Eluding the Gravitino Overproduction in Inflaton Decay

It is known that gravitinos are non-thermally produced in inflaton decay processes, which excludes many inflation models for a wide range of the gravitino mass. We find that the constraints from the gravitino overproduction can be greatly relaxed if the supersymmetry breaking field is much lighter than the inflaton, and if the dynamical scale of the supersymmetry breaking is higher than the inflaton mass. In particular, we show that many inflation models then become consistent with the pure gravity mediation with O(100)TeV gravitino which naturally explains the recently observed Higgs boson mass of about 125GeV.


Introduction
contribution in the pure gravity mediation model, even if z is charged under a certain symmetry. In fact, since the F-term of z develops VEV there is still a mixing between φ and z, which induces the inflaton decay into the gravitinos. The rate, however, is significantly suppressed if m z ≪ m φ , where m z and m φ denote the mass of z and φ, respectively [11,12].
The problem is that if the inflaton mass is larger than the dynamical SUSY breaking scale Λ, it can decay into hadrons in the hidden sector, which eventually produce many gravitinos [13,14]. Thus, a guess is that the gravitino production is suppressed if the following relation is satisfied : This requires a hierarchy between m z and Λ, which can be easily realized in some dynamical SUSY breaking scenarios, as we shall see later. Interestingly enough, the SUSY breaking scale Λ ∼ m 3/2 M P is close to the inflaton mass in many inflation models for m 3/2 ∼ O(100) TeV. Thus we have much chance to suppress the gravitino overproduction in the high-scale SUSY scenario.
We note however that, if the mass of z is too light, the gravitino production from the coherent oscillations of z becomes non-negligible. Therefore, it is important to take into account all these contributions to see to what extent the constraints on the inflation models can be relaxed.
Lastly let us clarify the difference of the present paper from Ref. [15]. In Ref. [15], the relation (1) was assumed to avoid the gravitino overproduction in the gravity and gauge mediation, and the allowed region for the single-field new inflation was studied. In the present work, we shall derive the constraints on the general inflation model parameters for the case of heavy gravitino.
This paper is organized as follows. In Sec. 2 we summarize the inflaton decay rate into the gravitino and the resulting gravitino abundance. In Sec. 3, we discuss the Polonyi problem in dynamical SUSY breaking models and show that the gravitino production can be indeed suppressed in an explicit SUSY breaking model. We conclude in Sec. 4.

Non-thermal gravitino production from inflaton decay
We assume dynamical SUSY breaking where SUSY is spontaneously broken by the strong dynamics at the scale Λ. A concrete model will be given later. Discussion in this section does not depend on details of the dynamical SUSY breaking models. Below the scale Λ, the SUSY breaking field z has a superpotential of the form where µ represents the SUSY breaking scale, and the constant W 0 ≃ m 3/2 M 2 P is fixed so that the cosmological constant almost vanishes. The F-term of z is given by F z ≃ −µ 2 ≃ √ 3m 3/2 M P , and SUSY is indeed broken. The z obtains a non-SUSY mass through the following non-renormalizable operator in the Kähler potential, HereΛ is some cutoff scale, which is roughly equal to Λ if z itself is involved in the strong dynamics, while it can be much larger than Λ if z is weakly coupled to the strong sector as shown explicitly in Sec. 3.2. It generates the mass of z as m 2 z = 4|F z | 2 /Λ 2 . We assume m z ≫ m 3/2 so that the VEV of z is suppressed by m 2 3/2 /m 2 z . Hereafter we assume that z is charged under some symmetry, such as global U(1), which is spontaneously broken by the strong dynamics in the hidden sector.
Let us consider the mixing of inflaton, which is denoted by X or φ in the following, and SUSY breaking field z. As an example, we consider the following Kähler and superpotentials: where the first term in W corresponds to the inflaton sector with g being the coupling constant and v the constant giving the inflation energy scale. At the potential minimum, φ develops a VEV, φ ≡ |v 2 /g| 1/n , while X sits near the origin. Note that φ n can be replaced with (φφ) n/2 , but the following discussion does not change due to this choice. This class of inflation models includes the hybrid (n = 2) [21] and smooth-hybrid inflation [22] as well as the new inflation model (n ≥ 4) [23,24]. Also, the following arguments can be applied to the chaotic inflation model [25] without a discrete symmetry on X and φ.
Around the potential minimum, φ and X get maximally mixed with each other to form mass eigenstates, Φ ± ≡ (φ ± X † )/ √ 2, in the presence of W 0 [9]. The inflaton mass is (approximately) given by m φ = ng φ n−1 . This mixing is meaningful as long as the decay rates of φ and X are smaller than m 3/2 , which is assumed in the following. 1 From the supergravity scalar potential, we find the mixing of X and z as The mixing angle between X and z is approximately given by Thus, the effective mixing angle between Φ ± and z is given by θ/ √ 2.
The inflaton decay into the gravitino is induced by the operator (3). It leads to the following term in the Lagrangian wherez denotes the goldstino, which is eaten by the gravitino through the super Higgs mechanism. This operator induces the z decay into the goldstino pair with the decay rate As far as the inflaton mass is much heavier than the gravitino, we can estimate the inflaton decay into gravitinos in the goldstino picture thanks to the equivalence theorem.
The inflaton decays into a pair of goldstinos via the mixing with z, and the rate is given 1 Otherwise, too many gravitinos are thermally produced. by where Φ collectively denotes the inflaton mass eigenstates Φ ± . Therefore, the decay rate is suppressed for m φ ≫ m z . The precise form of the decay rate is given in Appendix.
Note that z has a charge and hence terms such as K ⊃ |φ| 2 z and |φ| 2 zz are forbidden, which would otherwise induce the gravitino oveproduction. However, no symmetry forbids the following non-renormalizable interaction between the inflaton and z: where c is a constant of order unity. This induces the inflaton decay into the scalar component of the SUSY breaking field as Since z predominantly decays into the gravitino pair, this process yields gravitinos with the same order of those from (10). Note also that the operator like K ∼ (|φ| 2 /M 2 P )(|z| 4 /Λ 2 ) gives comparable rate with that given above. See Appendix for the details.
If the inflaton is heavier than the dynamical scale Λ, the inflaton decays into hadrons in the hidden sector, which also poses severe constraints on inflation models. The decay proceeds through both tree-level [13] and one-loop level [14], but the tree-level process depends on the details of the SUSY breaking models, while the decay via anomalies is more robust. Assuming that the hidden hadron masses are given by Λ, the decay rate at one-loop level is given by [14,16] where T G and T R are Dynkin index of the adjoint representation and and matter fields in the representation R, α h is the fine structure constant of the hidden gauge group and N g the number of generators of the gauge group. We have assumed the minimal coupling between the inflaton sector and the hidden sector in the Kähler potential. For simplicity, If this decay mode is open, the gravitino overproduction problem is severe since each hidden hadron jets finally produce gravitinos. As a result, we obtain the following condition for significantly relaxing the gravitino overproduction problem : Actually, this condition is easily satisfied in a dynamical SUSY breaking model explained in Sec. 3.2 (see Eq. (27)). However, one should note that too light m z may lead to the Polonyi problem as shown later.
The gravitino abundance, in terms of the number-to-entropy ratio, Y 3/2 ≡ n 3/2 /s, is given by where Γ tot is the total decay rate of the inflaton and it is related to the reheating temperature T R as Γ tot ≡ (π 2 g * /90) 1/2 T 2 R /M P , and N 3/2 represents the averaged number of gravitinos per hidden hadron jet. We will take N 3/2 = 1 for simplicity. It is clearly seen that the gravitino abundance is significantly reduced in the range m z ≪ m φ < Λ. At large m φ , three lines coincide since the gravitino production is dominated by the inflaton decay into hidden hadrons. One can read off the gravitino abundance for other values of φ and T R by noting that Y   mological problem associated with the z coherent oscillation is much weaker than the conventional Polonyi problem in gravity-mediation models [26]. Still, however, there may be significant contributions to the gravitino abundance from the decay of the z coherent oscillations. Let us go into details.
Below the dynamical scale Λ, the potential of the Polonyi field z can be written as 2 where H denotes the Hubble parameter and b is a constant of order unity assumed to be positive. Let us estimate the Polonyi abundance in the two cases : H inf ≫ m z and H inf ≪ m z , where H inf denotes the Hubble scale during inflation.
First we discuss the case of H inf ≫ m z . When H is large enough, the minimum of z is close to the origin. It is expected that the z begins to oscillate around the true minimum at H ≃ m z with an amplitude of Thus the Polonyi abundance is given by where T R is the reheating temperature and we have assumed T R √ m z M P .
Next we consider the opposite case, H inf ≪ m z . In this case, z already sits at the position close to the minimum during inflation. The deviation from the true minimum at the end of inflation is estimated as Since m φ ≫ m z , the Polonyi cannot track the change of the potential at the end of inflation and oscillation of the Polonyi field is induced [27]. Then the Polonyi abundance is given by 3 As shown in (9), the Polonyi dominantly decays into the gravitino pair. The gravitino abundance produced by the Polonyi decay is calculated as where Therefore, the contribution to the gravitino abundance from the z coherent oscillations is negligible for m z 10 9 GeV for m 3/2 ∼ 10 2 − 10 3 TeV. We assume this in the following.
Note that the VEV of z (17) is smaller than Λ in such a case, hence the discussion so far remains valid. This should be contrasted to the analysis of Ref. [15].

A model of dynamical SUSY breaking
Here we give an example of dynamical SUSY breaking model : the IYIT model [28] having a desired structure to suppress the gravitino overproduction. We introduce chiral superfields Q i (i = 1 − 4), each of which transforms as a doublet representation under an SP(1) gauge group, which becomes strong at the dynamical scale Λ. We also introduce six gauge singlets z ij (z ij = −z ji ) which couples to Q i as follows : This form of the coupling is ensured by SU(4) F flavor symmetry, under which both Q i and z ij are charged. The strong dynamics enforces a constraint on the QQ pair as Pf(Q i Q j ) = Λ 4 . This contradicts with the equation of motion of z ij , ∂W/∂z ij = 0. Hence, SUSY is broken dynamically. As a result, one of the combination of z ij , which we denote by z, obtains an F -term as where we have relied on the naive dimensional analysis [29]. Hereafter we assume that Notice that this is close to the inflaton mass scale for many inflation models. The mass of z is generated from the quantum corrected effective Kähler potential Therefore,Λ in (3) is related with Λ through the relationΛ = (4π/λ 2 )Λ. This yields Thus m z is much smaller than the dynamical scale Λ for λ ≪ 4π, while hadrons in hidden sector have masses of ∼ Λ. For fixed gravitino mass, Λ becomes larger and z becomes lighter as λ decreases, and so, the gravitino production rate is suppressed (see Eq. (10)). This hierarchy between m z and Λ has important implications on the gravitino overproduction problem from inflaton decay.
Note that the superpotential (23) induces the three-body inflaton decay into zQQ.
The decay rate is given by [13] where Q(Q) represents the scalar (fermionic) component. 5 Gravitinos are produced by these processes and they should be added to the estimate (15) as for m φ > 2Λ.

Constraint on inflation models
Now let us derive constraints on inflation models from the gravitino overproduction. We consider the following SUSY inflation models : new inflation [23,24,20], hybrid inflation [21,19], smooth-hybrid inflation [22] and chaotic inflation [25]. Since we are interested in the heavy gravitino scenario, gravitinos decay well before BBN. The constraint comes from the requirement that LSPs produced by the decay of (non-)thermal gravitino should not exceed the observed DM abundance : LSP denote the abundance of thermal gravitinos and the thermal relic abundance of the LSP, respectively [30] and m LSP the LSP mass. Hereafter we assume the Wino LSP. Then, for the Wino mass lighter than ∼ 2.7 TeV, the thermal relic density is too small to account for all the dark matter density. right. It is seen that the constraint is significantly relaxed for small λ since m z becomes small and the gravitino production rate gets suppressed by a factor of ∼ (m z /m φ ) 4 for m z ≪ m φ . It is remarkable that the hybrid inflation model and new inflation with n > 2, and even the chaotic inflation model without Z 2 -symmetry may be allowed.
The abundance of the non-thermal gravitino is proportional to mW φ 2 /T R . Thus, for the other parameters fixed, the constraints in the figure shift as T R /mW , as long as do not exceed about half of the observed dark matter abundance. For instance, if we decrease T R by a factor of 10 2 , the constraint on φ becomes severer by a factor of 10 for the fixed inflaton mass. Note also that we cannot reduce the value of λ further, since it tends to decrease the z mass and correspondingly the Polonyi-induced gravitino problem becomes severer (see Eq. (21)).

Conclusions
We have revisited the issue of gravitino overproduction in inflaton decay in light of the recent discovery of the 125 GeV Higgs boson, which implies relatively heavy gravitino : m 3/2 = 10 2 -10 3 TeV. It is found that gravitino production rate is significantly suppressed in a dynamical SUSY breaking scenario, if following conditions are met. (1) The SUSY breaking field z is charged under some symmetry, so that terms such as |φ| 2 z and |φ| 2 zz are forbidden. (2) There is hierarchy among the gravitino mass, the z mass, m z , and the dynamical scale Λ. Then, the gravitino overproduction in inflation models with m 3/2 ≪ m z ≪ m φ Λ are greatly relaxed. Thus many inflation models are consistent with the SUSY breaking scenario with m 3/2 = 10 2 -10 3 TeV. We have obtained the constraints on the inflation models in the pure gravity mediation assuming the IYIT SUSY breaking model. large SUSY mass, m φ . For simplicity we focus on a single-field inflation. In the presence of X as in Eq. (5), the mixing between the inflaton mass eigenstate(s) with z should be Therefore the decay rates in the text are half of the followings. We adopt the Planck unit, unless the Planck scale is explicitly shown.

A.1 Decay into a pair of gravitinos
We assume that (14) is satisfied, and that the z is charged under some symmetry so that its VEV is suppressed by m 2 3/2 /m 2 z . Then, the decay rate of the inflaton into a pair of gravitinos is given by [12], with |G (eff) We have assumed that the diagonal elements of the kinetic terms are normalized as and that the kinetic mixing is small, |K φz | ≪ 1. Thus, we obtain where we have defined In general, we expect c ′ = O(1) in the Planck unit. The first term in (33) is important only for light m z and heavy m φ , and so, we have focused on the second term in the text.

A.2 Decay into the scalar components of z
Let us estimate the inflaton decay into z and z † . The decay into zz is suppressed by the VEV of z. The effective interactions are obtained by expanding the kinetic term and the mass term of z as where G = K + ln |W | 2 and |G z | ≃ |G z | ≃ √ 3. Note that the second terms is obtained by expanding the mass term for zz † with respect to φ. Using the equation of motion for z, the effective interactions can be written as where we have used the fact that the mass of z is given by and we have defined In generalc = O(1). The decay rate is thus given by