Supersymmetric (SUSY) extension of the Randall-Sundrum (RS) model [1, 2, 3] provides an interesting setup for the physics beyond the standard model. For instance, localized wavefunctions in the extra dimension can be considered as a source of Yukawa hierarchy  as well as the large hierarchy between the weak and the Planck scales. The SUSY breaking sector can be sequestered from the visible sector in the extra dimension resulting in a flavor blind SUSY breaking patterns of the superparticle masses and couplings, i.e., anomaly mediation . Besides, the application of the so-called AdS/CFT correspondence  to the SUSY RS model might provide a way to analyze perturbatively the four-dimensional (4D) strongly coupled theories.
On the other hand, the superstring theories would provide a unified framework of the standard model and gravity. It is known that a low energy effective theory of the strongly coupled heterotic string theory  can be described by 5D supergravity on an orbifold, which is sometimes referred to as the 5D heterotic M-theory . Some three-brane solutions have been derived in such 5D model, which can be utilized to construct brane-world scenarios of our universe, where the visible sector and the hidden sector reside in different branes located at different orbifold fixed points.
All the above examples can be categorized into a unique theory, i.e., the 5D gauged supergravity, and share a common important subject, that is, an issue of moduli stabilization. 5D models have at least a radius modulus, i.e., the radion, whose vacuum value corresponds to the radius of the extra dimension. In 4D effective theory of 5D models, some couplings depend on the radius which is undetermined unless the radion is dynamically stabilized. If a 5D model is an effective theory of some higher-dimensional model, there might exist other moduli. These moduli fields including the radion form supermultiplets in the 4D effective theory if the compactification respects SUSY, and these multiplets can mediate SUSY breaking effects. Thus the moduli stabilization is one of the most important issues in the model building based on the higher-dimensional theory.
Several years ago, an interesting class of the moduli stabilization scheme was proposed by Kachru-Kallosh-Linde-Trivedi (KKLT)  based on the type IIB supergravity . In such framework, light moduli are first stabilized at a SUSY preserving anti-de Sitter (AdS) minimum of the scalar potential. Then it is uplifted to a Minkowski minimum by a SUSY breaking vacuum energy generated in the hidden sector such as anti D-branes [9, 11], or dynamically generated F-terms [12, 13] and D-terms , which are well sequestered from the light moduli as well as the visible sector. This scenario overcomes, in a controllable manner, a big difficulty existing in supergravity/string compactifications , that is, a realization of the SUSY breaking Minkowski vacuum where all the moduli are stabilized, which is required from the observation of our universe.
Moreover, it has been shown that this kind of moduli stabilization procedures generically yields an interesting pattern of soft SUSY breaking terms in the visible sector, that is, the mirage mediation [11, 16, 17]. In the mirage mediation, the modulus-mediated contribution is comparable to that of the anomaly mediation. The low energy superparticle spectrum is quite different from the other mediation schemes such as the pure modulus mediation, the pure anomaly mediation and the gauge mediations. These would be distinguished by high-energy experiments and cosmological observations in future.
In this paper we study 5D gauged supergravity with a universal hypermultiplet whose isometry group is . We consider various gaugings of the isometries and introduce some superpotential terms at the orbifold fixed points. Among eight independent isometries, we focus on three directions to gauge by the graviphoton. Thus the model we consider has three gauging parameters and includes both the SUSY RS model with an arbitrary bulk mass parameter for the hypermultiplet and 5D heterotic M-theory as different choices of the parameters. We investigate the vacuum structure of such models, especially the nature of moduli stabilization, assuming perturbative superpotential terms at the fixed points. Motivated by the KKLT moduli stabilization scheme, we also discuss the uplifting of SUSY AdS vacua in our models and the resultant SUSY breaking.
The following sections are arranged as follows. In Sec. 2, we review the off-shell description of 5D supergravity and see the representation of the isometries on the scalar manifold in such description. Then we derive the 4D effective action of a class of models obtained by gauging three independent isometries by the graviphoton with arbitrary superpotential terms at the orbifold fixed points. In Sec. 3, we study the moduli stabilization and the uplifting of the scalar potential in a case where the effective Kähler and superpotentials are expressed by analytic functions. In Sec. 4, we carry out a similar analysis in a model which corresponds to the generalized Luty-Sundrum model . Sec. 5 is devoted to the summary. In Appendix, we show how to derive the isometry transformations in the on-shell description of 5D supergravity from the off-shell formulation.
2 4D effective action of 5D gauged supergravity
2.1 N=1 off-shell description of 5D supergravity action
In this paper we consider 5D gauged supergravity compactified on an orbifold with a universal hypermultiplet and arbitrary superpotentials at the fixed points (or the boundaries) of the orbifold. The metric is written as
where is a warp factor. The off-diagonal components of the metric is gauged away. We take the fundamental region of the orbifold as , where is a constant,111 In principle, is nothing to do with the radius of the orbifold. It coincides with the latter only when the coordinate is redefined so that . and take the unit of , where is the 5D Planck mass.
The off-shell description of 5D supergravity is quite useful for our purpose. It enables us to treat the localized terms at the orbifold boundaries independently from the bulk action. Furthermore as will be seen in the next subsection, the isometries of the scalar manifold, some of which are to be gauged, are linearly realized in the off-shell description. Therefore we start from 5D off-shell (conformal) supergravity developed by Ref. -. 5D superconformal multiplets relevant to our study are the Weyl multiplet , the vector multiplets and the hypermultiplets , where and . Here , are the numbers of compensator and physical hypermultiplets, respectively. These 5D multiplets are decomposed into superconformal multiplets  as , and , where is the Weyl multiplet, is the general multiplet whose scalar component is , is the vector multiplet, and , , are chiral multiplets.
where is the warp factor of the background metric to be determined on-shell, , and is a cubic function defined by
where is a real constant tensor which is completely symmetric for indices, and is a gauge-invariant quantity. The ellipsis in (2.2) denotes the vector multiplet part. The boundary Lagrangian can be introduced independently of the bulk action. Note that (2.2) is a shorthand expression for the full supergravity action. We can always restore the full action by promoting the and integrals to the - and -term formulae of conformal supergravity formulation , which are compactly listed in Appendix C of Ref. . In the above expression of the 5D action, the hypermultiplet isometries are linearly realized. We can partially or fully gauge these isometries by the vector multiplets with the generator .
2.2 Gauged supergravity with universal hypermultiplet
Now we consider the gauged supergravity with a single universal hypermultiplet spanning the manifold . The universal hyperscalars are commonly denoted as and , which are even and odd under the orbifold parity [27, 28]. The scalar manifold has an isometry group, which is nonlinearly realized in the on-shell description . (See (A.6) in Appendix A, for example.) In the off-shell description (2.2), on the other hand, it is linearly realized. This greatly simplifies the analysis. The situation we consider is realized by taking . Then the bulk action in (2.2) has a symmetry. Since the superconformal gauge-fixing conditions can eliminate only one compensator multiplet, we introduce a nondynamical (auxiliary) Abelian vector multiplet to eliminate the other compensator multiplet. We gauge the overall subgroup of the symmetry group , which we refer to as , by . Namely the charges of the hypermultiplets for this gauging are assigned as . As a result the symmetry group is reduced to after eliminating the nondynamical vector multiplet . The -parities and the charges of the multiplets are listed in Table. 1.
Two compensator multiplets and must have the opposite -parities for consistency with the superconformal gauge-fixing. (See the appendix in Ref. .) For the vector multiplets, we divide the index into so that and are odd and even under the -parity respectively.
The following analyses are independent of the number of -even vector multiplets , and then we just choose to simplify the discussion (i.e., ). The gauged supergravity is obtained by gauging some directions within the isometry group by . In this paper we restrict ourselves to the simple case , where only the graviphoton takes part in the gauging of the isometries. Namely the function defined in (2.3) is now
The most general form of the gauging is parameterized by
acting on or , where () are matrix-valued generators of shown in Eq. (A.4). Here the real coefficients determine the gauging direction. Since the graviphoton is -odd, the parameters () are -even while the others are -odd and have kink profiles for .222 The -odd gauge couplings can be consistently introduced into supergravity by the so-called four-form mechanism proposed in Ref.. In the following, we consider a case that are parameterized by three parameters () as
This class of models contain the following interesting models as special limits of the gauge parameters. In the limit of , this model is reduced to the 5D effective theory of heterotic M-theory, which is derived in Ref.  on-shell. As shown in Ref. , it has a linearly warped BPS background geometry given by
where is a constant. In this case, the gauge parameter is related to a flux, that is, a curvature four-form integrated over a four-cycle of the Calabi-Yau manifold,
If we turn on the -gauging, the background metric (2.7) becomes
and the model has an exponentially warped geometry. Especially it is reduced to the SUSY RS model [2, 3, 28] when . This means that the -gauging induces a negative cosmological constant in the bulk, which leads to the AdS curvature . We can see from the 5D action that it also induces the bulk mass for the physical hypermultiplet. By introducing the -gauging to this case, the parameters and become independent and are given by
Note that these relations hold only when . If we turn on the -gauging, the model deviates from the SUSY RS model, and the above relations are modified to more complicated ones.
Thus the -gauging model, which we refer to as the hybrid model in this paper, can be regarded as a hybrid formulation of the SUSY RS model and the 5D heterotic M-theory. Although we have gauged only three directions among eight isometries within , this -gauging model contains most of important structures in the gauged supergravity with a universal hypermultiplet because both the linear and the exponential warp factors can be realized just by taking different limit of the the gauge parameters. The other gauge parameters in Eq. (2.5) generate a similar warping such as or which are just combinations of the exponential factors.
Therefore, in the following, we study the -gauging model in detail with boundary induced superpotentials. Since the boundary Lagrangian must be invariant for , it is written as
where () are the boundary superpotentials. The induced chiral multiplet has zero Weyl weight and is neutral for . Thus it is identified as
Note that only -even multiplets can appear in .
2.3 4D effective action
Now we derive the 4D effective theory for the -gauging with superpotentials at the orbifold fixed points. For this purpose, we adopt the off-shell dimensional reduction proposed by Refs. [24, 25], which are based on the superspace description  of the 5D off-shell supergravity and developed in subsequent studies . This method enables us to derive the 4D off-shell effective action directly from the 5D off-shell supergravity action keeping the off-shell structure. The procedure is straightforward. We start from the off-shell description of 5D action (2.2) with (2.11). After some gauge transformation, we drop kinetic terms for -odd multiplets because they are negligible in low energies. Then those multiplets play a role of the Lagrange multipliers and their equations of motion extract zero-modes from the -even multiplets. Note that only the -even multiplets have zero-modes that appear in the effective theory.
In our model the -even multiplets are , , and . Due to the -invariance, they appear in the action only through the combinations of , and which carry the zero-modes, the radion multiplet , 4D chiral compensator and the matter multiplet , respectively. Following the procedure of Ref. [24, 25], we obtain the 4D effective action as
where the Kähler potential and the superpotential are given by
Here, and are defined as
As the boundary superpotentials in (2.11), we consider the following polynomials.
where () are constants.
There are two simple cases to analyse. When , the Kähler potential can be expressed by an analytic function. When , on the other hand, the superpotential is reduced to a simple form, i.e., a polynomial for . Thus we will discuss these two cases in detail in the next two sections.
3 Moduli stabilization in hybrid model ()
In the case that , the -integration in the Kähler potential (2.14) can be carried out analytically. Such analytic expression allows us to study more about the 4D effective theory of the above hybrid model. In this section, we analyze the vacuum structure of the -gauging model assuming, for simplicity and concreteness, the boundary superpotentials consist of only constant and tadpole terms for the universal hypermultiplet, i.e., for in (2.18).
Here, is the incomplete gamma function, and the chiral multiplets are defined as
The parameters and in the superpotential are given by linear combinations of the constants in the boundary superpotentials (2.18) as
For given and , the scalar potential is calculated by the formula,
where , and run over all chiral multiplets. Note that the scalar potential (3.5) is that in the Einstein frame, which corresponds to a gauge where the chiral compensator scalar is fixed as
Here and henceforth, we take the unit of the 4D Planck mass, i.e., . (In the previous section, we took the unit of the 5D Planck mass.)
In our model, is calculated as
In this paper we use the same symbols for the scalar fields as the chiral multiplets they belong to.
3.1 Heterotic M-theory limit (pure -gauging)
In the limit , Eq.(3.1) becomes
The above Kähler potential reproduces the known result, i.e., the 4D effective Kähler potential of the heterotic M-theory  when as pointed out in Ref. . The superpotential originates from the boundary superpotentials (2.18). In this case, the vacuum values of , and determine the Calabi-Yau volume at and the radius of the compact 11th dimension, respectively. Here we assume333In the case of , we can repeat the same arguments by exchanging and . that , then the scalar fields have to satisfy in order for them to have the physical interpretation as the volumes.
The scalar potential (3.7) is now reduced to
From the SUSY preserving conditions: , we obtain
The first equation indicates that SUSY point exists only when is real and positive. Thus we assume that is real positive in the following. Solving (3.11), we find a SUSY point,
Since the scalars must satisfy , this point is in the physical region only when and . We focus on this parameter region. From Eq. (3.13), we find a flat direction in an imaginary direction of . The superpotential takes the following value at the SUSY point.
Thus from (3.5), the vacuum energy is negative at this SUSY point, that is, the geometry is AdS. By evaluating the second derivatives of the potential (3.10), we can see that this SUSY point is a saddle point. Here we should note that SUSY points are always stable in a sense that they satisfy the Breitenlohner-Freedman bound .444 For a compact proof, see Appendix C in Ref. , for example. As will be done in Sect. 3.4, we will uplift the negative vacuum energy of the SUSY AdS vacuum by a SUSY breaking vacuum energy in the hidden sector in order to obtain a SUSY breaking Minkowski vacuum. In general a SUSY saddle point remains to be a saddle point after the uplifting unless the uplifting potential is steep, and such saddle point after the uplifting is not stable any more. So we would like to look for a local minimum of the potential which is expected to be stable after the uplifting.
3.2 Randall-Sundrum limit (pure -gauging)
In the limit , the Kähler and the superpotentials (3.1) become
where is a warp factor superfield. The above reproduces the radion Kähler potential of the SUSY RS model . In the following, we assume . Then the scalar fields must satisfy and . Although is more conventional than for the SUSY RS model, we use as a matter chiral multiplet because we will interpolate this model and the Heterotic M-theory limit (, ). We can always translate to by the relation (3.3).
In this limit, the scalar potential (3.7) is reduced to
From the SUSY conditions: , we obtain
In the case that , we find a SUSY solution as
where is a solution of Eq. (3.20).
On the other hand, in the case that , that is,
where is a constant, the relation (3.20) is rewritten as
where , and the superpotential in this case is found as
For , the SUSY point is found as
At this point, and thus the vacuum energy vanishes, resulting a local Minkowski minimum. This corresponds to the SUSY Minkowski vacuum discussed in Ref. , in which the boundary superpotentials (2.18) consist of only the tadpole terms, i.e., (or .)
For , the SUSY solution is found as
where the superpotential does not vanish at this point,
This means that the vacuum energy is negative and the geometry becomes AdS. This SUSY solution is a saddle point.555 A case that only constant terms exist in the boundary superpotentials, i.e., , is studied in Ref.  where a consistent result with ours is obtained.
In either case, the SUSY point exists in the region and only when . For , and is undetermined.
Here we have decomposed the complex scalars and the parameters into real ones as
From the stationary conditions for and , we obtain
Focusing on the SUSY local minimum (3.25), we calculate the mass eigenvalues. Evaluating the second derivatives of the scalar potential (3.28), we can see that the four real scalars do not mix with each other. Then after normalizing them canonically, the mass eigenvalues are found as
We have assumed and .
In the previous subsections, we analyzed in detail the vacuum structures in the two typical limits of the hybrid model, i.e., the heterotic M-theory limit (pure -gauging) and the SUSY RS limit (pure -gauging). In this subsection, we study intermediate regions of the hybrid model. We assume that the parameters satisfy the relation (3.22) with for simplicity of the analysis.
3.3.1 Numerical result
First we show some numerical results. We focus on the SUSY preserving points (3.12) in the pure -gauging and (3.25) in the pure -gauging, and numerically interpolate these stationary points in the intermediate region, where both and are nonvanishing.
Fig. 1 plots the gravitino mass and the position of the SUSY point on the plane for various values of with the other parameters fixed. The left end-point of the curve corresponds to (near the M-theory limit) and the other end-point corresponds to (near the SUSY RS limit). The curve on the plane represents the projection on that plane. Since the vacuum energy at the SUSY point is related to the gravitino mass through , we can see from Fig. 1 that monotonically decreases as decreases. This is consistent with the fact that the SUSY point (3.25) is a Minkowski vacuum in the RS limit. Since corresponds to the size of the orbifold, it should be larger than the Planck length . Thus we can see from Fig. 1 that the region around the RS limit is favored for our parameter choice.
Fig. 2 plots the lightest modulus mass squared and the SUSY point on the plane. Again we vary from 100 (left end-point) to 0.1 (right end-point) while fix the other parameters. Now we can see from this figure that the SUSY point near the M-theory limit (left end-point) is not local minimum since the lightest modulus is tachyonic, but this tachyonic mass squared monotonically increases as decreases and becomes positive when . Thus the region around the SUSY RS limit (), where is realized without any tachyonic masses, is the best candidate for the KKLT-type uplifting. We will study this region analytically in the following.
3.3.2 Analytic result near the RS limit
In the vicinity of , i.e., the SUSY RS model, the Kähler and the superpotentials in (3.1) are expressed as
Here times is supposed to be large. Then from the SUSY conditions: , we find the SUSY point as
Here and . Since the SUSY point in the pure -gauging (3.25) is a local minimum of the potential, this point is also a local minimum when . Due to the correction from the pure -gauging case, the superpotential does not vanish at this point.
Thus the vacuum energy becomes negative, evaluated as
Namely this is an AdS SUSY vacuum.
So far we have found some stationary solutions of the scalar potential in the hybrid model assuming certain simple superpotential terms at the orbifold fixed points. In the heterotic M-theory limit (pure -gauging), the SUSY point is a saddle point. In the SUSY RS limit (pure -gauging), on the other hand, the local minimum of the potential is a SUSY Minkowski vacuum.
Finding a SUSY breaking Minkowski minimum, which is a candidate of our present universe, is indeed a hard task in supergravity models. The hybrid model studied in this section is also the case. As mentioned in the introduction, the KKLT model provides an interesting and systematic way of achieving a SUSY breaking minimum with vanishing vacuum energy, that is, uplifting SUSY AdS minimum by a SUSY breaking vacuum energy which is assumed to be well sequestered from the light moduli as well as the visible sector. Here we consider the uplifting of the AdS SUSY minimum in our hybrid model by a SUSY breaking sector which is assumed to be well sequestered from and (or ).
As mentioned at the end of Sec. 3.3.1, the region around the SUSY RS model () is the best candidate for the uplifting. So we consider such a parameter region which we discussed in Sec. 3.3.2. Following the KKLT model, the uplifting potential is assumed as [11, 36]
where is a constant. The typical value of for the sequestered SUSY breaking source is given by [11, 36]. The total scalar potential is then given by . Then the local minimum is shifted from the SUSY point (3.33) by
Here we have chosen as