F-term Moduli Stabilization and Uplifting

We study K\"ahler moduli stabilization in IIB superstring theory. We propose a new moduli stabilization mechanism by the supersymmetry-braking chiral superfield which is coupled to K\"ahler moduli in K\"ahler potential. We also study uplifting of the Large Volume Scenario (LVS) by it. In either case, the form of superpotential is crucial for moduli stabilization. We confirm that our uplifting mechanism does not destabilize the vacuum of the LVS drastically.


Introduction
Superstring theory is a promising candidate for a quantum theory of gravity. Also, it is a good candidate for a unified theory of all the gauge interactions and matter particles such as quarks and leptons as well as the Higgs particle. Superstring theory predicts six-dimensional (6D) compact space in addition to the four-dimensional (4D) space-time. From the theoretical and phenomenological viewpoints, moduli stabilization of the 6D compact space is one of the most serious problems. Without moduli stabilization, we cannot determine parameters of 4D low energy effective field theory of superstring theory, including the Kaluza-Klein scale, supersymmetry (SUSY) breaking scale, gauge couplings, Yukawa couplings and so on. (For phenomenological aspects of superstring theory, see [1,2] and reference therein.) In the mid 2000's, several moduli stabilization mechanisms were proposed. Among them, the Kachru-Kallosh-Linde-Trivedi (KKLT) scenario [3] and the Large Volume Scenario (LVS) [4,5] are two well-known mechanisms in type IIB superstring theory. In such scenarios of type IIB superstring theory, moduli stabilization is carried out in three steps. First, background 3-form fluxes are turned on, and they induce superpotential for the dilaton and complex structure moduli stabilization [6]. Second, some corrections, such as α ′ corrections, string 1-loop corrections, and non-perturbative corrections, are introduced. They generate potential including Kähler moduli and stabilize them. The potential minimum is an anti de Sitter vacuum. Finally, a source of SUSY-breaking such as anti D-branes is introduced and the vacuum energy is uplifted to the Minkowski vacuum. The KKLT scenario and the LVS have been actively investigated since they can realize de Sitter vacua in controllable schemes.
In the second step, both the KKLT and the LVS make use of non-perturbative effects, such as gaugino condensations and D-brane instanton effects to stabilize the Kähler moduli. However, there is no reason why the non-perturbative effects behave as the leading order contribution.
In this paper, we propose a new Kähler modulus stabilization mechanism. We study the modulus potential from the Kähler potential with α ′ correction and the superpotenital with a chiral superfield X which spontaneously breaks SUSY. When X couples to Kähler modulus in the Kähler potential, it effectively generates a modulus potential. We show that when the modulus dependence satisfies certain conditions, the Kähler modulus can be stabilized. However, the vacuum energy in our model is positive definite for a nonvanishing vacuum expectation value (VEV) of the superpotential, and it is quite large for a natural VEV of the superpotential compared with the cosmological constant. It is because the chiral superfield X uplifts the vacuum energy. In order to realize the Minkowski vacuum, we need some effects to depress the vacuum energy. Indeed, instead of anti D-branes, F-term uplifting by X was already studied in the KKLT scenario [7,8,9,10]. Here, we also study F-term uplifting for the LVS by the chiral superfield X.
This paper is organized as follows. In section 2, we study a new model for Kähler modulus stabilization by the chiral superfield. There, we consider Kähler potential, where the chiral superfield couples to the Kähler modulus. In section 3, we study another scenario for uplifting the AdS vacuum of the LVS by the chiral superfield. Section 4 is conclusion.

Kähler Moduli Stabilization
In this section, we study moduli stabilization mechanism by the SUSY breaking chiral superfield. We consider IIB flux compactification; Type IIB superstring theory compactified on a Calabi-Yau 3-folds with background 3-form fluxes. Moduli fields are classified to three types: the dilaton S, complex structure moduli U i and Kähler moduli T j , where i and j represent indeces of (1, 2)-cycle and a (1, 1)-cycle respectively [11]. Their effective theory is described by supergravity. The scalar potential is given by where K and W is the Kähler potential and the superpotential respectively. K ij is the inverse of K ij = ∂ 2 K/∂φ i ∂φ j , and D i W is the covariant derivative; D i W = ∂K/∂φ i W + ∂W/∂φ i , where φ i represent scalar components of all the chiral superfields in the model. The superpotential containing the dilaton and the complex structure moduli is induced by the background 3-form flux G 3 [6]. They are stabilized at the point satisfying D S,U i W = 0. On the other hand, a potential for the Kähler moduli is not generated at the tree level. After integrating the dilaton and the complex structures out, the Kähler potential for the Kähler moduli is given by where V denotes the volume of a compact space in units of ℓ s = 2π √ α ′ . The volume V is a function of Kähler moduli, i.e.
where τ i denotes the volume of the corresponding 4-cycle. The superpotential is constant; Suppose that there is a single Kähler modulus T and the whole volume is given by, In this setup, the tree level potential is calculated that Thus, the potential of T vanishes. This is known as the no-scale structure of supergravity. It is also true for the model including several Kähler moduli. Thus, we need some effects for moduli stabilization. 1 Non-perturbative effects can stabilize T successfully. Non-perturbative superpotential is typically written as, Such a superpotential is effectively induced by gaugino condensations and D-brane instanton effects. When W 0 is sufficiently small, D T W = 0 has nontrivial solution. Its solution is known as the KKLT vacuum [3]. Also, perturbative α ′ corrections can play an important role for the moduli stabilization. The corrected Kähler potential is given by [13],  [4,5], moduli fields are stabilized at a point where the nonperturbative effects and the α ′ corrections are balanced. This model has a SUSY-breaking vacuum, which means ∂V /∂τ i = 0, but D T i W = 0.
In the above two models, the no-scale structure is broken by the (non-)perturbative corrections. Here, we propose a new mechanism for Kähler moduli stabilization.

Potential by Chiral Superfield
Suppose that there is a chiral superfield X in addition to the Kähler modulus T . We assume that the Kähler potential is given by, This form of Kähler potential is given by expanding an original Kähler potential in terms of the chiral superfield X. This expansion is justified when X is much less than 1. The modular weight n, which would be fractional number, depends on the origin of X. 2 In this paper, we do not specify a concrete origin of X, but we treat n as a free parameter. We assume the following superpotential, The linear term X would be generated, e.g. from the Yukawa term, W (Y ) = Y XQQ after condensation QQ = 0 by strong dynamics 3 . When the Yukawa coupling Y depends only on the dilaton and complex structure moduli, i.e., the perturbative Yukawa coupling term, f (T ) is just constant, f . When this Yukawa coupling term is induced by non-perturbative effects, the function f (T ) would be written by f (T ) = Ae −bT . We also assume that X is coupled with other massive chiral fields φ in the superfield such as Xφ 2 , and then radiative corrections generate the mass of X like the O'Raifeartaigh model [15] as explicitly studied in section 2.2. Thus, the potential in our model is written by wherem X is the SUSY breaking mass of X generated by quantum corrections. The tilde indicates that the chiral superfield X is not canonically normalized yet. We assume that the mass of X is much larger than that of T . We justify this assumption later. For the moduli stabilization, the form of f (T ) is crucial. We expand the scalar potential as V = V 0 + V 1 + V 2 + ... in terms of X. When f is a real constant and sufficiently small, each V i is given as follows, where the ellipsis represents mass terms of O(V −2 ). The f 2 term comes from K XX ∂ X W ∂XW .
After integrating X out, the modulus potential is given by (2.12) We also assume that X is small enough and higher order terms of X are negligible. We will justify this assumption later, too. This potential has a local minimum V = V 0 , satisfying the following equations, where h(V) is given by When n is smaller than or equal to 1, h(V) is always negative, and the above conditions can not be satisfied.
When n is equal to 2, the local minimum conditions h = 0 and V T T > 0 are rewritten as, If ξ is negative, these equations have no solutions. If ξ is positive, the second inequality means ξ The necessary condition for a solution of (2.16) to exist is | f W 0 | 2 ( ξ 4 ) 4/3 6.83, and a range of solutions is As the result, the volume of the compact space is positive definite and stabilized. When n is equal to 3, the potential always has a nontrivial local minimum. If ξ is negative, the minimum V 0 is negative and it is not valid, since the volume of the compact space must be positive. If ξ is positive, the solution is given by V 0 = ξ 4 ± 3|W 0 | 2f , and the potential is minimized by, When n is larger than 3, the second term of (2.12) overcomes, and it diverges to positive infinity as V → ∞. If ξ is positive, (2.12) diverges positive infinity as V → ξ 4 , too, and we must have global minimum in the range of ξ 4 < V. If ξ is negative and 12) diverges positive infinity as V → ξ 4 and we have a nontrivial solution. Otherwise, we have no solutions.
Therefore, when n is equal to 2 or larger, the potential has nontrivial (local) minimum. T is successfully stabilized. In Figure 1, we show typical shapes of the potentials with n = 1, 2, 3. It shows that large n potential stabilizes the Kähler moduli.
On the other hand, if f is a function of T and f X is generated non-perturbatively, it is completely different. The superpotential is rewritten as, and the F-term potential is expanded as When X is sufficiently small, we can approximate the modulus potential by the above V 0 . T is stabilized at the point satisfying the following equataions, where g(V) is given by (2.26) When ξ > 0, to realize g(V) = 0, the first and second term of (2.26) must be balanced, which implies Thus, we need n > bV 2/3 . Here, bV 2/3 is considered as an instanton action and it should be much larger than 1 for the single instanton condition. n is given by dimensional reduction, and it is naturally of O(1). Therefore, we can not satisfy the stationary condition. When ξ < 0, there may be a stationary solution, but since there is the single Kähler modulus T in our model, it is natural to assume that ξ ∝ (h 1,2 − h 1,1 ) > 0. Thus, this is invalid solution. We conclude that this form of superpotential can not stabilize the whole volume.

Consistency
Here, we examine the consistency of our model. Hereafter, we assume that the prefactor f is constant and the volume can be stabilized.
For consistent moduli stabilization, the compact space should be large enough to justify the supergravity approximation. In our scenario, the size of the compact space is characterized by ξ and | f W 0 | 2 . Since ξ can be of O(10), the volume of the compact space is of O(10), or larger depending on | W 0 f |. When n is equal to 3, V 0 is given by (2.19).
In either case, the compact space is sufficiently large. We have large range of parameters to realize consistent moduli stabilization.
Next, we also have to justify our assumptions that the mass of X is much heavier than that of T and the modulus potential tems proportional to |X| are negligible. The mass of X is generated by quantum corrections [10,15]. For concrete discussion, we briefly review the mass of X generated by the O'Raifeartaigh-like model. Suppose that there are extra chiral superfields φ 1 , φ 2 and the superpotential is given by, and Kähler potential has the simple form; K = |φ 1 | 2 + |φ 2 | 2 + |X| 2 . For simplicity, we use the canonical form, although we can discuss the non-canonical Käher potential. The masses m of φ 1,2 are relatively heavier than that of X, and we can integrate them out in order to study the dynamics of X. We also assume λf ≪ m, which means the VEVs of φ 1 and φ 2 are sufficiently small. Integrating φ 1 , φ 2 out, we obtain the Coleman-Weinberg potential. It is interpreted as a correction to the Kähler potential and written as where we have assumed many φ 1 and φ 2 , and the constant parameter c denotes their multiplicity. Expanding the Kähler potential, we obtain where Λ 2 = 16π 2 m 2 cλ 4 . Then, the mass of X comes from F-term potential; e K (K XX D X W DXW − 3|W | 2 ). It is calculated as and the mass of X is 2f Λ . In our model, a similar mass term can be generated. We can expect thatm 2 To guarantee that φ 1,2 are heavier than X, Λ 2 is much larger than f . The canonically normalized masses are calculated as When n is less than 3, V 0 is characterized by ξ. We can estimate m T and m X as where the parameters, C 1 and C 2 , are of order O(1) 4 . When | f W 0 | is much less than 1, The mass ratio is given as, The mass ratio is given as, and it is much greater than 1. In either case, a small Λ justifies our assumption. When n is equal to 3, V 0 is given by (2.19). m 2 T is given by (2.37) 4 Here, we assume V 0 ∼ V 0 − ξ 4 ∼ ξ. When (V 0 − ξ 4 ) −1 diverges, our estimation may be invalid.
The ratio of the masses is written as When |W 0 | f ≫ 1 and a is of O(1), we can realize m 2 X ≫ m 2 T for Λ 2 ≤ 1. Now, we examine our assumption that X ≪ 1. From (2.10) and (2.11), X is given by, (2.39) Sincem X is given by (2.32), |X| is calculated as To realize small X , we need Λ 2 |W 0 | f ≪ 1. For either condition, the consistency condition is satisfied when Λ is much smaller than the string scale M s .
We should comment that too small Λ may be inconsistent with the mass generation mechanism of X. There, we assumed φ 1,2 are heavier than X and their masses are roughly estimated to Λ. However, even if it is difficult to set Λ sufficiently small, small |W 0 | also justify our assumptions. Thus, allowing fine tunings, our moduli stabilization mechanism always works well.
We summarize the consistency conditions of our model. For supergravity approximation, the volume of a compact space should be larger than 1, and it is realized when either ξ or W 0 f is of O(10) or larger. It is naturally satisfied. On the other hand, the consistency conditions corresponding to X is more subtle. The mass of X must be heavy enough not to disturb moduli stabilization. We also assume that the VEV of |X| is much less than 1. These assumptions are realized when Λ is sufficiently small or |W 0 | ≪ 1. If the above conditions are satisfied, we conclude that the chiral superfield X, whose modular weight n is equal to 2 or larger, can stabilize the Kähler modulus.

Cosmological Constant
In the above scenario, we can realize the modulus stabilization, where the potential minimum is given by (2.12), and the stationary point is given by (2.13). Then, we can approximate the vacuum energy as When |W 0 | is not suppressed, we can estimate On the other hands, as |W 0 | f → 0, the F X term becomes dominant in the moduli potential and the cosmological constant is approximated as (2.43) As a result, this model has a large positive cosmological constant. The vacuum energy is uplifted by the auxiliary component of X. It may be interesting that the vacuum energy is proportional to V −3 0 when |W 0 | is not suppressed; it is the same as that of the LVS. In order to realize the Minkowski vaccum, we need some effects to depress the vacuum energy. The LVS vacuum as well as the KKLT vacuum has a negative vacuum energy without upfilting by anti D-brane. In the next section, we will discuss the possibility of the F-term uplifting of the LVS by the F X .

F-term Uplifting and the Large Volume Scenario
In this section, we study uplifting the AdS vacuum of the LVS to the Minkowski vacuum by adding one chiral superfield. First, we briefly review the LVS, and then, we study F-term uplifting mechanism.

Large Volume Scenario
The LVS was proposed in [4,5] about 10 years ago. Here, we give a brief review on the LVS. In this scenario, the Kähler moduli are stabilized at the point where the α ′ corrections and the non-perturbative effects are balanced. In this paper, we study the LVS based on swiss cheese compactifications, which means that the dimensionless volume of the Calabi-Yau space is given like where T i represent the moduli corresponding to i-th 4-cycle on the Calabi-Yau manifold, and γ i is a geometrical parameter. The Kähler potential and the superpotential are assumed as follows, Calculating (2.1), the scalar potential is given as where A, B i , C i are given as, The minimum of the potential is given by the point satisfying the following equations, When a i τ i is much larger than 1, the solution is approximated by, As the results, all the Kähler moduli are stabilized successfully. The volume of the compact space is stabilized at an exponentially large value. The vacuum of the LVS breaks SUSY. In fact, the auxiliary fields of the Kähler moduli are not zero. However, its vacuum is a AdS vacuum. The minimum value of the potential is calculated as, and it is negative.
In the original paper, anti-D-branes are introduced for uplifting. Here, we study uplifting by the chiral superfield X.

F-term Uplifting
We study moduli stabilization and uplifting simultaneously. Suppose that there are two Kähler moduli, T 1 , T 2 and one chiral superfield X. Their Kähler potential and volume of the compact space are given by, 5 Similar to the previous section, we consider two forms of superpotential, and where f is a real constant. We assume that W 0 is real for simplicity. In either case, we expect that the mass of X is generated by radiative corrections and the scalar potential is given by, (3.12) 5 A similar model was considered in [16], too.
We also assume that X is much heavier than the other moduli, and we can integrate X out before studying the Kähler moduli stabilization. We confirm this assumption later. We expand V in terms of X as, where V i is the i-th order term of X. For the case of (3.10), we obtain (3.14) where V LV S is the moduli potential of the LVS and the ellipses represent the higher order terms of V −1 . We denote W ′ 0 = W 0 + A 2 e −a 2 T 2 . Using the (3.6), we can approximate V 1 as, Then, the approximated VEV of X is given by , O( X 2 ) term is negligible. After integrating X out, the moduli potentialṼ is expected tõ The potential is uplifted by f ′2 V 2n/3−2 (τ 2 ) m . If f ′ is sufficiently small, the stationary point of our model is approximated by that of the LVS. Its vacuum energy is approximated as where V 0 and τ 2,0 are the minimum of the LVS potential. The Minkowski vacuum can be realized by, More precisely, the stationary point of our model is perturbed from that of the LVS. The true minimum is represented by, (3.25) We assume δ V /V 0 , δ τ /τ 2,0 ≪ 1. Using (3.23), we can calculate the leading order deviations from the vacuum of the LVS as follows, That is, the deviations of the VEVs are estimated as, where x 0 = a 2 τ 2,0 . The vacuum of the LVS implies that a 2 τ 2,0 is of O(10). Thus, δ τ /τ 2,0 is suppressed and it would be small. On the other hand, there is no suppression factor for the deviation of the whole volume δ V /V 0 , and it seems to be of O(1). However, since the denominator of (3.29) is of O(10), δ V /V 0 is successfully suppressed and of O(10 −1 ). Therefore, the deviations are indeed small. The typical values of δ V /V 0 are summarized in Table 1, and we can confirm that the range of δ V /V 0 is small. The uplifting term does not destabilize the vacuum of the LVS drastically. Our rough estimation is valid and we can successfully uplift the vacuum energy of the LVS.  Table 1: Typical values of |δ V /V 0 |. The column represents m and the row is n.
Finally, we study the mass of X. The heaviest mode of the Kähler moduli is the small volume moduli τ 2 , and its canonically normalized mass m τ 2 is estimated [4,5] The mass squared of X is given by (2.33). Substituting f by (3.23), the mass of X is estimated as Roughly speaking, X is heavier than T , when Λ −2 > V 0 . For instance, when V 0 ∼ 10 6 , Λ should be smaller than 10 −3 M s . When the above condition is satisfied, we can safely integrate X out before the Kähler moduli stabilization, and succeed to uplift the vacuum energy.
On the other hand, if the superpotential of X includes Kähler moduli T i and superpotential is given as (3.11), it is completely different. Expanding its scalar potential in term of X and integrating X out, we obtain moduli potential, Its stationary point must satisfy the following conditions, where we used that τ 1 is much larger than 1. Since ∂V /∂τ 2 = 0 implies m = 2bτ 2 , there are no stable vacuum. Therefore, such a superpotential destabilizes the LVS vacuum.
Hereafter, we treat f as a constant parameter and the superpotential is written as (3.10).

Numerical Analysis
In the previous subsection, we considered the LVS model uplifted by the chiral superfield.
We confirmed that if f is constant and does not depend on moduli T i , the vacuum of the LVS can be uplifted to the Minkowski vacuum successfully, by expanding the moduli potential in terms of the deviations from the vacuum of the LVS. Here, we examine the previous conclusion numerically. In Fig.2, we show the shape of a potential of the LVS (left) and that of the uplifted LVS (right). In Fig.2, we set A = 1000, B = 10, C = 1, a 2 = 2π. In this case, the approximated minimum of the original LVS (3.6) is calculated at, The value of the potential minimum is well approximated. When n = 1, m = 0, the value of f ′ is calculated as Analytical calculation of the potential minimum of the uplifted LVS is difficult. We only illustrate the existence of the minimum of the uplifted LVS and its rough position by Fig.2. The orange surface represents the potential of the LVS and the uplifted LVS. The blue surface is V = 0. In the left figure, there is a large region where the moduli potential is negative around the curve of V ∼ 1 2 B i C i √ a i τ i e a i τ i . Thus its potential minimum is definitely negative. However, in the right figure, the orange surface is above the blue surface in almost all of the region, and the blue region vanishes. The potential minimum must be located in a small region at the upper-left corner of the right figure, where the blue surface overcomes the orange surface. This small region includes (V 0 , τ 2,0 ). The deviations from the vacuum of the LVS is small. The potential is almost uplifted by X. We conclude that our uplifting mechanism works well.

F-terms
In this subsection, we study the auxiliary components of Kähler moduli fields and X. F-term SUSY breaking is characterized by the VEV of the auxiliary components of chiral superfields. They are given by, (3.38) D T 1 W, D T 2 W, D X W are estimated by, We used that X is of O(V −2 0 ) and its liner term is a subleading term. K ij and K ij are calculated as, The VEVs of the F-terms are estimated as, (3.44) Thus, if n is larger than 7 2 , we obtain V On the other hand, the gravitino mass m 3/2 is independent of n, and it is given by, Using the above equations, we can calculate soft terms. Although F X can overcome or be comparable to F T 1 ,T 2 , the spectrum is not affected from those of the LVS.

Conclusion and Discussion
We have studied the new type of Kähler moduli stabilization, and F-term uplifting of the Large Volume Scenario has also been studied. For realistic string models, moduli stabilization is crucial. In addition, since our universe has positive cosmological constant, there must be a source of uplifting in superstring theory.
First, we have investigated whether the chiral superfield X which is coupled with the whole volume in the Kähler potential can stabilize the Kähler modulus or not. The mass of X is much heavier than the modulus and the VEV X is almost negligible, but its Fterm potential can affect moduli potential. We assumed that the superpotential is written as W = W 0 − f (T )X and Kähler potential is given as K = −2 ln(V + ξ 2 ) + (T +T ) −n |X| 2 . We showed that if f is constant and n is equal to 2 or larger, the whole volume of the extra dimension can be successfully stabilized. However, if f is a function of T and it is generated non-perturbatively, modulus potential has no stationary point and Kähler modulus can not be stabilized. Thus, the form of the superpotential is very important for modulus stabilization. We also studied the condition for heavy X. The mass scale of X must be heavier than that of Kähler modulus. It is realized by setting the scale of extra chiral superfield Λ is sufficiently small. Such a small Λ may be inconsistent for mass generation of X, but we can always make X heavy by fine tuning of |W 0 |. Our moduli stabilization mechanism works well. However, for a natural value of |W 0 |, the vacuum energy in our model is quite large compared with the experimental value of the cosmological constant. In order to realize the Minkowski vacuum, we need some effects to depress the vacuum energy.
Next, we have studied uplifting of the LVS. In the original LVS, the vacuum energy is uplifted by anti D-branes. In this paper, we studied F-term uplifting of the LVS vacuum by the chiral superfield. We found that F-term uplifting requires a certain form of the superpotential too. We need the constant f term in superpotential. Otherwise X destabilizes the LVS vacuum completely. It is the same as the moduli stabilization. If such a superpotential is induced, the vacuum energy of the LVS can be uplifted to the Minkowski vacuum (or de Sitter vacuum) by fine tuning the prefactor f .
In either case, the form of the superpotentials is crucial. The superpotential including X is written as (4.1) f must be constant for the moduli stabilization and F-term uplifting, otherwise it destabilizes the moduli stabilization mechanism completely. For example, such a constant prefactor may be induced by non-perturbative effects on D3-brane (or D(-1)-brane instanton). Since it is suppressed by e −S , where S is the dilaton. S is stabilized by a 3-form flux at the tree level, and it can be substituted by its VEV. Another possibility is non-perturbative effects on D7-branes (D3-brane instantons) wrapping 4-cycles whose sizes are already stabilized by other effects. Such a mechanism may be provided by Dterms in magnetized D-branes 6 or other flux effects. Studying concrete origin of such a superpotential in superstring theory would be interesting.