Pure Gravity Mediation and Chaotic Inflation in Supergravity
IIPMU 14-0154
Pure Gravity Mediation and Chaotic Inflation in Supergravity
Keisuke Harigaya and Tsutomu T. Yanagida
Kavli IPMU (WPI), TODIAS, University of Tokyo, Kashiwa, 277-8583, Japan (Dated: October 5, 2018)
Abstract
We investigate compatibility of the pure gravity mediation (or the minimal split supersymmetry)with chaotic inflation models in supergravity. We find that an approximate Z parity of the inflatonis useful to suppress gravitino production from thermal bath and to obtain consistent inflationdynamics. We discuss production of the lightest supersymmetric particle through the decay of theinflaton with the approximate Z symmetry and find that a large gravitino mass is favored to avoidthe overproduction of the lightest supersymmetric particle, while a lower gravitino mass requirestuning of parameters. This may explain why the nature has chosen the gravitino mass of O (100)TeV rather than O (100) GeV. a r X i v : . [ h e p - ph ] J u l . INTRODUCTION High scale supersymmetry (SUSY) with the gravitino mass m / = O (100) TeV is one ofthe most interesting models beyond the standard model. It not only explains the observedHiggs boson mass m h (cid:39)
126 GeV [1, 2] by stop and top-loop radiative corrections [3–5], butalso it is free from serious phenomenological and gravitino problems thanks to large sfermionand gravitino masses, m sfermion (cid:39) m / = O (100) TeV. Among high scale SUSY models, thepure gravity mediation (PGM) [6–8] is a particularly attractive scenario, for we do not needto introduce the Polonyi field to generate gaugino masses [9, 10] and the SUSY invariantmass (so-called µ ) term of higgs multiplets [11, 12] in the minimal SUSY standard model(MSSM). Thus, the model is completely free from the cosmological Polonyi problem [13, 14](see also the minimal split SUSY [15] whose basic structure is identical to the PGM.). On the other hand, the chaotic inflation [18] is one of the most attractive cosmic inflationscenarios [19, 20]. It is free from the initial condition problem [21]. That is, inflation takesplace for generic initial conditions of the inflaton field and the space-time. The chaoticinflation has been successfully realized in the supergravity (SUGRA) [22].In this paper, we investigate compatibility of the PGM with the chaotic inflation. Weshow that in the PGM, the inflaton should have a Z odd parity to suppress the reheatingtemperature, avoiding the gravitino overproduction from thermal bath [23–26]. We alsoshow that the Z symmetry is helpful for the inflaton to have consistent dynamics withouttuning of parameters in the inflaton sector.In order for the inflaton to decay, we argue that the Z symmetry is softly broken bya small amount. We discuss the reheating process assuming the small breaking of the Z symmetry, with paying attention to the gravitino overproduction problem. It is knownthat the inflaton in general decays into gravitinos, which leads to the overproduction of thelightest SUSY particle (LSP) [27–32]. We consider that the LSP is stable and a candidatefor dark matter (DM) in the universe. We discuss how the overproduction of the LSP canbe avoided. Assuming that leptogenesis [33] (for a review, see Ref. [34]) is responsible forthe origin of the baryon asymmetry in the universe, we show that our solution to the aboveproblem suggests a gravitino mass far larger than the electroweak scale, m / > ∼ O (100) TeV, High scale SUSY models are also discussed in Refs [16, 17]. In Ref. [16], a mediation scale other than thePlanck scale is introduced to generate soft scalar masses, and hence soft masses have a broader range thanin the case of the PGM. In Ref [17], the Polonyi field is introduced to generate the µ term, and hence itis essentially different from the PGM. (cid:72) m (cid:144) (cid:144) TeV (cid:76) D i s t r i bu ti ono f m (cid:144) Low energy bias byelectroweak scale High energy bias byLSP overproduction???
FIG. 1.
A sketch on possible distributions of the gravitino mass. while fine tuning of parameters in the SUSY breaking sector and the MSSM sector is requiredfor a smaller gravitino mass. We note that we do not use any constraints from the successfulBig Bang Nucleosynthesis (BBN) to derive the natural lower bound on the gravitino mass.This may answer to a fundamental question for the high scale SUSY; why the naturechooses the high scale SUSY with m / = O (100) TeV, but not so-called a natural SUSYwith m / = O (100) GeV. The gravitino mass was in fact expected of O (100) GeV beforethe Large Hadron Collider, for the electroweak scale is naturally obtained without tuning ofparameters in the MSSM when m / = O (100) GeV. In the landscape point of view [35–38],it seems to be difficult to understand why the nature chooses m / = O (100) TeV. As weshow in this paper, the gravitino mass of O (100) GeV requires fine tuning to avoid the LSPoverproduction, otherwise the DM density of the present universe is outside the anthropicwindow [39, 40]. Thus, m / = O (100) TeV may be as plausible as m / = O (100) GeV (seeFig. 1 for the schematic picture).This paper is organized as follows. In the next section, we review chaotic inflation modelsin the supergravity and show that the inflaton should have a Z odd parity in the PGM. InSec. III, we discuss the decay of the inflaton into gravitinos and show how the LSP over-production can be avoided. We show that the solution to the LSP overproduction problemfavors a gravitino mass far larger than the electroweak scale and smaller gravitino massesrequire tuning of parameters. The last section is devoted to discussion and conclusions.3 I. CHAOTIC INFLATION MODEL IN SUPERGRAVITY
The chaotic inflation [18] is an attractive inflation model, for it is free from the initialcondition problem [21]: Inflation takes place for generic initial conditions of the inflatonfield and the space-time. In this section, we review chaotic inflation in SUGRA proposedin Ref. [22]. We first discuss the inflaton dynamics in chaotic inflation model in SUGRA.Next, we show that the inflaton should have a Z odd parity in the PGM. In order for theinflaton to decay, we assume that the Z symmetry is explicitly broken by a small value ofa spurious field E . Then, we discuss the decay of the inflaton into MSSM fields. A. Supergravity chaotic inflation model
In SUGRA, the scalar potential is given by the Kahler potential K ( φ i , φ ∗ ¯ i ) and thesuperpotential W ( φ i ), where φ i and φ ∗ ¯ i are chiral multiplets and their conjugate anti-chiralmultiplets, respectively. The scalar potential is given by V = e K (cid:104) K ¯ ii D i W D ¯ i W ∗ − | W | (cid:105) ,D i W ≡ W i + K i W, (1)where subscripts i and ¯ i denote derivatives with respect to φ i and φ ∗ ¯ i , respectively. K ¯ ii isthe inverse of the matrix K i ¯ i . Here and hereafter, we use a unit of the reduced Planck mass M pl (cid:39) . × GeV being unity.The chaotic inflation requires a large field value of the inflaton during inflation. Withthe large field value, the slow-roll inflation seems to be difficult to take place in SUGRA,because of the exponential factor in the scalar potential, e K . This problem was naturallysolved in Ref. [22] by assuming a shift symmetry of the inflaton chiral multiplet Φ,Φ → Φ + iC, (2)where C is a real number. The inflaton is identified with the imaginary scalar componentof Φ, φ ≡ √ φ , and hence the slow-roll inflationis naturally realized for a large field value of φ . Note that the chiral multiplet Φ must havea vanishing R charge to be consistent with the shift symmetry.4he inflaton potential is obtained by breaking the shift symmetry softly in the super-potenial, W = mX Φ , (3)where X is a chiral multiplet with an R charge of 2. The explicite breaking of the shiftsymmetry is expressed by the parameter m . Here, we have eliminated the term allowed bythe R symmetry, W ⊃ M X , where M is a constant, by a redefinition Φ → Φ − M /m .Let us discuss the inflaton dynamics. The Kahler potential consistent with the shiftsymmetry is given by K = c (Φ + Φ † ) + 12 (Φ + Φ † ) + XX † + · · · , (4)where · · · denotes higher dimensional terms, which we neglect for simplicity. The scalarpotential is given by V ( φ, σ ) = exp (cid:16) σ + √ cσ (cid:17) m (cid:0) φ + σ (cid:1) , (5)where σ is the real scalar component of Φ, σ ≡ √ X is stabilized near the originduring inflation by a Hubble induced mass term, we have set X = 0 [22]. For given φ (cid:29) σ = − c/ √
2. Thus, the scalar potential of φ duringinflation is given by V inf ( φ ) (cid:39) m φ , m eff ≡ m × e − c / . (6)The observed magnitude of the curvature perturbation, P ζ (cid:39) . × − [49], determines m eff as m eff (cid:39) . × − = 1 . × GeV , (7)where we have assumed that the number of e-foldings corresponding to the pivot scale of0 .
002 Mpc − is as large as 50 − For discussion on the shift symmetry breaking in the Kahler potential, see Refs. [41–44]. ∂V∂φ ∝ φ = 0 , ∂V∂σ ∝ σ (cid:18) σ + c √ σ + 1 (cid:19) + φ (cid:18) σ + c √ (cid:19) = 0 . (8)For c <
8, Eq. (8) has a unique solution at the origin. For c >
8, Eq. (8) has threesolutions for σ . One of the solutions, σ = 0, is the minimum with a vanishing potentialand another solution, σ = − c/ (2 √ − sgn( c ) (cid:112) c / −
1, is a local minimum with a non-vanishing potential. The other is a local maximum. Since σ is trapped at σ = − c/ √ φ values, as φ becomes small, σ moves to the local minimum with a non-vanishingpotential, which prevents the inflation from ending. Thus, it is required that c < m eff . Since c <
8, the massof the inflaton at the origin, m , is within the range of . × GeV = m eff ≤ m < e m eff = 1 . × GeV . (9) B. Motivation of a Z symmetry Let us consider possible couplings of the inflaton to the MSSM particles. We first notethat the field X has an R charge of 2. This is mandatory because the inflaton multipletΦ must possess a shift symmetry, so its R charge must vanish. On the other hand, in thePGM, the higgsino Dirac mass term, so called the µ term, is generated by the tree levelcoupling of the higgs multiplets to the R symmetry breaking [11, 12]. This ensures the µ term to be of the same order as the soft scalar mass term, i.e., the gravitino mass. Thismechanism requires the combination H u H d , where H u and H d are the up and down typehiggs multiplets, to have vanishing charges under any symmetry. Therefore, the followingsuperpotential term is not forbidden by the R symmetry, W ⊃ gXH u H d , (10)where g is a constant.The inflaton decays into higgs pairs through the coupling in Eq. (10). The resultant This range is slightly widen by taking account of higher dimensional terms in the Kahler potential. Even ifthe mass of the inflaton is as large as 10 GeV and hence the decay of the inflaton after inflation producesparticles with extremely large momenta, the decay products thermalize soon after their production [45].Thus, the standard estimation of the reheating temperature in the following discussion is valid. T RH = 1 . × GeV gm (cid:18) m . × GeV (cid:19) / . (11)For g = O (1), the reheating temperature is so high that too many gravitinos are producedthrough thermal scatterings [23–26]. The coupling g must be extremely suppressed [22]. The suppression is easily achieved if X and Φ are odd under a Z symmetry. We notethat the Z symmetry is also helpful to have successful inflaton dynamics. As we havementioned in the previous subsection, the superpotential term of W ⊃ M X is allowed bythe R symmetry. The constant M is expected to be of order one without the Z symmetry.As we shift Φ, Φ → Φ − M /m , to eliminate the superpotential term, a large linear termin the Kahler potential, c (Φ + Φ † ) in Eq. (4), is induced. However, for inflation to end, theconstant c in Eq. (4) must be smaller than √
8, which requires tuning among parameters inthe Kahler potential. We can easily avoid the tuning if we have the Z symmetry.Taking those problems seriously, we assume, throughout this paper, the Z symmetryunder which X and Φ are odd. In order for the inflaton to decay into the MSSM particles,we assume that the Z symmetry is broken by a small amount, which we express by aspurious field E . Here, the spurion E is odd under the Z symmetry and a non-vanishingvalue of E represents the Z symmetry breaking. C. Decay of the inflaton into MSSM fields
Based on the assumption of the broken Z symmetry, we consider the following superand Kahler potential for the inflaton and the MSSM sectors, W = X ( m Φ − E ) + a E XH u H d + W MSSM ,K = XX † + 12 (Φ + Φ † ) + QQ † , (12) If g is not suppressed, the F term of X strongly depends on H u H d . The H u H d direction works as awaterfall field in the hybrid inflation [46], and thus inflation ends for | φ | (cid:29)
1. This changes the predictionon the spectral index and the tensor fraction. We note that during the waterfall phase, the instability of H u and H d grows and the reheating temperature becomes extremely high. The Z symmetry is consistent with the shift symmetry given in Eq. (2). g can be also suppressed if H u H d carries a Peccei-Quinn charge. For the PGM model with the Peccei-Quinn symmetry, see Refs. [47, 48]. Alternatively, the inflaton can decay into MSSM fields if MSSM fields are also charged under the Z symmetry [50, 51]. We do not consider this possibility in this paper. W MSSM is the superpotential of the MSSM, Q denotes MSSM fields collectively, and a is an order one coefficient. We take m to be real without loss of generality. To beconcrete, we have assumed the minimal form of the Kahler potential. For clarity, we shiftΦ as Φ → Φ + E /m . Then the super and the Kahler potential is given by W = mX Φ + a E XH u H d + W MSSM ,K = XX † + 12 (Φ + Φ † ) + c (Φ + Φ † ) + QQ † , (13)where c ≡ ( E + E † ) /m is a real constant. For a successful inflation, c must be smaller than √
8, which indicates that |E | < O ( m ).Let us discuss the decay of the inflaton into MSSM fields. First, the inflaton decays intohiggs pairs through the coupling in the superpotential in Eq. (12) with the width,Γ( φ → H u H d ) = 14 π | a E | m. (14)Second, the inflaton automatically decays through the linear term of the inflaton field inthe Kahler potential, if a non-vanishing superpotential of MSSM fields exists [31, 32]. As-suming the presence of right-handed neutrinos with Majorana masses to explain the neutrinomass [52], dominant decay modes are provided by the following superpotential, W = y t Q ¯ u H u + 12 M N N N, (15)where Q , ¯ u and N are the third-generation quark doublet, the third-generation up-typequark, and a right-handed neutrino, respectively. y t and M N are the top yukawa couplingand the right-handed neutrino mass, respectively. For simplicity, we assume that only oneright-handed neutrino is lighter than the inflaton. Decay widths of the inflaton by theseinteractions are Γ( φ → Q ¯ u H u ) = 3128 π c y t m , Γ( φ → N N ) = 116 π c mM N . (16)Third, the inflaton couples with gauge multiplets through radiative corrections [32]. Ra-8iative corrections induce couplings of the inflaton in kinetic functions, (cid:20) g + i θ YM π + 116 π c Φ( T G − T M ) (cid:21) W α W α , (17)where g , θ YM and W α are the gauge coupling constant, the theta angle, and the field strengthsuperfield, respectively. T G is the Dynkin index of the adjoint representation and T M is thetotal Dynkin index of matter fields. The decay width of the inflaton into the gauge multiplet V by the gauge kinetic function is given byΓ( φ → V V ) = α π N G ( T G − T M ) c m , (18)where α = g / π and N G is the number of the generator of the gauge symmetry. Due tothe suppression by a one-loop factor, this decay mode is sub-dominant in the MSSM. As wewill see, however, this decay mode plays an important role in considering the decay of theinflaton into the SUSY breaking sector in Sec. III.In Fig. 2, we show the relation between the reheating temperature T RH ≡ . √ Γ tot andthe parameter c , where Γ tot is the total decay width of the inflaton. Here, we assume a = 1and E is real.Let us put a restriction on the reheating temperature, which is crucial for the discussionon the gravitino problem in the next section. Throughout this paper, we assume thatleptogenesis [33] is responsible for the origin of the baryon asymmetry of the universe.The thermal leptogenesis requires T RH > ∼ × GeV [53, 54], and hence c > ∼ .
7. Sincethe inflaton decays into the right-handed neutrino, non thermal leptogenesis [55–57] is alsopossible. In Fig. 2, we also show T RH × Br( φ → N N ) by a dashed line. Here, it is assumedthat M N = m/
2, so that the decay width of the inflaton into the right-handed neutrinois maximum. Non thermal leptogenesis requires T RH × Br( φ → N N ) × (2 M N /m ) > ∼ GeV [56, 59], and hence c > ∼ . T RH × Br( φ → N N ) × (2 M N /m ) > GeV, thatis, c > . T RH > × GeV, that is, c > . When one moves on to the Einstein frame and canonicalizes fields, one encounters inflaton-dependentchiral rotations of fermions fields. Thus, in the Einstein frame with canonical normalization for matterand gauge fields, the shift symmetry also involves chiral rotations of fermions fields, which is anomalous.The coupling in Eq. (17) can be understood as the counter term for the anomaly. Leptogenesis from inflaton decay is also discussed in Ref. [58], where the mechanism of generating thelepton asymmetry depends on the grand unification scale spectrum, however. (cid:45) c (cid:61) (cid:72) (cid:69) (cid:43) (cid:69) † (cid:76)(cid:144) m T R H (cid:144) G e V T RH (cid:61) GeV T RH (cid:61) (cid:180) GeV T RH T RH (cid:180) Br (cid:72) Φ(cid:174) NN (cid:76) m (cid:61) m eff e c (cid:146) , M N (cid:61) m (cid:144) FIG. 2.
The real line shows the reheating temperature for a given parameter c . The dashed lineshows the reheating temperature times the branching ratio of the inflaton into the right-handedneutrino. This constraint should be satisfied when M N (cid:28) m and hence Br( φ → N N ) × (2 M N /m ) issuppressed. III. GRAVITINO PROBLEM AND THE GRAVITINO MASS
It is known that gravitinos are in general produced through the decay of the inflaton,which results in the overproduction of the LSP [27–32]. In this section, we first discuss howgravitinos are produced from the decay of the inflaton. Then we discuss how large gravitinomass is required to avoid the LSP overproduction.
A. Review on the decay of the inflaton into gravitinos
Let us consider the simplest SUSY breaking model with the following (effective) super-potential, W = µ Z, (19)where µ = √ m / is the SUSY breaking scale and Z is the SUSY breaking field. Sincethe SUSY breaking field Z does not obtain its mass from the superpotential, it should10btain its mass from the Kahler potential; otherwise, the SUSY breaking field obtains alarge amplitude in the early universe, causing the cosmological Polonyi problem [13, 14].The Kahler potential term which yields the mass term is K = − ZZ † ZZ † = − m Z m / ZZ † ZZ † , (20)where Λ (cid:28) m Z is the mass of the scalar component of Z . Thisterm is provided by interaction of the SUSY breaking field with other fields in the SUSYbreaking sector. The inflaton in general decays into those fields in the SUSY breaking sector,as is the case with MSSM fields. Since the SUSY breaking sector fields couple to the SUSYbreaking field Z , they eventually decay into the gravitino. We examine this issue for concreteexamples later.The inflaton also decays into a pair of gravitinos through the mass mixing between theinflaton and the scalar component of the SUSY breaking field Z [27]. For the Kahler andsuper potential in Eq. (13), the mass mixing is given by V mix = √ cm / mZX † + h . c ., (21)at around Z = Φ = X = 0. The mixing angle between the inflaton and the SUSY breakingfield is given by θ = (cid:114) c m / mm Z − m . (22)The coupling between the scalar component of Z and the gravitino, that is, the goldstino ψ is given by the Kahler potential in Eq. (20) as L = − √ m Z m / Z † ψψ + h . c .. (23)From Eqs. (22) and (23), we obtain the decay width of the inflaton into a pair of gravitinos,Γ( φ → ψ / ) = c π m (cid:18) m Z m Z − m (cid:19) (cid:39) c π m ( m Z (cid:29) m ) c π m Z m ( m Z (cid:28) m ) . (24)11he decay width is of the same order as that into MSSM fields if m Z (cid:29) m .Now it is clear that the inflaton in general decays into gravitinos. The gravitino eventuallydecays into the LSP. The density parameter of the LSP is given byΩ LSP h (cid:39) (cid:88) f n f Br( φ → f ) 3 T RH m m LSP . × − GeV . (25)Here, f denotes decay modes and n f is a number of gravitinos produced per the decay mode.For example, n f = 2 for f = 2 ψ / . B. Gravitino problem in strongly coupled SUSY breaking model
We first discuss a strongly coupled SUSY breaking model. To be concrete, let us considerthe SU (5) SUSY breaking model [60, 61]. The model is composed of an SU (5) gauge theorywith and ¯5 representations. Since there is no parameter expect for the gauge coupling,Λ and µ are as large as the dynamical scale of the SU (5) gauge theory, Λ . Assuming thenaive dimensional analysis [62, 63], the Kahler and the super potentials are evaluated as W = c Λ π Z ,K = ZZ † − c π Λ ZZ † ZZ † . (26)where c and c are order one coefficients, and Z is a composite field responsible for theSUSY breaking. Here, we have assumed that only one composite field has a non-vanishingSUSY breaking F term, for simplicity.As shown in Eq. (18), the inflaton decays into the SU (5) gauge multiplet V throughthe kinetic function if the dynamical scale is small enough, m > ∼ . Here, we assume thatmasses of hadrons of the SU (5) gauge theory is as large as Λ . The decay rate is given byΓ( φ → V V ) = 27 α π c m , (27)where α is the fine structure constant of the SU (5) gauge theory. Note that the decay rateis of the same order as the decay rate into MSSM particles (see Eqs. (14) and (16)), andhidden hadrons eventually decay into gravitinos.Even if the decay mode is kinematically closed, m < ∼ , then the direct decay into12ravitinos is unsuppressed since m Z ∼ Λ > ∼ m (see Eq. (24)). Thus, the decay of theinflaton inevitably produces gravitinos and the resultant density parameter of the LSP isΩ LSP h (cid:39) T RH m m LSP . × − GeV . (28)The universe is over closed by the LSP unless m LSP < ∼
10 MeV m . × GeV 10 GeV T RH . (29)When m LSP is such small, however, thermally produced LSPs over close the universe (recallthe Lee-Weinberg bound [64]) unless the LSP is degenerated with a charged SUSY particle.Such a light charged SUSY particle is already excluded by various experiments. We willnot consider the strongly coupled SUSY breaking model below.
C. Gravitino problem in a SUSY breaking model with weak coupling
The origin of the failure in the strongly coupled SUSY breaking model is that either thedecay of the inflaton into gravitinos or that into SUSY breaking sector fields is unsuppressed.Note that simultaneous suppression of these two decay modes is achieved by realizing thefollowing hierarchy, m Z (cid:28) m (cid:28) m SUSY − breaking , (30)where m SUSY − breaking is the mass scale of SUSY breaking sector fields. We show in thissubsection that this hierarchy is easily achieved if the SUSY breaking sector involves weakcouplings [65].To be concrete, let us consider the IYIT SUSY breaking model [66, 67] with the SU (2)gauge theory. We introduce four fundamental representation of the SU (2), Q i ( i = 1-4).Below the dynamical scale of the SU (2), Λ , the theory is described by meson fields withthe deformed moduli constraint [68], W dyn = 4 π Ξ(Pf M ij − Λ π ) , (31) In the PGM, the photino LSP of a mass of O (10) MeV is naturally obtained if m / = O (1) GeV. In thiscase, however, the electroweak symmetry breaking scale is also O (1) GeV. M ij = − M ji ∼ Q i Q j / Λ are meson fields and Ξ is a Lagrange multiplier field. Pfdenotes the Pfaffian over indices i, j . Here, we again assume the naive dimensional analysisand put order one coefficients to unity. It can be seen that there are flat directions, in whichPf M ij = Λ / π .To fix the flat directions, let us introduce five singlet chiral multiplets, Z a ( a = 1-5) andassume the following superpotential, W tree = λc a,ij Z a Q i Q j , (32)where λ and c a,ij are constants. To simplify our discussion, we assume a global SO (5)symmetry under which Z a and Q i are the vector and the spinor representation of the SO (5)symmetry. c a,ij should be appropriate Clebsh-Gordan coefficients. Adding Eqs. (31) and(32), we obtain the effective superpotential, W = λ π Λ Z a M a + 4 π Ξ( M a M a + M − Λ π ) , (33)where we take linear combinations of meson fields and form a vector representation of the SO (5), M a ( a = 1-5). M is the remaining independent linear combination. Now, flatdirections are fixed and the vacuum is given by Z a = M a = 0, M = Λ / π .To break the SUSY, we add an additional singlet chiral multiplet Z and add the super-potential, ∆ W = yZc ij Q i Q j , (34)where y is a constant and c ij is an appropriate Clebsh-Gordan coefficient to form a singletof the SO (5). Adding Eqs. (33) and (34), we obtain the superpotential, W = y π Λ ZM + λ π Λ Z a M a + 4 π Ξ( M a M a + M − Λ π ) . (35)Assuming y (cid:28) λ , the vacuum is given by Z a (cid:39) M a (cid:39) M (cid:39) Λ / π . The F term of Z isnon-zero and hence the SUSY is spontaneously broken.Let us discuss the decay of the inflaton into the SUSY breaking sector. If the dynamicalscale is small enough, m > ∼ , the inflaton decays into gauge multiplets of the SU (2), as14hown in Eq. (18). The decay rate and n f areΓ( φ → V V ) = 3 α π c m (for m > ) , n V V ≥ , (36)where α is the fine structure constant of the SU (2) gauge theory. As in the case of the SU (5) model, for m > , this decay mode is as dominant as decay modes into MSSMfields and hence the gravitino is overproduced.If m < , on the other hand, the mass of the inflaton is not far above the dynamicalscale, and hence we can treat the decay of the inflaton into SUSY breaking sector fields bycomposite picture. The decay rate through mass terms in Eq. (35) and n f areΓ( φ → Z a M a ) = 516 π c m (cid:18) λ π Λ (cid:19) (for m > λ π Λ ) , n Z a M a = 4 , Γ( φ → ZM ) = 116 π c m (cid:16) y π Λ (cid:17) (for m > Λ ) , n ZM = 4 . (37)To be conservative, we assume that λ (cid:39) π . In this case, the decay into Z a M a is kinemati-cally forbidden.Now, we are at the point to show that the desired hierarchy in Eq. (30) can be realized.After integrating out Z a , M a and M , we are left with the effective superpotential, W eff = y π c Λ Z, (38)where c = 1 is a constant, which we leave as a free parameter for later convenience. Thedynamical scale Λ is related with the gravitino mass asΛ = 3 / m / / πy − / c − / = 2 . × GeV y − / (cid:16) m /
10 TeV (cid:17) / c − / . (39)The mass of the scalar component of Z is given by the Kaher potential, K = − y π Λ ZZ † ZZ † , (40)and is as large as m Z = 2 y (4 π ) Λ c = 2 . × GeV y / (cid:16) m /
10 TeV (cid:17) / c / . (41)It can be seen that the hierarchy in Eq. (30) is achieved for small y , and hence the overpro-15uction of the LSP is avoided.For a small y , however, the scalar component of Z is light and the oscillation of the scalar Z is induced in the early universe [65]. The oscillation eventually decays into gravitinos,which may lead to the overproduction of the LSP. Let us estimate the abundance of theLSP from this contribution. The potential of the scalar component of Z during inflation isgiven by V ( Z ) = a H | Z | + m Z | Z | − (2 √ m / Z + h . c . ) , (42)where H inf is the Hubble scale during inflation and a is an order one constant, which weassume to be positive. Since H inf (cid:39) GeV is larger than m Z for the parameter of interest,the Hubble induced mass term traps Z to its origin during inflation. After inflation, as theHubble scale drops below m Z , Z begins its oscillation around the origin, Z = 2 √ m / /m Z = 1 . × GeV y − m /
10 TeV c − , (43)with an initial amplitude Z i = Z . As anticipated, the amplitude is larger for smaller y .The LSP abundance originated from the oscillation of Z is given byΩ osc h = T RH m Z Z i M m LSP . × − GeV . (44)Let us show how large gravitino mass is required. In Fig. 3, we show the constrainton m / and y . Here, we assume that c = 0 . m LSP = 3 × − m / . In the red-shaded region (Ω SUSY h > . m = Λ .In the blue shaded region (Ω / h > . osc h > . c = 0 .
008 (i.e. non thermal leptogenesis is possible). From both figures, we see the constraint16n the gravitino mass, m / > O (100) TeV . (45)It is remarkable that the constraint in Eq. (45) coincides with what is expected in thePGM [6–8].Let us discuss how we can avoid the constraint on the gravitino mass. First, we haveassumed that m LSP = 3 × − m / to obtain the constraint, since it is determined by theanomaly mediation [9, 10]. However, a lower mass for the LSP can be obtained by cancelingthe anomaly mediated contribution by the higgsino threshold correction [9]. In Fig. 5, weshow the constraint on m / and y for ( c, m LSP ) = (0 . , × − m / ). It can be seen thatregions with m / = O (10) TeV is allowed.Let us compare the plausibility of m / = O (10) TeV with that of m / = O (100) TeV inthe landscape point of view. Since we have no knowledge about distributions of parametersin the landscape, we discuss on our naive expectation in the following. We note that differentassumptions on the distribuions lead to different conclusions.For the electroweak scale, m / = O (10) TeV would be more natural than m / = O (100) TeV by a factor of (100 TeV) / (10TeV) = 100. For the LSP mass, since theLSP mass is a complex parameter, m LSP = 3 × − m / would requires tuning of (3 × − / × − ) ∼ − . Thus, we naively expect that the region with m / = O (100) TeVmay be more natural than the region with m / = O (10) TeV.Second, we have assumed the SO (5) symmetric IYIT model to simplify our discussion.Without the SO (5) symmetry, c in Eq. (38) is a constant which is determined by cou-pling constants in the SUSY breaking model. If there is fine-tuned cancellation betweencondensation of hidden quarks which couple to the SUSY breaking field, c can be muchsmaller than O (1). This cancellation further separates the SUSY breaking scale from thedynamical scale. For given m / and y , the constraints shown in Figs. 3 and 4 are relaxed.In Fig. 6, we show the constraint for ( c, c ) = (1 . , − ). It can be seen that the regionwith m / = O (1) TeV survives.Let us again naively compare the plausibility of m / = O (1) TeV with that of m / = O (100) TeV. For the electroweak scale, m / = O (1) TeV would be more natural than A similar conclusion is derived in Ref. [69] where the BBN constraints are used. Notice that we haveobtained Eq. (45) solely from constraints on the LSP DM density. / = O (100) TeV by a factor of (100 TeV) / (1TeV) = 10 . On the other hand, since c is a complex parameter, c = 10 − would require fine tuning of 10 − . These two regions, m / = O (1) TeV and O (100) TeV, may be equally plausible.Third, we have assumed the minimal form of the Kahler potential. By considering higherdimensional terms in the Kahler potential and tuning their coefficients, the decay of theinflaton into the SUSY breaking sector can be suppressed. In principle, the gravitino massof O (100) GeV survives by the tuning. However, to suppress all the decay modes, all thecoefficients of the higher dimensional terms must be carefully chosen, which may requiremore fine tuning.We should stress, finally, that all of the above arguments are merely a sketch on whatkinds of fine tuning is needed to have the gravitino mass below O (100) TeV. Since we donot know distributions of relevant parameters in landscape, it is impossible for us to drawany definite conclusion on the most plausible gravitino mass. However, the present analysisshows that it is not necessarily surprising that the nature has really chosen the gravitinomass of O (100) TeV, even if the SUSY breaking scale is low energy biased in order to obtainthe electroweak scale naturally. IV. DISCUSSION AND CONCLUSION
In this paper, we have investigated compatibility of the PGM with chaotic inflation insupergravity. We have shown that the inflaton should have a Z odd parity to suppressthe reheating temperature, avoiding the gravitino overproduction from thermal bath in thePGM. We have also shown that the Z symmetry is helpful for the inflaton to have consistentdynamics without tuning of parameters in the inflaton sector.In order for the inflaton to decay, we assume that the Z symmetry is broken by a smallamount. We have discussed the reheating process and the gravitino problem under theassumption of a small breaking of the Z symmetry. We have discussed how the gravitinooverproduction by the decay of the inflaton can be avoided, and shown that the solution tothe overproduction problem favors a gravitino mass far larger than the electroweak scale, m / > ∼ O (100) TeV.This consideration gives a new insight on the fine tuning problem in the high scale SUSY.It is usually assumed that the gravitino mass of O (100) GeV is natural, for the electroweak18cale is obtained without tuning of parameters in the MSSM. It can be hardly understoodwhy the nature chooses the gravitino mass of O (100) TeV. However, as we have shown inthis paper, the gravitino mass of O (100) GeV requires some amount of fine tuning to avoidthe LSP overproduction. Therefore, it may not be surprising even if the nature has chosena high scale SUSY with the gravitino mass of O (100) TeV.In this paper, we have assumed the Z symmetry to suppress the reheating temperature.Another option is to assume the spacial separation of the inflaton sector and the MSSMsector in a higher dimensional theory. Our discussion on the LSP overproduction is alsoapplicable to this case.We should note that we can replace the inflaton mass m in Eq. (12) by a vacuum expec-tation value of some field. Consider a B − L gauge symmetry, for example, which is brokenby a vacuum expectation value of a chiral multiplet S with a B − L charge of +1. We assumethat X carries a B − L charge of − W = k Φ SX, k (cid:104) S (cid:105) = m. (46)The Yukawa coupling k represents a shift symmetry breaking. We may take k = O (0 . (cid:104) S (cid:105) = O (10 − ) as an example. The unwanted linear term W = M X is replaced by W = M (cid:104) S (cid:105) X , and the required condition M = M (cid:104) S (cid:105) < ∼ m may be explained by M < ∼ . Z symmetry.In this paper, we have assumed that m LSP = O (10 − ) m / , which is the case of the PGM.In general gravity mediation models with a singlet SUSY breaking field (i.e. a Polonyi field),the LSP mass is expected to be of order the gravitino mass. If it is this case, thermallyproduced LSPs will easily over close the universe unless the reheating temperature is farsmaller than the LSP mass. For T RH > ∼ GeV, the gravitino mass smaller than 10 GeVis excluded.Finally, let us comment on other inflation models. The lower bound on the gravitinomass in Eq. (45) is basically obtained from the condition that masses of SUSY breakingsector fields are larger than the inflaton mass. Thus, for models with the inflation massof O (10 ) GeV, a similar bound on the gravitino mass to Eq. (45) will be obtained. Onthe contrary, if models have the maximal reheating temperature, the lower bound on thegravitino mass may be obtained [71] so that enough LSPs are produced through the gravitino19roduction in thermal bath to explain the DM density. ACKNOWLEDGMENTS
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Constraint on the gravitino mass m / and the coupling of the SUSY breaking field y .In the red-shaded region ( Ω SUSY h > . ), the blue shaded region ( Ω / h > . ) and the yellowshaded region ( Ω osc h > . ), the universe is over closed by the LSP due to the decay of the inflatoninto SUSY breaking sector fields, that into gravitino pairs, and that of the SUSY breaking fieldinto gravitinos, respectively. We assume c = 0 . , c = 1 and m LSP = 3 × − m / . (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) m (cid:144) (cid:144) GeV (cid:76) L og y (cid:87) SUSY h (cid:62) (cid:87) osc h (cid:62) (cid:87) (cid:144) h (cid:62) (cid:61) c (cid:61) m LSP (cid:61) (cid:180) (cid:45) m (cid:144) FIG. 4.
Same as Fig. 3 but with c = 0 . , c = 1 and m LSP = 3 × − m / . (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) m (cid:144) (cid:144) GeV (cid:76) L og y (cid:87) SUSY h (cid:62) (cid:87) osc h (cid:62) (cid:87) (cid:144) h (cid:62) (cid:61) c (cid:61) m LSP (cid:61) (cid:180) (cid:45) m (cid:144) FIG. 5.
Same as Fig. 3 but with c = 0 . , c = 1 and m LSP = 3 × − m / . (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) m (cid:144) (cid:144) GeV (cid:76) L og y (cid:87) SUSY h (cid:62) (cid:87) osc h (cid:62) (cid:87) (cid:144) h (cid:62) (cid:61) c (cid:61) (cid:45) m LSP (cid:61) (cid:180) (cid:45) m (cid:144) FIG. 6.
Same as Fig. 3 but with c = 0 . , c = 10 − and m LSP = 3 × − m / ..