Inflation, Gravity Mediated Supersymmetry Breaking, and de Sitter Vacua in Supergravity with a Kähler-Invariant FI Term
IInflation, Gravity Mediated SupersymmetryBreaking, and de Sitter Vacua in Supergravitywith a K¨ahler-Invariant FI Term
Hun Jang and Massimo Porrati
Center for Cosmology and Particle PhysicsDepartment of Physics, New York University726 Broadway, New York, NY 10003, USA
Abstract
We use a new mechanism for generating a Fayet-Iliopoulos term in supergravity, which is notassociated to an R symmetry, to construct a semi-realistic theory of slow-roll inflation for a theorywith 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 sectorto ensure that the cutoff of the effective field theory is above the Hubble scale of inflation. Thegravitino has a super-EeV mass and supersymmetry breaking is communicated to the observablesector through gravity mediation. Some mass scales of the supersymmetry-breaking soft termsin the observable sector can be parametrically smaller than the SUSY breaking scale. If a stringrealization of the new FI term were found, our model could be the basis for a low energy effectivesupergravity description of realistic superstring models of inflation. a r X i v : . [ h e p - t h ] M a r ontents N = 1 supergrav-ity 33 Hidden sector dynamics 1: a minimal supergravity model of inflation, high-scalesupersymmetry breaking, and de Sitter vacua 64 Hidden sector dynamics 2: super-EeV gravitino mass, weak gauge coupling, and ahierarchy of energy scales 105 Observable sector dynamics: low scale soft supersymmetry breaking interactions 136 Conclusions and outlook 18 It is rather challenging to describe inflation, supersymmetry (SUSY) breaking, and de Sitter (dS)vacua in simple supergravity models and even more so in string theory. In string theory, the Kachru-Kallosh-Linde-Trivedi (KKLT) model [1] is a prototype that can give de Sitter (dS) vacua, undercertain assumptions about moduli stabilization. The effective field theory description of the KKLTmodel is a supergravity with a no-scale K¨ahler potential for its volume modulus and with a super-potential that differs from its constant no-scale form because of two non-perturbative corrections .The superpotential produces a supersymmetric Anti-de-Sitter (AdS) vacuum. In ref. [1], a mechanismwas proposed for generating dS vacua through the addition of anti-D3 brane contributions to thesuperpotential, that uplifts the AdS vacuum to dS. While the additional correction by anti-D3 branescreates dS vacua, it also deforms the shape of the scalar potential creating a “bump” which gives riseto a moduli stabilization problem [1]. As an attempt to improve on KKLT, Kachru, Kallosh, Linde,Maldacena, McAllister and Trivedi (KKLMMT) proposed a model that modifies KKLT by introducinga contribution arising from the anti-D3 tension in a highly warped compactifications [2].Both models, KKLT and the KKLMMT, contain anti-D3 branes, whose known effective field the-ory description uses nonlinear realizations of supersymmetry. The presence of nonlinearly realized The corrections come from either Euclidean D3 branes in type IIB compactifications or from gaugino condensationdue to D7 branes. M string ,then the known description of KKLT cannot accurately describe the whole energy range E (cid:46) M string .On the other hand, nothing in principle forbids the existence of some effective field theory descriptioneven in that energy range, but such description must employ a linear realization of supersymmetry,which would necessarily employ only whole multiplets. A natural question to ask from an effectivefield theory point of view is whether such a description is possible. Said differently: does a supergrav-ity with the same K¨ahler potential and superpotential as KKLT exists, that breaks supersymmetry,gives rise to an inflationary potential and a dS post-inflationary vacuum, and is valid even at energyscales where supersymmetry is restored? We answer affirmatively to this question by adding to theKKLT effective theory a new Fayet-Iliopoulos (FI) term, in the form proposed by Antoniadis, Cha-trabhuti, Isono and Knoops [3]. We will show that this FI term also generates irrelevant operatorsthat introduce a cutoff scale for the effective theory. We will also show that differently from nonlinearrealizations, this cutoff can be made larger than the supersymmetry breaking scale –and in fact evenlarger than the Planck scale.Our construction begins with the observation that, in the absence of anti-D3 branes, the super-gravity scalar potential of the KKLT model has a supersymmetric AdS vacuum and it becomes flatfor large values of the volume modulus field. The flat direction could be used for constructing a viablemodel of inflation without eta problem, if the scalar potential minimum V could be simply translatedupward by a constant, V → V + constant. This could happen if a constant positive FI term existed.This term was long thought to be forbidden in supergravity, since the only possible FI terms werethought to arise from gauging the R-symmetry [4], require an R-invariant superpotential [5] and besubject to quantization conditions when the gauged R-symmetry is compact [6]. On the other hand,recently FI terms not associated with R-symmetry were proposed, starting with ref. [7]. We use herethe K¨ahler-invariant FI term proposed in [3] and we call it “ACIK-FI” to distinguish it from manyother new FI terms suggested in the literature (for instance in [7, 8, 9, 10]). To find an approxi-mately flat potential for inflation and a dS post-inflationary vacuum, we add an ACKI-FI term tothe N = 1 supergravity describing the KKLT model without anti-D3 branes. We must remark thata field-dependent generalization of the new K¨ahler-invariant FI term has been introduced recently inref. [11], which also studies the cosmological consequences of such a term.In our model supersymmetry is spontaneously broken in a hidden sector at a very high but stillsub-Planckian scale M pl (cid:29) M S (cid:29) − M pl . We employ gravity mediation (see e.g. the review [12])to communicate the SUSY breaking to the observable sector, where supersymmetry breaking manifestsitself through the existence of explicit soft SUSY breaking terms, characterized by an energy scale M observable (cid:28) M S . The reason for a high M S is that M S controls the magnitude of non-renormalizablefermionic terms that determine the cutoff of the effective theory. This is a feature that the ACIK-FIterm shares with liberated supergravity (see e.g. [13, 14]). We assume M string < M pl , with M pl the Planck scale. N = 1 supergravity effective theory of the KKLT model. Next we add matter, which we divide intoa hidden sector and an observable sector. Supersymmetry is broken in the hidden sector and theSUSY breaking is communicated to the observable sector via gravity mediation. In Secs. 3 and 4 weprobe the hidden-sector dynamics of our model. In Sec. 3, we construct a minimal supergravity modelof plateau-potential inflation –sometimes called in the literature “Starobinsky” or “Higgs” inflation–with high scale SUSY breaking and dS vacua, using the results from Sec. 2. In Sec. 4, we explore thegravitino mass, which is very high, being well above the EeV-scale. We also study possible constraintson the ACIK-FI term by investigating the nonrenormalizable fermionic terms in the Lagrangian. Theseconstraints can be satisfied if the gauge coupling for a certain U(1) necessarily present in our model issufficiently small. They also lead to a hierarchy of energy scales. In Sec. 5 we study the observable-sector dynamics of our model by computing its soft SUSY breaking terms. A few final observationsare collected in Sec. 6. N = 1 supergravity In this section, we propose an N = 1 supergravity model that can describe the low energy effectivefield theory of inflation and moduli stabilization in KKLT-type backgrounds [1]. To do so, we firstadd 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 agauged 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 . Inflationwill come from the same potential as in the KKLT scenario. KKLT [1] argues that in string theorysome moduli can develop a non-perturbative superpotential of the form W = W + Ae − aT , (1)where T is a “volume” modulus field, which is a chiral superfield, and W , A are constants. For ourconstruction it is sufficient to compute the component action of N = 1 supergravity characterized bythe superpotential (1) and by an ACIK-FI term. Notice that Antoniadis and Rondeau have recentlystudied cosmological applications of generalized ACIK-FI terms by considering no-scale models with3 constant superpotential W = W [11]. Differently from that model, ours uses the KKLT-typesuperpotential (1).The key assumption that we will use is that both the volume modulus T and the other matter fieldsthat may exist in the superpotential are gauge-invariant under the U(1) that is used to introduce theACIK-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 withrealistic moduli stabilization and supersymmetry breaking pattern. The study of irrelevant operatorsgenerated by the ACIK-FI term will show that a low energy supersymmetry breaking is incompatiblewith demanding that the cutoff for the effective field theory is higher than the Hubble constant duringinflation. So, we take an alternative approach and break supersymmetry at a high scale in the hiddensector (as in e.g. [17] ) while keeping some of the scales of supersymmetry breaking interactions in theobservable sector low [18].To do so, we first decompose matter into a hidden sector and an observable sector. We will discussthem separately in Sections 3, 4 and 5. So we separate the field coordinates y A into y A ≡ ( T, z ˆ I ) ≡ ( { T, z I } h , { z i } o ) , (2)where ˆ I ≡ ( 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 andobservable-sector term W o : W ( y A ) ≡ W h ( T, z I ) + W o ( z i ) . (3)We further assume that the hidden-sector superpotential carries a high energy scale compared to theobservable-sector one. This implies that we decompose the F-term scalar potential into two differentparts: a hidden sector F-term potential characterized by a high energy scale and observable-sectorF-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 transformas Z i → Z i , T → T, Z I → e − q I Ω Z I . (4)Here q I denote the U(1) gauge charges of the hidden-sector chiral multiplets Z I and Ω is the chiralmultiplet containing in its lowest component the ordinary gauge parameter. We make these choicesbecause we will introduce both a new FI term generated by a gauge vector multiplet and a KKLTsuperpotential, which depends on the volume modulus T and must be gauge invariant under all gaugesymmetries. 4he superconformal action of the ACIK-FI term [3, 11] is defined by L NEW FI ≡ − ξ (cid:20) ( S ¯ S e − K ( Ze qV , ¯ Z ) ) − ( W α ( V ) W α ( V ))( ¯ W ˙ α ( V ) ¯ W ˙ α ( V )) T ( ¯ w ) ¯ T ( w ) ( V ) D (cid:21) D , (5)and the corresponding superconformal action of N = 1 supergravity with superpotential (1) and thenew the FI term is L = − S ¯ S e − K ( Ze qV , ¯ Z ) / ] D + [ S W ( Z, Z (cid:48) )] F + 12 g [ W α ( V ) W α ( V )] F + c.c. − ξ (cid:20) ( S ¯ S e − K ( Ze qV , ¯ Z ) ) − ( W α ( V ) W α ( V ))( ¯ W ˙ α ( V ) ¯ W ˙ α ( V )) T ( ¯ w ) ¯ T ( w ) ( V ) D (cid:21) D . (6)In Eqs. (5,6) S 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 , K ( Ze qV , ¯ Z ) is a K¨ahler potential gauged bya vector multiplet V ; W ( Z, Z (cid:48) ) is a superpotential; W α ( V ) is the field strength of the vector multiplet V ; ξ is the constant coefficient of ACIK-FI term; w ≡ W α ( V ) W α ( V )( S ¯ S e − K ( Z, ¯ Z ) ) and ¯ w ≡ ¯ W ˙ α ( V ) ¯ W ˙ α ( V )( S ¯ S e − K ( Z, ¯ Z ) ) arecomposite multiplets, T ( X ) , ¯ T ( X ) are chiral projectors, and ( V ) D is a real multiplet, whose lowestcomponent is the auxiliary field D of the vector multiplet V .Next, we write the following K¨ahler potential, invariant under the same U(1) that generates theACIK-FI term K ( Z A e qV , ¯ Z ¯ A ) ≡ − T + ¯ T − Φ( Z I e qV , ¯ Z ¯ I ; Z i , ¯ Z ¯ i ) / , (7)where Φ is a real function of the matter multiplets Z i , Z I and the two terms in the superpotential W ≡ W h + W o are the hidden-sector term W h ( T ) ≡ W + Ae − aT (8)and the observable-sector superpotential W o ( Z i ) ≡ B + S i Z i + M ij Z i Z j + Y ijk Z i Z j Z k + · · · , (9)where B , S i , M ij , Y ijk are constant coefficients. We will choose Φ to be sum of a term containing onlyhidden-sector fields and one containing only those of the observable sectorΦ = Φ h ( Z I e qV , ¯ Z ¯ I ) + Φ o ( Z i , ¯ Z ¯ i ) . (10)The supergravity G-function corresponding to our model is then G ( y A , ¯ y ¯ A ) ≡ K ( y A , ¯ y ¯ A ) + ln | W ( y A ) | = − T + ¯ T − Φ( z ˆ I , ¯ z ¯ˆ I )3 ] + ln | W h ( T, z I ) + W o ( z i ) | . (11)5he F-term supergravity scalar potential is given by the formula V F ≡ e G ( G A G A ¯ B G ¯ B − V F = − X [( W h + W o ) ¯ W h ¯ T + ( ¯ W h + ¯ W o ) W hT ]+ 13 | W hT | X + 19 | W hT | X [Φ I Φ I ¯ J Φ ¯ J + Φ i Φ i ¯ j Φ ¯ j ]+ 13 1 X [ W hT (Φ I Φ I ¯ J ¯ W h ¯ J + Φ i Φ i ¯ j ¯ W o ¯ j ) + ¯ W h ¯ T ( W hI Φ I ¯ J Φ ¯ J + W oi Φ i ¯ j Φ ¯ j )]+ 1 X [ W hI Φ I ¯ J ¯ W h ¯ J + W oi Φ i ¯ j ¯ W o ¯ j ] . (12)When matter scalars are charged under a gauge group there exists also a D-term contribution tothe scalar potential, V D . In our model, we find it to be V D = 12 g (cid:16) ξ + (cid:88) I ( q I z I G I + q I ¯ z ¯ I G ¯ I ) (cid:17) = 12 g (cid:16) ξ + q I z I Φ I + q I ¯ z ¯ I Φ ¯ I X (cid:17) , (13)where X ≡ T + ¯ T − Φ / g is the gauge coupling constant and ξ is the ACIK-FI constant. Rememberthat only hidden-sector chiral matter multiplets are charged under the U(1). The scalar potential isthe sum of two terms. One, V h contains the D-term contribution and the F-term potential of thehidden sector, depends on the high mass scale M S and is O ( H M pl ) during inflation; the other, V soft contains the observable sector scalars and depends only on low mass scales: V = V h + V soft , (14)where V h ≡ V D − W hT ¯ W h + ¯ W h ¯ T W h X + | W hT | X (cid:16) X + 13 Φ I Φ I ¯ J Φ ¯ J (cid:17) + 13 1 X [ W hT Φ I Φ I ¯ J ¯ W h ¯ J + ¯ W h ¯ T W hI Φ I ¯ J Φ ¯ J ] + 1 X W hI Φ I ¯ J ¯ W h ¯ J , (15) V soft ≡ − X [ W o ¯ W h ¯ T + ¯ W o W hT ] + 19 | W hT | X Φ i Φ i ¯ j Φ ¯ j + 13 1 X [ W hT Φ i Φ i ¯ j ¯ W o ¯ j + ¯ W h ¯ T W oi Φ i ¯ j Φ ¯ j ] + 1 X W oi Φ i ¯ j ¯ W o ¯ j . (16) In this section, we explore a minimal supergravity model of high-scale supersymmetry breaking andplateau-potential inflation through gravity mediation and no-scale K¨ahler potential. We investigatefirst the hidden sector dynamics. We have assumed that the hidden-sector potential depends on a highenergy scale and dominates over the observable-sector one. Hence, it is reasonable to minimize the6idden-sector potential first. Let us compute now the F-term potential in the hidden sector. Recallingthat the KKLT superpotential is W h ( T ) ≡ W + Ae − aT , (17)and redefining W ≡ − cA , we rewrite it as W h ( T ) = A ( e − aT − c ) , (18)where a, c, A are positive constants. Note that W hI = ∂W h /∂z I = 0.Since we defined X ≡ T + ¯ T − Φ /
3, the KKLT superpotential gives W hT = − aAe − aT , | W hT | = a A e − a ( T + ¯ T ) = a A e − a ( X +Φ / , (19) W hT ¯ W h + ¯ W h ¯ T W h = − aA e − aT ( e − a ¯ T − c ) − aA e − a ¯ T ( e − aT − c )= − aA e − a ( T + ¯ T ) + aA c ( e − aT + e − a ¯ T )= − aA e − a ( X +Φ / + aA c ( e − a (Re T + i Im T ) + e − a (Re T − i Im T ) )= − aA e − a ( X +Φ / + 2 aA ce − a Re T cos( a Im T )= − aA e − a ( X +Φ / + 2 acA e − a ( X +Φ / / cos( a Im T ) (20) | W h | = A | e − aT − c | = A ( e − aT − c )( e − a ¯ T − c ) = A ( e − a ( T + ¯ T ) − c ( e − aT + e − a ¯ T ) + c )= A ( e − a ( X +Φ / − ce − a ( X +Φ / / cos( a Im T ) + c ) . (21)Here we have used the following transformation from the complex coordinate T to two real coordinates X, Im T : T = Re T + i Im T = 12 (cid:16) X + Φ3 (cid:17) + i Im T, (22)which gives e − aT = e − a ( X + Φ3 ) e − ai Im T . Remember that X ≡ T + ¯ T − Φ / W hI = 0, the corresponding hidden-sector F-term scalar potential is given by V hF = − W hT ¯ W h + ¯ W h ¯ T W h X + | W hT | X (cid:16) X + 13 Φ I Φ I ¯ J Φ ¯ J (cid:17) = − X (cid:18) − aA e − a ( X +Φ / + 2 acA e − a ( X +Φ / / cos( a Im T ) (cid:19) + 13 X (cid:16) X + 13 Φ I Φ I ¯ J Φ ¯ J (cid:17) a A e − a ( X +Φ / . (23)Since we assume that the D-term potential belongs to the hidden sector, the hidden-sector totalscalar potential can be written as V h = V D + V hF = 12 g (cid:16) ξ + q I z I Φ I + q I ¯ z ¯ I Φ ¯ I X (cid:17) − X (cid:18) − aA e − a ( X +Φ / + 2 acA e − a ( X +Φ / / cos( a Im T ) (cid:19) + 13 X (cid:16) X + 13 Φ I Φ I ¯ J Φ ¯ J (cid:17) a A e − a ( X +Φ / . (24)7e define the SUSY breaking scale M S in terms of the scalar potential V h and the gravitino mass m / as V + ≡ M S = V h + 3 m / = V h + 3 X A ( e − a ( X +Φ / − ce − a ( X +Φ / / cos( a Im T ) + c ) . (25)To investigate the moduli stabilization, we identify the canonically normalized fields by inspectionof the kinetic terms, which are given by L K = Φ ˆ I ¯ˆ J X g µν D µ z ˆ I D ν ¯ z ¯ˆ J + 34 X g µν ∂ µ X∂ ν X + 3 X g µν [ ∂ µ Im T − (Im D µ z ˆ I Φ ˆ I / ∂ ν Im T − (Im D ν z ˆ I Φ ˆ I / , (26)where D µ ≡ ∂ µ − iq ˆ I A µ is the U(1) gauge covariant derivative for the matter multiplets z ˆ I = ( z I , z i )with gauge charge q ˆ I = ( q I (cid:54) = 0 , q i = 0), and A µ is the corresponding gauge field.After performing another field redefinition X ≡ e √ / φ , we find L K = Φ I ¯ J e − √ / φ g µν D µ z I D ν ¯ z ¯ J + 12 g µν ∂ µ φ∂ ν φ +3 e − √ / φ g µν [ ∂ µ Im T − (Im D µ z I Φ I / ∂ ν Im T − (Im D ν z I Φ I / . (27)Notice that φ is canonically normalized, while the other fields z I , Im T are so only when φ is small.Now, let us investigate the scalar potential vacuum. First of all, we find the minimum with respectto the matter scalars z ˆ I , ∂V∂z ˆ I = 0 = ⇒ Φ ˆ I = 0 . (28)If we choose a real function such that Φ ˆ I = 0 implies Φ = 0 together with z ˆ I = 0 then at this vacuumthe scalar potential becomes V h = 12 g ξ − X (cid:18) − aA e − aX + 2 acA e − aX/ cos( a Im T ) (cid:19) + 13 X a A e − aX . (29)Next, we consider the vacuum with respect to the Im T field. We find the vacuum at a Im T = nπ ,where n is an even integer, leading to cos( a Im T ) = 1 and V h = 12 g ξ + 2 aA X e − aX − acA X e − aX/ + a A X e − aX . (30)Next, let us find the vacuum with respect to the φ field. Recalling that X = e √ / φ , calling (cid:104) φ (cid:105) the vacuum expectation value of φ and setting φ = (cid:104) φ (cid:105) = (cid:113) ln (cid:104) X (cid:105) = (cid:113) ln x , where X = (cid:104) X (cid:105) ≡ x , The observable-sector superpotential W o can shift the VEVs of the scalars in the observable sector z i , but sincethose VEVs must be in any case small compared to H and M pl we can approximately set z i = 0. Moreover, in our toyexample in Section 5 we will choose a superpotential that indeedgives a minimum at z i = 0. When cos( a Im T ) = 1, the second derivative of the potential can be positive, which means that the stationary pointis a minimum.
8e have ∂V h ∂φ (cid:12)(cid:12)(cid:12)(cid:12) φ = (cid:104) φ (cid:105) = ∂V h ∂X (cid:12)(cid:12)(cid:12)(cid:12) X = x ∂X∂φ (cid:12)(cid:12)(cid:12)(cid:12) φ = (cid:104) φ (cid:105) = 0 = ⇒ ∂V h ∂X (cid:12)(cid:12)(cid:12)(cid:12) X = x = 0 , (31)which gives ∂V h ∂X (cid:12)(cid:12)(cid:12)(cid:12) X = x = − aA x e − ax − a A x e − ax + 4 acA x e − ax/ + a cA x e − ax/ − a A x e − ax − a A x e − ax = 0 . (32)At first glance, this equation seems a little complicated, but after a short calculation, we can obtainthe following simple relation ∂V h ∂X (cid:12)(cid:12)(cid:12)(cid:12) X = x = 0 = ⇒ c = (cid:16) ax (cid:17) e − ax/ . (33)Inserting the value of c into V h , we obtain the following equation V h = 12 g ξ + 2 aA X e − aX − aA X (cid:16) ax (cid:17) e − ax/ e − aX/ + a A X e − aX , (34)where X = e √ / φ .Then, the vacuum energy at X = x is given by V h | X = x = 12 g ξ − a A e − ax x ≡ Λ , (35)where Λ is defined to be the post-inflationary cosmological constant, and the SUSY breaking scale isgiven by V + | X = x = V h | X = x + 3 X A ( e − aX − ce − aX/ + c ) (cid:12)(cid:12)(cid:12)(cid:12) X = x = Λ + 3 x A ( e − ax/ − c ) = Λ + 3 A x a x e − ax
9= Λ + a A e − ax x = 12 g ξ ≡ M S , (36)where M S is by definition the SUSY breaking scale. We can set Λ to any value we wish, in particularwe can choose it to be Λ ∼ − .Here, we point out that the term g ξ governs the magnitude of the total scalar potential, andsimultaneously controls the scale of spontaneously supersymmetry breaking. Hence, if we want thatthe scalar potential describes inflation, we need to impose M S = 12 g ξ = H M pl ≡ M I , (37)where H is the Hubble parameter, and M I is defined to be the mass scale of inflation.9e then identify A = (cid:114) x ( M I − Λ) e ax a , W = − cA = − (cid:16) ax (cid:17)(cid:114) x ( M I − Λ) a . (38)Substituting the above parameters A, W , M I into the hidden-sector potential, we can obtain aplateau inflation potential V h = M I − ( M I − Λ) x (cid:20) e − a ( X − x ) / aX (cid:16) − e − a ( X − x ) / + ax (cid:17) − e − a ( X − x ) X (cid:21) , (39)where X = e √ / φ and φ is defined to be the inflaton. Notice that the inflaton mass after inflation isof order of the Hubble scale, i.e. m φ ∼ H = 10 − M pl .We note that the hidden-sector scalar potential V h has a plateau, so it is of HI type (in thenotations of ref. [19]). Furthermore, it depends only on four parameters, which are: the vacuumexpectation value of X ( i.e. x ≡ (cid:104) X (cid:105) ); the KKLT parameter a in the superpotential, which will bedetermined according to the type of the nonperturbative correction we choose ; the inflation scale M I , and the post-inflationary cosmological constant Λ. At X = x , the potential indeed reducesto 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 KKLTsuperportential come from gaugino condensation [1], a smaller parameter a corresponds to more D7branes being stacked. In this section, we investigate some physical implications that can be obtained from our model. Firstof all, let us find the gravitino mass after inflation, which is generated by the high-scale SUSY breakingin the hidden sector. It is given by m / = e G = | W h | X = A X ( e − a ( X +Φ / − ce − a ( X +Φ / / cos( a Im T ) + c ) (cid:12)(cid:12)(cid:12)(cid:12) z I =0 ,a Im T =0 ,X = x = 3 x ( M I − Λ) e ax a x ( e − ax − ce − ax/ + c ) = 3 x ( M I − Λ) e ax a x ( c − e − ax/ ) = ( M I − Λ)3 = ⇒ m / ≈ H √ − M pl ∼ GeV = 10 EeV , (40)which is compatible with the case of EeV-scale gravitino cold dark matter candidates. This is notsurprising because we are considering the same high-scale supersymmetry breaking scale as in [20, 21, For example, if we consider a nonperturbative correction due to gaugino condensation, then we find a = πN c for anon-abelian gauge group SU ( N c ) where N c is interpreted as the number of coincident D7 branes being stacked [1]. O ( H ), irrespective of the ultraviolet cutoff.Next, we explore possible constraints on the FI term by analyzing the fermionic nonrenormalizableinteraction terms that are induced by such term (5). Schematically the general fermionic terms havethe form L F ⊃ ξM m +2 pl D − m − p O (10 − p ) F , (41)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 thecomposite chiral fields T ( ¯ w ) and ¯ T ( w ) defined after Eq. (6), and p = 0 ,
1. Detailed calculations willbe 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 thefollowing Lagrangian for them L aux D = 12 D − i ( G i k i − G ¯ i k ¯ i ) D − ξD ≡ D − ( ξ (cid:48) + ξ ) D, (42)where ξ is the new FI constant while ξ (cid:48) ≡ i ( G i K i − G ¯ i K ¯ i ) 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 symmetryon the scalar fields. G is the standard supergravity G-function [16]. Restoring the dependence on thegauge coupling constant g we have L aux D = 12 g D − ξ (cid:48) D − ξD = 12 g D − ( ξ (cid:48) + ξ ) D. (43)After solving the equation of motion for D , we find the solution D = g M pl ( ξ + ξ (cid:48) ) (= g ( ξ + ξ (cid:48) ) when M pl = 1) . (44)Plugging this solution into the fermionic terms in Eq. (41) we obtain L F ⊃ ξM m +2 pl ( g M pl ( ξ + ξ (cid:48) )) − m − p O (10 − p ) F = ξ ( g ( ξ + ξ (cid:48) )) − m − p M − ppl O (10 − p ) F . (45)The fermionic nonrenormalizable interactions generated by the ACIK-FI term introduce a strongcoupling scale that sets the limit of validity of the effective field theory description. If we demandthat the theory is valid up to some cutoff scale Λ cut , we find the folloiwing constraint on the ACIK-FIterm: ξ ( g ( ξ + ξ (cid:48) )) − m − p (cid:46) (cid:18) M pl Λ cut (cid:19) − p . (46)11e must also examine the constraints on the post-inflation vacuum, that is the true vacuum, inwhich ξ (cid:48) = 0. In this case, we obtain( g − ) (cid:18) Λ cut M pl (cid:19) − p < ( g − ) 2(2 m + 4 − p )(2 m + 3 − p ) (cid:18) Λ cut M pl (cid:19) − p (cid:46) ξ. (47)This inequality reduces to the following: for all Λ cut ≤ M pl , we obtain g − (cid:18) Λ cut M pl (cid:19) ≤ ξ. (48)Now we are ready to ask how does this constraint affect our supergravity model of inflation. To answerthis, let us get back to the definition of the inflation scale (restoring the mass dimension) M I = 12 g ξ M pl = ⇒ ξ = √ M I M pl g − ∼ − g − . (49)Inserting this equation into the constraints, we find that for Λ cut ≤ M pl :10 (cid:18) Λ cut M pl (cid:19) ≤ g (50)We see from this equation that it is easy to obtain g (cid:46)
1. Let us define Λ cut ≡ k M pl where k ∈ R and plug this into the constraint in Eq. (50). Then, we have10 k ≤ g. (51)If we demand a small gauge coupling such that g (cid:46)
1, then the constraint reduces to10 k ≤ g (cid:46) ⇒ k (cid:46) − .
25 = ⇒ Λ cut (cid:46) − . M pl < M pl . (52)Therefore, we note that requiring a small gauge coupling such that g (cid:46) cut is lower than the Planck scale M pl and enables us to choose any sub-Planckian cutoff scaleup to the upper bound in Eq. (52).As an example, if we assume that the cutoff of our theory is given by a Grand Unified (GUT)scale ( i.e. Λ cut ∼ − M pl (cid:29) H ≈ − M pl ), then we find that the gauge coupling must onlyobey 10 − ≤ g , so it can easily obey g (cid:46)
1. Remember that our SUSY breaking scale was givenby M S = M I = (cid:112) HM pl ∼ − . M pl , which is slightly below the GUT-scale cutoff, i.e. M S < Λ cut = Λ GUT = 10 − M pl . Consequently, we have to consider an effective theory with the followinghierarchy of scales: H (cid:28) M S (cid:46) Λ cut = Λ GUT < M pl , to ensure that the gauge coupling constantobeys O (10 − ) (cid:46) g (cid:46)
1. We may also set the cutoff at the string scale M string ∼ − M pl . In thiscase, the hierarchy of mass scales is given by H (cid:28) Λ cut = M string < M S < M pl .12 Observable sector dynamics: low scale soft supersymmetry break-ing interactions
In this section we investigate the mass scales of the soft supersymmetry-breaking interactions in theobservable sector. We need to find under which conditions our model could be phenomenologicallyrealistic. A full investigation of the detailed structure of the soft interactions in the observable sectorrequires a study that goes beyond the scope of this work, so here we will limit ourselves to generalremarks and a coarse-grained analysis of necessary conditions for the viability of our model. We focusour analysis on the soft masses.Restoring the mass dimension (so that the
T, z i have canonical mass dimension 1), the soft-termpotential becomes V soft ≡ − M pl X [ W o ¯ W h ¯ T + ¯ W o W hT ] + 19 | W hT | M pl X Φ i Φ i ¯ j Φ ¯ j + 13 1 M pl X [ W hT Φ i Φ i ¯ j ¯ W o ¯ j + ¯ W h ¯ T W oi Φ i ¯ j Φ ¯ j ] + 1 X W oi Φ i ¯ j ¯ W o ¯ j . (53)This formula is obtained by taking the following low-energy limit: F h , M pl → ∞ (where F h arethe hidden-sector auxiliary F-term fields) while m / =constant [12]. Elegant examples of gravitymediation and soft SUSY breaking are simply explained in e.g. [26].In addition, the hidden-sector superpotential can be written as W h = A ( e − aT/M pl − c ) = M pl (cid:114) x ( M I − Λ) e ax a ( e − aT/M pl − (1 + ax/ e − a/ )= M pl (cid:114) x ( M I − Λ) e ax a ( e − a ( X +Φ / M pl ) / e − ia Im T/M pl − (1 + ax/ e − ax/ ) , (54)where we have used e − aT = e − a ( X +Φ / M pl ) e − ai Im T/M pl .Then, using W hT = − M pl aAe − a ( X +Φ / M pl ) / e − ia Im T/M pl , | W hT | = 1 M pl a A e − a ( X +Φ / M pl ) , (55)we obtain V soft ≡ aAe − a ( X +Φ / M pl ) / M pl X [ W o e ia Im T/M pl + ¯ W o e − ia Im T/M pl ] + 19 a A e − a ( X +Φ / M pl ) M pl X Φ i Φ i ¯ j Φ ¯ j − aAe − a ( X +Φ / M pl ) / M pl X [ e − ia Im T/M pl Φ i Φ i ¯ j ¯ W o ¯ j + e ia Im T/M pl W oi Φ i ¯ j Φ ¯ j ] + 1 X W oi Φ i ¯ j ¯ W o ¯ j . (56)At the true vacuum we have a Im T /M pl = nπ, X = x, z I = 0 where n is an even integer, so the13oft terms become V soft ≡ aAe − ax/ M pl x [ W o + ¯ W o ] + 19 a A e − ax M pl x Φ i Φ i ¯ j Φ ¯ j − aAe − ax/ M pl x [Φ i Φ i ¯ j ¯ W o ¯ j + W oi Φ i ¯ j Φ ¯ j ] + 1 x W oi Φ i ¯ j ¯ W o ¯ j . (57)From A = (cid:113) x ( M I − Λ) e ax a M pl ≈ √ a x / e ax/ M I M pl , we find aAe − ax/ = √ x / M I M pl . Inserting thisexpression into the soft-terms potential, we get V soft ≡ √ x / M I M pl M pl x [ W o + ¯ W o ] + 19 ( √ x / M I M pl ) M pl x Φ i Φ i ¯ j Φ ¯ j − √ x / M I M pl M pl x [Φ i Φ i ¯ j ¯ W o ¯ j + W oi Φ i ¯ j Φ ¯ j ] + 1 x W oi Φ i ¯ j ¯ W o ¯ j . (58)The soft-terms potential thus reduces to V soft ≡ √ x − / M I M pl [ W o + ¯ W o ] + 13 M I M pl x Φ i Φ i ¯ j Φ ¯ j − √ x − / M I M pl [Φ i Φ i ¯ j ¯ W o ¯ j + W oi Φ i ¯ j Φ ¯ j ] + 1 x W oi Φ i ¯ j ¯ W o ¯ j . (59)Next, let us consider a general expansion of the observable-sector superpotential W o W o ( z i ) = (cid:88) n =0 W oi ··· k n ! z i · · · z k = B + S i z i + M ij z i z j + Y ijk z i z j z k + · · · , (60)where W oi ··· k ≡ ∂ n W o ( z i ) /∂z i · · · ∂z k and B , S i , M ij , Y ijk are constant parameters determining massesand interactions.We won’t perform a full analysis of all possible ranges of values for B , 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. The soft terms inthe scalar potential are then generated only by following terms in the expansion of W o [18]. W o ( z i ) = M ij z i z j + Y ijk z i z j z k , (61)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 cosmologicalconstant because all the minima of the z i are located at zero. We will choose the U(1) gauge-invariantK¨ahler function of matter fields as followsΦ = δ I ¯ J z I ¯ z ¯ J + δ i ¯ j z i ¯ z ¯ j , (62)where the first (second) term corresponds to hidden (observable) sector.14ith our simplifying assumptions we obtain V soft = 2 √ x − / M I M pl (cid:104) ( M ij z i z j + Y ijk z i z j z k ) + c.c. (cid:105) + M I x − M pl δ i ¯ j z i ¯ z ¯ j + x − [ M ij z j + Y ijk z j z k ] δ i ¯ j [ ¯ M ¯ i ¯ j ¯ z ¯ i + ¯ Y ¯ i ¯ j ¯ k ¯ z ¯ i ¯ z ¯ k ] . (63)We also find the magnitude of the corresponding soft parameters from Eq. (63) as2 √ M I M pl x − / | M ij | ≡ m s , √ M I M pl x − / | Y ijk | ≡ m s , M I M pl x − ≡ m s , | M ij | x − ≡ m s , | Y ijk | x − ≡ m s M pl , | M ij || Y ijk | x − ≡ m s . (64)We observe that during inflation (for large X or φ ) all the soft mass parameters are very small. Also,the above result give us the following relations x = M I m s M pl = H m s , | M ij | = 16 M I M pl m s m s = 16 H m s m s = √ m s H x / , | Y ijk | = 16 M I M pl m s m s ,m s = 14 m s m s , m s = 14 (cid:18) m s m s (cid:19) M pl , m s = 14 (cid:18) m s m s (cid:19) m s . (65)We note that only m s , m s , m s are free parameters. However, when we examine the kinetic term inEq. (27) we observe that at x = 1 the kinetic terms of the matter multiplets are canonically normalized.The condition x = 1 then gives m s = H √ = 10 − M pl ∼ m / . So in this case, the free parametersreduce to m and m only. Notice that in the regime m s ∼ m / , the parameter m s determines themagnitude of | M ij | and m s , while m s determines that of | Y ijk | , m s , and m s .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 massesand interactions to a future work, since the purpose of this section is to demonstrate the existence oflight scalars in the observable sector, whose masses can be smaller than that of the gravitino. Becauseof 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 expan-sion 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 i ¯ j z i ¯ z ¯ j . We thus consider the general expression for the total scalar potential, which is15ritten with the canonical mass dimensions by V = V D + V hF + V soft = 12 g M pl (cid:16) ξ + q I z I Φ I + q I ¯ z ¯ I Φ ¯ I XM pl (cid:17) − X M pl (cid:18) − aA e − a ( X +Φ / M pl ) + 2 acA e − a ( X +Φ / M pl ) / cos( a Im T /M pl ) (cid:19) + 13 X M pl (cid:16) X + 13 M pl Φ I Φ I ¯ J Φ ¯ J (cid:17) a A e − a ( X +Φ / M pl ) + aAe − a ( X +Φ / M pl ) / M pl X [ W o e ia Im T/M pl + ¯ W o e − ia Im T/M pl ] + 19 a A e − a ( X +Φ / M pl ) M pl X Φ i Φ i ¯ j Φ ¯ j − aAe − a ( X +Φ / M pl ) / M pl X [ e − ia Im T/M pl Φ i Φ i ¯ j ¯ W o ¯ j + e ia Im T/M pl W oi Φ i ¯ j Φ ¯ j ] + 1 X W oi Φ i ¯ j ¯ W o ¯ j . (66)First, we find that masses of the hidden-sector matter scalars z I and Im T can be independentlydefined by tuning the magnitude of the U(1) gauge charge ∀ I : q I ≡ q and the parameter a respectivelysuch that they are positive definite. This implies that the hidden-sector fields can be heavy as muchas we wish. Thus, to get an effective single-field slow-roll inflation we should make the hidden-sectormatter scalars much heavier than the Hubble scale during slow roll. Their masses can be lighter thanthe 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 thescalar potential is of “HI” or “Starobinsky”form and has a de Sitter vacuum, as we have seen in theprevious sections.Next, we investigate masses of the observable-sector fields. We can simplify further our analysisto 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 obs at the vacuum specified by the conditions that a Im T = nπ , z I = 0, and z i = 0 M obs ≡ (cid:32) V i ¯ j V ij V ¯ i ¯ j V ¯ ij (cid:33) (67)where V i ¯ j = − a A X δ i ¯ j e − aX + ca A X δ i ¯ j e − aX/ − a A X e − aX δ i ¯ j + a A X e − aX δ i ¯ j + 1 X W oil Φ l ¯ n ¯ W o ¯ n ¯ j = 1 X W oil Φ l ¯ n ¯ W o ¯ n ¯ j − a A X e − aX δ i ¯ j = (cid:16) M X − a A X e − aX (cid:17) δ i ¯ j ,V ij = aA X e − aX/ W oij = aA X e − aX/ M δ ij . (68)16estoring the mass dimension, the mass eigenvalues are m ± ≡ (cid:16) M X − a A X M pl e − aX (cid:17) ± aA X M pl e − aX/ M = 1 X (cid:32) M ± aA M pl e − aX/ (cid:33) − a A X M pl e − aX . (69)We observe that if M ∼ aA/M pl (which is equivalent to the condition that m s ∼ H ), then bothmasses m ± are positive definite for all X = e √ / φ > φ ), which means that during inflationthe matter scalar masses are well defined (and become very light for large values of X or φ ). This willbe confirmed in the following.Let us check the values of the scalar masses on the post-inflationary vacuum. Using the relations A = M pl (cid:113) x ( M I − Λ) e ax a , c = (1 + ax ) e − ax/ , and M = √ m s H x / in Eq. (65) and setting Λ ≈ X = x = 1, where the kinetic terms of the matter scalars are canonically normalized, we obtain m ± = 34 M pl M I m s − M I M pl ± m s = (cid:16) k ± k − (cid:17) H , (70)in which we define m s ≡ kH (where k > M I /M pl = H ). We notice that physical masses ofscalars are determined only by the “free” parameter m s (or k ) and the Hubble mass H . Positivity ofthe physical masses “ m ± ” imposes the inequality m + : −
23 + 23 √ < k, m − : 23 + 23 √ < k = ⇒
23 + 23 √ < k (71)Then, with this inequality, we can choose an arbitrary value of k such that23 + 23 √ < k = 23 + 23 (cid:113) m − /H ) = −
23 + 23 (cid:113) m /H ) (72)allowing one physical mass m − to be parametrically lighter than the other physical mass m + as m = 4 H (cid:16) (cid:113) m − /H ) (cid:17) + 3 m − . (73)We note that m s = √ kH ∼ H when m − (cid:28) H , implying that m ± > m − can be much smaller than the Hubble scale, while the otherphysical mass are of the order of the Hubble scale: m − (cid:28) H, m + (cid:38) H. (74)From this we note that in the observable sector after inflation (that is at x = 1) one physical mass m − can be lighter than that of the gravitino, while the other physical mass m + becomes of the sameorder of the gravitino mass. We also note that the matter scalar with masses of order of the super-EeVgravitino mass ( ∼ − M pl ) may be a candidate for heavy dark matter candidate, because it is in the17ass range 10 − M pl ≤ m χ ≤ M pl , which is outside the excluded region shown in Figs. 2, 3, and 4 ofref. [24]. To summarize, we found the following constraints on soft masses. First, m s must satisfyEq. (71) to allow for some light scalars while m s is of the same order as the gravitino mass m / and m s is determined by the chosen value of m s . Notice that all these mass parameters are subject tostrict constraints such as Eq. (71). Furthermore, m s , m s , and m s can be arbitrarily small, m s and m s are proportional to m s and m s respectively, and m s 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 robustunder the introduction of light scalars. The parameters characterizing the reheating scenario areΓ φ < Γ z i < m z i ∼ m − < H, (cid:10) z i (cid:11) M pl (cid:28) , (75)where Γ φ , Γ z i are the decay rates of φ and z i during the reheating phase and (cid:10) z i (cid:11) are the expectationvalues of matter scalars z i after inflation. We note that (cid:104) z i (cid:105) M pl (cid:28) M pl . Hence, as long as the above“Case 1” reheating-scenario condition is satisfied, the slow-roll inflation in our model will effectivelybe driven by a single inflaton field φ along the minima of the matter scalars. To summarize our findings, we have seen that our model can naturally produce plateau-potentialinflation at the Hubble scale with a high scale spontaneously supersymmetry breaking in the hiddensector and low scale soft supersymmetry breaking interactions with various soft masses in the observ-able sector. We also obtain naturally a super-EeV gravitino, which is compatible with constraint forheavy gravitino cold dark matter ( i.e. . (cid:46) m / (cid:46) Acknowledgments
M.P. is supported in part by NSF grant PHY-1915219.
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