From Hybrid to Quadratic Inflation With High-Scale Supersymmetry Breaking

Motivated by the reported discovery of inflationary gravity waves by the BICEP2 experiment, we propose an inflationary scenario in supergravity, based on the standard superpotential used in hybrid inflation. The new model yields a tensor-to-scalar ratio r ~ 0.14 and scalar spectral index ns ~ 0.964, corresponding to quadratic (chaotic) inflation. The important new ingredients are the high-scale, (1.6-10) x 10^13 GeV, soft supersymmetry breaking mass for the gauge singlet inflaton field and a shift symmetry imposed on the K\"ahler potential. The end of inflation is accompanied, as in the earlier hybrid inflation models, by the breaking of a gauge symmetry at (1.2-7.1) x 10^16 GeV, comparable to the grand-unification scale.


I. INTRODUCTION
The discovery of B-modes in the polarization of the cosmic microwave background radiation at large angular scales by the BICEP2 experiment [1] has created much excitement among inflationary model builders, since this effect can be caused by an early inflationary era with a large tensor-to-scalar ratio r = 0.16 +0.06 −0.05 -after substraction of a dust foreground. Although other interpretations [2,3] of this result are possible, it motivates us to explore how realistic supersymmetric (SUSY) inflation models can accommodate such large r values.
The textbook quadratic inflationary model [4] predicting r = 0.13 − 0.16, and a (scalar) spectral index n s = 0.96 − 0.967, seems to be in good agreement with BICEP2 (r) and the WMAP [5] and Planck [6] measurements (n s ). Quadratic inflation can be accompanied by a Grand Unified Theory (GUT) phase transition in non-supersymmetric inflation models, based either on the Coleman-Weinberg or Higgs [7] potential, which yield predictions for n s that more or less overlap with the prediction of the quadratic model [8,9]. However, significant differences appear between the predictions of r in these models which can be settled through precision measurements. The consistent supersymmetrization of these models is a highly non-trivial task due to the trans-Planckian values of the inflaton field which aggravate the well-known η-problem within supergravity (SUGRA).
One of the more elegant SUSY models which nicely combines inflation with a GUT phase transition is the model of F-term hybrid inflation [10,11] -referred as FHI. It is based on a unique renormalizable superpotential, dictated by a U (1) R-symmetry, employs sub-Planckian values for the inflaton field and can be naturally followed by the breaking of a GUT gauge 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 group of 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 gauge symmetry G 5X = SU (5) × U (1) X . The embedding of the simplest model of FHI within a GUT based on a higher gauge group may suffer from the production of disastrous cosmic defects which can be evaded, though, by using shifted [15] or smooth [16] FHI.
In the simplest realization of FHI the standard [10] superpotential is accompanied by a minimal (or canonical) Kähler potential. The resulting n s is found to be in good agreement with the WMAP and Planck data after including in the inflationary potential radiative corrections (RCs) [10] and the soft SUSY breaking (SSB) linear term [12,18] -with a mass parameter in the TeV range -a SSB mass term for the inflaton in the same energy region can be ignored in this analysis. This scenario yields [12] r values which lie many orders of magnitude below the measurement reported [1] by BICEP2. A more elaborate extension of this standard FHI scenario exploits nonminimal, quasi-canonical Kähler potentials [19,21] or SSB mass of magnitude as large as 10 10 GeV for the inflaton field [20]. Depending on the underlying assumptions, the predictions for r are considerably enhanced compared to the minimal scenario of Ref. [12,18]. Thus, r values as large as 0.01 to 0.03 have been reported [20,21]; this fact certainly puts r in the observable range, but it still remains an order of magnitude below the BICEP2 measurement -however, see Ref. [22] for models of FHI with Kähler potential not-respecting the Rsymmetry.
Motivated by this apparent discrepancy between the large r values reported by BICEP2 and the predictions of FHI models, we present here a modified scenario of F-term inflation in which a quadratic potential dictates the inflationary phase, thus replicating the predictions of quadratic inflation, employing the well-studied standard superpotential of FHI. The two key elements for successfully implementing this scenario include a judicious choice of the Kähler potential and a highscale SUSY breaking. In particular, following earlier similar attempts [23] a shift symmetry is imposed on the Kähler potential to protect the inflationary potential from SUGRA corrections which are dangerous due to trans-Planckian inflaton field values. Moreover, we assume that SUSY is broken at an intermediate scale, m ∼ 10 13 GeV, which can be identified with the SSB mass of the inflaton. In the context of high-scale SUSY [24,25], such a large SSB scale can become consistent with the LHC results [26] on the mass, m h ≃ 126 GeV, of the SM Higgs boson, h. The end of inflation can be accompanied by the breaking of some gauge symmetry such as G LR or G 5X with the gauge symmetry breaking scale M assuming values close to the SUSY GUT scale M GUT ≃ 2.86 · 10 16 GeV.
Below, we describe in Sec. II the basic ingredients of our inflationary scenario. Employing a number of constraints presented in Sec. III, we provide restrictions on the model parameters in Sec. IV. Our conclusions are summarized in Sec. V. Henceforth we use units where the reduced Planck scale m P = 2.44 · 10 18 GeV is taken equal to unity.

A. THE GUT SYMMETRY BREAKING
In the standard FHI we adopt the superpotential which is the most general renormalizable superpotential consistent with a continuous R-symmetry [10] under which Here S is a G-singlet left-handed superfield, and the parameters κ and M are made positive by field redefinitions. In our approachΦ, Φ are identified with a pair of left-handed superfields conjugate under G which break G down to G SM . Indeed, along the D-flat direction |Φ| = |Φ| the SUSY potential, V SUSY , extracted -see e.g. Ref.
[28] -from W in Eq. (1), reads From V SUSY in Eq. (3) we find that the SUSY vacuum lies at where the vacuum expectation values of Φ andΦ lie along their SM singlet components. As a consequence, W leads to the spontaneous breaking of G to G SM .

B. THE INFLATIONARY SET-UP
It is well-known [10] that W also gives rise to FHI since, for values of |S| ≫ M , there exist a flat direction which provides us with a constant potential energy κ 2 M 4 suitable for supporting FHI. The inclusion of SUGRA corrections with canonical (minimal) Kähler potential does not affect this result at the lowest order in the expansion of Sdue to a miraculus cancelation occuring. The SUGRA corrections with quasi-canonical Kähler potential [19,21] can be kept under control by mildly tuning the relevant coefficients thanks to sub-Planckian S values required by FHI. The resulting n s values can be fully compatible with the data [5,6] but the predicted r [20,21] remains well below the purported measurement reported by BICEP2.
In order to safely implement quadratic inflation, favored by BICEP2, within SUGRA and employing W in Eq. (1), we have to tame the η problem which is more challenging due to the trans-Planckian values needed for the inflaton superfield, S. To this end, we exploit a Kähler potential which respects the following symmetries: where c is a real number -cf. Ref. [23]. Namely we take Here k S , k Φ , kΦ, k SΦ and k SΦ are positive or negative constants of order unity -for simplicity we take k SΦ = k SΦ -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 αβ = δ αβnote that the complex scalar components of the various superfields are denoted by the same symbol. The F-term (tree level) SUGRA scalar potential, V I0 , of our model is obtained from W in Eq. (1) and K in Eq. (7) by applying the standard formula: We explicitly verify that the SUSY vacuum of Eq. (4) remains intact for the choice of K in Eq. (7). Along the field direction in Eq. (5) the only surviving terms of V I0 are where the canonically normalized inflaton, σ, is defined by As shown from Eq. (9), V I0 is not suitable to drive inflation mainly due to the minus sign which renders V I0 unbounded from below for large σ's -cf. Ref. [17]. On the other hand, the symmetries in Eq. (6) ensure a complete disappearance of the exponential prefactor in Eq. (9), which could ruin any inflationary solution for large σ's. A satisfactory solution can be achieved, if we consider an intermediate-scale SSB mass parameter m, whose contribution can exceed the negative contribution to V I0 for conveniently selected κ and M . Such a heavy mass parameter is normally generated following the usual SUSY breaking procedures -see e.g. Ref. [27] -provided that the gravitino mass is of similar size and the Polonyi field has canonical Kähler potential. The contributions to the inflationary potential from the SSB effects [12,18] can be parameterized as follows: where we assume for simplicity that there is a universal SSB mass m for all the superfields Φ α = S, Φ,Φ of our model. Also a S and A κ are mass parameters comparable to m. Along the field configuration in Eq. (5), V IS reads We note in passing that, due to Eq. (11a), | S | is shifted [13] from its value in Eq. (4) to where we selected conveniently the phases of A κ and a S so that V SUSY + V IS is minimized.

C. BEYOND THE TREE-LEVEL POTENTIAL
Expanding the various fields, besides S -see Eq. (10) -, in real and imaginary parts according to the prescription where X = Φ,Φ and x = φ,φ respectively, we are able to check the stability of the field directions in Eq. (5). Namely, we check the validity of the conditions where χ α = σ, s, φ i andφ i with i = 1, 2 and V tr stands for the tree-level inflationary potential with V I0 and V IS given in Eq. (9) and (11b). Note that the imposed Z 2 symmetry on K -see Eq. (6) -excludes the terms (S − S * ) or (S − S * ) 3 which could violate the first condition in Eq. (14a) for χ α = s. Moreover, in Eq. (14a), m 2 χ α are the eigenvalues of the mass squared matrix M 2 αβ = ∂ 2 V tr /∂χ α ∂χ β which are presented in Table I. Setting (where we employ Eq. (21a) and set a S ≪ 1 for the derivation of the latter expression above) and, neglecting M 4 terms, assists us to achieve the positivity of m 2 σ , m 2 s and m 2 φ+ , respectively. Note that the two first terms in the expression for m 2 φ± are neglected in the derivation of Eq. (15b), since their contribution is suppressed for k SΦ ∼ 1 and |A κ | ≃ 10 −6 −10 −5 . In Table I we also present the masses squared of the chiral fermions of the model along the trajectory in Eq. (5). We remark that the fermionic and bosonic degrees of freedom are equal to 2(1 + 2N). Inserting these masses into the wellknown Coleman-Weinberg formula, we can find the one-loop RCs, ∆V , which can be written as Here Q is a renormalization group mass scale and N is the dimensionality of the representations to whichΦ and Φ belong -we have [12,21] N = 1, 2, 10 for G = G B−L , G LR and G 5X , correspondingly. All in all, the full potential of our model is with V tr and ∆V given in Eq. (14b) and (16) respectively.

III. CONSTRAINING THE MODEL PARAMETERS
Based on V I in Eq. (17) we proceed to explore the allowed parameter space of our model employing the standard slowroll approximation [28]. The free parameters are κ, M, k S , k SΦ , Λ, m, a S , |A κ | and N .
The parameters k S , k SΦ and |A κ | exclusively influence the values of m 2 s and m 2 φ± -see Table I -and so, we take for them a convenient value, close to unity, which can assist us to achieve the positivity and heaviness -see below -of these masses squared, e.g., k S = −k SΦ = −5 and |A κ | = 10 −6 . The remaining parameters can be restricted by imposing a number of observational (1,3) and theoretical (2) restrictions specified below:

A. INFLATIONARY OBSERVABLES
The number of e-foldings, N ⋆ , that the pivot scale k ⋆ = 0.05/Mpc undergoes during inflation, and the amplitude A s of the power spectrum of the curvature perturbation can be calculated using the standard formulae where the prime denotes derivation with respect to σ, σ ⋆ is the value of σ when k ⋆ crosses outside the horizon of inflation, and σ f is the value of σ at the end of inflation which coincides with σ c , Eq. (15b), if ǫ(σ c ) ≤ 1 and η(σ c ) ≤ 1 or is determined by the condition: Here ǫ and η are the well-known [28] slow-roll parameters defined as follows: Agreement with the observations [5,6] requires which allow us to restrict σ ⋆ and m. Neglecting ∆V in Eq. (17) and assuming that a S is adequately suppressed we approach the quadratic inflationary model with Hence, inflation takes place for σ ≫ 1 with σ f ∼ 1 and σ c ≪ 1 -see Eq. (15b). Employing the last equalities in Eqs. (18) and (21a)  We can finally calculate n s , its running, α s , and r, via the relations: where ξ ≃ m 4 P V ′ I V ′′′ I /V 2 I and all the variables with the subscript ⋆ are evaluated at σ = σ ⋆ . These results are in agreement with the observational data [1,5,6] derived in the framework of the ΛCDM model.
Since there is no observational hint [6] for large non-Gaussianity in the cosmic microwave background, we should make sure that the masses squared of the scalar excitations in Table I, besides m 2 σ , are greater than the hubble parameter squared, H 2 I = V I /3m 2 P , during the last 50 − 60 e-foldings of inflation, so that the the observed curvature perturbation is generated wholly by σ as assumed in Eq. (20). The lowest m 2 χ α in Table I, by far, is the one for χ α = s and its ratio to employing Eq. (21b) and under the assumptions made above. Given that m 2 s /H 2 I increases as σ drops, we end up with the following condition: from which we can derive an upper bound, more restrictive than that of Eq. (15a), on Λ Λ 6|k S |N ⋆ κM 2 ranging from 0.74 to 0.3 as κ and M vary from 0.1 to 0.01 -recall that we use k S < 0, as dictated by Eq. (23a). The most natural scale close to these Λ values is the string scale, i.e., Λ = 0.1 · (5/2.44) ≃ 0.2; we thus confine ourselves to this choice for Λ onwards and restrict κ or M -with given Λ. E.g., Eq. (23b) implies: which turns out to be more restrictive than that of Eq. (15b) if we make use of Eq. (21b).

B. THE GUT PHASE TRANSITION
One outstanding feature of our proposal is that the inflationary scenario is followed by a GUT phase transition, in sharp contrast to the original quadratic inflation [4]. We should note, however, that V tr , Eq. (14b), develops along the track of Eq. (5) an absolute minimum at which has the sign of a S and a possible complication may be that σ gets trapped in this false vacuum and consequently no GUT phase transition takes place if σ c ≤ σ 0 for σ ⋆ > 0, or σ c ≥ σ 0 for σ ⋆ < 0. Note that the inflationary observables remain unchanged under the the replacements a S → −a S and σ → −σ , since V tr remains invariant. To assure a timely destabilization ofΦ − Φ system -in the φ 1+ or φ 2− direction -we impose the condition The structure of V tr for σ ⋆ > 0 [σ ⋆ < 0] is visualized in Fig. 1-(a) [Fig. 1-(b)], where we present V tr -conveniently normalized such that V tr (σ 0 ) = 0 -as a function of σ for the same κ and M (κ = 0.01 and M = 0.012) and two different a S values with constant |a S | taking into account Eq. (20). Namely, in Fig. 1-(a), we take a S = −[+]2 · 10 −5 -gray [light gray] line -corresponding to σ ⋆ = 13.95 [15.9] and σ f = 0.44 [2.4]. As anticipated from Eqs. (26) and (15b), V tr develops minima at the points |σ 0 | ≃ 0.97, whereas σ c ≃ 0.017 is constant in all cases since it is independent of a S . We observe that for a S < 0, we obtain σ 0 < σ c and so the GUT phase transition can proceed without doubt, whereas for a S > 0 we have σ 0 > σ c , making the destabilization of the φ + direction -see Table I -rather uncertain. 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 = +[−]2·10 −5 with σ ⋆ = −13.95 [−15.9] -gray [light gray] line. We remark that the case with a S < 0 remains problematic since σ meets first σ 0 = −0.97 < σ c = 0.017 and its trapping in the minimum is possible, whereas the case with a S > 0 is free from such a problem, since σ 0 = 0.97 > σ c = 0.017. Given this situation we henceforth concentrate on the case with σ ⋆ > 0. The results for the case with σ ⋆ < 0 are obtained by flipping the sign of a S as suggested by the symmetry of V tr , Eq. (25).

C. COMPATIBILITY WITH THE FORMATION OF COSMIC STRINGS
If G = G B−L , B − L cosmic strings are produced during the GUT phase transition, at the end of inflation. The tension µ cs of these defects has to respect the bound [12,29,30]: where β = κ 2 /8g 2 ≤ 10 −2 with g ≃ 0.7 being the gauge coupling constant close to M GUT . From Eq. (27), for κ = 0.1, 0.01 and 0.001, we obtain 10 3 M ≤ 1.33, 1.7 and 2, whereas Eq. (23d) entails 10 3 M ≥ 2.7, 8.7 and 27 respectively. As a consequence, our scheme is not compatible with the choice G = G B−L . This negative result can be, most probably, avoided if we invoke the superpotential employed in shifted [15] or smooth [16] FHI. In that cases, Φ andΦ are confined to some non-vanishing value during inflation; thus, the B − L strings can be easily inflated away.

IV. RESULTS
Following our previous discussion we henceforth concentrate our analysis on G = G LR or G 5X . 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 the SUSY vacuum, Eq. (4) -see Ref. [21]. However, in highscale SUSY [24,25] the GUT scale is model dependent and so any M value between 0.001 and 0.1 is, in principle, acceptable. For reference we mention that the conventional SUSY GUT scale corresponds to the choice gM = (2/2.44) · 10 −2 , i.e., M ≃ 0.012. Recall finally that we set k S = −k SΦ = −5 |A κ | = 10 −6 and Λ = 0.2 throughout.
In our numerical calculations, we use the complete formulae for V I , N ⋆ , A s and the slow-roll parameters -see Eqs. (17), (18) and (19b) -and not the approximate relations listed in Sec. 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 our results 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 5X .
Confronting our model with the imposed constraints, we depict the allowed (lightly gray shaded σ f = σ c and not the one in Eq. (19a) which exclusively gives σ f for a S > 0, and in the regions below the thick dashed lines for a S < 0. Note that for a S < 0 we have allowed parameters even for |a S | = m which are depicted by the gray lines. Finally, beyond the (thin and thick) dotted lines, our results become unstable with respect to the variations of Q; the model predictions are, thus, less trustable and we do not pursue it any further.
Summarizing our findings from Fig. 2  for a S < 0 [a S > 0]. The most natural framework of SSB in which our model can be embedded is that of high-scale SUSY since the m values encountered here are roughly consistent with m h ≃ 126 GeV [24]. On the other hand, split SUSY cannot be directly combined with our proposal since requiring m h ≃ 126 GeV implies [25] m ≤ 10 8 GeV, which is rather low to drive inflation. However, a possible coupling of S with the electroweak higgses of the minimal SUSY SM can modify this conclusion as outlined in Ref. [25].
It is worth noticing that, contrary to Ref. [20], κ and M are constrained so that the contribution to V I from Eq. (11b) exceeds that from Eq. (9). As a consequence, our model here shares identical predictions with the original quadratic inflationary model as regards n s , α s and r, and so it is consistent with BICEP2 findings [1]. Indeed, for N ⋆ = 55 we find 0.12 r 0.14 and 0.963 n s 0.969, 4.7 −α s /10 −4 6.8 (30) which are consistent with WMAP [5] and Planck [6] results within the ΛCDM model. Contrary to quadratic model, however, our model implies a built-in mechanism for spontaneous breaking of G at the scale M , Eq.
which allow for the decay of the inflaton to right-handed neutrinos, if the relevant couplings exist. Thus, a successful scenario of non-thermal leptogenesis, along the lines of Ref. [12,33], can be easily constructed.

V. CONCLUSIONS
We have presented a framework for implementing quadratic (chaotic) inflation in realistic SUSY models which have previously been used for FHI. Namely, we have retained a U (1) R-symmetry from earlier FHI which yields a unique superpotential, W , at renormalizable level, linear with respect the inflaton field. On the other hand, the Kähler potential, K, is judiciously chosen so that no extensive SUGRA corrections arise. Our model is thus protected against contributions from higher order terms in both K and W . We showed that the model displays a wide and natural range of the parameters κ, M and a S which allows quadratic inflation to be successfully implemented, provided that the SSB mass parameter 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 excellent agreement with the combined analysis of the Planck, WMAP and BICEP2 measurements.