Supersymmetric Baryogenesis in a Hybrid Inflation Model

We study baryogenesis in a hybrid inflation model which is embedded to the minimal supersymmetric model with right-handed neutrinos. Inflation is induced by a linear combination of the right-handed sneutrinos and its decay reheats the universe. The decay products are stored in conserved numbers, which are transported under the interactions in equilibrium as the temperature drops down. We find that at least a few percent of the initial lepton asymmetry is left under the strong wash-out due to the lighter right-handed (s)neutrinos. To account for the observed baryon number and the active neutrino masses after a successful inflation, the inflaton mass and the Majorana mass scale should be $10^{13}\,{\rm GeV}$ and ${\cal O}(10^{9}$-$10^{10})\,{\rm GeV}$, respectively.


Introduction
The baryon number of the universe is precisely determined by the observations. For example, the latest result by the Planck collaboration gives [1] (n B /s) obs = (8.70 ± 0.04) × 10 −11 , (1.1) where n B and s are the number density of the baryon and the entropy density of the present universe, respectively. The genesis of the baryon number is one of the mysteries of our universe since the standard model of particle physics can not explain the observed value of the baryon number. Leptogenesis [2] is a viable mechanism to generate the baryon number. The heavy right-handed neutrinos are introduced in addition to the standard model particles and their decay and scattering in the thermal plasma produce the sufficient lepton density. Finally the lepton number is converted to the baryon number by the sphaleron process. This mechanism is economical in a sense that the heavy right-handed neutrinos also explain the tiny neutrino masses naturally [3][4][5][6][7][8], called seesaw mechanism.
In this work, we study baryogenesis in a supersymmetric model motivated by the inflation. This model is based on the superconformal subcritical hybrid inflation model [9,10], where inflation continues even after the inflaton field becomes below the critical point value of the hybrid inflation. Such subcritical regime of inflation is originally considered in Refs. [11,12] with an approximate shift symmetry in Kähler potential. #1 Refs. [9,10] consider the superconformal model combined with the approximate shift symmetry and found the model gives a good fit with the observed spectral index of the scalar amplitude and the tensor-to-scalar ratio, and that the inflaton mass is predicted to be around 10 13 GeV. In the current study, we embed the inflation model to the minimal supersymmetric standard model with right-handed neutrinos. The model is the same as one considered in Ref. [16] and we extend the study to a more realistic scenario by taking into account the flavor effects [17][18][19][20][21][22] and the spectator effects [23][24][25]. After inflation, the inflaton decays to reheat the universe and produce the lepton numbers. The important points are i) not all the lepton numbers are washed out due to the lighter right-handed (s)neutrinos [26,27] and ii) the B − L, where B and L are baryon and lepton number respectively, #1 In the scenario, the waterfall field value is suppressed and inflation continues in the infaton direction, meanwhile Refs. [13][14][15] study the case where inflation along the direction of the waterfall field.
remains due to the conserved charges even in a case where the wash-out effect is most effective [28,29]. In the latter point the supersymmetry plays the crucial role.

The model
We consider an extended minimal supersymmetric standard model (MSSM) augmented by three right-handed neutrinos N i and two standard model singlet fields S ± , which are described by a superpotential where y ij and λ i are coupling constants and M ij are the Majorana masses. The indices i, j take 1, 2, 3. L i and H u are the left-handed lepton doublets and the up-type Higgs, respectively. In this model we assume that S ± has a local U (1) charge, ±q, and the other fields are the U(1) singlets, and that the gauge symmetry is spontaneously broken due to the D-term potential, which is studied in Ref. [16].
Then in (N i , S − ) T basis, we have the following 4 by 4 mass matrix, which leads to a 3 by 3 active neutrino mass matrix, This matrix is diagonalized by a unitary matrix U ν as It is known that the matrix M ν gives rise to one massless neutrino [16]. We follow the convention such that m 1 = 0 and m 3 > m 2 for the normal hierarchy (NH) and m 3 = 0 and m 2 > m 1 for the inverted hierarchy (IH). Another important fact is that M ν is independent of both the mass scale λ i ⟨S + ⟩ and y 3i . Therefore, λ i ⟨S + ⟩ and y 3i are not constrained by the observed neutrino masses. This is crucial in the later discussion.
For later convenience, we introduce two bases; inflaton basis and mass eigenstate basis. The former one, written as (N ′ 1 , N ′ 2 , N ′ 3 ) T , is a basis where N ′ 3 only couples to S + S − . Namely λ i N i ≡λN ′ 3 and the scalar component of N ′ 3 plays the role of the inflaton field. In (N ′ i , S − ) T basis, we have the following 4 by 4 mass matrix, where N ′ i = U inf ij N j and m ϕ =λ ⟨S + ⟩, which corresponds to the inflaton mass. In this basis,ỹ transforms asỹ The latter one is the basis where both M ′ N and charged lepton mass matrix are diagonalized. In the basis, the relevant terms in the superpotential are whereN I (I = 1, 2, 3, 4) andL i are the mass eigenstates of the heavy right-handed neutrinos plus S − and the charged leptons. #2 Namely, (2.10) Here we have defined a unitary matrix U N as 11) #2 We will sometimes use a notationL α where α = e, µ, τ in the later discussion.
where M 1 < M 2 < M 3 < M 4 and similar for T . Then the PMNS matrix is given by In order not to disturb the inflationary trajectory, we assume where Λ represents the typical scale of the Majorana masses. In this limit, the mass eigenvalues M I have a relation 13) and U N has a structure as where u 2×2 is O(1) 2 by 2 matrix. Eq. (2.12) is the feature of this model and Eqs. (2.12)-(2.14) are important in the estimation of the lepton asymmetry. As we described below Eq. (2.5), the active neutrino mass matrix is independent of m ϕ .
Thus m ϕ is not constrained by the observation of the neutrino masses. We will see in the next section thatŷ 3i are free from the constraint, which means thatŷ 3i are free parameters in this model.

Baryogenesis
The overview of the thermal history of our model is the following: At the stage a) we can take √ 2 ReÑ ′ 3 = ϕ as the inflaton without the loss of generality and ϕ drives inflation. Such inflation models are discussed in Refs. [16,31]. #3 The decays ofN 1 andN 2 might give comparable contributions by a tuning of the model parameters [30]. To be conservative we ignore them.
After inflation, the inflaton field decays to reheat the universe. The lepton number is produced simultaneously by the inflaton decay, which is given by where T R and ϵ ϕ are the reheating temperature and the lepton asymmetry of the inflaton decay, respectively. #4 This corresponds to the stage b). Assuming the instantaneous reheating and T R /m ϕ ≲ 1, the reheating temperature is given by the decay width Γ ϕ of the inflaton as

and
where we have used Λ/m ϕ ≪ 1, given in Eq. (2.12). Similarly, the asymmetry ϵ ϕ is given by  33 10 −6 m ϕ 10 13 GeV in the parameter space we are interested in.
To estimate T R and ϵ ϕ , it is convenient to introduce a 4 by 3 matrix R based on Ref. [41]: , 0) for the IH. R satisfies R T R = diag(0, 1, 1) and diag(1, 1, 0) for the NH and IH, respectively. From the equation, we writeŷ in terms of R asŷ which leads to In the expansion of Λ/m ϕ , we find where we have used Eq. (2.14). It is clear thatŷ 3i is not constrained by the observed results in the neutrino sector. We note that imaginary part of {(ŷŷ † ) 34 } 2 is suppressed by Λ/m ϕ compared with the naive expectation, which cancel a factor of m ϕ /∆M .

Thus, we get
Here we have introduced a coefficient a to take into account O(Λ/m ϕ ) term in Eq. (3.9). We expect a = O(1) without a fine-tuning. T R /m ϕ can be written in Thus the condition T R /m ϕ ≲ 1 is equivalent to K ϕ ≲ 1.
At the stage c), the generated lepton number suffers from the wash-out byN 1 andN 2 . To evaluate the wash-out effect, we estimate K 1,2 . From Eq. (3.7), it is straightforward to obtainm for I = 1, 2 where m min = m 2 ≃ 8.6 × 10 −3 eV and m 1 ≃ 4.9 × 10 −2 eV for the NH and IH cases, respectively [42]. Here we have taken Λ/m ϕ ≪ 1 to derive m min .
Therefore, K 1,2 have the minimum values as This means that the wash-out effect is strong. Even in the strong wash-out regime, not all lepton number is washed out [26,27]. At the production of the lepton number we assume This requirement is for a successful inflation, which will be quantified later in Eq. (3.23).
The mass scale of inflaton means that the all Yukawa interactions are out of equilibrium at the decay of the inflaton, except for the top Yukawa interaction. Consequently the produced lepton is a coherent state Iα |ℓ α ⟩ . As the temperature drops down, the spectator effects [23][24][25], the flavor effects [17][18][19][20][21][22] and the wash-out effect due toN 1 andN 2 become important.
Regarding the masses of the lighter right-handed neutrinos, we consider Here the upper bound is from the requirement (2.12), meanwhile the lower one is to ignore the µ term, i.e., µH u H d , and the gaugino masses. To be more quantitative, M 1,2 should be larger than roughly 2 × 10 7 GeV(µ/100 GeV) 2/3 and 8 × In the present case, left-handed (s)leptons and up-type Higgsinos (Higgses) are produced by the inflaton decay. Additionally, R and R χ 3q are expected to be generated. This is due to the R symmetry breaking during the coherent inflaton oscillation.
Therefore the inflaton decay gives an initial condition of the chemical potentials of the conserved charges are written as ) and the other chemical potentials are zero. µ L dec is the chemical potential of the total lepton number produced at the inflaton decay, i.e., n L | dec = µ L dec T 2 /6 and r α are the fractions of each flavor. µ R is the chemical potential of R number.
A finite value of µ R may come from inflation or the coherent oscillation of the inflaton field. In the VEV ⟨ϕ⟩ or the variance ⟨ϕ 2 ⟩ of the inflaton field induces additional one-loop diagram with R-breaking intermediate states appears, which leads to an asymmetry of R. With the variance, for instance, the asymmetry is estimated to be suppressed by at least (ŷŷ † ) 33 ⟨ϕ 2 ⟩ /m 2 ϕ , compared with ϵ ϕ . Here we found the same suppression factor Λ/m ϕ from the imaginary part of the Yukawa couplings as ϵ ϕ . Then, it is O(10 −10 ) suppression in our target parameter space. #7 Therefore it can be ignored in our current study and we omit the contribution from µ R in the discussion below. On the other hand, it may be an importnat contribution to the baryon assymmetry if (ŷŷ † ) 33 ∼ 1. In that case, we need to take into account the non-perturbative decay of inflaton during the coherent oscillation. Or if we consider a different type of the seesaw mechanism, the suppression factor Λ/m ϕ may be irrelevant to boost the asymetric parameter. We will leave these possibilities for the future research. out in the energy scale we are interested in. The mass scale of S − is the order of the inflaton mass.
Therefore, it is also integrated out below the energy scale of the reheating, as well as the inflaton field. #6 The chemical potential of the right-handed neutrinos are Boltzmann-suppressed and irrelevant for the conditions of the equilibrium [26,27,44], for instance, given as µ ℓα + µH u + µg = 0. See also Appendix B. #7 We use an estimation of the variance ⟨ϕ 2 ⟩ ∼ (ŷŷ † ) 33 /(4π)M pl ∼ 10 11 GeV × (ŷŷ † ) 33 where M pl is the reduced Planck mass.
Let us consider M 1,2 ∼ 10 9 -10 10 GeV. Using Eq. (B.6) in Appendix B, we get Referring to Ref. [27], µ Q ∆ 0 can be obtained as follows. At the temperature the QCD and electroweak sphaleron processes, b, τ , and c Yukawa interactions are in equilibrium. Due to the τ Yukawa interaction, |ℓ τ ⟩ component in |ℓ ϕ ⟩ is washed out since K τ I ≫ 1 for I = 1, 2, where K α I ≡ K I | ⟨ℓ α |ℓ I ⟩ | 2 (α = e, µ, τ ). #8 Let us call |ℓ ⊥ τ ⟩ as a state orthogonal to |ℓ τ ⟩. Next, we decompose |ℓ ⊥ τ ⟩ by states |ℓ τ ⊥ I ⟩ and |ℓ τ ⊥ I ⊥ ⟩ (I = 1, 2); the former is |ℓ I ⟩ projected on to a plane perpendicular to |ℓ τ ⟩ and the latter is one which is orthogonal to |ℓ τ ⊥ I ⟩ in that plane. Then |ℓ τ ⊥ I ⟩ component is washed out due to K e I + K µ I ≫ 1. To summarize, using the decomposition [27]. µ Q ∆ 0 depends on the details of the model parameters, such as y ij , M ij , and λ i . Here we take into account the spectator effects and the wash-out effect by introducing a coefficient d, which range from about 0.04 to 0.8. The result is plotted on ((ŷŷ † ) 33 , m ϕ ) plane in Fig. 1. Here we also indicate the region where inflation induced byÑ ′ 3 predicts the spectral index and the tensor-to-scalar ratio that are consistent with the Planck observation based on Refs. [9,10] (see Appendix C for details): 0.7 × 10 13 GeV < m ϕ < 11 × 10 13 GeV .
Therefore, the observed baryon number is obtained after the successful inflation in the region (ŷŷ † ) 33 ∼ 10 −7 -10 −6 and m ϕ ∼ 10 13 GeV. This value of the neutrino Yukawa coupling is desirable for the reason discussed below. Let us say that all y ij have the same order. Then we obtain (ŷŷ † ) ∼ m i Λ/ ⟨H u ⟩ 2 ∼ 10 −6 (Λ/10 10 GeV) from the observed neutrino masses. This is consistent with the assumption (3.16).
In addition, the baryon number is predicted to behave as Y B ∝ Λ 3/2 , which means that Λ ∼ 10 10 GeV, i.e. M 1,2 ∼ 10 10 GeV, is required to get the observed number.
Therefore, in a case where all y ij are the same order, the observed baryon asymetry can be obtained in the setup of this hybrid inflation model that is preferred by both the Planck observation and the neutrino masses.

Conclusion
We consider a model of supersymmetric hybrid inflation and study the reheating and baryogenesis after inflation. The model consists of three right-handed neutrinos

A The interactions and charges
We construct a set of U(1) charges in the MSSM based on the technique given in Ref. [28]. In our study we ignore the µ term for Higgses and the masses of gauginos by assuming µ and the supersymmetry breaking scale smaller than O(10 9 ) GeV.
−µ t + µ Q 3 + µH u + µg = 0 etc. To describe the transportation of the chemical potentials, we introduce a set of charges for fermions f based on Refs. [28,44].
With the chemical potentials µ Q i , the asymmetry number density for the conserved charges are given by For example, the asymmetry of B − L is given by α µ Q ∆α T 2 /6.

B B − L at the wash-out regime
With the interactions and the conserved charges introduced in the previous section, we can calculate the B − L for a given set of the equilibrium conditions. Here we give several temperature regime from 10 10 GeV to 10 5 GeV. Though we focus on the range [10 7 GeV, 10 10 GeV], we give the result as a reference. In out study, we consider tan β, the ratio of the VEV of the up-type and down-type Higgs, is ∼ 1 and adopt the equilibrium temperatures of the relevant interactions given in Refs. [29,47] (i) T ∼ 10 9 -10 10 GeV: t, b, c, τ Yukawa interactions and strong, weak sphaleron processes are in equilibrium. In this case, τ (including ℓ τ ) is distinguished. On the other hand, a linear combination of ℓ e and ℓ µ , which are tentatively denoted as ℓ ′ e and ℓ ′ µ , are disentangled if the interaction withN 1 andN 2 are in equilibrium, which will be discussed later.