From Hybrid to Quadratic Inflation With High-Scale Supersymmetry Breaking
aa r X i v : . [ h e p - ph ] J u l F ROM H YBRID TO Q UADRATIC I NFLATION W ITH H IGH -S CALE S UPERSYMMETRY B REAKING C ONSTANTINOS P ALLIS AND Q AISAR S HAFI Departament de F´ısica Te`orica and IFIC, Universitat de Val`encia-CSIC, E-46100 Burjassot, SPAIN e-mail address: [email protected] Bartol Research Institute, Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA e-mail address: [email protected] A BSTRACT : Motivated by the reported discovery of inflationary gravity waves by the B
ICEP r ≃ . and scalar spectral index n s ≃ . , correspondingto quadratic (chaotic) inflation. The important new ingredients are the high-scale, (1 . − · GeV, softsupersymmetry breaking mass for the gauge singlet inflaton field and a shift symmetry imposed on the K¨ahlerpotential. The end of inflation is accompanied, as in the earlier hybrid inflation models, by the breaking of agauge symmetry at (1 . − . · GeV , comparable to the grand-unification scale.
PACs numbers: 98.80.Cq, 12.60.Jv I. I NTRODUCTION
The discovery of B-modes in the polarization of the cosmicmicrowave background radiation at large angular scales by theB
ICEP r = 0 . +0 . − . – after substraction of a dust foreground. Al-though other interpretations [2, 3] of this result are possible, itmotivates us to explore how realistic supersymmetric ( SUSY )inflation models can accommodate such large r values.The textbook quadratic inflationary model [4] predicting r = 0 . − . , and a (scalar) spectral index n s = 0 . − . , seems to be in good agreement with B ICEP r ) andthe WMAP [5] and Planck [6] measurements ( n s ). Quadraticinflation can be accompanied by a Grand Unified Theory ( GUT ) phase transition in non-supersymmetric inflation mod-els, based either on the Coleman-Weinberg or Higgs [7] poten-tial, which yield predictions for n s that more or less overlapwith the prediction of the quadratic model [8, 9]. However,significant differences appear between the predictions of r inthese models which can be settled through precision measure-ments. The consistent supersymmetrization of these models isa highly non-trivial task due to the trans-Planckian values ofthe inflaton field which aggravate the well-known η -problemwithin supergravity ( SUGRA ).One of the more elegant SUSY models which nicely com-bines inflation with a GUT phase transition is the model of
F-term hybrid inflation [10, 11] – referred as
FHI . It is basedon a unique renormalizable superpotential, dictated by a U (1) R-symmetry, employs sub-Planckian values for the inflatonfield and can be naturally followed by the breaking of a GUTgauge symmetry, G , such as G B − L = G SM × U (1) B − L [12] –where G SM = SU (3) C × SU (2) L × U (1) Y is the gauge groupof the Standard Model ( SM ) – G LR = SU (3) C × SU (2) L × SU (2) R × U (1) B − L [13], and flipped SU (5) [14], with gaugesymmetry G X = SU (5) × U (1) X . The embedding of thesimplest model of FHI within a GUT based on a higher gaugegroup may suffer from the production of disastrous cosmicdefects which can be evaded, though, by using shifted [15] orsmooth [16] FHI. In the simplest realization of FHI the standard [10] super-potential is accompanied by a minimal (or canonical) K¨ahlerpotential. The resulting n s is found to be in good agreementwith the WMAP and Planck data after including in the infla-tionary potential radiative corrections ( RCs ) [10] and the softSUSY breaking ( SSB ) linear term [12, 18] – with a mass pa-rameter in the TeV range – a SSB mass term for the inflaton inthe same energy region can be ignored in this analysis. Thisscenario yields [12] r values which lie many orders of magni-tude below the measurement reported [1] by B ICEP
2. A moreelaborate extension of this standard FHI scenario exploits non-minimal, quasi-canonical K¨ahler potentials [19, 21] or SSBmass of magnitude as large as GeV for the inflaton field[20]. Depending on the underlying assumptions, the predic-tions for r are considerably enhanced compared to the mini-mal scenario of Ref. [12, 18]. Thus, r values as large as . to . have been reported [20, 21]; this fact certainly puts r in the observable range, but it still remains an order of magni-tude below the B ICEP r values reported by B ICEP e m ∼ GeV, which can be identifiedwith the SSB mass of the inflaton. In the context of high-scaleSUSY [24, 25], such a large SSB scale can become consistentwith the LHC results [26] on the mass, m h ≃ GeV, of theSM Higgs boson, h . The end of inflation can be accompaniedby the breaking of some gauge symmetry such as G LR or G X with the gauge symmetry breaking scale M assuming valuesclose to the SUSY GUT scale M GUT ≃ . · GeV .I TheInflationaryScenario Below, we describe in Sec. II the basic ingredients of ourinflationary scenario. Employing a number of constraints pre-sented in Sec. III, we provide restrictions on the model pa-rameters in Sec. IV. Our conclusions are summarized inSec. V. Henceforth we use units where the reduced Planckscale m P = 2 . · GeV is taken equal to unity.
II. T HE I NFLATIONARY S CENARIO A . T HE GUT S
YMMETRY B REAKING
In the standard FHI we adopt the superpotential W = κS (cid:0) ¯ΦΦ − M (cid:1) , (1)which is the most general renormalizable superpotential con-sistent with a continuous R-symmetry [10] under which S → e iα S, ¯ΦΦ → ¯ΦΦ , W → e iα W. (2)Here S is a G -singlet left-handed superfield, and the param-eters κ and M are made positive by field redefinitions. Inour approach ¯Φ , Φ are identified with a pair of left-handedsuperfields conjugate under G which break G down to G SM .Indeed, along the D-flat direction | ¯Φ | = | Φ | the SUSY poten-tial, V SUSY , extracted – see e.g. Ref. [28] – from W in Eq. (1),reads V SUSY = κ (cid:0) ( | Φ | − M ) + 2 | S | | Φ | (cid:1) . (3)From V SUSY in Eq. (3) we find that the SUSY vacuum lies at |h S i| = 0 and |h Φ i| = (cid:12)(cid:12) h ¯Φ i (cid:12)(cid:12) = M, (4)where the vacuum expectation values of Φ and ¯Φ lie alongtheir SM singlet components. As a consequence, W leads tothe spontaneous breaking of G to G SM . B . T HE I NFLATIONARY S ET - UP It is well-known [10] that W also gives rise to FHI since,for values of | S | ≫ M , there exist a flat direction s ≡ √ Im [ S ] = 0 and ¯Φ = Φ = 0 , (5)which provides us with a constant potential energy κ M suitable for supporting FHI. The inclusion of SUGRA cor-rections with canonical (minimal) K¨ahler potential does notaffect this result at the lowest order in the expansion of S –due to a miraculus cancelation occuring. The SUGRA cor-rections with quasi-canonical K¨ahler potential [19, 21] can bekept under control by mildly tuning the relevant coefficientsthanks to sub-Planckian S values required by FHI. The re-sulting n s values can be fully compatible with the data [5, 6]but the predicted r [20, 21] remains well below the purportedmeasurement reported by B ICEP
ICEP
2, within SUGRA and employing W in Eq. (1), we have to tame the η problem which is more challenging due tothe trans-Planckian values needed for the inflaton superfield, S . To this end, we exploit a K¨ahler potential which respectsthe following symmetries: S → S + c and S → − S, (6)where c is a real number – cf. Ref. [23]. Namely we take K = −
12 ( S − S ∗ ) + | Φ | + | ¯Φ | + ( S − S ∗ ) (cid:0) k S ( S − S ∗ ) + k S Φ | Φ | + k S ¯Φ | ¯Φ | (cid:1) + 1Λ (cid:0) k Φ | Φ | + k ¯Φ | ¯Φ | (cid:1) + · · · · (7)Here k S , k Φ , k ¯Φ , k S Φ and k S ¯Φ are positive or negative con-stants of order unity – for simplicity we take k S Φ = k S ¯Φ – and Λ is a cutoff scale determined below. Although K is not invariant under the R symmetry of Eq. (2), the fields Φ α = S, Φ , ¯Φ are canonically normalized, i.e., K α ¯ β = δ α ¯ β –note that the complex scalar components of the various super-fields are denoted by the same symbol.The F–term (tree level) SUGRA scalar potential, V I0 , ofour model is obtained from W in Eq. (1) and K in Eq. (7) byapplying the standard formula: V I0 = e K (cid:16) K α ¯ β F α F ¯ β − | W | (cid:17) , (8)with K α ¯ β = K , Φ α Φ ∗ ¯ β , K ¯ βα K α ¯ γ = δ ¯ β ¯ γ and F α = W , Φ α + K , Φ α W. We explicitly verify that the SUSY vacuum of Eq. (4)remains intact for the choice of K in Eq. (7). Along the fielddirection in Eq. (5) the only surviving terms of V I0 are V I0 = e K (cid:16) K SS ∗ | W ,S | − | W | (cid:17) = κ M (cid:18) − σ (cid:19) , (9)where the canonically normalized inflaton, σ , is defined by S = ( σ + is ) / √ . (10)As shown from Eq. (9), V I0 is not suitable to drive inflationmainly due to the minus sign which renders V I0 unboundedfrom below for large σ ’s – cf. Ref. [17]. On the other hand,the symmetries in Eq. (6) ensure a complete disappearanceof the exponential prefactor in Eq. (9), which could ruin anyinflationary solution for large σ ’s.A satisfactory solution can be achieved, if we consider anintermediate-scale SSB mass parameter e m , whose contribu-tion can exceed the negative contribution to V I0 for conve-niently selected κ and M . Such a heavy mass parameter isnormally generated following the usual SUSY breaking pro-cedures – see e.g. Ref. [27] – provided that the gravitino massis of similar size and the Polonyi field has canonical K¨ahlerpotential. The contributions to the inflationary potential fromthe SSB effects [12, 18] can be parameterized as follows: V IS = e m P α | Φ α | − (cid:0) a S κM S − κA κ S Φ ¯Φ + c . c . (cid:1) , (11 a ) FromHybridtoQuadraticInflationWithHigh-ScaleSUSYBreaking T ABLE I: The mass spectrum of the model along the path in Eq. (5).Fields Eingestates Masses SquaredBosons real scalar σ m σ = e m − κ M real scalar s m s = e m + κ M · (cid:0) (3 − σ ) − k S / Λ (cid:1) N complex φ i ± = ¯ φ i ± φ i √ m φ ± ≃ k S Φ κ M Λ ∓ κ | A κ | σ √ + scalars ( i = 1 , e m + κ (cid:16) (1 ± M ) σ ∓ M (cid:17) Fermions Weyl spinor ψ S m ψ S = κ M σ / N Weyl spinors ψ ± = ψ ¯Φ ± ψ Φ √ m ψ ± = κ σ / where we assume for simplicity that there is a universal SSBmass e m for all the superfields Φ α = S, Φ , ¯Φ of our model.Also a S and A κ are mass parameters comparable to e m . Alongthe field configuration in Eq. (5), V IS reads V IS = e m σ / − √ S κM σ. (11 b )We note in passing that, due to Eq. (11 a ), |h S i| is shifted [13]from its value in Eq. (4) to |h S i| ≃ ( | A κ | − | a S | ) / κ (1 + e m / κ M ) , (12)where we selected conveniently the phases of A κ and a S sothat h V SUSY + V IS i is minimized. C . B EYOND THE T REE -L EVEL P OTENTIAL
Expanding the various fields, besides S – see Eq. (10) –, inreal and imaginary parts according to the prescription X = ( x + ix ) / √ (13)where X = Φ , ¯Φ and x = φ, ¯ φ respectively, we are able tocheck the stability of the field directions in Eq. (5). Namely,we check the validity of the conditions ∂V tr /∂χ α = 0 and m χ α > , (14 a )where χ α = σ, s, φ i and ¯ φ i with i = 1 , and V tr stands forthe tree-level inflationary potential V tr = V I0 + V IS (14 b )with V I0 and V IS given in Eq. (9) and (11 b ). Note that theimposed Z symmetry on K – see Eq. (6) – excludes theterms ( S − S ∗ ) or ( S − S ∗ ) which could violate the firstcondition in Eq. (14 a ) for χ α = s . Moreover, in Eq. (14 a ), m χ α are the eigenvalues of the mass squared matrix M αβ = ∂ V tr /∂χ α ∂χ β which are presented in Table I. Setting e m ≥ √ κM , Λ ≤ p | k S |√ N ⋆ − (15 a ) (where we employ Eq. (21 a ) and set a S ≪ for the derivationof the latter expression above) and, neglecting M terms, σ ≥ σ c ≃ √ √ κ M − e m κ √ M with M > e mκ (15 b )assists us to achieve the positivity of m σ , m s and m φ + , re-spectively. Note that the two first terms in the expressionfor m φ ± are neglected in the derivation of Eq. (15 b ), sincetheir contribution is suppressed for k S Φ ∼ and | A κ | ≃ − − − . In Table I we also present the masses squared ofthe chiral fermions of the model along the trajectory in Eq. (5).We remark that the fermionic and bosonic degrees of freedomare equal to N ) . Inserting these masses into the well-known Coleman-Weinberg formula, we can find the one-loopRCs, ∆ V , which can be written as ∆ V = 164 π m σ ln m σ Q + m s ln m s Q − m ψ S ln m ψ S Q + 2 N P i = ± m φ i ln m φ i Q − m ψ ± ln m ψ ± Q !! . (16)Here Q is a renormalization group mass scale and N is thedimensionality of the representations to which ¯Φ and Φ belong– we have [12, 21] N = 1 , , for G = G B − L , G LR and G X , correspondingly.All in all, the full potential of our model is V I = V tr + ∆ V, (17)with V tr and ∆ V given in Eq. (14 b ) and (16) respectively. III. C ONSTRAINING THE M ODEL P ARAMETERS
Based on V I in Eq. (17) we proceed to explore the allowedparameter space of our model employing the standard slow-roll approximation [28]. The free parameters are κ, M, k S , k S Φ , Λ , e m, a S , | A κ | and N . The parameters k S , k S Φ and | A κ | exclusively influence thevalues of m s and m φ ± – see Table I – and so, we take forthem a convenient value, close to unity, which can assist usto achieve the positivity and heaviness – see below – of thesemasses squared, e.g., k S = − k S Φ = − and | A κ | = 10 − .The remaining parameters can be restricted by imposing anumber of observational (1,3) and theoretical (2) restrictionsspecified below: A . I NFLATIONARY O BSERVABLES
The number of e-foldings, N ⋆ , that the pivot scale k ⋆ =0 . / Mpc undergoes during inflation, and the amplitude A s of the power spectrum of the curvature perturbation can becalculated using the standard formulae N ⋆ = Z σ ⋆ σ f dσ V I V ′ I and p A s = 12 √ π V / ( σ ⋆ ) | V ′ I ( σ ⋆ ) | (18)II ConstrainingtheModelParameters where the prime denotes derivation with respect to σ , σ ⋆ isthe value of σ when k ⋆ crosses outside the horizon of infla-tion, and σ f is the value of σ at the end of inflation whichcoincides with σ c , Eq. (15 b ), if ǫ ( σ c ) ≤ and η ( σ c ) ≤ or isdetermined by the condition: max { ǫ ( σ ) , η ( σ ) } = 1 for σ ≥ σ c . (19 a )Here ǫ and η are the well-known [28] slow-roll parametersdefined as follows: ǫ = (cid:16) V ′ I / √ V I (cid:17) and η = V ′′ I /V I . (19 b )Agreement with the observations [5, 6] requires N ⋆ ≃ and p A s ≃ . · − , (20)which allow us to restrict σ ⋆ and e m . Neglecting ∆ V inEq. (17) and assuming that a S is adequately suppressed weapproach the quadratic inflationary model with ǫ = η = 2 /σ , σ f ≃ √ and σ ⋆ ≃ p N ⋆ . (21 a )Hence, inflation takes place for σ ≫ with σ f ∼ and σ c ≪ – see Eq. (15 b ). Employing the last equalities in Eqs. (18)and (21 a ) we find e m ≃ √ p κ M N ⋆ + 2 A s π /N ⋆ = (6 − · − , (21 b )for the values of Eq. (20) and κ and M of order 0.01. There-fore, the range of the e m values is somehow extended com-pared to those obtained in the quadratic model.We can finally calculate n s , its running, α s , and r , via therelations: n s = 1 − ǫ ⋆ + 2 η ⋆ ≃ − /N ⋆ = 0 . , (22 a ) α s = 23 (cid:0) η ⋆ − ( n s − (cid:1) − ξ ⋆ ≃ − N ⋆ = − · − , (22 b ) r = 16 ǫ ⋆ ≃ /N ⋆ = 0 . , (22 c )where ξ ≃ m V ′ I V ′′′ I /V and all the variables with the sub-script ⋆ are evaluated at σ = σ ⋆ . These results are in agree-ment with the observational data [1, 5, 6] derived in the frame-work of the Λ CDM model.Since there is no observational hint [6] for large non-Gaussianity in the cosmic microwave background, we shouldmake sure that the masses squared of the scalar excitationsin Table I, besides m σ , are greater than the hubble parametersquared, H = V I / m , during the last − e-foldingsof inflation, so that the the observed curvature perturbation isgenerated wholly by σ as assumed in Eq. (20). The lowest m χ α in Table I, by far, is the one for χ α = s and its ratio to H is estimated to be m s H ( σ ⋆ ) ≃ κ M N ⋆ (cid:0) Λ (3 − N ⋆ ) − k S (cid:1) A s Λ π + 32 N ⋆ , (23 a )employing Eq. (21 b ) and under the assumptions made above.Given that m s /H increases as σ drops, we end up with thefollowing condition: m s /H ( σ ⋆ ) ≥ , (23 b ) from which we can derive an upper bound, more restrictivethan that of Eq. (15 a ), on ΛΛ . p | k S | N ⋆ κM p κ M N ⋆ + A s π (23 c )ranging from . to . as κ and M vary from . to . – recall that we use k S < , as dictated by Eq. (23 a ). Themost natural scale close to these Λ values is the string scale,i.e., Λ = 0 . · (5 / . ≃ . ; we thus confine ourselves tothis choice for Λ onwards and restrict κ or M – with given Λ .E.g., Eq. (23 b ) implies: M & r Λ πκ s A s N ⋆ (Λ (3 − N ⋆ ) − k S ) , (23 d )which turns out to be more restrictive than that of Eq. (15 b ) ifwe make use of Eq. (21 b ). B . T HE GUT P
HASE T RANSITION
One outstanding feature of our proposal is that the inflation-ary scenario is followed by a GUT phase transition, in sharpcontrast to the original quadratic inflation [4]. We shouldnote, however, that V tr , Eq. (14 b ), develops along the trackof Eq. (5) an absolute minimum at σ = √ κ a S M e m − κ M , (24)which has the sign of a S and a possible complication may bethat σ gets trapped in this false vacuum and consequently noGUT phase transition takes place if σ c ≤ σ for σ ⋆ > , or σ c ≥ σ for σ ⋆ < . Note that the inflationary observablesremain unchanged under the the replacements a S → − a S and σ → − σ , (25)since V tr remains invariant. To assure a timely destabilizationof ¯Φ − Φ system – in the φ or φ − direction – we imposethe condition σ c ≥ σ for σ ⋆ > , or σ c ≤ σ for σ ⋆ < . (26)The structure of V tr for σ ⋆ > [ σ ⋆ < ] is visualized inFig. 1- (a) [Fig. 1- (b) ], where we present V tr – convenientlynormalized such that V tr ( σ ) = 0 – as a function of σ for thesame κ and M ( κ = 0 . and M = 0 . ) and two differ-ent a S values with constant | a S | taking into account Eq. (20).Namely, in Fig. 1- (a) , we take a S = − [+]2 · − – gray[light gray] line – corresponding to σ ⋆ = 13 .
95 [15 . and σ f = 0 .
44 [2 . . As anticipated from Eqs. (26) and (15 b ), V tr develops minima at the points | σ | ≃ . , whereas σ c ≃ . is constant in all cases since it is independentof a S . We observe that for a S < , we obtain σ < σ c and so the GUT phase transition can proceed without doubt,whereas for a S > we have σ > σ c , making the destabi-lization of the φ + direction – see Table I – rather uncertain. FromHybridtoQuadraticInflationWithHigh-ScaleSUSYBreaking -2 0 2 4 6 8 10 12 14 1601020304050 σ * κ = 0.01, M = 0.012a S = - 2 10 -5 , 2 10 -5 V t r σ (a) { -1 0 1 20.00.51.0 σ c σ f σ -16 -14 -12 -10 -8 -6 -4 -2 0 201020304050 σ * κ = 0.01, M = 0.012a S = 2 10 -5 , - 2 10 -5 V t r σ (b) { -2 -1 0 10.00.51.0 σ c σ f σ F IG . 1: Tree level inflationarypotential V tr as a functionof σ for σ ⋆ > and a S = − [+] 2 · − (a) or σ ⋆ < and a S = +[ − ] 2 · − (b)–gray[lightgray]line. Weset κ = 0 . , M = 0 . , k S = − and | A κ | = 10 − . Thevaluesof σ ⋆ , σ f , σ and σ c arealsodepicted. In Fig. 1- (b) , we present V tr versus σ changing the signs of a S and σ ⋆ according Eq. (25), i.e., we set a S = +[ − ]2 · − with σ ⋆ = − .
95 [ − . – gray [light gray] line. We remark thatthe case with a S < remains problematic since σ meets first σ = − . < σ c = 0 . and its trapping in the minimumis possible, whereas the case with a S > is free from such aproblem, since σ = 0 . > σ c = 0 . . Given this situa-tion we henceforth concentrate on the case with σ ⋆ > . Theresults for the case with σ ⋆ < are obtained by flipping thesign of a S as suggested by the symmetry of V tr , Eq. (25). C . C OMPATIBILITY W ITH THE F ORMATION OF C OSMIC S TRINGS If G = G B − L , B − L cosmic strings are produced duringthe GUT phase transition, at the end of inflation. The tension µ cs of these defects has to respect the bound [12, 29, 30]: µ cs = 9 . πM ln(2 /β ) ≤ · − ⇒ M ≤ . (cid:18) ln(2 /β )1 . π (cid:19) / , (27)where β = κ / g ≤ − with g ≃ . being the gaugecoupling constant close to M GUT . From Eq. (27), for κ =0 . , . and . , we obtain M ≤ . , . and ,whereas Eq. (23 d ) entails M ≥ . , . and respec-tively. As a consequence, our scheme is not compatible withthe choice G = G B − L . This negative result can be, mostprobably, avoided if we invoke the superpotential employedin shifted [15] or smooth [16] FHI. In that cases, Φ and ¯Φ areconfined to some non-vanishing value during inflation; thus,the B − L strings can be easily inflated away. IV. R ESULTS
Following our previous discussion we henceforth concen-trate our analysis on G = G LR or G X . For both selected G ’s, M can be related to the GUT scale since the non-singlet under G SM gauge bosons acquire mass equal to gM at theSUSY vacuum, Eq. (4) – see Ref. [21]. However, in high-scale SUSY [24, 25] the GUT scale is model dependent andso any M value between . and . is, in principle, accept-able. For reference we mention that the conventional SUSYGUT scale corresponds to the choice gM = (2 / . · − ,i.e., M ≃ . . Recall finally that we set k S = − k S Φ = − | A κ | = 10 − and Λ = 0 . throughout.In our numerical calculations, we use the complete formu-lae for V I , N ⋆ , A s and the slow-roll parameters – see Eqs. (17),(18) and (19 b ) – and not the approximate relations listed inSec. III for the sake of presentation. As regards Q in Eq. (16),we determine it by requiring [31] ∆ V ( σ ⋆ ) = 0 . Note that Q is not well-defined if we impose the alternative condition[31] ∆ V ( σ f ) = 0 since m φ + instantaneously vanishes when σ f = σ c . To reduce the possible [31, 32] dependence of ourresults on the choice of Q , we confine ourselves to values of κ, M and a S which do not enhance ∆ V . As a consequence,our findings are highly independent of the specific choice of G . For definiteness we mention that we take G = G X .Confronting our model with the imposed constraints, wedepict the allowed (lightly gray shaded [lined]) regions for a S < [ a S > ] in the κ − | a S | plane with M = 0 . and in the M − | a S | plane with κ = 0 . – see Fig. 2- (a) and (b) respectively. The left bounds in both plots come from thesaturation of Eq. (23 b ). It is straightforward to show that the(simplified) analytical expression in Eq. (23 d ) is in accordancewith the bound, . [ . ] depicted in Fig. 2- (a) [Fig. 2- (b) ]. Had we used k S = − , this bound in Fig. 2- (a) [Fig. 2- (b) ] would have been moved to . [ . ] cutting a minorslice of the allowed region. It is clear from Eq. (26) that theallowed region for a S > is considerably shrunk compared tothat for a S < , since a S < implies σ < , and so Eq. (26)is automatically fulfilled thanks to the positivity of σ c – seeEq. (15 b ). Indeed, the saturation of Eq. (26) gives the upperbound of the allowed (lined) regions for a S < . On the otherhand, for a S < no solution to Eq. (20) exists beyond thethin dashed line. In the shaded region between the thick andthin dashed lines the end of inflation is found by the condition Conclusions -3 -2 -1 | a S | ( - ) κ (10 - 2 ) (a) -3 -2 -1 | a S | ( - ) M (10 - 2 ) (b) F IG . 2: Allowed(shaded [lined])regions for σ ⋆ > and a S < [ a S > ]inthe κ − | a S | plane with M = 0 . (a) andinthe M − | a S | planewith κ = 0 . (b). Alongthegraylineweset a S = − e m . Wetake k S = − k S Φ = − , | A κ | = 10 − and Λ = 0 . . σ f = σ c and not the one in Eq. (19 a ) which exclusively gives σ f for a S > , and in the regions below the thick dashedlines for a S < . Note that for a S < we have allowedparameters even for | a S | = e m which are depicted by the graylines. Finally, beyond the (thin and thick) dotted lines, ourresults become unstable with respect to the variations of Q ;the model predictions are, thus, less trustable and we do notpursue it any further.Summarizing our findings from Fig. 2 the parameters of W in Eq. (1) are bounded as follows: . . κ/ − . and . . M/ − . . . (28)Moreover, the SSB mass parameters in Eq. (11 b ) are confinedin the following ranges: . . e m/ − . . and | a S | / − . . . (29)for a S < [ a S > ]. The most natural framework of SSB inwhich our model can be embedded is that of high-scale SUSYsince the e m values encountered here are roughly consistentwith m h ≃ GeV [24]. On the other hand, split SUSYcannot be directly combined with our proposal since requiring m h ≃ GeV implies [25] e m ≤ GeV, which is ratherlow to drive inflation. However, a possible coupling of S withthe electroweak higgses of the minimal SUSY SM can modifythis conclusion as outlined in Ref. [25].It is worth noticing that, contrary to Ref. [20], κ and M areconstrained so that the contribution to V I from Eq. (11 b ) ex-ceeds that from Eq. (9). As a consequence, our model hereshares identical predictions with the original quadratic infla-tionary model as regards n s , α s and r , and so it is consis-tent with B ICEP N ⋆ = 55 we find . . r . . and . . n s . . , . . − α s / − . . (30)which are consistent with WMAP [5] and Planck [6] resultswithin the Λ CDM model. Contrary to quadratic model, how-ever, our model implies a built-in mechanism for spontaneousbreaking of G at the scale M , Eq. (28), compatible with the SUSY GUT scale, M ≃ . . The resulting mass of theinflaton at the SUSY vacuum takes values . . m σ / − . . , (31)which allow for the decay of the inflaton to right-handedneutrinos, if the relevant couplings exist. Thus, a success-ful scenario of non-thermal leptogenesis, along the lines ofRef. [12, 33], can be easily constructed. V. C ONCLUSIONS
We have presented a framework for implementing quadratic(chaotic) inflation in realistic SUSY models which have pre-viously been used for FHI. Namely, we have retained a U (1) R-symmetry from earlier FHI which yields a unique super-potential, W , at renormalizable level, linear with respect theinflaton field. On the other hand, the K¨ahler potential, K , isjudiciously chosen so that no extensive SUGRA correctionsarise. Our model is thus protected against contributions fromhigher order terms in both K and W . We showed that themodel displays a wide and natural range of the parameters κ, M and a S which allows quadratic inflation to be success-fully implemented, provided that the SSB mass parameter e m lies at the intermediate energy scale motivated by high-scale(or, under some special circumstances, split) SUSY breaking.As a consequence the inflationary observables are in excellentagreement with the combined analysis of the Planck, WMAPand B ICEP A CKNOWLEDGMENTS
Q.S. thanks Gia Dvali, Ilia Gogoladze, Matt Civiletti, andTianjun Li for helpful discussions and acknowledges supportby the DOE grant No. DE-FG02-12ER41808. C.P. acknowl-edges support from the Generalitat Valenciana under grantPROMETEOII/2013/017.
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