Inflation and Leptogenesis in a U(1)-enhanced supersymmetric model
aa r X i v : . [ h e p - ph ] F e b IBS-CTPU-17-21
Inflation and Leptogenesisin a U (1) -enhanced supersymmetric model Y. H. Ahn
Center for Theoretical Physics of the Universe,Institute for Basic Science (IBS), Daejeon, 34051, Korea ∗ Abstract
Motivated by the flavored Peccei-Quinn symmetry for unifying flavor physics and string theory, weinvestigate a supersymmetric extension of standard model (SM) for a lucid explanation of infla-tion and leptogenesis by introducing U (1) symmetries such that the U (1)-[ gravity ] anomaly-freecondition together with the SM flavor structure demands additional sterile neutrinos as well as noaxionic domain-wall problem. Such additional neutrinos may play a crucial role as a bridge betweenleptogenesis and new neutrino oscillations along with high energy cosmic events. In the model grav-itational interactions explicitly break supersymmetry (SUSY) down to SUSY inf × SUSY vis , whereSUSY inf corresponds to the supergravity symmetry with its goldstino (mainly as inflatino) eatenby gravitino, while the orthogonal SUSY vis is approximate global symmetry with its correspond-ing uneaten goldstino giving masses to all the supersymmetric SM superpartners. In a realisticmoduli stabilization, we show that the moduli backreaction effect on the inflationary potentialleads to the energy scale of inflation with the inflaton mass in a way that the power spectrumof the curvature perturbation and the scalar spectral index are to be well fitted with the latestPlanck observation. We suggest that a new leptogenesis scenario could naturally be implementedvia Affleck-Dine mechanism. So we show that the resultant baryon asymmetry, constrained by thesum of active neutrino masses and new high energy neutrino oscillations, crucially depends on thereheating temperature T reh . And we show that the model has a prediction T reh ≃ (59 −
84) TeV,which is compatible with the required T reh to explain the baryon asymmetry of the Universe. ∗ Electronic address: [email protected] . INTRODUCTION The standard model (SM) of particle physics has been successful in describing propertiesof known matter and forces to a great precision until now, but we are far from satisfied sinceit suffers from some problems or theoretical arguments that have not been solved yet, whichfollows: inclusion of gravity in gauge theory, instability of the Higgs potential, cosmologicalpuzzles of matter-antimatter asymmetry, dark matter, dark energy, and inflation, and flavorpuzzle associated with the SM fermion mass hierarchies, their mixing patterns with the CPviolating phases, and the strong CP problem. The SM therefore cannot be the final answer.It is widely believed that the SM should be extended to a more fundamental underlyingtheory. If nature is stringy, string theory should give insight into all such fundamentalproblems or theoretical arguments . As indicated in Refs. [1, 2] , such several fundamentalchallenges strongly hint that a supersymmetric hybrid inflation framework with new gaugesymmetries as well as higher dimensional operators responsible for the SM flavor puzzlesmay be a promising way to proceed.Since astrophysical and cosmological observations have increasingly placed tight con-straints on parameters for axion, neutrino, and inflation including the amount of reheating,it is in time for a new scenario on axion and neutrino to mount such interesting challenges,see also Ref. [1, 4]. In a theoretical point of view axion physics including neutrino physicsrequires new gauge interactions and a set of new fields that are SM singlets. Thus in exten-sions of the SM, sterile neutrinos and axions could naturally be introduced, e.g. , in view of U (1) symmetry. As a new paradigm to explain the aforementioned fundamental challenges,in this paper we investigate a minimal and economic supersymmetric extension of SM for alucid explanation of inflation and leptogenesis, which can be realized within the framework of G ≡ SM × U (1) X × A . All renormalizable and nonrenormalizable operators allowed bysuch gauge symmetries, non-Abelian discrete symmetry, and R -parity exist in the superpo- In Ref. [2] a concrete model is designed to bridge between string theory as a fundamental theory and lowenergy flavor physics. Ref. [1] introduces a superpotential for unifying flavor and strong CP problems, the so-called flavored PQsymmetry model in a way that no axionic domain wall problem. Here the flavored Peccei-Quinn (PQ) symmetry U (1) X embedded in the non-Abelian A finite group [3]could economically explain the mass hierarchies of quarks and leptons including their peculiar mixingpatterns as well as provide a neat solution to the strong CP problem and its resulting axion [4]. θ ) and Dirac CPphase ( δ CP ) together with well-fitted solar θ and reactor θ mixing angles have a remark-able coincidence with the most recent data by the NO ν A [5] and/or T2K [6] experiments.We assume throughout that the model can be derived as consistent type IIB string vacuum.In such a vacuum, as shown in Ref. [4] the U (1) X -mixed anomalies such as U (1) X -[ U (1) Y ] , U (1) X -[ SU (2) L ] , U (1) X -[ SU (3) C ] , and U (1) Y -[ U (1) X ] have been cancelled by appropri-ate shifts of Ramond-Ramond axions in the bulk [7]. And since non-perturbative quantumgravitational effects spoil the axion solution to the strong CP problem [8, 9], in order to elim-inate such breaking effects of the axionic shift symmetry by gravity the author in Ref. [4] hasimposed an U (1) X × [ gravity ] anomaly cancellation condition [4] in a way that no axionicdomain-wall problem occurs, thereby additional sterile neutrinos are introduced. Such ster-ile neutrinos are light or heavy and do not participate in the weak interaction. Moreover,the latest results [10] from Planck and Baryon Acoustic Oscillations (BAO) show that thecontribution of light sterile neutrinos to N eff ν at the Big-Bang Nucleosynthesis (BBN) [11]era is negligible; such neutrinos may play a crucial role as a bridge between leptogenesis andnew neutrino oscillations along with high energy cosmic events. As demonstrated in Ref. [2],by introducing two gauged U (1) symmetries in the context of supersymmetric moduli sta-bilization based on type IIB string theory, three size moduli and one axionic partner withpositive masses are stabilized while leaving two axions massless. The two massless axiondirections are gauged by the U (1) gauge interactions, and such gauged flat directions areremoved through the Stuckelberg mechanism, leaving behind low energy symmetries whichare anomalous global U (1) X i .Supergravity (SUGRA) is a theory with local super-Poincare symmetry. As addressed inRef. [4] where two U (1) symmetries are imbedded, the U (1) X and U (1) X breaking scalesare separated by the Gibbons-Hawking temperature, T GH = H I / π , and both of which areto be much above the electroweak scale h Φ i < H I π < h Φ i , (1)where H I is the inflationary Hubble constant, and the fields Φ = { Φ S , Θ } and Φ = { Ψ , ˜Ψ } are charged under the U (1) X and U (1) X symmetries, respectively. Here we have assumedthat the electroweak symmetry is broken by some mechanism, such as radiative effects3hen SUSY (supersymmetry) is broken. So we can picture two secluded SUSY breakingsectors by the inflationary sector and by the visible sector in the present Universe, i.e. ,SUSY=SUSY inf × SUSY vis , respectively. Both sectors interact non-gravitationally via in-flaton field as well as gravitationally. In the absence of direct interactions, gravitationalor non-gravitational, the U (1) X -charged chiral superfields Φ have a two-fold enhancedSUSY inf × SUSY vis
Poincare symmetry, while the U (1) X -charged chiral superfields Φ havea SUSY vis Poincare symmetry. However, gravitational interactions explicitly break the SUSYdown to true
SUSY inf × SUSY vis , where SUSY inf corresponds to the genuine SUGRA sym-metry, while the orthogonal SUSY vis is approximate global symmetry. In each sector, spon-taneous breakdown of F -term occurs at a scale F i ( i = inf, vis) independently, producing acorresponding goldstino. Hence, in the presence of SUGRA, the SUSY inf is gauged and thusits corresponding goldstino is eaten by the gravitino via super-Higgs mechanism, leavingbehind the approximate global symmetry SUSY vis which is explicitly broken by SUGRAand thus its corresponding the uneaten goldstino as a physical degree of freedom. Duringinflation and the beginning of reheating (preheating) the SUSY inf is mainly broken by theinflaton implying the goldstino produced is mainly inflatino; the gravitino produced non-thermally is effectively massless as long as the Hubble parameter is larger than the gravitinomass, H > m / . However, this correspondence does not necessarily hold at late times, sincethe SUSY vis is broken by other fields in the true vacuum implying that the correspondinguneaten goldstino gives masses mainly to all the supersymmetric SM superpartners in thevisible sector; gravitinos are produced non-thermally by the decay of the inflaton.In this paper, in order to provide a lucid explanation for inflation we present a realisticmoduli stabilization, which is essential for the flavored PQ axions to be realized at lowenergy scale [4]. Such moduli stabilization has moduli backreaction effects on the inflationarypotential, which provides a lucid explanation for the cosmological inflation at high energyscale. The inflaton as a source of inflation is displaced from its minimum and whose slow-roll dynamics leads to an accelerated expansion of the early universe. During inflation theuniverse experiences an approximately de Sitter (dS) phase with the inflationary Hubbleconstant H I ≃ × GeV. Quantum fluctuations during this phase can lead to observablesignatures in cosmic microwave background (CMB) radiation temperature fluctuation, asthe form of density perturbation, in several ways [12], when they become much bigger thanthe Hubble radius long after inflation has been completed. When interpreted in this way,4nflation provides a causal mechanism to explain the observed nearly-scale invariant CMBspectrum. In the present inflation model which provides intriguing links to ultraviolet (UV)-complete theories like string theory, the PQ scalar fields Ψ( ˜Ψ) play a role of the waterfallfields, that is, the PQ phase transition takes place during inflation such that the PQ scale µ Ψ ( t I ) is fixed by the amplitude of the primordial curvature perturbation and turns out to bearound 0 . × GeV which is smaller than string moduli mass m T . In the present model,since SUSY breaking is transmitted by gravity, all scalar fields acquire an effective mass ofthe order of the expansion rate during inflation. So we expect that the inflaton acquiresa mass of order the Hubble constant H I , and which in turn indicates that the soft SUSYbreaking mass (the inflaton mass m Ψ ) during inflation strongly depends on the scale ofwaterfall fields by VEVs v Ψ ( t I ) and/or v ˜Ψ ( t I ) induced by tachyonic SUSY breaking masses.Thus such moduli stabilization with the moduli backreaction effects on the inflationarypotential leads to the energy scale of inflation with the inflaton mass, m Ψ = √ H I , in away that the power spectrum of the curvature perturbation and the scalar spectral indexare to be well fitted with the latest Planck observation [13]. Since the moduli masses aremuch larger than inflaton mass and accordingly are quickly stabilized to their minima atfinite moduli fields values separated by high barrier from the runaway direction duringinflation without perturbing the inflaton dynamics [14], the height of the barrier protectingmetastable Minkowski space ( ≃ dS space) are independent of the gravitino mass hence theHubble scale H I during inflation is also independent of the gravitino mass. The moduli-induced slope partially cancels the slope of the Coleman-Weinberg potential [15], whichflattens the inflationary trajectory and reduces the distances in field space correspondingto the N e ∼ e -folds of inflation. The number of e -foldings depends on the amount ofreheating which in turn depends on the decay rate of the inflaton and waterfall field fieldinto relativistic particles. And the amount of reheating could be strongly correlated withboth baryogenesis via leptogenesis and the yield of gravitinos. Note that after inflationthe inflaton and waterfall fields get mixed almost maximally to form mass eigenstates, andthe universe is dominated by both the inflaton and one of waterfall fields, while the otherwaterfall field gives negligible contribution to the total energy of the universe. And at thereheating epoch the inflation and waterfall field release their energy into a thermal plasmaby their decays, and the universe is reheated.Now, we suggest, interestingly enough, a new leptogenesis scenario which could natu-5ally be implemented through Affleck-Dine (AD) mechanism for baryogenesis [16] and itssubsequent leptonic version so-called AD leptogenesis [17]. The interaction between the ADfields and inflaton generates the potential for D -flat direction, and which in turn producescoherent oscillations along the supersymmetric flat directions, leading to dynamics in fieldspace that ultimately breaks CP and baryon number. Interestingly enough, the pseudo-Dirac mass splittings, suggested from the new neutrino oscillations along with high energycosmic events [4], strongly indicate the existence of lepton-number violation which is a cru-cial ingredient of the present leptogenesis scenario. Then the AD fields have large VEVsalong the flat directions during inflation in the early universe, in turn which together with H I provides a lower bound on the pseudo-Dirac mass splittings for the new neutrino oscil-lations [4]. The AD fields start their coherent oscillations after the inflation ends and theycreate a large net lepton number asymmetry, which is finally transferred to matter parti-cles when they eventually decay. So the resultant baryon asymmetry is constrained by thecosmological observable ( i.e. the sum of active neutrino masses) with the new high energyneutrino oscillations, and crucially depends on the reheating temperature which depends ongravitational and non-gravitational decays of the inflaton and waterfall field. Since all theparticles including photons and baryons in the present universe are ultimately originatedfrom the inflaton and waterfall field decays, it is crucial to reveal how the reheating proceeds.We show that the reheating temperature is mainly determined by the non-gravitational de-cay of the waterfall field, leading to a relatively low reheating temperature T reh ≃ (59 − Y ∆ B ≃ × − [13], together with the pseudo-Dirac mass split-tings responsible for new oscillations ∆ m i ≃ O (10 − − ) eV . And we show that, eventhe gravitational coupling is universal, it is too weak to cause the reheating with gravity inthe present model. Thus, the present model is very attractive in that with the predictivereheating temperature almost at around 70 TeV scale we can have the right value of theBAU which constrains the pseudo-Dirac mass splittings for new neutrino oscillations. Inaddition, since gravitinos are present in the supersymmetric model we are going to addressgravitino overabundance problem. We consider direct decays of the inflaton to gravitinoscompeting with the thermal production in the thermal plasma formed after reheating whensetting limits on the couplings governing inflaton decay, see Eq. (145). We stress that inthe present model the gravitino mass O (100) TeV is given by the process of supersymmet-6ic moduli stabilization in the Kallosh and Linde (KL)-type model [14], whose value givessuitable large gaugino masses [14]. Since the yield of gravitinos is proportional to T reh andinversely proportional to | g Ψ | , i.e. , Y / ≃ . × − ( T reh / . × − /g Ψ ) , alower bound on the Higgs-inflaton coupling can be derived as 2 . × − . g Ψ in Eq. (145)by the BBN constraints Y BBN3 / [18] with the reheating temperature T reh ∼
70 TeV for thesuccessful leptogenesis.The rest of this paper is organized as follows. In Sec. II we setup and review the modelbased on A × U (1) X symmetry in order to investigate an economic SUSY inflationaryscenario and a new leptogenesis via AD mechanism. In Sec. III, first we study a realisticmoduli stabilization in type IIB string theory with positive vacuum energy, which is essentialfor the flavored PQ axions at low energy as well as a lucid explanation for cosmologicalinflation at high energy scale. And we investigate how the size moduli stabilized at ascale close to Λ GUT significantly affect the dynamics of the inflation, as well as how the X -symmetry breaking scale during inflation is induced and its scale is fixed at ∼ . × GeV by the amplitude of the primordial curvature perturbation and the spectral index. Themain focus on Sec. IV is to show that a successful leptogenesis scenario could be naturallyimplemented through AD mechanism, and subsequently estimate the reheating temperaturethat is required to generate sufficient lepton number asymmetry following the hybrid F -term inflation. In turn, we show that the successful leptogenesis is closely correlated withthe neutrino oscillations available on high- and low-energy neutrinos, and how the amountof reheating could be strongly correlated with the successful leptogenesis and the yield ofgravitinos. And we show that it is too weak to cause the reheating with gravity in the presentmodel even the gravitational coupling is universal. Moreover, we discuss that it is reasonablefor the reheating temperature T reh ≃ (59 −
84) derived from the non-gravitational decaysof the inflaton and waterfall field to be compatible with the required reheating temperaturefor the successful leptogenesis. Finally, we discuss briefly on dynamics of the waterfall fieldsand how the uneaten goldstino gives masses to all the supersymmetric SM superpartners.What we have done is summarized in Sec. V, and we provide our conclusions.7
I. FLAVOR A × U (1) X SYMMETRY AND ITS REVIEW
Unless flavor symmetries are assumed, particle masses and mixings are generally unde-termined in the SM gauge theory. In order to provide an elegant solution to the strongCP problem and describe the present SM flavor puzzles associated with the fermion masshierarchies including their mixing patterns, the author in Ref. [1, 4] has introduced the non-Abelian discrete A flavor symmetry [19, 20] which is mainly responsible for the peculiarmixing patterns, as well as an additional continuous symmetry U (1) X which is mainly forvacuum configuration as well as for describing mass hierarchies of leptons and quarks. Alongwith Ref. [1] in a way that no axionic domain wall problem occurs, which plays a crucial rolein cosmology when the X -symmetry breaking occurs after inflation, this U (1) symmetry isreferred to as “flavored-PQ symmetry”. Then the symmetry group for matter fields (leptonsand quarks), flavon fields and driving fields is A × U (1) X , whose quantum numbers are as-signed in TABLE I and II. In addition, the superpotential W in the model (see, Eqs. (3) and(5)) is uniquely determined by the U (1) R symmetry, containing the usual R -parity as a sub-group : { matter f ields → e iξ/ matter f ields } and { driving f ields → e iξ driving f ields } ,with W → e iξ W , whereas flavon and Higgs fields remain invariant under an U (1) R symme-try. As a consequence of the R symmetry, the other superpotential term κ α L α H u and theterms violating the lepton and baryon number symmetries are not allowed .We take the U (1) X breaking scale corresponding to the A symmetry breaking scale andthe U (1) X breaking scale to be separated by Gibbons-Hawking temperature, T GH = H I / π ,and both of which are to be much above the electroweak scale in our scenario , that is, h H u,d i ≪ h Φ T i , h Φ i < H I π < h Φ i (2)where H I is the inflationary Hubble constant, and the fields Φ = { Φ S , Θ } and Φ = { Ψ , ˜Ψ } The flavon fields are responsible for the spontaneous breaking of the flavor symmetry, while the drivingfields are introduced to break the flavor group along required vacuum expectation value (VEV) directionsand to allow the flavons to get VEVs, which couple only to the flavons. It is likely that an exact continuous global symmetry is violated by quantum gravitational effects [21].Here the global U (1) X symmetry is a remnant of the broken U (1) X gauge symmetry which connectsstring theory with flavor physics [2, 4]. In addition, higher-dimensional supersymmetric operators like Q i Q j Q k L l ( i, j, k must not all be the same)are not allowed either, and stabilizing proton. See the symmetry breaking scales from the astrophysical constraints [4], and in more detail Sec. III D onthe PQ symmetry breaking scale during inflation. U (1) X and U (1) X symmetries, respectively. Here we assume that theelectroweak symmetry is broken by some mechanism, such as radiative effects when SUSY isbroken. So we can picture two secluded SUSY breaking sectors by the inflationary sector andby the visible sector in the present Universe, i.e. , SUSY=SUSY inf × SUSY vis , respectively.Both sectors interact non-gravitationally via inflaton field as well as gravitationally. Sincethe Kahler moduli superfields putting the GS mechanism into practice are not separatedfrom the SUSY inf during inflation, the U (1) X -charged matter fields develop a large VEVduring inflation by taking tachyonic SUSY breaking scalar masses m ∼ − H I induced‘dominantly’ by the U (1) X D -term, compared to the Hubble induced soft masses generatedby the F -term SUSY breaking. On the other hand, in the present Universe both the U (1) X i -charged matter fields Φ and Φ develop large VEVs by the soft-SUSY breaking mass. So,in the absence of direct interactions, gravitational or otherwise, the U (1) X -charged chiralsuperfields Φ have a two-fold enhanced SUSY inf × SUSY vis
Poincare symmetry. However,gravitational interactions explicitly break the SUSY down to true
SUSY inf × SUSY vis , whereSUSY inf corresponds to the genuine SUGRA symmetry, while the orthogonal SUSY vis is onlyapproximate global symmetry. In each sector, spontaneous breakdown of F -term occurs at ascale F i ( i = inf, vis) independently, producing a corresponding goldstino. In the presence ofSUGRA, SUSY inf is gauged and thus its corresponding goldstino is eaten by the gravitino viasuper-Higgs mechanism, leaving behind the approximate global symmetry SUSY vis which isexplicitly broken by SUGRA and thus its corresponding the uneaten goldstino as a physicaldegree of freedom. During inflation and the beginning of reheating (preheating) the SUSY inf is mainly broken by the inflaton implying the goldstino produced is mainly inflatino; thegravitino produced non-thermally is effectively massless as long as H > m / . However, thiscorrespondence does not necessarily hold at late times, since the SUSY vis is broken by otherfield in the true vacuum implying that the corresponding uneaten goldstino gives massesmainly to all the supersymmetric SM superpartners in the visible sector.The U (1) X invariance forbids renormalizable Yukawa couplings for the light families, butwould allow them through effective nonrenormalizable couplings suppressed by ( F / Λ) n with n being positive integers [22, 23]. Even with all couplings being of order unity, hierarchicalmasses for different flavors can be naturally realized. The flavon field F is a scalar fieldwhich acquires a vacuum expectation value (VEV) and breaks spontaneously the flavored-PQ symmetry U (1) X . Here Λ, above which there exists unknown physics, is the scale of flavor9ynamics, and is associated with heavy states which are integrated out. The effective theorybelow Λ is rather simple, while the full theory will have many heavy states. We assume thatthe cut-off scale Λ in the superpotential (5) is a scale where the complex structure and axio-dilaton moduli are stabilized through fluxes. So, in our framework, the hierarchy h H u,d i = v u,d ≪ Λ is maintained, and below the scale Λ the higher dimensional operators express theeffects from the unknown physics. Since the Yukawa couplings are eventually responsiblefor the fermion masses they must be related in a very simple way at a large scale in orderfor intermediate scale physics to produce all the interesting structure in the fermion massmatrices. On the other hand, cosmological observables, such as power spectrum of curvatureperturbations and spectral index, do not generically receive significant contributions frompossible higher-dimensional non-renormalizable operators, as these are suppressed by thePlanck mass M P . So inflationary dynamics is mainly governed by a few renormalizableoperators which might have observable implications for laboratory experiments. A. Superpotential dependent on driving fields
To impose the A flavor symmetry on our model properly, apart from the usual two Higgsdoublets H u,d responsible for electroweak symmetry breaking, which are invariant under A ( i.e. flavor singlets with no T -flavor), the scalar sector is extended by introducing two typesof new scalar multiplets, flavon fields Φ T , Φ S , Θ , ˜Θ , Ψ , ˜Ψ that are SU (2)-singlets and drivingfields Φ T , Φ S , Θ , Ψ that are associated to a nontrivial scalar potential in the symmetrybreaking sector: we take the flavon fields Φ T , Φ S to be A triplets, and Θ , ˜Θ , Ψ , ˜Ψ to be A singlets with no T -flavor ( representation), respectively, that are SU (2)-singlets, and drivingfields Φ T , Φ S to be A triplets and Θ , Ψ to be an A singlet. Under A × U (1) X × U (1) R , thedriving, flavon, and Higgs fields are assigned as in TABLE I. The superpotential dependenton the driving fields, which is invariant under SU (3) c × SU (2) L × U (1) Y × U (1) X × A , isgiven at leading order by W v = Φ T (˜ µ Φ T + ˜ g Φ T Φ T ) + Φ S (cid:16) g Φ S Φ S + g ˜ΘΦ S (cid:17) + Θ (cid:16) g Φ S Φ S + g ΘΘ + g Θ ˜Θ + g ˜Θ ˜Θ (cid:17) + g Ψ (cid:16) Ψ ˜Ψ − µ (cid:17) , (3)where the fields Ψ and ˜Ψ charged by − q, q , respectively, are ensured by the U (1) X symmetryextended to a complex U (1) due to the holomorphy of the supepotential. SUSY hybrid infla-10 ABLE I: Representations of the driving, flavon, and Higgs fields under A × U (1) X . Here U (1) X ≡ U (1) X × U (1) X symmetries which are generated by the charges X = − p and X = − q .Field Φ T Φ S Θ Ψ Φ S Φ T Θ ˜Θ Ψ ˜Ψ H d H u A U (1) X p p − p − p − p − q q U (1) R tion , defined by the last term in the above superpotential, provides a compelling frameworkfor the understanding of the early universe, where Ψ and Ψ( ˜Ψ) are identified as the inflatonand waterfall fields, respectively. Note here that the PQ scale µ Ψ ≡ p v Ψ v ˜Ψ / S Φ S in the superpotential W v [19]. Dueto the assignment of quantum numbers under A × U (1) X × U (1) R the usual superpotentialterm µH u H d is not allowed, while the following operators driven by Ψ and Φ T are allowedby g Ψ Ψ H u H d + g T Λ (Φ T Φ T ) H u H d , (4)which is to promote the µ -term µ eff ≡ g Ψ h Ψ i + g T h Φ T i v T / ( √ m S and/or m S v T / Λ (here h Ψ i and h Φ T i : the VEVs of the scalar components of the driving fields, m S :soft SUSY breaking mass). The inflaton field Ψ can predominantly decay into Higgses(and Higgsinos) through the first term after inflation , which is important for inflation andleptogenesis (see Sec. IV), while the second term is crucial for relating the sizable µ -term withthe low energy flavor physics. Here the supersymmetry of the model is assumed broken byall possible holomorphic soft terms which are invariant under A × U (1) X × U (1) R symmetry,where the soft breaking terms are already present at the scale relevant to flavor dynamics. See the details in Sec. III. As will be discussed in Sec. IV, the size of the renormalizable superpotential coupling of the inflaton toparticles of the SM is severely restricted by the reheating temperature, T reh , and in turn a successfulleptogenesis. Consequently, we have µ eff ≃ g T h Φ T i v T / Λ as in Ref. [1], which can describe the correctCKM mixing matrix with v T / Λ ∼ . ≃ λ / √ W (Φ T Φ S ) terms are absentdue to different U (1) X quantum number, which is crucial for relevant vacuum alignments inthe model to reproduce the present large leptonic mixing and small quark mixing [1, 4]. It isinteresting that at the leading order the electroweak scale does not mix with the potentiallylarge scales h Φ S i , h Φ T i , h Θ i and h ˜Ψ i . B. Review of Lepton sector
Before discussing a leptogenesis scenario, we briefly review the lepton part addressed inRef. [4]. Under A × U (1) X , the matter fields are assigned as in TABLE II. Because of thechiral structure of weak interactions, bare fermion masses are not allowed in the SM. Fermionmasses arise through Yukawa interactions . Recalling that v Ψ / Λ = v ˜Ψ / Λ ≡ λ in Eq. (A9)is used when the U (1) X quantum numbers of the SM charged fermions are assigned. TABLE II: Representations of the matter fields under A × U (1) X .Field L e , L µ , L τ e c , µ c , τ c N c S ce , S cµ , S cτ A , ′ , ′′ , ′′ , ′ , ′′ , ′ U (1) X − q − p p + 15 q, p + 13 q, p + 11 q p p + 25 qU (1) R In the lepton sector, based on the field contents in TABLE I and II the superpotential forYukawa interactions under SU (3) c × SU (2) L × U (1) Y × U (1) X × A reads at leading order W ℓν = y s L e S ce H u + y s L µ S cµ H u + y s L τ S cτ H u + 12 (cid:0) y ss S ce S ce + y ss S cµ S cτ + y ss S cτ S cµ (cid:1) ˜Ψ+ y ν L e ( N c Φ T ) H u Λ + y ν L µ ( N c Φ T ) ′′ H u Λ + y ν L τ ( N c Φ T ) ′ H u Λ+ 12 (ˆ y Θ Θ + ˆ y ˜Θ ˜Θ)( N c N c ) + ˆ y R N c N c ) s Φ S + y e L e e c H d + y µ L µ µ c H d + y τ L τ τ c H d . (5) Since the right-handed neutrinos N c ( S c ) having a mass scale much above (below) the weak interactionscale are complete singlets of the SM gauge symmetry, they can possess bare SM invariant mass terms.However, the flavored-PQ symmetry U (1) X guarantees the absence of bare mass terms M N c N c and µ s S c S c .
12n the above leptonic Yukawa superpotential , W ℓν , charged lepton sector has three inde-pendent Yukawa terms at the leading: apart from the Yukawa couplings, each term doesnot involve flavon fields. The left-handed lepton doublets L e , L µ , L τ transform as , ′ , and ′′ , respectively; the right-handed leptons e c ∼ , µ c ∼ ′′ , and τ c ∼ ′ . In neutrino sector,two right-handed Majorana neutrinos S and N are introduced, in a way that no axionicdomain-wall problem occurs and the mixed U (1) X -[ gravity ] anomaly is free [4], to makelight neutrinos pseudo-Dirac particles and to realize tribimaximal (TBM) pattern , respec-tively; S ce , S cµ , S cτ and N transform as , ′′ , ′ , and under A symmetry, respectively.They compose two Majorana mass terms; one is associated with an A singlet ˜Ψ, whilethe other one is associated with an A singlet Θ and an A triplet Φ S , in which all flavonfields associated with the Majorana mass terms are the SM gauge singlets. The two dif-ferent assignments of A quantum number to Majorana neutrinos guarantee the absence ofthe Yukawa terms S c N c × f lavon f ields . Correspondingly, two Dirac neutrino mass termsare generated; one is associated with S c , and the other is N c . Imposing the continuousglobal U (1) X symmetry in TABLE II explains the absence of the Yukawa terms LN c Φ S and N c N c Φ T as well as does not allow the interchange between Φ T and Φ S , both of which trans-form differently under U (1) X , so that bi-large θ , θ mixings with a non-zero θ mixing forthe leptonic mixing matrix could be obtained after seesawing [25] (as will be shown later,the effective mass matrix achieved by seesawing contributes to TBM mixing pattern andpseudo-Dirac mass splittings, except for active neutrino masses. Such pseudo-Dirac masssplittings are responsible for very long wavelength, which in turn connect to an axion decayconstant [4], see Eqs. (16) and (91).Since the U (1) X quantum numbers are assigned appropriately to the matter fields con-tent as in TABLE II, it is expected that the SM gauge singlet flavon fields derives higher-dimensional operators, which are eventually visualized into the Yukawa couplings of chargedleptons as a function of flavon field Ψ, i.e. , y e,µ,τ = y e,µ,τ (Ψ): y e = ˆ y e (cid:18) ΨΛ (cid:19) , y µ = ˆ y µ (cid:18) ΨΛ (cid:19) , y τ = ˆ y τ (cid:18) ΨΛ (cid:19) . (6)On the other hand, the neutrino Yukawa couplings in terms of the flavons Ψ( ˜Ψ) and Θ are Direct NG (Nambu-Goldstone) mode couplings to ordinary leptons through Yukawa interactions are dis-cussed in Ref. [4]. See Eq. (89) the exact TBM mixing [24]. y si = ˆ y si (cid:18) ΨΛ (cid:19) , y ssi = ˆ y ssi (cid:18) ΨΛ (cid:19) ΘΛ ,y νi = ˆ y νi ˜ΨΛ ! , ˆ y Θ ≈ ˆ y ˜Θ ≈ ˆ y R ≈ O (1) . (7)Here the hat Yukawa couplings ˆ y are complex numbers and of order unity, i.e. / √ . | ˆ y | . √
10. We note that the flavon fields Φ S and Φ T derive dimension-5 operators in theDirac neutrino sector, apart from the Yukawa couplings, while the flavon fields Ψ and ˜Ψderives higher dimensional operators through the Yukawa couplings with the U (1) X flavorsymmetry responsible for the hierarchical charged lepton masses as shown by Eqs. (6) and(7). C. A direct link between Low and High energy Neutrinos
Once the scalar fields Φ S , Θ , ˜Θ , Ψ and ˜Ψ get VEVs, the flavor symmetry U (1) X × A isspontaneously broken And at energies below the electroweak scale, all leptons obtain masses.Since the masses of Majorana neutrino N R are much larger than those of Dirac and lightMajorana ones, after integrating out the heavy Majorana neutrinos, we obtain the followingeffective Lagrangian for neutrinos −L νW ≃ (cid:16) ν cL S R (cid:17) M ν ν L S cR + 12 N R M R N cR + ℓ R M ℓ ℓ L + g √ W − µ ℓ L γ µ ν L + h.c. (8)with M ν = − m TD M − R m D m TDS m DS M S . (9)Here the Majorana neutrino mass terms M νν and M S , and the Dirac mass term m DS aregiven (see Appendix A 1) by M νν = U ∗ L ˆ M νν U † L = − m TD M − R m D , M S = U ∗ R ˆ M S U † R , m DS = U ∗ R ˆ M U † L , (10)where “hat” matrices represent diagonal mass matrices of their corresponding leptons, and U L ( R ) are their diagonal left(right)-mixing matrix. Since m DS is dominant over M νν and M S due to Eqs. (A15-A18), the low energy effective light neutrinos become pseudo-Dirac parti-cles. Keeping terms up to the first order in heavy Majorana mass, in the mass eigenstates14 , ν , ν , S c , S c , S c basis the Hermitian matrix M ν M † ν can be diagonalized as a real andpositive 6 × W ν in Eq. (A11) W Tν M ν M † ν W ∗ ν = | ˆ M | + | ˆ M || δ | | ˆ M | − | ˆ M || δ | ≡ diag( m ν , m ν , m ν , m s , m s m s ) , (11)where ˆ M ≡ diag( m , m , m ). Here the pseudo-Dirac mass splitting, δ , can be given by δ ≡ ˆ M νν + ˆ M † S ≃ ˆ M νν , (12)where the second equality is due to | ˆ M νν | ≫ | ˆ M S | . As is well-known, because of the observedhierarchy | ∆ m | = | m ν − ( m ν + m ν ) / | ≫ ∆ m ≡ m ν − m ν >
0, and the requirementof a Mikheyev-Smirnov-Wolfenstein resonance for solar neutrinos, there are two possibleneutrino mass spectra: (i) the normal mass ordering (NO) m ν < m ν < m ν , m s < m s 3. It is anticipated that ∆ m k ≪ ∆ m , | ∆ m | , otherwise the effects ofthe pseudo-Dirac neutrinos should have been detected. But in the limit that ∆ m k = 0, itis hard to discern the pseudo-Dirac nature of neutrinos. The pseudo-Dirac mass splittingscould be limited by several constraints, that is, the active neutrino mass hierarchy, the BBNconstraints on the effective number of species of light particles during nucleosynthesis, thesolar neutrino oscillations: we roughly estimate a bound for the tiny mass splittings6 × − . ∆ m k / eV . . × − , (14)where the upper bound comes form the solar neutrino oscillations [26], and the lower boundcomes from the inflationary (Sec. III) and leptogenesis (Sec. IV) scenarios by assuming m ν i ∼ . 01 eV. In the present model the lightest effective neutrino mass could not be extremely small because the valuesof δ k through the relation Eq. (13), are constrained by the µ − τ powered mass matrix in Eq. (87). m ν k = m k , then the sum of light neutrino massesgiven by X k m ν k = 12 (cid:18) ∆ m δ + ∆ m δ + ∆ m δ (cid:19) (15)is bounded by 0 . . P i m ν i / eV < . is given by Planck Collaboration [10] whichis subject to the cosmological bounds P i m ν i < . 194 eV at 95% CL (the CMB temperatureand polarization power spectrum from Planck 2015 in combination with the BAO data,assuming a standard ΛCDM cosmological model).The masses of the active neutrinos, m ν i , are determined in a completely independent waythat the neutrino mixing angles are obtained through the seesaw formula in Eq. (16) (seealso Eq. (87)), but they are tied to each other by the tiny mass splittings in Eq. (11). Fromthe basis rotations of weak to mass eigenstates, one of Majorana neutrino mass matrices, M νν = − m TD M − R m D in Eq. (9), can be diagonalized asˆ M νν = U TL M νν U L = − U TL m TD M − R m D U L ≃ δ . (16)The three neutrino active states emitted by weak interactions are described in terms of thesix mass eigenstates as ν α = U αk ξ k with ξ k = 1 √ (cid:16) i (cid:17) ν k S ck , (17)in which the redefinition of the fields ν k → e i π ν k and S ck → e − i π S ck is used. Since theactive neutrinos are massive and mixed, the weak eigenstates ν α (with flavor α = e, µ, τ )produced in a weak gauge interaction are linear combinations of the mass eigenstates withdefinite masses, given by | ν α i = P N ν k W ∗ αk | ξ k i where W αk are the matrix elements of theexplicit form of the matrix W ν . Note that even the number N ν of massive neutrinos canbe larger than three, in the present model the light fermions S α do not take part in thestandard weak interaction and thus are not excluded by LEP results according to which the Massive neutrinos could leave distinct signatures on the CMB and large-scale structure at different epochsof the universe’s evolution [27]. To a large extent, these signatures could be extracted from the availablecosmological observations, from which the total neutrino mass could be constrained. W ± and Z bosons is N ν = 2 . ± . 008 [28].The charged gauge interaction in Eq. (9) for the neutrino flavor production and detection iswritten in the charged lepton basis as −L c . c . = g √ W − µ ℓ α γ γ µ U αk ξ k + h . c . , (18)where g is the SU(2) coupling constant, and U ≡ U L is the 3 × U PMNS . Thus in the mass eigenstate basis the PMNS leptonicmixing matrix [29] at low energies is visualized in the charged weak interaction, which isexpressed in terms of three mixing angles, θ , θ , θ , and three CP -odd phases (one δ CP for the Dirac neutrino and two ϕ , for the Majorana neutrino) as U PMNS = c c c s s e − iδ CP − c s − s c s e iδ CP c c − s s s e iδ CP s c s s − c c s e iδ CP − s c − c s s e iδ CP c c P ν , (19)where s ij ≡ sin θ ij , c ij ≡ cos θ ij and P ν is a diagonal phase matrix what is that particles areMajorana ones. After the relatively large reactor angle θ measured in Daya Bay [30] andRENO [31] including Double Chooz, T2K and MINOS experiments [32], the recent analysisbased on global fits [33, 34] of the neutrino oscillations enters into a new phase of pre-cise determination of mixing angles and mass squared differences, indicating that the exactTBM [24] for three flavors should be corrected in the lepton sector. As shown in Ref.[4]by numerical analysis based on the present model, together with well-fitted solar θ andreactor θ mixing angles the values of atmospheric ( θ ) and Dirac CP phase ( δ CP ) have aremarkable coincidence with the most recent data by the NO ν A [5] and/or T2K [6] experi-ments.The pseudo-Dirac mass splittings in Eq. (13) will manifest themselves through very longwavelength oscillations characterized by the ∆ m k . Such new oscillation lengths far beyondthe earth-sun distance will be provided by astrophysical neutrinos, which fly galactic andextra galactic distances with very high energy neutrinos. Once very tiny mass splittingsare determined by performing astronomical-scale baseline experiments to uncover the os-cillation effects of very tiny mass splitting ∆ m k , the active neutrino mixing parameters( θ , θ , θ , δ CP and m ν , m ν , m ν ) are predicted in the model due to Eqs. (13) and (16).Thus we can possibly connect the pseudo-Dirac neutrino oscillations with the low energy17eutrino properties as well as a successful leptogenesis in Eq. (107). With the help of themixing matrix Eq. (A11), the flavor conversion probability between the active neutrinosfollows from the time evolution of the state ξ k as, P ν α → ν β ( W ν , L, E ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) W ∗ ν e − i ˆ M ν E L W Tν (cid:19) αβ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 14 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k =1 U βk (cid:26) e i m νkL E + e i m SkL E (cid:27) U ∗ αk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (20)in which L = flight length, E = neutrino energy, and ˆ M ν ≡ W Tν M ν W ν , see Eq. (A12). Forthe baseline, 4 πE/ ∆ m , Atm ≪ L , the probability of neutrino flavor conversion reads [35] P ν α → ν β = X k =1 | U αk | | U βk | cos (cid:18) ∆ m k L E (cid:19) , (21)where the oscillatory terms involving the atmospheric and solar mass-squared differences areaveraged out over these long distances. See the related experiments [36–38]. III. INFLATION The inflation that inflated the observable universe beyond the Hubble radius, and couldhave produced the seed inhomogeneities needed for galaxy formation and the anisotropiesobserved by COBE [39], must occur at an energy scale V / ≤ × GeV [40], wellbelow the Planck scale. At this relatively low energies, superstrings are described by aneffective N = 1 supergravity theory [41]. We work in the context of supersymmetric modulistabilization, in the sense that all moduli masses are independent of the gravitino mass andlarge compared to the scale of any other dynamics in the effective theory, e.g., the scaleof inflation, m T i > H I where H I = p V / M P is the Hubble scale during inflation. Asin Ref [2, 4], the size moduli with positive masses have been stabilized, while leaving twoaxions massless and one axion massive, i.e. m T ∼ m θ st ≫ m / . So we will discuss thatsuch moduli stabilization has moduli backreaction effects on the inflationary potential, inparticular, the spectral index of inflaton fluctuations, which provides a lucid explanation forthe cosmological inflation at high energy scale. We are going to see how the size modulistabilized at a scale close to Λ GUT significantly affect the dynamics of the inflation, as wellas how the X -symmetry breaking scale during inflation is induced and its scale is fixed at ∼ . × GeV, close to Λ GUT , by the amplitude of the primordial curvature perturbation.18he model addressed in Refs. [1, 2] naturally causes a hybrid inflation , in which theQCD axion and the lightest neutralino charged under a stabilizing symmetry could becomecomponents of dark mater. We work in a SUGRA framework based on type IIB string theory,and assume that the dilaton and complex structure moduli are fixed at semi-classical levelby turning on background fluxes [48]. Below the scale where the complex structure and theaxio-dilaton moduli are stabilized through fluxes as in Refs. [49, 50], in Einstein frame theSUGRA scalar potential is V = e G M P (cid:16) X α G α G α − (cid:17) + 12 f − ij D i D j , (22)where G α = G α ¯ β G ¯ β with G α ¯ β = M P K α ¯ β , M P = (8 πG N ) − / = 2 . × GeV is thereduced Planck mass with the Newton’s gravitational constant G N , and f ij is the gaugekinetic function. And the F -term potential is given by the first term in the right hand sideof Eq. (22); the D -term, the second term in the right hand side of Eq. (22), is quartic inthe charged fields under the gauge group, and in the model it is flat along the inflationarytrajectory so that it can be ignored during inflation . The generalized Kahler potential, G ,is given by G = KM P + ln | W | M P . (23)Here the low-energy Kahler potential K and superpotential W for moduli and matter su-perfields, invariant under U (1) X gauged symmetry, are given in type IIB string theory by [2] K = − M P ln n ( T + ¯ T ) Y i =1 (cid:0) T i + ¯ T i − δ GS i π V X i (cid:1)o + ˜ K + ... (24)with ˜ K = X i =1 Z i Φ † i e − X i V Xi Φ i + X k Z k | ϕ k | ,W = W Y + W v + W + W ( T ) , (25)in which Φ = { Φ S , Θ , ˜Θ } , Φ = { Ψ , ˜Ψ } , ϕ i = { Ψ , Φ T , Φ T } , dots represent higher-orderterms. W stands for the constant value of the flux superpotential at its minimum. Since Supersymmetirc realizations of F -term hybrid inflation were first studied in Ref. [42]. And the hybridinflation model in supergravity [43, 44] and the F -term hybrid inflation in supersymmetric moduli stabi-lization [45] were studied in detail. See also Refs. [46, 47] Assuming the FI D -terms do not appear during inflation, ξ FI i = 0, it is likely that D terms in the inflatonsector do not give a significant contribution to the inflaton potential. See Sec. III D. W at leading order, they are notfixed by the fluxes. So a non-perturbative superpotential W ( T ) is introduced to stabilizethe Kahler moduli [2], although W ( T ) in Eq. (25) is absent at tree level. The Kahler moduliin K of Eq. (24) control the overall size of the compact space, T = ρ + iθ, T i = ρ i + iθ i with i = 1 , , (26)where ρ ( ρ i ) are the size moduli of the internal manifold and θ ( θ i ) are the axionic parts.As can be seen from the Kahler potential above, the relevant fields participating in thefour-dimensional Green-Schwarz (GS) mechanism [51] are the U (1) X i charged chiral mattersuperfields Φ i , the vector superfields V X i of the gauged U (1) X i which is anomalous, andthe Kahler moduli T i . The matter superfields in K consist of all the scalar fields Φ i thatare not moduli and do not have Planck sized VEVs, and the chiral matter fields ϕ k areneutral under the U (1) X i symmetry. We take, for simplicity, the normalization factors Z i = Z k = 1, and the holomorphic gauge kinetic function f ij = δ ij (1 /g j + ia T j / π ), i.e. , T i = 1 /g X i + ia T i / π on the Kahler moduli in the 4-dimensional effective SUGRA where g X i are the four-dimensional gauge couplings of U (1) X i . Actually, gaugino masses requirea nontrivial dependence of the holomorphic gauge kinetic function on the Kahler moduli.This dependence is generic in most of the models of N = 1 SUGRA derived from extendedsupergravity and string theory [52]. And vector multiplets V X i in Eq. (24) are the U (1) X i gauge superfields including gauge bosons A µi . The GS parameter δ GS i characterizes thecoupling of the anomalous gauge boson to the axion.Non-minimal SUSY hybrid inflation can be defined by the superpotential W inf which isan analytic function, together with a Kahler potential K inf which is a real function W ⊃ W inf = g Ψ (cid:16) Ψ ˜Ψ − µ (cid:17) , (27)˜ K ⊃ K inf = | Ψ | + | Ψ | + | ˜Ψ | + k s | Ψ | M P + k | Ψ | | Ψ | M P + k | Ψ | | ˜Ψ | M P + k | Ψ | M P + ... (28)where Ψ and Ψ( ˜Ψ) denote the inflaton and PQ fields, respectively. The PQ scalar fieldsplay a role of the waterfall fields, that is, the PQ phase transition takes place during inflationsuch that the PQ scale µ Ψ = µ Ψ ( t I ) sets the energy scale during inflation.The kinetic terms of the Kahler moduli and scalar sectors in the flat space limit of the 4dimensional N = 1 supergravity are expressed as L kinetic = K T ¯ T ∂ µ T ∂ µ ¯ T + K T i ¯ T i ∂ µ T i ∂ µ ¯ T i + K Φ i ¯Φ i ∂ µ Φ i ∂ µ Φ † i . (29)20ere we set K Φ i ¯Φ i = 1 for canonically normalized scalar fields. In addition to the superpo-tential in Eq. (25) the Kahler potential in Eq. (24) deviates from the canonical form due tothe contributions of non-renormalizable terms scaled by an UV cutoff M P , invariant underthe both gauge and the flavor symmetries. A. Supersymmetric Moduli Stabilization In string theory, one must consider stabilization of the volume moduli to explain why ouruniverse is 4-dimensional rather than 10-dimensional. Since the three moduli all appear inthe Kahler potential Eq. (24), by solving the F -term equations the three size moduli and oneaxionic partner with positive masses are stabilized while leaving two axions massless throughan effective superpotential W ( T ) [2]. As will be seen later, the two massless axion directionswill be gauged by the U (1) gauge interactions associated with D -branes, and the gaugedflat directions of the F -term potential will be removed through the Stuckelberg mechanism.The F -term scalar potential has the form V F = e ˜ K/M P ( T + ¯ T )( T + ¯ T )( T + ¯ T ) n X I = T,T ,T K I ¯ I | D I W | − M P | W | + K i ¯ i | D i W | o (30)for V X i = 0, where K I ¯ J = 0 for I = J , and I, J stand for T, T i and i, j for the bosoniccomponents of the superfields Φ i , ϕ i . Here the Kahler covariant derivative and Kahler metricare defined as D I W ≡ ∂ I W + W ∂ I K/M P and K I ¯ J ≡ ∂ I ∂ ¯ J K , where D ¯ I W = ( D I W ), and K I ¯ J is the inverse Kahler metric ( K ) − I ¯ J . In order for the Kahler moduli T and T i to bestabilized certain non-perturbative terms are introduced as an effective superpotential [2] W ( T ) = A (Φ i ) e − a ( T + T + T ) + B (Φ i ) e − b ( T + T + T ) , (31)where the coefficients a = 2 π or 2 π/N and b = 2 π or 2 π/M are the corrections arising from D SU ( N ) × SU ( M ). Here A (Φ i ) and B (Φ i ) are analytic functions of Φ i transformingunder U (1) X i as A (Φ i ) → A (Φ i ) e i a π ( δ GS1 Λ + δ GS2 Λ ) , B (Φ i ) → B (Φ i ) e i b π ( δ GS1 Λ + δ GS2 Λ ) , (32)and invariant under the other gauge group. Since there are two non-perturbative superpo-tentials of the form W np = Ae − aT , the structure of the effective scalar potential has two21on-trivial minima at different values of finite T ( i ) . One corresponds to a supersymmetricMinkowski vacuum which could be done through the background fluxes W , while the othercorresponds to a negative cosmological constant which gives rise to a supersymmetric Antide Sitter (AdS) vacuum. So the height of the barrier separates the local Minkowski mini-mum from the global AdS minimum, and the gravitino mass vanishes at the supersymmetricMinkowski minimum. As will be seen in Eq. (70), inflaton mass ( m Ψ ∼ H I ) is much smallerthan the size moduli masses, and consequently the size moduli will be frozen quickly duringinflation without perturbing the inflation dynamics. And it is expected that H I ≪ Λ GUT asa consequence of the enormous flatness of inflaton potential, where Λ GUT ≃ × GeV isthe scale gauge coupling unification in the supersymmetric SM. The scalar potential of thefields ρ and ρ i has local minimum at σ , σ i which is supersymmetric, i.e., W ( σ , σ i ) = 0 , D T W ( σ , σ i ) = D T i W ( σ , σ i ) = 0 , (33)and Minkowski, i.e., V F ( σ , σ i ) = 0 , (34)where σ = σ i = a − b ln (cid:16) a A b B (cid:17) . And W is fine-tuned as W = − A (cid:18) a A b B (cid:19) − aa − b − B (cid:18) a A b B (cid:19) − ba − b , (35)where A and B are constant values of order O (1) of A (Φ i ) and B (Φ i ), respectively, at a setof VEVs h Φ i i that cancel all the D -terms, including the anomalous U (1) X i , see Ref. [4]. Herethe constant W is not analytic at the VEVs h Φ i i , where the moduli are stabilized at the localsupersymmetric Minkowski minumum. Moreover, since W ( T ) is an effective superpotentialits analyticity does not need to be guaranteed in the whole range of the Φ i fields, andso, as will be shown later, the anomalous FI terms at the global supersymmetric AdSminimum can not be cancelled and act as uplifting potentials. Restoration of supersymmetryin the supersymmetric local Minkowski minimum implies that all particles whose mass isprotected by supersymmetry are expected to light in the vicinity of the minimum. However,supersymmetry breaks down and all of these particles become heavy once one moves awayfrom the minimum of the effective potential. This is exactly the situation required for themoduli trapping near the enhanced symmetry points [53].22he F -term equations D T W = D T i W = 0, where we set the matter fields to zero, provide ρ = ρ i , and lead to aA e − aρ e − ia θ st + bB e − bρ e − ib θ st + W + A e − aρ e − ia θ st + B e − bρ e − ib θ st ρ = 0 (36)for V X i = 0, where θ st ≡ θ + θ + θ . This shows that the three size moduli ( ρ, ρ i ) andone axionic direction θ st are fixed, while the other two axionic directions ( θ st1 ≡ θ − θ and θ st2 ≡ θ − θ ) are independent of the above equation. So, without loss of generality, we rebasethe superfields T with θ st = Im[ T ] and T i with θ st i = Im[ T i ] as T = ρ + iθ → T = ρ + iθ st ,T i = ρ i + iθ i → T i = ρ i + iθ st i . (37)Then from the F -term scalar potential, while the gravitino mass in the supersymmetric localMinkowski minimum vanishes, the masses of the fields ρ , ρ , ρ , and θ st , respectively, areobtained as m T = 12 K T ¯ T ∂ T ∂ ¯ T V F (cid:12)(cid:12)(cid:12) T = ¯ T = σ = 3 ln (cid:16) a A b B (cid:17) M P ( a − b ) n A a (cid:16) a A b B (cid:17) − aa − b + B b (cid:16) a A b B (cid:17) − ba − b o ,m θ st = 12 K T ¯ T ∂ θ st ∂ θ st V F (cid:12)(cid:12)(cid:12) T = ¯ T = σ = 3 W M P n − A a (cid:16) a A b B (cid:17) − aa − b − B b (cid:16) a A b B (cid:17) − ba − b o + 6 ln (cid:16) a A b B (cid:17) M P ( a − b ) n − A B ( a − b ) (cid:16) a A b B (cid:17) − a + ba − b (cid:16) a − b (cid:0) a A b B (cid:1) + a b (cid:17)o . (38)Here the mass squared of the size moduli fields ρ ( i ) at the minimum is given by m T ≡ m ρ = m ρ i = 3 σ | W T T ( σ ) | /M P where W T T | all matter fields=0 = a A e − a ( T + T + T ) + b B e − b ( T + T + T ) with W T T ≡ ∂ W/ ( ∂T ) . With the conditions a < b > | a | < | b | ) and A > B < a, b are constants, while A , B are constants in M P units. For a simple choice of parameters, A = 1 . B = − . a = − . 022 and23 = 0 . 13, one has σ ≃ . m T ≃ . × GeV m θ st ≃ . × GeV . (39)As will be seen in Sec. III and in TABLE III, the moduli stabilized at a scale close to Λ GUT will significantly affect the dynamics of the inflation and well fit the cosmological observables. B. Supersymmetry breaking and Cosmological constant As discussed before, the supersymmetric local Minkowski vacuum at ρ = σ and ρ i = σ i is absolutely stable with respect to the tunneling to the vacuum with a negative cosmologicalconstant because the Minkowski minimum is separated from a global AdS minimum by ahigh barrier. This vacuum state becomes metastable after uplifting of a AdS minimum tothe dS minimum with Λ c ∼ − M P . The other supersymmetric global AdS minimum isdefined by W ( σ ˜0 , σ ˜ i ) = 0 D T W ( σ ˜0 , σ ˜ i ) = D T i W ( σ ˜0 , σ ˜ i ) = 0 , (40)corresponding to the minimum of the potential with V AdS < 0. And at this AdS minimumone can set the value of the superpotential ∆ W ≡ h W i AdS by tuning W at values of finite σ ˜0 , σ ˜ i . The existence of FI terms ξ FI i for the corresponding U (1) X i implies the existence ofuplifting potential which makes a nearly vanishing cosmological constant and induces SUSYbreaking. A small perturbation ∆ W to the superpotential [14, 54] is introduced in order todetermine SUSY breaking scale. Then the minimum of the potential is shifted from zero toa slightly negative value at σ ˜0 = σ + δρ and σ ˜ i = σ i + δρ i by the small constant ∆ W . Theresulting F -term potential has a supersymmetric AdS minimum and consequently the depthof this minimum is given by V AdS = − e ˜ K/M P | W | M P ; which can be approximated in terms of W ( σ + δρ, σ i + δρ i ) ≃ ∆ W + O (∆ W ) as V AdS (∆ W ) ≃ − M P (∆ W ) σ σ σ = − M P (cid:16) a − b ln aA bB (cid:17) (∆ W ) . (41) These values ensure m T ∼ GeV and | ˜ g | = O (1) × − through ˜ g = g / (2 σ ) in Eq. (53),satisfying the two observables, i.e. , the scalar spectral index n s and the power spectrum of the curvatureperturbations ∆ R ( k ) in TABLE III. 24t the shifted minimum SUSY is preserved, i.e. D T W ( σ + δρ ) = 0 and D T i W ( σ i + δρ i ) = 0,leading to W T ( σ + δρ ) = W T i ( σ + δρ i ) ≃ W/ σ . At this new minimum the displacements δρ = δρ i are obtained as δρ ( i ) ≃ W σ W T T ( σ ) = 3( a − b )∆ W (cid:16) aA bB (cid:17) n A a (cid:16) aA bB (cid:17) − aa − b + B b (cid:16) aA bB (cid:17) − ba − b o . (42)After adding the uplifting potentials SUSY is broken and then the gravitino in the upliftedminimum acquires a mass m / = h e ˜ K/M P i | W | /M P : m / = s | V AdS | M P ≃ | ∆ W | M P (cid:16) a − b aA bB (cid:17) . (43)The important point is that the masses m T and m θ st in Eq. (38), as well as the height ofbarrier from the runaway direction, do not have any relation to the gravitino mass, i.e. , m T ∼ m θ st ≫ m / . Thus we will consider the F -term hybrid inflation for H I ≫ m / inthe Sec. III.The uplifting of the AdS minimum to the dS minimum can be achieved by consideringnon-trivial fluxes for the gauge fields living on the D7 branes [55] which can be identified asfield-dependent FI D -terms in the N = 1, 4 D effective action [56]. As shown in Refs. [55],uplifting of the AdS minimum induces SUSY breaking and is achieved by adding to the po-tential two terms ∆ V i ≈ | V AdS | σ i /ρ if the uplifting term occurs due to a D -term. Similarly,we can parameterize the uplifting terms as∆ V i = 12 ( ξ FI i ) g X i ≃ | V AdS | (cid:18) σ ˜ i ρ i (cid:19) (44)such that the value of the potential at the new minimum become equal to the observed valueof the cosmological constant. So, the anomalous FI terms can not be cancelled, and act asuplifting potential. And expanding the Kahler potential K in components, the term linearin V X i produces the FI factors ξ FI i = ∂K∂V Xi (cid:12)(cid:12) V Xi =0 ∆ ρ i as ξ FI i = M P δ GS i π σ ˜ i ∆ ρ i . (45)Here the displacements ∆ ρ i ≡ ρ i − σ ˜ i in the moduli fields are induced by the uplifting terms,∆ ρ i ≃ M P | V AdS | W T T ( σ ) σ ˜ i ρ i , (46)25hich are achieved by ∂ ρ i ( V F + ∆ V i ) = 0. Since the uplifting terms by ∆ ρ i making thedS minimum induce SUSY breaking, all particles whose mass is protected from supersym-metry become massive. With the choice of parameters above Eq. (39), the gravitino masscorresponds to m / ≃ 104 TeV (47)implies | ∆ W | ≃ − M P , and which in turn means that the FI terms proportional to | V AdS | /m T are expected to be strongly suppressed.The cosmological constant Λ c has the same effect as an intrinsic energy density of thevacuum ρ vac = Λ c M P . The dark energy density of the universe, Ω Λ = ρ vac /ρ c , is expressed interms of the critical density required to keep the universe spatially flat ρ c = 3 H M P where H = 67 . ± . 46 km s − Mpc − is the present Hubble expansion rate [13]. Using the darkenergy density of the universe Ω Λ = 0 . ± . c ∼ . × − M P . From Eqs. (41) and (44), one canfine-tune the value of the potential in its minimum, V min , to be equal to the observed tinyvalues 7 . × − M P , V min = | V AdS | n − (cid:18) σ ˜1 ρ (cid:19) + 12 (cid:18) σ ˜2 ρ (cid:19) o . (48)The positive vacuum energy density resulting from a cosmological constant implies a negativepressure, and which drives an accelerated expansion of the universe, as observed. C. Moduli backreaction on inflation Since in general the interference between the moduli and inflaton sectors generates acorrection to the inflationary potential we consider the effect of string moduli backreactionon the inflation model which is linked to SUSY breaking scale . In small-field inflation,such as hybrid inflation, this produces a linear term in the inflaton at leading order as inRef. [57]. This is analogous to the effect of supersymmetry breaking which induces a linearterm proportional to the gravitino mass. Depending on its size such a linear term can have There are many studies [57, 58] on the moduli backreaction effect on the inflation and its link to SUSYbreaking. 26 significant effect on inflationary observables well fitted in CMB data, in particular, thespectral index of scalar fluctuations.At T ( i ) = ¯ T ( i ) = σ due to W ( σ ) = 0 = W T ( σ ) one can obtain V F (cid:12)(cid:12) σ = V inf (2 σ ) + 3 e ˜ K/M P (2 σ ) M P | W inf | , (49)where V inf is the inflation potential in the absence of moduli sectors V inf = e ˜ K/M P n K j ¯ j | D j W inf | − M P | W inf | o . (50)Since all powers of 2 σ in Eq. (49) can be absorbed by a redefinition of W inf the potentialis rescaled as V F (cid:12)(cid:12) σ → V inf + e ˜ K/M P M P | W inf | , indicating that there is no backreaction to theinflation on the moduli sector. However, due to the effect of the inflationary large positiveenergy density, see Eq. (57), the minimum of the moduli are shifted by δT and δT i , and atthis new shifted position the potential is minimized. The displacements are obtained byimposing ∂ T V | σ + δT = 0 and ∂ T i V | σ + δT i = 0, and the expression for δT and δT i can beexpanded in powers of H I /m T , δT ( i ) ≃ W inf √ √ σ m T M P + 12(2 σ ) m T M P n K j ¯ j D j W inf ∂ ¯ j ¯ W inf − M P | W inf | − W M P (cid:18) 32 + (3 σ ) / W T T T ( σ ) M P m T (cid:19) o + O (cid:18) H I m T (cid:19) . (51)This implies that there is a supersymmetry breakdown by the inflaton sector during inflation D T ( i ) W (cid:12)(cid:12) σ + δT ( i ) = 1 √ σ ) m T K j ¯ j D j W inf ∂ ¯ j ¯ W inf + O (cid:18) H I m T (cid:19) , (52) i.e. , D T ( i ) W (cid:12)(cid:12) σ + δT ( i ) are suppressed by one power of m T , which vanish in the limit of infinitelyheavy moduli.Since the moduli are very heavy they stabilize quickly to their minima and the inflationarypotential get corrected after setting T and T i to their minima as follows V F (cid:12)(cid:12) σ + δT ( i ) = V inf (2 σ ) − σ ) W T T ( σ ) h W inf (cid:8) V inf + e ˜ KM P K j ¯ j ∂ j W inf D ¯ j ¯ W inf (cid:9) + h . c . i + O (cid:18) H I m T (cid:19) . (53)Using | W T T ( σ ) | = q M P √ σ m T , and rescaling as V inf / (2 σ ) → V ( t I ) and W inf / (2 σ ) / → W inf ( t I ), it is evident that the inflationary potential due to the moduli backreaction induces27 linear term in the inflaton potential V F | σ + δT ( i ) = V ( t I ) n − √ √ m T M P ( W inf + h.c.) o + O (cid:18) | Ψ | m T (cid:19) (54)Clearly, as we can see here, in the limit m T → ∞ the interference term between stringmoduli and inflaton sectors is disappeared. D. Scale of PQ-symmetry breakdown during inflation In the following, let us consider the PQ phase transition scale during inflation. Due toEq. (2) during inflation we have v Θ ( t I ) = v S ( t I ) = v T ( t I ) = 0 . (55)And the Kahler moduli fields we consider are stabilized during inflation and their potentialhas a local minimum at finite moduli fields values separated by a high barrier from therunaway direction. Since the moduli masses are much larger than the inflaton mass andaccordingly will be frozen quickly during inflation without perturbing the inflaton dynamics,the height of barrier protecting metastable Minkowski ( ≃ dS) space are independent of thegravitino mass hence the inflationary Hubble constant is also independent of the gravitinomass [14].We consider the PQ symmetry breaking scale, µ Ψ ( t I ), during inflation. In the globalSUSY minima where V SUSY = 0, all the flavon and driving fields have trivial VEVs, whilethe waterfall fields Ψ( ˜Ψ) can have non-zero VEVs. The FI D -terms must then be zero, i.e. ξ FI1 = ξ FI2 = 0. During inflation, if | Ψ | takes a large value the waterfall fields stayat the origin of the field space (the local minimum appears at h Ψ i = h ˜Ψ i = 0); and thesuperpotential is effectively reduced to W inf ( t I ) = − ˜ g Ψ µ ( t I ) , (56)with ˜ g ≡ g / (2 σ ) and ˜ g < 0, which gives a positive contribution to the inflation energy V ( t I ) = 3 H I M P ≃ (cid:12)(cid:12)(cid:12) ∂W inf ( t I ) ∂ Ψ (cid:12)(cid:12)(cid:12) = ˜ g µ ( t I ) , (57)and in turn drives inflation. Since the potential for | Ψ | ≫ | Ψ c | ≡ µ Ψ ( t I ) with h Ψ i = h ˜Ψ i = 0is flat before the waterfall behavior occurs, inflation takes place there. And the waterfall28ehavior is triggered, when the inflaton Ψ reaches the critical value | Ψ c | . Once | Ψ | rollsdown from a large scale and approaches its critical value | Ψ c | , the inflaton and waterfallfields get almost maximally mixed to form mass eigenstates:Ψ ′ ≃ √ ± ˜Ψ) , Ψ ′ ≃ √ − Ψ ⊥ ) , ˜Ψ ′ ≃ − √ ⊥ ) , (58)where Ψ ⊥ ≃ ( ± Ψ − ˜Ψ) / √ ′ . And their corresponding mass eigenvaluesare given by m Ψ ′ ≃ | ˜ g | µ Ψ ( t I ) , m ˜Ψ ′ ≃ | ˜ g | µ Ψ ( t I ) , m Ψ ′ ≃ . (59)Let us schematically see this is the case. The potential at global SUSY limit V globalinf = ˜ g | Ψ ˜Ψ − µ ( t I ) | + ˜ g | Ψ | ( | Ψ | + | ˜Ψ | )= (cid:16) Ψ ′∗ ˜Ψ ′ (cid:17) ˜ g ( | Ψ | − µ ( t I )) 00 ˜ g ( | Ψ | + µ ( t I )) Ψ ′ ˜Ψ ′∗ + ... (60)implies that (i) when | Ψ | < µ Ψ ( t I ), one of the mass eigenstates, Ψ ′ , becomes tachyonic: thewaterfall fields fixed at h Ψ i = h ˜Ψ i = 0 is not stable since Ψ( ˜Ψ) have an opposite sign of U (1) X charges. As can be seen from Eq. (24) since the Kahler moduli superfields puttingthe GS mechanism into practice are not separated from the SUSY breaking by the inflatonsector during inflation, by taking tachyonic SUSY breaking scalar masses m ∼ − H I induceddominantly by the U (1) X D -term, the waterfall field Ψ ′ rolls down its true minimum froma large scale. (ii) The other ˜Ψ ′ stays positive definite throughout the inflationary trajectoryup to a critical value | Ψ c | ≈ µ Ψ ( t I ). (iii) After inflation the universe is dominated byboth the inflaton Ψ ′ and one of waterfall fields, ˜Ψ ′ , while the other waterfall field Ψ ′ givesnegligible contribution to the total energy of the universe. (iv) After inflation and thewaterfall transition mechanism has been completed Ψ ′ approaches to zero and Ψ ′ ( ˜Ψ ′ ) relaxto the flat direction of the field space given by Ψ ′ ˜Ψ ′ = µ ( t I ): the positive false vacuum ofthe inflaton field breaking the global SUSY spontaneously gets restored once inflation hasbeen completed.Now, we discuss how the inflation could be realized explicitly. The F -term scalar poten-tial, the first term in the right hand side of Eq. (22), can be expressed as V ( φ α ) = e ˜ K/M (X α K α ¯ α D α W inf D α ∗ W ∗ inf − | W inf | M ) (61)29ith α being the bosonic components of the superfields ˆ φ α ∈ { ˆΨ , ˆΦ T , ˆΦ S , ˆΘ , ˆΨ, ˆ˜Ψ, ˆΦ S ,ˆΘ, ˆ˜Θ, ˆΦ T } , and where the Kahler covariant derivative and Kahler metric are defined as D α W inf ≡ ∂W inf ∂φ α + M − ∂K∂φ α W inf , K α ¯ β ≡ ∂ K∂φ α ∂φ ∗ β (62)and D α ∗ W ∗ inf = ( D α W inf ) ∗ with ˜ K α ¯ β ≡ ( ˜ K α ¯ β ) − . The lowest order ( i.e. global supersym-metric) inflationary F -term potential V globalinf receives corrections for | φ α | ≪ M P . Duringinflation, working along the direction | Ψ | = | ˜Ψ | = 0, from Eqs. (28) and (61) a small curva-ture needed for the slow-roll can be represented by the inflationary potential V inf V inf = V treeinf + V sugra + ∆ V − loopinf . (63)The leading order potential, corrected by the interference term induced by the moduli back-reaction, can be written in Eq. (54) as V treeinf = V ( t I ) n − √ √ √ V m T M P (Ψ + Ψ ∗ ) o , (64)where V ( t I ) is the rescaled vacuum energy during inflation, see Eq. (54). Substituting K inf and W inf in Eq. (28) into V inf F in Eq. (50), and minimizing with respect to Ψ and ˜Ψ for | Ψ | > µ Ψ ( t I ) gives V inf F = ˜ g µ ( t I ) (cid:26) − k s | Ψ | M P + γ s | Ψ | M P + O (cid:16) | Ψ | M P (cid:17)(cid:27) , (65)where γ s ≡ − k s / − k . Such a supergravity induced mass squared is expected to havethe same form as the Ψ mass squared, namely ˜ g µ ( t I ) /M P = V ( t I ) /M P which is the orderof the Hubble constant squared H I = V ( t I ) / M P . Then the SUGRA contribution V sugra to V inf leads to V sugra = − c H H I | Ψ | + V γ s | Ψ | M P + O (cid:16) | Ψ | M P (cid:17) . (66)The inflaton Ψ also receives 1-loop radiative correction in the potential [15] due to themismatch between masses of the scalar and fermion components of Ψ( ˜Ψ), which are non-vanishing since SUSY is broken by ∂W int /∂ Ψ = 0. The corresponding 1-loop correction tothe scalar potential is analytically calculated as∆ V − loop = X i ( − f m i π ln m i Q = ˜ g µ ( t I )8 π F ( x ) (67)30here F ( x ) = (cid:8) ( x + 1) ln x − x + 2 x ln x +1 x − + 2 ln g µ x Q − (cid:9) and the sum is taken overthe field degrees of freedom and f = 0 for scalar and f = 1 for fermion. Here the Q isa renormalizable scale, x is defined as x ≡ | Ψ | /µ Ψ ( t I ) = ϕ/ ( √ µ Ψ ( t I )) where ϕ is thenormalized real scalar field. In the limit x ≫ 1, i.e. ϕ ≫ √ µ Ψ ( t I ), this is approximated as∆ V − loop ≃ ˜ g µ ( t I )16 π ln ˜ g ϕ Q . (68)If we let the inflaton field Ψ ≡ ϕ e iθ / √ 2, and during the inflation period, taking intoaccount the radiative correction, supergravity effects, and moduli backerction effects, theinflationary potential is of the following form V inf ( ϕ ) = V ( t I ) n − √ √ V m T M P ϕ cos θ + γ s ϕ M P + ˜ g π ln ˜ g ϕ Q o + ϕ (cid:18) m − k s V M P (cid:19) . (69)The moduli-induced slope partially cancels the slope of the Coleman-Weinberg potential,which flattens the inflationary trajectory and reduces the distance in field space correspond-ing to the N e ∼ e -folds of inflation. And the inflaton mass m Ψ can be given for k s = 1by m Ψ = | ˜ g | µ ( t I ) M P ; (70)since the inflaton acquires a mass of order the Hubble constant, m Ψ = H I √ 3, agreement oftheory’s prediction for spectral index n s with observation strongly suggests the presence ofa negative Hubble-induced mass-term, and the k s parameter term vanishes identically. Thisinflaton mass ( ≫ m / ) can directly be obtained from Eqs. (27) and (28) as m Ψ = (cid:12)(cid:12) M P h e G ∇ Ψ G Ψ i (cid:12)(cid:12) = √ H I , (71)where ∇ k G α = ∂ k G α − Γ jkα G j with the Christoffel symbol Γ jkα = G jℓ ∗ G kαℓ ∗ [79], and ∇ Ψ G Ψ ≃ − ( W Ψ /W ) is used. This inflaton mass is in agreement with the above pre-diction in Eq. (70).Inflation stops at | Ψ c | ≃ µ Ψ ( t I ), where the mass of Ψ becomes negative and the fieldacquires a non-vanishing expectation value. In order to develop the VEV of the waterfallfield Ψ, we destabilize the waterfall field Ψ by taking tachyonic Hubble induced masses of thePQ-breaking waterfall field, i.e. , m ∼ − H I < 0. Then, the VEV of the waterfall field could31e determined by considering both the SUSY breaking effect and a supersymmetric nextleading order term. The next leading Planck-suppressed operator invariant under A × U (1) X is given by ∆ W v ≃ ˆ αM P Ψ Ψ ˜Ψ , (72)where we set the VEVs of all other matter fields to zero except the waterfall field andneglected their corresponding trivial operators. Note that the constant ˆ α = O ( α/ π ) witha constant α being of order unity. Since the soft SUSY-breaking terms are already presentat the scale relevant to inflation dynamics, the scalar potential for the waterfall field Ψ atleading order reads V Ψ ( t I ) ≃ D X + ˆ α Ψ ˜ m | Ψ | + ˆ α ˜Ψ ˜ m | ˜Ψ | + | ˆ α | | Ψ | | ˜Ψ | M P + ... , (73)where | ˆ α Ψ ˜ m | , | ˆ α ˜Ψ ˜ m |≪ | D X ( t I ) | with | ˆ α Ψ , ˜Ψ | ≪ m Ψ , ˜Ψ ≃ | Ψ c | ∼O ( | F Ψ | /M P ) with F Ψ = K Ψ ¯Ψ D Ψ W inf ≃ √ H I M P represents the Hubble inducedsoft scalar masses generated by the F -term SUSY breaking, during inflation. If thetachyonic SUSY breaking scalar masses are dominantly induced by the U (1) X D -term, D X ( t I ) ∼ O ( H I ), compared to the Hubble induced soft masses generated by the F -termSUSY breaking, the soft SUSY breaking mass of Ψ during inflation are approximated by m ( t I ) = ˆ α Ψ ˜ m + D X ( t I ) ≃ − ˆ β Ψ H I , with ˆ β Ψ > . (74)Then the scalar potential in Eq. (73) for the waterfall field Ψ is good approximated as V Ψ ( t I ) ≃ − ˆ β Ψ H I | Ψ | + | ˆ α | | Ψ | | ˜Ψ | M P . (75)Here the constant ˆ β Ψ are of order unity, while ˆ α = α/ (8 π ) with α being of order unity. Wefind the minimum as v Ψ ( t I ) = s β Ψ | ˆ α | H I (cid:18) M P v ˜Ψ (cid:19) , (76)leading to M P ≫ µ Ψ ( t I ) ≫ H I and the PQ breaking scales during inflation µ ( t I ) ≡ v Ψ ( t I ) v ˜Ψ ( t I )2 = s ˆ β Ψ | ˆ α | (cid:18) H I v ˜Ψ ( t I ) M P (cid:19) . (77)In supersymmetric theories based on SUGRA, since SUSY breaking is transmitted by gravity,all scalar fields acquire an effective mass of the order of the expansion rate during inflation.32o, we expect that the inflaton acquires a mass of order the Hubble constant, and which inturn indicates that the soft SUSY breaking mass (the inflaton mass m Ψ ) during inflationstrongly depends on the scale of waterfall (or PQ) fields by the above Eq. (77); for example,for µ Ψ ( t I ) ∼ GeV one obtains H I ∼ × GeV (78)for ˆ β i ∼ α ∼ / (8 π ), see TABLE III. E. Cosmological observables The inflaton as a source of inflation is displaced from its minimum and whose slow-rolldynamics leads to an accelerated expansion of the early universe. During inflation the uni-verse experiences an approximately dS phase with the Hubble parameter H I . Quantumfluctuations during this phase can lead to observable signatures in CMB radiation tempera-ture fluctuation, as the form of density perturbation, in several ways [12], when the quantumfluctuations are crossing back inside the Hubble radius long after inflation has been com-pleted. When interpreted in this way, inflation provides a causal mechanism to explain theobserved nearly-scale invariant CMB spectrum. (i) Quantum fluctuations of the inflatonfield during inflation give rise to fluctuations in the scalar curvature and lead to the adia-batic fluctuations that have grown into our cosmologically observed large-scale structuremuch bigger than the Hubble radius and then eventually got frozen. Adiabatic density per-turbations seeded by the quantum fluctuations of the inflaton have a nearly scale-invariantspectrum, ∆ R ( k ), which is a cosmological observable of the curvature perturbations. Thepower spectrum of the curvature perturbations, ∆ R ( k ), reads in the Planck 2015 result at68% CL (for the base ΛCDM model) [13]∆ R ( k ) = (2 . +0 . − . ) × − , (79)at the pivot scale k = 0 . 002 Mpc − (wave number), which is compatible with the onesuggested for the COBE normalization [59]. (ii) Fluctuations of the metric lead to tensor- These correspond to fluctuations in the total energy density, δρ = 0, with no fluctuation in the localequation of state, δ ( n i /s ) = 0. On the other hand, isocurvature perturbations correspond to fluctuationsin the local equation of state of some species, δ ( n i /s ) = 0, with no fluctuation in the total energy density, δρ = 0 [12]. 33 mode fluctuations in the CMB radiation. Primordial gravitational waves are generatedwith a nearly scale-invariant spectrum, ∆ h ( k ), which reads in the Planck 2015 result [13]∆ h ( k ) < . × − . (iii) Quantum fluctuations are imprinted into every masslessscalar field in dS space during inflation, with an approximately scale-invariant spectrum, h| δφ ( k ) | i = ( H I / π ) / ( k / π ) for a canonically normalized scalar field φ , which is es-sentially a thermal spectrum at Gibbons-Hawking temperature T GH = H I / π . The otherimportant cosmological observables imprinted in the CMB spectrum are followings: theBAU (which will be discussed in Sec. IV), the fractions of relic abundance Ω DM (see Ref. [4])and dark energy Ω Λ (see Sec. III B).The slow-roll condition [60] is well satisfied up to the critical point ϕ c = √ µ Ψ ( t I ),beyond which the waterfall mechanism takes place. Here the slow-roll parameters, ǫ and η ,are approximately derived as ǫ ≡ M P (cid:18) V ϕ V (cid:19) ≃ (cid:16) ˜ g π M P ϕ (cid:17) n γ s π ˜ g (cid:16) ϕM P (cid:17) − √ 32 8 π | ˜ g | µ Ψ m T µ Ψ M P ϕM P cos θ o ≪ ,η ≡ M P V ϕϕ V ≃ ˜ g π (cid:16) M P ϕ (cid:17) n γ s π ˜ g (cid:16) ϕM P (cid:17) − o , | η | ≪ , (80)where V ϕ denotes a derivative with respect to the inflaton field ϕ = √ , and M P ≫| Ψ | ≫ | Ψ c | (or M P ≫ | ϕ | ≫ | ϕ c | ) is assumed. Recalling that ˜ g = g / (2 σ ) . The aboveequations clearly show that the curvature of the inflationary potential is dominantly affectedby the moduli backreaction in Eq. (54) as well as the 1-loop radiative correction. In the slow-roll approximation, the number of e -foldings after a comoving scale l has crossed the horizonis given by the inflationary potential through N ( ϕ ) = Z t l t ( ϕ c ) H I dt = 1 M P Z ϕ l ϕ c V ( ϕ ) V ϕ ( ϕ ) dϕ , (81)where ϕ l is the value of the field at the comoving scale l , and ϕ c is the one at the endof inflation. The field value ϕ c is determined from the condition Max { ǫ ( ϕ c ) , | η ( ϕ c ) |} =1 [61]. The power spectrum ∆ R ( k ) sensitively depends on the theoretical parameters of the34nflationary potential,∆ R ( k ) ≃ π M P V ( ϕ l ) | V ϕ ( ϕ l ) | ≃ (cid:16) π ˜ g (cid:17)(cid:16) µ Ψ M P (cid:17) (cid:16) ϕ l M P (cid:17) n √ 32 8 π ˜ g (cid:16) µ Ψ m T (cid:17)(cid:16) µ Ψ M P (cid:17)(cid:16) ϕ l M P (cid:17) cos θ o − (82)where the potential V ( ϕ l ) and its derivative V ϕ ( ϕ l ) are evaluated at the epoch of horizonexit for the comoving scale k . It should be compared with the Planck 2015 result Eq. (79).With the definition of the number of e -folds after a comoving scale k leaves the horizon,we can obtain the corresponding inflaton value ϕ l /M P from Eq. (81). And the number of e -folds N e corresponding to the comoving scale k is around 50 depending on the energyscales H I and T reh N e = 49 . (cid:18) . 002 Mpc − k (cid:19) + 13 ln (cid:18) T reh GeV (cid:19) + 13 ln (cid:18) H I GeV (cid:19) (83)where T reh represents the maximal temperature of the last radiation dominated era, so-calledthe reheating temperature. The tensor and scalar modes have spectrum A t = 2 H I / ( π M P )and A s ≡ ∆ R ( k ) [13], respectively. In the supergravity F -term inflation we consider, thetensor-to-scalar ratio r = A t /A s ≃ ǫ ( ϕ l ) is much lower than the Planck 2015 bound( r . < . i.e. well bellow 10 − , and the running of the spectral index dn s /d ln ˜ g isalways smaller than 10 − and so unobservable. And the scalar spectral index n s is approxi-mated as n s ≃ − ǫ ( ϕ l ) + 2 η ( ϕ l ) ≃ η ( ϕ l ) . (84)We can compare this quantity with the results of the Planck 2015 observation [13] n s = 0 . ± . . (85)In order for the power spectrum of the curvature perturbation and the spectral index to bewell fitted with the Planck 2015 observation, the four independent parameters m T , µ Ψ ( t I ), γ s , and | ˜ g | are needed and those parameters have predictions, m T = O (10 ) ≫ µ Ψ ( t I ) = O (10 ) GeV, γ s = O (1) and | ˜ g | = O (1) × − as in TABLE III, where we have set cos θ = 1in Eqs. (80) and (82). This table shows that the cosmological observables can be well fittedwhere both the moduli stabilized at a scale close to Λ GUT and the PQ symmetry breakingscale induced at µ Ψ ( t I ) ≃ . × GeV < m T . As shown in TABLE III, the number of35 ABLE III: Four independent input parameters m T , µ Ψ ( t I ), γ s , and | ˜ g | provide predictions on n s , N e , ∆ R ( k ) / − , and T reh / GeV. m T GeV µ Ψ ( t I )10 GeV γ s | ˜ g | − H I GeV ϕ l GeV ϕ c GeV n s N e ∆ R ( k )10 − T reh GeV . 173 0 . 673 0 . 544 1 . 189 1 . 281 1 . 051 0 . 952 0 . 968 53 . 210 2 . 095 1 . × . 592 0 . 685 0 . 568 0 . 963 1 . 074 1 . 047 0 . 969 0 . 969 51 . 285 2 . 187 6 . × . 057 0 . 672 0 . 448 1 . 077 1 . 155 1 . 027 0 . 950 0 . 968 50 . 010 2 . 149 1 . × . 381 0 . 674 0 . 321 1 . 160 1 . 254 1 . 024 0 . 954 0 . 967 49 . 408 2 . 121 2 . × . 287 0 . 673 0 . 671 1 . 025 1 . 102 1 . 046 0 . 951 0 . 968 48 . 726 2 . 154 3 . × e -foldings depends on the amount of reheating temperature which in turn depends on thedecay rate of the inflaton Ψ ′ and waterfall field ˜Ψ ′ into relativistic particles. In the followingsection we will see how the amount of reheating, T reh , could be strongly correlated with bothbaryogenesis via leptogenesis and the yield of gravitinos. IV. LEPTOGENESIS Let us discuss on how the matter-antimatter asymmetry of the universe could be realizedin the context of the present model. In order to account for a successful leptogenesis,we introduce the AD mechanism for baryogenesis [16] and its subsequent leptonic versionso-called AD leptogenesis [17]. In the global SUSY limit, i.e. M P → ∞ , as well as inthe energy scale where A × U (1) X is broken, some combinations of scalar fields do notenter the potential, composing flat directions of the scalar potential. So, taking the flatdirections H u = L i = ζ i / √ i = 1 , , ζ i = (2 e L i H u ) / where e L i are scalar components of the chiral multiplets L i of SU (2) L -doublet leptons. After integrating out the heavy Majorana neutrinos, N R , therelevant superpotential (5) induces the effective operator at low energies W eff ⊃ M i ( e L i H u ) , with M i ≡ v u ( ˆ M νν ) i . (86)where ( ˆ M νν ) i = ( U T PMNS M νν U PMNS ) ii ≃ δ i in Eq. (16). Recalling that the 3 × U L = U PMNS diagonalizing the mass matrix M νν = − m TD M − R m D participates inthe charged weak interaction, the active neutrino mixing angles ( θ , θ , θ , δ CP ) and the36seudo-Dirac mass splittings δ k responsible for new wavelength oscillations characterized bythe ∆ m k could be obtained from the mass matrix M νν formed by seesawing. Then, fromEqs. (A17) and (A18) we obtain the µ − τ powered mass matrix as in Refs. [1, 62] M νν = m e iπ F (1 − F ) y (1 − F ) y (1 − F ) y (1 + F +3 G ) y (1 + F − G ) y y (1 − F ) y (1 + F − G ) y y (1 + F +3 G ) y = U ∗ PMNS ˆ M νν U † PMNS , (87)where m ≡ (cid:12)(cid:12)(cid:12)(cid:12) ˆ y ν υ u M (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) v T √ (cid:19) (cid:18) v Ψ √ (cid:19) , F = (cid:0) ˜ κ e iφ + 1 (cid:1) − , G = (cid:0) ˜ κ e iφ − (cid:1) − . (88)In the limit y ν = y ν = y ν ( y , y → | δ k | :sin θ = 13 , sin θ = 12 , sin θ = 0 , | δ | = ∆ m m = 3 m | F | , | δ | = ∆ m m = 3 m , | δ | = ∆ m m = 3 m | G | . (89)These | δ k | are disconnected from the TBM mixing angles. It is in general expected thatdeviations of y , y from unity, leading to the non-zero reactor mixing angle [30, 31], i.e. θ ≃ . ◦ at 1 σ best-fit [34], and in turn opening a possibility to search for CP violation inneutrino oscillation experiments. These deviations generate relations between mixing anglesand eigenvalues | δ k | . Therefore Eq. (87) directly indicates that there could be deviationsfrom the exact TBM if the Dirac neutrino Yukawa couplings in m D of Eq. (A17) do not havethe same magnitude, and the pseudo-Dirac mass splittings are all of the same order | δ | ≃ | δ | ≃ | δ | ≃ O ( m ) . (90)As shown in Ref.[4] by numerical analysis, together with well-fitted θ and θ the values ofatmospheric ( θ ) and Dirac CP phase ( δ CP ) have a remarkable coincidence with the recentdata by the NO ν A [5] and/or T2K [6] experiments. From the overall scale of the mass matrixin Eq. (88) the pseudo-Dirac mass splitting, δ , is expected to be | δ | ≃ . × − (cid:18) . × GeV M (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ˆ y ν v T √ (cid:12)(cid:12)(cid:12)(cid:12) sin β eV , (91)37n which the scale of the heavy neutrino, M , can be estimated from Eq. (A19) through theastrophysical constraints as M = | ˆ y Θ | × . +1 . − . × GeV which is connected to the PQsymmetry breaking scale via the axion decay constant in Ref. [4]. Eq. (91) shows that thevalue of δ depends on the magnitude ˆ y ν v T / Λ since M is constrained by the axion decayconstraints: the smaller the ratio v T / Λ, the smaller becomes | δ k | responsible for the pseudo-Dirac mass splittings . However, the value of | δ k | is constrained from Eq. (14); for example,using tan β = 2 and v T / Λ ≃ λ / √ | δ | ≃ . × − | ˆ y ν | eV . (92)Since the potential is (almost) flat in these directions ζ i , they have large initial VEVs inthe early universe, see Eq. (97). Such flat directions are lifted by some effective operators ina later epoch, receiving soft-masses in the SUSY breaking vacuum. Then the potential ofthe flat directions, ζ i , is directly written as V ( ζ i ) = m ζ i | ζ i | + m / M i ( a m ζ i + h.c.) + | ζ i | M i . (93)Here in the mass terms m ζ i we have included soft scalar masses generated by the F -termSUSY breaking, that is, the contribution from the effective µ -term, W ⊃ µ eff H u H d , whichgives mass terms µ | ζ i | / 2. Since our model lies in the gravity-mediated SUSY breakingmechanism it is expected that m ζ i ∼ m / and | a m | ∼ A -term The potential for ζ i in Eq. (93) is D -flat, | ζ i | = 0, and also F -flat in the limit of δ i (or ∆ m i ) → 0. So, the ADfields ζ i can develop large VEVs during inflation. As discussed before, during inflation theenergy density of the universe is dominated by the inflaton Ψ , that is, V ( t I ) = 3 H I M P .The potential for D -flat direction is generated from the coupling between the AD fields ζ i and the inflaton Ψ , which generically takes the form K ⊃ K AD = | Ψ | + | ζ i | + (cid:18) k ζi | Ψ | M P ζ i + h . c . (cid:19) + γ ζi | Ψ | | ζ i | M P + ... , (94) Moreover, the overall scale of the heavy neutrino mass M is closely related with a successful leptogenesis(see the details in Sec. IV), constraints of the mass-squared differences in Eq. (13), and the CKM mixingparameters, therefore it is very important to fit the parameters v T / Λ and M . The value of v T / Λ is also related to the µ -term in Eq. (4). In the context of Kallosh-Linde (KL) type models the dominant contributions to A -term arise fromloop corrections [63] because at tree level A -terms are strongly suppressed by m / /m T , hence one needsrelatively large O (100) TeV gravitino mass in order to get properly large A -terms [64]. k ζi and γ ζi are complex and real constants, respectively, and the dots represent higherorder terms which are irrelevant for our discussion. Then, due to the finite energy densityof the inflaton Ψ during inflation the AD fields ζ i receive additional SUSY breaking effects.And such SUGRA contribution reads V sugra ( ζ i ) = − ˜ c H H I | ζ i | + H I M i ( a H ζ i + h.c.) . (95)Here by taking ˜ c H > c H being of order unity we assume that the AD fields ζ i canobtain negative Hubble-induced mass terms. From Eq. (93) and (95) the total effectivepotential for the AD fields ζ i relevant to the leptogenesis reads V ( ζ i ) = V ( ζ i ) + V sugra ( ζ i ) . (96)Then the minima of the potential are given by h| ζ i |i ≃ (cid:18) 43 ˜ c H (cid:19) (cid:18) m i ∆ m i H I v sin β (cid:19) . M P , (97)and arg( a H ) + 4 arg( ζ i ) ≃ π (2 n + 1) / n = 0 , 1, in which we have used m ζ i , m / | a m | ≪ H I . The AD fields ζ i at the origin are unstable due to the negative Hubble mass terms inEq. (95), and so roll down toward their global SUSY minima of the potential in Eq. (96)during inflation. Thus, the AD fields ζ i have large scales of ∼ p v u H I / | δ i | . M P in Eq. (97)during inflation. This is compatible with the fact that the Planck scale, M P , sets theuniverse’s minimum limit, beyond which the laws of physics break. If we set the initialminima of the AD fields to the (almost) Planck scale, the ratios m i / ∆ m i responsible forthe neutrino mass splittings δ i (relevant to the low energy neutrino oscillation as well as thehigh energy neutrino at the IceCube telescope) could be resricted as1 δ i = 2 m i ∆ m i . M P H I v sin β (cid:16) c H (cid:17) / . (98)Using H I ≃ GeV, sin β ≃ 1, and ˜ c H ≃ δ i & . × − eV (99)which is well compatible with the constraints from the neutrino data in Eq. (14) as well asa successful leptogenesis in Eq. (107).After inflation ends, the inflaton Ψ ′ and waterfall field ˜Ψ ′ (see Eq. (58)) begin to oscillatearound their VEVs, h ˜Ψ ′ i = µ Ψ and h Ψ i ≃ deviates from zero because39f the supergravity effect: h Ψ i ∼ m / / | ˜ g | at the true minimum, see Eq. (119)) and theirdecays produce a dilute thermal plasma formed by collisions of relativistic decay products.Since the energy density of the universe is still dominated by the inflaton Ψ ′ and waterfallfield ˜Ψ ′ during the inflaton and waterfall field oscillations epoch, the AD fields potential isstill governed by the Hubble-induced mass terms in Eq. (95) together with V ( ζ i ) in Eq. (93)at the first stage of oscillation. Thus, the AD fields ζ i are trapped in the minima determinedmainly by the Hubble A -term as in Eq. (97) because the curvatures around the minima alongboth radial and angular directions are of the order of H I also in this period. However, afterinflation the values of ζ i in Eq. (97) gradually decrease to the order of ζ i masses as the Hubbleparameter H ( T ) decreases, then the negative Hubble-induced mass terms are eventuallyexceeded by the Hubble parameter at T ∼ p m ζ i v sin β/ ∆ m i , i.e. , ˜ c H H ( T ) . m ζ i inthe potential Eq. (96). And the AD fields begin to oscillate around the potential minima h ζ i i ≃ m ζ i ) with H ( T ) = H osc when the Hubble parameter H ( T ) of the universebecomes comparable to the SUSY breaking mass m ζ i . (Hereafter “osc” labels the epochwhen the coherent oscillations commence.) Then the interactions of dimension-5 operatorscreate lepton number.Now we see how the lepton number is created. At the beginning of the oscillation, theAD fields have the initial values | ζ i ( t osc ) | ≃ (cid:16) 43 ˜ c H (cid:17) / (cid:16) m ζ i m i ∆ m i v sin β (cid:17) / ≪ M P . (100)in which m ζ i ≃ H osc is used. The evolution of the AD fields ζ i after H ≃ H osc is described ina Friedmann-Robertson-Walker(FRW) universe by the equation of motion with the potential V ( ζ i ) as ¨ ζ i + 3 H ( T ) ˙ ζ i + ∂V ( ζ i ) ∂ζ ∗ i ≃ , (101)where H ( T ) = ( π g ∗ ( T ) / M P ) / T ≈ . p π g ∗ ( T ) T /M P is the Hubble rate for aradiation-dominated era with the total number of effective degrees of freedom g ∗ ( T ) at atemperature T [69], ∂V ( ζ i ) /∂ζ ∗ i ≃ m ζ i ζ i , and dot indicates time derivative. It is clear thatthe AD fields ζ i oscillate around the origin ( h ζ i i ≃ 0, the VEVs of ζ i deviate from zero dueto the SUGRA effect) and the amplitude of the oscillation damps as | ζ i | ∝ H ∝ t − .Since the AD fields ζ i carry lepton number, the baryon number asymmetry will be createdduring coherent oscillation of the AD fields. The number density of the AD fields is related40o the lepton number density n L i as n L i = i ( ∂ζ ∗ i ∂t ζ i − ζ ∗ i ∂ζ i ∂t ), then from Eq. (101) the evolutionof n L i are given by ∂n L i ∂t + 3 H n L i − m / M i Im( a m ζ i ) − H M i Im( a H ζ i ) ≃ . (102)Since the Hubble parameter H ( T ) decreases as temperature decreases, the relative phasebetween a m and a H changes with time when the AD fields ζ i trace the valleys determinedmainly by the Hubble A -term . And during their rolling towards the true minima, thecontribution of Im( a H ζ i ) is suppressed compared with Im( a m ζ i ). Then the motion of ζ i inthe angular direction generating lepton number is expressed as ∂n L i ∂t + 3 H n L i ≃ m / M i Im( a m ζ i ) , (103)where H = ˙ R ( t ) /R ( t ), and R ( t ) stands for the scale factor of the expansion universe withcosmic time t . The produced lepton number asymmetry at a time t can be obtained byintegrating the above equation ∂ ( R n L i ) /∂t ≃ m / M i R Im( a m ζ i ) where R = R ( t ). Afterthe end of inflation, the inflaton field Ψ ′ and waterfall field ˜Ψ ′ begin to oscillate aroundthe potential minimum such that the universe is effectively matter dominated, which scalesas R ∝ H − ∝ t . And before the beginning of the ζ i oscillation, due to | ζ i | ∝ H / ∝ t − / , the net lepton number generated keeps constant for the period t < t osc . Duringmatter dominated epoch the Hubble parameter is related to the expansion time by H osc =(2 / t − . Then using Eq. (100) the generated lepton number at this stage ( t = t osc ) is givenapproximately by n L i ( t osc ) ≃ ˜ c H m i v sin β ∆ m i ( m / | a m | ) H osc δ eff , (104)where δ eff ≃ sin(4 arg ζ i + arg a m ) represents an effective CP violating phase. It is expectedthat the production of net lepton asymmetry occurs before the reheating process completes, i.e. , Γ all = Γ Ψ + Γ ˜Ψ < H osc , c.f., see Eq. (125) ; the production of lepton number is stronglysuppressed after the AD fields ζ i start their oscillations, because Im( a m ζ i ) change theirsign rapidly due to the oscillation of ζ i as well as the amplitude of ζ i oscillation is dampedwith expansion (see below Eq. (101)). Thus after inflation R n L i (cid:12)(cid:12) t = t osc = R n L i (cid:12)(cid:12) t = t R ∼ If there are no true minima, i.e. m / = 0, the AD fields get eternally trapped in the minima Eq. (98)and there is no motion of ζ i changing with time along the angular direction, leading to no lepton numberproduction. L i ( t R ) /ρ rad ( t R ) stays constant until the inflaton Ψ ′ and waterfall field ˜Ψ ′ decays into lightparticles. Here ρ rad ( t R ) = 3 M P Γ is the energy density of the inflaton. Then the generatedlepton number when the reheating process completes ( t = t R , H ≃ Γ all ) is given by n L i ( t R ) = n L i ( t osc ) (cid:18) Γ Ψ H osc (cid:19) . (105)The inflaton decays reheats the universe producing entropy s of radiation such that ρ rad ( t R ) = 3 T reh s ( t R ) / 4. Then the lepton number asymmetry is approximately expressed as n L i ( t R ) s = ˜ c H m i v sin βM P ∆ m i T reh (cid:18) m / | a m | H osc (cid:19) δ eff (106)when the reheating process of inflaton completes. Later, we will discuss the reheatingtemperature, see Sec. IV B, and its related gravitino problem, see Sec. IV A. Recalling thatthe H osc depends on M i as H osc ≃ m ζ i . Since M i is directly related to the pseudo-Dirac masssplittings δ i as M i = h H u i /δ i in Eq. (12) in addition to O ( δ ) ≃ O ( δ ) ≃ O ( δ ) = O ( m )in Eq. (90), there are three flat directions corresponding to the almost degenerate neutrinopairs ξ i in Eq. (17), i.e. , the three generation AD fields ζ i / √ e L i = H u with i = 1 , , A × U (1) X × SUSYis broken but the SM gauge group remains unbroken. So the baryon number produced isthermalized in a hot plasma into real baryons at a relatively low temperature. Therefore,the present baryon asymmetry can be expressed by n B s ≃ . X i =1 , , n L i s ≃ . × − × P i =1 m i ∆ m i . × eV − × (cid:18) δ eff . (cid:19) × (cid:18) T reh 70 TeV (cid:19) (cid:18) m / | a m | H osc (cid:19) . (107)where n B is the baryon number density and s is the entropy density, and we have usedsin β ≃ c H ≃ 1. Considering | a m | ≃ H osc ≃ m / ≃ m ζ i , the resultantbaryon asymmetry only depends on the neutrino parameters m i and ∆ m i , the reheatingtemperature T reh , and an effective CP violating phase δ eff . Quantitatively, the value of BAUis inferred from the two observations, m i ( ≃ m ν i ) and ∆ m i , independently: from Eqs. (14),(15), and (98) the following quantity could be extracted as10 eV − . X i m ν i ∆ m i = 12 (cid:18) δ + 1 δ + 1 δ (cid:19) . × eV − , (108)42n which the upper bound is derived from an initial condition of the AD fields in Eq. (98);the lower bound comes from the neutrino data in Eqs. (14) and (15). In terms of Y ∆ B ≡ ( n B − n ¯ B ) /s | today (which is conserved throughout the thermal evolution of the universe) theBBN results [70] and the CMB measurement [13] read at 95% CL Y BBN∆ B = (8 . ± . × − , Y CMB∆ B = (8 . ± . × − . (109)Taking into account δ eff ≃ O (0 . 1) and P i m ν i / ∆ m i ≃ O (10 ) eV − , for the baryon asymme-try in Eq. (107) to satisfy the BBN results and CMB measurement the reheating temperatureshould be T reh ∼ O (70) TeV . (110)Later, we will show that the bound of Eq. (110) could be consistent with the bound fromthat predicted from Sec. IV B. A. Gravitino production It is well known that thermal leptogensis in supersymmetric framework, which is oneof attractive mechanism for origin of matter, requires a large reheating temperature in theearly universe, T reh ∼ M > GeV, where M is a lightest heavy neutrino mass. Thegravitino, which appears in all models with local supersymmetry, is the superpartner of thegraviton. Gravitino is produced thermally [71] or non-thermally [72–76] in the cosmologicalhistory. The excessive production of gravitinos in the early universe may destroy the nu-cleosynthesis of the light elements for unstable gravitinos or overclose of universe for stablegravitinos [77]. Since the gravitino is present in the supersymmetric model, we are going toaddress (unstable) gravitino overabundance problem.After the inflation ends, the inflaton Ψ ′ and waterfall field ˜Ψ ′ release their energy into athermal plasma by the decays, and the universe is reheated. Since all the particles includingphotons and baryons in the present universe are ultimately originated from the decays, itis crucial to reveal how the reheating proceeds. For simplicity, hereafter we treat the mixedmass eigenstates as the single field eigenstates,Ψ ′ → Ψ , Ψ ′ → Ψ , ˜Ψ ′ → ˜Ψ . (111)43s mentioned in the introduction, there are two secluded SUSY breaking sectors, i.e. ,SUSY=SUSY inf × SUSY vis . Gravitational interactions explicitly break the SUSY down to true SUSY inf × SUSY vis , where SUSY inf corresponds to the genuine SUGRA symmetry, whilethe orthogonal SUSY vis is approximate global symmetry. In each sector, spontaneous break-down of F -term occurs at a scale F i ( i = inf, vis) independently, producing a correspondinggoldstino. Hence, in the presence of SUGRA, the SUSY inf is gauged and thus its corre-sponding goldstino is eaten by the gravitino via super-Higgs mechanism, leaving behind theapproximate global symmetry SUSY vis which is explicitly broken by SUGRA and thus itscorresponding the uneaten goldstino as a propagating degree of freedom.During inflation and the beginning of reheating (preheating) when SUSY is spontaneouslybroken there are possible productions of fermonic quanta which are strongly coupled to theinflaton field. During this stage the SUSY inf is mainly broken by the inflaton implying thatthe goldstino produced is mainly inflatino (instead of the gravitino in the low energy); thegravitino produced non-thermally is effectively massless as long as the Hubble parameteris larger than the gravitino mass, H > m / [75]. However, this correspondence does notnecessarily hold at late times, since the SUSY vis is broken by other fields in the true vacuum.In SUGRA framework, with the linear Kahler potential in Eq. (28) the inflaton field Ψ has a non-vanishing auxiliary field G Ψ . Such non-vanishing auxiliary field allows the inflatondecay into a pair of the gravitinos, whose decay process is crucial in the reheating process [73].The constraint on the inflaton potential G Ψ depending on the gravitino mass must besatisfied to avoid an overproduction of the gravitino keeping the success of the standardcosmology. In the unitary gauge in the Einstein frame, the goldstino (the longitudinalcomponent of the gravitino) can be gauged away through the super-Higgs mechanism leadingto vanishing of the gravitino-goldstino mixing. Then the relevant interactions for the inflatondecay into a pair of gravitinos reads [79] − e − L = 18 ǫ µνρσ ( G Ψ ∂ ρ Ψ − G ¯Ψ ∂ ρ Ψ ∗ ) ¯ ψ µ γ ν ψ σ + e G/ M P ( G Ψ Ψ + G ¯Ψ Ψ ∗ ) ¯ ψ µ [ γ µ , γ ν ] ψ ν (112) The inflatinos produced during inflation and preheating may be partially converted to the gravitinos in thelow energy, since G Ψ is generically non-zero in the true minimum [78]. At this stage, since the inflationarysector and the sector responsible for the low energy effective SUSY breaking are distinct, the gravitinosgenerated non-thermally are produced with a sufficiently low abundance. ψ µ is the gravitino field. The real and imaginary components of the inflaton field havethe same decay rate at leading order [74]Γ / ≡ Γ(Ψ → ψ / + ψ / ) ≃ π M P K Ψ ¯Ψ (cid:12)(cid:12) h G Ψ i (cid:12)(cid:12) (cid:16) m Ψ M P (cid:17) (cid:16) m Ψ m / (cid:17) m Ψ (113)in the limit of m Ψ ≫ m / after canonical normalization ˆΨ = p K Ψ ¯Ψ Ψ . The decay rate isenhanced by the gravitino mass in the denominator, which comes from the goldstino (mainlyas the inflatino) in the massless limit. The decay into the gravitinos only proceeds at thestage H < m / , when the SUSY breaking contribution of the inflaton is subdominant [73].Thus, the gravitinos produced at the reheating epoch by the inflaton decay through theinteraction (112) should coincide with those in the low energy.Now, we estimate how much the gravitinos are produced at the reheating epoch. Afterthe inflation ends both the inflaton Ψ and waterfall field ˜Ψ oscillate around the potentialminimum and dominate the universe until the reheating. We express the superpotential (27)relevantly W ⊃ W ( z ) + ˜ g Ψ ( ˜ΨΨ − µ ) (114)where W ( z ) is introduced to determine SUSY breaking scale, see Sec. III B, and ˜ g = g / (2 σ ) corrected by the string moduli backreaction. Then the scalar potential in Eq. (22)is extremized in the true vacuum if h ∂ i V i = 0, and the resulting cosmological constantshould vanish if h V i = 0. Together with, these conditions are satisfied if h G α G α i = 3 , h G α ∇ k G α + G k i = 0 . (115)Then the condition of the potential minimum read h M P { G Ψ Ψ G ¯Ψ + G ΨΨ G ¯Ψ + G ˜ΨΨ G ¯˜Ψ + G z Ψ G ¯ z } + G Ψ i = 0 , (116) h M P { G ΨΨ G ¯Ψ + G Ψ Ψ G ¯Ψ + G ˜ΨΨ G ¯˜Ψ + G z Ψ G ¯ z } + G Ψ i = 0 , (117)and the minimization condition for ˜Ψ is the same as for Ψ. The inflaton mass ( ≫ m / ),after the inflation, is given by m Ψ ≃ (cid:12)(cid:12) M P h e G ∇ Ψ G ˜Ψ ∇ Ψ G ˜Ψ i (cid:12)(cid:12) ≃ | ˜ g | µ Ψ ( t I ) , (118)where ∇ Ψ G ˜Ψ ≃ W ˜ΨΨ /W is used, which is almost equal to the mass of waterfall field ˜Ψ.This inflaton mass is in agreement with Eq. (59). Since the z field is responsible for the SUSY45reaking, one obtains | G z | ≃ √ /M P , and in turn the gravitino mass m / ≡ h M P e G/ i ≃| W | /M P ≃ | W z | / √ M P . Assuming | G Ψ | ≃ | G ˜Ψ | . | Ψ | /M P , one obtains G Ψ ≃ W Ψ /W ,leading to W Ψ /W ≃ Ψ /M P and W ˜Ψ /W ≃ ˜Ψ /M P . Using W Ψ = ˜ g Ψ ˜Ψ in Eq. (27) we obtain h Ψ i ≃ m / | ˜ g | . (119)Using | G Ψ | . | Ψ | /M P one obtains W Ψ /W ≃ Ψ /M P . Inserting G Ψ Ψ = − W /W , G ΨΨ ≃ − Ψ W Ψ / ( W M P ) ± ˜ g ˜Ψ / ( m / M P ), and G z Ψ ≃ √ W Ψ / ( W M P ) into Eqs. (116)and (117) h G Ψ i ∼ h Ψ i M P ≃ m / | ˜ g | M P , h G Ψ i ∼ m / | ˜ g | h Ψ i M P , (120)which indicates h G Ψ i is much larger than h G Ψ i . Then, from Eqs. (113) and (118) theinflaton decay width is roughly given byΓ / ≃ π (cid:16) m Ψ M P (cid:17) (cid:16) µ Ψ ( t I ) M P (cid:17) m Ψ . (121)At the reheating epoch, gravitinos are produced by the non-thermal inflaton decay process( Y Ψ / : the yield of the gravitinos by the inflaton decay) as well as by the thermal scattering( Y th3 / : the yield of the gravitinos produced by thermal scatterings); the ratio of gravitino-to-entropy density is given by Y / = Y Ψ / + Y th3 / , which remains constant as the universeexpands as long as there is no additional entropy production. Gravitinos thermally pro-duced in the early universe, predominantly via 2 → th3 / h , and contribution to the energy density, Y th3 / , growwith the reheating temperature after inflation, the yield of the gravitinos thermally pro-duced is estimated as Y th3 / ∼ − ( T reh / Y Ψ / ≫ Y th3 / by Y / ≃ Y Ψ / . (122)Here the gravitino yield produced by the inflaton decay process Ψ → Ψ / + Ψ / via theinteraction Eq. (112) is Y Ψ / ≡ n Ψ / s ≃ / Γ Ψ T reh m Ψ , (123) The production of gravitinos after inflation has been studied in some detail [81]. n Ψ / is the number density of gravitinos by the inflaton decay, and s =(2 π / g ∗ s ( T ) T is the entropy density with g ∗ s ( T ) being the effective number of the mass-less degrees of freedom at the temperature T .The gravitino yield is severely constrained by BBN, Y / < Y BBN3 / , in order to keep thesuccess of the standard scenario of BBN [81]. Otherwise, the decay products of the grav-itino would change the abundances of primordial light elements too much and consequentlyconflict with the observational data. Refs. [18, 82] shows that, when the hadronic branch-ing ratio of the gravitino decay is of order unity, Y BBN3 / ∼ − for m / ∼ Y BBN3 / ∼ − − for m / ∼ 10 TeV; for m / & 100 TeV the constraint disappears. Onthe other hand, in the context of supersymmetric moduli stabilization where moduli arestrongly stabilized, at tree level the gaugino masses and A -terms are strongly suppressed by m / /m T and as such effectively vanish [64], while the dominant contributions to the gaug-ino masses and A terms arise from loop corrections [63]: m / = b a g a / (16 π )( F C /C ) and A ijk = − ( γ ijk / π )( F C /C ) where b a = 11 , , − a = 1 , , γ ijk are the anomalous dimensions of the matter fields, and F C /C ∼ m / .Thus, in order to have suitably large gaugino masses, relatively large O (100) TeV gravitinomasses must be considered [64]. The relic abundance of nuetralino LSP (lightest supersym-metric particle) as dark matter will be considered in future work, see also Ref. [4].In order to estimate Y / we have to calculate the decay width of the inflaton, Γ Ψ , atreheating epoch. B. Reheating temperature Since inflation leaves the early universe cold and empty, the inflaton Ψ and waterfallfield ˜Ψ where all energy resides in must transfer their energy to a radiation dominatedplasma in local thermodynamic equilibrium at a temperature sufficient to allow standardnucleosynthesis T reh > T (BBN). So the universe must be reheated after inflation. Theenergy of the inflaton Ψ and waterfall field ˜Ψ are transferred to the SM sector throughtheir gravitational and/or non-gravitational decays once their fields acquire finite VEVs,which in turn produce SM matter. Their decay products thermalize.We are in the case where the inflaton Ψ and waterfall field ˜Ψ dominate the energy of theuniverse when they decay. The reheating temperature T reh resulting from the perturbative47ecays of the inflaton Ψ and waterfall field ˜Ψ may be estimated by using the relationΓ all = 3 H ( T reh ) (124)at the end of the reheating process, where the Hubble parameter H ( T ) is given in theradiation dominated era of the universe. Inflaton Ψ and waterfall field ˜Ψ decays reheat theuniverse, when Γ all & H ( T reh ): T reh = (cid:18) π g ∗ (cid:19) / p Γ all M P , with Γ all = Γ sugraΨ + Γ sugra˜Ψ + Γ visΨ + Γ vis˜Ψ (125)where g ∗ ( T ) is the number of the relativistic degrees of freedom in the plasma , and Γ sugraΨ +Γ sugra˜Ψ and Γ visΨ +Γ vis˜Ψ stand for gravitational and non-gravitational decay widths, respectively.Later, we will see that it is too weak to cause the reheating with gravity in the model eventhe gravitational coupling is universal.As in Ref. [4] (see Eq. (4) for lepton sector), in the supersymmetric visible sector theinflaton Ψ and waterfall field ˜Ψ couple to the SM particles via the following interactionsdominantly W ⊃ g Ψ Ψ H u H d + ˆ y c (cid:16) ˜ΨΛ (cid:17) Q c c H u (126)where g Ψ is a real and positive coupling constant, while the hat Yukawa coupling ˆ y c is oforder unity complex number. Here Q is the second generation left handed quark doublet,which transforms as ′′ under A symmetry; the right handed charm quark c c ∼ ′ under A . The first term is also associated with the µ -term in Eq. (4) since the VEV of Ψ is givenby h Ψ i ∼ m / / | ˜ g | . And so the inflaton with a non-zero VEV can decay into the visiblesector through the non-gravitational coupling of the inflaton to matter with the decay rateΓ visΨ = Γ(Ψ → → ≃ × | g Ψ | π m Ψ , (127) The energy transfer from the inflaton and waterfall field to the SM fields in general proceeds both throughnon-perturbative effects and perturbative decays [83] We estimate the total number of effectively massless degree of freedom of the radiation, g ∗ ( T ), at tem-perature of the order of the decay rate of the inflaton Γ Ψ , i.e., there are 17 bosons and 48 Weyl fermionsfor T EW < T < m / : g ∗ ( T ) = P j =bosons g j ( T j /T ) + (7 / P j =fermions g j ( T j /T ) = 34 + (7 / T j denotes the effective temperature of any species j . H u , withoutloss of generality, as˜Ψ = 1 √ (cid:16) v ˜Ψ + h ˜Ψ √ − i φ Ψ √ (cid:17) , H u = v u + h u √ , (128)the second term in Eq. (126) is expressed in terms of Lagrangian form as −L = ˆ y c (cid:16) v ˜Ψ √ (cid:17) v u n h u √ v u + √ v ˜Ψ ( h ˜Ψ − iφ Ψ ) o ¯ c L c R + h . c .. (129)Here the waterfall field ˜Ψ with a non-zero VEV can decay into the visible sector throughthe non-gravitational coupling of the waterfall field ˜Ψ to matter with the decay rateΓ vis˜Ψ ≃ Γ( ˜Ψ → c ¯ c ) ≃ | ˆ y c | π (cid:16) v ˜Ψ √ (cid:17) (cid:16) v u v ˜Ψ (cid:17) m ˜Ψ = | g ˜Ψ | π m ˜Ψ , (130)where g ˜Ψ ≡ ˆ y c ( v ˜Ψ / √ ( v u /v ˜Ψ ), and the mass of the final-state compared to that of thewaterfall field ˜Ψ is neglected. Using | ˆ y c | ≃ v ˜Ψ / √ 2Λ = λ/ √ v u /v ˜Ψ ≃ − where λ ≈ . β ≃ v ˜Ψ ≈ . × GeV [4], we obtain | g ˜Ψ | ≃ . × − . (131)Putting Eqs. (130) and (127) into Eq. (125), the reheating temperature can be expressed as T reh ≃ (cid:18) π g ∗ (cid:19) / q m Ψ M P ( | g Ψ | + | g ˜Ψ | ) . (132)Since there is no information on the size of the renormalizable superpotential coupling g Ψ of the inflaton to the Higgses and Higgssinos, first we consider the case of Γ reh ≃ Γ visΨ ≫ Γ vis˜Ψ + Γ sugra˜Ψ + Γ sugraΨ . In this case, that is, g Ψ ≫ | g ˜Ψ | , the size of the Higgs-inflaton couplingcan severely restrict the lower limit on T reh in Eq. (132) as T reh & TeV (cid:16) g Ψ − (cid:17) (cid:18) ˜ g . × − (cid:19) / (cid:18) µ Ψ ( t I )6 . × GeV (cid:19) / (133)where we have used m Ψ = | ˜ g | µ Ψ ( t I ) in Eqs. (59) and (118). This lower limit on T reh is conflict with the limit for the successful leptogenesis in Eqs. (107) and (110). Hence49e conclude that | g ˜Ψ | & g Ψ for Γ reh ≃ Γ visΨ + Γ vis˜Ψ ≫ Γ sugra˜Ψ + Γ sugraΨ ; then the reheatingtemperature is in a good approximation given by T reh ≃ (59 − µ -term sincethe coupling g Ψ ( ≪ ˜ g ) should be enormously suppressed for a successful leptogenesis to besatisfied, see Eqs. (107) and (134).Next, we consider the gravitational effects on the reheating temperature. For example,the inflaton Ψ with a non-zero VEV can also decay into the visible sector through theSUGRA effects [72]. Then the reheating can be induced by the inflaton decay through non-renormalizable interactions. The relevant interactions for the matter-fermion production areprovided in the Einstein frame as [79] e − L = i K ij ∗ ¯ χ j γ µ ∂ µ χ i + i M P K ij ∗ ( K σ ∂ µ φ σ − K σ ∗ ∂ µ φ ∗ σ ) ¯ χ j γ µ χ i − i M P K ij ∗ Γ iσρ ( ∂ µ φ σ ) ¯ χ j γ µ χ σ + 12 e K/ M P ( D i D j W ) χ i χ j + h.c. (135)where D i D j W = W ij + K ij M P W + K i M P D j W + K j M P D i W − K i K j M P W − Γ kij M P D k W . Here φ i and χ i stand for the matter fields, and φ i collectively denotes on arbitrary filed including theinflaton Ψ . And the matter-scalar production is represented by the kinetic term and thescalar potential − e − L = iK ij ∗ ∂ µ φ i ∂ µ φ ∗ j + e K/M P (cid:26) K ij ∗ ( D i W )( D ¯ j ¯ W ) − M P | W | (cid:27) . (136)In the model superpotential the supersymmetric visible sector contains the followingrenormalizable interactions W ⊃ y t Q t c H u + 12 M R N c N c , (137)where the first term is the top quark operator as in [1] and the second term comes fromEq. (5) after the U (1) X is spontaneously broken. The partial decay width of the inflatonthrough the neutrino Yukawa coupling is [72]Γ N (sugra)Ψ = Γ(Ψ → N c N c ) + Γ(Ψ → ˜ N c ˜ N c ) ≃ × c N π m Ψ (cid:18) − M m (cid:19) / , (138)50here c N ≃ e K/M P (cid:12)(cid:12)(cid:12) K Ψ0 M P W N c N c − k Ψ N c W Nck M P (cid:12)(cid:12)(cid:12) ; (sum over k ) and the heavy neutrino mass M given in Eq. (A19). For the minimal Kahler potential, for simplicity, using Eq. (119) theparameter c N can be approximately given by c N ≃ (cid:18) h Ψ i M P (cid:19) (cid:18) MM P (cid:19) = (cid:18) m / m Ψ (cid:19) (cid:18) µ Ψ ( t I ) M P (cid:19) (cid:18) MM P (cid:19) (139)where in the last equality the inflaton mass m Ψ in Eq. (59) or Eq. (118) is used. And thepartial decay width of the inflaton through the top quark Yukawa coupling is [72]Γ t (sugra)Ψ = Γ(Ψ → → ≃ c t π (cid:18) m Ψ M P (cid:19) m Ψ , (140)where the masses of the final state particles are neglected, the additional numerical factorcomes from SU (3) × SU (2), and c t ≃ e K/M P (cid:12)(cid:12)(cid:12) K Ψ0 M P W t c Q H u − ℓ Ψ H u W t c Q ℓ (cid:12)(cid:12)(cid:12) ; (sum over ℓ ).Similarly, the parameter c t is approximately given by c t ≃ (cid:18) h Ψ i M P (cid:19) | y t | = (cid:18) m / m Ψ (cid:19) (cid:18) µ Ψ ( t I ) M P (cid:19) | y t | . (141)In addition, the decay rate into the visible sector through the top and neutrino Yukawa cou-plings is much larger than that into the gluons and gluoinos via the anomalies of SUGRA [72].Then, from Eqs. (138) and (140) the inflaton decay rate through the gravitational couplingof the inflaton to matter is approximately given byΓ sugraΨ ≃ Γ t (sugra)Ψ + Γ N (sugra)Ψ ≃ m Ψ π (cid:16) m / m Ψ (cid:17) (cid:16) µ Ψ ( t I ) M P (cid:17) n | y t | π (cid:16) m Ψ M P (cid:17) + (cid:16) MM P (cid:17) (cid:16) − M m (cid:17) o . (142)Given that m Ψ ∼ GeV, µ Ψ ( t I ) ∼ GeV, M ∼ GeV, and m / ∼ O (100)TeV, we clearly have Γ visΨ ≫ Γ sugraΨ for g Ψ ∼ | g ˜Ψ | , and the total decay rate of the inflatonfield in Eq. (124) is approximately given byΓ Ψ ≃ Γ visΨ (143)which is much larger than Γ / in Eq. (121).Inserting Eqs. (121) and (143) into Eq. (122), the production of the gravitinos, dependingon the size of the Higgs-inflaton coupling, has a lower bound Y / ≃ . × − (cid:16) . × − g Ψ (cid:17) (cid:16) T reh 70 TeV (cid:17)(cid:16) | ˜ g | . × − (cid:17) (cid:16) µ Ψ ( t I )6 . × GeV (cid:17) . (144)51ince the total yield Y / ≃ Y Ψ / is inversely proportional to | g Ψ | and proportional to T reh ( Y th3 / is also proportional to T reh ), it can provide a lower bound on the size of the Higgs-inflaton coupling, | g Ψ | , with the given reheating temperature for the successful leptogenesis;2 . × − . | g Ψ | . | g ˜Ψ | ≃ . × − . (145)We conclude that it is reasonable for the reheating temperature Eq. (110) derived from thesuccessful leptogenesis to lie in the range T reh ≃ (59 − 84) TeV in Eq. (134). The above yieldof gravitino Y / in Eq. (144) with the reheating temperature for the successful letogenesisis well constrained by the BBN constraints Y BBN3 / in Ref. [18]. C. Dynamics of the waterfall fields and axino fields after inflation Finally, we roughly describe the dynamics of the waterfall (PQ) fields after inflation.After the inflation ends, the inflaton Ψ and waterfall field ˜Ψ start to oscillate and theirdecays produce a dilute thermal plasma formed by collisions of relativistic decay products.During the epoch when the energy density of the universe is still dominated by the oscillatinginflaton and waterfall field, their oscillations behave as matter, so their amplitudes | Ψ | and | ˜Ψ | decrease proportional to H ∝ /t ∝ R − / . As described Sec. III D, the waterfall field Ψquickly rolls down to the flat direction (see Eq. (60)), ˜ΨΨ = µ ; actually, the flat directionis not flat at this stage, because the waterfall field Ψ obtain mass of m Ψ ≃ | ˜ g || Ψ | beforethe inflaton Ψ and waterfall field ˜Ψ decay, see Eq. (59). Thus, the waterfall fields arestabilized at h Ψ i = h ˜Ψ i = µ Ψ at this epoch and both fields oscillate around µ Ψ . Since thegravitino mass is larger than | ˜ g || Ψ | , i.e. m / > | ˜ g || Ψ | , the saxion h Ψ begins to movetoward the true minimum; in addition, due to already H ( T ) < m / , and hence the frictionis not efficient, so the saxion adiabatically approaches to the true minima in Ref. [4] withoutoscillation. And also, even the inflaton Ψ can decay into axinos, but its process can not beused as a reheating process since the produced axinos could not thermalize. In the followingwe will see this is the case.In the gravity-mediated scenario, the axino mass is likely to be greater than the grav-itino mass [86]. Since the gravitino mass serves as the order parameter for the spontaneousbreaking of SUGRA when the cosmological constant is zero, one can estimate the axinomass. The goldstino field overlaps with a chiral superfield C i which acquires a VEV along52calar h c i i and auxiliary component F i . In the decoupling limit of SUSY inf × SUSY vis , when C i acquires a VEV equal to F i , SUSY is spontaneously broken and thus there should ex-ist a corresponding massless fermion, goldstino. The goldstino superfield C i is non-linearlyparameterized [86, 87] as C i = e Qη i / √ F i ( c i + ϑ F i )= c i + η i F i + √ ϑη i + ϑ F i , (146)where Q = ∂/∂ϑ is the generator of SUSY transformations, ϑ is a Grassmann variable, andall derivative couplings are neglected; η i is the goldstino associated with the F -term breakingof SUSY i ( i = f inf , f vis in the absence of direct gravitational couplings). The goldstinois electrically neutral, R = − 1, Majorana chiral fermion. When SUSY is broken in thepresence of SUGRA, the SUSY inf corresponding to the genuine SUGRA symmetry is gaugedand thus its corresponding goldstino χ (one linear combination of the fermionic componentof the chiral superfield C i ) is eaten by the gravitino via the super-Higgs mechanism, leavingbehind the approximate global symmetry SUSY vis which is explicitly broken by SUGRA andthus its corresponding the uneaten goldstino η as a propagating degree of freedom. Herethe physical state χ (= η f inf cos θ + η f vis sin θ ) and η (= − η f inf sin θ + η f vis cos θ ) could be linearcombinations of goldstinos, η f inf and η f vis , generated in each secluded sector. The interactionstate η ′ i = ( η f inf , η f vis ) T can be expressed in terms of the physical state η a = ( χ, η ) T with a2 × V ai , i.e. η ′ i = V ai η a .In the unitary gauge where all terms proportional to χ = G i η i vanish identically, theremaining fermions have a quadratic Lagrangian in the interaction basis ( η ′ , η ′ ) ≡ ( η f inf , η f vis ) − e − L = iK i ¯ j ¯ η ′ ¯ j ¯ σ µ D µ η ′ i + 12 m ′ ij η ′ i η ′ j + 12 m ′ ¯ i ¯ j ¯ η ′ ¯ i ¯ η ′ ¯ j + m / (cid:16) ψ µ σ µν ψ ν + ¯ ψ µ ¯ σ µν ¯ ψ ν − ǫ µνρσ ¯ ψ µ ¯ σ ν ˜ D ρ ψ σ (cid:17) , (147)where D µ and ˜ D ρ are general covariant derivatives and m ′ ij = m / M P {∇ i G j + G i G j / } isthe uneaten goldstino mass matrix [79]. The interaction state η ′ i could be expressed in termsof the physical state η with a relation tan θ = F f vis /F f inf : η ′ = − η sin θ and η ′ = η cos θ . Sincethe direction χ corresponding to the eaten goldstino has a zero mass eigenvalue, in a basiswhere χ couples only derivatively its Lagrangian can be written as L eff = − χ∂ µ ˜ J µ / | F z | + h . c . Note here the notation χ is different from the one used in Eq. (135). J µ is the supercurrent. And from Eqs. (43) and (114) since the z field is responsiblefor the SUSY breaking, from the condition for the vanishing cosmological constant (andhence flat space), i . e ., h G z G z i = 3, we obtain | G z | ≃ √ /M P (see above Eq. (71)), leadingto the effective SUSY breaking scale set by p | F z || F z | = q F f inf + F f vis = p | V AdS |≃ √ M P m / . (148)Then the Lagrangian (147) for goldstino can be expressed in terms of the physical state η − e − L = i ˜ K ¯ ab ¯ η ¯ a ¯ σ µ D µ η b − m ab η a η b − m † ¯ a ¯ b ¯ η ¯ a ¯ η ¯ b , (149)where ˜ K ¯ ab = [ V † ¯ j ¯ a K i ¯ j V ib ] is the Kahler metric with the true goldstino direction removed, and m ab = [ V T m ′ V ] ab . The remaining uneaten goldstino mass as axino mass can be determinedby the physical mass-squared matrix m a ¯ b = m ¯ ℓa m ¯ b ¯ ℓ , (150)with m ¯ ℓa = m ak G k ¯ ℓ = m / G k ¯ ℓ h∇ a G k + G a G k i and m ¯ b ¯ ℓ = m / M P h∇ ¯ b G ¯ ℓ + G ¯ b G ¯ ℓ i inRef. [79]. The condition for the potential minimum in Eq. (115) read h G z (cid:16) X za − W z W a W (cid:17) + G a i = 0 , (151)where X za = K za /M P + W za /W − Γ jaz G j . Using G z ≃ W z /W, G a ≃ W a /W , and h G z G z i = 3in Eq. (115), we obtain X za ≃ G a G z , ∇ a G z ≃ G a M P K z ¯ α G ¯ α − G z G a , (152)and consequently, m ¯ ℓa = m / h δ ¯ ℓa − M P K z ¯ ℓ G z G a i . Thus the mass matrix m ab can bewritten as m ab = m ¯ ℓa δ b ¯ ℓ ≃ m / h δ ab − G a G ¯ ℓ δ b ¯ ℓ i . (153)Since the uneaten goldstino η is orthogonal to χ = G a η a , the second term in the bracket isirrelevant. We obtain that the axino mass is equivalent to m ˜ a ≃ m / as in Ref. [86].Next we describe that, after the inflaton Ψ and waterfall field ˜Ψ decays, how the waterfallfields Ψ and ˜Ψ could remain trapped in the true minima. The inflaton Ψ and waterfall54eld ˜Ψ decays thermalize the universe and the decay products interact among others in thethermal bath. In the model, the waterfall fields Ψ and ˜Ψ interact with the SM particlesthrough the Yukawa couplings and QCD couplings. The PQ field Ψ decays as a result ofscattering with the thermalized decay products of the inflaton Ψ and waterfall field ˜Ψ. Andthe temperature of this dilute plasma behaves roughly as [69] T ≃ (cid:0) ( T reh ) H M P (cid:1) . (154)Then, the effective potential for ˜Ψ induced by thermal plasma (similarly for Ψ) is onlyprovided by [84, 85] V th = a g α s T ln | ˜Ψ | T ! , (155)where a g is a constant and α s ≡ g s / π , which lifts up the flat direction Ψ ˜Ψ = µ . Then weobtain an effective thermal mass for | ˜Ψ | m = 12 ∂ V th ( ∂ | Ψ | ) = α s T | Ψ | . (156)The evolution of Ψ is now described by the equation of motion¨Ψ + 3 H ( T ) ˙Ψ + ∂V tot ∂ ˜Ψ ∗ ≃ , (157)where V tot = V ( t I ) + V th . After the reheating process finishes, both Ψ and ˜Ψ fields stay at µ Ψ . Since the thermal mass is larger than the Hubble parameter, m th > H ( T ), Ψ rolls downthe thermal potential and | Ψ | increases until m th ∼ H ( T ) with H osc ≃ ( a g M P ) / ( α s T reh ) / .Then the Ψ stops rolling and gets trapped with the initial amplitude | Ψ | ∼ α s ( T ) M P . Atthis stage the gravitino mass is larger than the thermal mass m th ≃ a / g α / s ( T /M P ) / ,and hence the radial components of the fields Ψ and ˜Ψ are stabilized at the true minima. V. CONCLUSION The model is based on the SM × U (1) X × A symmetry, which is essential for the flavoredPQ axions at low energy. Note that the U (1) X -charged Kahler moduli superfields put theGS anomaly cancellation mechanism into practice. As the U (1) X breaking scales accordingto Ref. [4] are secluded by the Gibbons-Hawking temperature T GH = H I / π , the model is55esigned in a way that gravitational interactions explicitly break supersymmetry (SUSY)down to SUSY inf × SUSY vis , where SUSY inf corresponds to the supergravity symmetry, whilethe orthogonal SUSY vis is approximate global symmetry. Hence, in the presence of SUGRA,the SUSY inf is gauged and thus its corresponding goldstino is eaten by the gravitino viasuper-Higgs mechanism, leaving behind the approximate global symmetry SUSY vis which isexplicitly broken by SUGRA and thus its corresponding the uneaten goldstino as a physicaldegree of freedom giving masses to all the supersymmetric SM superpartners.In order to provide a lucid explanation for inflation we have considered a realistic super-symmetric moduli stabilization. Such moduli stabilization has moduli backreaction effectson the inflationary potential, in particular, the spectral index of inflaton fluctuations. Duringinflation the universe experiences an approximately dS phase with the inflationary Hubbleconstant H I ≃ × GeV. In the present inflation model which provides intriguing links toUV-complete theories like string theory, the PQ scalar fields Ψ( ˜Ψ) play a role of the waterfallfields, that is, the PQ phase transition takes place during inflation such that the PQ scale µ Ψ ( t I ) during inflation is fixed by the amplitude of the primordial curvature perturbationand turns out to be roughly 0 . × GeV. We have found that such moduli stabilizationwith the moduli backreaction effects on the inflationary potential could lead to the energyscale of inflation in a way that the power spectrum of the curvature perturbation and thescalar spectral index are to be well fitted with the Planck 2015 observation [13]. And wehave driven that the inflaton mass during inflation is given by m Ψ = √ H I which is muchlarger than the gravitino mass, and its mass is in agreement with its theory prediction forspectral index with observation.Through the introduction of U (1) X symmetry in a way that the U (1) X -[ gravity ] anomaly-free condition together with the SM flavor structure demands additional sterileneutrinos as well as no axionic domain-wall problem, the additional neutrinos may play acrucial role as a bridge between leptogenesis and new neutrino oscillations along with highenergy cosmic events. We have shown that a successful leptogenesis scenario could be nat-urally implemented through Affleck-Dine mechanism. The pseudo-Dirac mass splittings,which is suggested from new neutrino oscillations along with high energy cosmic events,strongly indicate the existence of lepton-number violation which is a crucial ingredient ofthe present leptogenesis scenario. The resultant baryon asymmetry is constrained by thecosmological observable ( i.e. the sum of active neutrino masses) with the new high energy56eutrino oscillations. In addition, the resultant baryon asymmetry, which crucially dependson the reheating temperature, is suppressed for relatively high reheating temperatures. Wehave shown that the right value of BAU, Y ∆ B ≃ × − prefers a relatively low reheatingtemperature with the well constrained pseudo-Dirac mass splittings responsible for new os-cillations ∆ m i . Moreover, we have shown that it is reasonable for the reheating temperature T reh ≃ (59 − 84) TeV derived from the non-gravitational decays of the inflaton and waterfallfield to be compatible with the required reheating temperature for the successful leptoge-nesis, leading to ∆ m i ∼ − − eV . And we have shown that, even the gravitationalcoupling is universal, it is too weak to cause the reheating in the present model. We havestressed that the present model requires m / ≃ O (100) TeV gravitino mass in order to havesuitable large gaugino masses. 57 ppendix A: The A Group The group A is the symmetry group of the tetrahedron, isomorphic to the finite groupof the even permutations of four objects. The group A has two generators, denoted S and T , satisfying the relations S = T = ( ST ) = 1. In the three-dimensional complexrepresentation, S and T are given by S = 13 − − − , T = ω 00 0 ω . (A1) A has four irreducible representations: one triplet and three singlets , ′ , ′′ . An A singlet a is invariant under the action of S ( Sa = a ), while the action of T produces T a = a for , T a = ωa for ′ , and T a = ω a for ′′ , where ω = e i π/ = − / i √ / A representations decompose into irreduciblerepresentations according to the following multiplication rules: ⊗ = s ⊕ a ⊕ ⊕ ′ ⊕ ′′ , ′ ⊗ ′′ = , ′ ⊗ ′ = ′′ and ′′ ⊗ ′′ = ′ . Explicitly, if ( a , a , a ) and ( b , b , b ) denotetwo A triplets, then we have Eq. (A2).Four irreducible representations are , , ′ , ′′ with ⊗ = s ⊕ a ⊕ ⊕ ′ ⊕ ′′ , and ′ ⊗ ′ = ′′ . The details of the A group are shown in Appendix A. Let ( a , a , a ) and( b , b , b ) denote the basis vectors for two ’s. Then, we have( a ⊗ b ) s = 1 √ a b − a b − a b , a b − a b − a b , a b − a b − a b ) , ( a ⊗ b c ) a = i ( a b − a b , a b − a b , a b − a b ) , ( a ⊗ b ) = a b + a b + a b , ( a ⊗ b ) ′ = a b + a b + a b , ( a ⊗ b ) ′′ = a b + a b + a b . (A2)To make the presentation of our model physically more transparent, we define the T -flavor quantum number T f through the eigenvalues of the operator T , for which T = 1. Indetail, we say that a field f has T -flavor T f = 0, +1, or -1 when it is an eigenfield of the T operator with eigenvalue 1, ω , ω , respectively (in short, with eigenvalue ω T f for T -flavor T f ,considering the cyclical properties of the cubic root of unity ω ). The T -flavor is an additivequantum number modulo 3. We also define the S -flavor-parity through the eigenvalues of58he operator S , which are +1 and -1 since S = 1, and we speak of S -flavor-even and S -flavor-odd fields. For A -singlets, which are all S -flavor-even, the representation has no T -flavor ( T f = 0), the ′ representation has T -flavor T f = +1, and the ′′ representation has T -flavor T f = − 1. Since for A -triplets, the operators S and T do not commute, A -tripletfields cannot simultaneously have a definite T -flavor and a definite S -flavor-parity.The real representation, in which S is diagonal, is obtained through the unitary trans-formation A → A ′ = U ω A U † ω , (A3)where A is any A matrix in the real representation and U ω = 1 √ ω ω ω ω . (A4)We have S ′ = − − , T ′ = . (A5)For reference, an A triplet field with T -flavor eigenfields ( a , a , a ) in the complex repre-sentation can be expressed in terms of components ( a R , a R , a R ) as a R = a + a + a √ , a R = a + ω a + ω a √ , a R = a + ω a + ω a √ . (A6)Inversely, a = a R + a R + a R √ , a = a R + ω a R + ω a R √ , a = a R + ω a R + ω a R √ . (A7)Now, in the S diagonal basis the product rules of two triplets ( a R , a R , a R ) and( b R , b R , b R ) according to ⊗ = s ⊕ a ⊕ ⊕ ′ ⊕ ′′ are as follows( a R ⊗ b R ) s = ( a R b R + a R b R , a R b R + a R b R , a R b R + a R b R ) , ( a R ⊗ b R ) a = ( a R b R − a R b R , a R b R − a R b R , a R b R − a R b R ) , ( a R ⊗ b R ) = a R b R + a R b R + a R b R , ( a R ⊗ b R ) ′ = a R b R + ω a R b R + ω a R b R , ( a R ⊗ b R ) ′′ = a R b R + ω a R b R + ω a R b R . (A8)59 . Lepton mass matrices The model implicitly has two U (1) X ≡ U (1) X × U (1) X symmetries which are generatedby the charges X = − p and X = − q . The A flavor symmetry along with the flavoredPQ symmetry U (1) X is spontaneously broken by two A -triplets Φ T , Φ S and by a singlet Θin TABLE I. And the U (1) X symmetry is spontaneously broken by Ψ , ˜Ψ, whose scales aredenoted as v Ψ and v ¯Ψ , respectively, and the VEV of Ψ (scaled by the cutoff Λ) is assumedas h Ψ i Λ = h ˜Ψ i Λ ≡ λ √ . (A9)Here the parameter λ ≈ . 225 stands for the Cabbibo parameter [29]. After getting VEVs h Θ i , h Φ S i 6 = 0 (which generates the heavy neutrino masses given by Eq. (A18)) and h Ψ i 6 = 0,the flavored PQ symmetry U (1) X is spontaneously broken at a scale much higher than theelectroweak scale and is realized by the existence of the NG modes A , that couples toordinary quarks and leptons at the tree level through the Yukawa couplings as in Ref. [4].According to the simple basis rotation by Lim and Kobayashi [88], we perform basisrotations from weak to mass eigenstates in the leptonic sector, ν L S cR −→ W † ν ν L S cR = ξ L . (A10)Here the transformation matrix W ν is unitary, which is given by W ν = U L U R V iV V − iV Z , with Z = e i π cos θ − e i π sin θe − i π sin θ e − i π cos θ (A11)where the 3 × U L participates in the leptonic mixing matrix, the 3 × U R isan unknown unitary matrix and V and V are the diagonal matrices, V = diag(1 , , / √ V = diag( e iφ , e iφ , e iφ ) / √ φ i being arbitrary phases. Then the 6 × W Tν M ν W ν = Z T ˆ M νν ˆ M ˆ M ˆ M S Z ≡ diag( m ν , m ν , m ν , m s , m s , m s ) (A12)with ˆ M νν = U TL M νν U L , ˆ M S = U TR M S U R , ˆ M = U TR m DS U L ≡ diag( m , m , m ) . (A13)60he charged lepton mass term and the Dirac and Majorana neutrino mass terms read M ℓ = y e y µ 00 0 y τ v d = ( λ √ ) ˆ y e λ √ ) ˆ y µ 00 0 ˆ y τ (cid:18) λ √ (cid:19) v d , (A14) m DS = ˆ y s y s 00 0 ˆ y s (cid:18) v Ψ √ (cid:19) v u , (A15) M S = ˆ y ss y ss y ss v ˜Ψ √ (cid:18) v Ψ √ (cid:19) v Θ √ , (A16) m D = ˆ y ν y ν y ν v T √ (cid:18) v ˜Ψ √ (cid:19) v u = ˆ y ν y y v T √ (cid:18) v ˜Ψ √ (cid:19) v u , (A17) M R = ˜ κ e iφ − ˜ κ e iφ − ˜ κ e iφ − ˜ κ e iφ ˜ κ e iφ − ˜ κ e iφ − ˜ κ e iφ − ˜ κ e iφ ˜ κ e iφ M , (A18)where v d ≡ h H d i = v cos β/ √ 2, and v u ≡ h H u i = v sin β/ √ v ≃ 246 GeV, and y ≡ ˆ y ν ˆ y ν , y ≡ ˆ y ν ˆ y ν , ˜ κ ≡ r (cid:12)(cid:12)(cid:12) ˆ y R v S M (cid:12)(cid:12)(cid:12) , φ ≡ arg (cid:18) ˆ y R ˆ y Θ (cid:19) with M ≡ (cid:12)(cid:12)(cid:12)(cid:12) ˆ y Θ v Θ √ (cid:12)(cid:12)(cid:12)(cid:12) . (A19) Acknowledgments We would like to give thanks to Eibun Senaha for useful conversations. 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