A model realizing inverse seesaw and resonant leptogenesis

We construct a model realizing the inverse seesaw mechanism. The model has two types of gauge singlet fermions in addition to right-handed neutrinos. A required Majorana mass scale (keV scale) for generating the light active neutrino mass in the conventional inverse seesaw can be naturally explained by a"seesaw"mechanism between the two singlet fermions in our model. We find that our model can decrease the magnitude of hierarchy among the mass parameters by $\mathcal{O}(10^4)$ from that in the conventional inverse seesaw model. We also show that a successful resonant leptogenesis occurs for generating the baryon asymmetry of the universe in our model. The desired mass degeneracy for the resonant leptogenesis can also be achieved by the"seesaw"between the two singlet fermions.


Introduction
Small neutrino masses are observed in neutrino oscillation experiments. One of simple mechanisms to generate the small neutrino masses is the conventional Type-I seesaw mechanism [1], in which right-handed neutrinos N i R (i denotes the generation) are introduced to the standard model ( There are several extensions of the Type-I seesaw model. One extension is the inverse (double) seesaw mechanism [2][3][4] with additional singlet fermions S α . In a basis of (ν c L , N R , S) T with three flavors (generations), a neutrino mass matrix is given by When one assigns the lepton number one unit to ν L , N R , and S (S has an opposite lepton number with respect to that of N R ), the Majorana mass terms of S do not conserve the lepton number. Note that the absence of (M) 13 and (M) 31 in Eq. (1) are ensured by a field redefinition. Assuming µ ≪ M D < M S , one can describe the neutrino mass as M ν ≃ µM 2 D /M 2 S by diagonalizing the above mass matrix. For M D = 10 GeV and M S = 1 TeV, µ ≃ 1 keV is required for generating the small active neutrino mass scale. In this model, one obtains the heavy Majorana neutrinos with masses as M S ± µ/2. Thus, when µ ≪ M S , such neutrinos are degenerate in mass and can realize the resonant leptogenesis mechanism to generate the baryon asymmetry of the universe (BAU) [5][6][7][8]. The explanations of neutrino experimental data and dark matter in the generic class of the inverse seesaw model have been discussed in [9] and [10], respectively.
In this work, we will discuss the inverse seesaw model realized by a "seesaw" mechanism in the TeV scale physics. Our model has two kinds of new gauge singlet fermions S 1 and S 2 in addition to N R , which corresponds to the n = 2 multiple seesaw mechanism in Ref. [11]. We will find that our model can naturally induce very small mass difference between heavy (∼ TeV scale) neutrino states, which can also be responsible for a successful resonant leptogenesis.

Inverse seesaw from "seesaw"
We discuss a realization of the inverse seesaw from the "seesaw" mechanism. The relevant Lagrangian is given by whereH ≡ iσ 2 H * , H is the SM Higgs doublet, L is the left-handed lepton doublet, N R is the right-handed neutrinos, Φ 1 and Φ 2 are gauge singlet scalars under the SM gauge groups, S 1 and S 2 are gauge singlet fermions, and M µ is a Majorana mass of S 2 . Note that S 2 is added to the original inverse seesaw mechanism. Here details of additional symmetries in our model are not specified, but discussed later. In order to reproduce two (solar and atmospheric) mass scales of the active neutrinos, one must introduce at least two generations for the right-handed neutrino or the gauge singlet fermions. We omit the generation and flavor indices for fermions in Eq. (2). After spontaneous gauge symmetry breaking, one can describe a neutrino mass matrix as in the basis of (ν c L , N R , S 1 , , and these are described by matrices. 1 If one adds three generations for each singlet fermion, the neutrino mass matrix M is a 12 × 12 matrix. When we assume that values of all matrix elements of M S 2 are much smaller than those of M µ ((M S 2 ) jk ≪ (M µ ) lm ), we can diagonarize lower right 2×2 sub-matrix (integrate out S 2 field), i.e. utilize a "seesaw" mechanism. Then, the block diagonalization gives in a basis. Notice thatM takes the same form of the original inverse seesaw as in Eq. (1), and the smallness of µ can be naturally realized by the "seesaw" mechanism, The mass matrix for the three active neutrinos M ν can be obtained after the inverse seesaw as in the flavor basis of active neutrinos. The magnitude of the matrix elements of active neutrino is realized as In the conventional (TeV scale) inverse seesaw mechanism, one should require µ in the mass matrix of Eq. (1)

Leptogenesis
Next, we discuss a generation of the BAU. Our model includes several singlet Majorana fermions, and masses of some of them can be taken as O(1) TeV. Thus, the resonant leptogenesis [13] might be possible in the model.
We start from the mass matrix Eq. (5). Since typical size of the matrix elements of M S 1 is much larger than that of µ, a mixing angle for a block diagonalization of lower right 2 × 2 sub-matrix of Eq. (5) is almost maximal. Thus,M is rotated as The relevant Lagrangian for the resonant leptogenesis is given from Eq. (2) as for the eigenstates of X ± . Note that typical size of matrix elements of Y N ν and Y S ν is the same order as that of Y ν in Eq.
2)Y ν at the leading order. Hereafter we assume that 3×3 matrices M S 1 and µ are diagonal matrices, for simplicity. We also assume the hierarchical structure for M S 1 as m S 1 ≡ (M S 1 ) 11 ≪ (M S 1 ) 22 , (M S 1 ) 33 so that the BAU can be induced by the decays of the first generation of X ± (≡ χ ± ) whose masses are obtained as The lepton asymmetry from the decays of χ − and χ + is calculated as [5,18] where and Γ ± = A ± m χ ± /(8π) is the decay width of χ ± . The baryon asymmetry is given by the lepton asymmetry as where g * = 106.75 is the relative degree of freedom and K ± = Γ ± /(2H(T ))| T =mχ ± with the Hubble constant H(T ) = 1.66 √ g * T 2 /m Pl . Note that the baryon asymmetry is enhanced for In order to obtain the baryon asymmetry by the decays of χ ± , the χ ± should be decoupled at T ∼ m χ ± , which is realized for the Yukawa couplings (Y N ν ) α1 and (Y S ν ) α1 being < O(10 −6 ). Under these conditions, the appropriate order of r in Eq. (11) for the resonant leptogenesis is r ∼ 10 −9 , which can also be naturally realized in our model. Regarding with masses of additional scalars Φ 1,2 , these must be larger than the masses of χ ± of O(1) TeV. If those masses are smaller than the TeV scale, χ ± decay into the scalars. As a result, the lepton asymmetry cannot be produced. On the other hand, the VEV of Φ 2 should be larger than O(10) MeV to realize the inverse seesaw when Y S 2 ≤ O(1). Such a hierarchy between the mass and VEV can be realized in the neutrino-philic Higgs model [19] (see also [20,21]). The relevant scalar potential for the realization is, for example, 2 Such a hierarchical mass structure among singlet Majorana fermions can be realized by several models [14][15][16][17]. where m Φ 1,2 , m, λ Φ 1,2 , and λ 3,4,5 are all assumed to be real and positive, for simplicity. The In addition, when one introduces the symmetry, which forbids the term (13), the hierarchy m ≪ m Φ 1,2 seems to be natural. As a result, | Φ 2 | ≪ | Φ 1 | can be realized, where an assumption λ Φ 2 | Φ 2 | 2 ≪ m 2 Φ 2 is consistent with this realization. Thus, one can have the hierarchy between the VEVs, | Φ 1 | ≃ m Φ 1 / λ Φ 1 O(1) TeV and Φ 2 = O(10) MeV, when one takes m = O(10) GeV and masses of Φ 1,2 as O(1) TeV. Here we take masses of Φ 1,2 are larger than the masses of χ ± . The above calculation is valid in this case. In our model, the other singlet fermions (≃ S 2 ) can also decay into L and H through the mixing between S 1 and S 2 . But the process cannot generate the sufficient magnitude of lepton asymmetry because S 2 is not degenerate with N R and S 1 state. Note that one does not need a fine tuning among masses of Majorana fermions to realize the BAU as seen below. Figure 1 shows the baryon asymmetry as a function of µ. We assume the hierarchical structures for Yukawa couplings Y N ν and Y S ν so as to realize the out of thermal equilibrium of χ ± at the . The observed baryon asymmetry η B = 6×10 −10 is also shown by the horizontal line in Fig. 1 As discussed above, the favored scale of µ for the active neutrino mass can be realized by the "seesaw" between S 1,2 fermions. In addition, the first generations of S 1 and N R (those mass eigenstates are χ ± ) play a role for generating the BAU via the resonant leptogenesis. In the case, the required size of mass degeneracy between χ ± for the resonant leptogenesis, µ ≃ O(1) keV, can also be realized by the "seesaw" between S 1,2 . The both realizations are non-trivial results in our model.

Signatures for LHC experiment
We discuss signatures of this model at the LHC experiment. This model can induce lepton number violating processes. One interesting process is the like-sign dilepton production, qq ′ → ℓ ± ℓ ± W ∓ , where the lepton number conservation is violated by two units, ∆L = 2, due to the Majorana nature of neutrinos. References [22][23][24][25][26][27][28][29] explore this process at the LHC experiment in the SM with the right-handed Majorana neutrinos (see also [30] for a review of the collider phenomenology with the right-handed and sterile Majorana neutrinos). 3 According to Ref. [25], it is found that there is 2σ (5σ) sensitivity for the µ ± µ ± modes in the mass range of a Majorana neutrino of 10 GeV ≤ m χ ≤ 350 (250) GeV at the 14 TeV LHC experiment with 100 fb −1 . Regarding with the inverse seesaw case, the singlet neutrinos and fermions are pseudo-Dirac neutrinos due to a small Majorana mass µ, and the neutrinos contain tiny Majorana state.
The ratio of the Majorana state is typically determined by µ/M S 1 ≃ 1 keV/1 TeV ≃ O(10 −9 ). Thus, since the like-sign dilepton production process in the inverse seesaw case is suppressed by (µ/M S 1 ) 2 ≃ O(10 −18 ) compared with the results of Refs. [22][23][24][25], the signatures of the process in the inverse seesaw case cannot reach at the sensitivity at the LHC experiment.
Similarly, for the other singlet fermions (≃ S 2 ) with the lepton number violating Majorana mass of O(1) TeV in our model, the result of analysis in Refs. [22][23][24][25] cannot simply be adopted.
Since the like-sign dilepton production process is induced through the mixing between S 1 and S 2 in addition to the mixing of the pseudo-Dirac states of N R and S 1 mentioned above, the amplitude is typically suppressed by (M S 2 /M µ ) 2 (µ/M S 1 ) 2 ≃ (10 MeV/1 TeV) 2 (1 keV/1 TeV) 2 ≃ O(10 −28 ). Therefore, the collider signatures of the these singlet fermions in our model cannot also reach at the sensitivity at the LHC experiment.
The above discussion can be generalized to the multiple seesaw models [11]. For the n = 2k+1 (k = 0, 1, 2, · · · ) multiple seesaw models (k = 0 is the conventional inverse seesaw model), the active neutrino mass matrix in the n = 2k + 1 (k ≥ 1) multiple seesaw models is given by where n denotes the number of gauge singlet fermions S without the number of generation (flavor) and M µ is the lower right element of the (n + 2) × (n + 2) generalized neutrino mass matrix. The like-sign dilepton production process is suppressed by (M µ /M Sn ) 2 in all models of On the other hand, for the n = 2k (k = 1, 2, · · · ) multiple seesaw models (k = 1 case is our model), the active neutrino mass matrix can be given by The amplitude of the like-sign dilepton production process is suppressed by (M Sn /M µ ) 2 × (1 keV/M S 1 ) 2 in all models of n = 2k multiple seesaw with M Sn ≪ M µ ≃ M S 1 ≃ · · · ≃ M S n−1 .
Note that since the n = 2k multiple seesaw model is reduced to the inverse seesaw model, there is an additional suppression (M Sn /M µ ) 2 in the n = 2k cases compared with the n = 2k + 1 multiple seesaw models. 4

Summary
We have discussed the inverse seesaw model realized by a "seesaw" mechanism. The conventional  (7)). Thus, the magnitude of mass hierarchy in the model can be decreased to M µ /M S 2 ≃ O(10 5 ), which is due to the "seesaw" mechanism between S 1 and S 2 singlet fermions.
We have also considered a leptogenesis scenario with a mass degeneracy for generating the BAU, the so-called resonant leptogenesis. The scenario can be realized by the keV scale mass degeneracy between the first generations of the right-handed neutrino and one of the singlet 4 In Refs. [32][33][34], the authors have discussed the Higgs signatures via the large Yukawa couplings in the inverse seesaw model at the LHC. The second and third generations of X ± in our model might be adopted to the discussion although the Yukawa couplings of the first generations are too small to lead the sufficient magnitude of the signals.
fermions. We have shown that such mass degeneracy can also be realized by the "seesaw" in our model, and thus the successful resonant leptogenesis is achieved. Regarding the signatures of qq ′ → ℓ ± ℓ ± W ∓ processes at the LHC experiment, our model cannot reach at the sensitivity at the LHC due to the significant suppression by the mixings between the singlet fermions. Finally, we comment on a realization of our model. One simple way to obtain our model is to introduce a symmetry. In Ref. [11], the global U(1) × Z 2N symmetry for realizing the multiple seesaw models have been discussed. Following that, our model (the n = 2 multiple seesaw model) can be obtained by imposing the global U(1) × Z 6 symmetry. Here the global U(1) symmetry is identified with the lepton number, U(1) L , and a charge assignment under the symmetry is given in Tab. 1. Note that the Majorana mass term (M µ /2)S c 2 S 2 induces the lepton number violation.