Induced-Gravity GUT-Scale Higgs Inflation in Supergravity
aa r X i v : . [ h e p - ph ] J un I NDUCED -G RAVITY
GUT-S
CALE H IGGS I NFLATION IN S UPERGRAVITY C ONSTANTINOS P ALLIS AND Q AISAR S HAFI School of Electrical & Computer Engineering, Faculty of Engineering,Aristotle University of Thessaloniki, GR-541 24 Thessaloniki, GREECE 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 : Models of induced-gravity inflation are formulated within Supergravity employing as inflaton theHiggs field which leads to a spontaneous breaking of a U (1) B − L symmetry at M GUT = 2 · GeV. We usea renormalizable superpotential, fixed by a U (1) R symmetry, and K¨ahler potentials which exhibit a quadraticnon-minimal coupling to gravity with or without an independent kinetic mixing in the inflaton sector. In bothcases we find inflationary solutions of Starobinsky type whereas in the latter case, others (more marginal)which resemble those of linear inflation arise too. In all cases the inflaton mass is predicted to be of the orderof GeV. Extending the superpotential of the model with suitable terms, we show how the MSSM µ parameter can be generated. Also, non-thermal leptogenesis can be successfully realized, provided that thegravitino is heavier than about TeV.
PACs numbers: 98.80.Cq, 04.50.Kd, 12.60.Jv, 04.65.+e
Publishedin
Eur.Phys.J.C , no. 6, 523 (2018). I. I NTRODUCTION
The idea of induced gravity ( IG ), according to which the(reduced) Planck mass m P is generated [1] via the vacuumexpectation value ( v.e.v ) that a scalar field acquires at the endof a phase transition in the early universe, has recently at-tracted a fair amount of attention. This is because it may fol-low an inflationary stage driven by a Starobisky-type potential[2] in Supergravity ( SUGRA ) [3–7] and in non-
Supersymmetric ( SUSY ) [8–12] settings, which turns out to be nicely compati-ble with the observational data [13]. As a bonus, the resultingeffective theories do not suffer from any problem with per-turbative unitarity [3, 5, 11, 14, 15] in sharp contrast to somemodels of non-minimal inflation [16–19] where the inflatonafter inflation assumes a v.e.v much smaller than m P .The simplest way to realize the idea of IG is to employ adouble-well potential, λ ( φ − v ) , for the inflaton φ [1, 3–5, 8–11] – scale invariant realizations of this idea are proposedin Ref. [12]. If we adopt a non-minimal coupling to gravity[9, 10] of the type f R = c R φ and set v = m P / √ c R , then h f R i = m , i.e., f R reduces to m at the vacuum generating,thereby, Einstein gravity at low energies. The implementationof inflation, on the other hand, which requires the emergenceof a sufficiently flat branch of the potential at large field val-ues constrains c R to sufficiently large values and λ as a func-tion of c R . An even more restrictive version of this scenariowould be achieved if φ is involved in a Higgs sector whichtriggers a Grand Unified Theory ( GUT ) phase transition in theearly Universe [7, 9]. The scale of a such transition is usu-ally related to the (field dependent) mass of the lightest gaugeboson and can be linked to some unification condition in su-persymmetric ( SUSY ) – most notably – settings [19–23]. Asa consequence, c R can be uniquely determined by the theo-retical requirements, giving rise to an economical, predictiveand well-motivated set-up, thereby called IG Higgs inflation ( IGHI ). To our knowledge, the unification hypothesis has notbeen previously employed in constraining IGHI. Since gauge coupling unification is elegantly achievedwithin the minimal supersymmetric standard model ( MSSM ),we need to formulate IGHI in the context of SUGRA. Namely,we employ a renormalizable superpotential, uniquely deter-mined by a gauge and a U (1) R symmetry, which realizes theHiggs mechanism in a SUSY framework. Actually, this is thesame superpotential widely used for the models of F-term hy-brid inflation [24–28]. Contrary to that case, though, wherethe inflaton typically is a gauge singlet and a pair of gaugenon-singlets are stabilized at zero, here the inflaton is involvedin the Higgs sector of the theory whereas the gauge singlet su-perfield is confined at the origin playing the role of a stabilizer – for a related scenario see Ref. [29]. For this reason we call it
Higgs inflation ( HI ). As regards the K¨ahler potentials, K , weconcentrate on semi-logarithmic ones which employ variablecoefficients for the logarithmic part and include only quadraticterms of the various fields, taking advantage of the recently es-tablished [6] stabilization mechanisms of the accompanyingnon-inflaton fields.More specifically, we distinguish two different classes of K ’s, depending whether we introduce an independent kineticmixing in the inflaton sector or not. In the latter case the non-minimal coupling to gravity reads f R ∼ c R φ and imposingthe IG and unification conditions allows us to fully determine c R . In the former case, apart from the non-minimal couplingto gravity expressed as f R = c + φ , the models exhibit a ki-netic mixing of the form f K ≃ c − f R , where the constants c − and c + can be interpreted as the coefficients of the principalshift-symmetric term ( c − ) and its violation ( c + ) in the K ’s.Obviously these models are inspired by the kinetically modi-fied non-minimal HI studied in Ref. [20–23]. The observablesnow depend on the ratio r ± = c + /c − which can be foundprecisely enforcing the IG and unification conditions. As aconsequence, for both classes of models more robust predic-tions can be here achieved than those presented in the originalpapers [19–23], where m P is included in f R from every be-ginning. Most notably, the level of the predicted primordialI InflationaryModels gravitational waves is about an order of magnitude lower thanthe present upped bound [13, 33] and may be detectable in thenext generation of experiments [34–37].We exemplify our proposal employing as “GUT” gaugesymmetry G B − L = G SM × U (1) B − L , where G SM = SU (3) C × SU (2) L × U (1) Y is the gauge symmetry of thestandard model, and B and L denote baryon and lepton num-ber respectively – cf. Ref. [20, 22, 23, 27]. The embeddingof IGHI within this particle model gives us the opportunity toconnect inflation with low energy phenomenology. In fact, theabsence of the gauge anomalies enforces the presence of threeright-handed neutrinos N ci which, in turn, generate the tinyneutrino masses via the type I seesaw mechanism. Further-more, the out-of-equilibrium decay of the N ci ’s provides uswith an explanation of the observed baryon asymmetry of theuniverse ( BAU ) [38] via non-thermal leptogenesis ( nTL ) [39]consistently with the gravitino ( e G ) constraint [40–43] and thedata [44, 45] on the neutrino oscillation parameters. Also,taking advantage of the adopted R symmetry, the parameter µ appearing in the mixing term between the two electroweakHiggs fields in the superpotential of MSSM is explained asin Refs. [3, 23, 25] via the v.e.v of the stabilizer field, pro-vided that the relevant coupling constant is appropriately sup-pressed. The post-inflationary completion induces more con-straints testing further the viability of our models.The remaining text is organized into three sections. Wefirst establish and analyze our inflationary scenarios in Sec. II.We then – in Sec. III – examine a possible post-inflationarycompletion of our setting. Our conclusions are summarized inSec. IV. Throughout the text, the subscript of type , z denotesderivation with respect to ( w.r.t ) the field z , and charge conju-gation is denoted by a star. Unless otherwise stated, we useunits where m P = 2 . · GeV is taken to be unity.
II. I NFLATIONARY M ODELS
In Sec. II A we describe the generic formulation of IG mod-els within SUGRA, in Sec. II B , we construct the inflationarypotential, and in Sec. II C we analyze the observational conse-quences of the models. A . E MBEDDING I NDUCED -G RAVITY HI IN SUGRA
The implementation of IGHI requires the determination ofthe relevant super- and K¨ahler potentials, which are specifiedin Sec. II
A 1 . In Sec. II
A 2 we present the form of the actionin the two relevant frames and in Sec. II
A 3 we impose the IGconstraint. . Set-up
As we already mentioned, we base the construction of ourmodels on the superpotential W HI = λS (cid:0) ¯ΦΦ − M / (cid:1) (1) which is already introduced in the context of models of F-term hybrid inflation [24]. Here ¯Φ , Φ denote a pair of left-handed chiral superfields oppositely charged under U (1) B − L ; S is a G B − L -singlet chiral superfield; λ and M are parameterswhich can be made positive by field redefinitions. W HI is themost general renormalizable superpotential consistent with acontinuous R symmetry [24] under which S → e iα S, ¯ΦΦ → ¯ΦΦ , W HI → e iα W HI . (2)Here and in the subsequent discussion the subscript HI is fre-quently used instead of IGHI to simplify the notation.As we verify below, W HI allows us to break the gauge sym-metry of the theory in a simple, elegant and restrictive way.The v.e.vs of these fields, though, have to be related with thesize of m P according to the IG requirement. To achieve this,together with the establishment of an inflationary era, we haveto combine W HI with a judiciously selected K¨ahler potential, K . We present two classes of such K ’s, which respect the(gauge and global) symmetries of W HI and incorporate onlyquadratic terms of the various fields. We distinguish theseclasses taking into account the origin of the kinetic mixing inthe inflaton sector. Namely:(a) K ’s without independent kinetic mixing. Having inmind the general recipe [19, 46] for the introduction of non-minimal couplings in SUGRA we include the gauge invariantfunction F R = ¯ΦΦ (3)in the following K ’s K R = − N ln (cid:18) c R ( F R + F ∗R ) − | Φ | + | ¯Φ | N + F S (cid:19) , (4 a )which is completely logarithmic, and K R = − N ln (cid:18) c R ( F R + F ∗R ) − | Φ | + | ¯Φ | N (cid:19) + F S , (4 b )which is polylogarithmic. In both cases we take N > . Thecrucial difference of the K ’s considered here, compared tothose employed in Ref. [19, 46], is that unity does not accom-pany the terms c R ( F R + F ∗R ) . As explained in Sec. II A 3 ,the identification of this quantity with unity at the vacuumof the theory essentially encapsulates the IG hypothesis –cf. Ref. [3, 5]. The existence of the real function | Φ | + | ¯Φ | in-side the argument of logarithm is vital for this scenario, sinceotherwise the K¨ahler metric is singular. These terms providecanonical kinetic terms for K = K R and N = 3 in the Jor-dan frame or c R -dependent kinetic mixing in the remainingcases, as we show in the next Section.(b) K ’s with independentkineticmixing. In this case weintroduce a softly broken shift symmetry for the Higgs fields– cf. Ref. [20, 31] – via the functions F ± = (cid:12)(cid:12) Φ ± ¯Φ ∗ (cid:12)(cid:12) . Inparticular, the dominant shift symmetry adopted here is Φ → Φ + c and ¯Φ → ¯Φ + c ∗ with c ∈ C , (5) Induced-GravityGUT-ScaleHIinSUGRAunder which F − remains unaltered whereas F + expresses theviolation of this symmetry and is placed in the argument of alogarithm with coefficient ( − N ) , whereas F − is set outside it.Namely, we propose the following K ’s K = − N ln (cid:0) c + F + + F S ( | S | ) (cid:1) + c − F − , (6 a ) K = − N ln ( c + F + ) + c − F − + F S ( | S | ) , (6 b ) K = − N ln ( c + F + ) + F S ( F − , | S | ) , (6 c )where N > . As in the case of the K ’s in Eqs. (4 a ) and(4 b ) unity is not included in the argument of the logarithm.In the present case, the identification of c + F + with unity –see Sec. II A 3 – at the vacuum of the theory incarnates the IGhypothesis – cf. Ref. [3, 5]. The degree of the violation of thesymmetry in Eq. (5) is expressed by r ± = c + /c − , which isconstrained by the unification condition to values of the order . – see Sec. II B 3 . Since this value is quite natural we are notforced here to invoke any argument regarding its naturalness– cf. Ref. [23].The models employing the K ’s in Eqs. (4 a ) and (4 b ) aremore economical compared to the models based on the K ’sin Eqs. (6 a ) – (6 c ). Indeed, the latter include two parame-ters ( c + and c − ) from which one ( c + ) enters f R and the other( c − ) dominates independently the kinetic mixing – see below.However, these parameters are related to the shift symmetry inEq. (5) which renders the relevant setting theoretically moreappealing. Indeed, this symmetry has a string theoretical ori-gin as shown in Ref. [47]. In this framework, mainly integer N ’s are considered which can be reconciled with the obser-vational data – see Sec. II C 3 . Namely, N = 3 [ N = 2 ] for K = K [ K = K or K ] yields completely acceptable re-sults. However, the deviation of the N ’s from these integervalues is also acceptable [5, 20, 22, 23, 48] and assist us tocover the whole allowed domain of the observables.Another possibility that could be inspected is what happensif we place the term c − F − inside the argument of the loga-rithm in Eqs. (6 a ) and (6 b ) – cf. Ref. [20] – considering theK¨ahler potentials K = − N ln ( c + F + − c − F − /N + F S ) , (7 a ) K = − N ln ( c + F + − c − F − /N ) + F S . (7 b )These K ’s, though, reduce to K R and K R respectively ifwe set c + = N c R − N and c − = N c R + 12 . (8)For c R ≫ the arrangement above results in r ± ≃ /N . Onthe other hand, the same r ± is found if we impose the unifi-cation constraint. Therefore, the observational predictions ofthe models based on the K ’s above are expected to be verysimilar to those obtained using Eqs. (4 a ) and (4 b ).The functions F lS with l = 1 , , encountered in Eqs. (4 a ),(4 b ) and (6 a ) – (6 c ) support canonical normalization and safestabilization of S during and after IGHI. Their possible formsare given in Ref. [23]. Just for definiteness, we adopt here only their logarithmic form, i.e., F S = − ln (cid:0) | S | /N (cid:1) , (9 a ) F S = N S ln (cid:0) | S | /N S (cid:1) , (9 b ) F S = N S ln (cid:0) | S | /N S + c − F − /N S (cid:1) , (9 c )with < N S < . Recall [6, 46] that the simplest term | S | leads to instabilities for K = K and light excitationsfor K = K and K . The heaviness of these modes is re-quired so that the observed curvature perturbation is generatedwholly by our inflaton in accordance with the lack of any ob-servational hint [53] for large non-Gaussianity in the cosmicmicrowave background. . From Einstein to Jordan Frame
With the ingredients above we can extract the part of the
Einstein frame ( EF ) action within SUGRA related to the com-plex scalars z α = S, Φ , ¯Φ – denoted by the same superfieldsymbol. This has the form [46] S = Z d x p − b g (cid:18) − b R + K α ¯ β b g µν D µ z α D ν z ∗ ¯ β − b V (cid:19) , (10 a )where b R is the EF Ricci scalar curvature, D µ is the gaugecovariant derivative, K α ¯ β = K ,z α z ∗ ¯ β , and K α ¯ β K ¯ βγ = δ αγ .Also, b V is the EF SUGRA potential which can be found interms of W HI in Eq. (1) and the K ’s in Eqs. (6 a ) – (6 c ) via theformula b V = e K (cid:16) K α ¯ β ( D α W HI ) D ∗ ¯ β W ∗ HI − | W HI | (cid:17) + g P a D , (10 b )where D α W HI = W HI ,z α + K ,z α W HI , D a = z α ( T a ) βα K β and the summation is applied over the generators T a of G B − L .In the right-hand side ( r.h.s ) of the equation above we clearlyrecognize the contribution from the D terms (proportional to g ) and the remaining one which comes from the F terms.If we perform a conformal transformation, along the linesof Ref. [20, 46], defining the frame function as − Ω /N = exp ( − K/N ) ⇒ K = − N ln ( − Ω /N ) , (11)we can obtain the form of S in the Jordan Frame ( JF ) whichis written as [20] S = Z d x √− g (cid:18) Ω2 N R − N Ω A µ A µ − V + (cid:18) Ω α ¯ β + 3 − NN Ω α Ω ¯ β Ω (cid:19) D µ z α D µ z ∗ ¯ β (cid:19) , (12 a )where we use the shorthand notation Ω α = Ω ,z α , and Ω ¯ α =Ω ,z ∗ ¯ α . We also set V = b V Ω /N and A µ = − iN (cid:0) Ω α D µ z α − Ω ¯ α D µ z ∗ ¯ α (cid:1) / . (12 b )Computing the expression in the parenthesis of the second linein Eq. (12 a ) for K = K R and K R , we can easily verify thatI InflationaryModels the choice for N = 3 ensures canonical kinetic terms – in ac-cordance with the findings in Ref. [19, 46] – whereas in theremaining cases a c R - (and not φ -) dependent kinetic mixingemerges. Indeed, in any case we have Ω α ¯ β = δ α ¯ β and for N = 3 the second term in the parenthesis vanishes. On thecontrary, for K = K , K and K , the same expression is notonly different than δ α ¯ β but also includes ( φ -dependent) entriesproportional to and dominated by c − ≫ c + . For this reason,the relevant models of IGHI may be more properly character-ized as kinetically modified. The non-renormalizability of thiskinetic mixing is under control since φ ≪ and the theory istrustable up to m P , as we show in Sec. II C 2 .Most importantly, though, the first term in the first line ofthe r.h.s of Eq. (12 a ) reveals that − Ω /N plays the role of anon-minimal coupling to gravity. Comparing Eq. (11) withthe K ’s in Eqs. (4 a ) – (6 c ) we can infer that − Ω N = ( N c R + 1) F R /N for K = K R and K R ,c + F + for K = K , K and K , (13)along the field configuration Φ = ¯Φ ∗ and S = 0 , (14)which is a honest inflationary trajectory, as shown inSec. II B 2 . The identification of the quantity in Eq. (13) with m at the vacuum, according to the IG conjecture, can be ac-commodated as described in the next section. . Induced-Gravity Requirement
The implementation of the IG scenario requires the genera-tion of m P at the vacuum of the theory, which thereby has tobe determined. To do this we have to compute V in Eq. (10 b )for small values of the various fields, expanding it in powers of /m P . Namely, we obtain the following low-energy effectivepotential V eff = e e K e K α ¯ β W HI α W ∗ HI ¯ β + g P a D + · · · , (15 a )where the ellipsis represents terms proportional to W HI or | W HI | which obviously vanish along the path in Eq. (68) –we assume here that the vacuum is contained in the inflari-onary trajectory. Also, e K is the limit of the K ’s in Eqs. (4 a )– (6 c ) for m P → ∞ . The absence of unity in the argumentsof the logarithms multiplied by N in these K ’s prevents thedrastic simplification of e K , especially for K = K R and K – cf. Ref. [23]. As a consequence, the expression of the result-ing V eff is rather lengthy. For this reason we confine ourselvesbelow to K = K or K where F lS with l = 2 , is placedoutside the first logarithm and so e K can be significantly sim-plified. Namely, we get e K = − N ln c + F + + c − F − + | S | , (15 b )from which we can then compute (cid:16) e K α ¯ β (cid:17) = diag (cid:16) f M ± , (cid:17) with f M ± = c − e K Φ¯Φ ∗ e K Φ¯Φ ∗ c − . (16 a ) Here, e K Φ¯Φ ∗ = N (Φ + ¯Φ ∗ ) and e K ¯ΦΦ ∗ = N (Φ ∗ + ¯Φ) , (16 b )since e K Φ = − N/ (Φ + ¯Φ ∗ ) + c − (Φ ∗ − ¯Φ) (16 c )and e K ¯Φ = − N/ (Φ ∗ + ¯Φ) − c − (Φ − ¯Φ ∗ ) . (16 d )To compute V eff we need to know (cid:16) e K α ¯ β (cid:17) = diag (cid:16) f M − ± , (cid:17) , (17 a )where f M − ± = 1 det f M ± c − − e K Φ¯Φ ∗ − e K ¯ΦΦ ∗ c − , (17 b )with det f M ± = c − − N /F . (17 c )Upon substitution of Eqs. (17 a ) and (17 b ) into Eq. (15 a ) weobtain V eff ≃ λ e e K + (cid:12)(cid:12)(cid:12)(cid:12) ¯ΦΦ − M (cid:12)(cid:12)(cid:12)(cid:12) + g (cid:16) Φ e K Φ − ¯Φ e K ¯Φ (cid:17) + λ e e K + | S | det f M ± (cid:16) c − (cid:0) | Φ | + | ¯Φ | (cid:1) − e K Φ¯Φ ∗ ¯Φ ∗ Φ − e K ¯ΦΦ ∗ ¯ΦΦ ∗ (cid:17) , (18)where e K + = − N ln c + F + . We remark that the direction inEq. (68) assures D-flatness since h Φ e K Φ i = h ¯Φ e K ¯Φ i and so thevacuum lies along it with h S i = 0 and |h Φ i| = |h ¯Φ i| = M/ . (19)The same result holds also for K = K R , K R and K aswe can verify after a more tedious computation. Eq. (19)means that h Φ i and h ¯Φ i spontaneously break U (1) B − L downto Z B − L . Note that U (1) B − L is already broken during IGHIand so no cosmic string are formed – contrary to what happensin the models of the standard F-term hybrid inflation [25, 26],which also employ W HI in Eq. (1).Inserting Eq. (19) into Eq. (13) we deduce that the conven-tional Einstein gravity can be recovered at the vacuum if M = (p N/ ( N c R − for K = K R and K R , / √ c + for K = K , K and K . (20)For c R ≃ or c + ∼ (10 − ) employed here, the re-sulting values of M are theoretically quite natural since theylie close to unity. Indeed, since the form of W HI in Eq. (1)is established around m P we expect that the scales entered byhand in the theory have comparable size. Induced-GravityGUT-ScaleHIinSUGRA B . I NFLATIONARY P OTENTIAL
Below we outline the derivation of the inflationary potentialin Sec. II
B 1 and check its stability by computing one-loopcorrections in Sec. II
B 2 . The last part of the analysis allowsus to determine the gauge-coupling unification condition (seeSec. II
B 3 ) which assists us to further constrain our models. . Tree-Level Result
If we express Φ , ¯Φ and S according to the parametrization Φ = φe iθ √ θ Φ , ¯Φ = φe i ¯ θ √ θ Φ and S = s + i ¯ s √ , (21)with ≤ θ Φ ≤ π/ , the trough in Eq. (68) can be written as ¯ s = s = θ = ¯ θ = 0 and θ Φ = π/ . (22)Along this the only surviving term in Eq. (10 b ) is b V HI = e K K SS ∗ | W HI ,S | , (23 a )which, for the choices of K ’s in Eqs. (6 a ) – (6 c ), reads b V HI = λ f W f − N R · ( f R for K = K R , K , for K = K R , K and K , (23 b )where f − N R = e K and we define the (inflationary) frame func-tion as f R = − Ω N (cid:12)(cid:12)(cid:12)(cid:12) Eq . (22) (23 c )which is translated as f R = ( ( N c R − φ / N for K = K R and K R ,c + φ for K = K , K and K . (23 d )The last factor in Eq. (23 b ) originates from the expression of K SS ∗ for the various K ’s. Also f W = ( ( N c R − φ − N for K = K R and K R ,c + φ − for K = K , K and K , (23 e )arises from the last factor in the r.h.s of Eq. (23 a ) togetherwith a W = ( N c R − for K = K R and K R and a W = c + for K = K , K and K . If we set N = ( n + 3 for K = K R , K , n + 1) for K = K R , K and K , (23 f )we arrive at a universal expression for b V HI which is b V HI = λ f W f n ) R · (24) The value n = 0 is special since we get N = 3 for K = K R and K or N = 2 for K = K R , K or K . Therefore, b V HI develops an inflationary plateau as in the original case ofStarobinsky model within no-scale SUGRA [3, 6] for large c R or c + . Contrary to that case, though, here we also have n and c − , whose variation may have an important impact onthe observables – cf. Ref. [20, 22]. In particular, for n < , b V HI remains an increasing function of φ , whereas for n > ,it develops a local maximum b V HI ( φ max ) = λ n n (1 + n ) n ) at φ max = r n a n , (25)where a = c R / for K = K R and K R whereas a = c + for K = K , K and K . In a such case we are forced to assumethat hilltop [49] IGHI occurs with φ rolling from the regionof the maximum down to smaller values. The relevant tuningof the initial conditions can be quantified by defining [26] thequantity ∆ max ⋆ = ( φ max − φ ⋆ ) /φ max , (26)where φ ⋆ is the value of φ when the pivot scale k ⋆ =0 . / Mpc crosses outside the inflationary horizon. The natu-ralness of the attainment of IGHI increases with ∆ max ⋆ , and itis maximized when φ max ≫ φ ⋆ which results in ∆ max ⋆ ≃ .To specify the EF canonically normalized inflaton, we notethat, for all choices of K in Eqs. (4 a ), (4 b ) and (6 a ) – (6 c ), K α ¯ β along the configuration in Eq. (22) takes the form (cid:0) K α ¯ β (cid:1) = diag ( M ± , K SS ∗ ) , (27)where K SS ∗ = 1 /f R [ K SS ∗ = 1 ] for K = K R , K [ K = K R , K and K ]. For K = K R and K R we find M ± = (1 + N c R ) / f R N/φ N/φ (1 + N c R ) / f R . (28)and upon diagonalization we obtain the following eigenvalues κ + = N c R f − R and κ − = f − R . (29)Note that the existence of the real function | Φ | + | ¯Φ | insidethe argument of logarithm is vital for this scenario, since oth-erwise M ± develops zero eigenvalue and so it is singular, i.e.,no K α ¯ β can be defined. On the other hand, for K = K , K and K we obtain M ± = c − N/φ N/φ c − , (30)with eigenvalues κ ± = c − ± N/φ . (31)Given that the lowest φ value is given in Eq. (20), we canimpose, in this case, a robust restriction on the parameters toassure the positivity of κ − during and after IGHI. Namely, κ − & ⇒ r ± . /N , (32)I InflationaryModels T ABLE I: Mass-squared spectrum of the inflaton sector for K = K R , K R , K , K and K along the path in Eq. (22).F IELDS E IGEN - M
ASSES S QUARED S TATES K = K R K = K R K = K K = K K = K b θ + b m θ + b H (1 − /N ) 6 b H b H /N S ) b H Scalars b θ Φ b m θ Φ M BL + 6 b H c R ( N − M BL + 6 b H c R ( N − M BL + 6 b H M BL + 6(1 + 1 /N S ) b H b s, b ¯ s b m s b H ( N − c R φ /N ) 3 b H (4 /N − N + 2 /N S ) 6 f W b H /N b H /N S A BL M BL Ng / ( Nc R − g (cid:0) c − φ − N (cid:1) Weyl b ψ ± b m ψ ± b H ( c R ( N − φ − N ) /N c R φ (cid:0) N − c + ( N − φ (cid:1) b H /c − f φ Spinors λ BL , b ψ Φ − M BL Ng / ( Nc R − g (cid:0) c − φ − N (cid:1) whereas we are not obliged to impose any condition for K = K R and K R .Inserting Eqs. (21) and (30) in the second term of the r.h.s ofEq. (10 a ) we can define the EF canonically normalized fields,denoted by hat, as follows d b φdφ = J = √ κ + , b θ + = Jφθ + √ , b θ − = r κ − φθ − , (33 a ) b θ Φ = φ √ κ − ( θ Φ − π/ , ( b s, b ¯ s ) = p K SS ∗ ( s, ¯ s ) , (33 b )where θ ± = (cid:0) ¯ θ ± θ (cid:1) / √ . Note, in passing, that the spinors ψ S and ψ Φ ± associated with the superfields S and Φ − ¯Φ are similarly normalized, i.e., b ψ S = √ K SS ∗ ψ S and b ψ Φ ± = √ κ ± ψ Φ ± with ψ Φ ± = ( ψ Φ ± ψ ¯Φ ) / √ . . Stability and Loop-Corrections
We can verify that the inflationary direction in Eq. (22) isstable w.r.t the fluctuations of the non-inflaton fields. To thisend, we construct the mass-squared spectrum of the variousscalars defined in Eqs. (33 a ) and (33 b ). Taking the limit c − ≫ c + we find the expressions of the masses squared b m χ α (with χ α = θ + , θ Φ and S ) arranged in Table I. For φ ≃ φ ⋆ thesefairly approach the quite lengthy, exact expressions taken intoaccount in our numerical computation. Given that φ < . for K = K R and f W ≫ for K = K we deduce that b m s > for N ≃ . Also for K = K R , K or K and < N S < , b m s > stays positive and heavy enough,i.e. b m z α ≫ b H = b V HI / . In Table I we also display themasses, M BL , of the gauge boson A BL – which signals thefact that G B − L is broken during IGHI – and the masses of thecorresponding fermions. Note that the unspecified eigestate b ψ ± is defined as b ψ ± = (cid:16) b ψ Φ+ ± b ψ S (cid:17) / √ . (34)As a consequence, let us again emphasize that no cosmicstring are produced at the end of IGHI.The derived mass spectrum can be employed in order to findthe one-loop radiative corrections, ∆ b V HI , to b V HI . Considering SUGRA as an effective theory with cutoff scale equal to m P ,the well-known Coleman-Weinberg formula [50] can be em-ployed taking into account only the masses which lie well be-low m P , i.e., all the masses arranged in Table I besides M BL and b m θ Φ – note that these contributions are cancelled out for K = K R and N = 3 or K = K R and N = 2 . The re-sulting ∆ b V HI leaves intact our inflationary outputs, providedthat the renormalization-group mass scale Λ , is determined byrequiring ∆ b V HI ( φ ⋆ ) = 0 or ∆ b V HI ( φ f ) = 0 . These conditionsyield Λ ≃ . · − − . · − and render our results practi-cally independent of Λ since these can be derived exclusivelyby using b V HI in Eq. (24) with the various quantities evaluatedat Λ – cf. Ref. [20]. Note that their renormalization-grouprunning is expected to be negligible because Λ is close to theinflationary scale b V / ≃ (3 − · − . Recall, here, that inthe case of F-term hybrid inflation [24–27] the SUSY poten-tial is classically flat and the radiative corrections contribute(together with the SUGRA corrections) in the inclination ofthe inflationary path. . SUSY Gauge Coupling Unification
The mass M BL listed in Table I of the gauge boson A BL may, in principle, be a free parameter since the U (1) B − L gauge symmetry does not disturb the unification of the MSSMgauge coupling constants. To be more specific, though, weprefer to determine M BL by requiring that it takes the value M GUT dictated by this unification at the vacuum of the theory.Namely, we impose h M BL i = M GUT ≃ / . · − = 8 . · − . (35)This simple principle has important consequences for bothclasses of models considered here. In particular:(a) For K = K R or K R . In this cases, the conditionabove completely determines c R since it implies via the find-ings of Table I c R = 1 N + 2 g M ≃ . · , (36) Induced-GravityGUT-ScaleHIinSUGRAleading to M ≃ . via Eq. (20). Here we take g ≃ . which is the value of the unified coupling constant withinMSSM. Although c R above is very large, there is no prob-lem with the validity of the effective theory, in accordancewith the results of earlier works [3, 5, 11] on IG inflation withgauge singlet inflaton. Indeed, expanding about h φ i = M –see Eq. (20) – the second term in the r.h.s of Eq. (10 a ) for µ = ν = 0 and b V HI in Eq. (24) we obtain J ˙ φ ≃ − r N c δφ + 32 N c δφ − r N c δφ + · · · ! ˙ c δφ , (37 a )where c δφ is the canonically normalized inflaton at the vacuum– see Sec. III C 1 – and b V HI ≃ λ c δφ N c R (cid:18) − N − √ N c δφ + 8 N − N + 18 N c δφ + · · · (cid:19) . (37 b )These expressions indicate that Λ UV = m P , since c R doesnot appear in any of their numerators. Although these expan-sions are valid only during reheating we consider Λ UV ex-tracted this way as the overall cut-off scale of the theory sincereheating is regarded [15] as an unavoidable stage of IGHI.(b) For K = K , K or K . In this cases, the conditionabove allows us to fix r ± since, substituting Eq. (20) in M BL shown in Table I, we obtain g (cid:0) c − h φ i − N (cid:1) = M ⇒ r ± = g N g + M . (38)Since M GUT > the condition above satisfies the restric-tion in Eq. (32) yielding r ± close to its upper bound because M GUT ≪ .As a bottom line, under the assumption in Eq. (35), c R for K = K R and K R or r ± for K = K , K and K cease to be free parameters, in sharp contrast to the modelsof Ref. [19–23] where the same assumption is employed toextract M ≪ as a function of the free parameters withoutany other theoretical constraint between them. Therefore, theinterplay of Eqs. (20) and (38) leads to the reduction of thefree parameters by one, thereby rendering the present set-upmore restrictive and predictive. C . I NFLATION A NALYSIS
In Secs. II
C 2 and II
C 3 below we inspect analyticallyand numerically respectively, if the potential in Eq. (24) en-dowed with the condition of Eqs. (20) and (38) may be con-sistent with a number of observational constraints introducedin Sec. II
C 1 . . General Framework
Given that the analysis of inflation in both EF and JF yieldsequivalent results [9], we carry it out exclusively in the EF. In particular, the period of slow-roll IGHI is determined in theEF by the condition max { b ǫ ( φ ) , | b η ( φ ) |} ≤ , (39 a )where the slow-roll parameters [51], b ǫ = (cid:16) b V HI , b φ / √ b V HI (cid:17) and b η = b V HI , b φ b φ / b V HI . (39 b )The number of e-foldings b N ⋆ that the scale k ⋆ = 0 . / Mpc experiences during IGHI and the amplitude A s of the powerspectrum of the curvature perturbations generated by φ can becomputed using the standard formulae [51] b N ⋆ = Z b φ ⋆ b φ f d b φ b V HI b V HI , b φ and A / = 12 √ π b V / ( b φ ⋆ ) | b V HI , b φ ( b φ ⋆ ) | , (40)where φ ⋆ [ b φ ⋆ ] is the value of φ [ b φ ] when k ⋆ crosses the infla-tionary horizon. These observables are to be confronted withthe requirements [53] b N ⋆ ≃ . b V HI ( φ ⋆ ) / b V HI ( φ f ) / + 12 f R ( φ ⋆ ); (41 a ) A / ≃ . · − . (41 b )Note that in Eq. (41 a ) we consider an equation-of-state pa-rameter w int = 1 / corresponding to quartic potential whichis expected to approximate b V HI rather well for φ ≪ – seeRef. [20]. We obtain b N ⋆ ≃ (57 . − .Then, we compute the remaining inflationary observables,i.e., the (scalar) spectral index n s , its running a s , and thescalar-to-tensor ratio r which are found from the relations[51] n s = 1 − b ǫ ⋆ + 2 b η ⋆ , r = 16 b ǫ ⋆ , (42 a ) a s = 2 (cid:0) b η ⋆ − ( n s − (cid:1) / − b ξ ⋆ , (42 b )where the variables with subscript ⋆ are evaluated at φ = φ ⋆ and b ξ = b V HI , b φ b V HI , b φ b φ b φ / b V .The resulting values of n s and r must be in agreement withthe fitting of the data [13, 33] with Λ CDM + r model. We takeinto account the data from Planck and
Baryon Acoustic Os-cillations ( BAO ) and the
BK14 data taken by the B ICEP / KeckArray
CMB polarization experiments up to and including the2014 observing season. The results are (a) n s = 0 . ± . and (b) r ≤ . , (43)at 95 % confidence level ( c.l. ) with | a s | ≪ . . . Analytic Results
A crucial difference of the present analysis w.r.t that for themodels in Ref. [19–23] is that M , given by Eq. (20), is notnegligible during the inflationary period and enters the rele-vant formulas via the function f W defined below Eq. (23 b ).We find it convenient to expose separately our results for thetwo basic classes of models introduced in Sec. II A 1 . Namely:I InflationaryModels (a) For K = K R and K R . The slow-roll parameterscan be derived employing J in Eq. (24), without explicitlyexpressing b V HI in terms of b φ . Our results are b ǫ = 4 ˜ f ( n ˜ f W − N c R φ and b η = 8 2 − ˜ f W − n ˜ f W + n ˜ f N ˜ f , (44)where ˜ f W = c R φ − . The condition Eq. (39 a ) is violatedfor φ = φ f , which is found to be φ f ≃ max √ c R s n n + √ N , r c R r − n n + N ! . (45)Then, b N ⋆ can be also computed from Eq. (40) as follows b N ⋆ ≃ ( N c R φ ⋆ / for n = 0 ,N ln (cid:16) n )2 − n ˜ f W ∗ (cid:17) / n (1 + n ) for n = 0 , (46)where ˜ f W ∗ = ˜ f W ( b φ ⋆ ) . Solving the above equations w.r.t φ ⋆ we obtain a unified expression φ ⋆ ≃ r f R ⋆ c R with f R ⋆ = 1 + nn (cid:16) − e − n (1+ n ) b N ⋆ /N (cid:17) (47)reducing to b N ⋆ /N in the limit n → . For c R in Eq. (36) wecan verify that φ ⋆ ∼ . and so the model is (automatically)well stabilized against corrections from higher order terms ofthe form (Φ ¯Φ) p with p > in W HI – see Eq. (1). Thanks toEq. (36), we can derive uniquely λ from the expression λ = 8 p A s πc R f n +1 R ⋆ n (1 − f R ⋆ ) + 1 √ N ( f R ⋆ − , (48)applying the second equation in Eq. (40). Upon substitutionof f R ⋆ into Eq. (42 a ) we obtain the predictions of the modelwhich are n s ≃ − n N + 16 N nf R ⋆ − − N f R ⋆ + 1( f R ⋆ − , (49 a ) r ≃ N (cid:18) n (1 − f R ⋆ ) f R ⋆ − (cid:19) . (49 b )Since only | n | ≪ are allowed, as we see below, the resultsabove, together with a s , can be further simplified as follows n s ≃ − b N ⋆ − nN − n b N ⋆ N , (50 a ) r ≃ N b N ⋆ − n N + 80 n N − n b N ⋆ N , (50 b ) a s ≃ − b N ⋆ + 3 n b N ⋆ + 8 n N − N b N ⋆ , (50 c )where, for n = 0 , the well-known predictions of the Starobin-sky model are recovered, i.e., n s ≃ . and r = 0 . [ r = 0 . ] for K = K R [ K = K R ]. On the other hand,contributions proportional to b N ⋆ can be tamed for sufficientlylow n as we can verify numerically. (b)For K = K , K and K . Working along the lines ofthe previous paragraph we estimate the slow-roll parametersas follows b ǫ = 8(1 + n − nc + φ ) c − φ f ; (51 a ) b η n − (3 + 10 n ) c + φ + 4 n f + nc φ c − φ f · (51 b )Given that φ ≪ , Eq. (39 a ) is saturated at the maximal φ value, φ f , from the following two values φ ≃ s c − r / ± and φ ≃ s c − (cid:18) r ± (cid:19) / , (52)where φ and φ are such that b ǫ ( φ ) ≃ and b η ( φ ) ≃ .The n dependence is not so crucial for this estimation. Since φ ⋆ ≫ φ f , from Eq. (40) we find b N ⋆ ≃ ( c − φ ⋆ (cid:0) c − r ± φ ⋆ / − (cid:1) / for n = 0 , − (cid:16) nc + φ ⋆ + ln (cid:16) − nc + φ ⋆ n (cid:17)(cid:17) / n r ± for n = 0 , (53)where b φ ⋆ is the value of b φ when k ⋆ crosses the inflationaryhorizon. As regards the consistency of the relation above for n > , we note that we get nc + φ ⋆ < n in all relevant casesand so, ln(1 − nc + φ ⋆ / (1 + n )) < assures the positivity of b N ⋆ . Solving the equations above w.r.t φ ⋆ , we can express φ ⋆ in terms of b N ⋆ as follows φ ⋆ ≃ f R ⋆ c + with f R ⋆ = (cid:16) r ± b N ⋆ (cid:17) for n = 0 , (1 + n + W k ( y )) /n for n = 0 , (54)where we make use of Eq. (23 d ). Also, W k is the Lambert W (or product logarithmic) function [52] with y = − (1 + n ) exp (cid:16) − − n (1 + 8 n b N ⋆ r ± ) (cid:17) . (55)We take k = 0 for n > and k = − for n < .Contrary to what happens for K = K R and K R , c − isnot uniquely determined here. Therefore, for any n we areobliged to impose a lower bound on it, above which φ ⋆ ≤ .Indeed, from Eq. (54) we have φ ⋆ ≤ ⇒ c − ≥ f R ⋆ /r ± , (56)and so our proposal can be stabilized against corrections fromhigher order terms. Despite the fact that c − may take rel-atively large values, the corresponding effective theories arevalid [14, 15] up to m P = 1 for r ± given by Eq. (38). To fur-ther clarify this point we have to identify the ultraviolet cut-offscale Λ UV of the theory by analyzing the small-field behaviorof our models. More specifically, adapting the expansions inEqs. (37 a ) and (37 b ) in our present case, we end up with theexpressions J ˙ φ ≃ (cid:16) − r ± c δφ + 3 N ¯ r ± c δφ − N ¯ r ± c δφ + · · · (cid:17) ˙ c δφ , (57 a ) Induced-GravityGUT-ScaleHIinSUGRAwhere we set ¯ r ± = p r ± / (1 + N r ± ) , and b V HI ≃ λ ¯ r ± c δφ c (cid:16) − (3 + 4 n )¯ r ± c δφ + (cid:18)
254 + 14 n + 8 n (cid:19) ¯ r ± c δφ + · · · (cid:19) . (57 b )From the expressions above we conclude that Λ UV = m P since r ± ≤ (and so ¯ r ± ≤ ) due to Eq. (38).From the second equation in Eq. (40) we can also concludethat λ is proportional to c − for fixed n . Indeed, pluggingEq. (54) into this equation and solving w.r.t λ , we find λ = 32 p A s πc − r / ± f n +1 / R ⋆ n (1 − f R ⋆ ) + 1( f R ⋆ − . (58)Numerically, – see below – we find that λ/c − develops a max-imum at n ≃ − . which signals a transition to a branchof inflationary solutions which deviate from those obtainedwithin the Starobinsky-like inflation.Inserting f R ⋆ from Eq. (54) into Eqs. (42 a ) and (42 b ) weobtain n s ≃ − f R ⋆ (cid:18) f R ⋆ + 1( f R ⋆ − − n f R ⋆ + 3 f R ⋆ − n (cid:19) , (59 a ) r ≃ r ± f R ⋆ (cid:18) − n ( f R ⋆ − f R ⋆ − (cid:19) , (59 b ) a s ≃ r ± f R ⋆ − f R ⋆ (cid:16) − f R ⋆ (2 f R ⋆ + 1)+ 3( f R ⋆ − f R ⋆ (7 f R ⋆ + 9) − n + 2( f R ⋆ − ( f R ⋆ ( f R ⋆ −
42) + 121) n (cid:17) . (59 c )where we can recognize the similarities with the formulasgiven in Eqs. (49 a ) and (49 b ). For | n | < . these formulasmay be expanded successively in series of n and / b N ⋆ withresults n s ≃ − n r ± − n r / ± b N / ⋆ − − n b N ⋆ − n b N ⋆ r ± ) / , (60 a ) r ≃ − n b N ⋆ − b N ⋆ r ± + 2(3 + 2 n )3( b N ⋆ r ± ) / + 32 n r / ± b N / ⋆ , (60 b ) a s ≃ − nr / ± b N / ⋆ − − n b N ⋆ . (60 c )From the expressions above, we can infer that there is a clear n (and r ± ) dependence of the observables which deviate some-what from those obtained in the pure Starobinsky-type infla-tion (or IG inflation) [3, 5, 6]. Note that the formulae, althoughsimilar, are not identical with those found in Ref. [23]. . Numerical Results
The approximate analytic expressions above can be verifiedby the numerical analysis. Namely, we apply the accurate ex-pressions in Eq. (40) and confront them with the requirements r . ( . ) n s (0.1) F IG . 1: Allowed curves in the n s − r . plane for K = K R (dashed line) and K = K R (solid line) – the n values in [out-side] squared brackets correspond to K = K R [ K = K R ]. Themarginalizedjoint [ ]regionsfrom Planck ,BAOandBK14dataaredepictedbythedark[light]shadedcontours. in Eqs. (41 a ) – (41 b ) adjusting c R and λ for K = K R and K R or c − and λ for with any selected n . Then, we com-pute the model predictions via Eqs. (42 a ) and (42 b ). Our re-sults are mainly displayed in Figs. 1 and 2, where we showa comparison of the models’ predictions against the observa-tional data [13, 33] in the n s − r . plane, where r . =16 b ǫ ( b φ . ) with b φ . being the value of b φ when the scale k = 0 . / Mpc , which undergoes b N . = b N ⋆ + 3 . e-foldings during IGHI, crosses the horizon of IGHI. Let us dis-cuss separately the results for the two classes of models. Inparticular:(a) For K = K R and K R . We depict in Fig. 1 bya dashed [solid] line the model predictions for K = K R [ K = K R ] against the observational data. We see thatthe whole observationally favored range at low r ’s is coveredvarying n which remains, though, rather close to zero. In fact n is tuned closer to zero and r is slightly lower compared tothose obtained for K = K , K and K – see below. Moreexplicitly, we find the allowed ranges . & n/ . & − and . . r/ − . . (61 a )for K = K R , whereas for K = K R we have . & n/ . & − and . . r/ − . . . (61 b )As n varies in its allowed ranges presented below, we obtain . . λ/ . . or . . λ/ . . . , (62)for K = K R or K = K R respectively. If we take n = 0 ,we find the central values of λ in the ranges above which are . and . correspondingly.(b) For K = K , K and K . In this case, let us clarifythat the (theoretically) free parameters of our models are n I InflationaryModels r . ( . ) n s (0.1) F IG . 2: The same as Fig. 1 but for K = K (dashed line) and K = K or K (solidline)withthe n valuesindicatedonthecurves(the n valuesinsquaredbracketscorrespondto K = K ). and λ/c − and not n , c − , and λ as naively expected – recallthat M and r ± are found from Eqs. (20) and (38). Indeed, ifwe perform the rescalings Φ → Φ / √ c − , ¯Φ → ¯Φ / √ c − and S → S, (63) W HI in Eq. (1) depends on λ/c − and r − ± , while the K ’s inEq. (6 a ) – (6 c ) depend on n and r ± . As a consequence, b V HI depends exclusively on λ/c − and n . Since the λ/c − varia-tion is rather trivial – see Eq. (58) – we focus below on thevariation of n .In Fig. 2 we depict the theoretically allowed values withsolid and dashed lines for K = K or K and K = K re-spectively. The variation of n is shown along each line. Insquared brackets we display the n values for K = K whenthese differ appreciably from those for K = K or K . Weremark that for n > there is a discrepancy of about changing K from K to K or K , which decreases as n de-creases below zero. This effect originates from the differencein J – see Eqs. (31) and (33 a ) – which becomes smaller andsmaller as n decreases or c − increases. We observe that n > values dominate the part of the curves with lower r values and n s ≤ . , whereas the n < values generate the part of thecurves with n s close to its upper bound in Eq. (43) and appre-ciably larger r values. Roughly speaking, the displayed curvescan be produced interconnecting the limiting points of the var-ious curves in Fig. 2- (a) of Ref. [23], although the curves for < n < . and n < − . are not depicted there. This is be-cause the r ± ’s resulting from Eq. (38) are close to their upperlimits induced by Eq. (32).Comparing these theoretical outputs with data depicted bythe dark [light] shaded contours at c.l. [ c.l.] we findthe allowed ranges. Especially, for K = K we obtain . & n/ . & − . , . . r ± / . . . . (64)On the other hand, for K = K or K , we find one branchlocalized in the ranges . & n/ . & − . , . . r ± / . . . , (65) T ABLE
II:
Parameters and observables for the points shown in Fig. 2with K = K and K . n/ . r ± / . λ/ − c − n s / . − a s / − r/ . .
51 4 .
76 1 . .
58 5 . . . .
81 2 9 .
62 5 . . . . .
55 9 .
69 5 0 . − .
15 5 .
07 3 . .
75 4 . . − . .
32 4 . .
78 3 . . − . .
88 4 . .
78 3 .
98 3 . − . .
66 3 . .
76 4 6 . − . .
35 3 . .
73 4 . . and another one − . & n/ . & − . , . . r ± / . . . . (66)The findings for K = K or K , can also be read-off fromTable II where we list the values of the input parameter ( n )depicted in Fig. 2, the corresponding output parameters ( r ± and λ/c − ) and the inflationary observables. We observe that n s and r are well confined in the allowed regions of Eq. (43),while a s varies in the range − (3 . − . · − and so,our models are consistent with the fitting of data with the Λ CDM+ r model [13]. Comparing these numerical valueswith those obtained by the analytic expressions in Eqs. (59 a )– (59 c ) we obtain complete agreement for any n . On the otherhand, the approximate formulas in Eqs. (60 a ) – (60 c ) are validonly for | n s | < . , i.e., the Starobinsky-like region. HilltopIGHI is attained for n > and there we find ∆ max ⋆ & . ,where ∆ max ⋆ increases as n drops. The required tuning is notsevere, mainly for n < . since ∆ max ⋆ & . Since ourmodels predict r & . , they are testable by the forthcom-ing experiments, like B ICEP r withan accuracy of − . We do not present in Table II φ ⋆ valuessince, as inferred by Eq. (58), every φ ⋆ satisfying Eq. (41 a )leads to the same ratio λ/c − . For the reasons mentioned be-low Eq. (56), we prefer φ ⋆ ≤ . To achieve this, we need c − & (30 − for K = K and c − & (40 − for K = K or K , where the variation of c − is given as n de-creases.For K = K we expect similar values for λ/c − and theinflationary observables. However, r ± will differ appreciablydue to the different relation between n and N – see Eq. (23 f ).To highlight it further, we present in Fig. 3 the r ± values,obtained by Eq. (38), as a function of n for K = K (dashedlines) or K = K and K (solid lines). The values of thecurves which are preferred by the observational data at c.l. [ c.l.] are included in the dark [light] gray segments– cf. Fig. 2. We observe that for tiny n values, r ± which isroughly /N lies close to / for K = K and / for K = K , K . For larger | n | values r ± deviates more drasticallyfrom these values.The rather different predictions attained for low ( | n | ≤ . )and large ( | n | > . ) n values hint that the structure of b V HI Induced-GravityGUT-ScaleHIinSUGRA -4 -3 -2 -1 03456789 K = K or K r + ( . ) n (0.1) _ K = K F IG . 3: Values of r ± allowed by Eq. (20) as a function of n for K = K (dashed lines) and K = K or K (solid lines). Thevalueswhicharealsopreferredbytheobservational dataat c.l.[ c.l.] areincludedinthedark[light]graysegments. changes drastically. To illuminate this fact we show b V HI as afunction of φ in Fig. 4 for K = K or K , n = 0 . (grayline) and n = − . (light gray line). We take in both cases φ ⋆ = 1 . Therefore, the corresponding c − and λ values areconfined to their lowest possible values enforcing Eqs. (41 a )and (41 b ). More specifically, we find λ = 9 . · − or . · − and c − = 38 . or , with M = 0 . or M = 0 . for n = 0 . or n = − . respectively. The corresponding r ± values and observable quantities are listed in Table II. Wesee that in both cases b V HI develops a singularity at φ = 0 contrary to the models of non-minimal inflation – cf. Ref. [20,23] – where b V HI exhibits a maximum. However, for n > and | n | ∼ . , b V HI resembles the potential of Starobinsky modelwith a maximum at φ max = 1 . . This does not affect muchthe inflationary evolution since we find ∆ max ⋆ = 39% , and sothe tuning of the initial conditions of IGHI is rather mild. Onthe contrary, b V HI increases monotonically and almost linearlywith φ for n = − . . Both behaviors can be interpreted fromEq. (24) taking into account that f W ∼ φ and f R ∼ φ . For n ∼ . , b V HI ∼ f /f R becomes more or less constant,whereas for n ≃ − . , b V HI ∼ f /f · / R ∼ φ /φ ∼ φ . Itis also remarkable that in the latter case r increases, thanks tothe increase of the inflationary scale, b V / . Similar region ofparameters is recently reported in Ref. [54]. III.
A P
OSSIBLE P OST -I NFLATIONARY C OMPLETION
Our discussion about IGHI is certainly incomplete withoutat least mentioning how the transition to the radiation domi-nated era is realized and the observed BAU is generated. Sincethese goals are related to the possible decay channels of the in-flaton, the connection of IGHI with some low energy theoryis unavoidable. A natural, popular and well motivated frame-work for particle physics at the TeV scale beyond the standardmodel is MSSM. A possible route to such a more complete , φ , φ f n c _ V H I ( - ) φ ^ * > * φ max F IG . 4: The inflationary potential b V HI as a function of φ for K = K or K , φ ⋆ = 1 and n = 0 . (gray line) or n = − . (lightgray line). The values corresponding to φ ⋆ , φ f and φ max (for n =0 . )arealsoindicated. scenario is described in Sec. III A . Next, Sec. III B is devotedto the connection of IGHI with MSSM through the generationof the µ term. In Sec. III C , finally, we analyze the scenario ofnTL exhibiting the relevant constraints and further restrictingthe parameters. Here and hereafter we restore units, i.e., wetake m P = 2 . · GeV. A . T HE R ELEVANT S ET -U P Following the post-inflationary setting of Ref. [23] we sup-plement the superpotential of the theory with the terms W RHN = λ iN c ¯Φ N c i + h Nij N ci L j H u , (67 a )which allows for the implementation of type I see-saw mech-anism (providing masses to light neutrinos) and supports a ro-bust baryogenesis scenario through nTL, and W µ = λ µ SH u H d , (67 b )inspired by Ref. [25], which offers a solution to one of themost tantalizing problems of MSSM, namely the generationof a µ term – for an alternative solution see Ref. [55]. Here weadopt the notation and the B − L and R charges of the varioussuperfields as displayed in Table 1 of Ref. [23]. Let us onlynote that L i denotes the i -th generation SU (2) L doublet left-handed lepton superfields, and H u [ H d ] is the SU (2) L dou-blet Higgs superfield which couples to the up [down] quarksuperfields. Also, we assume that the superfields N cj havebeen rotated in the family space so that the coupling constants λ i are real and positive. This is the so-called [3, 23] N ci basis,where the N ci masses, M iN c , are diagonal, real, and positive.We assume that the extra fields X β = H u , H d , e N ci haveidentical kinetic terms as the stabilizer field S expressed bythe functions F lS with l = 1 , , in Eqs. (9 a ) – (9 c ) – seeRef. [23]. Therefore, N S may be renamed N X henceforth.II APossiblePost-InflationaryCompletion T ABLE
III:
Mass-squared spectrum of the non-inflaton sector for K = K R and K R along the path in Eqs. (22) and (68).F IELDS E IGEN - M
ASSES S QUAREDSTATES K = K R K = K R
10 Real b h ± , b ¯ h ± b m h ± b H c R (cid:0) φ / N ± λ µ /λ (cid:1) b H (cid:0) /N X ± λ µ /λφ (cid:1) Scalars b ˜ ν ci , b ¯˜ ν ci b m i ˜ ν c b H c R (cid:0) φ / N + 8 λ iN c /λ (cid:1) b H (cid:0) /N X + 16 λ iN c /λ φ (cid:1) Weyl Spinors b N ci b m iN c b H λ iN c /λ φ T ABLE
IV:
The same as Table III but for K = K , K and K .F IELDS E IGEN - M
ASSES S QUAREDSTATES K = K K = K K = K
10 Real b h ± , b ¯ h ± b m h ± b H (cid:0) f W /N ± λ µ c φ /λf W (cid:1) b H (1 + 1 /N X ± λ µ c + /λf W ) Scalars b ˜ ν ci , b ¯˜ ν ci b m i ˜ ν c b H (cid:0) (1 + f W /N ) /c + φ + 16 λ iN c c φ /λ f (cid:1) b H (cid:0) /N X + 16 λ iN c c φ /λ f (cid:1) Weyl Spinors b N ci m iN c λ iN c c φ b H /λ f The inflationary trajectory in Eq. (22) has to be supplementedby the conditions H u = H d = e N ci = 0 , (68)and the stability of this path has to be checked, parameterizingthe complex fields above as we do for S in Eq. (21). Therelevant masses squared are listed in Table III for K = K R and K R and Table IV for K = K , K and K , where wesee that b m i ˜ ν c > and b m h + > for every φ . On the otherhand, the positivity of the eigenvalues b m h − associated withthe eigenstates b h − and b ¯ h − , where b h ± = ( b h u ± b h d ) / √ and b ¯ h ± = ( b ¯ h u ± b ¯ h d ) / √ , (69)with the hatted fields being defined as b s and b ¯ s in Eq. (33 b ),requires the establishment of the inequalities λ µ . λφ / N for K = K R , (70 a ) λ µ . λφ (1 + 1 /N X ) / for K = K R , (70 b ) λ µ . λf W (1 + f W /N ) / λ µ c φ for K = K , (70 c ) λ µ . λf W (1 + 1 /N X ) / c + for K = K , K . (70 d )In all cases, the inequalities are fulfilled for λ µ . · − .Similar numbers are obtained in Ref. [3, 23]. We do notconsider such a condition on λ µ as unnatural, given that theYukawa coupling constant h U , which provides masses to theup-type quarks, is of the same order of magnitude too at a highscale – cf. Ref. [56]. Note that the hierarchy in Eqs. (70 a ) –(70 d ) between λ µ and λ differs from that imposed in the mod-els [25] of F-term hybrid inflation, where S plays the role ofinflaton and Φ , ¯Φ , H u and H d are confined at zero. Indeed,in that case we demand [25] λ µ > λ so that the tachyonicinstability in the Φ − ¯Φ direction occurs first, and the Φ − ¯Φ system start evolving towards its v.e.v, whereas H u and H d continue to be confined to zero. In our case, though, the infla-ton is included in the ¯Φ − Φ system while S and the H u − H d system are safely stabilized at the origin both during and afterIGHI. Therefore, φ settles in its vacuum and S , H u and H d take their non-vanishing electroweak scale v.e.vs afterwards. B . A S
OLUTION TO THE µ P ROBLEM OF
MSSM
A byproduct of the R symmetry associated with our modelsis that it assists us to understand the origin of the µ term ofMSSM – see Sec. III B 1 – connecting thereby the high withthe low energy phenomenology as described in Sec. III
B 2 . . Generating the µ Parameter
Working along the lines of Sec. II
A 2 we can verify that thepresence of the terms in Eqs. (67 a ) and (67 b ) leave the v.e.vsin Eq. (19) unaltered whereas those of X β are found to be h H u i = h H d i = h e N ci i = 0 . (71)On the other hand, the contributions from the soft SUSYbreaking terms, although negligible during IGHI – since theseare expected to be much smaller than φ –, may slightly shift[3, 23, 25] h S i from zero in Eq. (19). Indeed, the relevantpotential terms are V soft = (cid:0) λA λ S ¯ΦΦ − a S SλM / . c . (cid:1) + m γ | X γ | , (72)where X γ = Φ , ¯Φ , S, H u , H d , e N ci , and m γ , A λ and a S aresoft SUSY breaking mass parameters of the order of TeV.The emergence of these terms depend on the mechanism ofSUSY breaking which is not specified here. We restrict our-selves to the assumption that this extra sector of the theorymay be included in the present set-up without disturbing thestatus of inflation – cf. Ref. [57]. Confining Φ , ¯Φ , H u , H d and N ci in their v.e.vs in Eqs. (19) and (71), b V in Eq. (10 b ) reduces Induced-GravityGUT-ScaleHIinSUGRAagain to V eff in Eq. (15 a ) with vanishing terms represented byellipsis since h W HI i = 0 . We then rotate S to the real axis byan appropriate R -transformation and choose conveniently thephases of A λ and a S so as the total low-energy potential V tot = V eff + V soft (73)to be minimized. Since the form of V eff depends on theadopted K , we single out the cases:(a) K = K , K and K . Focusing on K = K or K we obtain h V tot ( S ) i ≃ λ S h det M ± i (cid:0) c − M − N m (cid:1) − λ a / m / M S, (74 a )where the first term in the r.h.s originates from the second lineof Eq. (18) for e e K + /m ≃ , and Φ and ¯Φ equal to their v.e.vsin Eq. (19). Also, we take into account that m S ≪ M , andwe set | A λ | + | a S | = 2a / m / , (74 b )where m / is the e G mass and a / > a parameter of orderunity which parameterizes our ignorance of the dependenceof | A λ | and | a S | on m / . The minimization condition for thetotal potential in Eq. (74 a ) w.r.t S leads to a non vanishing h S i as follows ddS h V tot ( S ) i = 0 ⇒ h S i ≃ a / m / c − (1 + N r ± ) /λ, (74 c )since from Eqs. (17 c ), (19) and (20) we infer h det M ± i = c − − N c = c − (1 + N r ± )(1 − N r ± ) . (74 d )At this S value, h V tot ( S ) i develops a minimum since d dS h V tot ( S ) i = λ /c + ( c − + N c + ) > . (74 e )For K = K Eq. (74 c ) can be obtained again by doing anexpansion of the relevant expressions in powers /m P .The µ term generated from Eq. (67 b ) exhibits the mixingparameter µ = λ µ h S i ≃ λ µ a / m / c − (1 + N r ± ) /λ . (75 a )Comparing this result with the corresponding one in Ref. [23],we deduce a crucial difference regarding the sign of the ex-pression in the parenthesis which originates from the terms inthe second line of Eq. (18). With the aid of Eq. (58) we mayeliminate c − and λ from the above result which then reads µ ≃ . · λ µ a / m / (1 + N r ± )( f R ⋆ − r / ± f n +1 / R ⋆ ( n (1 − f R ⋆ ) + 1) , (75 b )where Eq. (41 b ) is employed to obtain the numerical prefac-tor. Taking into account Eqs. (38) and (54), we infer that theresulting µ depends only on n and not on λ, c − and r ± – cf. Ref. [23, 25]. For the λ µ values allowed by Eqs. (70 c ) and(70 d ), any | µ | value is accessible with a mild hierarchy be-tween m / and µ – from Table IV we see that both signs of λ µ (and so µ ) are possible without altering the stability anal-ysis of the inflationary system. To understand this, let us firstremark that Eq. (20) implies r ± ≃ /N and f R ⋆ varies fromabout to [ to ] for K = K [ K = K and K ],as n varies in the allowed ranges of Eqs. (64) – (66). A roughestimation gives µ ∼ λ µ f / R ⋆ = 10 − . m / and so weexpect that µ is about one order of magnitude less than m / .(b) K = K R and K R . In this case, V tot in Eq. (73)with all the fields except S equal to their v.e.vs in Eqs. (19)and (71) is written as h V tot ( S ) i = λ m S c R ( N c R − − λ a / m / M S . (76 a )The minimization of h V tot ( S ) i w.r.t S leads to a new non van-ishing h S i , h S i ≃ N a / m / c R /λ, (76 b )where M is replaced by Eq. (20). Therefore, the µ parameterinvolved in Eq. (67 b ) is µ = λ µ h S i = N λ µ a / m / c R /λ . (76 c )This still depends only n thanks to the condition in Eq. (48)which fixes λ/c R as a function of n – see Eq. (47).To highlight further the conclusions above, we can em-ploy Eq. (75 a ) to derive the m / values required so as toobtain a specific µ value. Given that Eq. (75 a ) depends on n , which crucially influences n s and r , we expect that the re-quired m / is a function of n s and r as depicted in Fig. 5- (a) and Fig. 5- (b) respectively. We take λ µ = 10 − , in ac-cordance with Eqs. (70 c ) and (70 d ), a / = 1 , K = K or K with N X = 2 and µ = 0 . TeV (dot-dashed line), µ = 1 TeV (solid line), or µ = 2 TeV (dashed line). Varying n in the allowed range indicated in Fig. 2- (a) we obtain thevariation of m / solving Eq. (75 a ) w.r.t m / . The valuesof the curves which are preferred by the observational data at c.l. [ c.l.] are included in the dark [light] gray seg-ments – cf. Fig. 2. We see that m / increases with µ and itslowest value m / ≃ TeV is obtained for µ = 0 . TeV. Aswe anticipated above, m / is almost one order of magnitudelarger than the corresponding µ . Moreover, for fixed µ , eachcurve develops a maximum at n ≃ − . , which coincideswith the right corner of the curve in Fig. 2. This behavior de-viates a lot from the one found in Ref. [23] and comes fromthe different sign in the parenthesis of Eq. (75 a ). . Connection with the Parameters of CMSSM
The SUSY breaking effects, considered in Eq. (72), explic-itly break U (1) R to a subgroup, Z R which can be identifiedII APossiblePost-InflationaryCompletion µ = 2 TeV µ = 1 TeV µ = 0.5 TeV m / ( T e V ) n s (0.1) (a) m / ( T e V ) r (0.1) (b) F IG . 5: The gravitino mass m / versus n s (a) and r (b) for λ µ = 10 − , a / = 1 , K = K or K with N X = 2 and µ = 0 . TeV(dot-dashedline), µ = 1 TeV (solidline),or µ = 2 TeV (dashedline). ThecolorcodingisasinFig.3. T ABLE V: Therequired λ µ valuesrenderingourmodelscompatiblewiththebest-fitpointsoftheCMSSMasfoundinRef.[58]withtheassumptionsinEq.(77).CMSSM A/H ˜ τ − χ ˜ t − χ ˜ χ ± − χ R EGION : F
UNNEL C OANNIHILATION | A | ( TeV ) 9 .
924 1 .
227 9 .
965 9 . m ( TeV ) 9 .
136 1 .
476 4 .
269 9 . | µ | ( TeV ) 1 .
409 2 .
62 4 .
073 0 . / .
086 0 .
831 2 .
33 1 . λ µ (10 − ) for K = K R n = 0 0 . . .
91 0 . λ µ (10 − ) for K = K R n = 0 1 .
184 17 .
81 3 .
41 0 . λ µ (10 − ) for K = K n = 0 .
02 1 . . .
063 1 . n = − .
25 1 .
87 28 .
11 5 .
39 1 . λ µ (10 − ) for K = K and K n = 0 .
02 1 . . .
23 1 . n = − .
25 2 . . .
025 2 . with a matter parity. Under this discrete symmetry all the mat-ter (quark and lepton) superfields change sign – see Table 1 ofRef. [23]. Since S has the R symmetry of the total superpoten-tial of the theory, h S i in Eq. (74 c ) also breaks spontaneously U (1) R to Z R . Thanks to this fact, Z R remains unbroken andso no disastrous domain walls are formed. Combining Z R with the Z f2 fermion parity, under which all fermions changesign, yields the well-known R -parity. This residual symme-try prevents rapid proton decay and guarantees the stabilityof the lightest SUSY particle ( LSP ), providing thereby a well-motivated cold dark matter ( CDM ) candidate.The candidacy of LSP may be successful, if it generates thecorrect CDM abundance [53] within a concrete low energyframework, which in our case is the MSSM or one of its vari- ants – for an alternative approach within high-scale SUSY seeRef. [59]. Here, we adopt the
Constrained MSSM ( CMSSM )which is the most restrictive, predictive and well-motivatedversion of MSSM, which allows the lightest neutralino to playthe role of LSP in a sizable portion of the parametric space.This is based on the free parameters sign µ, tan β = h H u i / h H d i , M / , m , and A , where sign µ is the sign of µ , and the three last mass pa-rameters denote the common gaugino mass, scalar mass andtrilinear coupling constant, respectively, defined (normally)at M GUT . The parameter | µ | is not free, since it is com-puted at low scale by enforcing the conditions for the elec-troweak symmetry breaking. The values of these parame-ters can be tightly restricted imposing a number of cosmo-phenomenological constraints from which the consistency ofLSP relic density with observations plays a central role. Someupdated results are recently presented in Ref. [58], where wecan also find the best-fit values of | A | , m and | µ | listed in thefirst four lines of Table V. We see that there are four allowedregions characterized by the mechanism applied for accom-modating an acceptable CDM abundance.Taking advantage of this investigation, we can checkwhether the µ and m / values satisfying Eq. (75 a ) are consis-tent with these values. Selecting some representative n valuesand adopting the identifications m = m / and | A λ | = | a S | = | A | , (77)we can first derive a / from Eq. (74 b ) and then the λ µ valuesfrom Eqs. (75 a ) – (76 c ), which yield the phenomenologicallydesired | µ | shown in the third line of Table V. Here we assumethat renormalization effects in the derivation of µ are negligi-ble. The outputs of our computation are assembled in the lastten lines of Table V. As inputs, we take n = 0 . for K = K R and K R n = 0 . and − . for K = K , K and K .These are central values in the regions compatible with theinflationary observations as found in Sec. II C 3 . The λ µ val-ues for K = K R and K R are lower than those obtained for K = K , K and K , larger than those found in Ref. [23], Induced-GravityGUT-ScaleHIinSUGRAand similar to those in Ref. [3] especially for K = K R . Onthe other hand, the λ µ values found for K = K , K and K are larger compared to those found in Refs. [3, 23].From the outputs we infer that the required λ µ values arecomfortably compatible with Eqs. (70 a ) – (70 d ) for N X =2 , in all cases besides the one corresponding to the ˜ τ − χ coannihilation region. In that case, m is lower than | µ | andso marginally large λ µ values are required. In the cases wherenumbers are written in italics, we obtain instability along theinflationary path for K = K R and K , whereas for K = K R , K and K we need ≤ N X ≤ to avoid this effect.In sharp contrast to the model of Ref. [23], only the A/H funnel and ˜ χ ± − χ coannihilation regions can be consistentwith the e G limit on T rh – see Sec. III C 2 . Indeed, m / & TeV become cosmologically safe under the assumption ofan unstable e G , for the T rh values necessitated for satisfactoryleptogenesis – see Sec. III C 3 . C . N ON -T HERMAL L EPTOGENESIS
Our next task is to specify how our inflationary scenariomakes a transition to the radiation dominated era – seeSec. III
C 1 – and offers an explanation of the observed BAUconsistent with the e G constraint and the low energy neu-trino data – see Sec. III C 2 . Our results are summarized inSec. III
C 3 . . Inflaton Mass and Decay
When IGHI is over, the inflaton continues to roll down to-wards the SUSY vacuum, Eq. (19). Soon after, it settles intoa phase of damped oscillations around the minimum of b V HI .The (canonically normalized) inflaton, c δφ = h J i δφ with δφ = φ − M, (78)and h J i = ( √ N c R for K = K R and K R , p c − (1 + N r ± ) for K = K , K and K , (79)acquires mass, at the SUSY vacuum in Eq. (19), which isgiven by b m δφ λm P = (p c R ( N c R − for K = K R and K R ,c − p N r ± ) for K = K , K and K . (80)From the last expression we can infer that b m δφ remains con-stant for fixed n since λ/c R [ λ/c − ] is fixed too – see Eqs. (48)and (58). More specifically, for the allowed range of n inEqs. (61 a ) and (61 b ) we obtain . . b m δφ / GeV . . for K = K R , (81 a ) . . b m δφ / GeV . . for K = K R , (81 b ) with the value b m δφ = 2 . · GeV corresponding to n =0 . Furthermore, for K = K and the allowed range of n inEq. (64) we obtain . . b m δφ / GeV . . (82 a )For K = K and K , in the allowed ranges of Eqs. (65) and(66), we obtain . . b m δφ / GeV . . (82 b ) . . b m δφ / GeV . . . (82 c )We remark that b m δφ is somewhat affected by the choice of K ’s in Eqs. (6 a ) – (6 c ). For n = 0 , b m δφ = 4 . · GeVfor K = K , and b m δφ = 5 . · GeV for K = K and K , which are both somewhat larger than the value obtainedwithin Starobinsky inflation [3, 6]. On the other hand, thesevalues are close to the maximal ones found in Ref. [23], sincehere r ± approaches its maximal value.The inflaton can decay [60] perturbatively into:(a) A pair of N ci with Majorana masses M iN c = λ iN c M with the decay width b Γ δφ → N ci N ci = g iN c π b m δφ − M iN c b m δφ ! / , (83 a )where the relevant coupling constant g iN c = ( N − λ iN c h J i (83 b )arises from the lagrangian term L c δφ → N ci N ci = − e K/ m W RHN ,N ci N ci N ci N ci + h . c . = g iN c c δφ N ci N ci + h . c . (83 c )This decay channel activates the mechanism of nTL, assketched in Sec. III C 2 .(b) Higgses H u and H d with the decay width b Γ δφ → H u H d = 28 π g H b m δφ where g H = λ µ √ (84 a )arises from the lagrangian term L c δφ → H u H d = − e K/m K SS ∗ | W µ,S | = − g H b m δφ c δφ ( H ∗ u H ∗ d + h . c . ) + · · · (84 b )Thanks to the upper bounds on λ µ from Eqs. (70 c ) and (70 d ), g H turns out to be comparable with g iN c .II APossiblePost-InflationaryCompletion (c) MSSM (s)-particles XY Z with the following c + -dependent 3-body decay width b Γ δφ → XY Z = g y n f π b m δφ m , (85 a )where for the third generation we take y ≃ (0 . − . , com-puted at the b m δφ scale, and n f = 14 for b m δφ < M N c . Also, g y = y · (p ( N c R − / c R for K = K R and K R ,N p r ± / (1 + N r ± ) for K = K , K and K (85 b )and y = h t,b,τ ( b m δφ ) ≃ . . Since r ± ≃ /N we can easilyinfer that g y above is enhanced compared to the correspondingone in Ref. [23] where r ± ≃ . , and an additional suppres-sion through a ratio M/m P exists. We therefore expect that b Γ δφ → XY Z contributes sizably to the total decay width of c δφ .Each individual decay width arises from the langrangian terms L c δφ → Xψ Y ψ Z = − e K/ m ( W y,Y Z ψ Y ψ Z ) + h . c . = − g y c δφm P ( Xψ Y ψ Z ) + h . c ., (85 c )where W y = yXY Z is a typical trilinear superpotential termof MSSM with y a Yukawa coupling constant, and ψ X , ψ Y and ψ Z are the chiral fermions associated with the superfields X, Y and Z whose scalar components are denoted with thesuperfield symbols.The resulting reheat temperature is given by [61] T rh = (cid:0) / π g ∗ (cid:1) / b Γ / δφ m / , (86 a )with the total decay width of c δφ being b Γ δφ = b Γ δφ → N ci N ci + b Γ δφ → H u H d + b Γ δφ → XY Z . (86 b )Here, g ∗ = 228 . counts the MSSM effective number of rel-ativistic degrees of freedom at temperature T rh . Let us clarifyhere that in our models there is no decay of a scalaron as in theoriginal (non-SUSY) [2] Starobinsky inflation and some [62]of its SUGRA realizations; thus, T rh in our case is slightlylower than that obtained there. . Lepton-Number and Gravitino Abundances
For T rh < M iN c , the out-of-equilibrium decay of ν ci gen-erates a lepton-number asymmetry (per ν ci decay), ε i – see,e.g., Ref. [38, 39]. The resulting ε i is partially convertedthrough sphaleron effects into a yield of the observed BAU[23, 38, 39], Y B = − . · · T rh b m δφ P i b Γ δφ → N ci N ci b Γ δφ ε i , (87) which has to reproduce the observational result [53] Y B = (cid:0) . +0 . − . (cid:1) · − . (88)The validity of Eq. (87) requires that the c δφ decay into a pairof N ci ’s is kinematically allowed for at least one species of the N ci ’s and also that there is no erasure of the produced Y L dueto N c mediated inverse decays and ∆ L = 1 scatterings [64].These prerequisites are ensured if we impose ( a ) b m δφ ≥ M N c and ( b ) M N c & T rh . (89)The quantity ε i can be expressed in terms of the Dirac massesof ν i , m i D , arising from the second term of Eq. (67 a ) –see Ref. [23]. Moreover, employing the seesaw formula wecan then obtain the light-neutrino mass matrix m ν in termsof m i D and M iN c . As a consequence, nTL can be nicelylinked to low energy neutrino data. We take as inputs thebest-fit values [44] – see also Ref. [45] – of the neutrinomass-squared differences, ∆ m = 7 . · − eV and ∆ m = (2 .
48 [ − . · − eV , of the mixing angles, sin θ = 0 . , sin θ = 0 . (cid:2) sin θ = 0 . (cid:3) and sin θ = 0 . (cid:2) sin θ = 0 . (cid:3) , and of the CP-violatingDirac phase δ = 1 . π [ δ = 1 . π ] for normal [inverted] or-dered (NO [IO]) neutrino masses , m iν ’s. The sum of m iν ’s isbounded from above by the data [53], P i m iν ≤ . eV (90)at 95% c.l. This is more restrictive than the c.l. upperbound arising from the effective electron neutrino mass in β -decay [65]: m β ≤ (0 . − . eV , (91)where the range accounts for nuclear matrix element uncer-tainties.The required T rh in Eq. (87) must be compatible with con-straints on the e G abundance, Y / , at the onset of nucleosyn-thesis (BBN), which are [42] given approximately by Y / . − − − for m / ≃ . TeV , . TeV , . TeV . (92)Here we consider the conservative case where e G decays witha tiny hadronic branching ratio. The bounds above can besomehow relaxed in the case of a stable e G – see e.g. Ref. [28].In our models Y / is estimated to be [41, 42]: Y / ≃ . · − T rh / GeV , (93)where we take into account only thermal production of e G ,and assume that e G is much heavier than the MSSM gauginos.Non-thermal contributions to Y / [60] are also possible butstrongly dependent on the mechanism of soft SUSY breaking.Moreover, no precise computation of this contribution existswithin IGHI adopting the simplest Polonyi model of SUSYbreaking [43]. It is notable, though, that the non-thermal con-tribution to Y / in models with stabilizer field, as in our case,is significantly suppressed compared to the thermal one. Induced-GravityGUT-ScaleHIinSUGRA . Results
It is worthwhile to test the applicability of the frameworkabove in the case of IGHI. Namely, following a bottom-up ap-proach detailed in Ref. [23], we find the M iN c ’s by using asinputs the m i D ’s, a reference mass of the ν i ’s – m ν for NO m iν ’s, or m ν for IO m iν ’s –, the two Majorana phases ϕ and ϕ of the PMNS matrix, and the best-fit values, mentioned inSec. III C 2 , for the low energy parameters of neutrino physics– note that there are no experimental constraints on ϕ and ϕ up to now. In our numerical code we also estimate, followingRef. [63], the renormalization group evolved values of the lat-ter parameters at the scale of nTL, Λ L = b m δφ , by consideringthe MSSM with tan β ≃ as an effective theory between Λ L and the soft SUSY breaking scale, M SUSY = 1 . TeV.We evaluate the M iN c ’s at Λ L , and we neglect any possiblerunning of the m i D ’s and M iN c ’s. Therefore, we present theirvalues at Λ L .We start the exposition of our results arranging in Table VIfor K = K or K and VII for K = K R some representativevalues of the parameters which yield Y B and Y / compatiblewith Eqs. (88) and (92), respectively. Throughout our compu-tation we take λ µ = 10 − , in accordance with Eqs. (70 c ) and(70 d ), and y = 0 . , which is a typical value encountered [56]in various MSSM settings with large tan β . Also, we select n = 0 . in Table VI and n = 0 in Table VII. These valuesyield n s and r in the “sweet” spot of the present data – seeFigs. 2 and 1. We obtain M = 2 . · GeV and b m δφ =2 . · GeV for K = K R or K R , M = 6 . · GeV and b m δφ = 4 . · GeV for K = K , or M = 5 . · GeVand b m δφ = 4 . · GeV for K = K or K . Althoughthe uncertainties from the choice of K ’s are negligible as re-gards the quantities above, the decay widths in Sec. III C 1 depend on N (and r ± ) which take slightly different values for K = K R or K and K = K R , K or K – see e.g. Fig. 3– discriminating somehow the various choices. For this rea-son, we clarify that we adopt K = K R in Table VII and K = K or K in Table VI. Had we employed K = K R or K , we would have obtained almost two times larger Y B ’swith the same values of the free parameters. Therefore a mildreadjustment is needed.In both Tables we consider NO (cases A and B), almostdegenerate (cases C, D and E) and IO (cases F and G) m iν ’s.In all cases Eq. (90) is safely met – the case D saturates it –whereas Eq. (91) is comfortably satisfied. The gauge groupadopted here, G B − L , does not predict any relation betweenthe Yukawa couplings constants h iN entering the second termof Eq. (67 a ) and the other Yukawa couplings in the MSSM.As a consequence, the m i D ’s are free parameters. However,for the sake of comparison, for cases A – F, we take m = m t (Λ L ) ≃ GeV, where m t denotes the mass of the topquark. Similar conditions for the lighter generations do nothold, though, in our data sample.Besides case A, where only the channel c δφ → N c N c iskinematically unblocked, c δφ decays into N c ’s and N c ’s. Inthe latter cases ε yields the dominant contribution to the cal-culation Y B from Eq. (87). From our computation, we also T ABLE
VI:
Parameters yielding the correct BAU for K = K or K , n = 0 . , λ µ = 10 − , y = 0 . and various neutrino massschemes.P ARAMETERS C ASES
A B C D E F GNormal Almost InvertedHierarchy Degeneracy HierarchyLow Scale Parameters (Masses in eV) m ν / . .
01 0 . . . . . . m ν / . .
09 0 .
13 0 .
51 1 . .
705 0 .
51 0 . m ν / . . .
51 0 .
71 1 .
12 0 . . . P i m iν / . . .
74 1 . . . . m β / .
01 0 .
22 0 .
98 3 . . . . . ϕ π/ π/ − π/ ϕ − π/ π/ − π − π/ π/ − π/ Leptogenesis-Scale Mass Parameters in GeV m .
98 1 . . .
16 5 . . m
38 16 . . . m /
100 1 1 1 1 1 1 0 . M N c / . . . . . . . M N c /
27 6 . . . . . . M N c /
22 4 . .
89 0 .
22 0 .
69 2 . . Decay channels of the Inflaton c δφ c δφ → N c N c , N c , N c , N c , N c , N c , Resulting B -Yield Y B / − .
63 8 8 . . . . . Y B / − .
67 8 .
59 8 .
69 8 .
56 8 .
65 8 .
67 8 . Resulting T rh (in GeV) and e G -Yield T rh / . . Y / .
91 2 . . . . . remark that b Γ δφ → N ci N ci < b Γ δφ → H u H d < b Γ δφ → XY Z , and sothe ratios b Γ δφ → N ci N ci / b Γ δφ introduce a considerable reduction( . − . ) in the derivation of Y B . As a consequence, theattainment of the correct Y B requires relatively large m i D ’swith i = 1 , in order to achieve sizable enough b Γ δφ → N ci N ci .Namely, m & GeV and m & . GeV. Besides caseA, the first inequality is necessary, in order to fulfill the sec-ond inequality in Eq. (89), given that m heavily influences M N c . In Table VII we list only m in case A or m inthe other cases which are adjusted so as to accommodate Y B within the range of Eq. (88) with the others m i D remainingas shown in Table VI. As a consequence, M iN c deviate verylittle from the values shown in Table VI.In both Tables we also display, for comparison, the B -yieldwith ( Y B ) or without ( Y B ) taking into account the renormal-ization group effects. We observe that the two results aremostly close to each other. Shown also are values for T rh , theV Conclusions T ABLE
VII:
SameasinTableVIbutfor K = K R and n = 0 .P ARAMETERS C ASES
A B C D E F GLow Scale Parameters as in Table VILeptogenesis-Scale Mass Parameters in GeV m ( ∗ ) i D .
91 16 . . .
15 9 .
25 6 .
37 9 . ( ∗ ) Where i = 1 for case A and i = 2 for the others.The remaining m i D and M iN c are as in Table VI.Resulting B -Yield Y B / − . . . . . . . Y B / − .
64 8 .
61 8 .
72 8 . .
73 8 . . Resulting T rh (in GeV) and e G -Yield T rh / . . . . . . . Y / / − .
44 2 . . . .
45 1 .
45 1 . majority of which are close to GeV, and the correspond-ing Y / ’s, with the results for K = K R being a little lower.Thanks to our non-thermal set-up, successful leptogenesis canbe accommodated with T rh ’s lower than those necessitated inthe thermal regime – cf. Ref. [66]. The resulting large Y / ’smay be consistent with Eq. (92) mostly for m / & TeV.These are marginally tolerated with the m / ’s appearing inTable V and Figs. 2 and 3 of Ref. [58] in the A/H funnel and χ ± − χ coannihilation regions – see also Ref. [67]. These m / ’s though are more easily reconciled with low energydata in less restrictive versions of MSSM – see e.g. Ref. [68].In order to extend the conclusions inferred from Table VIto the case of variable n , we can examine how the centralvalue of Y B in Eq. (88) can be achieved by varying m as afunction of n . The resulting contours in the n − m planeare presented in Fig. 6 – since the range of Y B in Eq. (88) isvery narrow, the c.l. width of these contours is negligi-ble. The convention adopted for these lines is also describedin the figure. In particular, we use solid, dashed, or dot-dashed line for m iν , m , m , ϕ , and ϕ correspondingto the cases B, D, or F of Table VI respectively. For n withinits allowed margins in Eqs. (65) and (66) we obtain . . T rh / GeV . . , which is perfectly acceptable fromEq. (92) for m / & TeV. Along the depicted contours, theresulting M N c ’s vary in the ranges (5 . − . · GeV, (1 . − . · GeV, (1 . − . · GeV for cases B, Cand F respectively, whereas M N c and M N c remain close totheir values presented in the corresponding cases of Table VI.Comparing, finally, our results above with those presentedin Ref. [23], we can deduce that here b m δφ and T rh gain almosttheir maximal allowed values since r ± is also maximized dueto the hypothesis of Eq. (38). As a consequence, m / also hasto be enhanced to avoid problems with BBN, whereas m , and M , N c are also constrained to larger values. On the otherhand, our results are closer to those obtained employing themodel of IG in Ref. [3] with gauge singlet inflaton and withoutunification constraint. -3 -2 -1 0510152025 m l ν , m , m , φ , φ as in Table V.Case B Case C
Case F m D ( G e V ) n (0.1) F IG . 6: Contours in the n − m plane yielding the central Y B inEq.(88)consistentlywiththeinflationaryrequirementsfor K = K or K , λ µ = 10 − , y = 0 . andthevaluesof m iν , m , m , ϕ ,and ϕ whichcorrespondtothecasesB(solidline),C(dashedline),andF(dot-dashedline)ofTableVI. ThecolorcodingisasinFig.3. IV. C ONCLUSIONS
We have proposed a class of novel inflationary models, inwhich a Higgs field plays the role of the inflaton, before set-tling in its final vacuum state where it generates the Planckscale and gives rise to a mass for the gauge boson consistentwith gauge coupling unification within MSSM. These two hy-potheses allow us to determine the mass scale M , entering W HI in Eq. (1), and c R for the K ’s in Eqs. (4 a ) and (4 b ) or r ± = c + /c − for the K ’s in Eqs. (6 a ) – (6 c ). In the lattercases, r ± expresses the amount of violation of a shift sym-metry. As a consequence, the inflationary scenario dependsessentially on two free parameters – n and λ or λ/c − for thefirst or second group of K ’s, respectively – leading naturallyto observationally acceptable results. Namely, for the K ’s inEqs. (6 a ) – (6 c ) we obtained slightly larger r ’s and two dis-tinct allowed regions of parameters with n values one order ofmagnitude larger than those needed for the K ’s in Eqs. (4 a )and (4 b ). As an example, the model for K = K or K , n = 0 and λ/c − = 3 · − yields n s ≃ . and r ≃ . with negligibly small a s . In all cases, inflation is attained forsubplanckian inflaton values, thereby stabilizing our predic-tions from possible higher order corrections, whereas the cor-responding effective theories remain trustable up to m P .The models were further extended to generate the MSSM µ parameter, consistently with the low energy phenomenol-ogy. Successful baryogenesis is achieved via primordial lep-togenesis, in agreement with the data on neutrino masses andmixing. More specifically, our post-inflationary setting favorsthe A/H funnel and the ˜ χ ± − χ coannihilation regions ofCMSSM with gravitino heavier than about TeV. Lepto-genesis is realized through the out-of equilibrium decay ofthe inflaton to the right-handed neutrinos N c and/or N c , withmasses lower than . · GeV, and reheat temperature T rh close to GeV. Induced-GravityGUT-ScaleHIinSUGRA A CKNOWLEDGMENTS
C.P. acknowledges the Bartol Research Institute and theDepartment of Physics and Astronomy of the University of Delaware for its warm hospitality, during which this work hasbeen initiated. He also acknowledges useful discussions withG. Lazarides and S. Martin. Q.S. acknowledges support bythe DOE grant No. DE-SC0013880. R EFERENCES [1] A. Zee, Phys. Rev. Lett. , 417 (1979); H. Terazawa, Phys.Lett.B , 43 (1981).[2] A.A. Starobinsky, Phys.Lett.B , 99 (1980).[3] C. Pallis, J.Cosmol.Astropart.Phys. , 024 (2014); , 01(E)(2017) [ arXiv:1312.3623 ].[4] R. Kallosh, Phys. Rev. D , 087703 (2014) [ arXiv:1402.3286 ].[5] C. Pallis, J. Cosmol. Astropart. Phys. , 057 (2014)[ arXiv:1403.5486 ]; C. Pallis, J.Cosmol.Astropart.Phys. ,058 (2014) [ arXiv:1407.8522 ]; C. Pallis, PoSCORFU ,156 (2015) [ arXiv:1506.03731 ].[6] C. Pallis and N. Toumbas, J. Cosmol. Astropart. Phys. ,no. 05, 015 (2016) [ arXiv:1512.05657 ]; C. Pallis andN. Toumbas, Adv. High Energy Phys. , 6759267 (2017)[ arXiv:1612.09202 ]; C. Pallis, PoS EPS-HEP , 047(2017) [ arXiv:1710.04641 ].[7] M.B. Einhorn and D.R.T. Jones, J.Cosmol.Astropart.Phys. ,049 (2012) [ arXiv:1207.1710 ].[8] F.S. Accetta, D.J. Zoller, and M.S. Turner, Phys. Rev. D , 3046 (1985); R. Fakir and W.G. Unruh, Phys. Rev. D , 1792 (1990); D.I. Kaiser, Phys. Rev. D , 4295 (1995)[ astro-ph/9408044 ].[9] D.S. Salopek, J.R. Bond and J.M. Bardeen, Phys. Rev. D ,1753 (1989); J.L. Cervantes-Cota and H. Dehnen, Phys.Rev.D , 395 (1995) [ astro-ph/9412032 ].[10] N. Kaloper, L. Sorbo and J. Yokoyama, Phys. Rev. D ,043527 (2008) [ arXiv:0803.3809 ].[11] G.F. Giudice and H.M. Lee, Phys. Lett. B , 58 (2014)[ arXiv:1402.2129 ].[12] K. Kannike et al. , J. High Energy Phys. , 065 (2015)[ arXiv:1502.01334 ]; M.B. Einhorn and D.R.T. Jones, J.HighEnergyPhys. , 019 (2016) [ arXiv:1511.01481 ].[13] P.A.R. Ade et al. [ Planck
Collaboration], Astron. Astrophys. , A20 (2016) [ arXiv:1502.02114 ].[14] J.L.F. Barbon and J.R. Espinosa, Phys. Rev. D ,081302 (2009) [ arXiv:0903.0355 ]; C.P. Burgess, H.M. Leeand M. Trott, J. High Energy Phys. , 007 (2010)[ arXiv:1002.2730 ].[15] A. Kehagias, A.M. Dizgah and A. Riotto, Phys. Rev. D ,043527 (2014) [ arXiv:1312.1155 ].[16] C. Pallis, Phys.Lett.B , 287 (2010) [ arXiv:1002.4765 ].[17] R. Kallosh, A. Linde and D. Roest, Phys. Rev. Lett. , 011303(2014) [ arXiv:1310.3950 ].[18] N. Okada, M.U. Rehman, and Q. Shafi, Phys. Rev. D ,043502 (2010) [ arXiv:1005.5161 ].[19] C. Pallis and N. Toumbas, J. Cosmol. Astropart. Phys. ,002 (2011) [ arXiv:1108.1771 ]; C. Pallis and N. Toum-bas, “Open Questions in Cosmology” (InTech, 2012)[ arXiv:1207.3730 ].[20] G. Lazarides and C. Pallis, J.HighEnergyPhys. , 114 (2015)[ arXiv:1508.06682 ].[21] C. Pallis, Phys. Rev. D , no. 12, 121305(R) (2015) [ arXiv: 1511.01456 ].[22] C. Pallis, J. Cosmol. Astropart. Phys. , no. 10, 037 (2016)[ arXiv:1606.09607 ].[23] C. Pallis, Universe , no. 1, 13 (2018) [ arXiv:1510.05759 ].[24] G.R. Dvali, Q. Shafi and R.K. Schaefer, Phys. Rev. Lett. , 1886 (1994) [ hep-ph/9406319 ].[25] G.R. Dvali, G. Lazarides and Q. Shafi, Phys. Lett. B , 259(1998) [ hep-ph/9710314 ].[26] M. Bastero-Gil, S.F. King and Q. Shafi, Phys. Lett. B , 345 (2007) [ hep-ph/0604198 ]; B. Garbrecht, C. Pal-lis and A. Pilaftsis, J. High Energy Phys. , 038 (2006)[ hep-ph/0605264 ]; M.U. Rehman, V.N. S¸ eno˘guz andQ. Shafi, Phys.Rev.D , 043522 (2007) [ hep-ph/0612023 ];C. Pallis, J. Cosmol. Astropart. Phys. , 024 (2009)[ arXiv:0902.0334 ]; M. Civiletti, C. Pallis and Q. Shafi, Phys.Lett.B , 276 (2014) [ arXiv:1402.6254 ].[27] V.N. S¸eno˘guz and Q. Shafi, hep-ph/0512170 ; W. Buchm¨uller,V. Domcke and K. Schmitz, Nucl. Phys. B862 , 587 (2012)[ arXiv:1202.6679 ]; C. Pallis and Q. Shafi, Phys.Lett.B ,327 (2013) [ arXiv:1304.5202 ].[28] N. Okada and Q. Shafi, Phys.Lett.B , 348 (2017) [ arXiv:1506.01410 ].[29] N. Okada and Q. Shafi, arXiv:1709.04610 .[30] M. Kawasaki, M. Yamaguchi and T. Yanagida, Phys.Rev.Lett. , 3572 (2000) [ hep-ph/0004243 ]; P. Brax and J. Martin, Phys. Rev. D , 023518 (2005) [ hep-th/0504168 ]; S. An-tusch, K. Dutta and P.M. Kostka, Phys.Lett.B , 221 (2009)[ arXiv:0902.2934 ]; R. Kallosh, A. Linde and T. Rube, Phys.Rev. D , 043507 (2011) [ arXiv:1011.5945 ]; T. Li, Z. Liand D.V. Nanopoulos, J. Cosmol. Astropart. Phys. , 028(2014) [ arXiv:1311.6770 ]; K. Harigaya and T.T. Yanagida,Phys.Lett.B , 13 (2014) [ arXiv:1403.4729 ]; A. Mazum-dar, T. Noumi and M. Yamaguchi, Phys. Rev. D , 043519(2014) [ arXiv:1405.3959 ]; C. Pallis and Q. Shafi, Phys.Lett.B , 261 (2014) [ arXiv:1405.7645 ].[31] I. Ben-Dayan and M.B. Einhorn, J. Cosmol. Astropart. Phys. , 002 (2010) [ arXiv:1009.2276 ].[32] C. Pallis, Phys. Rev. D , no. 12, 123508 (2015)[ arXiv:1503.05887 ]; C. Pallis, PoS PLANCK , 095(2015) [ arXiv:1510.02306 ].[33] P.A.R. Ade et al. [BICEP2/ Keck Array
Collaborations], Phys.Rev.Lett. , 031302 (2016) [ arXiv:1510.09217 ].[34] W.L.K. Wu et al. , J.Low.Temp.Phys. , no. 3-4, 765 (2016)[ arXiv:1601.00125 ].[35] P. Andre et al. [PRISM Collaboration], arXiv:1306.2259 .[36] T. Matsumura et al. , J. Low. Temp. Phys. , 733 (2014)[ arXiv:1311.2847 ].[37] F. Finelli et al. [CORE Collaboration] arXiv:1612.08270 .[38] K. Hamaguchi,
Phd Thesis [ hep-ph/0212305 ]; W. Buch-muller, R.D. Peccei and T. Yanagida, Ann.Rev.Nucl.Part.Sci. , 311 (2005) [ hep-ph/0502169 ].[39] G. Lazarides and Q. Shafi, Phys. Lett. B , 305 (1991);K. Kumekawa, T. Moroi and T. Yanagida, Prog. Theor.Phys. , 437 (1994) [ hep-ph/9405337 ]; G. Lazarides,R.K. Schaefer and Q. Shafi, Phys. Rev. D , 1324 (1997) eferences [ hep-ph/9608256 ]; V.N. S¸ eno˘guz and Q. Shafi, Phys. Rev.D , 043514 (2005) [ hep-ph/0412102 ].[40] M.Yu. Khlopov and A.D. Linde, Phys.Lett.B , 265 (1984);J. Ellis, J.E. Kim, and D.V. Nanopoulos, Phys.Lett.B , 181(1984).[41] M. Bolz, A.Brandenburg and W. Buchm¨uller, Nucl. Phys. B606 , 518 (2001); , 336(E) (2008) [ hep-ph/0012052 ];J. Pradler and F.D. Steffen, Phys. Rev. D , 023509 (2007)[ hep-ph/0608344 ].[42] M.Kawasaki, K.Kohri and T.Moroi, Phys.Lett.B , 7 (2005)[ astro-ph/0402490 ]; M. Kawasaki, K. Kohri and T. Mo-roi, Phys. Rev. D , 083502 (2005) [ astro-ph/0408426 ];J.R. Ellis, K.A. Olive and E. Vangioni, Phys. Lett. B , 30(2005) [ astro-ph/0503023 ]; M. Kawasaki, K. Kohri, T. Mo-roi and Y. Takaesu, Phys. Rev. D , no. 2, 023502 (2018)[ arXiv:1709.01211 ].[43] J. Ellis et al. , J.Cosmol.Astropart.Phys. , no. 03, 008 (2016)[ arXiv:1512.05701 ]; Y. Ema et al. , J.HighEnergyPhys. ,184 (2016) [ arXiv:1609.04716 ].[44] D.V. Forero, M. Tortola and J.W.F. Valle, Phys. Rev. D , no.9, 093006 (2014) [ arXiv:1405.7540 ].[45] M.C. Gonzalez-Garcia, M. Maltoni and T. Schwetz, J.HighEn-ergyPhys. , 052 (2014) [ arXiv:1409.5439 ]; F. Capozzi etal. , Nucl.Phys.B , 218 (2016) [ arXiv:1601.07777 ].‘[46] M.B. Einhorn and D.R.T. Jones, J. High Energy Phys. , 026(2010) [ arXiv:0912.2718 ]; H.M. Lee, J. Cosmol. Astropart.Phys. , 003 (2010) [ arXiv:1005.2735 ]; S. Ferrara et al. ,Phys.Rev.D , 025008 (2011) [ arXiv:1008.2942 ]; C. Pallisand N. Toumbas, J. Cosmol. Astropart. Phys. , 019 (2011)[ arXiv:1101.0325 ].[47] G. Lopes Cardoso, D. L¨ust and T. Mohaupt, Nucl. Phys. B432
68 (1994) [ hep-th/9405002 ]; I. Antoniadis, E. Gava,K.S. Narain and T.R. Taylor, Nucl. Phys.
B432
187 (1994)[ hep-th/9405024 ].[48] R. Kallosh, A. Linde and D. Roest, J. High Energy Phys. , 198 (2013) [ arXiv:1311.0472 ]; R. Kallosh, A. Lindeand D. Roest, J. High Energy Phys. , 052 (2014)[ arXiv:1405.3646 ].[49] L. Boubekeur and D. Lyth, J.Cosmol.Astropart.Phys. , 010(2005) [ hep-ph/0502047 ].[50] S.R. Coleman and E.J. Weinberg, Phys.Rev.D , 1888 (1973).[51] D.H. Lyth and A. Riotto, Phys. Rept. , 1 (1999) [ hep-ph/9807278 ]; J. Martin, C. Ringeval and V. Vennin, Physicsofthe DarkUniverse , 75 (2014) [ arXiv:1303.3787 ].[52] http://functions.wolfram.com .[53] P.A.R. Ade et al. [Planck Collaboration], Astron. Astrophys. , A13 (2016) [ arXiv:1502.01589 ].[54] A. Racioppi, arXiv:1801.08810 .[55] G. Lazarides and Q. Shafi, Phys. Rev. D , 071702 (1998)[ hep-ph/9803397 ].[56] S. Antusch and M. Spinrath, Phys. Rev. D , 075020 (2008)[ arXiv:0804.0717 ].[57] W. Buchm¨uller et al. , J. High Energy Phys. , 053 (2014)[ arXiv:1407.0253 ]; J. Ellis, M. Garcia, D. Nanopoulosand K. Olive, J. Cosmol. Astropart. Phys. , 003 (2015)[ arXiv:1503.08867 ]; E. Dudas, T. Gherghetta, Y. Mambriniand K.A. Olive, Phys. Rev. D , no. 11, 115032 (2017)[ arXiv:1710.07341 ].[58] P. Athron et al. [GAMBIT Collaboration], Eur. Phys. J. C ,no. 12, 824 (2017) [ arXiv:1705.07935 ].[59] A. Addazi, S.V. Ketov and M.Y. Khlopov, arXiv:1708.05393 .[60] M. Endo, F. Takahashi and T.T. Yanagida, Phys. Rev. D ,083509 (2007) [ arXiv:0706.0986 ].[61] C. Pallis, Nucl.Phys. B751 , 129 (2006) [ hep-ph/0510234 ].[62] S.V. Ketov and A.A. Starobinsky, Phys. Rev. D , 063512(2011) [ arXiv:1011.0240 ]; S. V. Ketov and N. Watanabe, J.Cosmol.Astropart.Phys. , 011 (2011) [ arXiv:1101.0450 ];S.V. Ketov and A.A. Starobinsky, J. Cosmol. Astropart. Phys. , 022 (2012) [ arXiv:1203.0805 ]; S.V. Ketov and S. Tsu-jikawa, Phys.Rev.D , 023529 (2012) [ arXiv:1205.2918 ].[63] S. Antusch, J. Kersten, M. Lindner and M. Ratz, Nucl. Phys. B674 , 401 (2003) [ hep-ph/0305273 ].[64] V.N. S¸eno˘guz, Phys. Rev. D , 013005 (2007) [ arXiv:0704.3048 ].[65] A. Gando et al. [KamLAND-Zen Collaboration], Phys. Rev.Lett , no.8, 082503.(2016); ibid , no.10, 109903 (2016)[ arXiv:1605.02889 ].[66] S. Antusch and A.M. Teixeira, J. Cosmol. Astropart. Phys. ,024 (2007) [ hep-ph/0611232 ].[67] N. Karagiannakis, G. Lazarides and C. Pallis, Phys.Rev.D ,no. 8, 085018 (2015) [ arXiv:1503.06186 ].[68] H. Baer et al. , Phys. Rev. Lett. , 161802 (2012) [ arXiv:1207.3343arXiv:1207.3343