Low-Scale Seesaw and the CP Violation in Neutrino Oscillations

We consider a version of the low-scale type I seesaw mechanism for generating small neutrino masses, as an alternative to the standard seesaw scenario. It involves two right-handed (RH) neutrinos $\nu_{1R}$ and $\nu_{2R}$ having a Majorana mass term with mass $M$, which conserves the lepton charge $L$. The RH neutrino $\nu_{2R}$ has lepton-charge conserving Yukawa couplings $g_{\ell 2}$ to the lepton and Higgs doublet fields, while small lepton-charge breaking effects are assumed to induce tiny lepton-charge violating Yukawa couplings $g_{\ell 1}$ for $\nu_{1R}$, $l=e,\mu,\tau$. In this approach the smallness of neutrino masses is related to the smallness of the Yukawa coupling of $\nu_{1R}$ and not to the large value of $M$: the RH neutrinos can have masses in the few GeV to a few TeV range. The Yukawa couplings $|g_{\ell 2}|$ can be much larger than $|g_{\ell 1}|$, of the order $|g_{\ell 2}| \sim 10^{-4} - 10^{-2}$, leading to interesting low-energy phenomenology. We consider a specific realisation of this scenario within the Froggatt-Nielsen approach to fermion masses. In this model the Dirac CP violation phase $\delta$ is predicted to have approximately one of the values $\delta \simeq \pi/4,\, 3\pi/4$, or $5\pi/4,\, 7\pi/4$, or to lie in a narrow interval around one of these values. The low-energy phenomenology of the considered low-scale seesaw scenario of neutrino mass generation is also briefly discussed.


Introduction
The seesaw mechanism [1] of neutrino mass generation is a very attractive mechanism which explains naturally the small masses of the neutrinos. According to the standard seesaw scenario the smallness of neutrino masses has its origin from large lepton-number violating Majorana masses of right-handed (RH) neutrinos. A very appealing aspect of the seesaw scenario is that we can relate the existence of large Majorana masses of the RH neutrinos to a spontaneous breaking of some high scale symmetry, for example, GUT symmetry. However, direct tests of the standard seesaw mechanism are almost impossible due to the exceedingly large masses of the RH neutrinos.
In the present article we consider an alternative mechanism for generating small neutrino masses. It involves two RH neutrinos ν 1R and ν 2R which have a Majorana mass M ν T 1R C −1 ν 2R , where C is the charge conjugation matrix. Assuming that ν 1R and ν 2R carry total lepton charges L(ν 1R ) = −1 and L(ν 2R ) = +1, respectively, this mass term conserves L. This implies that, as long as L is conserved, ν 1R and ν 2R (more precisely, We assume further that some small lepton-charge breaking effects induce tiny lepton--charge violating Yukawa couplings for ν 1R , namely −L ⊃ g 1 ν 1R H c † L , = e, µ, τ , with |g 1 | |g 2 |, , = e, µ, τ . Our setup will imply that the lepton-charge breaking diagonal Majorana mass terms are either forbidden or suppressed. In this case ν 1R and ν 2R (i.e., ν 1R and ν C 2L ) form a pseudo-Dirac pair. In this scenario the smallness of neutrino masses is due to the small Yukawa coupling |g 1 | 1 and hence we do not have to introduce the large Majorana mass M of the standard seesaw scenario. The mass M of the ν T 1R C −1 ν 2R mass term can be at the weak scale.
The strong hierarchy |g 1 | |g 2 | between the two sets of Yukawa couplings can be realised rather naturally, for example, within the Froggatt-Nielsen (FN) scenario [2].
Employing this scenario we will additionally consider that the Yukawa couplings g 2 obey a standard FN hierarchy [3], |g e2 | : |g µ2 | : |g τ 2 | ∼ : 1 : 1, ∼ 0.2. The magnitude of the Yukawa couplings of ν 1R should be completely different from that of the Yukawa coupling of ν 2R . However, due to the usual O(1) ambiguity in the FN approach, it is impossible to predict unambiguously the flavour dependence of g 1 and thus the ratios |g e1 | : |g µ1 | : |g τ 1 |.
We show in the present article, in particular, that in the model of neutrino mass generation with two RH neutrinos with the hierarchy and flavour structure of their Yukawa couplings and the mass term outlined above the Dirac CP-violating (CPV) phase is predicted to have one of the values δ π/4, 3π/4, or 5π/4, 7π/4.

General setup
We minimally extend the Standard Model (SM) by adding two RH neutrinos, i.e., two chiral fields ν aR (x), a = 1, 2, which are singlets under the SM gauge symmetry group.
The full 5 × 5 neutrino Dirac-Majorana mass matrix, given below in the (ν L , ν C L ) basis, can be made block-diagonal by use of a unitary matrix Ω, matrix Ω can be parametrised as [4,8]: under the assumption that the elements of the 3 × 2 complex matrix R are small, which will be justified later. At leading order in R, the following relations hold [4]: where 1 we have used eq. (2.6) to get the last equality in eq. (2.7). From the first two we recover the well-known seesaw formula for the light neutrino mass matrix, We are interested in the case where only the L-conserving Majorana mass term of ν 1R (x) and ν 2R (x), M ν T 1R C −1 ν 2R , with M > 0 and, e.g., L(ν 1R ) = −1 and L(ν 2R ) = +1, L being the total lepton charge, is present in the Lagrangian. In this case the Majorana mass matrix of RH neutrinos ν 1R (x) and ν 2R (x) reads: The factors 1/2 in the two terms ∝ R T R * M N and ∝ M N R † R in eq. (2.8) are missing in the corresponding expression in Ref. [4]. These two terms provide a sub-leading correction to the leading term M N and have been neglected in the discussion of the phenomenology in Ref. [4]. We will also neglect them in the phenomenological analysis we will perform. Using eqs. (2.2), (2.3) and eq. (2.9), we get the following expression for the light neutrino Majorana mass matrix m ν : With the assignments L(ν 1R ) = −1 and L(ν 2R ) = +1 made, the requirement of conservation of the total lepton charge L leads to g 1 = 0, = e, µ, τ . In this limit of g 1 = 0, we have m ν = 0, the light neutrino masses vanish and ν 1R and ν C 2L combine to form a Dirac fermion N D of massM ≡ M 2 + v 2 |g 2 | 2 2 , Thus, the massive fields N k (x) are related to the fields ν aR (x) by where the upper (lower) signs correspond to the case with the upper (lower) signs in eq. (2.12) and in the expressions for ν 1R and ν C 2L given after it. Small L-violating couplings g 1 = 0 split the Dirac fermion N D into the two Majorana fermions N 1 and N 2 which have very close but different masses, M 1 = M 2 , |M 2 − M 1 | M 1,2 . As a consequence, N D becomes a pseudo-Dirac particle [10,11]. Of the three light massive neutrinos one remains massless (at tree level), while the other two acquire non--zero and different masses. The splitting between the masses of N 1 and N 2 is of the order of one of the light neutrino mass differences and thus is extremely difficult to observe in practice.
More specifically, in the case of a neutrino mass spectrum with normal ordering (see, e.g., [12]) we have (at tree level) keeping terms up to 4th power in the Yukawa couplings 2 These general results can be inferred just from the form of the conserved "non-standard" lepton charge L [9] which is expressed in terms of the individual lepton charges L , = e, µ, τ , and L a (ν bR ) = − δ ab , a, b = 1, 2: L = L e + L µ + L τ + L 1 − L 2 (L (ν 1R ) = L 1 (ν 1R ) = −1 and L (ν 2R ) = −L 2 (ν 2R ) = +1). Then min(n + , n − ) and |n + − n − | are the numbers of massive Dirac and massless neutrinos, respectively, n + (n − ) being the number of charges entering into the expression for L with positive (negative) sign. g 1 and g 2 and taking g a to be real for simplicity: where The heavy neutrino mass spectrum is given by: The values of m 2,3 and M 1,2 given in eqs. (2.14) and (2.18) can be obtained as approximate solutions of the exact mass-eigenvalue equation: Note that, as it follows from eqs. where we have used the best fit values of ∆m 2 21 and ∆m 2 31 determined in the recent global analysis of the neutrino oscillation data [13] (see also Table 2). The corrections to the matrix V which diagonalises M N are of the order of AD/M 4 and are negligible, as was noticed also in [4].
To leading order in (real) g 1 and g 2 , the expressions in eqs. (2.14) and (2.18) simplify significantly [4]: The low-energy phenomenology involving the pseudo-Dirac neutrino N D , or equivalently the Majorana neutrinos N 1 and N 2 , is controlled by the matrix RV of couplings of N 1 and N 2 to the charged leptons in the weak charged lepton current (see Section 6).
When both g 1 and g 2 couplings are present, this matrix is given by: where we have used the expression for the matrix V in eq. (2.13) with the upper signs.
We will adhere to this convention further on.
It follows from the preceding discussion that the generation of non-zero light neutrino masses may be directly related to the generation of the L-non-conserving neutrino Yukawa couplings g 1 = 0, = e, µ, τ . Among the many possible mechanisms leading to g 1 = 0 there is at least one we will discuss further, that could lead to exceedingly small g 1 , say The low-scale type I seesaw scenario of interest with two RH neutrinos ν 1R and ν 2R with L-conserving Majorana mass term and L-conserving (L-non-conserving) neutrino Yukawa couplings g 2 (g 1 ) of ν 2R (of ν 1R ) was considered in [4] on purely phenomenological grounds (see also, e.g., [14]). It was pointed out in [4], in particular, that the strong

Froggatt-Nielsen Scenario
We work in a supersymmetric (SUSY) framework and consider a global broken U(1) FN Froggatt-Nielsen flavour symmetry, whose charge assignments we motivate below. We will show how an approximate U(1) L symmetry, related to the L-conservation, may arise in such a model, with g 1 = 0 as the leading L-breaking effect responsible for neutrino masses.
In our setup, one of the RH neutrino chiral superfields has a negative charge under to be close to the sine of the Cabibbo angle λ C , specifically = 0.2, in order to reproduce the hierarchies between charged lepton masses (see also [15,16]  The effective superpotential 3 for the neutrino sector reads where M 0 ∼ Λ and g 2 is an a priori O(1) coupling. Due to the condition of holomorphicity of the superpotential, no quadratic term forN 2 is allowed, justifying the absence of the Majorana mass term M ν T 2R C −1 ν 2R . This framework may naturally arrange for the suppression (M N ) 11 (M N ) 12 , as well as for a hierarchy between RH masses and the FN scale, M ∼ n−1 Λ Λ, provided the charge n is sufficiently large.
The limit of a largeN 1 charge, n 1, is quite interesting. In this limit, one finds an accidental (approximate) U(1) L symmetry, with assignments L(N 1,2 ) = ±1. Furthermore, the desired hierarchy between (would-be) L-breaking and (would-be) L-conserving Yukawa couplings, |g 1 | ∼ n+1 |g 2 |, is manifestly achieved. Finally, the mass term for Thus, in the present setup, the Yukawa matrix Y D obeys the following structure (up to phases): with sin β = H 0 u /v, and where g 1 , g 2 > 0, and the hierarchy g 1 g 2 ∼ < 1 is naturally realised. We see from eq. (2.11) that the scale of light neutrino masses depends on the size of the product g 1 g 2 , namely Despite being suppressed, the quadratic term forN 1 , and thus the Majorana mass term µ ν T 1R C −1 ν 1R , may still play a non-negligible role, for instance, in studies of leptogenesis [17].

Neutrino Mixing
The addition of the terms of eq. (2.1) to the SM Lagrangian leads to a Pontecorvo-Maki--Nakagawa-Sakata (PMNS) neutrino mixing matrix, U PMNS , which is not unitary. Indeed, the charged and neutral current weak interactions involving the light Majorana neutrinos χ i read: where = e, µ, τ and U l is a unitary matrix which originates from the diagonalisation of the charged lepton mass matrix and η ≡ − R R † /2. The transformation U l does not affect the power counting in the structure of eq. (3.2), though it may provide a source of deviations. We then choose to work in the charged lepton mass basis, in which U l = 1.
In this basis the PMNS neutrino mixing matrix is given by: is the unitary matrix diagonalising the Majorana neutrino mass matrix generated by the seesaw mechanism and η describes the deviation from unitarity of the PMNS matrix. As we will see further, the experimental constraints on the elements of η imply |η | ∼ < 10 −3 , , = e, µ, τ .
Due to the structure of the matrix of Yukawa couplings Y D given in eq.  where c ij ≡ cos θ ij and s ij ≡ sin θ ij , with θ ij ∈ [0, π/2], while δ and α denote the Dirac and Majorana [18] CP violation (CPV) phases, respectively, δ, α ∈ [0, 2π]. The current best fit values and 3σ allowed ranges for the neutrino mixing parameters and mass squared differences for NO spectrum are summarised in Table 2

Predictions for the CPV phases
It proves convenient for our further analysis to use the Casas-Ibarra parametrisation [19] of the Dirac mass matrix M D (neutrino Yukawa matrix Y D ): The Dirac neutrino mass matrix can be presented accordingly as M D = M D+ + M D− , with obvious notation. For the elements of a , = e, µ, τ, a = 1, 2 , where v g Given the fact that ( 2 . Unless otherwise stated we will employ this sign choice in the discussion which follows. Taking for definiteness ξ < 0, it follows from eqs. (5.6) -(5.9) that |g (−) a | (|g (+) a |) grows (decreases) exponentially with |ξ| 6 . Therefore, for sufficiently large |ξ| we will have , = e, µ, τ . (5.10) Using the 3σ allowed ranges of the neutrino oscillation parameters found in the global analysis of the neutrino oscillation data in [13] and given in Table 2  2 |. We will show next that, given the present neutrino oscillation data, enforcing the flavour pattern specified in eq. (3.2) results in a prediction for the Dirac phase δ close to π/4, 3π/4, 5π/4, 7π/4, and for the Majorana phase α close to zero.
As we have seen, the matrix of neutrino Yukawa couplings Y D can be reconstructed up to normalization, a complex parameter, and a sign using eqs.  One sees that the dependence on the complex parameterθ drops out in the ratios R (1,2) , which are determined by the light neutrino masses m 2 and m 3 and by neutrino mixing parameters only, once the sign in O in eq. (5.4) (or equivalently in eqs. (5.6) -(5.9)) is fixed. In particular, the flavour structure depends on the elements U 2 and U 3 of the PMNS matrix. Given the fact that m 2 = ∆m 2 21 , m 3 = ∆m 2 31 , and that ∆m 2 21 , ∆m 2 31 and the three neutrino mixing angles θ 12 , θ 23 and θ 13 have been determined in neutrino oscillation experiments with a rather high precision, the quantities R (1) and R (2) depend only on the CPV phases δ and α once the sign of ξ is fixed. This means that knowing any two of the ratios |g 1 |/|g 1 | or |g 2 |/|g 2 |, = = e, µ, τ allows to determine both δ and α.
In Figs. 1 and 2 we present the ratios R (1,2) as a function of δ for the case ξ < 0 and two representative values of α. Figure 1 is obtained using the best fit values of ∆m 2

21,31
and sin 2 θ ij taken from Table 2. In Fig. 2 we show the ranges in which R (1,2) vary when  and sin 2 θ ij quoted in Table 2. The vertical grey band indicates values of δ which are disfavoured at 3σ. The case ξ > 0 is obtained by exchanging R (1) and R (2) . (For interpretation of the colours in the figure(s), the reader is referred to the web version of this article.) ∆m 2 21,31 and the sin 2 θ ij are varied in their respective 3σ allowed intervals given in Table  2. In Table 3 we report the respective intervals in which each of the six ratios can lie. As Table 3 indicates, certain specific simple patterns cannot be realised within the scheme considered. Among those are, for example, the patterns |g e1 | : |g µ1 | : |g τ 1 | 1 : 1 : 1 and |g e2 | : |g µ2 | : |g τ 2 | 1 : 1 : 1.
The flavour structure of eq. (3.2), which is naturally realised in the model of Section 3, corresponds to the pattern |g e2 | : |g µ2 | : |g τ 2 | : 1 : 1, and thus to R

The requirement of having R
(2) µτ 1 favours α close to zero 7 . As can be inferred from Fig. 1, given the current best fit values of neutrino mass squared differences 7 Marginalizing over δ (either in its defining or in its 3σ range) and varying ∆m 2 21,31 and the sin 2 θ ij in their respective 3σ allowed ranges, the requirement that |R (2) µτ − 1| < 0.1 implies α < 0.36π ∨ α > 1.64π, independently of the sign of ξ. However, if we require that the relative probability of α having a given value in the indicated intervals is not less than 0.15, then we have α < 0.2π or α > 1.8π. For these values of α and = 0.2, the predictions for δ can be read off from the plots where α = 0.  Figure 2: Ratios R (1,2) of (absolute values of) Yukawa couplings for a NO neutrino spectrum as a function of the CPV phase δ for α = 0 (left panel) and α = π (right panel), in the case ξ < 0. Bands are obtained by varying ∆m 2 21,31 and the sin 2 θ ij in their respective 3σ allowed ranges given in Table 2. In the case α = π, the upper boundary of the R (1) and R (2) . and mixing parameters, the requirement of R (2) eµ R (2) eτ = 0.2 leads, for ξ < 0, to the prediction of δ 5π/4, 7π/4 8 . Taking into account the 3σ allowed ranges of ∆m 2 21,31 and sin 2 θ ij leads, as Fig. 2 shows, to δ lying in narrow intervals around the values 5π/4 and 7π/4. Allowing for a somewhat smaller value of , e.g., = 0.15, we find that δ should lie in the interval δ [5π/4, 7π/4] which includes the value 3π/2 (see Fig. 2).

Phenomenology
The low-energy phenomenology of the model of interest resembles that of the model with two heavy Majorana neutrinos N 1,2 forming a pseudo-Dirac pair considered in [4][5][6], in which the splitting between the masses of N 1,2 is exceedingly small. For this model direct and indirect constraints on the model's parameters, which do not depend on the splitting between the masses of N 1 and N 2 , as well as expected sensitivities of future lepton colliders have been analysed, e.g., in Refs. [4-6, 24, 25] (see also [26,27]).
Due to the mixing of LH and RH neutrino fields, i) the PMNS neutrino mixing matrix, Due to the Yukawa interactions, cf. eq. (2.2), there are interactions of the heavy Majorana neutrinos N 1,2 with the SM Higgs boson h as well (see [7]):
In [4,5] the constraint in eq. (6.4) is satisfied by finding a region, in the general parameter space of the model considered, in which to leading order a=1,2 (RV ) * a M a (RV ) * a = 0, i.e., the two terms in the sum cancel. In the version of the low-scale type I seesaw model with two RH neutrinos we are considering the constraint in eq. (6.4) is satisfied due to smallness of the product of Yukawa couplings |g 1 | and |g 2 |. In the model under consideration one gets a=1,2 (RV ) * a M a (RV ) * a = 0 in the limit of negligible couplings g 1 . Indeed, setting g 1 = 0 we get M 1 = M 2 and the expression for the matrix RV takes the form: The upper bound on the e−µ elements is relaxed to |η eµ | < 3.4×10 −4 for heavy Majorana neutrino masses below the electroweak scale (but still above the kaon mass, M k ∼ > 500 MeV) due to the restoration of a GIM cancellation [30]. The above constraints on η justify the assumption made in Section 2 regarding the smallness of the elements of R.
Using the expression for RV given in eq. (2.23) we find that, to leading order in g 1 ,

LFV Observables and Higgs Decays
The predictions of the model under discussion for the rates of the lepton flavour violating (LFV) µ → eγ and µ → eee decays and µ − e conversion in nuclei, as can be shown, and BR(µ → eee), and for the relative µ − e conversion in a nucleus X, CR(µX → eX), version of the TeV scale type I seesaw model considered in [5,6].
coincide with those given in Refs. [5,6] and we are not going to reproduce them here. The best experimental limits on BR(µ → eγ), BR(µ → eee) and CR(µX → eX) have been obtained by the MEG [31], SINDRUM [32] and SINDRUM II [33,34] Collaborations: The planned MEG II update of the MEG experiment [35] is expected to reach sensitivity to BR(µ → eγ) 4 × 10 −14 . The sensitivity to BR(µ → eee) is expected to experience a dramatic increase of up to four orders of magnitude with the realisation of the Mu3e Project [36], which aims at probing values down to BR(µ → eee) ∼ 10 −16 in its phase II of operation. Using an aluminium target, the Mu2e [37] and COMET [38] collaborations plan to ultimately be sensitive to CR(µ Al → e Al) ∼ 6 × 10 −17 . The PRISM/PRIME project [39] aims at an impressive increase of sensitivity to the µ − e conversion rate in titanium, planning to probe values down to CR(µ Ti → e Ti) ∼ 10 −18 , an improvement of six orders of magnitude with respect to the bound of eq. (6.12).
We show in Fig. 3 the limits on |g µ2 g e2 | implied by the experimental bounds in eqs. (6.10) -(6.13), as a function of the mass M , as well as the prospective sensitivity of the future planned experiments MEG II, Mu3e, Mu2e, COMET and PRISM/PRIME. The data from these experiments, as Fig. 3 indicates, will allow to test for values of |g µ2 g e2 | significantly smaller than the existing limits, with a significant potential for a discovery.
The interactions given in eq. h → ν L N k in the model considered in the present article is similar to that of the same decay investigated in detail in [7] in the model discussed in [5]. The rate of the decay h → ν L N 1,2 to any ν L and N 1 or N 2 is given in Ref. [7] and in the limit of zero mass  14) where in the model considered by us and we have used eqs. (6.6) and (6.7). The dominant decay mode of the SM Higgs boson is into bottom quark-antiquark pair, b −b. The decay rate is given by: The upper bound on ( |g 2 | 2 ) is determined essentially by the upper bound on |g τ 2 | 2 = 2|η τ τ |M 2 /v 2 , which is less stringent than the upper bounds on |g e2 | 2 and |g µ2 | 2 .
Using the bound |η τ τ | < 2.8 × 10 −3 quoted in eq. (6.8), we get for M = 100 GeV the upper bound |g τ 2 | 2 < 1.8 × 10 −3 . For the Higgs decay rate Γ(h → ν N ) in the case of M = 100 GeV and, e.g., ( |g 2 | 2 ) = 10 −3 , we get Γ(h → ν N ) = 3.2 × 10 −4 GeV. This decay rate would lead to an increase of the total SM decay width of the Higgs boson by approximately 8%. Thus, the presence of the h → ν N decay would modify the SM prediction for the branching ratio for any generic (allowed in the SM) decay of the Higgs particle [7], decreasing it.
We finally comment on neutrinoless double beta ((ββ) 0ν -) decay (see, e.g., [12]). The relevant observable is the absolute value of the effective neutrino Majorana mass | m | (see, e.g., [41]), which receives an extra contribution from the exchange of heavy Majorana neutrinos N 1 and N 2 . This contribution should be added to that due to the light Majorana neutrino exchange [42,43] (see also [4,44]). The sum of the two contributions can lead, in principle, to | m | that differs significantly from that due to the light Majorana neutrino exchange. The contribution due to the N 1,2 exchange in | m | in the model considered is proportional, in particular, to the difference between the masses of N 1 and N 2 , which form a pseudo-Dirac pair. For M ∼ > 1 GeV, as can be shown, it is strongly suppressed in the present setup due to the extremely small N 1 − N 2 mass difference, the stringent upper limit on |g e2 | 2 , and the values of the relevant nuclear matrix elements (NME), which at M = 1 GeV are smaller approximately by a factor of 6 × 10 −2 than the NME for the light neutrino exchange and scale with M as (0.9 GeV/M ) 2 . As a consequence, the contribution to | m | due to the exchange of N 1 and N 2 is significantly smaller than the contribution from the exchange of light Majorana neutrinos χ j .

Summary and Conclusions
In the present paper we have explored a symmetry-protected scenario of neutrino mass generation, where two RH neutrinos are added to the SM. In the class of models considered, the main source of L-violation responsible for the neutrino masses are small lepton-charge violating Yukawa couplings g 1 ( = e, µ, τ ) to one of the RH neutrinos, It is interesting to point out that, given the exceedingly small splitting between heavy neutrinos, the dependence on the Casas-Ibarra complex parameter drops out in the ratios between absolute values of Yukawa couplings to the same RH neutrino. These ratios are then determined (up to the exchange of g 1 and g 2 ) by neutrino low-energy parameters alone, namely, by neutrino masses, mixing angles and CPV phases δ and α. Given the Yukawa structure of our model, |g e2 | : |g µ2 | : |g τ 2 | : 1 : 1 with λ C 0.2, the Dirac CPV phase δ is predicted to have approximately one of the values δ π/4, 3π/4, or 5π/4, 7π/4, or to lie in a narrow interval around one of these values, while a Majorana CPV phase α 0 is preferred ( Figs. 1 and 2).
In the considered scenario, the maximal values of the elements of the neutrino mass matrix lead to constraints on the combinations |g 1 g 2 + g 1 g 2 |, , = e, µ, τ , which depend on products of L-conserving and L-violating Yukawa couplings (see Section 6.1).
Deviations from unitarity of the PMNS matrix constrain instead the products |g 2 g 2 |, , = e, µ, τ , of L-conserving couplings alone. In particular, the product |g µ2 g e2 | is constrained by data on muon lepton flavour violating (LFV) processes. Data from future LFV experiments (MEG II, Mu3e, Mu2e, COMET, PRISM/PRIME) will allow to probe values of |g µ2 g e2 | significantly smaller than the existing limits (Fig. 3). The decay of the Higgs boson into one light and one heavy neutrino can have a rate Γ(h → νN ) as large as 8% of the total SM Higgs decay width. This decay mode can lead to a change of the Higgs branching ratios with respect to the SM predictions. Concerning neutrinoless double beta decay in the considered model, the contribution due to N 1,2 exchange in the absolute value of the effective neutrino Majorana mass | m | is found to be negligible when compared to the contribution from the exchange of light Majorana neutrinos.
Finally, we comment on the issue of leptogenesis. For temperatures above the electroweak phase transition (EWPT), the Higgs VEV vanishes and thus, in the considered setup, the splitting between the masses of heavy neutrinos originates from the (suppressed) Majorana mass term µ ν T 1R C −1 ν 1R , with µ ∼ n+1 M ∼ |g 1 |M . This component of the heavy neutrino mass matrix -which in our case presents a subleading contribution to neutrino masses -is then crucial for resonant leptogenesis to proceed (see, e.g., [45]). The resonant condition reads µ Γ/2, where Γ denotes the average heavy neutrino decay width. However, the values of µ, Γ and neutrino masses are tightly connected in the FN model we analyse, which, together with the required smallness of µ, prevents reproducing the observed baryon asymmetry of the Universe (BAU), η obs B (6.09 ± 0.06) × 10 −10 [46].
One may instead successfully generate the observed BAU through the mechanism of anti-leptogenesis [47] (also known as "neutrino assisted GUT baryogenesis"). In this case, leptogenesis, this mechanism relies on a suppression of theL-violating heavy neutrino mass splitting above the EWPT, in order not to wash-out the asymmetry generated at a high scale. Modifying our setup as detailed in the end of Section 3, the Majorana mass term µ ν T 1R C −1 ν 1R is forbidden and the heavy neutrinos are degenerate above the EWPT. One then adds a third RH neutrino in the bulk with (B −L)(ν 3R ) = −1 and vanishing U(1) L charge, such that its Yukawa couplings, which violate lepton number, are allowed, and such that the mass term M 3 ν T 3R C −1 ν 3R is generated, M 3 ∼ Φ . Notice that only one such RH neutrino is needed to erase lepton number at high temperatures (M 3 ∼ (10 12 − 10 13 ) GeV), and that there is a large region of parameter space where the new contribution to the neutrino mass matrix is negligible [48]. Given these conditions, successful anti-leptogenesis may proceed.