Direct Search for Right-handed Neutrinos and Neutrinoless Double Beta Decay

We consider an extension of the Standard Model by two right-handed neutrinos, especially with masses lighter than charged $K$ meson. This simple model can realize the seesaw mechanism for neutrino masses and also the baryogenesis by flavor oscillations of right-handed neutrinos. We summarize the constraints on right-handed neutrinos from direct searches as well as the big bang nucleosynthesis. It is then found that the possible range for the quasi-degenerate mass of right-handed neutrinos is $M_N \geq 163 \MeV$ for normal hierarchy of neutrino masses, while $M_N = 188 \text{--} 269 \MeV$ and $M_N \geq 285 \MeV$ for inverted hierarchy case. Furthermore, we find in the latter case that the possible value of the Majorana phase is restricted for $M_N = 188 \text{--} 350 \MeV$, which leads to the fact that the rate of neutrinoless double beta decay is also limited.


Introduction
Various oscillation experiments have revealed non-zero masses of neutrinos. The observation shows that there exist two mass scales of neutrinos, the differences of mass squared ∆m 2 atm ≃ 2.43 × 10 −3 eV 2 and ∆m 2 sol ≃ 7.54 × 10 −5 eV 2 [1] , related to the so-called atmospheric and solar neutrinos, respectively. In the (canonical) Standard Model neutrinos are exactly massless, and new physics beyond the Standard Model is indicated. The crucial questions are then (i) what is the origin of neutrino masses? and (ii) how do we verify it experimentally?
One of the simplest and most attractive ways to generate neutrino masses is to introduce right-handed neutrinos ν R 's into the Standard Model. In this case neutrinos can obtain the Dirac masses as quarks and charged leptons. Furthermore, since these neutral fermions are singlets under the gauge group of the Standard Model, the Majorana masses of right-handed neutrinos are also allowed. Notice that we should require at least two right-handed neutrinos in order to explain ∆m 2 atm and ∆m 2 sol . When the Majorana masses are much heavier than the Dirac ones, the smallness of neutrino masses can be explained by the seesaw mechanism [2]. The mass eigenstates are then separated into three lighter and two heavier ones. The former ones are responsible to oscillation phenomena while the latter ones have not bean confirmed by experiments.
In this framework all the neutrinos are Majorana particles, and the (total) lepton number is violated which may be tested by the neutrinoless double beta (0ν2β) decay. Especially, when only two right-handed neutrinos are present, lightest (active) neutrino becomes massless and the predicted range of the rate for 0ν2β decay is very limited. Interestingly, the future 0ν2β experiments may probe such a range in the inverted hierarchy of neutrino masses. (See, for example, a review [3]) On the other hand, right-handed neutrinos can also play important roles to generate the baryon asymmetry of the universe. The concrete scenario for this baryogenesis strongly depends on the mass scales of right-handed neutrinos. In the canonical leptogenesis scenario [4] the lightest right-handed neutrino should be heavier than O(10 9 ) GeV [5,6] when they have hierarchical masses. If ν R 's are produced non-thermally at the reheating of the inflation, the required mass can be small as O(10 6 ) GeV [7]. Moreover, the resonant leptogenesis [8] by quasi-degenerate right-handed neutrinos is possible even for smaller masses of ν R 's. Interestingly, the successful scenario of baryogenesis can be realized even if the masses are smaller than the electroweak scale [9,10]. In these extensions of the Standard Model, therefore, the detailed examination of right-handed neutrinos is crucial in order to elucidate the mechanism to generate neutrino masses as well as the cosmic baryon asymmetry. For this purpose, the scenario with lighter ν R 's is more promising.
In this paper, we consider the Standard Model with two right-handed neutrinos which is probably the minimal extension to explain the neutrino oscillation results and the baryon asymmetry. Especially, we assume masses of these neutral leptons are smaller than K ± meson.
Such light ν R 's are good targets of the search experiments by using K ± and π ± decays (see Refs. [11,12,13,14,15]). It should be noted that, even when ν R 's are lighter than m K ± , ∆m 2 atm and ∆m 2 sol from the oscillation experiments can be explained via the seesaw mechanism by requiring that the Yukawa coupling constants of neutrinos are sufficiently small, and also the enough baryon asymmetry can be generated via the mechanism [9,10] by requiring that the ν R 's are sufficiently degenerate.
Under the above situation, we shall summarize the constraints on such light ν R 's from the direct search experiments as well as the cosmological observations, and then identify the allowed region in the parameter space of the model. In particular, the possible values of ν R masses are presented, as pointed out by Ref. [12].
In addition, we shall discuss the implications of the allowed parameter space. Especially, we find that the Majorana phase (which is the one of the CP violating parameter in the lepton sector) is restricted from the search and cosmological constraints when the inverted hierarchy of neutrino masses is considered. It will be then discussed that this leads to the important impact on the 0ν2β experiments in future.

Extension by two right-handed neutrinos
Let us start with the framework of the present analysis. We consider the Standard Model extended by two right-handed neutrinos ν R 's #1 together with where Φ and L α = (e L , ν L ) T are Higgs and lepton doublets, respectively. Here and hereafter, the indices of flavor are implicit unless otherwise mentioned. The 3 × 2 Yukawa matrix of neutrinos is denoted by F and the 2 × 2 matrix of Majorana masses by M M . Notice that we choose the basis in which the mass matrix of charged leptons and M M are diagonal. We assume that the Dirac masses of neutrinos M D = F Φ are much smaller than the Majorana ones M M for the seesaw mechanism. In this case, the mass matrix of light neutrinos participating the observed flavor oscillation is given by By using this relation, we can parameterize, without loss of generality, the Yukawa matrix F as follows [18,19] where M diag This parametrization is the same as that in Ref. [20]. The mixing matrix of light neutrinos, called #1 By adding the keV right-handed neutrino, it can play a role of the cosmic dark matter [17]. The results in the present analysis can be applied to the neutrino Minimal Standard Model (νMSM) [17,10] if the dark matter physics requires no limitation to the parameters of the considering two right-handed neutrinos.
as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [21], is written as with s ij = sin θ ij and c ij = cos θ ij . It is then found that the flavor neutrino ν L is written in terms of mass eigenstates ν's and N's as in the NH case, while in the IH case. Here ω is an arbitrary complex number and ξ = ±1 is the sign parameter. Here we apply the convention Imω ≥ 0, ξ = ±1 and η = 0-π.
for the NH (IH) case (at the 3σ level) [1]. Hereafter, we shall adopt the central values of these observables. It should be stressed that these observational data from the oscillation experiments can be reproduced being independent on the parameters of right-handed neutrinos, i.e., M N , ∆M, ω and ξ. In practice, we shall consider the case when M N < m K ± and ∆M ≪ M N (see the discussions below).
It is interesting to note that two right-handed neutrinos with M N < m K ± introduced above can be responsible to the baryon asymmetry of the universe by invoking the mechanism proposed in Ref. [10]. By originating the CP violations in the flavor oscillation as well as the production/destruction processes of right-handed neutrinos, the asymmetries of left-handed leptons are generated which is partially converted into the baryon asymmetry due to the rapid sphaleron transition [22]. See    We perform the numerical study of the generation of the baryon asymmetry #2 and identify the parameter region in which the observed value (the baryon to entropy ratio) n B /s = 8.8 × 10 −11 [27] can be explained. Here, we use the central values for the neutrino oscillation parameters and vary over all the possible value for the unknown parameters.
The region for the successful baryogenesis in the M N -∆M plane is shown in Fig. 1. It is then found that the enough baryon asymmetry can be generated if the masses of right-handed neutrinos are M N ≥ 2.1 MeV for the NH case 0.7 MeV for the IH case .
It is also found that the mass difference of two right-handed neutrinos should be ∆M ≪ M N . In these regions, we can explain the neutrino masses in the oscillation experiments and the baryon asymmetry at the same time only by using the two right-handed neutrinos. Such a mass bound had already been obtained as shown in Fig. 1 of Ref. [16]. Our obtained region is wider than theirs, especially the lower bound on M N in the NH case is smaller by a factor of five. This is mainly because the different values for θ ij and ∆m ij are chosen.

Constraints from direct search and cosmology
The heavy neutrinos (N ≃ ν R ) have the weak interaction due to the mixing induced by the seesaw mechanism, which strength is suppressed by the mixing elements Θ compared with #2 To estimate BAU, we solve numerically the kinetic Eqs. (5.2) and (5.3) in Ref. [28] by neglecting the momentum dependence in the density matrix of ν R 's, for simplicity. The details are found in Ref. [28] ordinary neutrinos. Thereby, they can be produced by meson decays (e.g. π → Ne, K → Nµ), and can decay to Standard Model particles (e.g. N → ννν, N → νe + e − , N → eπ). Using such processes various experiments have been conducted to search heavy neutrinos directly. Since such neutrinos have not been discovered, the upper bounds of the mixing elements have been imposed. In the discussing mass range, the beam dump experiment, PS191 experiment [29,30], have placed the strongest bounds on Θ. In this experiment heavy neutrinos are produced by π and/or K decays and charged particles from N decays in the far detector are searched as signal events. These processes are induced as the Θ 4 effect, therefore the experiment sets the upper bounds of the mixing elements as the following form, |Θ α I | 2 (a|Θ e I | 2 + b|Θ µ I | 2 + c|Θ τ I | 2 ), where α and I are the flavor indices of left-handed neutrinos and heavy neutrinos, respectively. a, b and c are coefficients depending on the mass and decay channel. In Refs. [29,30], such bounds had been derived by assuming that heavy neutrino is a Dirac particle and that it has only the charged current interaction. As pointed out in Ref. [11] (see also Ref. [31]), however, the bounds in the considering scenario must be evaluated with two Majorana (heavy) neutrinos including the neutral current interaction.
On the other hand, heavy neutrinos are also restricted from cosmological observation. To keep the success of the Big Bang Nucleosynthesis (BBN), the lifetime of N's, τ N , is required to be shorter than 0.1 sec for M N > m π , and τ N /sec < t 1 (M N /MeV) β + t 2 with t 1 = 128.7, t 2 = 0.04179, and β = −1.828 for M N ≤ m π [32]. Note that the lifetime bound for M N ≤ m π is discussed recently in Ref. [33]. Here we apply the bounds in Ref. [32] to make the conservative analysis.
From these constraints we evaluate the possible region of the heavy neutrinos as performed in previous work [12,23,31]. The direct search experiments put the upper bounds on |Θ| which result in the lower bound on τ N . On the other hand, the BBN puts the upper bound on τ N . Thus, we can identify the possible region of M N and τ N . Fig. 2 shows the results of this work. It is found that the following masses of N's are allowed.
It should be noted that a specific element of the mixing matrix Θ can be very suppressed compared with other elements by choosing the parameters carefully [20,31]. The possible component of this suppression is |Θ e | 2 in the NH case, while that is |Θ µ | 2 or |Θ τ | 2 in the IH case. For the current data from oscillation experiments [1], the suppression condition in the NH case cannot be satisfied exactly. However, |Θ e | 2 can be smaller by a few orders of magnitude compared with |Θ µ | 2 and |Θ τ | 2 near the suppression point. In the IH case the suppression condition can be satisfied even for the current data. The suppression is, however, forbidden to escape the constraints from search experiments and the BBN for M N ≤ 350 MeV. On the other hand, the experimental bounds on |Θ e | 2 are stronger than that on |Θ µ | 2 in almost all interesting mass range. Due to the feature of mixing elements |Θ| 2 and the present experimental bounds, the allowed range of M N in the NH case is wider than that in the IH case as shown in Fig. 2.  Before closing this section, it should be mentioned that we have numerically confirmed the enough baryon asymmetry can be generated in all the allowed regions shown in Fig. 2. As a result, we have found the parameter space of heavy neutrinos which can explain the neutrino masses as well as the observed values of BAU without conflicting with the experimental and cosmological constraints.

Implication to neutrinoless double beta decay
In the allowed parameter space, we find a distinctive feature in the IH case, namely the possible value of the Majorana phase η is restricted by the present search experiments and the BBN, as shown in Fig. 3. In the figure, the shaded region is excluded, and the (red) solid and (blue) dashed lines are derived by the experimental bound from the search mode K → eN → e(eπ) and that from the mode K → µN → µ(µπ) in PS191, respectively. The former one puts essentially the upper bound on |Θ e | 2 while the latter puts the bound on |Θ µ | 2 . In the left side of the (green) dotted line the sign parameter ξ is allowed to be only +1, while both signs ξ = ±1 are allowed in the right side. It can be seen that, when the mass is M N = 188-269 MeV, the Majorana phase close to π/2 is possible and η ≃ 0 and π are forbidden. When the mass becomes heavy as M N = 285-350 MeV, the possible range of η is changed, and η ≃ 0 and π as well as η ≃ π/2 are disfavored. If N's are heavy enough as M N > 350 MeV, the full range of η is consistent with the constraints.
Notice that the typical value of X ω is O(10) in the allowed region of Fig. 2. We can see from Eq. (9) that the mixing elements depend crucially on the combination ξ sin η. For M N ≤ 332 MeV, the constraint on |Θ e | 2 favors ξ sin η ∼ 1, and hence ξ = −1 and also η ∼ 0, π are excluded. In addition, when M N = 269-350 MeV, the constraint on |Θ µ | 2 favors ξ sin η ∼ −1, which excludes the possibility η ∼ π/2. We have found that the possible range of η is restricted in the IH case when M N ≤ 350 MeV. We should comment that the similar analysis can be done for the NH case. However, the present observational data of neutrino oscillation and search experiments cannot constrain the value of the Majorana phase. If the future experiments will improve the data we may also have the limitation on η even in the NH case.
Next, we turn to discuss the impact on the 0ν2β decay from the result in the IH case. The decay rate of 0ν2β decay is characterized by the effective neutrino mass m eff (see, e.g., Refs. [36,3]). In the model under consideration, it is given by [20] where m ν eff ≡ i m i U 2 ei and M I denotes the mass eigenvalue of the Ith heavy neutrino. f β (M I ) is a function which represent the suppression of the nuclear matrix element to the contribution of heavy neutrinos. The function is unity for M I ≪ 1GeV, and decrease as 1/M 2 I for M I ≫ 1GeV. The details of f β are described in Ref. [20] (see also Refs. [34,35]).
As shown in this equation, |m ν eff | have a significant dependence on η. By varying η from 0 to π we find |m ν eff | = (1.82-4.79) × 10 −2 eV. This range is indeed for the conventional case when only light neutrinos contribute to the 0ν2β decay.
There are two, important impacts on |m eff | from heavy neutrinos when the masses are smaller than about 500 MeV. The first is the destructive contribution from these particles [20], which can be easily seen from Eq. (11). The second is the impact from Majorana phase restricted by experimental and cosmological constraints. In Fig. 4 it is shown the predicted range of |m eff | being consistent with the constraints on heavy neutrinos.
We can see the correlation between the allowed region of η in Fig. 3 and the predicted range of |m eff | in Fig. 4. For M N = 188-269 MeV η ∼ π/2 is favored and then the lower value of |m eff | is predicted, while η ∼ π/2 is disfavored for M N = 285-350 MeV and then the larger value of |m eff | is predicted. When M N > 350 MeV, there is no limitation of η but the presence of heavy neutrinos induces the smaller value of |m eff | compared with |m ν eff |.

Conclusions
We have considered right-handed neutrinos which are responsible to neutrino masses and BAU. Especially, we have discussed the case when they are quasi-degenerate and lighter than charged kaon. It has been found that the constraints from BBN and direct search experiments can be avoided when the degenerate mass is and M N = 188-350 MeV. In such a case, the rate of 0ν2β decay is also limited which may be different from the prediction by the conventional scenario with three active neutrinos. Thus, the future experiments of 0ν2β decay may give us an important hint for the right-handed neutrinos considered in this analysis.