# RG-Induced Modulus Stabilization: Perturbative
de Sitter Vacua and Improved D3-${\overline{\hbox{D3}}}$ Inflation

C.P. Burgess1,2,3 and F. Quevedo4
1 Department of Physics & Astronomy, McMaster University, Hamilton ON, Canada.
2 Perimeter Institute for Theoretical Physics, Waterloo ON, Canada.
3 CERN, Theoretical Physics Department, Genève 23, Switzerland.
4 DAMTP, Cambridge University, Wilberforce Road, Cambridge, CB3 0WA, UK.

###### Abstract:

We propose a new mechanism that adapts to string theory a perturbative method
for stabilizing moduli without leaving the domain of perturbative control,
thereby evading the ‘Dine-Seiberg’ problem. The only required nonperturbative
information comes from the standard renormalization-group resummation of
leading logarithms that allow us simultaneously to work to a fixed order in
the perturbative parameter $\alpha$ and to all orders in $\alpha\ln\tau$ where
$\tau$ is a large extra-dimensional modulus. The resulting potential is
naturally minimized for moduli of order $\tau\sim e^{1/\alpha}$ and so can be
exponentially large given ${\cal O}(10)$ input parameters. The mechanism
relies on accidental low-energy scaling symmetries known to be generic and so
is robust against UV details. The resulting compactifications generically
break supersymmetry and 4D de Sitter solutions are relatively easy to achieve
without additional uplifting. Variations on the theme lead to inflationary
scenarios for which the size of the stabilized moduli differ significantly
before and after inflation and so provide a dynamical mechanism whereby
inflationary scales are much larger than late-time physical (e.g.
supersymmetry breaking) scales, with this hierarchy contingent on past cosmic
evolution with the inflaton playing a secondary late-time role as a relaxation
field. We apply this formalism to warped D3-${\overline{\hbox{D3}}}$ inflation
using non-linearly realized supersymmetry to describe the antibrane tension
and the Coulomb interaction, and show how doing so our perturbative modulus
stabilization mechanism evades the $\eta$-problem that usually plagues this
scenario. We speculate about the relevance of our formalism to tachyon
condensation at later stages of brane-antibrane annihilation.

††preprint: CERN-TH-2022-009

##  1 Introduction

If string theory is right there may be a problem: a simple argument suggests
string vacua are generically strongly coupled and without hierarchies, yet we
find ourselves in a world populated by different scales where experiments
often reveal weak interactions at play. How can these be reconciled? As it
turns out, modulus stabilization is the key. Modulus stabilization is to
string theory what logistics is to warfare: it is the difference between
winning and losing. Our goal in this paper is to add a new stabilization
mechanism to the string theory toolbox but we first start with a fuller
statement of the issue and why our mechanism helps.

The issue turns about a central string-theory feature: scarcity of free
parameters. Things that would be coupling constants in other theories arise as
fields in string theory, making all expansions ultimately field expansions.
For instance weak string coupling is an expansion in powers of the string
dilaton $e^{\hat{\phi}}=1/s$ and a 4D world only emerges from higher
dimensions through an expansion in inverse powers of fields that express
extra-dimensional size (such as the volume modulus ${\cal V}:=\tau^{3/2}$ that
measures its overall volume in string units).

#### The ‘Dine-Seiberg’ problem

Crucially, fields like $s$ and ${\cal V}$ are moduli, in that their values are
often not fixed by the leading classical field equations. This means their
vacuum values can be determined by energetic arguments purely within a four-
dimensional low-energy effective theory through their appearance within a
scalar potential that is to be minimized. Performing this stabilization is a
prerequisite for making practical predictions because particle masses and
couplings depend on the resulting stabilized values in an important way.
However this potential itself typically arises as an expansion in these
fields, as in

$V(s,\tau)=\sum_{nm}A_{nm}s^{-n}\tau^{-m}\,,$ (1)

and (as first articulated in early searches for realistic string
compactifications by Dine and Seiberg [1]) this leads to the problem.

On one hand, if the leading term is positive then the scalar potential slopes
off towards zero in the limit of vanishing string coupling $1/s\to 0$ and
infinite volume $\tau\to\infty$. But this stationary point of the potential
corresponds to 10D flat space and so does not describe what we see around us,
and lies beyond the reach of the 4D EFT. Ref. [1] then argues that if the
potential has a non-trivial minimum, as is required to avoid the runaway to
infinity, different orders in these expansions must compete with one another
(e.g. quantum effects must compete with the classical results) signalling the
breakdown of the perturbative expansion itself. They concluded that the
generic weak-coupling situation is a runaway without a non-trivial minimum.
Conversely, if a non-trivial minimum exists then it should generically arise
at strong coupling with an extra-dimensional volume of order the string scale:
$s\sim\tau\sim 1$. This argument can also be cast in terms of two accidental
approximate scale invariances that turn out to be shared by all string vacua,
for which $s$ and $\tau$ play the role of pseudo Goldstone modes and 10D flat
space corresponds to the scale invariant point (see for instance [2]).

Of course the key word in this argument is ‘generic’. Over the years many
efforts were made to overcome this general problem and obtain weak couplings
and large hierarchies in controlled ways. The solutions usually exploit the
few parameters that are not $vevs$ of moduli that can be adjusted to provide
non-generic solutions with weak coupling and large volume. These parameters
include the curvature of the extra dimensions, non-critical dimensionality,
the ranks of the various symmetry groups, or integer flux quantum numbers for
antisymmetric tensor fields that thread compact extra dimensions (similar to
magnetic flux threading a sphere – see [3] for a review).

In particular IIB string compactifications have been much explored with
successful scenarios using a combination of the huge number of possible fluxes
and various small non-perturbative effects, leading to two main approaches for
stabilizing moduli in IIB vacua. The first of these – the ‘KKLT’ scenario –
exploits the vast number of fluxes to tune the tree-level superpotential to be
exponentially small (so as to compete with small non-perturbative
contributions to the superpotential [4, 5]). The second class – the ‘large-
volume scenario’ or LVS – instead finds solutions with stabilized moduli by
balancing different orders of different expansions, exploiting the fact that
generically there are many moduli and many perturbative expansions going on at
the same time. In particular non-trivial vacua are found for which the non-
perturbative corrections to the superpotential $W_{np}\simeq e^{-a\tau_{s}}$
for one modulus $\tau_{s}$ compete with the perturbative corrections in the
much larger volume modulus ${\cal V}=\tau^{3/2}$, resulting in minima for
which $\tau\simeq e^{a\tau_{s}}$ and so give exponentially large volumes [6,
7]. For both scenarios the resulting potential is minimized with $V_{\rm
min}<0$ and so additional ‘uplift’ mechanisms are required to obtain flat or
de Sitter space, using extra ingredients such as antibranes, T-branes, etc.
[5, 8, 9]. These scenarios have been explored in considerable detail and so
far represent the state of the art for moduli stabilization in general and de
Sitter string solutions in particular.

In §2 we present an alternative mechanism for moduli stabilization that shares
some of the attractive properties of KKLT and LVS pictures, but also evades
some of their difficulties. Our proposal differs substantially from both of
them by being purely based on perturbative corrections within the
corresponding effective field theory (EFT), but doing so in a way consistent
with the Dine-Seiberg problem. Our scenario adapts a proposal made for higher-
dimensional theories in [10] (and further elaborated for 4D supergravity in
[11]).

The idea is very simple and can be illustrated with a concrete toy model. To
this end, consider the perturbative expansion for the low-energy scalar
potential $V$ in powers of the volume field $\tau$,

$V(\tau)=\sum_{n}A_{n}(s)\tau^{-n}$ (2)

where naively the coefficients $A_{n}$ depend on all of the other moduli such
as $s$. It is tempting to think that the $A_{n}$ should also be independent of
$\tau$, but this need not be true because $A_{n}$ can depend logarithmically
on $\tau$. It is generic that quantum corrections can introduce anomalous
scaling into effective interactions, which become logarithmic dependence on
ratios of particle masses in a perturbative regime for which a small expansion
parameter $\alpha$ exists. (This expansion parameter might simply be another
modulus, like $\alpha\simeq 1/s$.) But in string theory particle masses
generically depend on $\tau$ since this field determines the ratio of
fundamental scales like the string, Planck and Kaluza-Klein (KK) masses
($M_{s}$, $M_{p}$ and $M_{\scriptscriptstyle KK}$). Consequently any
logarithmic dependence on ratios of masses can also imply a logarithmic
dependence on $\tau$ (and on any other moduli that appear in mass ratios).

Because this dependence has its roots in anomalous scaling, renormalization
group (RG) techniques can be used to resum leading-log effects (i.e. they
allow one to work to all orders in the expansion $\alpha\ln\tau$ while still
neglecting subdominant terms like $\alpha^{2}\ln\tau$). Therefore, even though
$A_{n}$ might only be known perturbatively in $\alpha$, RG reasoning gives
this expansion to all orders in $\alpha\ln\tau$. The resulting potential can
be minimized with respect to $\tau$ without going beyond leading order in the
$1/\tau$ expansion, and naturally leads $\ln\tau$ to be fixed at a size
$\ln\tau\simeq 1/\alpha$. For weak coupling $\alpha\ll 1$, this stabilizes
$\tau$ at exponentially large values (providing a novel explanation for large
hierarchies), and the RG allows this to be done without losing perturbative
control. This is a key part of why we can evade the Dine-Seiberg conclusion,
and suggests the name RG stabilization (see [12] for a related proposal
without the RG overlay).

We see that the modulus $\tau$ can easily be exponentially large in RG
stabilization (similar to LVS stabilization), and this in turn reinforces the
logic of working within the $1/\tau$ expansion. Also like the LVS case, the RG
proposal requires generically at least two expansions: the $1/\tau$ and the
$\alpha$ expansions. Because the perturbative corrections modify the
supergravity Kähler potential (rather than the superpotential), the non-
trivial minima we obtain can equally well be de Sitter or anti de Sitter and
so a separate uplifting mechanism is not necessary (unlike for KKLT and LVS
pictures).

### Application to inflationary dynamics

Where de Sitter solutions exist, can inflationary solutions be far behind?
Interest in inflation is driven by its successful description of observed
primordial fluctuations in terms of quantum effects amplified by inflationary
expansion. The observation that the fluctuation amplitude points to energy
scales not too far below the Planck scale has raised hopes that these
primordial fluctuations might eventually provide an observational window on
very-high-energy physics.

This has stimulated the development of a great variety of string-inflationary
scenarios over the past decades (see [13, 14] for reviews), for which modulus-
stabilization again plays a crucial role. The various inflationary scenarios
choose different moduli to be the inflaton, with both the scalar and axionic
components of supersymmetric complex scalars playing important roles in
different pictures. Many of these proposals are quite promising, with some
contenders rising to the top as observations have become more constraining on
theoretical models [15]. They also tend to share the following three
challenges.

  * •

Modulus stabilization. A theory of modulus stabilization is always a
prerequisite for any microscopic inflationary model. After all, there is no
point arranging the scalar potential to be very shallow along a putative
inflaton direction in field space if the potential also turns out to be much
steeper in other directions; a slowly rolling field prefers to evolve in the
steepest direction available. One way to avoid motion in other steeper
directions is to arrange for local minima in these directions into which
noninflationary moduli can be trapped.

  * •

Fragility. The extreme shallowness of the potential required for slow-roll
inflation can easily be overwhelmed by corrections. In particular, classical
scalar masses are notoriously sensitive to quantum effects, but having a small
slow-roll parameter $\eta=M_{p}^{2}\,V^{\prime\prime}/V\ll 1$ implies the
squared-mass of the inflaton, $m^{2}\sim V^{\prime\prime}$, must be much
smaller than the Planck-suppressed Hubble scale, $H_{\scriptscriptstyle
I}^{2}\sim V/M_{p}^{2}$. This makes a slow roll hostage to the many
corrections to the inflaton mass that are of Hubble size or larger. Many
string inflation scenarios are in particular plagued by a specific version of
this fragility – called the ‘$\eta$ problem’ – that arises when inflation
occurs within a 4D supergravity framework (and is discussed in more detail
below).

  * •

Inflation vs SUSY breaking scales. String models often produce inflation at
high scales, such as the GUT scale $M_{{\scriptscriptstyle GUT}}\simeq
10^{17}$ GeV, particularly if they are designed to maximize tensor-mode
signals [16, 17]. However such constructions also tend to give a very large
supersymmetry breaking scale. This need not be a serious problem, but makes
specific realizations of low-energy supersymmetry breaking difficult if
combined with inflation since it creates a tension between the large
inflationary scale and the low scale of supersymmetry breaking [18, 19, 20].

In §3 we explore some of the implications of RG modulus stabilization for
string-inflationary models, with the encouraging news that it helps resolve
all three of these challenges.

We do so using the specific example of the brane-antibrane inflationary
scenario, for which the inflaton is proposed to be the extra-dimensional
separation between a mutually attracting brane and antibrane. Separations
between supersymmetric BPS branes were proposed as candidates for the inflaton
some time ago [21], with slow roll occurring because of the absence of inter-
brane forces implied by the BPS condition. The difficulty breaking
supersymmetry in these models made it hard to compute the inflaton potential,
however, though this was solved by instead exploiting the Coulomb attraction
of non-supersymmetric brane-antibrane configurations [22, 23], though it was
realized early on that slow roll remained difficult to achieve in simple
geometries because the branes could not be sufficiently separated within the
extra dimensions to allow them to experience a weak enough force [22].

More detailed string constructions required modulus-stabilization techniques
[4, 5], since only these allow the calculation of the scalar potential for all
low-energy moduli. Although it was initially hoped that extra-dimensional
warping might potentially resolve the ‘runway-length’ problem [5, 24], it
turned out that modulus stabilization carried a sting because it also robustly
introduced the $\eta$ problem [24], requiring parameters to be tuned in a way
that undermined warping’s utility. We find the $\eta$ problem is evaded within
our stabilization mechanism – largely because it does not rely on the non-
perturbative superpotential as do both KKLT and LVS stabilization – and so
allows the utility of warping for inflation to be resurrected as originally
intended.

### The inflaton and late-time relaxation

The advantages of RG stabilization for inflation are not tied to the details
of specific string constructions, however. To emphasize this, much of the
inflationary discussion of §3 is cast purely in terms of approximate
symmetries of the low-energy 4D effective theory. Although the required
symmetries are in particular generic to string vacua, phrasing the analysis in
terms of low-energy symmetries helps identify what is required (and what is
not) in any particular UV completion. We believe it also undermines the
evidence for the swampland hypothesis [25, 26] because it shows how properties
of the low-energy theory (like putative difficulty obtaining de Sitter vacua)
can be seen as relatively mundane consequences of symmetries rather than
indicating any deep failure of EFT methods [2].

Our inflationary analysis relies on the following two crucial components:

  1. 1.

Constrained supersymmetry: We use the lagrangian for nonlinearly realized 4D
supergravity: i.e. a supersymmetric gravity sector coupled to a matter sector
within which supersymmetry is badly broken (i.e. whose superpartners have been
integrated out, and so supersymmetry is nonlinearly realized using a goldstino
field $G$). In practice this goldstino can be represented using a nilpotent
superfield ($G\in X$, with $X^{2}=0$) [27]. The coupling of such a sector to
supergravity can be found in [28].

This kind of EFT is of interest in situations where mass splittings within
gravitationally coupled supermultiplets are much smaller than splittings in
other supermultiplets [29], such as often occurs in extra dimensional models
[30] and string vacua [31, 32, 33] when supersymmetry is badly broken on a
brane. It also plausibly applies to Standard Model phenomenology within models
that are supersymmetric in the UV, given the evidence for the absence of weak-
scale supersymmetry [34].

  2. 2.

Accidental approximate scale invariance. We take our representative modulus to
be a dilaton $\tau\in T$ that sits within a supermultiplet $T$ and acts as the
pseudo-Goldstone boson for an approximate accidental scale invariances. Such
scaling symmetries have long been known to be present for specific string
vacua [35, 36], although we now know them to be generic for extra-dimensional
supergravity [37], which in turn very robustly inherit them from string theory
[2].

To these we also add the inflaton field, $\phi$, assuming it to be part of the
sector that badly breaks supersymmetry (and so to realize supersymmetry only
nonlinearly). An attentive reader might be struck by the similarity between
this list of ingredients and those used in ref. [11], for which the only
difference is that there the field $\phi$ instead enters as a ‘relaxation’
field whose presence dynamically helps suppress the size of the scalar
potential at its minimum.

The appearance of $\phi$ in [11] is at first sight a bit jarring, since it
appears to arise in a bespoke way with nothing to do with any other physics.
We argue here that one way to understand the presence of $\phi$ is as an
inflaton: its very definition requires it to interpolate between a potential
that is dominated by a large positive vacuum energy and one where the
potential energy is small. (The only surprise in [11] is in just how small the
potential at this minimum turns out to be.) All that is needed to make early-
universe evolution of $\phi$ into an inflationary mechanism is a reason why
this transition should happen slowly (and showing how this occurs is the baton
we take up here).

Furthermore, the brane-antibrane system lends itself perfectly to this
scenario. First, it does so because it is well known that the supersymmetry
breaking of an antibrane is captured precisely by the nilpotent superfield
$X$. Second, the brane-antibrane Coulomb interaction is also easily captured
by a superpotential that couples the inter-brane distance $\phi$ to the
nilpotent superfield $X$. Brane-antibrane attraction thereby provides a
natural UV interpretation for the relaxation/inflaton field $\phi$.

In what follows we describe in detail in §3 how RG stabilization resurrects
warped brane-antibrane inflation, after first describing the RG stabilization
mechanism in §2. §2 also includes discussions of how to realize the RG
mechanism in IIB string theory; a determination of the relevant scales, like
gravitino mass and soft-breaking terms for the matter sector and their
implications for the size of the volume modulus $\tau$; the relevance of RG
stabilization to the Dine-Seiberg problem; and why runaway regions can be
sensibly addressed using EFT methods (including comparisons with calculations
in other areas of physics).

##  2 de Sitter vacua

In this section we explore the simplest stabilization example. Our goal is to
illustrate how standard renormalization-group methods can allow modulus
stabilization without losing perturbative control. We also use this example to
show how combining this stabilization mechanism with accidental low-energy
symmetries (in particular accidental, approximate scale invariance) leads to a
novel kind of spontaneous supersymmetry breaking that generates de Sitter
solutions with large hierarchies driven by the exponentially large value of
the stabilized moduli. We first present the mechanism in a stripped-down model
and then show how it naturally embeds into low-energy string vacua.

###  2.1 Accidental symmetries and dilaton dynamics

Consider a general low-energy 4D effective theory that is both supersymmetric
and enjoys an accidental approximate scaling symmetry (of the form argued in
[2] to be generic in low-energy string vacua and in [37] to be generic to
higher-dimensional supergravities more generally). The minimal such a model
involves the gravity supermultiplet and the chiral superfield $T\ni\\{{\cal
T},\xi\\}$ that contains a complex scalar
$T=\frac{1}{2}(\tau+i{\mathfrak{a}})$ whose real part ($\tau$) is the dilaton
required by the approximate scale invariance. Because the EFT is a 4D
supergravity it is determined at the two-derivative level in terms of standard
supergravity ingredients: by specifying how the Kähler potential $K$, the
superpotential $W$ and (should gauge multiplets also be present) the gauge
kinetic function ${\mathfrak{f}}_{\alpha\beta}$ depend on the one chiral
superfield $T$.

We take $W=w_{0}$ to be independent of $T$ (as can be enforced with the
axionic symmetry under shifts of ${\mathfrak{a}}$). Accidental approximate
scale invariance is implemented by demanding that $e^{-K/3}$ arises as a
series in powers of $1/\tau$, as in:

$e^{-K/3}=\tau-k+\frac{h}{\tau}+\cdots\,,$ (3)

where the ellipses denote higher orders in $1/\tau$. This ensures that the
lagrangian density comes as a series of terms, ${\cal L}=\sum_{n}{\cal
L}_{n}$, each of which scales homogeneously, ${\cal
L}_{n}\to\lambda^{p_{n}}{\cal L}_{n}$ for some $p_{n}$, when $g_{\mu\nu}$ and
$\tau$ are scaled by powers of the constant parameter $\lambda$. This kind of
expansion ensures that semiclassical methods arise as expansions in powers of
$1/\tau$ and so are good approximations in the regime $\tau\gg 1$ — and so
this is where we seek our minima once we compute a potential $V(\tau)$. It
represents a scale invariance because these rescalings of $\tau$ and the
metric are symmetries of the classical field equations to leading order in
$1/\tau$.

String theorists will recognize this system: in Type IIB string
compactifications the role of the field $\tau$ is played by the one always-
present Kähler modulus: the extra-dimensional volume (in string units), ${\cal
V}\propto\tau^{3/2}$. In a string context the reliance of semiclassical
arguments on large $\tau$ expresses how semiclassical supergravities provide
reliable EFTs for string vacua only for geometries that are much larger than
the string scale.

Crucially, although powers of $\tau$ are explicit in (3), in general quantum
effects complicate the scaling properties of subdominant terms in the
lagrangian. We return below to why this is so, but just record now that it
allows the functions $k$, $h$ to be rational functions of logarithms of
$\tau$: $k=k(\ln\tau)$, $h=h(\ln\tau)$ etc..

The lagrangian obtained with these choices for $K$ and $W$ have the familiar
supergravity form, with the Einstein-frame kinetic term for the bosons given
(in Planck units) by

$-\frac{{\cal L}_{\rm kin}}{\sqrt{-g}}=\frac{1}{2}\,{\cal
R}+K_{{\scriptscriptstyle T}{\overline{{\scriptscriptstyle
T}}}}\,\partial_{\mu}{\cal T}\,\partial^{\mu}{\overline{{\cal
T}}}\simeq\frac{1}{2}\,{\cal
R}+\left(\frac{3}{\tau^{2}}+\cdots\right)\partial_{\mu}{\cal
T}\,\partial^{\mu}{\overline{{\cal T}}}\,,$ (4)

where (as usual) ${\cal R}$ denotes the Ricci scalar built from $g_{\mu\nu}$
and subscripts on functions like $K$ and $k$ denote differentiation with
respect to the fields: e.g. $K_{{\scriptscriptstyle
T}{\overline{{\scriptscriptstyle T}}}}=\partial_{\scriptscriptstyle
T}\partial_{{\overline{{\scriptscriptstyle T}}}}K$.

The scalar potential is similarly given by

$V=e^{K}\Bigl{[}K^{{\overline{{\scriptscriptstyle T}}}{\scriptscriptstyle
T}}\,{\overline{D_{\scriptscriptstyle T}W}}D_{\scriptscriptstyle
T}W-3|W|^{2}\Bigr{]}\,.$ (5)

where $K^{{\overline{{\scriptscriptstyle T}}}{\scriptscriptstyle
T}}=1/K_{{\scriptscriptstyle T}{\overline{{\scriptscriptstyle T}}}}$ and

$D_{\scriptscriptstyle T}W=W_{\scriptscriptstyle T}+K_{\scriptscriptstyle
T}W\simeq\left(-\frac{3}{\tau}+\cdots\right)w_{0}\,.$ (6)

The last equality uses (3) for $K$. The leading parts of the scalar potential
then are

$V\simeq-\frac{3\,k_{{\scriptscriptstyle T}{\overline{{\scriptscriptstyle
T}}}}}{{\cal
P}^{2}}\,|w_{0}|^{2}+\cdots=\frac{3\,(k^{\prime}-k^{\prime\prime})}{\tau^{4}}\,|w_{0}|^{2}+{\cal
O}(\tau^{-5})\,,$ (7)

where ${\cal P}:=e^{-K/3}=\tau-k+\cdots$ and primes denote differentiation
with respect to $x=\ln\tau$. Notice that expression (7) vanishes whenever $k$
is independent of $T$, as it must do on general grounds because (3) becomes a
no-scale model [38] in the limit that $h$ (and higher terms) vanish and $k$ is
$T$-independent. For later purposes recall also that $k_{{\scriptscriptstyle
T}{\overline{{\scriptscriptstyle T}}}}$ can have either sign since it does not
control the sign of the kinetic energy for ${\cal T}$ in (4). Contributions
involving $h$ and other subdominant terms in (3) first arise at order ${\cal
O}(\tau^{-5})$.

###  2.2 Controlled perturbative stabilization

Eq. (7) reveals that the leading contribution to the potential for large
$\tau$ has the form

$V(\tau)\simeq\frac{U(\ln\tau)}{\tau^{4}}\,,$ (8)

with $U(\ln\tau)=-3\tau^{2}k_{{\scriptscriptstyle
T}{\overline{{\scriptscriptstyle
T}}}}|w_{0}|^{2}=3(k^{\prime}-k^{\prime\prime})|w_{0}|^{2}$. The minima of (8)
depend on the functional form of $U$ and so requires more information about
how $k$ acquires its dependence on $\ln\tau$.

To this end, following the ideas of [10] and [11], we imagine that $k$
acquires its dependence on $\ln\tau$ through the running of some dimensionless
coupling $\alpha_{g}$, due to a perturbative expansion of the form

$k\simeq k_{0}+k_{1}\,\alpha_{g}+\frac{k_{2}}{2}\,\alpha_{g}^{2}+\cdots$ (9)

with a dimensionless coupling $\alpha_{g}\ll 1$. In general the running of a
dimensionless coupling like $\alpha_{g}$ introduces logarithms of mass ratios,
such as when its renormalization-group evolution is integrated to give

$\frac{1}{\alpha_{g}(m_{1})}=\frac{1}{\alpha_{g}(m_{2})}-b_{1}\ln\left(\frac{m_{1}}{m_{2}}\right)\,.$
(10)

The main observation is that this can become a dependence111More precisely
masses actually develop a dependence on ${\cal P}$ rather than just $\tau$
because they typically acquire their leading dependence on $\tau$ through
powers of the Weyl rescaling factor $e^{-K/3}={\cal P}=\tau-k+\cdots$. This
makes $k$ a function of $\ln{\cal P}$ rather than $\ln\tau$ in the discussions
to follow; a distinction that often does not matter, but plays an important
role when discussing the $\eta$ problem for the inflationary scenarios of
§3.4. on $\ln\tau$ if there are multiple fields coupling to this interaction
whose masses222Notice that it is only ratios of physical masses that matter
here and not ratios of masses to the RG running parameter $\mu$. This is
because any $\tau$-dependence associated with $\mu$ ultimately cancels from
physical observables for the same reason that all $\mu$-dependence also
cancels. depend differently on $\tau$.

Such interactions do plausibly arise in string compactifications. For instance
in IIB compactifications particles localized on D3 and D7 branes have masses
that depend differently on the volume modulus and when such branes intersect
they can both couple to light open-string 4D gauge fields (whose gauge
coupling could be the $\alpha_{g}$ considered here).

In such a situation (9) predicts a logarithmic $\tau$ dependence for $k$ that
emerges through the $\tau$-dependence of $\alpha_{g}$, which in turn can be
expressed through a renormalization-group evolution like

$\tau\,\frac{{\rm d}\alpha_{g}}{{\rm
d}\tau}=\beta(\alpha_{g})=b_{1}\alpha_{g}^{2}+b_{2}\,\alpha_{g}^{3}+\cdots\,.$
(11)

For $\alpha_{g}$ small enough to neglect all but the leading term in $\beta$
this has solution

$\alpha_{g}(\tau)=\frac{\alpha_{g0}}{1-b_{1}\,\alpha_{g0}\,\ln\tau}\,,$ (12)

for some integration constant $\alpha_{g0}=\alpha_{g}(\tau=1)$. For the
present purposes what is important about the $\ln\tau$ dependence given in
(12) is that its derivation neglects only additional powers of $\alpha_{g}$ in
(11). Consequently for large $\tau$ it remains valid to all orders in
$\alpha_{g}\ln\tau$ while dropping contributions of order
$\alpha_{g}^{2}\ln\tau$. It is this renormalization-group resummation that
ultimately allows us to trust minima of the potential that occur in the regime
$\ln\tau\sim 1/\alpha_{g}$.

Now comes the main point. Using (9) and (11) to evaluate the $T$-derivatives
of $k$ then gives
$k^{\prime}=(k_{1}+k_{2}\,\alpha_{g}+\cdots)\beta(\alpha_{g})$ and similarly
for $k^{\prime\prime}$, and using these in (8) then leads to the expression

$U\simeq
U_{1}\,\alpha_{g}^{2}-U_{2}\,\alpha_{g}^{3}+U_{3}\,\alpha_{g}^{4}+\cdots\,,$
(13)

where $U_{1}=3k_{1}b_{1}|w_{0}|^{2}$ and so on. Furthermore, the Dine-Seiberg
argument leads one to expect that any minima $\tau=\tau_{0}$ of this potential
generically occur in the regime where $\alpha(\tau_{0})\sim{\cal O}(1)$. But
if stabilization of other moduli make $\alpha_{g0}$ small, then inspection of
(12) shows that $\tau_{0}$ must be very large because
$\alpha_{g0}\ln\tau_{0}\simeq{\cal O}(1)$.

This general argument can be made explicit purely using perturbative methods
if we arrange that the coefficients $U_{1}$, $U_{2}$ and $U_{3}$ appearing in
the potential (8) with $U$ given by (13) are all positive and satisfy the mild
hierarchy

$\left|\frac{U_{1}}{U_{2}}\right|\sim\left|\frac{U_{2}}{U_{3}}\right|\sim{\cal
O}(\epsilon)$ (14)

for some smallish $\epsilon\ll 1$. Such a hierarchy allows solutions to
$\left.\partial V/\partial\tau\right|_{\tau_{0}}=0$ for $\alpha_{0}\sim{\cal
O}(\epsilon)$ and so

$b_{1}\ln\tau_{0}=\alpha_{g0}^{-1}-\epsilon^{-1}$ (15)

can easily be order $1/\epsilon$ if $\epsilon\ll\alpha_{g0}$ and $b_{1}<0$.
For $\epsilon\mathrel{\raise 1.29167pt\hbox{$<$\kern-8.50006pt\lower
4.30554pt\hbox{$\sim$}}}1/10$ the value predicted for $\tau_{0}$ can be
enormous $\tau_{0}\sim e^{1/\epsilon}$, justifying the validity of the
$1/\tau$ expansion ex post facto. As is easy to check, when
$9\,U_{2}^{2}>32\,U_{1}U_{3}$ the potential has a local minimum at $\tau_{0}$
that is separated from the runaway to $\tau\to\infty$ by a local maximum at
$\tau_{1}>\tau_{0}$ (see Fig. 1).

Figure 1: A plot of $V$ vs $\tau$ for the scalar potential
$V=U(\ln\tau)/\tau^{4}$, revealing a de Sitter or anti-de Sitter minimum
separated from a runaway by a local maximum. The plots are obtained using the
representative values $k_{1}/k_{3}=0.01$ and $k_{2}/k_{3}=-0.133$ (arbitrary
scale). The main text describes the precise parameter range required to get de
Sitter rather than anti-de Sitter or a runaway.

The value of the potential at this minimum is positive if
$U_{2}^{2}<4\,U_{1}U_{3}$ and negative otherwise. Although (13) and (14) might
naively lead one to expect $U(\tau_{0})\sim{\cal O}(\epsilon^{4})$ when
$U_{3}\sim{\cal O}(1)$, it happens that the condition $V^{\prime}(\tau_{0})=0$
ensures that this leading contribution cancels, making the result at the
minimum instead $U(\tau_{0})\sim{\cal O}(\epsilon^{5})$. As a result both
$V(\tau_{0})$ and
$\left.\tau^{2}(\partial^{2}V/\partial\tau^{2})\right|_{\tau_{0}}$ are ${\cal
O}(\epsilon^{5}|w_{0}|^{2}/\tau_{0}^{4})$, and this can be extremely small
given that $\tau_{0}$ can be an exponential of $1/\epsilon$. Ref. [11]
explores some of the implications if this suppression were to explain the size
of the present-day Dark Energy density.

Because $U(\tau_{0})$ can have either sign both de Sitter and anti-de Sitter
solutions can be generated in this way depending on the values of the
coefficients $U_{0}$, $U_{1}$ and $U_{2}$. Both signs are allowed because (6)
shows that supersymmetry is broken for any finite $\tau$. It breaks because
the auxiliary field $F^{\scriptscriptstyle T}$ for the $T$ supermultiplet is
nonzero, since $w_{0}\neq 0$, even though $W_{\scriptscriptstyle T}$ vanishes.
Its size is instead controlled by the Planck suppressed term
$K_{\scriptscriptstyle T}W/M_{p}^{2}\in D_{\scriptscriptstyle T}W$. This type
of supersymmetry breaking is common in no-scale models and is responsible for
many of the unusual properties encountered in [11]. This source of
supersymmetry breaking is easily missed in global supersymmetry because it
disappears in the $M_{p}\to\infty$ limit.

###  2.3 Type IIB string theory realization

We next expand on how the above mechanism arises in the low-energy limit of
Type IIB string vacua. One purpose in doing so is to identify the scales to
which this stabilization mechanism points. Another purpose is to see how such
an explicitly perturbative mechanism evades the well-known challenges posed by
the Dine-Seiberg problem [1]. We discuss each of these issues after first
making the connection to IIB vacua more explicit.

The massless bosonic fields in the 10D supergravity relevant to Type IIB vacua
below the string scale are

$\tilde{g}_{{\scriptscriptstyle M}{\scriptscriptstyle N}},\quad{\cal
S}=s-iC,\quad G_{(3)}=H_{(3)}+i{\cal S}F_{(3)},\quad\tilde{F}_{(5)}={\rm
d}C_{(4)}+\frac{1}{2}\,C_{(2)}\wedge H_{(3)}+\frac{1}{2}\,B_{(2)}\wedge
F_{(3)}$ (16)

where a subscript $(p)$ indicates that the corresponding field is a $p$-form,
$s=e^{-\hat{\phi}}$ is the 10D dilaton333The hat on $\hat{\phi}$ distinguishes
the string dilaton from the inflaton field $\phi$ used everywhere else in this
paper. that controls the local string coupling and $C$ is an axionic scalar
while $H_{(3)}={\rm d}B_{(2)}$ and $F_{(3)}={\rm d}C_{(2)}$ are field
strengths for 2-form gauge potentials. At the two-derivative level the action
for these fields takes the schematic form

$S_{\rm bulk}=\int{\rm
d}^{10}x\,\sqrt{-\tilde{g}}\;\left\\{\tilde{R}-\frac{\lvert\partial{\cal
S}\rvert^{2}}{(\mathrm{Re}\,{\cal S})^{2}}-\frac{\lvert
G_{(3)}\rvert^{2}}{\mathrm{Re}\,{\cal
S}}-\tilde{F}_{(5)}^{2}\right\\}+\int\frac{1}{\mathrm{Re}\,S}\,C_{(4)}\wedge
G_{(3)}\wedge{\overline{G}}_{(3)}\ ,$ (17)

This action has two accidental symmetries that are important for our present
purposes:

  * •

An $SL(2,\mathbb{R})$ symmetry under which

$S\to\frac{a\,{\cal S}-ib}{ic\,{\cal S}+d}\quad\hbox{and}\quad
G_{(3)}\to\frac{G_{(3)}}{ic\,{\cal S}+d}\,,$ (18)

where $ad-bc=1$. Note that the special case $b=c=0$ and $a=1/d$ reduces to a
classical scaling symmetry

$\tilde{g}_{{\scriptscriptstyle M}{\scriptscriptstyle
N}}\rightarrow\tilde{g}_{{\scriptscriptstyle M}{\scriptscriptstyle N}}\
,\qquad{\cal S}\rightarrow a^{2}{\cal S}\ ,\qquad G_{(3)}\rightarrow
aG_{(3)}\,,\qquad\tilde{F}_{(5)}\rightarrow\tilde{F}_{(5)}\,.$ (19)

  * •

An approximate accidental scale invariance

$\tilde{g}_{{\scriptscriptstyle M}{\scriptscriptstyle
N}}\rightarrow\lambda\tilde{g}_{{\scriptscriptstyle M}{\scriptscriptstyle N}}\
,\qquad{\cal S}\rightarrow{\cal S}\ ,\qquad B_{(2)}\rightarrow\lambda
B_{(2)}\,,\qquad C_{(2)}\rightarrow\lambda C_{(2)}\,.\qquad
C_{(4)}\rightarrow\lambda^{2}C_{(4)}\,.$ (20)

under which the tree level action scales as
$S_{bulk}\rightarrow\lambda^{4}S_{bulk}$. Upon compactification to four
dimensions the non-trivial scaling of the 10D metric implies an overall
scaling of the volume modulus ${\cal V}\rightarrow\lambda^{3}{\cal V}$.

These two approximate symmetries are accidental in the sense that they are
broken by $\alpha^{\prime}$ and loop corrections to the effective action.
Indeed, how terms scale under these two transformations can be used to
identify how the 10D action depends on these two expansions [2]. For the 4D
theory, the $\alpha^{\prime}$ expansion becomes an expansion in inverse powers
of the volume ${\cal V}:=\tau^{3/2}$ while the string-loop expansion is in
powers of $(\hbox{Re}\,{\cal S})^{-1}=e^{\hat{\phi}}$.

Both scaling symmetries are spontaneously broken inasmuch as neither leaves
generic background fields unchanged and the volume modulus and the string
dilaton can be regarded as their pseudo-Goldstone dilaton modes. From this
point of view 10D flat space is special inasmuch as it leaves a scale
invariance unbroken because (20) does not act on ${\cal S}$ and scale
transformations of the flat metric can be compensated by a diffeomorphism. 10D
flat space corresponds in 4D to ${\cal V}\to\infty$ and $s\to\infty$ and the
scale-invariance of this point anchors the asymptotic value of the 4D scalar
potential to zero.

####  2.3.1 IIB modulus stabilization

String theory famously has no parameters, but if so what are the choices that
lead to differently shaped compactifications? For IIB Calabi-Yau orientifold
compactifications the choices made are the quantized fluxes of the three-form
fields whose presence and stress-energy stabilizes the complex structure
moduli $U$ and string dilaton ${\cal S}$, as pioneered in [4]. For
supersymmetric flux configurations these moduli are fixed in the 4D effective
description by the supersymmetric conditions $D_{\scriptscriptstyle
S}W=D_{\scriptscriptstyle U}W=0$.

The choice of higher-dimensional fluxes shows up in the low-energy 4D theory
in several ways. First, $(0,3)$ fluxes induce a non-trivial superpotential
$w_{0}\in W$ that is generically order unity (in Planck units) but can be
arranged to take larger – or extremely small [39] – values. Second, since
fluxes fix the string dilaton field ${\cal S}$ they provide a ‘discretuum’ of
possible values for the string coupling constant $g_{s}\sim s^{-1}$ in the 4D
theory. Third, fluxes can fix the complex structure moduli in such a way that
the corresponding three-cycles in the extra-dimensional geometry become long
throats along which 4D geometries are naturally warped with warp factor
$e^{{\cal A}}\sim e^{8\pi K/g_{s}M}$, where $K,M$ are integers. The three
quantities $w_{0}$, $g_{s}$ and $e^{{\cal A}}$ play important roles defining
the different scales that arise within the 4D theory.

The Calabi-Yau space’s Kähler moduli are not similarly fixed by these fluxes
and so their potential is naturally explored within the 4D theory. The
simplest case arises for Calabi-Yau orientifiolds that have the fewest
possible Kähler moduli: the single complex modulus
$T=\frac{1}{2}(\tau+i{\mathfrak{a}})$ whose real part describes the overall
volume ${\cal V}\propto\tau^{3/2}$ of the Calabi-Yau and whose imaginary part
${\mathfrak{a}}$ is an axionic partner. The shift symmetry for this axion
${\mathfrak{a}}\rightarrow{\mathfrak{a}}+c$ can forbid444Whether it does or
not depends on whether the corresponding symmetry has an anomaly. If so $W$
can depend exponentially on $T$. $T$-dependent corrections to $W$ that are
perturbative in $1/T$ are forbidden by the supersymmetric non-renormalization
theorems [40, 35, 36]. its appearance in the superpotential $W$. The leading
expression for the Kähler potential for $T$ is well known to be of the no-
scale type

$K(T,{\overline{T}})=-2\ln{\cal V}=-3\ln\tau\,,$ (21)

as can be derived either from explicit dimensional reduction or using the
transformation properties of the 4D action under the approximate accidental
scaling symmetries (19) and (20).

As discussed earlier, the condition $W_{\scriptscriptstyle T}=0$ together with
the no-scale identity $K^{{\overline{{\scriptscriptstyle
A}}}{\scriptscriptstyle B}}K_{{\overline{{\scriptscriptstyle
A}}}}K_{\scriptscriptstyle B}=3$ satisfied by the Kähler potential (21)
ensures the scalar potential (5) is independent of $\tau$ and this precisely
reproduces the microscopic statement that Kähler moduli are not fixed in the
underlying flux construction at leading order in string coupling and
$\alpha^{\prime}$. Kähler modulus stabilization proceeds because higher-order
corrections lift this flatness and so can stabilize fields like $T$. At
present the main approaches to modulus stabilization drive this stabilization
by introducing a $T$-dependent contribution to the superpotential, which can
arise nonperturbatively in $1/\tau$ through contributions of the form $\delta
W=W_{np}\propto e^{-\xi\,T}$ for some $\xi$. Introducing $T$-dependence to $W$
lifts the flatness of the no-scale potential, and can be consistent with the
underlying $1/\tau$ expansion either if $w_{0}$ happens to be extremely small
[5] or by considering multiple Kähler moduli, $\tau_{v}$ and $\tau_{s}$ and
having $\tau_{v}\sim e^{\,\xi\tau_{s}}$ so that powers of $1/\tau_{v}$ can
compete with $\delta W\propto e^{-\xi\,T_{s}}$ [6, 7].

We instead here do not introduce a $T$-dependence to $W$ at all, and
considering only perturbative corrections to $K$ (that would in any case
normally dominate over non-perturbative effects). Denoting $s=\hbox{Re}\,{\cal
S}=e^{-\phi}$, in general [2] perturbative corrections to $K$ in powers of
$1/s$ and $1/\tau$ can be written as

$e^{-K/3}=s^{1/3}\tau\sum_{nmr}{\cal
A}_{nmr}\left(\frac{1}{s}\right)^{n}\left(\frac{s}{\tau}\right)^{(m+r)/2}\,,$
(22)

with $n$ counting string loops and the $\alpha^{\prime}$ expansion receives
contributions from $r$ powers of extra-dimensional curvature and $m+1$ powers
of 3-form flux $G_{(3)}$. The coefficients ${\cal A}_{nmr}$ here are to be
regarded as functions of all other moduli555A crucial difference in our
approach is to consider that the coefficients ${\cal A}_{nmr}$ can have a
$\ln\tau$ dependence., but the powers of $s$ and $\tau$ associated with the
string-loop and $\alpha^{\prime}$ expansions are explicit. Tracking only the
volume dependence then shows that the Kähler potential can be written as the
following expansion in powers of $1/\tau$

$K(T,{\overline{T}})=-3\ln{\cal P},\quad\hbox{with}\quad{\cal
P}(\tau)=\tau\left[1-\frac{k}{\tau}+\frac{h}{\tau^{3/2}}+{\cal
O}\left(\frac{1}{\tau^{2}}\right)\right]\,.$ (23)

This has a form very similar to (3), differing666Notice that in principle (22)
allows a contribution with $m+r=1$ that, if present, would change (23) to
${\cal P}=\tau(1+g/\sqrt{\tau}-k/\tau+\cdots)$. This in turn would lead to a
leading correction to the scalar potential of order $\delta V\sim{\cal
O}(\tau^{-7/2})$ that would dominate the contributions we consider here.
However no known Calabi Yau produces these terms and it has recently been
shown [41] that such corrections are generally absent at least to leading
order in string loops. Including $m+r=1$ contributions still allows our
stabilization mechanism, but the logarithmic terms in $K$ would arise at order
$\tau^{-7/2}$ rather than order $\tau^{-4}$ as we use here. only by being an
expansion in powers of $\tau^{-1/2}\propto{\cal V}^{-1/3}$ rather than
$\tau^{-1}\propto{\cal V}^{-2/3}$.

As discussed above, the corrections in (23) generically lift the no-scale flat
direction, but when the coefficients $k$ and $h$ are $\tau$-independent the
leading contribution to $V$ comes from $h$ and arises at order $\delta
V\sim{\cal O}(\tau^{-9/2})$, corresponding to what are $(\alpha^{\prime})^{3}$
corrections in the underlying string construction. The persistence of the
potential’s flatness in the presence of a $T$-independent $k$ is known as the
compactification’s extended no-scale property, and follows because the first
two terms of (23) still satisfy the no-scale identity
$K^{{\overline{{\scriptscriptstyle A}}}{\scriptscriptstyle
B}}K_{{\overline{{\scriptscriptstyle A}}}}K_{\scriptscriptstyle B}=3$.

The new ingredient here is not to try to stabilize $\tau$ by balancing
different powers of $1/\tau$ in the expansion of $V$ that follow from (23),
but instead to recognize that the coefficients $k$ and $h$ can generically
contain a $\ln\tau$ dependence $k=k(\ln\tau)$. This gives the leading
contribution to the scalar potential as in (8), and the minimum is instead
obtained by balancing different powers of $\alpha_{g}\ln\tau$ against one
another. As discussed earlier this has the advantage that this balancing can
be done without undermining either the $1/\tau$ or $\alpha_{g}$ expansions.

We see in this way that the stabilization scenario proposed in §2.2 can apply
directly to IIB string theory. It easily gives exponentially large volumes,
similar to the large-volume scenario (LVS), and like the LVS requires a second
expansion modulus.777As mentioned earlier, exponentially large volumes arise
in LVS through the introduction of a second Kähler ‘blow-up’ modulus
$\tau_{s}$ that appears exponentially in the superpotential, with the
potential minimized when powers of $1/{\cal V}$ balance against this non-
perturbative contribution to $W$ (which occurs when ${\cal V}\simeq
e^{a\tau_{s}}$). It is the coupling $\alpha_{g}$ itself that would be the
required second modulus in the approach we follow here (and so need not be a
Kähler modulus, such as if it is the inverse string dilaton itself). Here,
naturally, the dilaton $s$ can be fixed by fluxes to give small enough
coupling (and so $1/s$ can play the role of $\alpha_{g}$ in our analysis).
This is similar to LVS. But, unlike LVS models, the mechanism presented here
does not need to add new uplifting mechanisms to obtain de Sitter space.888Of
course obtaining anti-de Sitter space at tree level need not mean a solution
fails to describe our universe because quantum corrections to $V$ often
dominate the classical prediction and so can be the source of positive energy
that allows de Sitter solutions (for a recent suggestion along these lines see
[42]). The scenario described here also extends straightforwardly to the more
general case with more than one Kähler modulus (ref. [11] explores some multi-
modulus examples) inasmuch as the volume is stabilized by leading order
contributions of order $\tau^{-4}$ whereas the smaller ‘fibre’ moduli can be
stabilized using the next order contributions of order $\tau^{-9/2}$ (such as
is done in LVS constructions, and exploited for inflationary purposes in
[17]).

Whether $\ln\tau$-dependence actually appears in $k$ in specific string
constructions is of course model dependent. For gauge interactions such log
dependence requires the existence of more than one type of matter field for
which the masses scale differently with $\tau$ since the logs are sensitive
only to ratios of masses. As mentioned earlier, in IIB models this is the case
whenever there are chiral states charged under both D3 and D7 gauge groups. A
general study of concrete models for which these logs are present is beyond
the scope of this article, but we refer the reader to recent discussions on
the appearance of logs in IIB EFTs [43, 44, 45, 46, 47, 12]. In particular
[12] computes amplitudes that lead to logarithmic corrections to the Kähler
potential of order $\delta K={\cal O}(\ln{\cal V}/{\cal V})$ and use them as a
mechanism to stabilize moduli (although without the renormalization-group
resummation used here).

####  2.3.2 Scales and Soft terms

We have seen the value of $\tau$ can be fixed at a wide range of exponentially
large values using only a relatively small range of parameters $k_{i}$, but
precisely how big should we like $\tau=\tau_{0}$ to be at the minimum? There
are two classes of regimes that are natural to consider.

#### Yoga models

The ambitious point of view asks $\tau_{0}$ to be large enough that $V_{\rm
min}\propto\tau_{0}^{-4}$ can be as small as the observed dark energy density.
This requires $\tau_{0}\mathrel{\raise 1.29167pt\hbox{$>$\kern-8.50006pt\lower
4.30554pt\hbox{$\sim$}}}10^{26}$ and is the regime explored in some detail in
[11]. For $\tau_{0}$ this large the mass of the $\tau$ field and its axionic
partner are light enough to be cosmologically active in the recent universe.
Although one might imagine such light scalars to be ruled out by solar-system
and cosmological tests of gravity, ref. [11] shows that they are surprisingly
hard to constrain, partly due to the appearance of surprising new mechanisms
for screening [48].

For the present purposes the main problem with choosing $\tau_{0}$ this large
is that the 4D theory near this minimum requires a UV completion at scales of
order $M_{p}/\tau_{0}$ because this is where the axion decay constant lies.
This in itself need not be a problem because this occurs at eV scales when
$\tau_{0}\sim 10^{26}$ and so the required UV physics could plausibly be
extra-dimensional. Whether this kind of a picture is viable then depends on
precisely how $\tau$ arises in the UV completion, but if it does so as a
volume modulus along the lines described here then there is a problem.

The problem arises because the string and Kaluza-Klein scales are related to
$\tau$ by

$M_{s}=\frac{M_{p}}{{\cal
V}^{1/2}}\sim\frac{M_{p}}{\tau^{3/4}}\quad\hbox{and}\quad
M_{\scriptscriptstyle KK}\sim\frac{M_{s}}{{\cal V}^{1/6}}=\frac{M_{p}}{{\cal
V}^{2/3}}\sim\frac{M_{p}}{\tau}\,.$ (24)

Requiring $M_{\scriptscriptstyle KK}\mathrel{\raise
1.29167pt\hbox{$>$\kern-8.50006pt\lower 4.30554pt\hbox{$\sim$}}}10$ TeV near
the minimum implies $\tau_{0}\mathrel{\raise
1.29167pt\hbox{$<$\kern-8.50006pt\lower 4.30554pt\hbox{$\sim$}}}10^{14}$, but
this bound can be model-dependent because it can be evaded by having some
extra dimensions be smaller than others. More robust is the condition
$M_{s}\mathrel{\raise 1.29167pt\hbox{$>$\kern-8.50006pt\lower
4.30554pt\hbox{$\sim$}}}10$ TeV, which implies $\tau_{0}\mathrel{\raise
1.29167pt\hbox{$<$\kern-8.50006pt\lower 4.30554pt\hbox{$\sim$}}}10^{20}$.
Because our focus here is on string embeddings we do not pursue values of
$\tau_{0}$ larger than this any further. See [11] for more in-depth discussion
of these issues.

#### SUSY Breaking at TeV and higher scales

The alternative point of view is to ignore (as most do) the cosmological
constant problem and ask how other scales depend on $\tau$. In this case
having $V_{\rm min}\propto\tau_{0}^{-4}$ still suppresses its present-day
value relative to other approaches, particularly when $\tau_{0}$ takes its
largest allowed values. In this case the main constraint on the present-day
value of $\tau$ actually comes from demanding the field $\tau$ to be heavy
enough to avoid the cosmological-modulus problem. This problem is a general
constraint on gravitationally coupled relics and requires $\tau$ to be heavy
enough to decay before nucleosynthesis, so as not to destroy its successes,
which for gravitational-strength decays requires $m_{\tau}\mathrel{\raise
1.29167pt\hbox{$>$\kern-8.50006pt\lower 4.30554pt\hbox{$\sim$}}}30$ TeV [49].

Re-introducing factors of $M_{p}$, the mass of $\tau$ and of the gravitino are
related by

$m_{\tau}=\left(\frac{\tau^{2}}{M_{p}^{2}}\frac{\partial^{2}V}{\partial\tau^{2}}\right)^{1/2}\sim\frac{\epsilon^{5/2}|w_{0}|}{\tau^{2}M_{p}^{2}}\sim\frac{\epsilon^{5/2}m_{3/2}}{\tau^{1/2}}\quad\hbox{where}\quad
m_{3/2}\sim\frac{|w_{0}|}{\tau^{3/2}M_{p}^{2}}\,.$ (25)

In these expressions we take $\epsilon\sim{\cal O}(1/10)$ since $\tau_{0}\sim
e^{1/\epsilon}$, but even once this is done the implications for $\tau_{0}$ of
the condition $m_{\tau}(\tau_{0})\mathrel{\raise
1.29167pt\hbox{$>$\kern-8.50006pt\lower 4.30554pt\hbox{$\sim$}}}30$ TeV
depends on the value of $|w_{0}|$. We choose two representative benchmarks:
$|w_{0}|\sim M_{p}^{3}$ (as is most commonly found in string
compactifications) or $|w_{0}|\sim M_{p}^{3}\,\tau_{0}^{1/2}$ (which is the
upper limit on what is possible for a 4D supergravity EFT, since for larger
$w_{0}$ the gravitino mass becomes larger than the Kaluza-Klein scale given in
(24) [50, 11]). Choosing $m_{\tau}\sim 30$ TeV for each of these cases implies

$\displaystyle\tau_{0}\sim 10^{6}\,,\quad m_{3/2}\sim
10^{9}\;\hbox{GeV}\,,\quad M_{\scriptscriptstyle KK}\sim
10^{12}\;\hbox{GeV}\,,\quad M_{s}\sim
10^{14}\;\hbox{GeV}\quad\hbox{if}\quad|w_{0}|\sim M_{p}^{3}$
$\displaystyle\qquad\tau_{0}\sim 10^{8}\,,\quad m_{3/2}\sim
M_{\scriptscriptstyle KK}\sim 10^{10}\;\hbox{GeV}\,,\quad M_{s}\sim
10^{12}\;\hbox{GeV}\quad\hbox{if}\quad|w_{0}|\sim
M_{p}^{3}\,\tau_{0}^{1/2}\,.$

Given a value for $\tau_{0}$, the size of soft supersymmetry-breaking terms,
superpartner masses and trilinear couplings for any Standard Model like sector
can also be estimated, under the assumption that their dominant source of
supersymmetry breaking comes from the $T$ auxiliary field, although the result
depends somewhat on the particular microscopic realization of the Standard
Model and hidden sectors. For instance, suppose a Standard Model multiplet
$\psi^{i}$ appears in the quantity $k(\psi,{\overline{\psi}})$ of eq. (23).
This would arise, for example, for states sequestered in local D3 or D7 branes
and predicts soft supersymmetry-breaking masses in a manner similar, although
slightly different dependence, to what is found for the large modulus in LVS
(see for instance [56]):

$m_{\psi}^{2}=m_{3/2}^{2}-F^{i}F^{{\overline{j}}}\partial_{i}\partial_{{\overline{j}}}\ln
Z_{\psi}\quad\hbox{which implies}\quad
m_{\psi}\sim\frac{w_{0}}{\tau^{2}}\sim\frac{m_{3/2}}{\tau^{1/2}}\,.$ (26)

Here $Z_{\psi}\sim\partial_{i\bar{\jmath}}K\sim-k_{i\bar{\jmath}}/\tau$ and
the $F$-term for $T$ is given by

$F^{\scriptscriptstyle T}=e^{K/2}K^{{\scriptscriptstyle
T}{\overline{{\scriptscriptstyle T}}}}K_{\scriptscriptstyle
T}W\sim\frac{w_{0}}{\tau^{1/2}}+{\cal O}(\tau^{-3/2})\,.$ (27)

For gaugino masses in this type of scenario it is instead the $F$-term of the
dilaton that plays the key role. This is true (as in LVS) even though to
leading order the dilaton ${\cal S}$ does not break supersymmetry
$F^{S}\propto D_{S}W=0$, since to next order in the $1/\tau$ expansion we have
$F^{\scriptscriptstyle S}\simeq e^{K/2}K^{{\scriptscriptstyle
S}{\overline{{\scriptscriptstyle S}}}}K_{\scriptscriptstyle S}w_{0}\sim
w_{0}/\tau^{5/2}$. This gives gaugino masses of order

$M_{\scriptscriptstyle
G}=\frac{F^{i}\partial_{i}f}{{\mathrm{R}e}f}\sim\frac{w_{0}}{\tau^{5/2}}\sim\frac{m_{3/2}}{\tau}\,.$
(28)

Similarly for the scalar trilinear soft-couplings in the potential ($A$ terms)
which are $A\sim M_{\scriptscriptstyle G}$.

Unless further cancellations happen, this gives a split-SUSY spectrum in the
sense that gauginos are lighter than scalar masses and both of these are
lighter than the gravitino mass and similar to the mass of $\tau$. Since
$\tau$ has to be heavier than $30$ TeV, this allows for gaugino masses of
order the TeV scale for $\tau_{0}\simeq 10^{6}$. Note however that we have
only included the contribution of the overall volume modulus $\tau$ and the
dilaton $s$ to supersymmetry breaking, the contribution of the Standard Model
cycle, if present, may dominate and wash out the sequestering making all soft
terms of order the gravitino mass and leading to intermediate scale
supersymmetry breaking. Therefore, similar to LVS but with different volume
dependence, our scenario may lead either to sequestered split supersymmetry or
intermediate scale supersymmetry breaking in both cases with intermediate
scale gravitino mass. A more detailed study of the structure of soft terms
requires constructing concrete realizations of the Standard Model sector and
is beyond the scope of this article.

###  2.4 The Dine-Seiberg problem

With the broad picture of the string embedding in place we can further comment
on how this construction bears on the challenges posed by the Dine-Seiberg
problem. These challenges come in two separate forms, each of which we discuss
in turn.

#### Field expansions

At its most basic the problem starts with the observation that expansion
parameters are fields in string theory, and for any potential of the form

$V(\tau)=\sum_{n}\frac{V_{n}}{\tau^{n}}$ (29)

the condition $V^{\prime}(\tau_{0})=0$ requires at least two terms of this
series to have a similar size. How can this be consistent with the underlying
expansion in $1/\tau$? Taken at face value this means stationary points of $V$
must occur outside the perturbative domain.

There are several well-known ways to evade this argument. One such observes
that perturbation theory need not break down if the first coefficient $V_{0}$
(or the first few) is for some reason unusually small. If it happens that
$V_{0}/V_{1}={\cal O}(\epsilon)$ for some $\epsilon\ll 1$ with all
coefficients except $V_{0}$ having roughly the same size, then the stationary
point $\tau_{0}$ of the series
$V(\tau)=\tau^{-p}(V_{0}+V_{1}\tau^{-1}+\cdots)$ satisfies
$V^{\prime}(\tau_{0})=0=-\tau_{0}^{-p-1}[p\,V_{0}+(p+1)V_{1}\tau_{0}^{-1}+\cdots]$
and so has a solution well-approximated by

$\frac{1}{\tau_{0}}\simeq-\frac{p\,V_{0}}{(p+1)V_{1}}={\cal O}(\epsilon)\,.$
(30)

Precisely this argument is applied to the series in powers of $\alpha_{g}$ in
(14) to find stationary points $V^{\prime}(\tau_{0})=0$ consistent with the
condition $\alpha_{g}(\tau_{0})\ll 1$. It is also the argument ultimately used
in single-modulus KKLT models, for which $|w_{0}|$ must be assumed small in
order to balance against terms like $\delta W\propto e^{-aT}$ within the
context of an overall expansion in power of $1/\tau$.

The large-volume scenario (LVS) of string vacua works with a variation of this
theme that requires the existence of at least two expansion parameters. In
this case the potential arises as a multiple expansion of the form

$V(\tau_{1},\tau_{2})=\sum_{mn}V_{mn}\,\varepsilon_{1}^{m}\,\varepsilon_{2}^{n}$
(31)

where $\varepsilon_{i}(\tau_{1},\tau_{2})$ are two small functions of the two
independent moduli (and in practice $\varepsilon_{1}=\tau_{1}^{-1/2}$ and
$\varepsilon_{2}=e^{-a\tau_{2}}$). In this case different terms in
$\partial_{\tau_{i}}V=0$ can balance against one another provided the
$\varepsilon_{i}$ are similar in size, but now this can be consistent with
both of them being small without having to assume special properties for the
coefficients $V_{mn}$. It is noteworthy from this point of view that multiple
moduli is the rule for the underlying Calabi-Yau spaces of interest, not the
exception.

The stabilization mechanism used here adds a third way to evade this problem.
In this case two expansion parameters are present but the stabilization occurs
completely using a fixed order in $1/\tau$ since the potential has the form
$V(\tau)\simeq U(\ln\tau)/\tau^{4}$, with $U$ a rational function of $\ln\tau$
arising due to its expansion in powers of $\alpha_{g}(\tau)$. The existence of
a minimum for $V$ with respect to variations of $\tau$ then also requires
terms at different orders in $\alpha_{g}$ to balance, and this is achieved
consistent with $\alpha_{g}(\tau_{0})\ll 1$ by using the assumption (14) –
similar to the reasoning leading to (30). The new ingredient arises because
the $\tau$ dependence embedded in $\alpha_{g}(\tau)$ itself comes as a series
in $\alpha_{g}\ln\tau$, and solutions come with $\tau$ large enough that
$\alpha_{g}\ln\tau$ need not be small even if $\alpha_{g}$ is. Nevertheless
the magic of the renormalization group ensures that the solution can be
computed reliably even if $\alpha_{g}\ln\tau\sim{\cal O}(1)$ without assuming
anything special about the coefficients $b_{i}$ in eq. (11).

#### Perturbation theory and the runaway

A stronger claim is sometimes superimposed on the Dine-Seiberg problem within
string theory. This claim (emphasised in particular by advocates of swampland
hypotheses and de Sitter conjectures [26, 51]) states that full control over
perturbative expansions in $1/\tau$ and/or $1/s$ are only valid for $\tau$ and
$s$ large enough to be in the runaway region, for which standard EFT methods
break down because there is no static string vacua exist about which to
perturb.

Different versions of this argument involve two separate claims:

  * •

In one the objection is that any effective 4D description inevitably breaks
down for large enough values of fields like $\tau$ because for large $\tau$ a
tower of high-energy states descend into the low-energy theory and ruin its
validity. This tower is not hypothetical in the case where $\tau$ is the
volume modulus because the solution in the limit $\tau\to\infty$ is 10D flat
space and the dangerous tower in question consists of the higher-dimensional
Kaluza-Klein modes.

  * •

In the other the objection is that the state of the string-theory art only
provides tools for studying static vacua and these do not allow comparison
with the runaway region for which $\tau$ and $s$ are asymptotically large but
not actually infinite.

We argue here that neither of these objections need be that worrisome.

In one sense the first objection is simply true: the first nonzero Kaluza-
Klein mass provides the UV scale above which any 4D EFT must break down, and
it is also true that $M_{\scriptscriptstyle KK}\to 0$ as $\tau\to\infty$. As
argued in [52] it is never a good approximation to include even a few KK
states with nonzero masses in the 4D EFT while neglecting the rest because the
underlying EFT expansion is in $M_{\rm low}/M_{\rm high}$ where $M_{\rm low}$
is the highest nonzero energy scale appearing in the low-energy theory and
$M_{\rm high}$ is the lowest energy scale intrinsic to the high-energy theory.
But Kaluza-Klein masses come in quantized towers, such as when $M_{n}=n/L$ for
$n$ some integer and $L$ some extra-dimesional length. Keeping the $n=1$ state
in the low-energy EFT while integrating out the $n=2$ state is only justified
within an expansion in powers of $M_{1}/M_{2}=\frac{1}{2}$; never a
parametrically small variable.

The same problem does not arise if only $n=0$ states (i.e. the moduli) are
included in the 4D theory because in this case the low-energy mass need not be
tied as rigidly to the KK scale $1/L$. Whether a 4D theory makes sense depends
on the masses acquired by the moduli, and in the examples of interest here we
have e.g. $m_{\tau}/M_{\scriptscriptstyle
KK}\propto\tau^{-1}\sim(M_{s}L)^{-4}$, which can be parameterically small
precisely when $\tau\gg 1$ because then $M_{\scriptscriptstyle KK}\sim 1/L\ll
M_{s}$. Although it is true that the UV cutoff of the 4D theory declines
monotonically as $\tau\to\infty$, the 4D theory can have a nontrivial domain
of validity for any large but finite $\tau$. One must of course check that
this suffices to describe the physical process of interest.

The second objection ultimately puts a premium on static solutions when
justifying using EFT methods. There is no evidence, however, that this is
required elsewhere in the many areas of physics for which EFT methods apply.
It is true that EFTs have additional conditions for validity when applied to
time-dependent systems: most notably the motion must be adiabatic in the sense
that $\dot{\phi}/\phi$ must be a low-energy scale (i.e. be much smaller than
UV scales like $M_{\scriptscriptstyle KK}$) for any moving low-energy field
$\phi$ (see [52, 53] for a more detailed discussion). But once these are
satisfied the usual rules for EFTs apply and there is no necessity to expand
around a strictly static vacuum solution (for examples where 4D effective
evolution is compared to explicit higher-dimension evolution see [54]). In the
end the only issue is whether the 4D EFT is being used within its domain of
validity, including for applications to time-dependent problems.

Many examples from other types of physics parallel the runaway situation
encountered for fields like $\tau$ and $s$ in string theory (and extra-
dimensional models more generally). One of these is the interaction energy of
two nearby atoms regarded as functions of their centre-of-mass positions,
$V({\bf r}_{1},{\bf r}_{2})$. For spinless atoms the low-energy EFT variables
can simply be the ${\bf r}_{i}$ if internal atomic size and structure define
the UV scale.

It can happen that van der Waals forces can make atoms attract for large
separations until more microscopic (e.g. exchange) forces eventually intervene
to convert this to strong repulsion. For specific atomic properties a stable
static (molecular) solution can exist for a specific separation, $r:=|{\bf
r}_{1}-{\bf r}_{2}|=a_{0}$, with no static solution possible for any other
finite $r$. This does not forbid the use of EFT methods (such as the Born-
Oppenheimer approximation) for values $r\neq a_{0}$, such as is done to map
out the shape of the potential $V({\bf r}_{1},{\bf r}_{2})$. Neither does it
make expansion around the free theory (in powers of $1/r$ around $r=\infty$)
useless. We see no reason why the large-$\tau$ and large-$s$ limits of string
theory should not be as benign as is this everyday analog.

Simpler atomic systems also shed light on arguments that non-perturbative
effects require a stable vacuum in order to be computable [55] (which we
mention despite our stabilization mechanism not requiring the use of non-
perturbative effects). For instance, suppose one or both of the underlying
nuclei were to be unstable to $\alpha$ decay with a very long half-life (such
as Uranium). Since $\alpha$-decay proceeds through the tunneling of He nuclei
through a Coulomb barrier it can be regarded as a nonperturbative effect. It
is hard to argue that one cannot compute – even in principle – the decay
lifetime of the uranium atom except at the one place where the molecule is
static. Notice that because the tunnelling rate is through an electromagnetic
potential it depends in principle on the position of the external electrons
and so at some small level also depends on the inter-atomic separation since
the electrons adjust to the presence of the other atom in the molecule.

We conclude that even though string theory has no free parameters and weak
couplings are related to runaway directions, the underlying issues of control
are not unique to string theory. Experience with reliable calculations under
similar conditions elsewhere in physics suggest that trustable perturbative
calculations should also be reliable in string theory.

##  3 Non-linear SUSY and Inflation

de Sitter solutions enter practical considerations in one of two ways: as
descriptions of our cosmological future or as descriptions of our distant
inflationary cosmological past. Although the solution described above might
conceivably describe the future universe, the necessity for inflation
eventually to end means that it cannot in itself do so in the past. We next
argue that minor extensions of the previous section’s discussion can include
both cases, along the way potentially explaining why each involves such
different scales.

###  3.1 The goldstino and the inflaton

To incorporate inflation we require two new ingredients. We first require a
large source of positive potential energy and because this necessarily breaks
supersymmetry we require a sector that breaks supersymmetry more dramatically
than does the dilaton multiplet999One might ask whether inflation can be
obtained directly from motion driven by the $\tau$ potential (8) itself. We
have been unable to do so within a regime under EFT control, largely because
the $\tau^{-4}$ behaviour of the potential is too steep. $T$. The minimum
number of degrees of freedom such a sector can introduce at low energy is the
goldstone fermion, $G$, for supersymmetry breaking. We follow [27] and
incorporate this fermion into the present discussion using a chiral field $X$
that satisfies a nilpotentcy constraint:

$X^{2}=0\,.$ (32)

This constraint removes any scalar superpartners of $G$ and ensures that it
nonlinearly realizes supersymmetry. The couplings of such fields to
supergravity are explored in [28].

The second new ingredient is an inflaton field $\phi$ that can interpolate
between a region where the large supersymmetry-breaking energy dominates and
one where it does not. Inflation is then imagined to take place as the
gravitational byproduct of the slow evolution of the field $\phi$ between
these different regimes. With later applications to brane-antibrane inflation
in mind we imagine $\phi$ also to arise within the sector for which
supersymmetry is badly broken. A nonsupersymmetric scalar $\phi$ can also be
represented by a chiral superfield $\Phi$ subject to a constraint [27, 28],
which in this case becomes:

${\overline{{\cal D}}}(X{\overline{\Phi}})=0\,.$ (33)

This states that $X{\overline{\Phi}}$ is left-chiral. If $\phi$ is also real
then the left-chiral field it is equal to is $X\Phi$, in which case (33)
strengthens to the constraint $X(\Phi-{\overline{\Phi}})=0$. In either case
the constraint removes the fermionic and auxiliary-field components of $\Phi$
in a way consistent with nonlinearly realized supersymmetry.

To incorporate these fields into a supersymmetric framework with accidental
approximate scale invariance we repeat the previous section’s construction but
now include these two new fields. For example, the Kähler potential built only
from the minimal superfields $X$, $T$ and $\Phi$ is, as before,

$e^{-K/3}=\tau-k+\frac{h}{\tau}+\cdots\,,$ (34)

where the ellipses denote higher orders in $1/\tau$, but now

$k={\mathfrak{K}}(\Phi,{\overline{\Phi}},\ln\tau)+(X+{\overline{X}}){\mathfrak{K}}_{\scriptscriptstyle
X}(\Phi,{\overline{\Phi}},\ln\tau)+{\overline{X}}X{\mathfrak{K}}_{{\scriptscriptstyle
X}{\overline{{\scriptscriptstyle X}}}}(\Phi,{\overline{\Phi}},\ln\tau)\,,$
(35)

and similarly for $h$ and higher-order terms (although these are not needed in
what follows). The most general superpotential similarly is

$W\simeq w_{0}(\Phi)+Xw_{{\scriptscriptstyle X}}(\Phi,{\overline{\Phi}})\,,$
(36)

where the unusual dependence of $W$ on ${\overline{\Phi}}$ is allowed because
the constraint (33) ensures that the result is chiral once multiplied by $X$.

The component lagrangian obtained from $K$ and $W$ is as given in [28]. The
constraint (32) ensures there is no independent propagating scalar for the $X$
multiplet, but the kinetic terms for the scalar parts of the remaining fields
$z^{\scriptscriptstyle I}:=\\{T,\Phi\\}$ are given by the standard form

$\displaystyle-\frac{{\cal L}_{\rm kin}}{\sqrt{-g}}$ $\displaystyle=$
$\displaystyle K_{{\scriptscriptstyle I}{\overline{{\scriptscriptstyle
J}}}}\,\partial_{\mu}z^{\scriptscriptstyle
I}\,\partial^{\mu}{\overline{z}}^{\scriptscriptstyle J}$ $\displaystyle\simeq$
$\displaystyle\frac{3}{{\cal
P}^{2}}\left(1+\frac{k^{\prime\prime}-2k^{\prime}}{{\cal
P}}\right)\partial_{\mu}{\cal T}\,\partial^{\mu}{\overline{{\cal
T}}}-\left[\frac{3}{{\cal
P}^{2}}\left(k_{\phi}-k_{\phi}^{\prime}\right)\partial_{\mu}\phi\,\partial^{\mu}{\overline{{\cal
T}}}+\hbox{h.c.}\right]$ $\displaystyle\qquad\qquad\qquad+\frac{3}{{\cal
P}}\left(k_{\phi{\overline{\phi}}}+\frac{k_{\phi}k_{{\overline{\phi}}}}{{\cal
P}}\right)\;\partial_{\mu}\phi\,\partial^{\mu}{\overline{\phi}}+\cdots\,,$

where the ellipses denote terms involving higher-order coefficients like $h$
and primes denote derivatives with respect to $\ln{\cal P}$.

The scalar potential resembles the standard supergravity form, although the
absence of auxiliary fields in $\Phi$ also makes it slightly different.
Writing $z^{\scriptscriptstyle A}:=\\{T,X\\}$ and adopting the standard
notation where $K^{{\overline{{\scriptscriptstyle A}}}{\scriptscriptstyle B}}$
is the inverse to the matrix $K_{{\scriptscriptstyle
A}{\overline{{\scriptscriptstyle B}}}}$, the scalar potential works out to
have the familiar supergravity form [28]

$V=e^{K}\Bigl{[}K^{{\overline{{\scriptscriptstyle A}}}{\scriptscriptstyle
B}}{\overline{D_{\scriptscriptstyle A}W}}D_{\scriptscriptstyle
B}W-3|W|^{2}\Bigr{]}\,.$ (38)

with the important proviso that the constrained field $\Phi$ is not included
in the index sums over $A$ and $B$. $X=0$ must be chosen (after
differentiation) when tracking the dependence on scalar fields because (32)
ensures the scalar part of $X$ is built from fermion bilinears.

The scalar potential again comes as a series in inverse powers of $\tau$,
whose leading terms turn out to be

$V=\frac{A|w{\scriptscriptstyle X}|^{2}}{{\cal P}^{2}}-\frac{2{\rm
Re}(B\,{\overline{w_{\scriptscriptstyle X}}}w_{0})}{{\cal
P}^{3}}+\frac{C|w_{0}|^{2}}{{\cal P}^{4}}\,,$ (39)

where (as before) ${\cal P}:=\tau-k+\cdots$ and we keep in mind that each $T$
derivative of $k$ costs a power of $1/{\cal P}$ because $k$ is a function of
$\ln{\cal P}$ rather than just $\ln\tau$ (see the observation in the footnote
below eq. (10)). The coefficients appearing in (39) are given explicitly by

$A\simeq\frac{1}{3}\,{\mathfrak{K}}^{{\overline{{\scriptscriptstyle
X}}}{\scriptscriptstyle X}}\,,\quad\frac{B}{{\cal
P}}\simeq{\mathfrak{K}}^{{\overline{{\scriptscriptstyle
X}}}{\scriptscriptstyle X}}{\mathfrak{K}}_{{\scriptscriptstyle
X}{\overline{{\scriptscriptstyle T}}}}\quad\hbox{and}\quad\frac{C}{{\cal
P}^{2}}\simeq-\frac{3({\mathfrak{K}}_{{\scriptscriptstyle
T}{\overline{{\scriptscriptstyle
T}}}}-{\mathfrak{K}}^{{\overline{{\scriptscriptstyle X}}}{\scriptscriptstyle
X}}{\mathfrak{K}}_{{\scriptscriptstyle T}{\overline{{\scriptscriptstyle
X}}}}{\mathfrak{K}}_{{\scriptscriptstyle X}{\overline{{\scriptscriptstyle
T}}}})}{1+2{\mathfrak{K}}^{{\scriptscriptstyle
X}{\overline{{\scriptscriptstyle X}}}}{\mathfrak{K}}_{\scriptscriptstyle
X}{\mathfrak{K}}_{{\overline{{\scriptscriptstyle X}}}}}\,,$ (40)

and we assume $A>0$ so the leading $|w_{\scriptscriptstyle X}|^{2}$ term is
positive. Notice that (39) reduces to (8) in the limit $w_{\scriptscriptstyle
X}=k_{\scriptscriptstyle X}=k_{{\scriptscriptstyle
X}{\overline{{\scriptscriptstyle T}}}}=0$. Notice also that
${\mathfrak{K}}_{{\scriptscriptstyle T}{\overline{{\scriptscriptstyle T}}}}$
and ${\mathfrak{K}}_{{\scriptscriptstyle X}{\overline{{\scriptscriptstyle
T}}}}$ would both be ${\cal O}(\alpha_{g}^{2})$ if $k$ inherits its $T$
dependence through the ${\cal P}$-dependence of a perturbative coupling
$\alpha_{g}({\cal P})$, as in §2.2, and this suggests that $B/A,C/A\sim{\cal
O}(\alpha_{g}^{2})$. The inflationary implications of this potential are the
subject of the remainder of this section.

###  3.2 Inflaton potential

The potential (39) is a function of the two scalars $\tau$ and $\phi$, and we
turn now to exploring its properties. The first step is to make contact with
previous sections, which can be done if there exists a configuration
$\phi_{0}$ for which $w_{\scriptscriptstyle X}(\phi_{0})=0$. When this exists
it is very close to a minium of the potential, since it effectively turns off
the leading ${\cal P}^{-2}$ and ${\cal P}^{-3}$ terms in $V$, leaving terms of
order $|w_{0}|^{2}/{\cal P}^{4}$ to dominate. $\tau$ can be minimized in this
regime using the $\ln{\cal P}$ dependence of $k$ very much along the lines
discussed in §2.

Within an inflationary perspective this minimum with $\phi$ near $\phi_{0}$ is
the endpoint of any $\phi$ evolution and so represents our present-day vacuum.
The inflationary regime instead occurs for $\phi$ far from $\phi_{0}$, where
the potential is dominated by the large positive contribution coming from the
$|w_{\scriptscriptstyle X}|^{2}/{\cal P}^{2}$ term in (39). The next sections
compute how the solution for $\tau(\phi)$ obtained by minimizing
$V(\tau,\phi)$ changes between these inflationary and post-inflationary
regimes.

#### Minimization at late times

Suppose first that there exists a field configuration $\phi_{0}$ that
satisfies $w_{\scriptscriptstyle X}(\phi_{0})=0$, and consider the
extremization problem near this point. This can be done very simply in the
particular case where none of ${\mathfrak{K}}^{{\overline{{\scriptscriptstyle
X}}}{\scriptscriptstyle X}}$, ${\mathfrak{K}}_{\scriptscriptstyle X}$ and
${\mathfrak{K}}_{{\scriptscriptstyle T}{\overline{{\scriptscriptstyle X}}}}$
depend on $\phi$ (though this assumption of $\phi$-independence can also be
relaxed – see [11]). In this case $\phi$ enters the potential only through
$w_{\scriptscriptstyle X}$ and so minimizing $V$ with respect to $\phi$
amounts to doing so with respect to $w_{\scriptscriptstyle X}$.

Since $V$ is quadratic in $w_{\scriptscriptstyle X}$ it is extremized by
evaluating at the saddle point

$w_{\scriptscriptstyle X}=3{\mathfrak{K}}_{{\scriptscriptstyle
X}{\overline{{\scriptscriptstyle T}}}}\,w_{0}\,,$ (41)

leading to

$V[\phi(\tau),\tau]\simeq-\frac{3|w_{0}|^{2}}{{\cal
P}^{2}}\left[{\mathfrak{K}}^{{\overline{{\scriptscriptstyle
X}}}{\scriptscriptstyle X}}{\mathfrak{K}}_{{\scriptscriptstyle
X}{\overline{{\scriptscriptstyle T}}}}{\mathfrak{K}}_{{\scriptscriptstyle
T}{\overline{{\scriptscriptstyle
X}}}}+\frac{{\mathfrak{K}}_{{\scriptscriptstyle
T}{\overline{{\scriptscriptstyle
T}}}}-{\mathfrak{K}}^{{\overline{{\scriptscriptstyle X}}}{\scriptscriptstyle
X}}{\mathfrak{K}}_{{\scriptscriptstyle T}{\overline{{\scriptscriptstyle
X}}}}{\mathfrak{K}}_{{\scriptscriptstyle X}{\overline{{\scriptscriptstyle
T}}}}}{1+2{\mathfrak{K}}^{{\scriptscriptstyle X}{\overline{{\scriptscriptstyle
X}}}}{\mathfrak{K}}_{\scriptscriptstyle
X}{\mathfrak{K}}_{{\overline{{\scriptscriptstyle X}}}}}\right]=:\frac{U}{{\cal
P}^{4}}\,.$ (42)

Notice that this vanishes when $k$ is independent of $T$. As described in the
footnote below eq. (10), we write the potential as a function of ${\cal P}$
(rather than just $\tau$) because this plays an important role in our later
discussion of the $\eta$ problem.

The potential (42) now has precisely the same form considered in §2, though
with $U=U(\ln{\cal P})$ given by a slightly different function of $k$ and its
derivatives. Minimization with respect to $\tau$ can be carried over in whole
cloth provided that $k$ depends on $\ln{\cal P}$ in the way described in §2.2,
such as if

${\mathfrak{K}}\simeq{\mathfrak{K}}_{0}+{\mathfrak{K}}_{1}\alpha_{g}+\frac{{\mathfrak{K}}_{2}}{2}\,\alpha_{g}^{2}+\cdots\quad\hbox{and}\quad\frac{1}{\alpha_{g}}=b_{0}-b_{1}\ln{\cal
P}$ (43)

and so on (for constants $b_{i}$). The arguments of §2 then go through as
before, showing that a minimum is possible at ${\cal P}={\cal P}_{0}$ with
$\alpha_{0}=\alpha_{g}({\cal P}_{0})\sim{\cal O}(\epsilon)$ and $\ln{\cal
P}_{0}\sim 1/\epsilon$, with $\epsilon\ll 1$. The potential evaluated at this
minimum is again order $V\sim\epsilon^{5}|w_{0}|^{2}/{\cal P}_{0}^{4}$ and so
can be extremely small. The field $\phi$, introduced here as an inflaton,
plays the role of a relaxation field by dynamically minimizing the
$|w_{\scriptscriptstyle X}|^{2}$ term (providing a simple rationale for the
relaxation mechanism as applied in [11] to present-day Dark Energy).

#### Inflationary regime

The other region of interest for the potential (39) is when $\phi$ is far from
$\phi_{0}$ and so $w_{\scriptscriptstyle X}$ is not small. This is the regime
likely appropriate for inflationary evolution since the potential is dominated
by the large positive contribution proportional to $|w_{\scriptscriptstyle
X}|^{2}$. In general the fields $\tau$ and $\phi$ can evolve independently in
this regime and it is not a priori necessary that either should sit at a local
minimum of the potential [57]. However when seeking the conditions for slow-
roll inflation it is instructive to first extremize the potential (39) with
respect to $\tau$ and ask how the result looks as a function of $\phi$.

To this end, suppose that $\phi$ lies in a region for which
$\delta:=|w_{\scriptscriptstyle X}/w_{0}|\mathrel{\raise
1.29167pt\hbox{$<$\kern-8.50006pt\lower 4.30554pt\hbox{$\sim$}}}0.01$ is small
(a regime that actually arises for large $\phi$ for the brane-antibrane
example described below because the brane tensions are parametrically smaller
than the extra-dimensional Planck scale). In this case $V$ is a sum of terms
of relative order $\delta^{2}$, $\delta/{\cal P}$ and $1/{\cal P}^{2}$, and so
can be extremized for values $1/{\cal P}\sim\delta$. For example, for real
$B$, $w_{\scriptscriptstyle X}$ and $w_{0}$ the extrema are
$\tau=\tau_{\pm}({\overline{\phi}},\phi)$ where101010For this we neglect for
simplicity any dependence of $A$, $B$ and $C$ on $\ln{\cal P}$ that is
implicit through their dependence on $\alpha_{g}({\cal P})$. This neglect is
justified because $\alpha_{g}({\cal
P})=\alpha_{g0}/(1-b_{1}\,\alpha_{g0}\ln{\cal P})$ ensures that all $\ln{\cal
P}$-dependence is subleading in powers of $\alpha_{g0}$. This argument is
consistent with keeping all orders in $\alpha_{g0}\ln{\cal P}_{0}$ for the
present-day stabilization of ${\cal P}_{0}$ because ${\cal P}$ turns out to be
much smaller during inflation than at present.

$\frac{1}{{\cal P}_{\pm}}\simeq\frac{D_{\pm}w_{\scriptscriptstyle
X}}{w_{0}}\sim{\cal O}(\delta)\quad\hbox{with}\quad
D_{\pm}:=\frac{3B}{4C}\pm\sqrt{\frac{9B^{2}}{16C^{2}}-\frac{A}{2C}}\,.$ (44)

This expression predicts several regimes:

  * •

When $A/C<0$ only one of these roots is positive, corresponding to a local
maximum with $V_{\rm max}>0$ when $A>0$ (and a local minimum with $V_{\rm
min}<0$ when $A<0$).

  * •

If $A/C>0$ then no stationary points exist at all for positive ${\cal P}$ if
$9\,B^{2}<8AC$. For fixed $\phi$ the potential $V$ is then a monotonically
decreasing positive function if $A>0$.

  * •

Both roots are real and positive (with ${\cal P}_{-}$ being a local maximum
and ${\cal P}_{+}$ a local minimum) when $A/C$ and $B/C$ are both positive and
$9B^{2}>8AC$. The potential evaluated at the minimum $\tau_{+}$ is also
positive provided $B^{2}<AC$ and negative otherwise.

When a local minimum exists and $A/C$ and $B/C$ are ${\cal O}(1)$ and
$\epsilon\sim\delta$ are similar in size (and small) then $\tau_{+}(\phi)\sim
1/\delta\gg 1$ is large but is much smaller than its value
$\tau_{0}\sim\exp(1/\epsilon)$ at the global minimum. If on the other hand
$|A/C|\gg|B/C|$ – as might be expected if $A\sim{\cal O}(\alpha_{g}^{0})$ and
$B,C\sim{\cal O}(\alpha_{g}^{2})$ – then stationary points only arise for
positive $A$ if $C<0$, corresponding to a local maximum with $V_{\rm max}>0$.
Notice that having $B/A,C/A\propto\alpha_{g}^{2}\sim{\cal O}(\epsilon^{2})$
need not be inconsistent with the existence of two roots $\tau_{\pm}$ (and so
having a local minimum) but only if the numerical coefficients in $C/A$ are
adjusted to be ${\cal O}(\epsilon^{2})$, since then $B^{2}/C^{2}$ and $A/C$
are both ${\cal O}(\epsilon^{-4})$. This can be done, for example, by
arranging ${\mathfrak{K}}_{{\scriptscriptstyle
T}{\overline{{\scriptscriptstyle T}}}}$ to be numerically suppressed by
$\epsilon^{2}$ so that ${\mathfrak{K}}_{{\scriptscriptstyle
T}{\overline{{\scriptscriptstyle T}}}}\sim{\cal
O}(\epsilon^{2}\alpha_{g}^{2})$ is similar in size to
$({\mathfrak{K}}_{{\scriptscriptstyle X}{\overline{{\scriptscriptstyle
T}}}})^{2}\sim{\cal O}(\alpha_{g}^{4})$. Using this in (44) then shows that
${\cal P}_{\pm}\sim\epsilon^{2}/\delta$ are only large if
$\delta\ll\epsilon^{2}$.

In the special case where a local minimum exists we can compute an effective
potential for $\phi$ defined by $V_{\rm eff}(\phi)=V[\phi,\tau_{+}(\phi)]$,
for which $\tau$ is assumed to remain at its local minimum as $\phi$ changes.
This evaluates to

$V_{\rm eff}(\phi)\simeq\frac{(A-BD_{+})}{2{\cal
P}_{+}^{2}}\,|w_{\scriptscriptstyle
X}(\phi)|^{2}=\frac{(A-BD_{+})D_{+}^{2}}{2|w_{0}|^{2}}\,|w_{\scriptscriptstyle
X}(\phi)|^{4}$ (45)

###  3.3 Inflationary evolution

For inflation we are interested in how the fields $\phi$ and $\tau$ evolve and
so must keep track of their kinetic terms given in (3.1). Although in general
$\tau$ and $\phi$ are mixed by the kinetic terms, this mixing arises at
subleading order in $1/\tau$. The leading form for the target-space metric is
$K_{{\scriptscriptstyle I}{\overline{{\scriptscriptstyle J}}}}\,{\rm
d}z^{\scriptscriptstyle I}\bar{z}^{\scriptscriptstyle J}\sim({\rm
d}\tau/\tau)^{2}+k_{\phi{\overline{\phi}}}|{\rm d}\phi|^{2}/\tau$ and so for
$k_{\phi{\overline{\phi}}}\sim 1$ the canonical fields are ${\rm d}\chi\sim
M_{p}\,{\rm d}\tau/\tau$ and ${\rm d}\varphi\sim{\rm d}\phi/\sqrt{\bar{\tau}}$
near a semiclassical background $\tau=\bar{\tau}$.

In general, motion along the $\tau$ direction is not a slow roll if inflation
is dominated by the term $V\sim|w_{\scriptscriptstyle X}|^{2}/\tau^{2}$ since
the slow-roll parameter in the $\tau$ direction is then

$\varepsilon_{(\tau)}\sim\left(\frac{M_{p}V_{\chi}}{V}\right)^{2}\sim\left(\frac{\tau
V_{\tau}}{V}\right)^{2}\sim{\cal O}(1)\,.$ (46)

The $\tau$ ‘mass’ can be similarly estimated when $V\sim|w_{\scriptscriptstyle
X}|^{2}/\tau^{2}$ dominates, giving

$m_{\tau}^{2}=\left(\frac{\partial^{2}V}{\partial\chi^{2}}\right)_{\tau_{+}}\sim\frac{\tau^{2}}{M_{p}^{2}}\frac{\partial^{2}V}{\partial\tau^{2}}\sim\frac{V}{M_{p}^{2}}\sim
H_{\scriptscriptstyle I}^{2}\,,$ (47)

where $H_{\scriptscriptstyle I}$ is the inflationary Hubble scale, during an
epoch when $V$ dominates the gravitating energy density. This is also
insufficiently small to justify a slow roll.

This makes it important to choose parameters so that $V$ has a local minimum
in the $\tau$ direction that stabilizes $\tau=\tau(\phi)$, in which case we
can see whether motion in the $\phi$ direction can be sufficiently slow. Even
if this can be done it must be asked whether it is a good approximation to
have $\tau$ remain trapped at its local minimum as $\phi$ evolves. This
depends on whether the volume modulus is heavy enough to integrate out the
transverse $\tau$ field to obtain an effectively single-field description. A
brief estimate arguing that it is heavy enough is given in Appendix §A.

Evaluated at the local minimum for $\tau=\tau_{+}(\phi)$ described above we
can evaluate the slow-roll parameters for evolving in the direction of the
canonically normalized field $\varphi$, and this is particularly simple when
$\phi$ dominantly enters through $w_{\scriptscriptstyle X}$, since

$\frac{\partial
V}{\partial\varphi}\simeq\frac{\partial}{\partial\varphi}\left(\frac{A|w_{\scriptscriptstyle
X}|^{2}}{\tau_{+}^{2}}-\frac{2\hbox{Re}\,(B{\overline{w_{\scriptscriptstyle
X}}}w_{0})}{M_{p}^{2}\tau_{+}^{3}}+\frac{C|w_{0}|^{2}}{M_{p}^{4}\tau_{+}^{4}}\right)\simeq\left(\frac{Aw_{\scriptscriptstyle
X}}{\tau_{+}^{2}}-\frac{Bw_{0}}{M_{p}^{2}\tau_{+}^{3}}\right)\frac{\partial{\overline{w}}_{\scriptscriptstyle
X}}{\partial\varphi}+\hbox{h.c.}\,,$ (48)

where factors of $M_{p}$ are re-instated for later convenience. Evaluating at
$\tau=\tau_{+}(\phi)$ ensures each term in $V$ and $\partial
V/\partial\varphi$ has a comparable size and so ensures $w_{\scriptscriptstyle
X}\sim(Bw_{0})/(AM_{p}^{2}\tau_{+})$ and so we may estimate the right-hand
side of (48) as being of order $[Bw_{0}/(M_{p}^{2}\tau_{+}^{3})](\partial
w_{\scriptscriptstyle X}/\partial\varphi)$. For comparison, with these same
estimates the potential energy itself is $V\sim
C|w_{0}|^{2}/(M_{p}^{4}\tau_{+}^{4})$ and the inflationary Hubble scale is
$H_{\scriptscriptstyle
I}\sim\sqrt{V}/M_{p}\sim\sqrt{C}\,|w_{0}|/(M_{p}^{3}\tau_{+}^{2})$.

The first slow-roll parameter for motion in the $\phi$-direction then becomes

$\varepsilon=\frac{1}{2}\left(\frac{M_{p}\,\partial
V/\partial\varphi}{V}\right)^{2}\sim\left(\frac{BM_{p}^{3}\tau_{+}}{C|w_{0}|}\,\frac{\partial
w_{\scriptscriptstyle
X}}{\partial\varphi}\right)^{2}\sim\tau_{+}^{3}\left(\frac{BM_{p}^{3}}{C|w_{0}|}\,\frac{\partial
w_{\scriptscriptstyle X}}{\partial\phi}\right)^{2}\,,$ (49)

which shows that slow roll requires $\partial w_{\scriptscriptstyle
X}/\partial\phi$ to be much smaller than order
$C|w_{0}|/(BM_{p}^{3}\tau_{+}^{3/2})$. The second slow-roll parameter is
similarly estimated to be

$\displaystyle\eta=\frac{M_{p}^{2}}{V}\,\frac{\partial^{2}V}{\partial\varphi^{2}}$
$\displaystyle\sim$ $\displaystyle\frac{1}{H_{\scriptscriptstyle
I}^{2}}\left[\frac{B|w_{0}|}{M_{p}^{2}\tau_{+}^{3}}\left(\frac{\partial^{2}w_{\scriptscriptstyle
X}}{\partial\varphi^{2}}\right)+\frac{A}{\tau_{+}^{2}}\left(\frac{\partial
w_{\scriptscriptstyle X}}{\partial\varphi}\right)^{2}\right]$ (50)
$\displaystyle\sim$
$\displaystyle\tau_{+}\left[\frac{BM_{p}^{4}\tau_{+}}{C|w_{0}|}\left(\frac{\partial^{2}w_{\scriptscriptstyle
X}}{\partial\phi^{2}}\right)+\frac{AM_{p}^{6}\tau_{+}^{2}}{C|w_{0}|^{2}}\,\left(\frac{\partial
w_{\scriptscriptstyle X}}{\partial\phi}\right)^{2}\right]\,.$

The property $B^{2}\sim AC$ – required for the existence of a minimum
$\tau_{+}(\phi)$ – ensures the second term of this expression is small when
$\varepsilon$ is small, so requiring small $\eta$ implies
$\partial^{2}w_{\scriptscriptstyle X}/\partial\phi^{2}$ is much smaller than
order $C|w_{0}|/(BM_{p}^{4}\tau_{+}^{2})$. Using $w_{0}\sim M_{p}^{3}$ and
$(B/A)^{2}\sim C/A\sim\epsilon^{4}M_{p}^{2}$ and
$\tau_{+}\sim\epsilon^{2}/\delta$ (as found above when
$B,C\propto\alpha_{g}^{2}$) then shows that slow roll requires the derivatives
of $w_{\scriptscriptstyle X}$ to satisfy

$\left|\frac{\partial w_{\scriptscriptstyle
X}}{\partial\phi}\right|\ll\frac{C|w_{0}|}{BM_{p}^{3}\tau_{+}^{3/2}}\sim\frac{\epsilon^{2}M_{p}}{\tau_{+}^{3/2}}\quad\hbox{and}\quad\left|\frac{\partial^{2}w_{\scriptscriptstyle
X}}{\partial\phi^{2}}\right|\ll\frac{C|w_{0}|}{BM_{p}^{4}\tau_{+}^{2}}\sim\frac{\epsilon^{2}}{\tau_{+}^{2}}\,.$
(51)

To go further we must specify in more detail how the functions $k$ and $W$
depend on $\phi$. We next identify a promising choice for these functions by
re-examining the brane-antibrane inflationary scenario to see how it is
affected by the new modulus stabilization mechanism.

###  3.4 Warped D3-${\overline{\hbox{D3}}}$ inflation revisited

Brane-antibrane inflation was the first attempt to derive inflation from a
string theory construction within a framework in which the inflaton potential
could be explicitly calculated [22, 23] (see [58] for a recent discussion).
The inflaton field is the separation between a brane and an antibrane and for
large separations the corresponding potential is the sum of two terms: the
brane tension and the brane-antibrane interaction generated by their couplings
to the various bulk fields. At large distances the inter-brane force takes a
‘Coulomb’ form111111We take the Coulomb energy for sources separated by a
distance $y$ in $d$ transverse dimensions to be an interaction energy that
falls like $y^{-p}$ with $p=d-2$ (with $p=4$ – corresponding to $d=6$ – being
the case relevant to space-filling 3-branes in 10D string vacua).  (and so
weakens at large separations) but the challenge was to find how to separate
the branes sufficiently within a finite-sized manifold and to compute how
brane motion changes the energetics of modulus stabilization.

Let us briefly recall the main ideas. Fluxes in IIB compactifications back-
react on the metric in such a way that the resulting compactification is a
conformal Calabi-Yau threefold with metric of the form [22, 23, 24]

${\rm d}s^{2}=\left(1+\frac{e^{4{\cal A}}}{{\cal V}^{2/3}}\right)^{-1/2}{\rm
d}s_{4}^{2}+\left(1+\frac{e^{4{\cal A}}}{{\cal V}^{2/3}}\right)^{1/2}{\rm
d}s^{2}_{{\scriptscriptstyle C}{\scriptscriptstyle Y}}$ (52)

with ${\cal A}(y)$ a calculable function of position within the extra
dimensions. The warp factor ${\cal W}:=\left(1+{e^{4{\cal A}}}{{\cal
V}^{-2/3}}\right)^{-1/2}$ plays a significant role in highly warped regions,
defined by the condition $e^{4{\cal A}}\gg{\cal V}^{2/3}\gg 1$. For future use
we note in passing that in order to have the warped string scale be larger
than the Kaluza-Klein scale (as required to have a reliable low-energy
effective field theory) the warping must be constrained to satisfy [59]

$e^{{\cal A}}\mathrel{\raise 1.29167pt\hbox{$<$\kern-8.50006pt\lower
4.30554pt\hbox{$\sim$}}}{\cal V}^{2/3}\,.$ (53)

A space-filling D3 brane sits at a particular point in the extra dimensions
and experiences no position-dependent forces due to supersymmetric BPS
cancellation of bulk forces, and so are free to move within the Calabi-Yau
space. Anti-D3 branes by contrast energetically prefer to minimize the warp
factor and so move to the tip of any warped throat for which $e^{4{\cal A}}$
takes its largest value, which turns out to be

$e^{4{\cal A}_{\rm tip}}:=e^{4\rho}=e^{{8\pi K}/({3g_{s}M})}$ (54)

where $K$ and $M$ are the integer flux quantum numbers that fix the relevant
complex structure moduli. Depending on the values of $K$ and $M$ the warp
factor can be significant and so can naturally lead to a source of hierarchies
within the theory.

Keeping in mind that $\tilde{g}_{\mu\nu}={\cal W}g_{\mu\nu}$ implies
$\sqrt{-\tilde{g}_{4}}={\cal W}^{2}\sqrt{-g_{4}}$, the tension of an anti-D3
brane localized at the tip of a strongly warped throat contributes the
following positive contribution to the low-energy 4D scalar potential,

$2T_{3}{\cal W}^{2}=\frac{2T_{3}}{1+(e^{4\rho}/{\cal V}^{2/3})}\simeq
2M_{s}^{4}e^{-4\rho}{\cal V}^{2/3}\sim\frac{e^{-4\rho}M_{p}^{4}}{{\cal
V}^{4/3}}$ (55)

which uses $e^{4\rho}\gg{\cal V}^{2/3}$ for strongly warped regions as well as
the value of the brane tension $T_{3}\propto M_{s}^{4}$ and the relation (24)
between the string scale and the 4D Planck scale:

$T_{3}=\frac{1}{8\pi^{3}g_{s}\alpha^{\prime
2}}=\frac{(2\pi)^{11}g_{s}^{3}M_{p}^{4}}{4{\cal V}^{2}}\,.$ (56)

The exponential dependence of the warp factor appearing within this brane
tension is used in [5] (with a volume dependence later corrected by [24]) to
uplift the AdS minimum found in previous modulus-stabilization mechanisms in
order to obtain a dS solution rather than AdS or the more generic runaway.

Combining this brane tension term with the Coulomb interaction between a
mobile D3 brane and the anti-D3 brane sitting in a warped environment gives
the candidate brane-antibrane inflation potential [22, 23, 24] (again in the
Einstein frame):

$V=2T_{3}(e^{-4\rho}{\cal
V}^{2/3})\left(1-\frac{27}{64\pi^{2}}\frac{2T_{3}(e^{-4\rho}{\cal
V}^{2/3})}{|\varphi|^{4}}\right)=:\Omega\left(1-\frac{{\mathfrak{b}}\,\Omega}{|\varphi|^{4}}\right)$
(57)

where $\varphi$ is the canonically normalised field determining the brane
separation $y$: $\varphi=\sqrt{T_{3}}\,y$ and the last equality evaluates
$T_{3}$ in Planck units using (56), and so

$\Omega=\frac{{\mathfrak{c}}\,e^{-4\rho}M_{p}^{4}}{{\cal
V}^{4/3}},\quad{\mathfrak{c}}=\frac{(2\pi)^{11}g_{s}^{3}}{2}\quad\hbox{and}\quad{\mathfrak{b}}=\frac{27}{64\pi^{2}}\,.$
(58)

#### Naive inflationary analysis

Putting aside for the moment how ${\cal V}$ evolves given this potential,
consider first the naive single-field inflationary picture that emerges for
$\varphi$ evolution at fixed ${\cal V}$. The slow-roll parameters for this
motion in the regime ${\mathfrak{b}}\,\Omega\ll|\varphi|^{4}$ are

$\varepsilon=\frac{M_{p}^{2}}{2}\left(\frac{V_{\varphi}}{V}\right)^{2}\simeq
8{\mathfrak{b}}^{2}\left(\frac{\Omega
M_{p}}{|\varphi|^{5}}\right)^{2}\quad\hbox{and}\quad\eta=\frac{M_{p}^{2}V_{\varphi\varphi}}{V}\simeq-\frac{20{\mathfrak{b}}\,\Omega
M_{p}^{2}}{|\varphi|^{6}}\,.$ (59)

Although these can be made arbitrarily small by making $|\varphi|$
sufficiently large, as noted in [22] inflation does not work (without the warp
factors) because it would require the brane separation to be larger than the
typical linear extent of the extra dimensions. But if the warp factors buried
in $\Omega$ are small enough the slow roll conditions $\varepsilon\ll 1$ and
$\eta\ll 1$ can be satisfied. Notice that these also imply that the ratio

$-\frac{\varepsilon}{\eta}\simeq\frac{2{\mathfrak{b}}\,\Omega}{5|\varphi|^{4}}\ll
1\,,$ (60)

is deep into the regime where quantum effects are dominated by stochastic
methods [60].

In terms of these the number of inflationary $e$-foldings between horizon exit
and inflation’s end is

$N_{e}=\frac{1}{M_{p}}\int_{\varphi_{end}}^{\varphi_{*}}\frac{d\varphi}{\sqrt{2\varepsilon}}\simeq\frac{\varphi_{*}^{6}}{24{\mathfrak{b}}\,\Omega
M_{p}^{2}}$ (61)

and the amplitude of primordial scalar density perturbations becomes

$\delta_{\scriptscriptstyle
H}=\frac{1}{\pi\sqrt{75}}\left(\frac{V^{3/2}}{M_{p}^{3}\,V_{\varphi}}\right)_{\varphi_{*}}\simeq\frac{\varphi_{*}^{5}}{4\pi{\mathfrak{b}}\sqrt{75}M_{p}^{3}\sqrt{\Omega}}\,.$
(62)

In these expressions $\varphi_{*}$ is the inflaton position at horizon exit,
relative to which its value at inflation’s end is neglected:
$\varphi_{*}\gg\varphi_{\rm end}$. The slow-roll parameters evaluated at
horizon exit then become

$\eta_{*}=-\frac{5}{6N_{e}},\qquad\varepsilon_{*}=\frac{20\pi\delta_{\scriptscriptstyle
H}}{9\sqrt{2}\,N_{e}^{5/2}}\simeq\frac{16\pi}{5}\sqrt{\frac{3}{5}}\,\delta_{\scriptscriptstyle
H}\,|\eta_{*}|^{5/2}\,.$ (63)

The spectral index $n_{s}$ and tensor-to-scalar ratio $r$ are given by the
usual expressions

$n_{s}=1+2\eta_{*}-6\varepsilon_{*}\simeq 1+2\eta_{*}\quad\hbox{and}\quad
r=16\varepsilon_{*}$ (64)

in which $\varepsilon_{*}\ll|\eta_{*}|$ is used in $n_{s}$, showing that the
measured value for $n_{s}$ fixes
$\eta_{*}\simeq\frac{1}{2}(n_{s}-1)\simeq-0.015$ and so $N_{e}\simeq 56$.
Combining (63) and (64) and using the measured amplitude
$\delta_{\scriptscriptstyle H}=1.9\times 10^{-5}$ then gives the following
prediction for the scalar-to-tensor ratio

$r=16\varepsilon_{*}\simeq\frac{64\pi}{5}\sqrt{\frac{3}{10}}\,\delta_{\scriptscriptstyle
H}|n_{s}-1|^{5/2}\simeq 2\times 10^{-8}\,,$ (65)

which is too small to be observable in the foreseeable future.

Although at face value the warp factors (buried in $\Omega$) allow a potential
as flat as desired, this assumes that the physics that stabilizes the overall
volume modulus appearing in $\Omega$ has been fixed in a way that does not
significantly alter the potential for $\phi$. However, the same shallowness
that makes (57) attractive for inflation also makes it fragile to changes
associated with modulus stabilization, as can be most clearly seen by
embedding the brane-antibrane dynamics into a full 4D supergravity EFT that
allows a consistent description of both inflaton and modulus-stabilization. As
argued in [24] this exercise opens up a new problem (the $\eta$ problem) that
generically ruins the shallowness of the potential (57). We repeat this
exercise here to show why the RG stabilization mechanism avoids this problem.

####  3.4.1 The Nilpotent Superfield and anti-D3 Branes

An interesting feature of the nilpotent superfield formalism of §3.3 is that
it captures very efficiently the physics of anti-D3 branes at the tip of a
Calabi-Yau throat as described above in this section. We now explore this
connection and determine the choices that it implies for quantities like
${\mathfrak{K}}$, $w_{0}$ and $w_{\scriptscriptstyle X}$.

#### Antibrane tension from the nilpotent superfield

First we recall how the ${\cal P}$-dependence of the leading part of the
potential built using a nilpotent superfield reproduces the volume dependence
of the anti-D3 brane tension at the tip of the warped throat. For this recall
that when $W=w_{0}+w_{\scriptscriptstyle X}X$ then the leading term in the
scalar potential (39) is

$V\simeq\frac{{\mathfrak{K}}^{{\overline{{\scriptscriptstyle
X}}}{\scriptscriptstyle X}}|w_{\scriptscriptstyle X}|^{2}}{3{\cal P}^{2}}$
(66)

which for $w_{\scriptscriptstyle X}\propto e^{-2\rho}$ reproduces the KKLT
expression (55) once the volume modulus is identified in the usual way: ${\cal
P}={\cal V}^{2/3}$.

It is noteworthy that this agreement between the volume-dependence of the
nilpotent potential and the brane tension works only when one uses the
warping-corrected volume-dependence given in [24] rather than the original
expression of [5] that does not include warping. Only in the warped case is
supersymmetry breaking sufficiently sequestered to be captured using only the
single goldstino field $X$. From the string point of view the fact the $X$ has
no independent scalar degree of freedom corresponds to the fact that the
isolated anti-D3 brane has no position modulus because it is energetically
stuck at the tip of the warped throat.

There is also additional evidence that warping can be captured by the
superpotential in this way, since it can also be derived within the 4D
effective supergravity as the expectation value of the throat’s complex
structure modulus $Y$, which schematically contributes to the superpotential
in the form $W(Y)=Y(n_{1}\log Y+n_{2})+YX$ with $n_{i}$ being integer flux
quantum numbers. Eliminating $Y$ from this superpotential in a supersymmetric
way gives rise to the required warp factor multiplying $X$ in $W$ [4, 33].

#### Inter-brane dynamics

To obtain a supergravity representation for the dynamics of brane-antibrane
motion we require a supermultiplet for the inter-brane separation field
$\phi$. We do so here by representing the inter-brane separation using the
constrained inflaton field $\Phi$ of §3.3.

To describe the kinetic energy of these fields we choose the function
${\mathfrak{K}}$ defined in (35) to have the form

${\mathfrak{K}}(\phi,{\overline{\phi}},\ln{\cal
P})\simeq\gamma({\overline{\phi}},\phi)+\hat{\mathfrak{K}}(\ln{\cal P})\,,$
(67)

where $\phi^{i}$ is proportional to a complex coordinate describing the
3-brane position within the extra dimensions. The quantity
$\hat{\mathfrak{K}}$ is the $\phi$-independent function of $\ln{\cal P}$
described in (43) whose presence stabilizes ${\cal P}$ at exponentially large
values in the present-day vacuum. In terms of this the kinetic term for
changes to $\phi$ become

$-\frac{{\cal L}_{\rm
kin}}{\sqrt{-g}}=K_{i\bar{\jmath}}\,\partial_{\mu}\bar{\phi}^{j}\,\partial^{\mu}\phi^{i}\simeq\frac{3\gamma_{i\bar{\jmath}}}{{\cal
P}}\,\,\partial_{\mu}\bar{\phi}^{j}\,\partial^{\mu}\phi^{i}$ (68)

showing that $\gamma_{i\bar{\jmath}}$ is naturally proportional to the extra-
dimensional metric $g_{i\bar{\jmath}}$ and so $\gamma({\overline{\phi}},\phi)$
is proportional to this metric’s Kähler potential. Coordinates can be chosen
without loss of generality so that $\gamma\simeq{\overline{\phi}}\,\phi$ near
$\phi=0$.

To capture the antibrane tension and the separation-dependent Coulomb
interaction we use the following superpotential121212See [32] for a first
supersymmetric discussion of the Coulomb potential.

$W=w_{0}+Xw_{\scriptscriptstyle
X}(\Phi,{\overline{\Phi}})\quad\hbox{with}\quad w_{\scriptscriptstyle
X}(\Phi,{\overline{\Phi}})={\mathfrak{t}}-\frac{{\mathfrak{g}}}{|\Phi|^{4}}+\cdots$
(69)

where the ellipses denote terms suppressed by even higher powers of
$|\Phi|^{-1}$.

With this choice the leading term in the scalar potential (39) then is

$V=\frac{{\mathfrak{K}}^{{\overline{{\scriptscriptstyle
X}}}{\scriptscriptstyle X}}|w_{\scriptscriptstyle X}|^{2}}{3{\cal
P}^{2}}=\frac{{\mathfrak{K}}^{{\overline{{\scriptscriptstyle
X}}}{\scriptscriptstyle X}}}{3{\cal
P}^{2}}\left[|{\mathfrak{t}}|^{2}-\frac{2\hbox{Re}({\overline{{\mathfrak{t}}}}{\mathfrak{g}})}{|\phi|^{4}}+\cdots\right]\,,$
(70)

which is to be compared to (57) and (58). This comparison is most easily done
using the canonically normalized field – see (68) – $\varphi\propto{\cal
P}^{-1/2}\phi\propto{\cal V}^{-1/3}\phi$, for which
$|\phi|^{4}\propto|\varphi|^{4}{\cal P}^{2}\propto|\varphi|^{4}{\cal
V}^{4/3}$131313Note that it is not trivial that the volume-dependence in the
scalar potential comes out as required given the fact that the volume cannot
appear in the superpotential.. Once expressed in terms of $\varphi$ both terms
of (70) have the same dependence on ${\cal P}$ (or ${\cal V})$ as in (57) and
(58), and so it becomes possible to read off the warping dependence of the
coefficients ${\mathfrak{t}}$ and ${\mathfrak{g}}$ in $w_{\scriptscriptstyle
X}$, leading to ${\mathfrak{K}}^{{\overline{{\scriptscriptstyle
X}}}{\scriptscriptstyle
X}}|{\mathfrak{t}}|^{2}\propto{\mathfrak{c}}e^{-4\rho}$ and
${\mathfrak{K}}^{{\overline{{\scriptscriptstyle X}}}{\scriptscriptstyle
X}}2\hbox{Re}\,{\overline{{\mathfrak{t}}}}{\mathfrak{g}}\propto{\mathfrak{b}}{\mathfrak{c}}^{2}e^{-8\rho}$.
The freedom to rescale $X$ allows the warping dependence to be moved around
somewhat, but if this is used to ensure
${\mathfrak{K}}^{{\overline{{\scriptscriptstyle X}}}{\scriptscriptstyle X}}$
is warping free then (for real ${\mathfrak{t}}$ and ${\mathfrak{g}}$) it
implies

${\mathfrak{t}}\propto e^{-2\rho}\quad\hbox{and}\quad{\mathfrak{g}}\propto
e^{-6\rho}\,.$ (71)

As mentioned earlier, it is a good thing that both ${\mathfrak{t}}$ and
${\mathfrak{g}}$ are suppressed by warping because we have seen – c.f. eq.
(51) – that inflationary slow roll can be ensured by making the $\varphi$
derivatives of $w_{\scriptscriptstyle X}$ sufficiently small. The subtlety in
this argument is that $V$ is really a function of two fields, $\phi$ and
$\tau$, and one must check that $\tau$ evolution does not ruin the desired
inflationary behaviour.

The above discussion ignores the $|{\mathfrak{g}}|^{2}/|\phi|^{8}$ term in $V$
because this must compete with higher order terms in $w_{\scriptscriptstyle
X}$ that are already dropped in (69). It is noteworthy that some things can
nonetheless be said for smaller $|\phi|$ even if the detailed form of
$w_{\scriptscriptstyle X}$ is not known in this limit. The main observation is
that the supergravity structure of the leading $1/{\cal P}^{2}$ term of $V$
comes proportional to $|w_{\scriptscriptstyle X}|^{2}$ and so cannot decrease
without bound as $|\phi|$ decreases. The lowest it can get is zero, which
would be obtained if there exists a $\phi=\phi_{0}$ for which
$w_{\scriptscriptstyle X}(\phi_{0})=0$. If such a field exists within the
domain of validity of the 4D EFT then this point is a local minimum of the
potential; unlike the standard Coulomb interaction $V$ would reach a minimum
value beyond which the interaction becomes repulsive rather than attractive —
see figure (2). Although in the simplest scenario this would usually be
expected only to occur at scales of order the warped string scale or less –
and so be beyond the domain of 4D methods – it remains to be seen whether more
complicated examples exist for which $\phi_{0}$ can lie within the domain of
4D methods.

Figure 2: The Coulomb potential from the superpotential $w_{\scriptscriptstyle
X}(\phi)$. Since $V\propto|w_{\scriptscriptstyle X}|^{2}$ this potential
reproduces the Coulomb interaction for large values of the brane separation
$\phi$ but at smaller distances the interaction potential has a minimum and
becomes repulsive. At large field values, and due to the large amount of
warping the potential is flat enough to give rise to inflation in a natural
way. The minimum of the potential lies outside the domain of validity of the
EFT and only for large values of $\phi$ is this potential under control.

####  3.4.2 Inflation, modulus stabilization and the $\eta$ problem

The problem is that the $\tau$-dependence of $V$ does tend to ruin inflation,
at least when $\tau$ is stabilized using the superpotential. To see why we now
repeat the argument given in [24] that shows why stabilizing the volume
modulus often introduces new issues. The problem is that the mechanism used in
[24] to fix the volume modulus also induces a mass term for the would-be
inflaton field $\varphi$ that makes the slow-roll parameter $\eta$ of order
one. This is a special case of a more general problem for supergravity-based
inflationary models, called the $\eta$ problem.

#### The $\eta$ problem

The problem arises because the Kähler potential very generally depends on both
$\tau$ and $\phi$: $K=-3\ln[\tau-k(\phi,\bar{\phi})+\cdots]$, where
$k(\phi,\bar{\phi})\simeq\bar{\phi}\,\phi+\cdots$ is responsible for the
kinetic term for $\phi$. So once $\tau$ is fixed by adding a holomorphic non-
perturbative superpotential $W_{np}(T)$, the dependence of $K$ on $\phi$
introduces a potential energy that generates a mass for $\phi$ because of the
potential’s overall dependence on $e^{K}$:

$V=e^{K}\,\widehat{V}_{0}\simeq\frac{\widehat{V}_{0}}{[\tau-\bar{\phi}\,\phi+\cdots]^{3}}\simeq\frac{\widehat{V}_{0}}{\tau^{3}}\left[1+\frac{3\bar{\phi}\,\phi}{\tau}+\cdots\right]\simeq\frac{\widehat{V}_{0}}{\tau^{3}}\Bigl{[}1+\bar{\varphi}\,\varphi+\cdots\Bigr{]}\,.$
(72)

It is the superpotential terms within $\widehat{V}_{0}$ that contain the small
warp factors that allow $\widehat{V}_{0}$ to depend so weakly on $\phi$ that
inflation can be possible. The value of $\widehat{V}_{0}$ also fixes the value
of the Hubble scale whenever the universal energy density is dominated by $V$,
since then $H_{\scriptscriptstyle I}^{2}\simeq
V/M_{p}^{2}\simeq\widehat{V}_{0}/(\tau^{3}M_{p}^{2})$. But when this is so eq.
(72) shows (once the $M_{p}$ factors are reinstated) that $\phi$ inevitably
has a mass contribution that is of order
$m^{2}_{\phi}\sim\widehat{V}_{0}/(\tau^{3}M_{p}^{2})\sim
H^{2}_{\scriptscriptstyle I}$ which therefore contributes a factor of order
unity to the second slow-roll parameter
$\eta=M_{p}^{2}\,V_{\varphi\varphi}/V\simeq m_{\phi}^{2}/H_{\scriptscriptstyle
I}^{2}$. Slow roll is only achieved in the standard construction by including
a large (unwarped) $\bar{\phi}\,\phi$ contribution into $\widehat{V}_{0}$ and
tuning this to cancel against the term coming from $e^{K}$. Even though
inflation is achievable in this way, it needs a very particular fine tuning
and the Coulomb potential is essentially replaced by a tuned inflection-point
inflation [61, 62].

The problem is quite generic because the Kähler potential very generally
depends only on ${\cal P}=\tau-k+\cdots$ but because the superpotential must
be a holomorphic function it cannot depend on ${\cal P}$ and must only depend
on $T$ and $\phi$ separately. But – as already pointed out in [24] – this also
shows that it is potentially evaded if the modulus-stabilization mechanism can
arise from corrections to $K$ rather than to $W$, provided these directly
stabilize ${\cal P}$ rather than just $\tau$. The modulus-stabilization
mechanism presented here evades the $\eta$-problem in precisely this way: the
stabilizing potential naturally arises as a function of ${\cal P}$ directly.
Inflation is therefore driven by an expression like (72) regarded as a
function of $\phi$ for fixed ${\cal P}$ rather than for fixed $\tau$ (see Fig.
3).

Figure 3: A plot showing the inflationary region in which the potential is
temporarily fixed at a minimum in the modulus $\tau$ direction while the
inflaton $\varphi$ slowly rolls. At the end of inflation the two fields run to
their minimum at $\tau_{0},\varphi_{0}$ with $\tau_{0}\gg\tau_{\rm inf}$.

#### Inflating with Kähler stabilization

We now return to the combined $\tau$ and $\phi$ dynamics when $W$ is
independent of $T$ and modulus stabilization instead arises through RG
stabilization, with the potential generated using (67). There are two ways to
proceed. The first simply uses the analysis given in §3.3, which explicitly
follows the potential as a function of $\tau$ and $\phi$, specializing it to
the superpotential (69) and Kähler function (67). This leads to expressions
(49) and (50) for the slow-roll parameters, and to the condition (51) for
inflation, which notably translates into the following conditions that really
can now be satisfied by choosing sufficient warping

$\frac{4{\mathfrak{g}}}{|\phi|^{5}}\ll\frac{C|w_{0}|}{BM_{p}^{3}\tau_{+}^{3/2}}\sim\frac{\epsilon^{2}M_{p}}{\tau_{+}^{3/2}}\quad\hbox{and}\quad\frac{20{\mathfrak{g}}}{|\phi|^{6}}\ll\frac{C|w_{0}|}{BM_{p}^{4}\tau_{+}^{2}}\sim\frac{\epsilon^{2}}{\tau_{+}^{2}}\,.$
(73)

The second approach uses the effective potential $V_{\rm
eff}(\phi)=V[\phi,\tau_{+}(\phi)]$ given in (45) obtained by evaluating
$\tau=\tau_{+}(\phi)$ at its $\phi$-dependent minimum. Single-field methods
are then directly applicable. Writing (45) as

$V_{\rm eff}(\phi)\simeq\frac{{\mathfrak{D}}|w_{\scriptscriptstyle
X}(\phi)|^{4}}{|w_{0}|^{2}}\,,$ (74)

we can directly differentiate to evaluate the slow-roll parameters. For these
purposes we can neglect the $\phi$-dependence of ${\mathfrak{D}}$ because this
is inherited from the $\ln{\cal P}$-dependence of $A$, $B$ and $C$, which have
been argued to involve subdominant powers of $\alpha_{g}$. Differentiating
with respect to the canonically normalized field
$\varphi\simeq\phi/\sqrt{3{\cal P}}$ gives (for real $w_{\scriptscriptstyle
X}$) the slow-roll parameters

$\varepsilon=\frac{1}{2}\left(\frac{M_{p}V_{\varphi}}{V}\right)^{2}=8\left(\frac{M_{p}w_{{\scriptscriptstyle
X}\varphi}}{w_{\scriptscriptstyle X}}\right)^{2}\simeq\frac{8{\cal
P}}{3}\left(\frac{4M_{p}{\mathfrak{g}}}{{\mathfrak{t}}|\phi|^{5}}\right)^{2}\simeq
8\left(\frac{3}{{\cal
P}}\right)^{4}\left(\frac{4M_{p}{\mathfrak{g}}}{{\mathfrak{t}}|\varphi|^{5}}\right)^{2}\,,$
(75)

and

$\displaystyle\eta=\frac{M_{p}^{2}V_{\varphi\varphi}}{V}$
$\displaystyle\simeq$ $\displaystyle\frac{4M_{p}^{2}w_{{\scriptscriptstyle
X}\varphi\varphi}}{w_{\scriptscriptstyle
X}}+12\left(\frac{M_{p}w_{{\scriptscriptstyle
X}\varphi}}{w_{\scriptscriptstyle X}}\right)^{2}$ $\displaystyle\simeq$
$\displaystyle-4\left(\frac{3}{{\cal
P}}\right)^{2}\left(\frac{20M_{p}^{2}{\mathfrak{g}}}{{\mathfrak{t}}|\varphi|^{6}}\right)+12\left(\frac{3}{{\cal
P}}\right)^{4}\left(\frac{4M_{p}{\mathfrak{g}}}{{\mathfrak{t}}|\varphi|^{5}}\right)^{2}$

where the warping suppression enters through the factor
${\mathfrak{g}}/{\mathfrak{t}}\propto e^{-4\rho}$.

To see the need for warping it is useful to estimate the size of the factors
that enter into $\eta$ and $\varepsilon$. To this end consider the following
factor

$\eta\ni 4\left(\frac{3}{{\cal
P}}\right)^{2}\left(\frac{20M_{p}^{2}{\mathfrak{g}}}{{\mathfrak{t}}|\varphi|^{6}}\right)\simeq\frac{80{\cal
P}}{3}\left(\frac{{\mathfrak{g}}}{{\mathfrak{t}}|\phi|^{4}}\right)\left(\frac{M_{p}^{2}}{|\phi|^{2}}\right)\,.$
(77)

Estimating ${\mathfrak{t}}\sim e^{-2\rho}M_{s}^{2}$ and ${\mathfrak{g}}\sim
e^{-6\rho}M_{s}^{6}$ and taking the inter-brane separation to be no bigger
than the extra dimensions, which means

$\phi\sim M_{s}^{2}y\mathrel{\raise 1.29167pt\hbox{$<$\kern-8.50006pt\lower
4.30554pt\hbox{$\sim$}}}M^{2}_{s}/M_{\scriptscriptstyle KK}\sim M_{s}{\cal
V}^{1/6}\sim M_{s}\,{\cal P}^{1/4}\sim M_{p}\,{\cal P}^{-1/2}$ (78)

then allows the lower bound on (77) to be written

$\eta\mathrel{\raise 1.29167pt\hbox{$>$\kern-8.50006pt\lower
4.30554pt\hbox{$\sim$}}}\frac{80{\cal P}}{3}\left(\frac{e^{-4\rho}}{{\cal
P}}\right){\cal P}\,.$ (79)

Although the requirement ${\cal P}\gg 1$ precludes $\eta$ being small for
unwarped geometries [22], a slow roll is possible provided $e^{-4\rho}\ll{\cal
O}({\cal P}^{-1})\sim{\cal O}({\cal V}^{-2/3})$, which is consisent with the
lower bound $e^{-\rho}\mathrel{\raise 1.29167pt\hbox{$>$\kern-8.50006pt\lower
4.30554pt\hbox{$\sim$}}}{\cal V}^{-2/3}$ given in (53).

We see that large enough warping now leads to the predictions $\eta<0$ and
$\varepsilon\ll|\eta|\ll 1$, and this is consistent with the stabilization of
the modulus $\tau$ without the $\eta$ problem because it is actually the full
quantity ${\cal P}$ that is stabilized. Because $\varepsilon$ is
hierarchically smaller than $\eta$ it is clear that the tensor to scalar ratio
is so small that there should be no observable primordial tensor fluctuations.
If these should be observed in the next few years this inflationary scenario
would be decisively ruled out.

As inflation proceeds the value of $w_{\scriptscriptstyle X}$ decreases until
at some point the above slow-roll analysis breaks down. Then the fields roll
more quickly until they are captured by the minimum or the 4D EFT breaks down
and we become unable to predict what happens (such as by having brane and
antibrane annihilate and release prodigious amounts of energy). One scenario
would be to have $\phi$ reach the zero of $w_{\scriptscriptstyle X}$ before
the EFT breaks down (such as in Fig. 2), in which case the fields can be
trapped by the late-time modulus-stabilization solution near
$w_{\scriptscriptstyle X}\sim 0$ described in §3.2. It is not in general
guaranteed that the volume modulus must get trapped at its new minimum, since
it can also overshoot and crest the nearby local maximum, followed by a
decompactification runaway to infinity.

Although trapping by the local minimum is not guaranteed, given efficient
enough reheating the ruthless efficiency of Hubble friction often makes it
much more robust than might naively be expected (as can be seen by numerical
evolution in the presence of a thermal background in similar examples [10, 19,
11]). When such trapping occurs the value of the $\tau$ modulus stabilized in
its late time minimum $\tau=\tau_{0}$ can be exponentially larger than its
values during inflation. Since $\tau$ determines the sizes of the string and
Kaluza-Klein scales relative to the Planck scale such a change can allow the
possibility of inflation being controlled by a much larger energy scale than
is associated with low-energy supersymmetry breaking and the later universe.

Any large excursion by $\tau$ between inflation and now can easily require the
canonical field $\chi\sim M_{p}\ln\tau$ to run a distance larger than $M_{p}$.
This need not be in contradiction with the distance conjecture [26, 51] since
the Kaluza-Klein levels provide explicit realizations of the hypothesized
infinite tower of states that descend into (and so ruin) the low-energy
theory. The only control issue concerns whether the evolution can be described
purely within the low-energy 4D EFT used here. As we have checked, this is
easy to ensure during inflation because the fields roll so slowly. It is also
fairly easy to arrange for cosmological evolution at later times since the
late-time 4D EFT breaks down at scales of order $M_{p}/\tau_{0}$ [11]. Whether
a 4D description suffices in between depends somewhat on the nature of the
post-inflationary evolution that intervenes between inflation and now, and for
some choices of this its evolution might require a more comprehensive UV
treatment (such as perhaps along the lines of [54]).

###  3.5 Annihilation and the tachyon superpotential

We close with more speculative remarks about the small-$\phi$ limit. Typically
the anti D3 brane also hosts other matter fields, including Higgs-like scalar
fields ${\cal H}$. These again appear in the low-energy supergravity in a
nonsupersymmetric way, appearing in the superpotential only coupled to the
goldstino superfield $X$ such as through terms like $W({\cal H})=X|{\cal
H}|^{2}$.

Including this kind of coupling in the superpotential together with the
Coulomb interaction gives rise to a superpotential like:141414Recall that
$X{\overline{{\cal H}}}$ is chiral for a constrained superfield representing a
spinless state, allowing terms like $X|{\cal H}|^{2}$ to appear in the
superpotential.

$W=w_{0}+Xw_{\scriptscriptstyle X}\quad\hbox{with}\quad w_{\scriptscriptstyle
X}={\mathfrak{t}}-\frac{{\mathfrak{g}}}{|\Phi|^{4}}-\lambda|{\cal H}|^{2}\,.$
(80)

In this case the scalar potential is151515Notice that when two fields appear
in a potential $V\propto|w_{\scriptscriptstyle X}(\Phi,{\cal H})|^{2}$ there
is generically a flat direction ${\cal H}(\Phi)$ defined by the condition
$w_{\scriptscriptstyle X}(\Phi,{\cal H})=0$. This direction is generically
lifted by the $D$-term potential when the fields carry charge (as would the
brane-antibrane tachyon). (assuming all constants real and positive)

$V\propto\frac{|w_{\scriptscriptstyle X}|^{2}}{{\cal
P}^{2}}=\frac{\left({\mathfrak{t}}|\phi|^{4}-{\mathfrak{g}}\right)^{2}}{{\cal
P}^{2}|\phi|^{8}}+\frac{2\lambda\left({\mathfrak{t}}|\phi|^{4}-{\mathfrak{g}}\right)}{{\cal
P}^{2}|\phi|^{4}}|{\cal H}|^{2}+\frac{\lambda^{2}|{\cal H}|^{4}}{{\cal
P}^{2}}$ (81)

and so as long as the combination
${\mathfrak{t}}-{{\mathfrak{g}}}/{|\phi|^{4}}$ is positive, the field ${\cal
H}$ has a positive mass and the potential is minimized at ${\cal H}=0$. But
once $\phi$ is small enough that
${\mathfrak{t}}-{{\mathfrak{g}}}/{|\phi|^{4}}<0$ the canonically normalised
field $H\propto{\cal H}/\sqrt{{\cal P}}$ acquires a tachyonic mass

$m^{2}=\frac{2\lambda}{{\cal
P}}\left({\mathfrak{t}}-\frac{{\mathfrak{g}}}{|\phi|^{4}}\right)<0\,.$ (82)

This captures the same behaviour as would be expected within string theory for
a mode of an open string stretching between the brane and the antibrane
161616The standard discussion assumes a mass term for the tachyon proportional
to $(\phi^{2}-m^{2}){\cal H}^{2}$ which becomes tachyonic once $\phi$ reaches
the mass scale $m$. A similar expression can be obtained from (82) by
expanding $\phi=\phi_{0}-\delta\phi$ with $\delta\phi\ll\phi_{0}$ as done in
[32].. Such a state becomes lighter as the branes approach one other until at
a critical distance it becomes tachyonic (believed to herald the onset of the
brane-antibrane annihilation instability). Indeed for large separations,
$\phi$, the mass for ${\cal H}$ predicted by (82) becomes proportional to
${\mathfrak{t}}/{\cal P}\propto e^{-2\rho}/{\cal P}$, which has the same
warping and volume dependence as does the square of the warped string scale.
See Figure 4 for a plot of the potential as a function of $\phi$ and ${\cal
H}$, showing in particular a flat inflaton direction with a waterfall-style
end of inflation occuring when the ${\cal H}$ field becomes tachyonic.171717We
here consider only the simplest potential for the tachyon. Further options can
be chosen in order to match the different proposals in the literature [63].
Notice that such a tachyonic field is easily ‘integrated in’ within our
inflationary picture simply by using (80) when evaluating
$|w_{\scriptscriptstyle X}|^{4}$ in the scalar potential (74). Most of our
discussion goes through unchanged because most of our conclusions are
independent of the detailed functional dependence of $w_{\scriptscriptstyle
X}(\phi)$.

Figure 4: The 4D scalar potential as a function of brane separation $\phi$ and
the tachyon ${\cal H}$.

This simple supersymmetric 4D EFT provides a transparent toy model that
captures many features of the full string brane-antibrane annihilation picture
[63]. Since the tachyon field’s expectation value breaks the gauge symmetries
to which it couples this evolution of $\phi$ to an ${\cal H}$ waterfall
provides a dynamical description of symmetry breaking in the antibrane gauge
sector.181818The current intuition about the end-point of brane-antibrane
annihilation within the full string theory is that no perturbative states
remain after tachyon condensation. A puzzle arises because the tachyon field
only breaks one combination of the two $U(1)$ gauge symmetries that live on
the two branes. It has been conjectured that the second $U(1)$ survives, but
in a confining phase that is not manifested perturbatively [64, 65, 66, 67].
It is tantalising to propose that this late-time behaviour is instead governed
by a brane-antibrane bound state – similar to branonium [68] but corresponding
to a nontrivial zero of $w_{\scriptscriptstyle X}$ – rather than continuing to
the singularity at $\phi=0$. Depending on the scales chosen one might build
models for which ${\cal H}$ breaks a Grand Unified symmetry at very high
scale, or perhaps break the symmetry group of the standard model itself. As
noted in [22] the tachyon plays other key roles in the physics of the brane-
antibrane inflation (besides providing its waterfall finish) such as by giving
rise to topological defects such as cosmic strings [69, 70]. For a discussion
on reheating after brane inflation see [71].

##  4 Conclusions

In this paper we explore the consequences of the RG modulus-stabilization
mechanism of [10, 11] to string vacua, showing that it can generate both de
Sitter and non-supersymmetric anti-de Sitter solutions. For the de Sitter
vacua no particular uplifting mechanisms is required (unlike for KKLT or LVS
stabilization191919It is worth emphasising that even though we introduced anti
D3 branes to address inflation, we do not need them to get de Sitter.
Furthermore, unlike KKLT, the contribution of the brane tension is only the
constant contribution of a field dependent $w_{\scriptscriptstyle X}$ which
essentially vanishes at the overall minimum.). We further explore its
implications for relatively simple inflationary scenarios, for which we find
it can evade other commonly encountered problems (such as the $\eta$ problem).

Because the RG mechanism is at heart a perturbative process, it must confront
the generic no-go arguments whose roots lie within the old Dine-Seiberg
problem [1] (and have been recently revived in the context of swampland
conjectures [26, 51]). We argue:

  * •

The Dine-Seiberg issue relies on the scalar potential’s runaway behaviour in
the weak-coupling limit, and this runaway itself can be interpreted in terms
of approximate symmetries of the EFT that are inherited from symmetries of the
underlying 10D supergravity (and string theory) underlying the robustness of
the problem.

  * •

The behaviour of the potential and of evolution in its presence can be
addressed within the runaway region using semiclassical expansions without
losing calculational control. We argue that the control issues encountered –
such as domain of validity of EFT methods in time-dependent situations for
both perturbative and nonperturbative physics – arise more generally in, and
do not undermine conclusions for, other areas of physics and that there is no
evidence that string theory effective actions behave differently.

  * •

Although the Dine-Seiberg argument is true generically – there are doubtless
many solutions in regions of strong coupling ($g_{s}\simeq{\cal O}(1)$) and
small volume ($\tau\simeq{\cal O}(1)$) – it can be evaded in specific
situations. The RG approach exploits mild hiearchies of coefficients in
perturbative expansions of the action (which allows minima to arise with small
expansion parameter, $\alpha\ll 1$) together with RG methods that both
identify a dependence on $\ln\tau$ and allow controlled resummations that work
to all orders in $\alpha\ln\tau$. As a bonus solutions naturally arise at very
large values $\ln\tau\simeq 1/\alpha$.

Although the RG scenario provides alternatives to the standard KKLT and LVS in
IIB string theory, there are plenty of moduli to go around and specific
solutions might exploit several of these mechanisms at the same time. For
instance, our mechanism may be used to stabilise the overall volume modulus
whereas the other Kähler moduli may be fixed by non-perturbative effects as in
KKLT or LVS. Furthermore, the ${\cal O}(1/10)$ hierarchy required in the RG
approach to obtain exponentially large volumes might itself be obtained if
these parameters are functions of other fields, such as the complex structure
moduli, fixed in other ways.

There is also nothing intrinsically IIB about the RG mechanism, which can in
principle also be put to work in other types of string vacua, such as
heterotic models, for in which modulus stabilization has proven to be more
challenging (see for instance [72]) than for IIB. We leave it as an open
question to explore this scenario through more explicit constructions, both in