Subcritical regime of hybrid inflation with modular $A_4$ symmetry

We consider a supergravity model that has the modular $A_4$ symmetry and discuss the interplay between the neutrino mixing and inflation. The model contains right-handed neutrinos that have the Majorana masses and additional Yukawa couplings to the waterfall field. In the model an active neutrino is massless and we find that only the inverted hierarchy is allowed and the Majorana phase is predicted to be around $\pm (120\text{--}180)^\circ$ from the observed neutrino mixing data. In the early universe, one of right-handed sneutrinos plays the role of the inflaton field. Focusing on the subcritical regime of the hybrid inflation that is consistent with the cosmic microwave background data, we analyze the dynamics of the scalar sector and derive an upper bound $\mathcal{O}(10^{10})~{\rm GeV}$ on the scale of the Majorana mass.


Introduction
Inflation, a hypothesis of accelerated expansion of the Universe, is supported by the observation of the cosmic microwave background (CMB) radiation. It is driven by the vacuum energy at the early state of the Universe and it is realized by a slowlyrolling scalar field, called inflaton. According to decades of the observations and analysis of the CMB, some properties of inflation have been revealed. The latest results by Planck collaboration [1] report that the amplitude A s and spectral index n s of the scalar perturbation and the ratio r of the tensor mode to the scalar mode are given by A s = (2.0989±0.029)×10 −9 (68% CL), n s = 0.9649±0.0042 (68% CL), and r < 0.056 (95% CL). The results already exclude some of inflation models and future experiments are expected to measure the observables with more precision.
Besides the accelerated expansion of the Universe, the inflaton field plays another important role, i.e., reheating to create the radiation dominated Universe. Then the production of dark matter and baryons follows. Thus, it is tempting to consider an inflation model that provide the sequence of the thermal history after inflation. Moreover, the model would be motivated if it is controlled by an underlying symmetry. The modular symmetry and supersymmetry are the promising candidates.
In the present study we apply the modular A 4 symmetry to inflation. Introducing three right-handed (s)neutrinos, we consider one of right-handed sneutrinos plays the role of the inflaton field. Since the typical model of the right-handed sneutrino inflation tends to be chaotic inflation due to the quadratic term, which is already excluded by the CMB observations, we extend the model to the D-term hybrid inflation by introducing a new Yukawa term. These days variety of new Dterm hybrid inflation models have been proposed, depending on the symmetry of the Kähler potential; R 2 Starobinsky model [96,97] appears from the superconformal symmetry [98], the chaotic regime emerges from the shift symmetry [99,100], and α-attractor [101] comes from their combination, which is called superconformal subcritical hybrid inflation [102]. A generalized version of the inflation model proposed in Ref. [102] is intensitively analyzed in Ref. [103]. In our paper we study the neutrino mixing and the dynamics of inflation based on the model in Ref. [103]. In the framework, an unconventional neutrino mixing pattern and the CP phases are obtained. We find that only the inverted hierarchy is consistent with the data and in the valid parameters the upper limit for the scale of the Majorana mass is obtained for successful inflation.
This paper is organized as follows. In Sec. 2, we introduce the supergravity model to study. Then neutrino masses and mixing pattern are discussed in Sec. 3. Here we show the predicted CP phases and the effective neutrino mass for the neutrinoless double beta decay. In Sec. 4, the dynamics of inflation is studied both analytically and numerically, especially focusing on the impact of the Majorana mass terms. Sec. 5 contains our conclusion.

The model 2.1 Brief review of modular symmetry
The modular symmetry is the geometric symmetry of the two dimensional torus. The two dimensional torus T 2 is defined by C/Λ, i.e., the complex plane C divided by the two dimensional lattice Λ = { 2 i=1 n i ω i | n i ∈ Z} with basis vectors ω i ∈ C. Here, the basis vectors are given by ω 1 = 2πR and ω 2 = 2πRτ with R ∈ R, and τ = ω 2 /ω 1 is the modulus defined in the upper half plane H = {τ ∈ C | Imτ > 0}. The same lattice is constructed using the basis vectors transformed as Then, the modulus is transformed as Eq. (2.3) is called modular transformation. It is seen that the transformation law of τ is the same for γ and −γ. Then, the group of modular transformation is isomorphic to P SL(2, Z) ≡ SL(2, Z)/ ± I ≡Γ, which is called modular group. The modular group is generated by two generators, The modulus τ is transformed by S and T as For a positive integer N , the principal congruence subgroup of level N is defined by Note that Γ(1) = Γ and −I ∈ Γ(N ) for N > 2. Then, we introduce the groupsΓ(N ) asΓ(2) = Γ(2)/ ± I andΓ(N ) = Γ(N ) (N > 2). The quotient groups Γ N ≡Γ/Γ(N ) are called finite modular groups and the generators of Γ N have the additional relation, It is known that the finite modular groups for N = 2, 3, 4, 5 are isomorphic to the non-Abelian discrete groups, [104]. Under the modular transformation, the holomorphic functions of τ , called the modular forms are transformed as (2.9) Here ρ(γ) is a unitary transformation of Γ N (N > 2) and k is non-negative even integer, called modular weight.
In this study, we focus on Γ 3 , i.e., A 4 . Then, the modular forms Y = (Y 1 , Y 2 , Y 3 ) T with representation of A 4 triplet and modular weight 2 are given by [2] where ω = e 2πi/3 and η(τ ) is the Dedekind eta-function defined by (2.11) The modular forms Y i (i = 1-3) satisfy the relation, They are transformed under T and S transformations as where 14) It is assumed that the superfields, denoted as Z I , are also transformed under A 4 as where the modular weight k I will be determined later.

The superpotential and Kähler potential
We consider a supergravity model of inflation inspired by modular A 4 symmetry. In addition to the fields in the minimal supersymmetric standard model (MSSM), we introduce three right-handed neutrinos N c i (i = 1-3) and two new fields S ± . Here S ± are gauge singlets under the MSSM gauge but are charged under a local U(1) with charge ±q (q > 0). They play important roles during inflation.
The superpotential allowed under A 4 symmetry is given by where The superpotential is similar to one considered in Ref. [105], but it is extended to accommodate the modular A 4 symmetry. The contents of the fields are listed in Table 1. The representation and the modular weights of the fields are also shown in the table, which is based on the model considered in Ref. [2,11]; the right-handed neutrinos and the left-handed lepton doublets L i = (ν Li , l Li ) T (i = e, µ, τ ) are the A 4 triplets and the others are the singlets. Here the three different singlet representations (1, 1 , 1 ) are assigned to the right-handed charged leptons E c = (e c , µ c , τ c ), respectively. We will discuss the assignment of the modular weights soon later. In the superpotential, α i (i = 1-3), g i (i = 1, 2), and λ are Yukawa coupling constants, and Λ determines the mass scale of the right-handed neutrinos. By redefinition of the fields, the parameters α i , λ, Λ, and g 1 can be taken to be real without the loss of generality. For Kähler potential, we adopt a class of the canonical superconformal supergravity model [106]. This type of model can be extended to the model with a parameter α, which corresponds to the parameter of the superconformal α attractor model [101]. Recently the dynamics of inflation has been analyzed in a generalized version of the model [103]. In order to consider the similar inflation model, we consider the Kähler potential based on Ref. [103]: #1 where Here we have introduced a positive parameter n in order to discuss the modular weights of the field generically. We take Z 0 as Z 0 =Z 0 = 2 Im τ , In general, Φ includes additional terms, such as (Y N c N c ) 1 or (ȲLY L) 1 [2,19]. In the current study we ignore them to focus on the simple setting. #2 The Kähler potential is further divided into the modular and matter parts as The matter part is constructed to be modular invariant. Then, under the modular transformation (2.3), the Kähler potential is transformed as In the supergravity the combination of the Kähler potential and superpotential, which is defined by should be modular invariant. Due to the invariance, the transformation of the superpotential is determined as, Namely, its modular weight is −3n. Consequently, the weight of fields should satisfy Those are the generic conditions for the modular weights in the present model. For instance, if we further impose the following conditions: the modular weights are uniquely determined for a given n as #2 This is also motivated by the results given by Ref. [103]. In the literature it is shown such a simple Käher potential gives a consistent result with the CMB observations. The value of n is a free parameter and it may restrict possible terms in the Kähler potential. However, we do not consider this direction seriously in the current study.
After the modulus parameter τ is fixed, we redefine the chiral superfields as Accordingly it is convenient to reparametrize the Yukawa couplings (α i , g i , λ) and Λ as (α i ,ǧ i ,λ) andΛ to give rise to the same form of W E +W N +W hyb . Hereafter, we will write the model in this field basis and omit the 'check' symbol for a simple notation unless otherwise noticed. Namely, we use the superpotential given in Eqs. (2.17) -(2.19) and the Kähler potential K m of the matter part with the function Φ For later calculation, we rewrite W hyb as

Neutrino masses and mixing pattern
In this section, we discuss the lepton sector, especially focusing on the neutrino mixing pattern. From the superpotential, the light neutrinos acquire masses in the seesaw mechanism [107][108][109][110][111][112]. In our model, however, S + obtains a vacuum expectation value (VEV), denoted as S + , at the global minimum, which leads to an unconventional mass matrix for the light neutrinos [105]. In addition, the components of the mass matrices are given in a limited number of parameters due to A 4 symmetry, which results in the characteristic pattern of the neutrino mixing and CP phases in the lepton sector.

Neutrino masses and PMNS matrices
To give the mass matrices of the leptons, it is convenient to adopt canonically normalized field basis. The canonically normalized fieldẐ I is obtained bŷ Here we have used −Φ/3 1 at the global minimum. The validity of the approximation is guaranteed by S + 1, which will be shown in Sec. 4. Accordingly we absorb the factor √ α by introducing Then, the mass matrix of the neutrinos in the basis of (N c HereM andM D are 4 × 4 and 4 × 3 matrices, respectively, written bỹ where Ĥ 0 u is the VEV of up-type neutral Higgs. We note thatλ Ŝ + corresponds to the scale of the inflaton mass, which will be shown later. Consequently, the light neutrinos mass matrix is obtained by the seesaw mechanism as (3.8) Using Eq. (2.12), it is straightforward to obtain the mass matrix M ν as #3 We use the same notation for the fermionic part as the chiral superfield for N c i and the leptons in the MSSM whileS − is the fermionic part of S − .  Table 2: Observed data of the neutrino mass squared differences and the neutrino mixing angles from NuFIT 5.0 [115]. Here, ∆m 2 3l = ∆m 2 31 > 0 (NH) and ∆m 2 3l = ∆m 2 32 It is worth notifying that the mass matrix is independent ofλ Ŝ + . Therefore, the scale of the inflaton mass is not affected by the observations of the neutrino sector. Finally M ν is diagonalized by a unitary matrix U ν as The obtained neutrino masses are then compared with the observed values shown in Table 2. We notice that the lightest neutrino mass becomes zero since rank of M ν is two [105]. Therefore, by imposing the condition in which the neutrino mass squared differences are within 3σ range of the observed values, the cosmological upper bound on the sum of light neutrino masses [113,114] i m i ≤ 120 meV , is automatically satisfied in our model for both the normal hierarchy (NH) and inverted hierarchy (IH). Another observable is the PMNS matrix, which is defined as U ≡ U † l U ν . Here U l is a unitary matrix that diagonalize the charged lepton mass matrix M E . M E is the same as one studied in Ref. [11]. The result is parametrized in terms of three mixing angles θ 12 , θ 13 , θ 23 , a Dirac phase δ CP , and a Majorana phase α 21 ; where s ij = sin θ ij and c ij = cos θ ij . The mixing angles are determined by the observations and they are summarized in Table 2. We note that there is only one Majorana phase since one of the light neutrinos is massless. Then, the invariant quantities regarding the CP phases are given by [116][117][118][119][120] J CP = Im U e1 U µ2 U * e2 U * µ1 = s 23 c 23 s 12 c 12 s 13 c 2 13 sin δ CP , (3.14) In addition, the following relations are useful to determine the CP phases [91]:

Observational consequences
Based on the arguments in the previous subsection, we compute the mixing angles and CP phases for a given set of parameters. The relevant parameters are τ andĝ and we parametrize them as where g = |ĝ| and φ g is argument ofĝ. Considering the fundamental domain F ofΓ for τ , we scan the following parameter ranges, We use the 3σ data of the mixing angles and mass squared differences from NuFIT 5.0 [115], which are listed in Table 2. In the analysis, we take the VEVs of the up-and down-type Higgs bosons as v/2 (v 246.7 GeV) by considering so-called high-scale SUSY to give 125 GeV Higgs mass [121,122].
First of all, we have found no allowed region for the NH case. Fig. 1 shows the allowed regions for the τ andĝ for the IH case. We found a specific pattern of the allowed values of τ andĝ is seen. The allowed value of τ is limited in 1.0 Im τ 1.1 and |Re τ | 0.1, or Im τ ∼ 1.2 and |Re τ | ∼ 0.5. This is a different feature compared to the previous work, e.g., Ref. [11], where the allowed values of τ is distributed more widely. This difference comes from the unconventional pattern of the active neutrino mass matrix (3.8) given by Eqs. (3.6) and (3.7), which comes from additional Yukawa  Table 2. couplings between the right-handed neutrinos and S + in Eq. (2.31) and the VEV of S + . Regardingĝ, g is distributed as 0.4 g 5. The phase φ g can take any value in [−180 • , 180 • ], but it is given by a smooth function of g. Such behavior is in contrast to the results in Ref. [11]; in the literature, the allowed region for both g and φ g are more restricted.
The discovered sets of the parameters lead to a new prediction for the CP phases, which is shown in Fig. 2. In the plot we show the correlation between sin 2 θ 23 and the CP phases. For δ CP , we found |δ CP | 120 • for any value of sin 2 θ 23 in the 3σ range. If sin 2 θ 23 0.55, on the other hand, then δ CP can take any values. Regarding α 21 , we found |α 21 | ∼ 120-180 • for any value of sin 2 θ 23 in 3σ range. An exception is |α 21 | ∼ 35 • for sin 2 θ 23 0.42-0.43. Finally, we found no specific correlation between δ CP and α 21 , i.e., |α 21 | ∼ 120-180 • is predicted while various value of δ CP is possible. To summarize, only the IH is allowed and Fig. 2 is the prediction for the CP phases in our model.
Another observable consequence of this model is the neutrinoless double beta decay (A, Z) → (A, Z + 2) + 2e − , which is a lepton number violating process in lowenergy phenomena. See, e.g., Ref. [123] for review. The decay rate of the process is proportional to the effective neutrino mass-squared, which is given by (3.23) The current upper limit on the effective mass of neutrinoless double beta decay from KamLAND-Zen is 36-156 meV [124]. Note that the experimental upper bound on the effective mass is affected by the uncertainty of the nuclear matrix elements in   Fig. 2 the decay process. In the present model, one of the light neutrinos is massless and we have seen that the IH is allowed. Therefore, the effective neutrino mass is given by [125]  The result is shown in Fig. 3. We found 14 meV m eff 28 meV for the most parameter sets, where α 21 ∼ ±180 • . One can see that it becomes as large as around 45-47 meV for α 21 ∼ ±35 • , which can be understood from Eq. (3.24). Namely the term proportional to cos α 21 is constructive in this case. The resultant value O(10) meV of m eff is the same as one shown in Ref. [125]. Since the model predicts a relatively large effective mass, neutrinoless double beta decay events may be observed in future experiments. If the CP phases are constrained, for instance, by considering the leptogenesis [126] in the present framework, then m eff may be more restrictive. That would be another future work to pursue (see also the discussion in Sec. 4.5).

Inflation
W hyb induces the D-term hybrid inflation. Motivated by the previous studies [102,103], we consider the subcritical regime of hybrid inflation, where one of right-handed sneutrinos plays the role of the inflaton field. In the present case, however, the additional Majorana mass terms may disturb the dynamics during inflation. We will derive the inflaton potential and see how the additional terms affect the dynamics. In this section, we examine the impact of the heavy Majorana masses on the subcritical hybrid inflation scenario. To be concrete, we assume Under the assumption, we expect that the same inflation model is obtained as Ref. [103]. We will derive the upper bound for M ij quantitatively.

New field basis
We start with the basis given in the last paragraph of Sec. 2.2. At the global minimum, the mass matrix squared M 2 sc of the scalar sector in the basis (Ñ c whereM is equal to Eq. (3.6) without 'hat' symbol. Under the condition (4.1), the mass matrix is approximated as and it can be approximately diagonalized by a unitary matrix V as Therefore, it is legitimate to use a new basis of (N c In this basis, it is straightforward to find that S ± only couple to N c 3 . Then, the superpotential (2.31) is written as #4Ñ c i and S ± are scalar components of the chiral superfield N c i and S ± , respectively.
becomes the inflaton field [103,105]. The parameter α can be taken to various values and different predictions for n s and r are obtained accordingly, which is intensively analyzed in Ref. [103]. In the current study, we take α = 2/3 ,λ = 0.959 × 10 −3 , ξ = (0.604 × 10 16 GeV) 2 , (4.8) and q = y = 1. Then the best-fit value of observed spectral index n s , along with the tensor-to-scalar ratio r 6 × 10 −4 , is obtained at the last 60 e-folds of the subcritical regime of the hybrid inflation [103]. With the parameters, the condition (4.1) gives where S + = (ξ/(2/3)q(1 +ξ)) 1/2 andξ ≡ ξ/2q has been used, which will be shown in the next subsection. In the following subsections, we will quantitatively examine the condition (4.9) to keep the inflaton dynamics unchanged. To keep the readable analytic expressions, we leaveλ, ξ, q and y as they are.

The scalar potential
The scalar potential is given by [103], where V F and V D are F and D-term potentials, respectively. Here W I ≡ ∂W/∂z I , Φ I ≡ ∂Φ/∂z I , and ∆ ≡ Φ − δ IJ Φ I ΦJ , and z I shows the scalar component of Z I . In the D-term potential, y and ξ (> 0) are the gauge coupling and the Fayet-Iliopoulos (FI) term related to the local U(1). Due to the FI-term, S + acquires a VEV as S + = (ξ/(2/3)q(1 +ξ)) 1/2 at the global minimum. #5 Then, s ≡ √ 2|S + | is identified with the waterfall field.
During inflation, we expect that the fields except for the inflaton and waterfall fields are stabilized at the origin. Thus the scalar potential during inflation is given as where Φ(φ, s) ≡ −3 + (s 2 + φ 2 )/2. ∆M 2 (φ, s) is the term that originates in the Majorana mass term, which is given by (4.14)

Critical point
As in the literature [99,100,102,103,105], we focus on the dynamics of the subcritical regime of the hybrid inflation. First of all, the Majorana mass term shifts the critical point value of the hybrid inflation. The critical point value is determined by the mass squared of the waterfall field, which is given by where Here we have given the mass in a canonically normalized basis in accordance with Ref. [103]. Consequently, the critical point value φ c is obtained by where φ 2 c,0 = 2(2/3) 2 qy 2 ξ/λ 2 [103] and (4.20) Imposing |δ| 1, the upper bound on |M i3 | is obtained as Those are weaker bounds compared to Eq. (4.9). Therefore, the critical point value is merely affected by the Majorana mass term.

Subcritical regime
Below the critical point, the waterfall field grows due to the tachyonic instability [99,127] and soon relaxes to the local minimum s min of the classical path. #6 s min is obtained by ∂V (φ, s)/∂s 2 = 0 as Consequently the potential in the subcritical regime can be expressed by a single field effective potential In the last step, we have used ξ 1. Therefore, V sub reduces to the inflaton potential in Ref. [103] when is satisfied. This condition gives rise to upper bounds on |M i3 | as Those are a bit tighter than the condition (4.9). When the above conditions are satisfied, s min is also in approximate agreement with the local minimum value of the waterfall field in Ref. [103], To summarize the inflaton dynamics is not affected if |M i3 | 10 12 GeV , (4.30) is satisfied. #7 In the next subsection, we will confirm the condition numerically.

Dynamics of scalar fields: numerical study
The upper bound (4.30) is obtained under the assumption that the fields except for the inflaton and waterfall fields are stabilized at the origin during inflation. However, we need to investigate the validity of this assumption since the values ofÑ c 1,2 and ImÑ c 3 may grow to affect the dynamics of inflation depending on the value of the Majorana masses. We will check the stability ofÑ c 1,2 and ImÑ c 3 by solving the equations of motion ofÑ c i and S + numerically and examine the condition (4.30) more quantitatively.
To this end, we define the relevant scalar fields as Then, the metric in terms of the field space of ϕ A is given by where K Z IZJ = ∂ 2 K ∂z I ∂z J . The equations of motion of ϕ A arë Here, dot denotes the time derivative and H is the Hubble parameter that depends on ϕ A andφ A . G AB is the inverse of G AB and Γ A BC is the connection defined by For the numerical analysis, we take a benchmark point from the allowed region of τ andĝ, which is given by With the parameters the corresponding neutrino mixing parameters are sin 2 θ 12 = 0.305 , sin 2 θ 13 = 0.0224 , sin 2 θ 23 = 0.572 , δ CP = −160 • , α 21 = −163 • , (4.36) #7 We have checked that the stability of theL i H u direction, pointed out by Ref. [128], is guaranteed if |M ij | 10 18 GeV.
#8 Namely, φ 3 = φ is the inflaton field in the notation of this subsection.  Since we are considering the subcritical regime, the initial values of φ and s are set to φ 2 init ≡ 0.98φ 2 c,0 and s init ≡ s min (φ init ), respectively, at the time t = 0, while the other initial field values are set to zero. We have confirmed that similar trajectories of the scalar fields are obtained for slightly different the initial values of the inflaton and waterfall fields. #9 Fig. 4 shows the time evolution of ϕ A before the end of inflation as a function of the e-folds N e defined by N e (t) = t end t dt H , (4.38) where t end is the time at the end of inflation. In the plot,Λ = 10 10 GeV is taken to satisfy the condition (4.30) at a percent level. We confirmed that the trajectory of the waterfall field well agrees with the local minimum s min , given in Eq. (4.29). In addition, we found that the τ i (i = 1-3) merely move during the inflation. On the other hand, φ 1 and φ 2 grow as large as O( s ) as seen in the figure. #10 They are, however, still subdominant components in the scalar potential and they do not have #9 The waterfall field value is much smaller than the inflaton one during inflation and the trajectory is almost straight along the inflaton field direction. Therefore, the trajectory of φ 3 -s system has no additional impact on the curvature perturbation, which is already studied in Ref. [103]. #10 φ i (i = 1, 2) are kicked by the term proportional to j Re(M * ij M j3 )φ i φ 3 . The direction that φ i is heading for depends on the sign of j Re(M * ij M j3 ). any impact on the waterfall-inflaton dynamics. For smaller value ofΛ we found that the field values of φ 1 and φ 2 reduce almost linearly, which is summarized in Fig. 5.
Having confirmed the condition (4.30), we derived more quantitative bound, Λ 10 10 GeV , (4.39) or equivalently |M i3 | 10 10 GeV . In the bottom panels of Fig. 4, the field growths of φ 1 and φ 2 get accelerated. This is because the Hubble parameter begins to decrease after the end of inflation since the inflaton field oscillates around the minimum to behave as a matter component. After that, the φ 1 and φ 2 start to oscillate with an angular frequency O(|M ij |). We checked this behavior numerically, which is shown in Fig. 6. Here we have assumed that the inflaton field keeps the coherent oscillation. Realistically the inflaton field decays to leptons and Higgsinos or sleptons and Higgses and it reheats the universe. Then the radiation domination follows. Although φ 1 and φ 2 continue the coherent oscillation, we expect that they do not become the dominant component of the universe because their energy density is highly suppressed. In that case, non-thermal leptogenesis by the inflaton field works and it is expected to provide a sufficient number of lepton asymmetry [105,126,[128][129][130][131][132][133][134][135][136][137][138]. The details depend on the model parameters and they are beyond the scope of our current study. We leave it for our future study.

Conclusion
We consider a supergravity model that has the modular A 4 symmetry. This model accommodates the MSSM augmented by three right-handed neutrino fields that have the Majorana masses. Additionally, two fields that are charged under a gauged U(1) are introduced. They couple to the right-handed neutrinos via the Yukawa interaction and consequently one of the scalar components plays a role of the waterfall field during inflation and acquires a VEV at the global minimum. With the extension, the pattern of the light neutrino mass, generated by the seesaw mechanism, changes and one of the light neutrinos becomes massless. On top of that, the modular A 4 symmetry restricts the mixing pattern. Comparing with the current observations regarding the neutrino mixings, we have found that only the IH case is allowed. The predicted Majorana phase is around ±35 The effective neutrino mass, which determines the decay rate of the neutrinoless double beta decay, is found to be around 14-28 meV and 45-47 meV. Such a relatively large effective mass of O(10) meV will be explored in future experiments.
The supergravity model we consider is based on a hybrid inflation model that has the subcritical inflation regime. Namely, inflation continues below the critical point. In the present model, the three right-handed sneutrinos are the candidates for the inflaton field. Compared to the inflation model studied in Ref. [103], the existence of the Majorana mass terms is a crucial difference, which may affect the inflation dynamics. Imposing the Majorana mass not to affect the inflation dynamics, we have revealed that only one of the right-handed sneutrinos turns out to couple to the waterfall field, then we have derived the upper bound for the Majorana mass scale analytically. We have confirmed the results numerically by solving the equations of motion for scalar fields. It is found the other right-handed sneutrinos grow as large as the VEV of the waterfall field. They are, however, negligible in the total energy density during the inflation if the Majorana mass scale is smaller than O(10 10 ) GeV. Though their field values are much suppressed during inflation, the scalar fields continue to oscillate after inflation, which may affect the subsequent thermal history. For instance, they may contribute to the generation of the lepton asymmetry. The details depend on the parameter and we leave it for the future work.