Inflation, Gravity Mediated Supersymmetry Breaking, and de Sitter Vacua in Supergravity with a K\"{a}hler-Invariant FI Term

We use a new mechanism for generating a Fayet-Iliopoulos term in supergravity, which is not associated to an R symmetry, to construct a semi-realistic theory of slow-roll inflation for a theory with the same K\"ahler potential and superpotential as the KKLT string background (without anti-D3 branes). In our model, supersymmetry must be broken at a high scale in a hidden sector to ensure that the cutoff of the effective field theory is above the Hubble scale of inflation. The gravitino has a super-EeV mass and supersymmetry breaking is communicated to the observable sector through gravity mediation. Some mass scales of the supersymmetry-breaking soft terms in the observable sector can be parametrically smaller than the SUSY breaking scale. If a string realization of the new FI term were found, our model could be the basis for a low energy effective supergravity description of realistic superstring models of inflation.

The purpose of our work is to find an effective field theory of inflation, de-Sitter moduli stabilization, and supersymmetry breaking as a cosmological application to the effective theory of KKLT of the ACIK-FI term proposed in [3]. The string theory origin of one particular type of the new FI terms has recently been investigated through a supersymmetric Born-Infeld action [15], so it would be of clear interest to study a possible string-theoretical origin of the ACIK-FI term.
This paper is organized as follows. In Sec. 2 we show how to add an ACIK-FI term to the N = 1 supergravity effective theory of the KKLT model. Next we add matter, which we divide into a hidden sector and an observable sector. Supersymmetry is broken in the hidden sector and the SUSY breaking is communicated to the observable sector via gravity mediation. In Secs. 3 and 4 we probe the hidden-sector dynamics of our model. In Sec. 3, we construct a minimal supergravity model of plateau-potential inflation -sometimes called in the literature "Starobinsky" or "Higgs" inflationwith high scale SUSY breaking and dS vacua, using the results from Sec. 2. In Sec. 4, we explore the gravitino mass, which is very high, being well above the EeV-scale. We also study possible constraints on the ACIK-FI term by investigating the nonrenormalizable fermionic terms in the Lagrangian. These constraints can be satisfied if the gauge coupling for a certain U(1) necessarily present in our model is sufficiently small. They also lead to a hierarchy of energy scales. In Sec. 5 we study the observablesector dynamics of our model by computing its soft SUSY breaking terms. A few final observations are collected in Sec. 6.
2 Adding a Kähler-invariant Fayet-Iliopoulous term to KKLT-type N = 1 supergravity In this section, we propose an N = 1 supergravity model that can describe the low energy effective field theory of inflation and moduli stabilization in KKLT-type backgrounds [1]. To do so, we first add an ACIK-FI term to an N = 1 supergravity that is compatible with the KKLT model.
In general, an ACIK-FI term can be introduced into an N = 1 supergravity without requiring a gauged R-symmetry [3,11]. In our proposal, we will introduce instead only an ordinary U(1) symmetry (under which the superpotential is invariant) which will be gauged by a vector multiplet V . Inflation will come from the same potential as in the KKLT scenario. KKLT [1] argues that in string theory some moduli can develop a non-perturbative superpotential of the form where T is a "volume" modulus field, which is a chiral superfield, and W 0 , A are constants. For our construction it is sufficient to compute the component action of N = 1 supergravity characterized by the superpotential (1) and by an ACIK-FI term. Notice that Antoniadis and Rondeau have recently studied cosmological applications of generalized ACIK-FI terms by considering no-scale models with a constant superpotential W = W 0 [11]. Differently from that model, ours uses the KKLT-type superpotential (1).
The key assumption that we will use is that both the volume modulus T and the other matter fields that may exist in the superpotential are gauge-invariant under the U(1) that is used to introduce the ACIK-FI term. In this paper, we use superconformal tensor calculus [16] to calculate the action.
The goal of this work is to find a modestly realistic minimal supergravity model of inflation with realistic moduli stabilization and supersymmetry breaking pattern. The study of irrelevant operators generated by the ACIK-FI term will show that a low energy supersymmetry breaking is incompatible with demanding that the cutoff for the effective field theory is higher than the Hubble constant during inflation. So, we take an alternative approach and break supersymmetry at a high scale in the hidden sector (as in e.g. [17] ) while keeping some of the scales of supersymmetry breaking interactions in the observable sector low [18].
To do so, we first decompose matter into a hidden sector and an observable sector. We will discuss them separately in Sections 3, 4 and 5. So we separate the field coordinates y A into whereÎ ≡ (I, i) and {T, z I } h are hidden-sector fields, while {z i } o are the observable-sector ones.
In addition to this, we write a generic superpotential W as a sum of a hidden-sector term W h and observable-sector term W o : We further assume that the hidden-sector superpotential carries a high energy scale compared to the observable-sector one. This implies that we decompose the F-term scalar potential into two different parts: a hidden sector F-term potential characterized by a high energy scale and observable-sector F-term potential containing only low scale SUSY-breaking soft terms.
Next, to introduce an ACIK-FI term into our theory we suppose that the volume modulus multiplet T and all observable-sector chiral matter multiplets Z i are neutral under an ordinary (non-R) U(1) gauge symmetry, while the hidden-sector chiral matter multiplets z I are charged, i.e. they transform Here q I denote the U(1) gauge charges of the hidden-sector chiral multiplets Z I and Ω is the chiral multiplet containing in its lowest component the ordinary gauge parameter. We make these choices because we will introduce both a new FI term generated by a gauge vector multiplet and a KKLT superpotential, which depends on the volume modulus T and must be gauge invariant under all gauge symmetries.
The superconformal action of the ACIK-FI term [3,11] is defined by and the corresponding superconformal action of N = 1 supergravity with superpotential (1) and the new the FI term is In Eqs. (5,6) S 0 is the conformal compensator with Weyl/chiral weights (1,1); Z A = (T, Z I ; Z i ) and V are chiral matter and vector multiplets with weights (0, 0); K(Ze qV ,Z) is a Kähler potential gauged by ) 2 are composite multiplets, T (X),T (X) are chiral projectors, and (V ) D is a real multiplet, whose lowest component is the auxiliary field D of the vector multiplet V .
Next, we write the following Kähler potential, invariant under the same U(1) that generates the ACIK-FI term where Φ is a real function of the matter multiplets Z i , Z I and the two terms in the superpotential and the observable-sector superpotential where B 0 , S i , M ij , Y ijk are constant coefficients. We will choose Φ to be sum of a term containing only hidden-sector fields and one containing only those of the observable sector The supergravity G-function corresponding to our model is then The F-term supergravity scalar potential is given by the formula V F ≡ e G (G A G AB GB − 3), which in our case reads When matter scalars are charged under a gauge group there exists also a D-term contribution to the scalar potential, V D . In our model, we find it to be where X ≡ T +T − Φ/3, g is the gauge coupling constant and ξ is the ACIK-FI constant. Remember that only hidden-sector chiral matter multiplets are charged under the U(1). The scalar potential is the sum of two terms. One, V h contains the D-term contribution and the F-term potential of the hidden sector, depends on the high mass scale M S and is O(H 2 M 2 pl ) during inflation; the other, V sof t contains the observable sector scalars and depends only on low mass scales: where 3 Hidden sector dynamics 1: a minimal supergravity model of inflation, high-scale supersymmetry breaking, and de Sitter vacua In this section, we explore a minimal supergravity model of high-scale supersymmetry breaking and plateau-potential inflation through gravity mediation and no-scale Kähler potential. We investigate first the hidden sector dynamics. We have assumed that the hidden-sector potential depends on a high energy scale and dominates over the observable-sector one. Hence, it is reasonable to minimize the hidden-sector potential first. Let us compute now the F-term potential in the hidden sector. Recalling that the KKLT superpotential is and redefining W 0 ≡ −cA, we rewrite it as where a, c, A are positive constants. Note that W h I = ∂W h /∂z I = 0.
Since we defined X ≡ T +T − Φ/3, the KKLT superpotential gives Here we have used the following transformation from the complex coordinate T to two real coordinates X, ImT : which gives e −aT = e − a Then, since W h I = 0, the corresponding hidden-sector F-term scalar potential is given by Since we assume that the D-term potential belongs to the hidden sector, the hidden-sector total scalar potential can be written as To investigate the moduli stabilization, we identify the canonically normalized fields by inspection of the kinetic terms, which are given by where D µ ≡ ∂ µ − iqÎ A µ is the U(1) gauge covariant derivative for the matter multiplets zÎ = (z I , z i ) with gauge charge qÎ = (q I = 0, q i = 0), and A µ is the corresponding gauge field.
After performing another field redefinition X ≡ e √ 2/3φ , we find Notice that φ is canonically normalized, while the other fields z I , ImT are so only when φ is small. Now, let us investigate the scalar potential vacuum. First of all, we find the minimum with respect to the matter scalars zÎ , If we choose a real function such that ΦÎ = 0 implies Φ = 0 together with zÎ = 0 then at this vacuum the scalar potential becomes 3 Next, we consider the vacuum with respect to the ImT field. We find the vacuum at aImT = nπ, where n is an even integer, leading to cos(aImT ) = 1 4 and Next, let us find the vacuum with respect to the φ field. Recalling that X = e √ 2/3φ , calling φ the vacuum expectation value of φ and setting φ = φ = 3 2 ln X = 3 2 ln x, where X = X ≡ x, 3 The observable-sector superpotential W o can shift the VEVs of the scalars in the observable sector z i , but since those VEVs must be in any case small compared to H and M pl we can approximately set z i = 0. Moreover, in our toy example in Section 5 we will choose a superpotential that indeedgives a minimum at z i = 0. 4 When cos(aImT ) = 1, the second derivative of the potential can be positive, which means that the stationary point is a minimum.
we have which gives At first glance, this equation seems a little complicated, but after a short calculation, we can obtain the following simple relation Inserting the value of c into V h , we obtain the following equation where Then, the vacuum energy at X = x is given by where Λ is defined to be the post-inflationary cosmological constant, and the SUSY breaking scale is given by where M S is by definition the SUSY breaking scale. We can set Λ to any value we wish, in particular we can choose it to be Λ ∼ 10 −120 .
Here, we point out that the term 1 2 g 2 ξ 2 governs the magnitude of the total scalar potential, and simultaneously controls the scale of spontaneously supersymmetry breaking. Hence, if we want that the scalar potential describes inflation, we need to impose where H is the Hubble parameter, and M I is defined to be the mass scale of inflation.
We then identify Substituting the above parameters A, W 0 , M I into the hidden-sector potential, we can obtain a plateau inflation potential where X = e √ 2/3φ and φ is defined to be the inflaton. Notice that the inflaton mass after inflation is of order of the Hubble scale, i.e. m 2 φ ∼ H 2 = 10 −10 M 2 pl .
We note that the hidden-sector scalar potential V h has a plateau, so it is of HI type (in the notations of ref. [19]). Furthermore, it depends only on four parameters, which are: the vacuum expectation value of X (i.e. x ≡ X ); the KKLT parameter a in the superpotential, which will be determined according to the type of the nonperturbative correction we choose 5 ; the inflation scale M I , and the post-inflationary cosmological constant Λ. At X = x, the potential indeed reduces to the post-inflationary cosmological constant. As an additional remark, we observe that for fixed x, M I , Λ inflation ends earlier when a is smaller. When the nonperturbative corrections to the KKLT superportential come from gaugino condensation [1], a smaller parameter a corresponds to more D7 branes being stacked.
4 Hidden sector dynamics 2: super-EeV gravitino mass, weak gauge coupling, and a hierarchy of energy scales In this section, we investigate some physical implications that can be obtained from our model. First of all, let us find the gravitino mass after inflation, which is generated by the high-scale SUSY breaking in the hidden sector. It is given by which is compatible with the case of EeV-scale gravitino cold dark matter candidates. This is not surprising because we are considering the same high-scale supersymmetry breaking scale as in [20, 21, matter candidates has recently been studied in ref. [24]. Notice that in our model the gravitino mass is always O(H), irrespective of the ultraviolet cutoff.
Next, we explore possible constraints on the FI term by analyzing the fermionic nonrenormalizable interaction terms that are induced by such term (5). Schematically the general fermionic terms have the form where ξ is the dimensionless ACIK-FI constant; O (δ) F is an effective field operator of dimension δ, which does not contain any power of D; m is the total order of derivatives with respect to the the composite chiral fields T (w 2 ) andT (w 2 ) defined after Eq. (6), and p = 0, 1. Detailed calculations will be given in [25], here we will only briefly summarize the main points.
To evaluate the fermionic terms we need to solve for the auxiliary field D. Equation (5) gives the following Lagrangian for them where ξ is the new FI constant while ξ ≡ i(G i K i −GīKī) is the standard field-dependent linear term in D. It is written in terms of the Killing vector K = K i ∂ i giving the action of our U(1) gauge symmetry on the scalar fields. G is the standard supergravity G-function [16]. Restoring the dependence on the gauge coupling constant g we have After solving the equation of motion for D, we find the solution D = g 2 M 2 pl (ξ + ξ ) (= g 2 (ξ + ξ ) when M pl = 1).
Plugging this solution into the fermionic terms in Eq. (41) we obtain The fermionic nonrenormalizable interactions generated by the ACIK-FI term introduce a strong coupling scale that sets the limit of validity of the effective field theory description. If we demand that the theory is valid up to some cutoff scale Λ cut , we find the folloiwing constraint on the ACIK-FI term: which ξ = 0. In this case, we obtain This inequality reduces to the following: for all Λ cut ≤ M pl , we obtain Now we are ready to ask how does this constraint affect our supergravity model of inflation. To answer this, let us get back to the definition of the inflation scale (restoring the mass dimension) Inserting this equation into the constraints, we find that for Λ cut ≤ M pl : We see from this equation that it is easy to obtain g 1. Let us define Λ cut ≡ 10 k M pl where k ∈ R and plug this into the constraint in Eq. (50). Then, we have If we demand a small gauge coupling such that g 1, then the constraint reduces to Therefore, we note that requiring a small gauge coupling such that g 1 guarantees that the cutoff scale Λ cut is lower than the Planck scale M pl and enables us to choose any sub-Planckian cutoff scale up to the upper bound in Eq. (52).
As an example, if we assume that the cutoff of our theory is given by a

Observable sector dynamics: low scale soft supersymmetry breaking interactions
In this section we investigate the mass scales of the soft supersymmetry-breaking interactions in the observable sector. We need to find under which conditions our model could be phenomenologically realistic. A full investigation of the detailed structure of the soft interactions in the observable sector requires a study that goes beyond the scope of this work, so here we will limit ourselves to general remarks and a coarse-grained analysis of necessary conditions for the viability of our model. We focus our analysis on the soft masses.
Restoring the mass dimension (so that the T, z i have canonical mass dimension 1), the soft-term potential becomes This formula is obtained by taking the following low-energy limit: F h , M pl → ∞ (where F h are the hidden-sector auxiliary F-term fields) while m 3/2 =constant [12]. Elegant examples of gravity mediation and soft SUSY breaking are simply explained in e.g. [26].
In addition, the hidden-sector superpotential can be written as Then, using we obtain At the true vacuum we have aImT /M pl = nπ, X = x, z I = 0 where n is an even integer, so the soft terms become Inserting this expression into the soft-terms potential, we get The soft-terms potential thus reduces to Next, let us consider a general expansion of the observable-sector superpotential W o where W o i···k ≡ ∂ n W o (z i )/∂z i · · · ∂z k and B 0 , S i , M ij , Y ijk are constant parameters determining masses and interactions.
We won't perform a full analysis of all possible ranges of values for B 0 , S i , M ij and Y ijk ; instead, we will simplify out analysis by setting S i = 0, so that the vacuum of the observable sector is at z i = 0, assume that for all i, j, k all M ij and Y ijk are of the same order, and set B 0 = 0. The soft terms in the scalar potential are then generated only by following terms in the expansion of W o [18].
where the M ij have mass dimension one and the Y ijk are dimensionless.
This choice also implies that such superpotential does not significantly change the cosmological constant because all the minima of the z i are located at zero. We will choose the U(1) gauge-invariant Kähler function of matter fields as follows where the first (second) term corresponds to hidden (observable) sector.
We note that only m s1 , m s2 , m s3 are free parameters. However, when we examine the kinetic term in Eq. (27) we observe that at x = 1 the kinetic terms of the matter multiplets are canonically normalized.
The condition x = 1 then gives m s3 = H √ 3 = 10 −6 M pl ∼ m 3/2 . So in this case, the free parameters reduce to m 1 and m 2 only. Notice that in the regime m s3 ∼ m 3/2 , the parameter m s1 determines the magnitude of |M ij | and m s4 , while m s2 determines that of |Y ijk |, m s5 , and m s6 .
Finally, let us investigate further the physical masses of matter scalars in the observable sector.
Here, we are going to look only at the matter scalar masses and leave a detailed study of fermion masses and interactions to a future work, since the purpose of this section is to demonstrate the existence of light scalars in the observable sector, whose masses can be smaller than that of the gravitino. Because of the soft mass parameters we found, we expect that some scalars will be as heavy as the gravitino, while other scalars could be much lighter.
To compute the scalar masses we must remember to include contributions coming from the expansion of the hidden-sector potential to second order in the observable-sector scalars z i : V h (z I , z i ) = V h (0.0) + V h ij z izj . We thus consider the general expression for the total scalar potential, which is written with the canonical mass dimensions by First, we find that masses of the hidden-sector matter scalars z I and ImT can be independently defined by tuning the magnitude of the U(1) gauge charge ∀I : q I ≡ q and the parameter a respectively such that they are positive definite. This implies that the hidden-sector fields can be heavy as much as we wish. Thus, to get an effective single-field slow-roll inflation we should make the hidden-sector matter scalars much heavier than the Hubble scale during slow roll. Their masses can be lighter than the Hubble scale before the onset of the slow-roll period, that is for very large values of X. Second, it is obvious that the inflaton mass is of the same order as the Hubble scale, i.e. m φ ∼ H, since the scalar potential is of "HI" or "Starobinsky"form and has a de Sitter vacuum, as we have seen in the previous sections.
Next, we investigate masses of the observable-sector fields. We can simplify further our analysis to make our point clearer by assuming that the quadratic term in the superpotential is diagonal M ij = δ ij M . From the total scalar potential, we find the observable-sector squared mass matrix M 2 obs at the vacuum specified by the conditions that aImT = nπ, z I = 0, and z i = 0 where Restoring the mass dimension, the mass eigenvalues are We observe that if M ∼ aA/M 2 pl (which is equivalent to the condition that m s1 ∼ H), then both masses m ± are positive definite for all X = e √ 2/3φ > 0 (or all φ), which means that during inflation the matter scalar masses are well defined (and become very light for large values of X or φ). This will be confirmed in the following.
Let us check the values of the scalar masses on the post-inflationary vacuum. Using the relations mass range 10 −8 M pl ≤ m χ ≤ M pl , which is outside the excluded region shown in Figs. 2, 3, and 4 of ref. [24]. To summarize, we found the following constraints on soft masses. First, m s1 must satisfy Eq. (71) to allow for some light scalars while m s3 is of the same order as the gravitino mass m 3/2 and m s4 is determined by the chosen value of m s1 . Notice that all these mass parameters are subject to strict constraints such as Eq. (71). Furthermore, m s2 , m s5 , and m s6 can be arbitrarily small, m s5 and m s6 are proportional to m 2 s2 and m s2 respectively, and m s2 is a free parameter.
It is worth noticing that the observable sector masses m − are compatible with the "Case 1" reheating-scenario condition of ref. [19], for which single-field plateau-potential inflation is robust under the introduction of light scalars. The parameters characterizing the reheating scenario are where Γ φ , Γ z i are the decay rates of φ and z i during the reheating phase and z i are the expectation values of matter scalars z i after inflation. We note that z i M pl 1 implies that the slow-roll inflation should begin around the minima of matter scalars, so that at the end of inflation the corresponding vacuum expectation values will be much smaller than the Planck scale M pl . Hence, as long as the above "Case 1" reheating-scenario condition is satisfied, the slow-roll inflation in our model will effectively be driven by a single inflaton field φ along the minima of the matter scalars.

Conclusions and outlook
To summarize our findings, we have seen that our model can naturally produce plateau-potential inflation at the Hubble scale with a high scale spontaneously supersymmetry breaking in the hidden sector and low scale soft supersymmetry breaking interactions with various soft masses in the observable sector. We also obtain naturally a super-EeV gravitino, which is compatible with constraint for heavy gravitino cold dark matter (i.e. 0.1 EeV m 3/2 1000 EeV) [21]. In this work, we have not investigated the specific structure and dynamics of observable-sector interactions or the detailed construction of a realistic low energy effective theory of the observable sector. It would be of obvious interest to see how far this scenario could be pursued and how to incorporate in it a supersymmetric extension of the Standard Model or a Grand Unified Theory. Models with a dynamically generated FI term, realistic observable sector, D-term inflation, and high-scale supersymmetry breaking have been studied in [27]; other uses of FI terms for inflation were presented in [28]. It would be interesting to reproduce the phenomenologically desirable features of [27,28] and other models proposed in the literature in our scenario. On a different note, it would be extremely interesting to see if the new FI term in general and in our KKLT-type scenario in particular can be obtained in string theory. In other words, it would be interesting to see if our model belongs to the string landscape or the string swampland [29] (See refs. [30,31] for recent reviews of swampland conjectures). Here we shall just mention that the magnetic Weak Gravity Conjecture given e.g. in eq. (3.7) of ref. [31] can be easily satisfied in our model.
On a more concrete note, a detailed analysis and derivation of the bounds on the new FI terms that were crucial in this paper is highly desirable. This will be the main focus of our forthcoming paper [25].