Hiding neutrinoless double beta decay in the minimal seesaw mechanism

We present a possibility that the neutrinoless double beta decay can be hidden in the minimal seesaw mechanism where the standard model is extended by two right-handed neutrinos which have a hierarchical mass structure. In this framework, the lepton number is violated due to the massive Majorana neutrinos. Especially, we investigate the case that the heavier right-handed neutrino is sufficiently heavy to decouple from the decay while the lighter one is lighter enough than the typical Fermi-momentum scale of nuclei and gives a sizable contribution to the decay. Under the specific condition on mixing elements, the lighter right-handed neutrino can give a significant destructive contribution which suppresses or even hides to the effective mass of the neutrinoless double beta decay. In this case, the flavor structure of the mixing element of the lighter right-handed neutrino with ordinary neutrinos is predicted depending on the Majorana CP violating phase of active neutrinos.


I. INTRODUCTION
Neutrino oscillation experiments have been developed so far and have provided many properties of three active neutrinos, including mass square differences, mixing angles, and even CP violating phase which has been getting to be revealed recently. Unfortunately, all the fascinated neutrino oscillation experiments can give no hint of whether neutrinos are Dirac or Majorana fermions, which is one of the most interesting missing pieces of neutrinos. One promising possibility to attack this issue is finding the phenomenon of the neutrinoless double beta (0νββ) decay. (See as a theoretical review, for instance Ref. [1].) It violates the lepton number by two units and shows a clear signal for physics beyond the Standard Model (SM). The decay can be mediated by massive Majorana neutrinos and the rate is characterized by the so-called effective mass m eff of Majorana neutrinos, which has been constrained by various 0νββ decay experiments until today .
The seesaw mechanism [28][29][30][31][32][33][34] by introducing right-handed neutrinos with Majorana masses is the one of the most attractive scenarios for explaining the origin and the observed smallness of neutrino masses. In this case the lepton number is violated by the Majorana masses and the 0νββ decay is possible to occur. The effective mass is expressed in terms of neutrino masses, mixing angles, and CP violating phases. In the effective SM with three massive neutrinos, there is a possibility where the effective mass is highly suppressed or even vanishes by tuning the lightest active neutrino mass (see, e.g., [1]).
When right-handed neutrinos are much lighter than the unification scale ∼ 10 16 GeV, or even lighter than the weak scale ∼ 100 GeV, they can give a sizable contribution to m eff in addition to active neutrinos' one. In such cases, the masses and mixing elements of right-handed neutrinos must be chosen appropriately without conflicting with the experimental limits on m eff . When all right-handed neutrinos are lighter than a typical scale of Fermi momentum of a nucleus Λ β (∼ O(100) MeV), the contributions of active neutrinos and heavier ones exactly cancel out each other due to the intrinsic property of the seesaw mechanism [35]. Further, it is shown that, when right-handed neutrinos are degenerate, m eff becomes smaller than the one solely from active neutrinos [36]. Although such right-handed neutrinos are attractive to realize the baryogenesis via the oscillation mechanism [37,38], there is no concrete reason to constrain ourselves to keep the degeneracy in general.
In this paper, we present another possibility to suppress the 0νββ decay by hierarchical right-handed neutrinos. As the simplest example, we consider the extended SM with two right-handed neutrinos where one is sufficiently heavier than Λ β to decouple from the system while the other is lighter than Λ β giving a destructive contribution to the decay rate. It is shown that m eff = 0 is possible due to the exact cancellation of the contributions between active neutrinos and the lighter right-handed neutrino if the mixing elements are chosen to be specific values. We then discuss the impacts of this cancellation conditions, especially, on the Majorana CP violating phase and the mass hierarchy of active neutrinos.

II. SEESAW MODEL WITH TWO RIGHT-HANDED NEUTRINOS
We consider here the simplest extension of the SM to explain the observed neutrino masses by adding two righthanded neutrinos #1 ν RI (I = 1, 2) where Φ and α (α = e, µ, τ ) are the Higgs and lepton doublets of the weak SU (2 M I , the seesaw mechanism for neutrino masses is realized. In addition to massive active neutrinos ν i (i = 1, 2, 3) there are heavy neutrinos N I with masses M I which almost correspond to right-handed neutrino states (we simply call them as right-handed neutrinos from now on). These states take part in weak gauge interactions through the mixing as where U αi is the mixing matrix of active neutrinos [39,40] while the mixing elements of N I are given by Θ αI = [M D ] αI M −1 I . Based on the parametrization proposed by Casas and Ibarra [41,42], the Yukawa couplings are written as Here with s ij = sin θ ij and c ij = cos θ ij . δ and η are the Dirac and Majorana CP violating phases, respectively. The 3 × 2 matrix Ω can be expressed as where s ω = sin ω and c ω = cos ω, respectively. ξ = ±1 is sign parameter and ω is a complex parameter, i.e., ω = ω r + iω i . Further, we introduce #1 The extension to the case with three right-handed neutrinos is straightforwardly possible. However, since the number of parameters is increased, the impacts discussed below would be blurred. This issue is beyond our scope.
since it represents the overall strength of the Yukawa couplings (see, e.g., the discussion in Ref. [36]). In practice, the Yukawa couplings scale as F ∝ X ω or X −1 ω for X ω 1 or 1. Throughout this analysis, we choose the convention in which ξ is selected to be positive, and fix θ ij and δ in the mixing matrix U to be the central values of the latest global fit of neutrino oscillation data [43,44].

III. NEUTRINOLESS DOUBLE BETA DECAY
In the considering model, the effective mass in the 0νββ decay is given by where the contribution from active neutrinos is Note that, since only two right-handed neutrinos are introduced, it is impossible to cancel the effective mass from active neutrinos by tuning the lightest neutrino mass. The contribution from right-handed neutrinos is The function f β represents the suppression by the propagator effect of right-handed neutrinos, and we use the approximate formula where Λ β is a typical scale of Fermi momentum of a nucleus which is evaluated as a few hundred MeV varied depending on nucleus and modelings [45][46][47][48]. We consider the case where M 1 < Λ β M 2 and take f β (M 1 ) = 1 and f β (M 2 ) = 0 approximately. Thus, the effective mass becomes independent of Λ β . In this case the effective neutrino mass is expressed as Importantly, we find out that the effective mass vanishes if the complex parameter ω satisfies In Fig. 1 we show the real and imaginary parts of ω satisfying the cancellation condition (12). It is found from Eq. (12) that the maximal value of X ω is achieved by the CP violating phases δ + η = −π/2 for the NH case while η = π/2 in the IH case. Notice that X ω becomes unity (i.e., no imaginary part of ω) when η = −δ in the NH case and η = 0 (π) in the IH case, respectively.

IV. DISCUSSIONS AND CONCLUSIONS
It is, therefore, found that the contribution to m eff from active neutrinos can be obscured by the light right-handed neutrino when Eq. (12) is fulfilled. In this case, we can determine the mixing elements of of N 1 for a given mass depending on the Majorana phase. This point is illustrated in Fig. 2.  Interestingly, the flavor structure of the mixing elements highly depends on the values of Majorana phase η. This pattern of the mixing elements may be tested by future direct search experiments of right-handed neutrinos. In the NH case |Θ µ1 | 2 |Θ e1 | 2 and then the experiments like the peak search in K → µ + N 1 , for instance, would help the observation. On the other hand, in the IH case |Θ e1 | 2 or |Θ µ1 | 2 becomes dominant depending on η, and K → e + N 1 or K → µ + N 1 would be the golden channel for the discovery. Since the relative sizes of the mixing elements are not so much identical we can extract important information of the mass hierarchy and the Majorana phase η from the combination of |Θ e1 | 2 and |Θ µ1 | 2 under the situation of no 0νββ decay is observed.
When the effective mass vanishes, the lifetime of N 1 can be predicted by M 1 and η. In the mass region of interest the possible decay channels are N 1 → ννν and N 1 → νe + e − (when M 1 > 2m e ) and we find the range of the lifetime is The suggested values of the lifetime are so long that N 1 decays after the onset of the big bang nucleosynthesis (BBN) and would destroy the success of the BBN and/or conflict with the observational data of the cosmic microwave background radiation. One possibility to avoid this difficulty is the dilution of the N 1 abundance by the late time entropy production. Such an additional production may be realized by the decay of the heavier right-handed neutrino N 2 [49].
To summarize we have examined the neutrinoless double beta decay in the seesaw mechanism by two right-handed neutrinos. The Majorana nature of active neutrinos and right-handed neutrinos breaks the lepton number of the theory, which may lead to the neutrinoless double beta decay. When the masses of right-handed neutrinos, however, are lighter than or comparable to the scale Λ β , they can give a significant effect, and the effective mass can vanish in some cases.
In this paper we have found a possibility when M 1 Λ β M 2 . It has been shown that N 1 contribution can obliterate the neutrinoless double beta decay even if active neutrinos do contribute it. If this is the case, the unknown parameters related to right-handed neutrinos appeared in Ω are highly restricted and the mixing elements of N 1 can be determined by the Majorana phase and the mass hierarchy of active neutrinos. Inversely speaking, we may obtain important information of the missing piece of neutrino properties, namely the Majorana phase and the mass hierarchy, from the relative sizes among the mixing elements of right-handed neutrinos measured at the future terrestrial experiments together with no neutrinoless double beta decay.