\frac{K}{g_{s}}z\right]+Ae^{-\frac{a}{g_{s}}T_{s}}\right)\,,$
(4.7)

where $z=\zeta e^{i\theta}$. From these definitions we have that the gravitino
mass is111111Note that the exact factors of $g_{s}$ in the gravitino mass are
convention dependent when expressed in terms of the Einstein frame volume
$\mathcal{V}$ (see footnote 3) ValeixoBento:2023afn .

$\displaystyle
m_{3/2}=e^{\mathcal{K}/2}|W|\approx\frac{g_{s}^{2}W_{0}}{\sqrt{8\pi}||\Omega||\mathcal{V}}M_{p}\,.$
(4.8)

Computing the scalar potential $V$ in the limit $\mathcal{V}\gg 1$ and
$\zeta\ll 1$ gives121212One must be careful with the term
$K^{z\bar{z}}(D_{z}W)(D_{\bar{z}}\bar{W})$, since there is a competition
between these two limits, whose result depends on the $\beta$-regime we are
working in. Usually, it is assumed that the warp factor completely dominates
the bracket ($\beta\ll 1$) in the second line, which is equivalent to taking
the limit with the constraint $\zeta^{4/3}\mathcal{V}^{2/3}\ll 1$. By
introducing $\beta$ before expanding the potential around $1/\mathcal{V}$, one
may take the two regimes into account.

$\displaystyle V$
$\displaystyle=\frac{g_{s}^{4}}{8\pi||\Omega||^{2}}\Bigg{(}\frac{8a^{2}A^{2}\sqrt{\tau_{s}}e^{-2\frac{a}{g_{s}}\tau_{s}}}{3\kappa_{s}g_{s}^{2}\mathcal{V}}+\frac{4aA\tau_{s}e^{-\frac{a}{g_{s}}\tau_{s}}}{g_{s}\mathcal{V}^{2}}W_{0}\cos(\frac{a}{g_{s}}\theta_{s}+\sigma)+\frac{3W_{0}^{2}}{4\mathcal{V}^{3}}\xi$
(4.9) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{\pi||\Omega||^{2}}{\mathcal{V}^{2}}\Big{(}\log\frac{\Lambda_{0}^{3}}{\zeta}+\frac{1}{(2\pi)^{4}}\frac{c^{\prime}(g_{s}M)^{2}}{\mathcal{V}^{2/3}\zeta^{4/3}}\Big{)}^{-1}\Bigg{[}\frac{M^{2}}{(2\pi)^{2}}\theta^{2}+\left(\frac{M}{2\pi}\log\frac{\Lambda_{0}^{3}}{\zeta}-\frac{K}{g_{s}}\right)^{2}\Bigg{]}\Bigg{)}\,.$

Notice that the $T_{b}$ axion, $\theta_{b}$, remains a flat direction at
leading order and would be stabilised by subleading non-perturbative effects.
Looking at $\partial_{\theta}V=\partial_{\theta_{s}}V=0$, we find the
solutions for the remaining axions

$\displaystyle\langle\theta\rangle=0\,,$
$\displaystyle\langle\theta_{s}\rangle=\frac{n\pi-\sigma}{a}g_{s},\quad
n\in\mathbb{Z}\,,$ (4.10)

and choose $n=1$, such that $\cos(\frac{a}{g_{s}}\theta_{s}+\sigma)=-1$. By
inspecting the Hessian matrix in the axion directions,
$\partial_{j}\partial_{\theta}V$ and $\partial_{j}\partial_{\theta_{s}}V$,
where $j$ runs through all fields, we conclude that these completely decouple
from the other moduli and therefore we can fix the axions to their minima and
then analyse the 3-field system: $(\mathcal{V},\tau_{s},\zeta)$. In
particular, the axion masses are always positive, making these stable
directions. The potential then becomes

$\displaystyle V=$
$\displaystyle\frac{g_{s}^{4}}{8\pi||\Omega||^{2}}\Bigg{(}\frac{8a^{2}A^{2}\sqrt{\tau_{s}}e^{-2\frac{a}{g_{s}}\tau_{s}}}{3\kappa_{s}g_{s}^{2}\mathcal{V}}-\frac{4aAW_{0}\tau_{s}e^{-\frac{a}{g_{s}}\tau_{s}}}{g_{s}\mathcal{V}^{2}}+\frac{3W_{0}^{2}}{4\mathcal{V}^{3}}\xi$
$\displaystyle+\frac{\pi||\Omega||^{2}}{c^{\prime}}\frac{(2\pi)^{4}}{\mathcal{V}^{4/3}}\frac{\zeta^{4/3}}{(g_{s}M)^{2}}\Big{(}1+\beta\Big{)}^{-1}\left(\frac{M}{2\pi}\log\frac{\Lambda_{0}^{3}}{\zeta}-\frac{K}{g_{s}}\right)^{2}\Bigg{)}$
$\displaystyle+c_{\rm
D3}\left(\frac{g_{s}^{3}}{8\pi}\right)\frac{2}{\mathcal{V}^{2}}\Bigg{\\{}1+\frac{1}{(2\pi)^{4}}\frac{2}{c^{\prime\prime}}\frac{(g_{s}M)^{2}}{\mathcal{V}^{2/3}\zeta^{4/3}}\Bigg{\\}}^{-1}\,,$
(4.11)

where we introduced our variable $\beta$, defined in (3.3), and the brane
potential (3.2). It turns out that the solution for $\tau_{b}$ does not depend
on the choice of $\beta$ regime, giving

$\displaystyle\mathcal{V}\approx\tau_{b}^{3/2}=\frac{3W_{0}g_{s}\kappa_{s}\sqrt{\tau_{s}}e^{\frac{a}{g_{s}}\tau_{s}}}{aA}\frac{a\tau_{s}-g_{s}}{4a\tau_{s}-g_{s}}$
(4.12)

in both cases. However, we see that the solution is given in terms of
$\tau_{s}$, which means that there is an implicit dependence on the choice for
$\beta$ hiding in the solution for $\tau_{s}$. In turn, both $\zeta$ and
$\tau_{s}$ have different solutions depending on the regime of $\beta$ that we
look at.

In the weakly-warped regime $(\beta\gg 1)$, the potential becomes131313Note
that the warp factor in $V_{\overline{\text{D3}}}$ can be written as
$h=1+\frac{2}{c^{\prime}c^{\prime\prime}}\frac{\log\frac{\Lambda_{0}^{3}}{\zeta}}{\beta}\,.$
(4.13) Here we assume that $\beta\gg\log\frac{\Lambda_{0}^{3}}{\zeta}$ and
expand $h$ accordingly.

$\displaystyle V$
$\displaystyle\approx\frac{g_{s}^{4}}{8\pi||\Omega||^{2}}\Bigg{\\{}\frac{8a^{2}A^{2}\sqrt{\tau_{s}}e^{-2\frac{a}{g_{s}}\tau_{s}}}{3\kappa_{s}g_{s}^{2}\mathcal{V}}-\frac{4aAW_{0}\tau_{s}e^{-\frac{a}{g_{s}}\tau_{s}}}{g_{s}\mathcal{V}^{2}}+\frac{3W_{0}^{2}}{4\mathcal{V}^{3}}\xi$
$\displaystyle+\frac{\pi||\Omega||^{2}}{g_{s}}\frac{1}{\mathcal{V}^{2}\log^{2}\frac{\Lambda_{0}^{3}}{\zeta}}\Bigg{(}\log\frac{\Lambda_{0}^{3}}{\zeta}-\frac{1}{(2\pi)^{4}}\frac{c^{\prime}(g_{s}M)^{2}}{\mathcal{V}^{2/3}\zeta^{4/3}}\Bigg{)}\left(\frac{M}{2\pi}\log\frac{\Lambda_{0}^{3}}{\zeta}-\frac{K}{g_{s}}\right)^{2}$
$\displaystyle+c_{\rm
D3}\frac{||\Omega||^{2}}{g_{s}}\frac{2}{\mathcal{V}^{2}}\Bigg{(}1-\frac{2}{c^{\prime}c^{\prime\prime}}\frac{1}{(2\pi)^{4}}\frac{c^{\prime}(g_{s}M)^{2}}{\mathcal{V}^{2/3}\zeta^{4/3}}\Bigg{)}\Bigg{\\}}\,,$
(4.14)

which has a minimum at

$\displaystyle\mathcal{V}$
$\displaystyle\approx\tau_{b}^{3/2}=\frac{3(a\tau_{s}-g_{s})}{4a\tau_{s}-g_{s}}\cdot\frac{W_{0}g_{s}\kappa_{s}\sqrt{\tau_{s}}}{aA}\cdot
e^{\frac{a}{g_{s}}\tau_{s}}\,,$ (4.15) $\displaystyle\tau_{s}^{3/2}$
$\displaystyle\approx\frac{\xi}{2\kappa_{s}}+\frac{g_{s}}{3a}+\frac{8||\Omega||^{2}\mathcal{V}}{9g_{s}\kappa_{s}W_{0}^{2}}+\mathcal{O}\Big{(}\frac{1}{\beta}\Big{)}\,,$
(4.16) $\displaystyle\zeta_{min}$ $\displaystyle\approx\zeta_{\rm
GKP}\cdot\exp\left\\{-c_{\rm D3}\frac{4K}{3\pi^{2}c^{\prime}M\zeta_{\rm
GKP}^{4/3}\mathcal{V}^{2/3}}\right\\}\,.$ (4.17)

In order for the third term in (4.16) to be a correction to the leading order
solution $(\tau_{s}^{(0)})^{3/2}=\xi/(2\kappa_{s})$ in a consistent way, we
must have

$\frac{8||\Omega||^{2}\mathcal{V}}{9g_{s}\kappa_{s}W_{0}^{2}}\ll\frac{\xi}{2\kappa_{s}}\,.$
(4.18)

One can also see this by noticing that the solution for $\tau_{s}$ is
implicit, since there is a dependence on $\tau_{s}$ in the third term through
$\mathcal{V}$. If (4.16) is a solution, the function

$F(\tau_{s})=-\tau_{s}^{3/2}+\frac{\xi}{2\kappa_{s}}+\frac{2||\Omega||^{2}}{3aAW_{0}}\sqrt{\tau_{s}}\leavevmode\nobreak\
e^{\frac{a}{g_{s}}\tau_{s}}\,,$ (4.19)

where we have substituted $\mathcal{V}$ by its solution in the limit
$a\tau_{s}\gg g_{s}$, must have a root. In particular, the exponential term
cannot be too large, since it has to balance against the $\tau_{s}^{3/2}$,
which suggests that the combination $AW_{0}$ must be exponentially large. We
can make a better estimate by noting that $F(\tau_{s})$ has a minimum, which
must be non-positive for a root to exist. Again in the limit $a\tau_{s}\gg
g_{s}$, this leads to the condition

$AW_{0}>\frac{4||\Omega||^{2}}{9g_{s}}\exp\left\\{\frac{a}{g_{s}}\Big{(}\frac{\xi}{2\kappa_{s}}\Big{)}^{2/3}\right\\}\,,$
(4.20)

which indeed corresponds to having $AW_{0}$ exponentially large. Note that
this condition is stronger than (4.18), which can be seen by rewriting (4.20)
in terms of the volume modulus vev using (4.15),

$\frac{8||\Omega||^{2}\mathcal{V}}{9g_{s}\kappa_{s}W_{0}^{2}}<\frac{3g_{s}}{2a}\cdot\left(\frac{\xi}{2\kappa_{s}}\right)^{1/3}\,.$
(4.21)

Figure 3: The plot shows (4.19) for different choices of the parameters
$W_{0}$ and $A$, with all other parameters fixed. Critical points for the
scalar potential (4.14) exist when $F(\tau_{s})=0$. We see that the product
$AW_{0}$ needs to be large enough for a solution to exist (4.20).

At this solution the vacuum energy is given by

$\displaystyle
V_{min}=\frac{g_{s}^{4}}{8\pi||\Omega||^{2}}\cdot\frac{3g_{s}\kappa_{s}W_{0}^{2}\sqrt{\tau_{s}}}{4a\mathcal{V}^{3}}\Big{(}-1+\alpha+\mathcal{O}\left(g_{s}\right)+\mathcal{O}\left(1/\beta\right)\Big{)}\,,$
(4.22)

where we define the uplift parameter $\alpha$ as141414Note that although the
definition of $\alpha$ differs from the one in Junghans:2022exo , it still
encodes the brane uplift and its backreaction on the solution just as it did
in the strongly-warped regime. It is here simply adapted to the weakly-warped
regime, making explicit the need for large $W_{0}$ in order to suppress the
effects of the brane. Note also that the $\tau_{s}$ in this expression
corresponds to the leading term in the solution, i.e.
$\tau_{s}^{(0)}=\frac{\xi^{2/3}}{(2\kappa_{s})^{2/3}}$.

$\displaystyle\alpha=c_{\rm D3}\leavevmode\nobreak\
\frac{8a||\Omega||^{2}}{9\kappa_{s}\sqrt{\tau_{s}}}\frac{\mathcal{V}}{g_{s}^{2}W_{0}^{2}}\,.$
(4.23)

Therefore a dS vacuum requires $\alpha>1$.

### Example

We choose a set of parameters which guarantees both $V_{crit}>0$ and
$m_{3}^{2}>0$, and thus a dS minimum. We have parameters associated with the
fluxes for the remaining complex structure moduli and the axio-dilaton
$(W_{0},g_{s})$, with the conifold $(M,K,\Lambda_{0})$, and with the CY and
Kähler moduli $(\kappa_{s},\xi,a,A)$, where
$\xi=-\frac{\chi\zeta(3)}{2(2\pi)^{3}}$. The fixed parameters in the potential
are $c^{\prime}=1.18$, $c^{\prime\prime}=1.75$, $||\Omega||^{2}=8$.

In Table 1 we present a set of parameters that serve to provide a concrete
example of the stabilisation mechanism presented in this subsection. As we
have seen, we need large values of $AW_{0}$. It may be possible to accomodate
these through the specific values of the complex structure moduli that have
been stabilised and integrated out. For example, we will see in the next
section how one-loop corrections to the gauge kinetic function on wrapped
D7-branes effectively increase the value of $A$. From the instanton point of
view, this corresponds to a one-loop corrected instanton action $W_{np}\sim
e^{-S_{\rm inst}}$, with $S_{\rm inst}\sim\frac{a}{g_{s}}\tau_{s}-\log
A\approx 4.3>1$ for our numerical example, which keeps the instanton expansion
under control. It is also worth emphasising that the solution is sourced by
flux numbers $MK=32$, which straightforwardly allows tadpole cancellation
using only orientifold planes within 4d perturbative type IIB.

$W_{0}$ | $\sigma$ | $g_{s}$ | $M$ | $K$ | $\Lambda_{0}$ | $\kappa_{s}$ | $\chi$ | $a$ | $A$
---|---|---|---|---|---|---|---|---|---
$2000$ | $0$ | $0.17$ | $16$ | $2$ | $0.43$ | $\frac{\sqrt{2}}{9}$ | $-280$ | $\frac{\pi}{3}$ | $870$
Table 1: Choice of parameters for the potential (4.11), with $\beta\gg 1$.

For the parameter set in Table 1, we determine the critical points of the full
potential (4.11), summarised in Table 2. While one critical point is a
minimum, the second is a saddle point with one unstable direction. Note that
for both cases we indeed find $\beta>1$ ($\beta_{min}\approx 7$ and
$\beta_{saddle}\approx 15$).

$\tau_{s}$ | $\tau_{b}$ | $\zeta$ | $V_{crit}$ | $m_{1}^{2}\sim m_{\zeta}^{2}$ | $m_{2}^{2}\sim m_{\tau_{s}}^{2}$ | $m_{3}^{2}$
---|---|---|---|---|---|---
1.80 | 239 | $4.17\times 10^{-4}$ | $1.70\times 10^{-13}$ | $9.23\times 10^{-5}$ | $4.87\times 10^{-4}$ | $4.32\times 10^{-11}$

$\mathcal{V}$ | $M_{s}$ | $m_{KK}$ | $m_{3/2}$ | $M_{s}^{\rm w}$ | $m_{KK}^{\rm w}$ | $m_{3/2}^{\rm w}$
---|---|---|---|---|---|---
$3684$ | $4.96\times 10^{-3}$ | $1.26\times 10^{-3}$ | $1.11\times 10^{-3}$ | $4.33\times 10^{-3}$ | $1.10\times 10^{-3}$ | $0.96\times 10^{-3}$

Table 2: Solution and masses for the fields $(\tau_{s},\tau_{b},\zeta)$ for
the parameter set in Table 1, and corresponding physical scales (in units of
$M_{p}$). The scales $M_{s}^{\rm w}$, $m_{KK}^{\rm w}$ and $m_{3/2}^{\rm w}$
are warped down with respect to their bulk counterparts by the factor
$h_{tip}^{-1/4}=h^{-1/4}_{IR}$ (see equation (2.19)); in the present weakly-
warped case the distinction is not significant. Since all mass-squareds are
positive, the solution is metastable. For comparison, $\zeta_{\rm GKP}\approx
7.83\times 10^{-4}$.

From Table 2 we see that the cutoff for the EFT is $m_{KK}\sim m_{KK}^{\rm
w}$, which reflects the fact that this solution has weak warping. Note that
the moduli masses are much lighter than the KK scale, so that the KK modes can
be safely integrated out. At the same time, $\frac{m_{3/2}}{m_{KK}}\approx
0.88$, which is at the limit of a consistent 4d supergravity description (see
also Cicoli:2013swa ). Also $\Lambda_{0}<1$, so that the conifold fits into
the bulk as in the strongly-warped case Carta_2019 ; Gao:2020xqh . Similarly
to the usual LVS, this solution avoids the bulk singularity problem
Gao:2020xqh ; Carta:2021lqg  due to the hierarchy
$\tau_{s}\ll\tau_{b}\sim\mathcal{V}^{2/3}$.

In Figs. 4–5 we show the plots of the potential in each direction and in the
plane $(\tau_{s},\tau_{b})$.

Figure 4: Plot of the potential (4.11) in the $\zeta$ direction (the
$\zeta$-axis is presented in log scale), for the parameter set in Table 1,
with $(\tau_{s},\tau_{b})$ fixed to the values given in Table 2. For
comparison, $\zeta_{\rm GKP}\approx 7.83\times 10^{-4}$. Figure 5: Plot of the
potential (4.11) in the $(\tau_{1},\tau_{2})$ plane (a rotation of the
$(\tau_{b},\tau_{s})$ plane aligned with the eigenvectors of the Hessian
matrix), for the parameters in Table 1 (in Planck units).

##  5 Dangerous corrections?

Although there seem to exist explicit constructions of dS vacua with an
$\overline{\text{D3}}$-brane uplift in the context of LVS, both in the weakly-
warped regime Bento:2021nbb , as we just reviewed, and in the strongly-warped
one LVSdS:2010.15903 , these implicitly rely on the assumption that all
putative subleading corrections are indeed subleading compared to the
contributions that have already been included in the 4d low energy effective
field theory. The question of whether this assumption is justified was
analysed in Junghans:2022exo  for the strongly-warped regime studied in
LVSdS:2010.15903 , by taking into account various types of corrections to the
LVS potential. The conclusion for the strongly-warped regime was that some
corrections do not have any parametric suppression, and moreover even those
that do can never be _simultaneously_ parametrically suppressed.

It is worth fleshing out the main point in the argument of Junghans:2022exo .
If a correction to the LVS potential can arise from some string theory effect
(e.g. one-loop or curvature corrections), one of two things must happen:
either this correction is parametrically suppressed for all quantities in the
solution of interest (moduli vevs, masses, vacuum energy), and can therefore
be neglected in the limit of weak coupling and/or large volume within the EFT,
or else one _must_ compute this correction explicitly and include it in the
analysis, and potentially worry about next-to-next-to-leading order
corrections. While the result of Junghans:2022exo  does not show that finding
a stable strongly-warped dS solution in LVS is impossible, it means that one
must take at least some of these corrections into account, compute their
coefficients explicitly, and keep them in the EFT unless their numerical
coefficients can be made small. This constitutes a challenge because, not only
are these coefficients typically hard to compute explicitly, but also the
known solutions in the LVS may be absent or at least corrected in significant
ways by these contributions.

It therefore becomes relevant to ask whether this is also the case in the
weakly-warped de Sitter vacua just discussed Bento:2021nbb . In order to
address this question, we will briefly outline the known subleading
contributions to the 4d low energy effective theory (for more details see
Junghans:2022exo  and references therein) and compute how they correct the
off-shell scalar potential in the weakly-warped regime. We will then
investigate the consequences of these corrections for the moduli vevs and
vacuum energy, and work out how they can be suppressed in order to achieve
self-consistent weakly-warped LVS dS vacua.

###  5.1 Corrections to the off-shell potential

Let us group the known subleading contributions to the 4d EFT according to
their origin and work out how they correct the off-shell scalar potential in
the weakly-warped regime.

### Curvature/loop corrections to the Kähler potential

String-loop corrections at order $\alpha^{\prime 2}$ from the exchange of KK
modes between D7/D3 branes or O7/O3 planes Berg:2005ja ; Berg:2014ama ;
Haack:2015pbv ; Haack:2018ufg ; Garcia-Etxebarria:2012bio  enter the Kähler
potential (4.7) as Cicoli:2007xp ; Berg:2007wt ; vonGersdorff:2005bf ;
Berg:2005yu

$\displaystyle\delta K=C^{\rm
KK}_{s}\cdot\frac{g_{s}^{2}\sqrt{\tau_{s}}}{\mathcal{V}}+C^{\rm
KK}_{b}\cdot\frac{g_{s}^{2}\sqrt{\tau_{b}}}{\mathcal{V}}\,,$ (5.1)

and thus the scalar potential is corrected as (cf. first line of eq. (4.14))

$\displaystyle\left(\frac{g_{s}^{4}}{8\pi||\Omega||^{2}}\right)^{-1}\delta V=$
$\displaystyle\leavevmode\nobreak\ C^{\rm
KK}_{s}\cdot\frac{g_{s}^{2}}{6\kappa_{s}\tau_{s}}\cdot\frac{8a^{2}A^{2}\sqrt{\tau_{s}}e^{-2\frac{a\tau_{s}}{g_{s}}}}{3g_{s}^{2}\kappa_{s}\mathcal{V}}+C^{\rm
KK}_{s}\cdot\frac{g_{s}^{2}}{3\kappa_{s}\tau_{s}}\cdot\frac{4aAW_{0}\tau_{s}e^{-\frac{a\tau_{s}}{g_{s}}}}{g_{s}\mathcal{V}^{2}}$
$\displaystyle+\left\\{(C^{\rm
KK}_{s})^{2}\cdot\frac{2g_{s}^{4}}{9\kappa_{s}\sqrt{\tau_{s}}}+C^{\rm
KK}_{b}\cdot\frac{8c^{\prime}g_{s}^{2}(g_{s}M)^{2}\zeta^{2/3}}{\mathcal{V}^{1/3}}\right\\}\cdot\frac{3W_{0}^{2}}{4\mathcal{V}^{3}}\,.$
(5.2)

Note that in principle the coefficients $\\{C^{\rm KK}_{s},C^{\rm KK}_{b}\\}$
depend on the complex structure moduli; however, as this dependence is unkown,
we will simply assume that $C^{\rm KK}_{s},C^{\rm
KK}_{b}\sim\mathcal{O}\left(1\right)$ and neglect the derivative
contributions. For easy identification, we have written each term isolating
clearly each suppression factor with respect to the corresponding uncorrected
term in the LVS potential. For example, all the terms above are suppressed by
a factor of $g_{s}^{2}$ compared to their LVS counterparts (see eq. (4.14)).
The $C^{\rm KK}_{b}$ term is also suppressed by the ratio
$\zeta^{2/3}/\mathcal{V}^{1/3}$ but enhanced by $(g_{s}M)^{2}$.

On the other hand, threshold corrections to gauge couplings lead to the one-
loop field redefinition of the Kähler modulus $\tau_{s}^{\rm
new}=\tau_{s}^{\rm old}+C^{\rm log}_{s}\log\mathcal{V}$, which implies a
correction to the Kähler potential once the physical volume is expressed in
terms of the corrected modulus,
$\mathcal{V}=\tau_{b}^{3/2}-\kappa_{s}(\tau_{s}-C^{\rm
log}_{s}\log\mathcal{V})^{3/2}$ Conlon:2010ji . It contributes to the scalar
potential as151515This field redefinition leads to a non-linear relation
between the volume $\mathcal{V}$ and the Kähler moduli ($\tau_{b},\tau_{s}$).
Treating this redefinition as a small correction, one can solve it
perturbatively for
$\mathcal{V}=\tau_{b}^{3/2}-\kappa_{s}\tau_{s}^{3/2}+\mathcal{O}\left(C^{\rm
log}_{s}\right)$ and use this leading order solution in the correction. Hence
at leading order the volume becomes
$\mathcal{V}\approx\tau_{b}^{3/2}-\kappa_{s}\Big{(}\tau_{s}-C^{\rm
log}_{s}\log(\tau_{b}^{3/2}-\kappa_{s}^{3/2})\Big{)}^{3/2}$.

$\displaystyle\left(\frac{g_{s}^{4}}{8\pi||\Omega||^{2}}\right)^{-1}\delta V=$
$\displaystyle\leavevmode\nobreak\ C^{\rm
log}_{s}\cdot\frac{g_{s}\log\mathcal{V}}{2\tau_{s}}\cdot\frac{8a^{2}A^{2}\sqrt{\tau_{s}}e^{-2\frac{a\tau_{s}}{g_{s}}}}{3g_{s}^{2}\kappa_{s}\mathcal{V}}+C^{\rm
log}_{s}\cdot\frac{g_{s}\log\mathcal{V}}{\tau_{s}}\cdot\frac{4aAW_{0}\tau_{s}e^{-\frac{a\tau_{s}}{g_{s}}}}{g_{s}\mathcal{V}^{2}}$
$\displaystyle-C^{\rm log}_{s}\cdot
6g_{s}\kappa_{s}\sqrt{\tau_{s}}\cdot\frac{3W_{0}^{2}}{4\mathcal{V}^{3}}\,.$
(5.3)

Once again, notice the relative factors with respect to the terms in the LVS
potential (4.14). Recalling that the LVS solution is
$\log\mathcal{V}\sim\frac{a\tau_{s}}{g_{s}}$, there seems to be no parametric
suppression in the correction terms in the first line.

At order $\alpha^{\prime 3}$ there are corrections from different sources
Berg:2014ama ; Haack:2015pbv ; Haack:2018ufg ; Minasian:2015bxa ;
Antoniadis:2019rkh  (such as the backreaction of 7-branes Minasian:2015bxa
and the exchange of KK modes with D7 branes/O7 planes Antoniadis:2019rkh ),
which contribute at leading order as a redefinition of
$\xi\to\xi-\Delta\xi+C^{\xi}_{1}g_{s}\log\mathcal{V}+C^{\xi}_{2}g_{s}$ in $K$
(4.7), correcting the scalar potential as

$\displaystyle\left(\frac{g_{s}^{4}}{8\pi||\Omega||^{2}}\right)^{-1}\delta V=$
$\displaystyle\leavevmode\nobreak\
(-\Delta\xi+C^{\xi}_{1}g_{s}\log\mathcal{V}+C^{\xi}_{2}g_{s})\cdot\frac{3W_{0}^{2}}{4\mathcal{V}^{3}}\,.$
(5.4)

Notice again that there seems to be no suppression for the $\Delta\xi$ and the
$C^{\xi}_{1}$ corrections. However, $\Delta\xi$ can be computed explicitly in
terms of the Poincaré dual $D_{\rm O7}$ of the divisor wrapped by O7 planes
Minasian:2015bxa ,

$\Delta\xi=\frac{\zeta(3)}{(2\pi)^{3}}\int_{X}D_{\rm O7}^{3}\,,$ (5.5)

and it amounts simply to a shift in the numerical parameter $\xi$.

### Curvature corrections to the gauge-kinetic function

The non-perturbative superpotential which is a key ingredient in the
stabilisation of the Kähler moduli arises from gaugino condensation on a stack
of D7-branes Jockers:2004yj ; Jockers:2005zy ; Haack:2006cy . Therefore
curvature corrections to the D7-brane action Bachas:1999um ; Bershadsky:1995qy
; Green:1996dd ; Cheung:1997az ; Minasian:1997mm , which correct the gauge-
kinetic function of the D7-branes will affect this non-perturbative
contribution. The gauge-kinetic function becomes
$f_{s}=T_{s}-\frac{\chi_{s}}{24}\tau$, where $\chi_{s}$ is the Euler number of
the 4-cycle wrapped by the D7-branes whose size is controlled by $\tau_{s}$,
so that the generated superpotential becomes

$\displaystyle W_{np}=A\cdot
e^{-\frac{a}{g_{s}}\big{(}\tau_{s}-\frac{\chi_{s}}{24g_{s}}\big{)}}\,,$ (5.6)

which is equivalent to a redefinition of $A\to A\cdot
e^{\frac{a\chi_{s}}{24g_{s}^{2}}}$. It is worth recalling that our weakly-
warped solution required somewhat large values of the parameter $A$ (4.20) —
one might therefore expect that this correction in particular will benefit,
rather than obstruct, these weakly-warped solutions.

### Higher F-terms

Further corrections arise from higher F-terms associated with 4 superspace
derivatives in the 4d EFT deAlwis:2012vp ; Cicoli:2013swa ; Ciupke:2015msa .
These could come for example from integrating out KK modes Cicoli:2013swa  or
from 10d 8-derivative terms with powers of $G_{3}$ Ciupke:2015msa , which
would lead to such a contribution in the potential. The scalar potential gets
corrected as

$\displaystyle\left(\frac{g_{s}^{4}}{8\pi||\Omega||^{2}}\right)^{-1}\delta
V=C^{F}\cdot\frac{8g_{s}^{2}W_{0}^{2}}{3\mathcal{V}^{2/3}}\cdot\frac{3W_{0}^{2}}{4\mathcal{V}^{3}}\,.$
(5.7)

This correction is suppressed relative to the LVS potential by the factor

$\displaystyle\frac{m_{3/2}^{2}}{m_{KK}^{2}}\sim\frac{g_{s}^{2}W_{0}^{2}}{\mathcal{V}^{2/3}}\,,$
(5.8)

which one would generically want to be small in order to keep the gravitino in
the EFT valid at energies $E\ll m_{KK}$ (and therefore preserve
$\mathcal{N}=1$ supersymmetry).

So far only one of the corrections is warping-related, the correction linear
in $C^{\rm KK}_{b}$ which depends on the warping due to its dependence on
$\zeta$. It does indeed arise from the warping correction to the deformation
modulus metric, which mixes this modulus with the volume $\mathcal{V}$ — this
was precisely the crucial term in the discussion of our weakly-warped
solutions. One would not expect the effect of the other corrections thus-far
considered to be dependent on which warping regime we consider, although, as
we will see below, having weak-warping rather than strong-warping can still
make a difference due to the different region of interest in parameter space.
The two remaining corrections to be considered, $\\{C^{\rm flux},C^{\rm
con}\\}$, are intrinsically warping-related, being associated with the tip of
the warped conifold, and one would therefore expect them to play a more
important role in the distinction between the two regimes.

### Curvature/warping corrections from conifold-flux backreaction

The charges responsible for the warping (such as branes, O-planes and fluxes)
can have large contributions to the 10d curvature even at large volumes and
result in large curvature corrections Gao:2020xqh ; Carta:2019rhx . For the KS
throat, the charge responsible is the flux contribution $KM$, which in the
strongly-warped regime is required to be large in order that the warping is
sufficient to avoid runaway in the volume modulus after uplifting. This leads
to corrections proportional to the ratio $\frac{KM}{\mathcal{V}^{2/3}}$, e.g.
from dimensionally reducing the 10d $R^{4}$ terms of type IIB, which appear in
the scalar potential as Junghans:2022exo

$\displaystyle\left(\frac{g_{s}^{4}}{8\pi||\Omega||^{2}}\right)^{-1}\delta
V=C^{\rm
flux}\cdot\frac{5g_{s}KM}{\mathcal{V}^{2/3}}\cdot\frac{3W_{0}^{2}}{4\mathcal{V}^{3}}\,.$
(5.9)

Since in the weakly-warped regime, one does not require very large charges
$KM$, this correction is expected to be less dangerous than in the strongly-
warped case.161616The form of the correction used in Gao:2022fdi  for the LVS
parametric tadpole constraint differs by a factor of $\xi$, which is just
$\mathcal{O}(1)$ in our example. The discrepancy follows from the assumption
in Gao:2022fdi  that the warp factor is a slowly varying function, in which
case one should expect this warping correction to scale with the Euler number
$\chi(X_{6})$. See, however, Junghans:2022kxg  for further discussion.

### Curvature corrections in the conifold region

Since the radius of the $S^{3}$ at the tip of the conifold is
$R^{2}_{S^{3}}\sim(g_{s}M)\alpha^{\prime}$, one expects $\alpha^{\prime}$
curvature corrections suppressed by $(g_{s}M)^{-1}$. E.g. the $R^{2}$
corrections to the DBI action of the $\overline{\text{D3}}$-brane in the
warped deformed conifold background correct the brane tension with a term
suppressed by $(g_{s}M)^{-2}$ Junghans:2022exo .171717These correspond to the
$(\alpha^{\prime})^{2}$ curvature corrections to the brane action studied in
Hebecker:2022zme  that lower the tension of the brane and provide an
alternative for warped throats. In practice, this corresponds to the
redefinition

$\displaystyle c_{\rm D3}\to c_{\rm D3}\Big{(}1+\frac{C^{\rm
con}}{(g_{s}M)^{2}}\Big{)}\,,$ (5.10)

which is easily incorporated in both the potential and the solutions. Although
the weakly-warped solutions do not require large $M$ and would hence prefer
smaller values of $M$ (e.g. by also taking into account the Tadpole Conjecture
tadpoleProblem ), they still require the supergravity approximation to be
under control near the tip of the conifold, which is controlled by the radius
of the $S^{3}$ at the tip $R^{2}_{S^{3}}\gg\alpha^{\prime}$ — this means that
even though the flux number $M$ is in general lower in the weakly-warped case,
it must still be large enough to make $g_{s}M\gg 1$ and keep this correction
term suppressed in the brane action.

Let us now look at the solutions for the corrected weakly-warped LVS
potential, as well as at the vacuum energy associated with these vevs, and
examine the suppression of each correction on these quantities.

###  5.2 Corrections to the solutions

We now compute how the solutions are modified due to the various corrections
outlined above.

####  5.2.1 Conifold modulus stabilisation

The only corrections involving the conifold modulus, and therefore affecting
its stabilisation, are those associated with $\\{C^{\rm KK}_{b},C^{\rm
con}\\}$. While $C^{\rm con}$ is simply a shift of $c_{\rm D3}$, and thus can
be trivially included in the solution (4.17), the correction $C^{\rm KK}_{b}$
can be taken into account by solving perturbatively in a similar way to how
the $1/\beta$ corrections were taken into account. We find that the vev of
$\zeta$ gets corrected as (see (4.17))

$\displaystyle\zeta\approx\zeta_{\rm GKP}\Bigg{(}1$ $\displaystyle-c_{\rm
D3}\Big{(}1+\frac{C^{\rm
con}}{(g_{s}M)^{2}}\Big{)}\frac{4K}{3\pi^{2}c^{\prime\prime}M\leavevmode\nobreak\
\zeta_{\rm GKP}^{4/3}\mathcal{V}^{2/3}}$ $\displaystyle-C^{\rm
KK}_{b}\cdot\frac{2c^{\prime}(2\pi)^{2}}{\pi||\Omega||^{2}}\frac{2\pi
K}{g_{s}M}\,\zeta_{\rm
GKP}^{2/3}\,\frac{g_{s}^{4}W_{0}^{2}}{\mathcal{V}^{4/3}}\Bigg{)}\,.$ (5.11)

Notice that the $C^{\rm con}$ correction is not only suppressed by its own
control parameter $(g_{s}M)^{-1}$, but also by $1/\beta$ since it arises from
the $\overline{\text{D3}}$-brane backreaction which itself contributes at
$\mathcal{O}\left(1/\beta\right)$. As for $C^{\rm KK}_{b}$, it appears to be
heavily suppressed by the factor $\frac{\zeta_{\rm
GKP}^{2/3}}{\mathcal{V}^{2/3}}\cdot\frac{g_{s}^{4}W_{0}^{2}}{\mathcal{V}^{2/3}}$.

####  5.2.2 Kähler moduli

Since the formal solution for $\mathcal{V}$ follows directly from
$\partial_{\tau_{s}}V=0$, one can also infer that only a few of the
corrections will appear — in particular, it will receive explicit corrections
from $\\{C^{\rm KK}_{s},C^{\rm log}_{s},\chi_{s}\\}$ only. Of course, due to
its dependence on $\tau_{s}$, which is sensitive to all corrections
considered, $\mathcal{V}$ is also implicitly sensitive to all corrections.
This is also true for $\zeta$, since the subleading terms in its vev (5.2.1)
depend on $\mathcal{V}$.

Solving perturbatively to leading order in all corrections, we find (cf.
(4.15), (4.16))

$\displaystyle\mathcal{V}$
$\displaystyle\approx\Bigg{(}\frac{3(a\tau_{s}-g_{s})}{4a\tau_{s}-g_{s}}+C^{\rm
KK}_{s}\cdot\frac{g_{s}^{2}}{8\kappa_{s}\tau_{s}}-C^{\rm
log}_{s}\cdot\frac{9a}{8}\Bigg{)}\frac{g_{s}W_{0}\kappa_{s}\sqrt{\tau_{s}}}{aA}\cdot
e^{\frac{a}{g_{s}}\left(\tau_{s}-\frac{\chi_{s}}{24g_{s}}\right)}\,,$ (5.12)
$\displaystyle\tau_{s}$
$\displaystyle\approx\frac{\hat{\xi}^{2/3}}{(2\kappa_{s})^{2/3}}+\frac{(1+2\alpha)g_{s}}{3a}+\mathcal{O}\left(\frac{1}{\beta}\right)+\mathcal{O}\left(g_{s}^{2}\right)$
$\displaystyle-C^{\rm KK}_{s}\cdot\frac{g_{s}^{2}}{3\kappa_{s}}+C^{\rm
log}_{s}\cdot\frac{5a}{3}\Big{(}\frac{\hat{\xi}}{2\kappa_{s}}\Big{)}^{2/3}+C^{\xi}_{1}\cdot\frac{a}{3\kappa_{s}}\Big{(}\frac{\hat{\xi}}{2\kappa_{s}}\Big{)}^{1/3}+C^{\xi}_{2}\cdot\frac{g_{s}}{3\kappa_{s}}\Big{(}\frac{\hat{\xi}}{2\kappa_{s}}\Big{)}^{-1/3}$
$\displaystyle+C^{\rm
flux}\cdot\frac{55}{27\kappa_{s}}\frac{g_{s}KM}{\mathcal{V}^{2/3}}\Big{(}\frac{\hat{\xi}}{2\kappa_{s}}\Big{)}^{-1/3}+C^{F}\cdot\frac{88}{81\kappa_{s}}\frac{g_{s}^{2}W_{0}^{2}}{\mathcal{V}^{2/3}}\Big{(}\frac{\hat{\xi}}{2\kappa_{s}}\Big{)}^{-1/3}$
$\displaystyle-C^{\rm con}\cdot\frac{2g_{s}\alpha}{3a(g_{s}M)^{2}}+C^{\rm
KK}_{b}\cdot\frac{80c^{\prime}(g_{s}M)^{2}}{27\kappa_{s}}\cdot\frac{g_{s}^{2}\zeta^{2/3}}{\mathcal{V}^{1/3}}\Big{(}\frac{\hat{\xi}}{2\kappa_{s}}\Big{)}^{-1/3}\,,$
(5.13)

with $\alpha$ given by (4.23) and $\hat{\xi}\equiv\xi-\Delta\xi$. The solution
for $\mathcal{V}$ (5.12) matches181818Apart from a factor of $3$ in the
$C^{\rm log}_{s}$ term. the one found in Junghans:2022exo  up to factors of
$g_{s}$ due to the use of different conventions (see below). This agrees with
our expectation that the explicit form of the $\\{C^{\rm KK}_{s},C^{\rm
log}_{s},\chi_{s}\\}$ corrections is independent of the warping regime. One
can immediately see that, although the $C^{\rm KK}_{s}$ correction is
suppressed by $g_{s}^{2}$, neither the $C^{\rm log}_{s}$ nor the $\chi_{s}$
corrections are suppressed — following the logic presented above, one must
therefore include these corrections in the EFT analysis. Although the
$\chi_{s}$ correction is not only not suppressed but actually enhanced by a
factor of $1/g_{s}$, the fact that we know its explicit form means that it can
be taken into account when looking for a solution and, as we saw in (5.6), it
can be seen as a rescaling of the parameter $A$ in the non-perturbative
contribution to the superpotential. In fact, given the need for exponentially
large $AW_{0}$ (4.20), this seems to help rather than hurt weakly-warped
solutions.191919Indeed, for the CY3 of LVSdS:2010.15903 , we have $\chi_{s}=3$
and hence the correction represents a rescaling of $A\to
Ae^{\frac{\chi_{s}}{24g_{s}}}$ by a factor of $12$, which would naïvely allow
for the much lower value $A\sim 70$. The $C^{\rm log}_{s}$ correction, on the
other hand, is not generically known explicitly and therefore one cannot
follow the same procedure — this coefficient would have to be computed for the
compact geometry of interest in order to be included in the EFT analysis.

The different $g_{s}$ factors in the solution with respect to Junghans:2022exo
arise from our choice of frame conventions, with $\tau_{b}$ and $\tau_{s}$
differing from the ones in Junghans:2022exo  by a factor of $g_{s}$, i.e.
$\tau_{i}^{\rm(ours)}=g_{s}\tau_{i}^{\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Junghans:2022exo}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
and
$\mathcal{V}^{\rm(ours)}=g_{s}^{3/2}\mathcal{V}^{\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Junghans:2022exo}{\@@citephrase{(}}{\@@citephrase{)}}}}}$.
Recall that in these conventions, the Einstein-frame volumes and the string-
frame volumes are the same at the vev, so that deciding e.g. whether
$\alpha^{\prime}$-corrections are under control for a given solution can be
done directly in terms of the Einstein-frame volumes.202020Note that checking
whether the $\alpha^{\prime}$-corrections are under control, i.e. whether a
given solution is consistent with the supergravity description being used,
should be done using string-frame volumes, which are parametrically (in
$g_{s}$) smaller than the Einstein-frame volumes when the convention
$\tau_{i}^{(S)}=g_{s}\tau_{i}^{(E)}$ is used for the 10d change of frames,
rather than the one we are using in this work for which these volumes are the
same. Therefore, one might naively think that the volumes are large “enough”
in Einstein-frame, while the string-frame volumes are actually inconsistent
with the $\alpha^{\prime}$-expansion. It also makes explicit the $1/g_{s}$
factor in the non-perturbative superpotential, rather than it appearing in the
solution for $\tau_{s}$ — both pictures are ultimately consistent since this
factor of $1/g_{s}$ will appear in the potential one way or another. Because
of this, however, our solution for $\tau_{s}$ (5.2.2) differs from the one in
Junghans:2022exo  by an overall factor of $g_{s}$, as can be most easily seen
by looking at the terms that behave similarly in both warping regimes.

The first line of (5.2.2) shows the expected LVS solution for $\tau_{s}$ which
already includes an $\mathcal{O}\left(g_{s}\right)$ correction (cf. (4.16)),
with the $\overline{\text{D3}}$-brane backreaction on $\tau_{s}$ encoded by
the uplift parameter $\alpha$. The corrections in the second and third lines
are suppressed in the same way as in the strongly-warped regime212121The
$C^{\rm log}_{s}$ factor again differs by a factor of $5/3$., as expected from
the fact that they do not involve the interplay between $\zeta$ and
$\mathcal{V}$. Whereas the $\\{C^{\rm KK}_{s},C^{\rm flux},C^{F}\\}$ terms are
suppressed by powers of $g_{s}$ and/or $\mathcal{V}$, $\\{C^{\rm
log}_{s},C^{\xi}_{1}\\}$ have no suppression whatsoever, and
$\\{C^{\xi}_{2}\\}$ is only suppressed by one power of $g_{s}$; the latter is
still dangerous because of the way the vev of the volume $\mathcal{V}$ depends
on $\tau_{s}$, i.e.

$\displaystyle\mathcal{V}\propto e^{\frac{a}{g_{s}}\tau_{s}}\approx
e^{\frac{a}{g_{s}}\tau_{s}^{(0)}}\cdot\exp\bigg(1+C^{\xi}_{2}\frac{a}{3\kappa_{s}}\Big{(}\frac{\hat{\xi}}{2\kappa_{s}}\Big{)}^{-1}+...\bigg{missing})\,.$
(5.14)

It also follows that the $\\{C^{\rm log}_{s},C^{\xi}_{1}\\}$ terms will both
appear with a $1/g_{s}$ power relative to the leading solution and therefore
grow at small couplings.

Finally, while the $C^{\rm con}$ correction is suppressed by $g_{s}\ll 1$ and
$g_{s}M\gg 1$, in the $C^{\rm KK}_{b}$ term one should pay closer attention to
the $g_{s}M\gg 1$ factor possibly enhancing it.

####  5.2.3 Vacuum Energy

With the vevs of the moduli in (5.2.1–5.2.2), we can compute the vacuum energy
given by the potential $V$ at the minimum, and look at the effect of each
correction. We can write it as (see (4.22))

$\displaystyle
V_{min}=\frac{g_{s}^{4}}{8\pi||\Omega||^{2}}\cdot\frac{3g_{s}\kappa_{s}W_{0}^{2}\sqrt{\tau_{s}}}{4a\mathcal{V}^{3}}\rho\,,$
(5.15)

with $\rho$ defined as

$\displaystyle\rho=$ $\displaystyle\leavevmode\nobreak\
\alpha-1+\mathcal{O}\left(g_{s}\right)+\mathcal{O}\left(1/\beta\right)$ (5.16)
$\displaystyle-C^{\rm
KK}_{s}\cdot\frac{g_{s}^{2}}{6\kappa_{s}}\Big{(}\frac{\hat{\xi}}{2\kappa_{s}}\Big{)}^{-2/3}+C^{\rm
log}_{s}\cdot\frac{a}{6}(11+12\log\nu)+C^{\xi}_{1}\cdot\frac{a\log\nu}{\kappa_{s}}\Big{(}\frac{\hat{\xi}}{2\kappa_{s}}\Big{)}^{-1/3}-C^{\xi}_{2}\cdot\frac{g_{s}}{3\hat{\xi}}$
$\displaystyle-C^{\rm
flux}\cdot\frac{10a}{9\kappa_{s}}\frac{KM}{\mathcal{V}^{2/3}}\Big{(}\frac{\hat{\xi}}{2\kappa_{s}}\Big{)}^{-1/3}-C^{F}\cdot\frac{16a}{27\kappa_{s}}\frac{g_{s}W_{0}^{2}}{\mathcal{V}^{2/3}}\Big{(}\frac{\hat{\xi}}{2\kappa_{s}}\Big{)}^{-1/3}$
$\displaystyle-C^{\rm con}\cdot\frac{5\alpha}{(g_{s}M)^{2}}-C^{\rm
KK}_{b}\cdot\frac{8c^{\prime}a(g_{s}M)^{2}}{9\kappa_{s}}\frac{g_{s}\zeta^{2/3}}{\mathcal{V}^{1/3}}\Big{(}\frac{\hat{\xi}}{2\kappa_{s}}\Big{)}^{-1/3}\,.$

where

$\nu=\frac{3g_{s}\kappa_{s}W_{0}}{8aA}\left(\frac{\hat{\xi}}{2\kappa_{s}}\right)^{1/3}\,.$
(5.17)

The first line reproduces the leading order result (4.22) and requires
$\alpha>1$ provided all corrections can be safely neglected. The correction
terms in the second and third lines are suppressed by the same factors as in
the strongly-warped case. In particular, the terms $\\{C^{\rm
log}_{s},C^{\xi}_{1}\\}$ appear unsuppressed and are thus the most dangerous.
On the other hand the corrections $\\{C^{\rm KK}_{b},C^{\rm con}\\}$ are
suppressed by factors of $\zeta,g_{s}\ll 1$ and $\mathcal{V},g_{s}M\gg 1$,
suggesting that they will not be among the most dangerous for the solution.

Note that the non-perturbative no-scale behaviour Junghans:2022exo  is
manifesting itself in the way the factors $\\{C^{\rm flux},C^{F}\\}$ are
suppressed relative to the leading LVS solution in $\rho$ compared to their
suppression in the off-shell potential, (5.7) and (5.9) respectively, having
acquired a $1/g_{s}$ enhancement in the vacuum energy (see discussion below).
It is also worth pointing out that the unsuppressed corrections $\\{C^{\rm
log}_{s},C^{\xi}_{1},C^{\xi}_{2}\\}$ are independent of the presence of the
conifold modulus and the brane — these are potentially dangerous corrections
even for pure LVS.

One might wonder whether it is possible to choose a specific model (i.e. a
specific compact space geometry) such that the unsuppressed corrections to the
vevs and vacuum energy $\\{C^{\rm log}_{s},C^{\xi}_{1},C^{\xi}_{2}\\}$ are not
present, the remaining corrections are all consistently suppressed, and the
solution is parametrically under control. Note that both the $C^{\rm KK}_{s}$
and $C^{\rm con}$ corrections are automatically suppressed as long as the
string loop expansion $(g_{s}\ll 1)$ and the supergravity approximation
$(g_{s}M\gg 1)$ are under control, which we must require for consistency of
our analysis. In order to suppress the corrections $\\{C^{\rm KK}_{b},C^{\rm
flux}\\}$, we require

$\displaystyle(g_{s}M)^{2}\frac{g_{s}\zeta^{2/3}}{\mathcal{V}^{1/3}}\,,\leavevmode\nobreak\
\frac{KM}{\mathcal{V}^{2/3}}\ll 1\,,$ (5.18)

simultaneously. These conditions can easily be satisfied in the weakly-warped
region of parameter space, which does not require large flux contributions.
However, suppression of the final correction $\\{C^{F}\\}$ requires

$\displaystyle\frac{g_{s}W_{0}^{2}}{\mathcal{V}^{2/3}}\ll 1\,.$ (5.19)

Recall that a solution in the weakly-warped regime must also satisfy (4.21),
which gives, parametrically,222222Note that although this condition from
(4.21) may seem to imply that we need $W_{0}\gg 1$, and not just $AW_{0}\gg
1$, when replacing the volume modulus vev (4.15), $\mathcal{V}\sim
W_{0}g_{s}/A$, we recover (4.20) and the large $AW_{0}$ bound.

$\displaystyle\frac{\mathcal{V}}{g_{s}^{2}W_{0}^{2}}<\mathcal{O}(1)\,.$ (5.20)

These two conditions together imply

$\displaystyle\mathcal{V}^{1/3}\ll g_{s}\,,$ (5.21)

which is never consistent. One can check explicitly using the parameter set of
Table 1 that taking into account all numerical coefficients
$\\{a,\kappa_{s},\hat{\xi},||\Omega||^{2}\\}$ does not change the conclusion.
Therefore the $C^{F}$ correction is unsuppressed unless $C^{F}\ll 1$.

It is worth noting that already in the strongly-warped case, the $C^{F}$
correction is dangerous for small $g_{s}$, which may not be obvious from the
suppression factor $\propto g_{s}W_{0}^{2}/\mathcal{V}^{2/3}$. One would
naïvely expect that since $\mathcal{V}\propto g_{s}\cdot e^{1/g_{s}}$, smaller
values of $g_{s}$ would suppress this correction the strongest. However, it is
important to note that one is not free to choose $W_{0}$ independently — in
particular, in the strongly-warped case Junghans:2022exo  (adapted to our
conventions)

$W_{0}^{2}\propto\frac{\zeta^{2/3}\mathcal{V}^{5/3}}{\tilde{\alpha}\leavevmode\nobreak\
g_{s}^{3}}\implies
W_{0}^{1/3}\propto\frac{\zeta^{2/3}}{\tilde{\alpha}\leavevmode\nobreak\
g_{s}^{4/3}}\cdot e^{\frac{5}{3}\frac{a}{g_{s}}\tau_{s}}\,,$ (5.22)

with $\tilde{\alpha}\sim\mathcal{O}\left(1\right)$. Plugging this into the
$C^{F}$ suppression factor,

$\frac{g_{s}W_{0}^{2}}{\mathcal{V}^{2/3}}\sim\frac{\zeta^{8/3}}{\tilde{\alpha}^{4}g_{s}^{5}}e^{6\cdot\frac{a}{g_{s}}\tau_{s}}\,,$
(5.23)

shows that indeed it grows for small $g_{s}$ contrary to our initial
expectation.

Just as in the strongly-warped regime, one concludes that several corrections
could be dangerous in the context of weakly-warped LVS de Sitter vacua. One
would at least need the corrections $\\{C^{\rm
log}_{s},C^{\xi}_{1},C^{\xi}_{2},C^{F}\\}$ to either vanish or be suppressed
through the explicit form of these numerical coefficients. This would require
explicit computation of these coefficients in order to choose a construction
for which they can be small, though they are not currently under control for
general Calabi-Yau geometries and setups. Alternatively, if they are
$\mathcal{O}(1)$, then the corresponding corrections would need to be included
in the 4d low energy effective field theory and new searches for de Sitter
solutions would need to be performed.

###  5.3 Comment on the NPNS

In Junghans:2022exo , a property of the LVS potential called the non-
perturbative no-scale (NPNS) structure was emphasised due to its effect on the
balance between leading and subleading (correction) terms in a given solution.
The root of the NPNS is the fact that the minimum in LVS — which is to be
considered the leading contribution to which subleading corrections are added
— itself appears at subleading order in $g_{s}$. To see this, consider the
change of variables $\mathcal{V}=\nu\cdot g_{s}\cdot e^{t/g_{s}}$ and
$\tau_{s}=t/a$, for which the LVS potential takes the form

$V=g_{s}\cdot e^{-\frac{3}{g_{s}}t}f(t,\nu)\,,$ (5.24)

where $f(t,\nu)$ does not depend on $g_{s}$.232323 Explicitly, we have
$f(t,\nu)=\frac{8a^{3/2}A^{2}\sqrt{t}}{3\kappa_{s}\nu}-\frac{4AW_{0}t}{\nu^{2}}+\frac{3W_{0}^{2}}{4\nu^{3}}\xi\,.$
(5.25) One might wonder whether $W_{0}$ and $A$ in particular can be important
for the argument we outline below. Note however that rescaling
$\nu\to\frac{W_{0}}{A}\cdot\nu$, the function becomes
$f(t,\nu)=\frac{A^{3}}{W_{0}}\left\\{\frac{8a^{3/2}\sqrt{t}}{3\kappa_{s}\nu}-\frac{4t}{\nu^{2}}+\frac{3\xi}{4\nu^{3}}\right\\}\,.$
(5.26) Since the ratio $A^{3}/W_{0}$ only appears as an overall factor, these
parameters will not affect the argument for the NPNS.  A minimum of this
potential therefore requires

$\displaystyle\partial_{t}V$ $\displaystyle=-\frac{3}{g_{s}}\cdot g_{s}\cdot
e^{-\frac{3}{g_{s}}t}f(t,\nu)+g_{s}\cdot e^{-\frac{3}{g_{s}}t}(\partial_{t}f)$
(5.27) $\displaystyle=-\frac{3}{g_{s}}V+g_{s}\cdot
e^{-\frac{3}{g_{s}}t}(\partial_{t}f)\overset{!}{=}0\,,$ (5.28)

which implies that the scalar potential at the minimum is suppressed by a
factor of $g_{s}$ compared to the off-shell potential (5.24),

$V_{min}=\frac{g_{s}}{3}\cdot g_{s}\cdot
e^{-\frac{3}{g_{s}}t}(\partial_{t}f)\,.$ (5.29)

Another way of thinking about this result, which is useful once we add
corrections, is that solving equation (5.27) perturbatively in $g_{s}$, i.e.
$t=t_{0}+t_{1}g_{s}+\mathcal{O}\left(g_{s}^{2}\right)$, the leading order
solution $t=t_{0}$ would result in a vanishing potential (which would in turn
leave $\nu$ as a flat direction).

Now consider adding a correction suppressed by $\epsilon\cdot
e^{-\frac{\lambda}{g_{s}}t}$,

$V=g_{s}\cdot e^{-\frac{3}{g_{s}}t}f(t,\nu)+g_{s}\cdot\epsilon\cdot
e^{-\frac{\lambda}{g_{s}}t}g(t,\nu)\,,$ (5.30)

where $g(t,\nu)$ does not contain $g_{s}$ or any other small parameters.
Minimising the potential now requires

$\displaystyle\partial_{t}V=$ $\displaystyle-\frac{3}{g_{s}}\cdot g_{s}\cdot
e^{-\frac{3}{g_{s}}t}f(t,\nu)+g_{s}\cdot e^{-\frac{3}{g_{s}}t}(\partial_{t}f)$
(5.31) $\displaystyle-\frac{\lambda}{g_{s}}\cdot g_{s}\cdot\epsilon\cdot
e^{-\frac{\lambda}{g_{s}}t}g(t,\nu)+g_{s}\cdot\epsilon\cdot
e^{-\frac{\lambda}{g_{s}}t}(\partial_{t}g)$ $\displaystyle=$
$\displaystyle-\frac{3}{g_{s}}\cdot\left\\{g_{s}\cdot
e^{-\frac{3}{g_{s}}t}f(t,\nu)+g_{s}\cdot\epsilon\cdot
e^{-\frac{\lambda}{g_{s}}t}g(t,\nu)\right\\}$ (5.32) $\displaystyle+g_{s}\cdot
e^{-\frac{3}{g_{s}}t}(\partial_{t}f)+g_{s}\cdot\epsilon\cdot
e^{-\frac{\lambda}{g_{s}}t}(\partial_{t}g)+(3-\lambda)\cdot\epsilon\cdot
e^{-\frac{\lambda}{g_{s}}t}\cdot g(t,\nu)$ $\displaystyle=$
$\displaystyle-\frac{3}{g_{s}}V+g_{s}\cdot
e^{-\frac{3}{g_{s}}t}(\partial_{t}f)+g_{s}\cdot\epsilon\cdot
e^{-\frac{\lambda}{g_{s}}t}(\partial_{t}g)$
$\displaystyle+(3-\lambda)\cdot\epsilon\cdot e^{-\frac{\lambda}{g_{s}}t}\cdot
g(t,\nu)\overset{!}{=}0\,,$ (5.33)

which now implies for the vacuum energy

$\displaystyle V_{min}=$ $\displaystyle\frac{g_{s}^{2}}{3}\cdot
e^{-\frac{3}{g_{s}}t}(\partial_{t}f)+\frac{g_{s}^{2}}{3}\cdot\epsilon\cdot
e^{-\frac{\lambda}{g_{s}}t}(\partial_{t}g)+\frac{3-\lambda}{3}\cdot
g_{s}\cdot\epsilon\cdot e^{-\frac{\lambda}{g_{s}}t}\cdot g(t,\nu)\,.$ (5.34)

Note that, unless $\lambda=3$ (i.e. the correction satisfies the NPNS
structure), the last term is the leading contribution from the corrections and
_it is not suppressed by the extra factor of_ $g_{s}$ that appears in the
first two terms — the extra $g_{s}$ suppression characteristic of LVS (5.29).
Therefore, to safely neglect this contribution from the correction one should
have242424Note that the suppression factor must include the exponential
$e^{-\frac{\lambda-3}{g_{s}}t}$, which corresponds to the volume suppression
(whose power is given by $\lambda-3$). In practice, $\epsilon$ counts all
factors apart from powers of the volume.

$\displaystyle g_{s}\cdot\epsilon\cdot e^{-\frac{\lambda}{g_{s}}t}\ll
g_{s}^{2}\cdot
e^{-\frac{3}{g_{s}}t}\quad\Rightarrow\quad\frac{\epsilon}{g_{s}}\cdot
e^{-\frac{(\lambda-3)}{g_{s}}t}\ll 1\,,$ (5.35)

rather than the naively expected $\epsilon\cdot e^{-\frac{\lambda}{g_{s}}t}\ll
1$.

To compare with (5.16) one should look at $\rho$, by taking out the
suppression factors $g_{s}^{2}\cdot e^{-\frac{3}{g_{s}}t}$ as in (4.22),

$\rho\propto\frac{1}{3}(\partial_{t}f)+\frac{1}{3}\cdot\epsilon\cdot
e^{-\frac{(\lambda-3)}{g_{s}}t}(\partial_{t}g)+\frac{(3-\lambda)}{3}\cdot\frac{\epsilon}{g_{s}}\cdot
e^{-\frac{(\lambda-3)}{g_{s}}t}\cdot g(t,\nu)\,,$ (5.36)

where we can see explicitly the factor of (5.35).

This is why the $\\{C^{\rm KK}_{b},C^{F},C^{\rm con},C^{\rm flux}\\}$ terms,
which do not satisfy the NPNS structure $(\lambda\neq 3)$, appear to be
enhanced by $1/g_{s}$ in (5.16) compared to their suppression factors in the
off-shell potential, while the $\\{C^{\rm KK}_{s},C^{\rm
log}_{s},C^{\xi}_{1},C^{\xi}_{2}\\}$ terms, which do satisfy the NPNS
$(\lambda=3)$, show the same suppression in $V_{min}$ as they did in the off-
shell potential (5)–(5.4). As an explicit example, for $\\{C^{\rm flux}\\}$ we
have

$\displaystyle\lambda=3+\frac{2}{3}\,,$ $\displaystyle\delta V\propto C^{\rm
flux}\cdot\frac{g_{s}KM}{\mathcal{V}^{2/3}}\,,$
$\displaystyle\delta\rho\propto C^{\rm
flux}\cdot\frac{KM}{\mathcal{V}^{2/3}}\,,$ (5.37)

breaking the NPNS, while for $\\{C^{\xi}_{2}\\}$

$\displaystyle\lambda=3\,,$ $\displaystyle\delta V\propto C^{\xi}_{2}\cdot
g_{s}\,,$ $\displaystyle\delta\rho\propto C^{\xi}_{2}\cdot g_{s}\,.$ (5.38)

As a final remark, note that whether a correction breaks the NPNS structure or
not, does not determine how dangerous it will be, even if it gives some
information on the relative $g_{s}$ suppression in the off-shell potential
compared to the moduli vevs and vacuum energy. Indeed $\\{C^{\rm
KK}_{s},C^{\rm log}_{s}\\}$ both have an NPNS structure, but only $C^{\rm
KK}_{s}$ turns out to be safe since $C^{\rm log}_{s}$ is anyway leading in the
off-shell potential; conversely, $\\{C^{\rm flux},C^{F}\\}$ both break the
NPNS structure, but only $C^{F}$ turns out to be dangerous in the parameter
space of weak warping.

##  6 Conclusions

In this paper, we have studied the consequences of subleading string-loop and
$\alpha^{\prime}$-corrections on candidate de Sitter vacua in string effective
field theories. Consistent constructions aim to stabilise moduli, by balancing
tree-level and leading-order perturbative and/or non-perturbative effects, at
positive vacuum energy density, weak coupling and large volume. Then
subleading corrections, in the weak coupling and large volume expansions, may
be expected to be sufficiently suppressed so as to only cause small
corrections to the leading order solution. We test this approach in the
context of the recently discovered type IIB metastable de Sitter vacua that
arise when an anti-D3-brane sits at the tip of a _weakly-warped_ conifold
Bento:2021nbb .

The weakly-warped de Sitter vacua involve an interesting interplay between the
light moduli — the conifold deformation modulus and the volume modulus. They
represent a small deviation from the famous KS/GKP/LVS solution
KS2000supergravity ; Giddings:2005ff ; LV1 , where KS/GKP fluxes stabilise the
deformation modulus and leading perturbative and non-perturbative corrections
stabilise the otherwise-flat volume modulus into a scale-separated weakly-
warped LVS AdS vacuum. The deviation is sourced by the
$\rm\overline{D3}$-brane, with conifold modulus
$|z|\sim|z_{GKP}|\left(1-\frac{4Kc_{\rm
D3}}{3\pi^{2}c^{\prime}M\Lambda_{0}^{4}\mathcal{V}^{2/3}}\right)$, and yields
a total vacuum energy that is positive. The weak-warping is a previously
unexplored regime of parameter space. The balance in the
$\rm\overline{D3}$-brane tension with respect to the volume stabilisation,
necessary to prevent volume destabilisation and usually ensured by strong-
warping, is here provided by relatively large coefficients in the
superpotential parameters $AW_{0}$. This allows for solutions with low
conifold flux numbers, $MK\lesssim 32$, and thus a standard description within
perturbative type IIB (unless large $W_{0}$ brings back large flux numbers
from the bulk). Overall, the weakly-warped de Sitter solution has an
ameliorated tadpole problem tadpoleProblem , no bulk-singularity problem
Carta_2019 ; Gao:2020xqh , no problem with the conifold fitting into the bulk
Carta:2021lqg  and is consistent with the Kaluza-Klein truncation,
$\alpha^{\prime}$ and string-loop expansions and supergravity description.

We compute the various subleading corrections that are expected to contribute
to the 4d low energy effective field theory and work out their impact on the
leading order metastable de Sitter vacuum. This is summarised in Table 3.
Analogous computations for the strongly-warped Large Volume Scenario have been
performed recently in Junghans:2022exo  and we use the same notation as in
that paper for ease of comparison. It turns out that the parametric dependence
of the subleading corrections takes the same form in the strongly-warped and
weakly-warped regimes, which was not obvious a priori given that different
terms dominate in the Kähler potential, and thus the scalar potential, in the
two regimes. The differences are then all due to the particular regime of
parameter space considered. In particular $C^{\rm flux}$ and $C^{\rm con}$ are
expected to be sensitive to the warping regime, being associated with the tip
of the conifold. Due to their parametric dependence, $C^{F}$ and $C^{\rm
KK}_{b}$ also end up being sensitive to the warping regime.

Correction |  Parametric suppression |  Origin | Dangerous? | c.f. strong-warping
---|---|---|---|---
$C^{\text{log}}_{s}$ |  $g_{s}\log\mathcal{V}$ |  threshold corrections to gauge couplings | yes | similar
$C^{\xi}_{1}$ |  $g_{s}\log\mathcal{V}$ |  backreaction of 7-branes and exchange of KK modes with D7/O7 planes | yes | similar
$C^{\xi}_{2}$ |  $g_{s}$ |  | yes | similar
$C^{F}$ |  $\frac{g_{s}^{2}W_{0}^{2}}{\mathcal{V}^{2/3}}$ |  integrating out KK modes yielding 4 superspace derivatives in EFT | yes | similar
$C^{\text{KK}}_{b}$ |  $\frac{g_{s}^{2}(g_{s}M)^{2}\zeta^{2/3}}{\mathcal{V}^{1/3}}$ |  exchange of KK modes between D7/D3 branes or O7/O3 planes | can be not | similar
$C^{\text{flux}}$ |  $\frac{g_{s}KM}{\mathcal{V}^{2/3}}$ |  backreaction from fluxes, branes and O-planes that source warping | can be not | better
$C^{\text{KK}}_{s}$ |  $g_{s}^{2}$ |  exchange of KK modes between D7/D3 branes or O7/O3 planes | no | similar
$C^{\text{con}}$ |  $\frac{1}{(g_{s}M)^{2}}$ |  curvature correction in conifold region | no | better
$\chi_{s}$ |  $Ae^{-\frac{a}{g_{s}}(\tau_{s}-\frac{\chi_{s}}{24g_{s}})}$ |  curvature corrections to D7 gauge kinetic function | no – it helps! | better

Table 3: Summary of subleading corrections considered, their origin and their
parametric suppression (or not) in the off-shell scalar potential, for the
weakly-warped de Sitter vacuum. We also compare with the strongly-warped case,
considering the danger of the corrections to candidate de Sitter vacua. We do
not include $\Delta\xi$ as it amounts simply to a shift in the numerical
parameter $\xi$; both $\Delta\xi$ and $\chi_{s}$ can be computed explicitly
and taken into account in the 4d EFT. Note that — in both weak and strong
cases — the LVS volume stabilisation satisfies $\log\mathcal{V}\sim 1/g_{s}$.

The implications for the leading order weakly-warped de Sitter vacuum — that
is, the moduli vevs $\zeta,\mathcal{V}$ and $\tau_{s}$, and vacuum energy —
are as follows. The corrections
$\left\\{C^{\text{KK}}_{s},C^{\text{con}}\right\\}$ are automatically
suppressed by the string-loop $g_{s}\ll 1$ and supergravity $g_{s}M\gg 1$
expansions. The corrections
$\left\\{C^{\text{KK}}_{b},C^{\text{flux}}\right\\}$ can also be
parametrically suppressed consistently with the weakly-warped regime of
parameter space. However, the
$\left\\{C^{\text{log}}_{s},C^{\xi}_{1},C^{\xi}_{2}\right\\}$ corrections are
unsuppressed due their $\log\mathcal{V}$ enhancements; this is unsurprising
given that the corrections are unsuppressed in the off-shell potential and
overall they would be better considered as a leading correction. The other
dangerous correction is $C^{F}$; it turns out that although this correction
has a $g_{s}$ and $1/\mathcal{V}^{2/3}$ suppression, it also suffers from a
$W_{0}$ enhancement, and overall it cannot be suppressed consistently within
the weakly-warped regime that requires large $W_{0}A$. The $\chi_{s}$
corrections are unsuppressed, but they actually help in going to the weakly-
warped regime, effectively increasing the superpotential parameter $A$. Thus
we can conclude that in order for the weakly-warped de Sitter solution to be
consistent, the higher dimensional construction must be such that the
numerical coefficients
$\left\\{C^{\text{log}}_{s},C^{\xi}_{1},C^{\xi}_{2},C^{F}\right\\}$ are
themselves suppressed or vanishing. These are those associated with the
backreaction, KK exchange and threshold corrections for D7-branes, and those
associated with integrating out KK modes and the consequent derivative
expansion in the 4d supersymmetric EFT. It would be important to understand
whether or not such higher dimensional constructions are possible, both for
the weakly-warped and strongly-warped constructions.

An interesting phenomenon that was identified in Junghans:2022exo  is the non-
perturbative no-scale behaviour in some of the subleading corrections. In
detail, this occurs because in the LVS solution, the vacuum energy and one of
the eigenvalues of the mass matrix turn out to have an additional $g_{s}$
suppression compared to the off-shell potential. Those corrections, $\\{C^{\rm
KK}_{s},C^{\rm log}_{s},C^{\xi}_{1},C^{\xi}_{2},\chi_{s}\\}$, which enter in
the off-shell potential with the same LVS dependence,
$e^{-\lambda\frac{a\tau_{s}}{g_{s}}}$ with $\lambda=3$, have this same $g_{s}$
suppression; those that break the non-perturbative no-scale behaviour,
$\\{C^{\rm KK}_{b},C^{\rm flux},C^{\rm con},C^{F}\\}$ for which $\lambda\neq
3$, have one less $g_{s}$ suppression in the vacuum energy – or, effectively,
a $g_{s}$ enhancement. We find that the non-perturbative no-scale behaviour
turns out to be not decisive in whether or not corrections are dangerous; one
has to recall that the NPNS
$\left\\{C^{\text{log}}_{s},C^{\xi}_{1},C^{\xi}_{2}\right\\}$ are anyway
leading in the off-shell potential with respect to the large volume expansion,
and one generally has to consider the overall balance of parameters,
$1/g_{s}$, $\mathcal{V}$, $g_{s}M$, $MK$ and $W_{0}A$.

Our approach in this paper has been to work out under which conditions
metastable de Sitter vacua found within weakly-warped Large Volume Scenarios
may be robust against expected subleading corrections. Having established that
there are well-motivated corrections that are in fact not parametrically
suppressed, it would be interesting to study the corrected moduli potentials
from scratch, to determine whether or not they themselves have metastable de
Sitter vacua. If they exist, they are likely not to be simply small
perturbations of earlier candidate vacua and one would then have to check
their own robustness against next-to-next-to-leading order corrections. Some
interesting studies on metastable de Sitter vacua in the presence of
$\log\mathcal{V}$ corrections can be found for example in Antoniadis:2019rkh ;
Burgess:2022nbx . A proposal that curvature corrections on the brane could
provide an alternative to warping to effectively suppress the uplifting
anti-D3-brane tension was put forward in Hebecker:2022zme .

## Acknowledgements

We are grateful to Daniel Junghans and Severin Lüst for discussions. IZ is
partially supported by STFC, grant ST/T000813/1.

For the purpose of open access, the authors have applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising.
Data access statement: no new data were generated for this work.

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