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Mixing of Active and Sterile Neutrinos Takehiko Asaka1,2, Shintaro Eijima3 and Hiroyuki Ishida3 1 1 0 1Department of Physics, Niigata University, Niigata 950-2181, Japan 2 2Max-Planck-Institut fu¨r Kernphysik, Postfach 103980, 69029 Heidelberg, Germany n a 3Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan J 7 January 10, 2011 ] h p - p e h [ 1 v Abstract 2 8 We investigate mixing of neutrinos in the νMSM (neutrino Minimal Standard Model), 3 which is the MSM extended by three right-handed neutrinos. Especially, we study ele- 1 . ments of the mixing matrix Θ between three left-handed neutrinos ν (α = e,µ,τ) and 1 αI α 0 two sterile neutrinos NI (I = 2,3) which are responsible to the seesaw mechanism gen- 1 erating the suppressed masses of active neutrinos as well as the generation of the baryon 1 asymmetry of the universe (BAU). It is shown that Θ can be suppressed by many or- : eI v ders of magnitude compared with Θ and Θ , when the Chooz angle θ is large in the µI τI 13 i X normal hierarchy of active neutrino masses. We then discuss the neutrinoless double beta r decay in this framework by taking into account the contributions not only from active a neutrinos but also from all the three sterile neutrinos. It is shown that N and N give 2 3 substantial, destructive contributions when their masses are smaller than a few 100 MeV, and as a results Θ receive no stringent constraint from the current bounds on such eI decay. Finally, we discuss the impacts of the obtained results on the direct searches of N in meson decays for the case when N are lighter than pion mass. We show that 2,3 2,3 there exists the allowed region for N with such small masses in the normal hierarchy 2,3 case even if the current bound on the lifetimes of N from the big bang nucleosynthesis 2,3 is imposed. It is also pointed out that the direct search by using π+ e+ +N and 2,3 → K+ e++N might miss such N since the branching ratios can be extremely small 2,3 2,3 → due to the cancellation in Θ , but the search by K+ µ+ +N can cover the whole eI 2,3 → allowed region by improving the measurement of the branching ratio by a factor of 5. 1 Introduction The extension by right-handed neutrinos is one of the most interesting physics beyond the Minimal Standard Model (MSM), since it gives a simple solution to the problem of the neu- trino masses confirmed by various oscillation experiments. Usually, right-handed neutrinos are introduced with superheavy Majorana masses and sizable Yukawa coupling constants in order to realize the seesaw mechanism [1], which accounts naturally for the smallness of neutrino masses. Furthermore, the decays of such right-handed neutrinos can be a source of the baryon asymmetry of the universe (BAU) through the leptogenesis mechanism [2, 3]. When the masses of right-handed neutrinos are hierarchical, the observed BAU requires the mass of the lightest one should be heavier than about 109 GeV [4]. Although such singlet fermions provide simple and natural solution to the origins of neutrino masses and BAU at the same time, it is almost impossible to test them at experiments in the future. Itshould, however, benotedthatright-handedneutrinoscanbringaboutimportantphysical phenomena, even when the scale of Majorana masses are so light to be produced in terrestrial experiments. OneattractivepossibilityistheνMSM[5,6]inwhichthreeright-handedneutrinos are introduced with masses below the electroweak scale.#1 Interestingly, this simple model can explain the origins of neutrino masses, BAU and also dark matter of the universe at the same time. The Yukawa coupling constants are so small that the seesaw mechanism still works, and mass eigenstates of neutrinos are divided into two groups, active and sterile neutrinos. The flavour mixing of active neutrinos accounts for neutrino oscillations observed in experiments. On the other hand, three sterile neutrinos, N , N , and N , solve cosmological problems in the 1 2 3 MSM. One of the sterile neutrinos, N , plays a role of dark matter [14].#2 When its mass is in 1 the keV order, it can be produced by the so-called Dodelson-Widrow mechanism [17], i.e., by the thermal scatterings through its mixing with the left-handed neutrinos. See Refs. [18]–[27]. This dark-matter particle receives severe astrophysical constraints [28]. One important bound comes from the X-ray background [19]–[21] and the other comes from the cosmic structure at small scales like the Lyman α forest [29]–[32]. Even when these constraints are imposed, as shown in Ref. [27], the correct abundance of the dark matter can be obtained through the Dodelson-Widrow mechanism by invoking the resonant production [18] in the presence of the large lepton asymmetry. Notice that N plays no significant role in the seesaw mechanism [5]. 1 This is because Yukawa coupling constants of N is so small that its contribution to the mass 1 matrix of active neutrinos is negligible. Due to the very suppressed interaction the direct search of N at experiments is very difficult. However, it can be observed by the specific spectrum in 1 #1TheexplanationoftheLSNDanomalyinthisframeworkhadbeeninvestigatedinRef.[7]. Thenon-minimal coupling of the Higgs field to gravity allows to realize the cosmic inflation [8]. Further, various extensions of the model have been discussed [9]-[13]. #2It has been investigated various phenomenon in astrophysics by the sterile neutrino dark-matter, e.g., the explanation of the pulsar kick [15]. See, for example, a review of this issue [16] and references therein. 1 the X-ray background coming from the decay of N into active neutrino and photon. It has 1 also been discussed the search in laboratory [33]. The rest two sterile neutrinos, N and N , are responsible to generate not only the seesaw 2 3 masses of active neutrinos but also BAU through the mechanism [34]. The flavour oscillation between N and N in the early universe induces the separation of lepton asymmetry between 2 3 left- and right-handed leptons and the asymmetry in the left-handed sector is partially con- verted into the baryon asymmetry by the sphaleron processes [35]. The νMSM realizes this baryogenesis scenario without conflict with the observational data of the neutrino oscillations when their masses are quasi-degenerate and in the range (0.1)– (10) GeV [6]. See also the O O recent analysis in Refs. [36, 37]. It isinteresting tonotethat sterileneutrinos N andN canbetestedinvariousexperiments 2 3 as pointed out in Ref. [38]. This is crucially important to reveal the origins of the neutrino masses as well as the cosmic baryon asymmetry. For this purpose, we would like to study the mixing of sterile neutrinos with left-handed neutrinos ν (α = e,µ,τ) in this paper. The α elements of such mixing matrix, Θ , are vital to discuss phenomenology of the νMSM, since αI the strength of the interactions of sterile neutrinos is determined by them. Sterile neutrinos in the model possess the Yukawa interactions and also the weak gauge interactions via the above mixing after the electroweak symmetry breaking. Since the elements Θ are proportional to αI the Yukawa coupling constants F , both interactions are controlled by Θ . αI αI It should be noted that the mixing elements of N and N can take values varying by many 2 3 orders of magnitude. This point had already been discussed by using the model with the lepton symmetry [39]. As we will show in Sec. 3, the mixing elements increase exponentially as Θ exp(Imω) for large Imω (ω is a complex parameter in the neutrino Yukawa matrix) αI ∝ keeping the masses and mixing angles of active neutrinos unchanged. This enhancement leads to the various significant impacts on phenomenology of sterile neutrinos. For example, it gives the larger production/detection rates in the search experiments, the larger CP asymmetry in baryogenesis, and the shorter lifetime of N making them cosmologically harmless. 2,3 The purpose of this paper is, thus, to investigate the mixing elements Θ in detail, and to αI reveal how the elements of N and N depend on mass hierarchy, mixing angles and CP phases 2 3 of active neutrinos in addition to the parameters of sterile neutrinos. Our analysis will show that there can be strong hierarchy among the mixing elements Θ , Θ and Θ depending on eI µI τI choice of the parameters. Especially, it will been pointed out that the strong suppression in the mixing elements of electron type Θ can happen for the normal hierarchy of active neutrino eI masses, which enlarges the allowed region of the model. Although we shall consider the νMSM, the results of the mixing elements Θ in this paper can be applied to the general models of αI the seesaw mechanism with two right-handed neutrinos. Further, we would like to discuss the implications to two phenomena of sterile neutrinos. The first one is the neutrinoless double beta decay in which the mixing elements of active and sterile neutrinos play the crucial roles. This problem had already been discussed in Ref. [40]. 2 We will extend the analysis especially when the masses of N and N are smaller than a few 2 3 100 MeV, and show that the rates of the decays in the νMSM is smaller than those in the usual case where only active neutrinos give the contribution. Thus, the model is free from significant constraints discussed in Ref. [41]. The other one is the search of N produced in the decays 2,3 of charged pions and/or kaons where the elements Θ 2 or Θ 2 determine the production eI µI | | | | rates. Especially, we will point out that N and N which masses are smaller than pion mass 2 3 are still allowed by the constraints from the direct searches as well as that from the big bang nucleosynthesis [42, 43] for the normal hierarchy case. This is different from a conclusion from Ref. [38]. The reason for it lies in the cancellation in the Θ mentioned above. In addition, we eI shall present some implications to the future searches by using the decays of π+ and K+. This paper is organized as follows. In Sec. 2 we briefly review the framework of the present analysis, i.e., the νMSM. In Sec. 3 we study the mixing of sterile neutrinos N and N in the 2 3 charged current interactions. Especially, we investigate how the mixing elements depend on the parameters of active neutrinos, i.e., the mass hierarchy, mixing angles and CP violating phases. We then apply the obtained results in phenomenology of N and N . In Sec. 4 we estimate 2 3 the contributions of sterile neutrinos to the neutrinoless double beta decay and address the importance of such contributions when the masses of N and N are smaller than about the 2 3 order of 100 MeV. In Sec. 5 we discuss search of N and N in the charged pion and kaon 2 3 decays for the case when their masses are lighter than the pion mass. Finally, our results are summarized in Sec. 6. We add App. A to present the expressions for the mixing elements. 2 The νMSM First of all, we review the νMSM, which is the MSM extended by three right-handed neutrinos ν (I = 1,2,3) with Lagrangian RI M = +iν γµ∂ ν F L Φν + I ν c ν +h.c. , (1) LνMSM LMSM RI µ RI − αI α RI 2 RI RI (cid:16) (cid:17) where is the MSM Lagrangian. Φ and L = (e ,ν )T (α = e,µ,τ) are Higgs and MSM α Lα Lα L lepton weak-doublets, respectively. We denote Yukawa coupling constants of neutrinos by F . αI Here and hereafter we work in a basis in which the mass matrix of charged leptons is diagonal. In this model neutrinos receive the Majorana masses [M ] = M δ (which are taken to be M IJ I IJ real and positive without loss of generality) and the Dirac masses [M ] = F Φ ( Φ is a D αI αI h i h i vacuum expectation value of the Higgs field). The distinctive feature of the model is the region of the parameter space of Eq. (1), i.e., we restrict ourselves in the region [M ] M . Λ . (2) D αI I EW | | ≪ Notice that the seesaw mechanism still works even if the Majorana masses are smaller than or comparable to the weak scale Λ = (102) GeV. This is simply because neutrino Yukawa EW O coupling constants of interest are extremely small. (See the discussion below.) 3 ˆ The mass matrix of neutrinos M, which is a 6 6 symmetric matrix, is given by × 0 M Mˆ = D . (3) (cid:18) MT M (cid:19) D M We can diagonalize it by using the unitary matrix Uˆ as Uˆ†Mˆ Uˆ∗ = Mˆdiag. The seesaw mecha- nism shows that Uˆ at the leading order takes the form U Θ Uˆ = . (4) (cid:18) Θ†U 1 (cid:19) − Here U is the 3 3 Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [44]; × U†M U∗ = diag(m ,m ,m ), (5) ν 1 2 3 where M = M M−1MT is the seesaw mass matrix. We call the eigenstates having masses ν − D M D m as active neutrinos ν (i = 1,2,3). The rest three mass eigenstates, denoted by N , are i i I almost corresponding to right-handed neutrinos N ν having masses M . The neutrino I RI I ≃ mixing in the charge current is then induced through ν = U ν +Θ Nc, (6) Lα αi i αI I where the 3 3 mixing matrix Θ is found at the leading order as × [M ] D αI Θ = , (7) αI M I and hence Θ 1 due to Eq. (2). We shall call N as sterile neutrinos since they possess very αI I | | ≪ suppressed gauge interactions. It should be stressed that sterile neutrinos here are originated from right-handed neutrinos in the seesaw mechanism. In the νMSM three right-handed neutrinos play important roles in cosmology. One of them, say N , is a candidate for dark matter of the universe. This dark-matter particle receives severe 1 astrophysical constraints as mentioned in Sec. 1. Even then, thecorrect dark-matter abundance can be obtained through the mechanism [17] with the resonant production [18] in the presence ofthelargeleptonasymmetry. The recent study[27]shows thattherequired mass isM = 4–50 1 keV and the Yukawa coupling constants are typically F = 5 10−15–4 10−13. As a result, α1 | | × × N gives no significant contribution to the seesaw mass matrix M [5]. 1 ν The otherright-handedneutrinos, N andN , arethenresponsible tothemasses andmixing 2 3 of active neutrinos. Notice that in this case the mass of the lightest active neutrino becomes m < (10−6) eV. Further, N and N can explain the origin of BAU. The flavour oscillation 1 2 3 O between them in the early universe can generate BAU through the mechanism proposed in Ref. [34]. In the νMSM the correct amount of BAU can be obtained when N and N are 2 3 quasi-degenerate in mass [6, 36, 37]. The main purpose of this paper is to study the mixing elements Θ (I = 2,3) of sterile αI neutrinos N and N with flavour neutrinos. To do this, let us express their Yukawa coupling 2 3 4 constants by using mixing angles and masses of active neutrinos in oscillation experiments. As mentioned before, the successful dark matter scenario requires very small Yukawa couplings of N and its contribution to M can be neglected. Thus, we set F = 0 here for simplicity. (See, 1 ν α1 however, the discussion in Sec. 4.) In this case the neutrino Yukawa matrix F for N and N , 2 3 which is a 3 2 matrix, can be expressed without loss of generality as [45, 46] × i F = U D1/2ΩD1/2. (8) Φ ν N h i Here parameters of active neutrinos are their masses D = diag(m ,m ,m ) and the mixing ν 1 2 3 matrix c c s c s e−iδ 12 13 12 13 13 U =  c s s c s eiδ c c s s s eiδ s c  diag(1, eiη, 1), (9) 23 12 23 12 13 23 12 23 12 13 23 13 − − − × s s c c s eiδ s c c s s eiδ c c 23 12 23 12 13 23 12 23 12 13 23 13  − − −  with s = sinθ and c = cosθ . Note that there is one Majorana phase η in addition to ij ij ij ij Dirac phase δ under the considering situation. Because we have set F = 0, masses of active α1 neutrinos are m = m > m = m > m = 0 in the NH case, 3 atm 2 sol 1 m = m2 +m2 > m = m2 > m = 0 in the IH case, (10) 2 atm sol 1 atm 3 q p The observational data of mixing angles are s2 = 0.318+0.062, s2 = 0.50+0.17, and s2 0.053, 12 −0.048 23 −0.14 13 ≤ respectively, and masses are m2 = ∆m2 = (7.59+0.68) 10−5 eV2 and m2 = ∆m2 = sol 21 −0.56 × atm | 31| (2.40+0.35) 10−3 eV2 (at the 3σ level) [47]. Hereafter, we shall adopt the central values unless −0.33 × otherwise stated. On the other hand, parameters of N and N are their masses D = diag(M ,M ) and the 2 3 N 2 3 3 2 matrix × 0 0 Ω =  cosω sinω  in the NH case, − ξsinω ξcosω   cosω sinω − Ω =  ξsinω ξcosω  in the IH case, (11) 0 0   where ξ = 1 and ω is an arbitrary complex number. Notice that the change of the sign ξ can ± be compensated by ω ω together with the redefinition of N as ξN N [46]. 3 3 3 → − → 3 Mixing matrix of sterile neutrinos The important parameters for phenomenology of sterile neutrinos N and N are their masses 2 3 M and mixing matrix Θ. Especially, the latter one is crucial to specify the strength of 2,3 5 interactions with other particles. Here we would like to discuss how they depends on the parameters of active neutrinos. Before discussing the νMSM, let us consider a toy model with one pair of left- and right- handed neutrinos. In this case, the mixing of sterile neutrino is given by [cf. Eq. (4)] M 2 M 1 GeV M2 1/2 Θ 2 = | D| = ν = 4.9 10−11 ν , (12) | | M2 M × (cid:18) M (cid:19)(cid:18)2.4 10−3 eV2(cid:19) N N N × where M and M are the Dirac and Majorana masses, and we have used the seesaw formula D N for active neutrino mass M = M2/M . Thus, the mixing is determined from the masses ν |− D N| of active and sterile neutrinos. In the νMSM, since the parameter space is larger, the mixing matrix of N and N is more 2 3 complicated and its elements can be much different from Eq. (12). Especially, as pointed out in Ref. [39], the larger mixing can be obtained in the model with U(1) symmetry. We shall reanalyze this point by using the parametrization of the Yukawa matrix F presented in Eq. (8). The key for this issue is the complex parameter ω in the Ω matrix. It can be seen that the Yukawa coupling constants as well as the elements of mixing matrix become exponentially large as F ,Θ exp( Imω ) for Imω 1, as long as the seesaw approximation is valid. It should αI αI ∝ | | | | ≫ be noted that the tiny neutrino masses observed in the oscillation experiments can be obtained even in this case. To express this enhancement factor, we introduce a parameter X by ω X = exp(Imω). (13) ω Before going into details, let us here summarize the general properties of mixing elements Θ . αI (i) Θ 2 can be divided into X2, X0 and X−2 terms. | αI| ω ω ω (ii) TheX2 termin Θ 2M isexactlythesameasthatin Θ 2M forα = e,µ,τ. Similarly, ω | α2| 2 | α3| 3 the X−2 term in Θ 2M is exactly the same as that in Θ 2M . ω | α2| 2 | α3| 3 (iii) The X0 term in Θ 2M is opposite to that in Θ 2M for α = e,µ,τ. ω | α2| 2 | α3| 3 (iv) The coefficient of the X−2 term in Θ 2M is obtained from the X2 term by changing ω | αI| I ω ξ ξ for α = e,µ,τ and I = 2,3. → − We have confirmed these properties by direct calculations. From now on, we will present the expressions of Θ 2 for Imω 1 and discuss how they depend on the neutrino parameters, αI | | | | ≫ namely mass hierarchy, mixing angles, and CP violating phases of active neutrinos. We first consider Θ 2 for X 1 (i.e., Imω 1) in the NH case. The leading (X2) | αI| ω ≫ ≫ O ω terms are found as m Θ 2 =X2 atm cos2θ tan2θ +2√r ξsin(δ +η)sinθ tanθ +r sin2θ ,(14) | eI| (cid:12)Xω2 ω 4MI 13(cid:2) 13 m 12 13 m 12(cid:3) m Θ 2(cid:12) =X2 atm sin2θ cos2θ [1+ (√r )] , (15) | µI| (cid:12)Xω2 ω 4MI 23 13 O m m Θ 2(cid:12) =X2 atm cos2θ cos2θ [1+ (√r )] , (16) | τI| (cid:12)Xω2 ω 4MI 23 13 O m (cid:12) 6 where r = m /m 0.18. (The complete expressions for the (X2) terms are collected m sol atm ≃ O ω in App. A.) Notice that these expressions hold for both I = 2 and 3 thanks to the general property (ii). It is seen that all the elements are proportional to X2m /M , and hence they ω atm I can be much larger than Eq. (12) for X 1. ω ≫ Since the experiments show that θ is small and θ is close to π/4, Θ 2 and Θ 2 can 13 23 µI τI | | | | be determined as m 1 GeV Θ 2 Θ 2 X2 atm = 6.1 10−12X2 . (17) | µI| ≃ | τI| ≃ ω 8M × ω (cid:18) M (cid:19) I I On the other hand, the element Θ behaves quite differently. Indeed, it is interesting to note eI that the X2 terms in Θ 2 and Θ 2 vanish at the same time, when ω | e2| | e3| ξsin(δ +η) = 1, (18) − and the mixing angle θ takes its critical value θcr: 13 13 tanθcr = √r sinθ . (19) 13 m 12 The experimental data of θ , m and m with 3σ errors gives the critical value of θ as 12 atm sol 13 sin2θcr = 0.041–0.070, which can be below the 3σ upper bound sin2θ < 0.053.#3 Moreover, 13 13 we find in this case that the (X0) term also vanishes and only the (X−2) term is left as O ω O ω m r 1 GeV Θ 2 X−2 atm m sin2θ 2.8 10−12X−2 , (20) | eI| ≃ ω M 12 ≃ × ω (cid:18) M (cid:19) I I which becomes much smaller thanother elements in Eq. (17) forX 1. Weshould mentioned ω ≫ that the above cancellation in Θ 2 can be realized for any choice of masses M and M . As eI 2 3 | | we will show in Fig. 1, the strong suppression in Θ 2 is still possible when θ is close to θcr. | eI| 13 13 On the other hand, when θ = 0, Θ 2 receives no strong suppression described above, but it 13 eI | | satisfies the relation Θ 2 Θ 2 | eI| | eI| 2r sin2θ 0.11. (21) Θ 2 ≃ Θ 2 ≃ m 12 ≃ µI τI | | | | This relation had already been obtained in Ref. [26]. In Fig. 1 we show the mixing elements Θ 2 in terms of X when M = 120 MeV. First, we α2 ω 2 | | observe that Θ 2 and Θ 2 scales as X2 for X 1, and they can take much larger values | µ2| | τ2| ω ω ≫ than the naive result in Eq. (12). This behavior does not change much as long as θ lies in the 13 experimentally allowed region. Second, when sin2θ = 0, Θ 2 behaves similar to Θ 2 and 13 e2 µ2 | | | | Θ 2 for X 1, but is smaller by one order of magnitude as shown in Eq. (21). Finally, it is τ2 ω | | ≫ clearly seen that Θ 2 can be suppressed by many orders of magnitude for X 1 when θ e2 ω 13 | | ≫ becomes close to its critical value with a suitable parameter choice. Moreover, if θ = θcr, we 13 13 #3When we use the data at 2σ level, sin2θcr =0.046–0.065, which exceeds the 2σ bound sin2θ <0.039. 13 13 7 Figure 1: Mixing elements Θ 2 in the normal hierarchy in terms of X . We take sin2θ = 0 α2 ω 13 | | (left), 0.05 (center) and sin2θcr (right), respectively. The red solid, green dashed and blue 13 dotted lines correspond to Θ 2, Θ 2 and Θ 2, respectively. Here we take M = 120 MeV, e2 µ2 τ2 2 | | | | | | Reω = π/4, δ = π/2, η = π, and ξ = +1. can see that Θ 2 is proportional to X−2 and it can be extremely suppressed for X 1. The | e2| ω ω ≫ cancellation of Θ 2 in the NH case is one of the most important observation in this analysis eI | | and we will discuss its impacts on the experimental signatures of N and N later. 2 3 On the other hand, when X 1 (i.e., Imω 1), the leading order contribution is ω ≪ ≪ − proportional to X−2 and their expressions are given by Eqs.(14), (15) and (16) by replacing X2 ω ω by X−2 and ξ by ξ due to the general property (iv). Therefore, the above arguments with ω − the opposite sign of ξ exactly hold and we will discard this case in the followings. Next, we turn to discuss the IH case when X 1. In this case, the leading term of the ω ≫ mixing element of electron type is found as X2m m m Θ 2 = ω 2 cos2θ cos2θ tan2θ 2ξsinη 1 tanθ + 1 , (22) | eI| (cid:12)Xω2 4MI 12 13(cid:20) 12 − rm2 12 m2(cid:21) (cid:12) and the expressions for Θ 2 and Θ 2 are so long and they are collected in App. A. It is µI τI | | | | interesting to note that the X2 term as well as the X0 term in Θ 2 vanishes when ω ω | eI| m ξsinη = +1 and tanθ = tanθcr = 1 = (1+r2 )−1/4, (23) 12 12 rm m 2 and then θcr is close to the maximal angle π/4. Unfortunately, it is far beyond the current 12 data of 3σ range, and the cancellation in Θ 2 cannot be realized in the IH case, which is eI | | different from the NH case.#4 This point gives significant effects on the discussions given in the following sections. However, “ξsinη” plays a crucial role to determine the mixing element Θ . To see this eI | | point, let us take θ = π/4 and θ = 0 for simplicity. In this case, the mixing elements are 23 13 #4 When ξsinη = 1,tanθ = m /m , andsinθ =0,the X2 terms in both Θ 2 and Θ 2 vanishat − 12 2 1 13 ω | µI| | τI| the same time. However, the requipred value of θ12 is not allowed by the current data. 8 Figure 2: Mixing elements Θ 2 in the inverted hierarchy in terms of X . We take sinη = α2 ω | | π/2 (left) and 3π/2 (right), respectively. The red solid, green dashed and blue dotted lines correspond to Θ 2, Θ 2 and Θ 2, respectively. Here we take M = 120 MeV, Reω = π/4, e2 µ2 τ2 2 | | | | | | δ = π/2, θ = 0, and ξ = +1. 13 given by X2m Θ 2 ω atm(1 ξsinηsin2θ ), (24) | eI| (cid:12)Xω2 ≃ 4MI − 12 (cid:12) X2m Θ 2 Θ 2 ω atm(1+ξsinηsin2θ ), (25) | µI| (cid:12)Xω2 ≃ | τI| (cid:12)Xω2 ≃ 8MI 12 (cid:12) (cid:12) which gives the relation [26] Θ 2 Θ 2 1 ξsinηsin2θ 0.071 for ξsinη = +1 eI eI 12 | | | | 2 − = . (26) Θ 2 ≃ Θ 2 ≃ 1+ξsinηsin2θ (cid:26) 56 for ξsinη = 1 | µI| | τI| 12 − Therefore, the mixing element of electron type can be smaller or larger than others depending on the choice of “ξsinη”. This property is represented in Fig. 2. It is seen that Θ 2 and µI | | Θ 2 are almost the same, but Θ 2 can be different from others. τI eI | | | | Before closing this section, we would like to stress again that the above results of the hierarchy between the mixing elements Θ are independent on the masses of sterile neutrinos. αI Therefore, they can be applied to the general seesaw models with two right-handed neutrinos. 4 Neutrinoless Double Beta Decay The neutrinoless double beta (0ν2β) decay is one important phenomenon in which the mixing of active and sterile neutrinos, U and Θ , plays a crucial role. The 0ν2β decay in the νMSM αi αI had already been investigated in Ref. [40], in which the contributions from active neutrinos and dark-matter sterile neutrino N are estimated and those from N are neglected since their 1 2,3 masses are assumed to be so heavy that they decouple from the considering decay processes. Further, it had been discussed in Ref. [41] that the 0ν2β decay gives the stringent constraint on 9

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