Advances in High Energy Physics
Volume 2016 (2016), Article ID 4303752, 17 pages
http://dx.doi.org/10.1155/2016/4303752

Review Article

Aspects of Moduli Stabilization in Type IIB String Theory

Shaaban Khalil, Ahmad Moursy, and Ali Nassar

Center for Fundamental Physics, Zewail City of Science and Technology, Giza 12588, Egypt

Received 1 October 2015; Revised 28 December 2015; Accepted 30 December 2015

Academic Editor: Elias C. Vagenas

Copyright © 2016 Shaaban Khalil et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

We review moduli stabilization in type IIB string theory compactification with fluxes. We focus on KKLT and Large Volume Scenario (LVS). We show that the predicted soft SUSY breaking terms in KKLT model are not phenomenological viable. In LVS, the following result for scalar mass, gaugino mass, and trilinear term is obtained: , which may account for Higgs mass limit if  TeV. However, in this case, the relic abundance of the lightest neutralino cannot be consistent with the measured limits. We also study the cosmological consequences of moduli stabilization in both models. In particular, the associated inflation models such as racetrack inflation and Kähler inflation are analyzed. Finally, the problem of moduli destabilization and the effect of string moduli backreaction on the inflation models are discussed.

1. Introduction

Ever since the invention of the Kaluza-Klein mechanism, it was realized that extradimensional models are plagued with massless scalar fields when compactified to 4D. In the original Kaluza-Klein construction, the radius of the -circle, is not fixed by the dynamics and appears as a scalar field with no potential in the effective 4D theory. This is a generic feature of most compactifications of higher-dimensional gravitational/Yang-Mills theories. The parameters which describe the shape and size of the compactification manifold give rise to massless scalar moduli with no potential at tree level in 4D (flat directions), that is, nonstabilized moduli fields. These moduli are gravitationally coupled and as such their existence would be in conflict with experiment (see [1] for a review).

String theory being a candidate for a unified theory of the forces of nature must be able to reproduce the physics of our real world which is 4-dimensional with a small positive cosmological constant and chiral gauge interactions. This amounts to finding de Sitter vacua of string theory with all moduli stabilized. The seminal work in [2] constructed the first models of Grand Unified Theories (GUTs) from Calabi-Yau compactification of the heterotic string. In these models, the 10D gauge group ( or ) is broken to Standard-Model-like gauge groups by turning on background gauge fields on the internal space. The chiral fermions are obtained from the dimensional reduction of the 10D gaugino and the number of the generations is half the Euler characteristic of the internal space [2]. These compactifications are purely geometric and give rise to an supersymmetric (SUSY) 4D theories with large number of moduli. Other compactifications which leads to SUSY in 4D are type II theories on Calabi-Yau orientifolds with D-branes and fluxes, M-theory on manifolds with holonomy, and F-theory on Calabi-Yau fourfold (see [3] for a review and references therein).

One essential fact about our universe is that it has a small positive cosmological constant, that is, a de Sitter space-time. The moduli scalar potential should be such that its minimum produces the observed value of the cosmological constant. This turns out to be a very difficult problem since de Sitter vacua are known to break supersymmetry. The first attempt at finding realistic vacua of string theory was mainly concerned with Minkowski and Anti-de Sitter vacua [2, 4–6]. It was shown that turning on magnetic fluxes in the internal manifold leads to a nontrivial warp factor and a non-Kähler geometry. The situation improved drastically since the introduction of D-branes as nonperturbative objects in string theory. This resulted in the celebrated KKLT scenario in which a moduli fixing mechanism was introduced [7]. The Large Volume Scenario followed after [8]. Therein, one can turn on a vacuum expectation value for the fluxes in the internal space without breaking the 4D Lorentz invariance.

In this paper, we review the flux compactifications and moduli stabilization in type IIB string theory. The Kachru-Kallosh-Linde-Trivedi (KKLT) [7] scenario was a major step in constructing dS vacua with all the moduli stabilized including the volume modulus. The Large Volume Scenario (LVS) by Quevedo et al. [9, 10] was proposed as an alternative to stabilize all the moduli with the volume moduli stabilized at extremely large volume. Realistic models of moduli stabilization must come as close as possible to the observed phenomenology at low energy and also to account for cosmological inflation at high energy scales. From this point of view, we analyze the low energy phenomenology of both KKLT and LVS. We emphasize that these models provide specific set of soft SUSY breaking terms, which are not phenomenologically viable. In addition, we study the impact of these scenarios on inflation. It turns out that inflation could destabilize the moduli again. This problem has been analyzed in details in [11, 12]. On the other hand, moduli stabilization may have backreaction effects on the inflationary potential, which could change the inflationary parameters.

This paper is organized as follows. In Section 2, we review the flux compactification and moduli stabilization in type IIB string theory. Section 3 is devoted to KKLT model and its variants, where we discussed several examples for vacuum uplifting. In Section 4, we present the LVS as an alternative scenario that overcomes some of the KKLT drawbacks. The phenomenological implications and SUSY breaking soft terms of these two models are studied in Section 5. In Section 6, we highlight the cosmological implications of these two models. Finally, we state our conclusions in Section 7.

2. Flux Compactifications and Moduli Stabilization in String Theory

We start by reviewing the heterotic string compactification which has been considered to connect string theory to four-dimensional physics [2]. Then, we discuss type IIB flux compactifications, where the complex structure moduli (CSM) and the dilaton are stabilized by the RR and NS-NS 3-form fluxes.

2.1. Heterotic String Compactification

To compactify string theory down to 4D, one looks for vacuum solutions of the form , where is assumed to have 4D Poincaré invariance and (or simply ) is a compact internal 6D Euclidean space. The most general metric compatible with these requirements can be written as [2]

where are the coordinates on with the metric and the requirement of Poincaré symmetry of still allows for a warp factor which depends on only.

We also require an supersymmetry in 4D since this is phenomenologically appealing and gives more analytic control. For , one considers homogenous and isotropic maximally symmetric solutions which implies that the Riemann tensor takes the form

where is fixed by contracting both sides with and turns out to be equal to . The constant scalar curvature could be (Minkowski), (AdS), or (dS).

When the radius of curvature of is large compared to the Planck scale, one can use the supergravity approximation of string theory. In order to have a supersymmetric background, the supergravity transformations of the fermions must vanish [2]

For heterotic string theory, the supergravity variations of the fermions are given by [2]

where , and are the gravitino, dilatino, and the gaugino, respectively. The -flux is and ; and is Yang-Mills field strength of the or gauge fields in 10D.

The Bianchi identity of is given by

Since is exact, that is, zero in cohomology, then the cohomology classes of and are the same.

The vanishing of the above variations for a given spinor will put some restrictions on the background fields and in particular on the geometry and topology of . The compactification of heterotic string theory with vanishing -flux (or vanishing torsion) was first studied in [2, 5] and it leads to the following conditions on the string background:

That is, the external space is Minkowski with a constant warp factor.

The dilatino variation gives

That is, the dilaton is constant over .

The gravitino variation gives

This equation says that admits a covariantly constant spinor. The integrability condition resulting from the above equation implies that is Ricci flat

Hence, the first Chern class of vanishes

It was conjectured by Calabi and proved by Yau that Ricci-flat compact Kähler manifolds with admit a metric with holonomy. These metrics come in families and are parameterized by continuous parameters which defines the shape and sizes of . The parameters appear as scalar fields (moduli) in 4D with no potential and a major goal in string theory is to generate a potential which stabilizes these moduli in a way that is consistent with observations.

One can describe the 4D models resulting from the heterotic string compactification in terms of an effective SUSY theory. This theory is characterized by a Kähler potential , a gauge kinetic function , and a superpotential . The tree-level superpotential does not fix the moduli. Due to nonrenormalization theorems, is not renormalized at any order in perturbation theory [14, 15]. This means that if supersymmetry is unbroken at tree level, it will remain unbroken at all orders of perturbation theory. Nonperturbative effects such as gaugino condensation [16] can correct the superpotential and fix some of the moduli.

It is worth mentioning that there is an alternative way to stabilize the moduli if the string model is nonsupersymmetric. In this case, perturbative corrections can generate a one-loop potential for the moduli, for example, a vacuum energy. For example, in the nonsupersymmetric tachyon-free heterotic string models studied in [17–19], the one-loop vacuum energy was shown to be finite and extremized at the symmetry-enhancement points in the moduli space. The one-loop potential can stabilize some of the moduli fields. Another way to generate masses for the moduli is by the breaking of supersymmetry through a Scherk-Schwartz mechanism [20].

2.2. Type IIB Compactification

We now turn our discussion to type IIB string theory. The massless bosonic spectrum of type IIB consists of the metric , RR 0-form , and scalar dilaton which are combined into the axiodilaton , where the string coupling is given by . In addition, the spectrum contains RR 2-form and 4-form , as well as the NS 2-form . It is convenient to combine the RR and NS 3-forms and into . The classical action of type IIB supergravity is divided into a bulk action , the Chern-Simons action , and contributions from the D-brane sources [21] (more precisely, represents the action of the localized sources for the case of a D-brane wrapping a ()-cycle )

In the string frame, , , and are given by

where and are, respectively, the tension and charge of the -brane. The string tension is expressed in terms of string length as

The 5-form is defined as

which is self dual and satisfies Bianchi identity

One would like to consider warped compactifications of type IIB on a compact manifold . The metric ansatz for a 4D warped compactification is given by [22]

The 10D Einstein equation of motion is

where

is the total stress tensor of supergravity plus the localized objects; that is

In this regard, the space-time components of the latter action reduce to

where is a function on the compact space. Integrating both sides of this equation over the compact manifold , the left-hand side gives zero since it is a total derivative. If there are no localized sources, then the right-hand side is a sum of positive terms and vanishes only if and are constants and . This is the familiar no-go theorem of flux compactifications [23, 24]. However, the existence of localized sources in string theory like orientifold planes can balance the contribution coming from fluxes to give a nontrivial warp factor. This was realized in string theory in [22]. The setup in [22] allows for a stabilization of complex structure moduli by turning on RR and NS fluxes in the internal space [25, 26].

2.3. Type IIB Fluxes and Moduli Stabilization

One way to see the problem in a simple setting is nicely reviewed in [25, 27] in a toy model and we review it here. One can generate a potential for the moduli by turning on fluxes in the internal space. The potential in 4D results from the Maxwell term of the fluxes

where is the metric of the internal space. The metric will depend on the moduli of the internal space and, after doing the integral on , one gets a potential for the moduli . For example, consider a 6D Maxwell-Einstein theory compactified on a two-sphere with a nonzero flux of piercing

This flux contributes a positive energy to the effective 4D potential which can then balance the negative contribution coming from the curvature of . More specifically, the contribution to the effective potential coming from the flux originates from the Maxwell term in 6D

where the determinant of the metric contributes a factor of and two metric contractions contribute a factor of while the transformation to the Einstein frame gives a factor . Therefore, total 4D potential takes the form

which is minimized at , and if is large the curvature is small and the supergravity approximation is reliable [25, 27]. In string theory, additional ingredients beside the fluxes are needed to construct stable vacua. These ingredients are the D-branes and orientifold planes [28].

The main idea of flux compactification is that there are solutions of the string tree-level equations in which some of the -form fields are nonzero in the vacuum. In these constructions, one needs to make sure that the backreaction of the flux on the geometry does not take us outside the supergravity approximation. This turns out to be possible [5] with the introduction of a warp factor varying over the internal manifold and hence the new geometry is conformal to the nonflux case. The fluxes, which can be turned on, are the RR fluxes of type II and the flux. In this case, the quantization condition on the fluxes is

where the integrality of the cohomology classes of is due to Dirac’s charge quantization. The nonvanishing of the cohomology classes of these -form fields leads to obstructions which lifts some of the flat directions of the compactification; that is, it leads to potential which freezes some of the moduli.

In the presence of sources, the modified Bianchi identity now reads [28]

Integrating this equation over the compact internal manifold, one gets the tadpole cancelation condition [28]

where

The type IIB string theory will be compactified on Calabi-Yau orientifolds in order to obtain a 4D model. It turned out that one needs to make an orientifold projection in order to have supersymmetric compactification [28]. The orientifold action projects out one of the two gravitinos and breaks the SUSY down to . The orientifold projection also introduces O-planes with a background charge and a negative energy density which balances the contribution of D-branes and leads to stable compactification. The RR and NS-NS 3-form fluxes are restricted via the integral cohomology which determines the quantized background fluxes as follows:

where are 3 cycles of the Calabi-Yau manifold. In this case, tadpole condition (27) reads

where is the net number of () branes filling the noncompact dimensions and is the Euler characteristic of the elliptically fibred Calabi-Yau fourfold .

The compactification to 4D on a Calabi-Yau manifold with orientifold planes will give rise to an supergravity theory which is characterized by a Kähler potential , a superpotential , and a gauge kinetic function [29–31]. The tree-level Kähler potential is given by the Weil-Petersson metric using the Kaluza-Klein reduction of type IIB supergravity

Here, represents the volume modulus and is one of the Kähler moduli. The conditions on the fluxes in type IIB are

Since the hodge depends on the metric, then one expects the above conditions can fix the geometric moduli except for the overall scale of the metric since the hodge is conformally invariant in six dimensions. This leaves the overall volume of the compactification manifold undetermined. These conditions can be derived from a superpotential given by the Gukov-Vafa-Witten (GVW) form [32]

This superpotential depends on the complex structure moduli through and is independent of the Kähler moduli.

The supergravity scalar potential is given by

where run over all the moduli. Due to the no-scale structure of the Kähler potential (31), the sum over Kähler moduli cancels the term and the potential (34) reduces to the no-scale structure [33–35]

where run over the dilaton and complex structure moduli. Accordingly, the dilaton and complex structure moduli can be stabilized in a supersymmetric minimum by solving the equation which may have a solution for generic choice of the flux. In this case, at the minimum.

The above discussion shows that the no-scale structure does not fix the value of the volume modulus ; that is, the modulus is a flat direction. The stabilization of is of uttermost importance in order for string theory to make contact with realistic models. This issue will be addressed in the upcoming sections.

3. KKLT and Its Variants

In order to stabilize the volume modulus , a nonperturbative superpotential was considered by Kachru, Kallosh, Linde, and Trivedi (KKLT) [7]. The source of these nonperturbative terms could be either D3-brane instantons or gaugino condensation from the nonabelian gauge sector on the D7-branes. As advocated in the previous section, the dilaton and the complex structure moduli are stabilized at a high scale by the flux induced superpotential and hence their contribution to the superpotential is a constant . Thus, the total effective superpotential is given by

The coefficient or is the correction arising from instantons or gaugino condensation, and is constant of order . In addition, the Kähler potential is given by

Here, , where is the volume modulus of the internal manifold and is the axionic part. A supersymmetric minimum is obtained by solving the equation

where is the value of that minimizes the scalar potential.

Substituting this solution in the potential

one finds the following negative minimum:

The scalar potential as a function of is given by (the imaginary part is frozen at zero)

It is important to uplift this AdS minimum to a Minkowski or a dS minimum, as shown in Figure 1, in order to have realistic models. The uplift of the above AdS vacuum to a dS one will break SUSY where one needs another contribution to the potential which usually has dependence like [36]

In this case, a new minimum is obtained due to shifting to , where is given by

Since the consistency of the KKLT requires that [7], the shift in the minima is much small and we can calculate physical quantities such as masses in terms of .

Figure 1: The scalar potential (multiplied by ) with , , and . The blue curve shows the AdS minimum, while the green and red curves exhibit the uplifting to dS minimum via , , respectively, with and .

There are many proposals for such uplifting, for example, adding anti-D3-branes [7], D-term uplift [37–43], F-term uplift [13, 44], and Kähler uplift [41, 45–47]. In the original KKLT scenario [7], some anti-D3-branes were added which contributes along with additional part to the scalar potential

One of the drawbacks of this mechanism is that SUSY is broken explicitly due to the addition of the anti-D3-branes. In this case, the effective 4D theory cannot be recast into the standard form of 4D supergravity and this in turn makes it very difficult to have a low energy effective theory [37]. An uplifting mechanism via a D-term scalar potential was proposed in [37] where the possible fluxes of gauge fields living on the D7-branes were used. In this case, the fluxes induce a term in the 4D effective action of the form

where is the 4 cycles on which D7-branes wrap, is the tension of D7-branes, and measures the strength of the flux. Accordingly, the D-term scalar potential is given by

where are charged matter fields with charges . These fields can be minimized at and accordingly the full potential will be

with and we end up with similar behavior to the KKLT uplifted potential.

Another approach for uplifting uses the F-term uplifting [13, 44]. In this case, SUSY will be broken spontaneously in the F-term moduli sector which in turn generates an uplift term for the AdS KKLT stabilized volume. For example, in [13], the Kähler potential contains a modulus and a matter field and has the form

The effective superpotential [13] is given by

where and are functions of the matter fields resulting due to integrating out heavy fields with index that runs over the gaugino condensates. The scalar potential will be minimized at [13]

Figure 2 shows the shape of the potential near the minimum , . This model is different from the approach studied in [48], where the effects of charged chiral fields that reside on D3-branes and D7-branes [49] were considered. In that case, new -dependence will be generated in the Kähler potential

where , are charged chiral matter fields. In addition, the superpotential does not contain any nonperturbative effects

Therefore, an AdS minimum is obtained which is uplifted by a D-term associated with the gauge symmetry of the matter fields.

(a)

(b)

Figure 2: The minimum of the potential of the scenario [13] at , . (a) corresponds to rescaled by and (b) corresponds to rescaled by .

Another way for uplifting to dS vacua of the volume stabilized moduli is by Kähler uplift models where the perturbative corrections to the Kähler potential will paly an important role in constructing dS vacua [41, 45–47]. Now, we consider a model proposed in [46] where and so that the Euler characteristic is . In this case, the Kähler potential and superpotential are

where is the normalized volume, with and

Accordingly, the dependence of supergravity scalar potential is given by [46, 50, 51]

where stabilizes at for . Using the approximations and and defining the quantities

the scalar potential will be simplified to the form

In order to have stable dS vacuum, must satisfy the constraint [46]

This is clarified in Figure 3, where, for values of , we have AdS minimum and, for values of , the volume is destabilized.

Figure 3: Potential of one modulus model for the Kähler uplift. The red curves correspond to the limits on the parameter for a dS vacuum, while the dotted one corresponds to in the destabilization region.

4. Moduli Stabilization in Large Volume Scenario

Another alternative scenario for moduli stabilization based on a Large Volume Scenario has been proposed by Quevedo et al. [9, 10] in order to overcome some of the drawbacks of the KKLT model. Basically, increasing the number of Kähler moduli will worsen the situation when corrections are neglected. The LVS was built on the proposal that the number of complex structure moduli is bigger than the number of Kähler moduli; that is, , as well as the inclusion of corrections. contribution to the Kähler potential (after integrating out the dilaton and the complex structure moduli) is given by [50–52]

with and is the type IIB dilaton. Here, is defined as the classical volume of the manifold which is given by

where is the Kähler class and are the moduli that measure the areas of 2 cycles with . The complexified Kähler moduli are defined as , where are the four-cycle moduli defined by the relation

Since the superpotential is not renormalized at any order in perturbation theory, it will not receive corrections. But there is a possibility of nonperturbative corrections, which may depend on the Kähler moduli (as in the KKLT model) via D3-brane instantons or gaugino condensation from wrapped D7-branes. Accordingly, the superpotential is given by

where are model dependent constants and again is the value of the superpotential due to the geometric flux after stabilizing the dilaton and the complex structure moduli. In this respect, the scalar potential will take the form [50, 51]

where correction is encoded in . The simplest example that can realize the notion of LVS [9, 10] is the orientifold of for which and and therefore the volume is given by

where the volume moduli are and and the link to is given by and . In this respect, the Kähler potential and the superpotential, after fixing the dilaton and complex structure moduli, are given in the string frame by

In the large volume limit, , the behavior of the scalar potential is given by [9, 10]

Minimizing the potential (66), one finds the following:

and solving the two equations in the two variables, , one can get one equation in

where

Using the assumption , which is necessary to neglect higher order instanton corrections [9], the solution is given by

Substituting for by their expressions, we have

Therefore, the potential possesses an AdS minimum at exponentially large volume since it approaches zero from below in the limit and . Namely, in the latter limit, the potential has the form

Therefore, the negativity of the potential requires a very large . Still one has to uplift this minimum by one of the mechanisms mentioned in Section 3. This result can be generalized to more than two moduli where one of them takes a large limit while other moduli stay small. This structure will form what is called the Swiss-cheese form of the CY manifold, as depicted in Figure 4. In this case, the volume will take the form

Figure 4: Swiss-cheese structure in Large Volume Scenario.

5. SUSY Breaking and Phenomenological Consequences

In this section, we will study SUSY breaking in the moduli sector and the properties of the corresponding soft terms (more general expressions of soft term for generic superpotentials and Kähler potentials in supergravity and string models are given in [53, 54].) that are induced in the observable sector, and then we summarize the phenomenological consequences. SUSY breaking in models of KKLT compactification type with phenomenological consequences has been extensively studied in [36, 40, 55–59], while, for LVS type, SUSY breaking was studied in [10, 60–65]. We would like to remark that in studying MSSM-like theories arising from large volume compactifications one faces the issue of the validity of perturbation theory. In large volume models, one has to worry about the Kaluza-Klein towers which become light when the compactification volume is large. If some of these KK modes are charged under the gauge symmetry, they can push the gauge and Yukawa couplings to large values and the perturbative control is lost.

(i) Soft Terms in KKLT Scenario. As shown in models of KKLT type in Section 3, SUSY is broken by one of the uplifting mechanisms. The gravitino mass at the dS minimum is given by

Therefore, we have gravitino mass of order TeV if and hence . In terms of the shifts (given in (43)) of , an approximate expression for near the dS minimum is given by

which is of the same order as the gravitino mass. Accordingly, the soft SUSY breaking terms are given by

where the gauge kinetic function can be chosen such that , which will lead to universal gaugino masses. The gauge kinetic function is considered to have a linear dependance on the modulus field which can be derived from the reduction of Dirac-Born-Infeld action for an unmagnetized brane [62, 63].

The soft terms indicate that SUSY breaking in KKLT is a special example of the constrained MSSM (CMSSM), where all the soft terms are given in terms of one free parameter () which is of order TeV. Note that is fixed as . It is well known that this type of soft terms cannot account for the experimental constraints imposed by the Large Hadron Collider (LHC) and relic abundance of the lightest SUSY particle. Even if we relax the Dark matter constraints, the Higgs mass limit (~125 GeV) and gluino mass limit (1.4 TeV) will imply TeV. Thus, all SUSY spectrum will be quite heavy which is beyond the LHC sensitivity. A feature of this model is the fact that gauginos are lighter than the sfermions by at least one order of magnitude. However, if one checks the parameter space for such set of soft terms, he finds that tadpole equations at the TeV scale are not satisfied. Namely, condition [66]

cannot account for positive . Therefore, the above set of soft terms failed to describe TeV scale phenomenology.

(ii) Soft Terms in LVS Scenario. Let us now consider the same scenario for SUSY breaking in LVS model, namely, with geometry. In this case, we have

Therefore, the gravitino mass will be given by

It is remarkable that depending on the large volume , the gravitino mass could be TeV or much larger. Considering the Kähler metric of the observable sector to have the form , with being constant, we have scalar soft masses of the following form:

where we have neglected the very tiny cosmological constant value . In the large volume limit, , the Kähler metric and its inverse are given by

The F-terms can be calculated from

Hence, the approximate dependence of the F-terms on the volume is given by

Consider a linear dependence of the gauge kinetic function on the Kähler moduli [62, 63]; therefore, the gaugino masses are given by

Also, the universal A-term is given by

Accordingly, at large volumes , the soft masses are related to the gravitino mass as follows:

These soft terms are generated at string scale; therefore, we have to run them to the electroweak (EW) scale and impose the EW symmetry breaking conditions to analyze the corresponding spectrum. In this case, condition (77) reads

which can be satisfied easily at the TeV scale. The above set of soft terms is special case of CMSSM. Therefore, the Higgs mass limit requires to be of order 1.5 TeV, as shown in Figure 5, where the Higgs mass is plotted versus the gravitino mass . With such heavy values of and , the SUSY spectrum will be quite heavy and the relic abundance of the lightest neutralino is not consistent with Planck’s results [67].

Figure 5: Higgs mass as a function of the gravitino mass . The region in left to the green line is disallowed by the gluino mass constraint, while the area between the red lines shows the region for which the Higgs mass lies in the range of  GeV.

6. Cosmological Consequences

One of the important consequences of the moduli stabilization is its effect on inflation. It turned out that the scalar potential of the modulus field in KKLT is not suitable to account for single-field inflation since the modulus potential is not flat enough to allow for slow roll [68, 69]. Namely, the real part is not protected by a shift symmetry which will result in an -problem. Although the axionic part is subjected to shift symmetry, the field dependent mass matrix is not diagonal with a significant mixing. Therefore, the shift symmetry is violated and the -problem persists (for a review for string inflation in the light of recent observations, see, e.g., [70]).

In this section, we discuss how this problem can be evaded in what is called the racetrack superpotential. Also, inflation via Kähler moduli is discussed. In addition, we explain the destabilization problem that arises in KKLT and LVS. Finally, we study the effect of moduli backreaction on inflationary scenarios.

6.1. Racetrack Inflationary Model

In the racetrack model, the superpotential is given by [68, 69]

where the additional nonperturbative term may be obtained from gaugino condensation in a theory with a product of gauge groups such as gauge groups; hence, and . This is known as the racetrack model. In this case, the global SUSY minimum is located at

The inflationary potential is given by the F-term scalar potential added to the uplifting term as follows:

Figure 6 shows the scalar potential with two degenerate minima and one saddle point located at . At this saddle point, the potential has a minimum in the direction and a maximum in the direction. The inflaton is considered to be rolling slowly from initial conditions near the saddle point on the inflationary trajectory. Namely, the initial motion is in direction and the inflationary path is determined numerically [68, 69].

(a)

(b)

Figure 6: (a) corresponds to the racetrack scalar potential and (b) corresponds to the variation of the potential in the neighborhood of the saddle point (all are multiplied by ). The parameter values are , , , , , and .

The racetrack inflation model predicts spectral index , inflation scale GeV, and tiny tensor to scalar ratio .

6.2. Kähler Moduli Inflation in the LVS

The idea of Kähler moduli inflation [71] is to produce an inflationary potential similar to the form

where and are positive constants.

In the case of multimoduli Calabi-Yau geometries, the Calabi-Yau volume can take the form

where is responsible for the large volume and , where are the blow-ups. Here, and are model dependent positive constants. In this case, the Kähler potential is given by

while the superpotential is given by

where . The minimum of the scalar potential exists at the large volume limit and the scalar potential has the form [71]

The inflation can occur away from the minimum and the inflaton is considered to be one of the small moduli . The volume and the small moduli, other than the inflaton, are guaranteed to be stabilized to their minima during the inflation [71]. Therefore, the inflationary potential is given by [71]

with which is constant during the inflation. This model predicts and for the number of -folding with inflationary scale  GeV.

6.3. Destabilization Problem

In the KKLT scenario, the modulus mass is given by [36]

Therefore, it is linked to by the relation

which is approximately two orders of magnitude greater than the TeV scale gravitino mass. In case there is an inflaton field different from modulus field with mass ~10¹³ GeV, the modulus may perturb GUT scale inflation. It turns out that a problem arises due to the conflict between the requirement of a high energy scale (GUT scale) inflationary phenomenon and low energy (TeV scale) SUSY phenomenology in the context of KKLT. The conflict is stemming from the constraint [11, 12]

where is the Hubble parameter. This constraint originates from the fact that the barrier from the runaway direction, as exhibited in Figure 7, is a bit less than the magnitude of the AdS minimum since the uplift effect is negligible for large . Hence, we have

Therefore, recovering the Planck mass, we have

If we included an inflaton, there will be a contribution to the overall inflationary potential by a term of the form due to the fact that the modulus couples to all sources of energy. Accordingly, the total inflationary potential at the KKLT dS minimum is given by

In this case, there will be a competition between the runaway dependence and the barrier . Therefore, protecting the volume modulus from destabilization will impose the condition

Figure 7: The purple curve shows the dS vacuum, while the green one displays the destabilization of the modulus due to large contribution of vacuum energy density.

On the other hand, in the LVS case, there is an extra suppression by large volume [72]

and correspondingly we have the constraint

which will not improve the situation much. In this regard, TeV scale gravitino mass which is favored by low energy phenomenology implies disfavored nontraditional low scale inflation. The property of seems to be a common property between many inflationary models in stabilization scenarios in string theories [72–74]. This problem is sometimes called the Kallosh-Linde problem. The latter originates from the fact that compatible large scale models of inflation require a very large gravitino mass, where

The KL Model and Strong Moduli Stabilization. The problem of destabilization can be evaded in models of strong moduli stabilization like the KL model [11, 12]. In this scenario, the volume is still determined by one modulus , but there is an additional nonperturbative term contributing to the racetrack superpotential

In this respect, the imaginary part stabilizes at the origin again, whereas the potential of the real part is given by

This potential possesses two minima; one is a metastable supersymmetric Minkowski vacuum at

and the other is a deeper AdS one as shown in Figure 8. The KL modulus squared mass is given by [36]

Therefore, for , , and , we find  GeV which means that the volume modulus mass is much larger than the inflaton mass ( GeV) and accordingly will be frozen quickly during the inflation without perturbing the inflaton dynamics. Clearly, this is much featured than the KKLT case since the gravitino mass vanishes at the SUSY minimum. Therefore, the hight of the barrier from the runaway direction is independent of the gravitino mass; hence, the Hubble parameter is also independent of the gravitino mass. In this case, KL model can account for high scale inflation.

Figure 8: The F-term scalar potential (multiplied by ) for KL model possesses both metastable Minkowski vacuum which is supersymmetric and another AdS minimum, for , , and .

In order to obtain interesting phenomenology at the TeV scale for SUSY breaking without conflict with the high scale inflation requirements, a constant shift is added to the superpotential [36] and it is supposed to be of the order of the weak scale. In this case, the value of minimizing the potential will be shifted to . The SUSY vacuum is obtained by solving the equation of motion

which implies

Correspondingly, the AdS minimum is independent of the sign of and is approximated to

where the above expressions are obtained using , , and . The AdS vacuum can be uplifted to a dS one by mechanisms similar to those in the KKLT model and hence SUSY will be broken with the gravitino becoming massive. In this respect, the gravitino mass is given in terms of as follows:

It is worth mentioning that the uplifting effect here is so small and cannot exceed the barrier between the dS and the AdS vacua which is one of the strengths of this model. On the other hand, a weakness of the approach is the lack of interpretation of the origin of the scale of the shift which should be around , if we are seeking TeV scale gravitino mass.

6.4. Impact of String Moduli Backreaction

Here, we give a brief overview of the effect of string moduli backreaction on the inflationary models and its link to SUSY breaking. To study such effects on the inflation, models of strong moduli stabilization are favored such as the KL model. In the latter model, the moduli are heavy; in particular, the modulus mass is larger than the Hubble parameter. The moduli backreaction effect on the inflation and its link to SUSY breaking were studied in many research papers [75–79]. For a single modulus field case, the impact of the stabilized volume modulus field on large and small field inflation was studied in [75]. In this respect, the total Kähler potential and superpotential are given by

where are chiral superfields related to the inflationary scenario and the constant for heterotic dilaton and for type IIB Kähler modulus. Accordingly, the F-term scalar potential can be written as

where

In this case, the potential is minimized at , due to the effect of the inflationary large positive energy density, where corresponds to the SUSY Minkowski minimum, if the scenario contains moduli only. Since the modulus is very heavy, it stabilizes quickly to its minimum and the inflationary potential gets corrections after setting to its minimum as follows [75]:

Now, consider a case of small field inflation such as the F-term Hybrid inflation [80, 81], where the Kähler potential and superpotential are given by

where is the inflaton and , correspond to the waterfall fields. The hybrid inflation scenario supposes that, for values of , the potential is minimized at and, therefore, the inflationary superpotential is effectively given by

Accordingly, the corrected inflationary potential due to the moduli backreaction will contain a linear term in the inflaton [75]

where is the one-loop correction to the inflation potential and represents the contribution due to supergravity, which has small effect and can be neglected. Here, is the rescaled vacuum energy during inflation given by , with . In this case, the value of the spectral index can be improved; , compared to its value in the pure hybrid scenarios; . So, we do not need to go to nonminimal Kähler scenario [80, 81].

Another example is the large field inflation which was successfully embedded in supergravity. In this case, a shift symmetry is used to avoid the -problem as well as considering stabilizer field [82–87]. In this regard, consider a form of the Kähler potential and superpotential given by

where the quartic term in the Kähler potential causes stabilizer to get mass much larger than the Hubble parameter and so stabilizes quickly to the origin. The Kähler potential is independent of the real part , due to the shift symmetry and, hence, it will correspond to the inflaton. The field and the imaginary part of will be stabilized at zero and hence the inflationary potential (which is given in pure inflation scenario case by ) get corrections as follows [75]:

where . For , we get the case of chaotic inflation. It is worth noting that the leading correction term here is proportional to , whereas in the case of hybrid inflation it is proportional to .

7. Conclusions

In this minireview, we have analyzed the problem of moduli stabilization in type IIB string theory with positive vacuum energy. We focused on KKLT and Large Volume Scenarios, where geometrical fluxes and nonperturbative superpotentials are required to stabilize complex structure moduli, dilation, and Kähler moduli. We also discussed some possible mechanisms for uplifting the AdS minimum and making it a metastable de Sitter ground state. We have derived the soft SUSY breaking terms in these models. We showed that in KKLT these terms are not consistent with electroweak breaking conditions and hence they are not phenomenologically viable, while in LVS we found that the scalar masses and gaugino masses and trilinear terms are universal and are given in terms of gravitino mass . We emphasized that a very heavy spectrum with  TeV is required to account for the lightest Higgs mass limit. However, in this case, the relic abundance of the lightest neutralino is not consistent with the measured limits. We also studied inflation scenarios associated with the moduli stabilization. We considered two examples of racetrack and Kähler inflation. Finally, we commented on the problem of moduli destabilization and moduli backreaction effects on inflation.

The recent developments in the KKLT and the LVS are worth mentioning. One of the updates is the anti-D3-brane uplift via a nilpotent superfield [88]. In the case of the LVS, the universal gaugino masses are of the same order as the scalar masses and have lower limits  TeV implied from the cosmological moduli problem. This will imply heavy SUSY spectrum which is far from the current LHC detectability. A similar situation for the KKLT model holds. Another modification is the sequestered scenario analyzed in [65]. In this scenario, the TeV scale MSSM and split SUSY model are obtained. For the MSSM scenario, . For split SUSY scenario, .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by the STDF Project 13858 and the ICTP Grant AC-80. Ahmad Moursy would like to thank W. Abdallah for the useful discussions.

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