Discrete symmetries for electroweak natural type-I seesaw mechanism
DDiscrete symmetries for electroweak natural type-I seesawmechanism
Pratik Chattopadhyay ∗ and Ketan M. Patel † Indian Institute of Science Education and Research Mohali,Knowledge City, Sector 81, S A S Nagar, Manauli 140306, India.
Abstract
The naturalness of electroweak scale in the models of type-I seesaw mechanism with O (1) Yukawacouplings requires TeV scale masses for the fermion singlets. In this case, the tiny neutrino masseshave to arise from the cancellations within the seesaw formula which are arranged by fine-tunedcorrelations between the Yukawa couplings and the masses of fermion singlets. We motivate suchcorrelations through the framework of discrete symmetries. In the case of three Majorana fermionsinglets, it is shown that the exact cancellation arranged by the discrete symmetries in seesawformula necessarily leads to two mass degenerate fermion singlets. The remaining fermion singletdecouples completely from the standard model. We provide two candidate models based on thegroups A and Σ(81) and discuss the generic perturbations to this approach which can lead to theviable neutrino masses. ∗ [email protected] † [email protected] a r X i v : . [ h e p - ph ] J un . INTRODUCTION The type-I seesaw mechanism is considered to be one of the simplest and minimal exten-sion of the Standard Model (SM) that can naturally generate small neutrino masses [1–7].It requires existence of new fermions, often called as right handed (RH) neutrinos, which aresinglets under the SM gauge symmetry. These fermions can have large Majorana mass andthey couple to the SM only through their Yukawa interactions with the leptons and Higgs.If such couplings are taken to be of the same order of other Yukawa couplings in the SM, thesinglet fermions are required to be as massive as 10 -10 GeV in order to comply with theneutrino mass scale that governs solar and atmospheric neutrino oscillations. This makes italmost impossible to verify the existence of RH neutrinos experimentally and in turn alsothe validity of type-I seesaw mechanism. The low scale versions of type-I seesaw mechanismhave been also put forward in which the masses of fermion singlets are assumed to be atexperimentally accessible scales [8–16]. For the recent study of the phenomenology of lightfermion singlets, see [17, 18] and references therein. In these versions, the smallness of neu-trino mass is arranged by assuming either very small Yukawa couplings or cancellations inthe seesaw mass formula [19–21] which arise due to some very particular choices of Yukawacouplings and masses of the fermion singlets. The tiny Yukawa couplings lead to very smallmixing between the SM neutrinos and fermion singlets making their production suppressedin the direct search experiments. While the possibility of seesaw cancellations with O (1)Yukawa couplings and light fermion singlets seems quite promising from experimental pointof view, it remains very highly fine-tuned if such cancellations are not motivated from somesymmetry or dynamical mechanisms. In any case, demanding low-scale seesaw mechanismbased on the grounds of only experimental accessibility is not well motivated as it goesagainst the very basic idea of seesaw mechanism for which it was actually proposed, namelyto naturally suppress neutrino masses through introducing a heavy scale in the theory.A more profound constraint on type-I seesaw mechanism comes from the requirement ofelectroweak naturalness. The discovery of boson with mass 126 GeV which seems very muchlike the SM Higgs [22–24] validates the idea of spontaneous breaking of electroweak symmetrythrough BroutEnglertHiggs mechanism. The Higgs field φ has potential V = − µ | φ | + λ | φ | where µ is a dimensionful and λ is a dimensionless parameter. They set vacuum expectationvalue (VEV) of Higgs field v = (cid:112) µ /λ ≡
246 GeV which is determined from the masses of W and Z bosons. They also determine the mass of physical Higgs boson, namely M φ = 2 λv .The measurement of Higgs mass therefore completely determines the Higgs potential andimplies the value for renormalized dimensionfull parameter, µ = M φ / √ ≈
89 GeV. Thehierarchy or electroweak naturalness problem refers to the higher order corrections to the µ parameter and concerns to the stability of this scale under such corrections. If SM is theonly fundamental theory extendable to any arbitrary high scale then there is no naturalnessproblem. However in the presence of any new physics beyond the SM, one must take intoaccount the corrections to the µ parameter, namely δµ , induced by such new physics[25, 26]. The scale of new physics and how it couples to the SM Higgs sector determines2he magnitude of δµ and the requirement that δµ should be of the order of µ (or TeV in a more conservative approach [26]) implies constraints on the scale and couplings of newphysics. In the case of type-I seesaw mechanism, one finds similar issue because of theexistence of right handed neutrinos and their couplings to the SM fields . The one-loop FIG. 1. The one loop correction to µ parameter in the presence of Majorana fermion singlet N . correction to the renormalized µ parameter, represented by the Feynman diagram shownin Fig. 1, was first calculated by Vissani in [27] in case of simple one-flavor type-I seesaw(see also [28–30]). Such correction is estimated as δµ ≈ π | y ν | M N = 12 π m ν v M N , (1)where y ν is a coupling between the fermion singlet with SM leptons and Higgs and M N is mass of the fermion singlet. The m ν = | y ν | v / (2 M N ) is the seesaw mass for the SMneutrino. Clearly, m ν = √ m atm = 0 .
05 eV leads to M N ≤ GeV if δµ is required to besmaller than (1 TeV) . Similar analysis for three flavoured type-I seesaw mechanism wascarried out in [31] and it was found that at least two of the three fermion singlets are requiredto be lighter than 10 GeV to maintain the electroweak naturalness. If the fermion singletsare close to the upper bound set by electroweak naturalness then it generically requires theYukawa couplings of O (10 − ) in order to produce viable light neutrino masses. Even smallerYukawa couplings are required when the masses of fermion singlets are further reduced.In this paper we argue that type-I seesaw mechanism loses its inherent naturalness whenthe criteria of elecroweak naturalness is imposed on it. The naturalness of both the Yukawacouplings and electroweak scale requires the masses of fermion singlets to be as light as of O (TeV). The standard seesaw mechanism then can no longer be considered as the source ofsmall neutrino masses. In this paper we modify the seesaw mechanism by incorporating itinto the framework of discrete symmetries which give rise to massless SM neutrinos despite O (1) Yukawa couplings and TeV scale masses for the fermion singlets. Discrete symmetrieshave been widely used to predict flavour mixing patterns in the lepton sector, see [32–36] forsome recent reviews. It is shown recently in [37–41] that a class of discrete symmetries canalso provide restrictions on the neutrino masses. We use this basic idea in order to suppressneutrino masses in electroweak natural seesaw setup. For this, we assume that the three In general, the naturalness criteria gets modified in the presence of other new physics beyond the SM. Inthis paper, we however restrict ourselves to the study of type-I seesaw mechanism only. Z m × Z n × Z p symmetry with m, n, p ≥
3. This symmetry of SM neutrinos together with their Majorana nature implies allof them to be massless. The fermion singlets are assigned appropriate discrete symmetries insuch a way that there exist three massive states of them and atleast one linear combinationof fermion singlets couples with the SM leptons through Yukawa interactions. As we showin this paper, these conditions necessarily lead to two degenerate fermion singlets and onemassive fermion singlet which completely decouples from the SM. The residual symmetriesof leptons and fermion singlets can be combined into a discrete group G f which should be thesymmetry of leptonic Lagrangian. The G f can be discrete subgroup (DSG) of SU(3) or U(3)depending on the representation to be chosen for the leptons. We provide specific modelfor each of this class of symmetries and discuss the phenomenology of generic perturbationswhich produces tiny neutrino masses. Our results open a new category of models in whichviable neutrino masses in the low-scale seesaw frameworks are naturally realized through amildly broken discrete symmetry.The paper is organized as the following. In the next section, we revisit the constraints ontype-I seesaw mechanism arising from the criteria of electroweak naturalness. In section III,we formulate the general discrete symmetries which lead to the massless neutrinos throughtype-I seesaw despite O (1) Yukawa couplings and TeV scale fermion singlets. We considerspecific examples of such symmetries in section IV. The perturbations required to generatetiny neutrino masses are studied in section V. Finally, we summarize in section VI. II. ELECTROWEAK NATURALNESS AND TYPE-I SEESAW MECHANISM
Consider an extension of the SM by n number of gauge singlet Majorana fermions N α .Their complete renormalizable interactions can be written as L = L SM + N α σ µ ∂ µ N α −
12 ( M N ) αβ N cα N β − ( Y D ) iα L i N α ˜ φ + h . c . , (2)where i = 1 , , α, β = 1 , ..., n are flavour indices, L = ( ν L , e L ) T , ˜ φ = iτ φ ∗ and φ is the SM Higgs doublet with vacuum expectation value (VEV) of its electrically neutralcomponent, (cid:104) φ (cid:105) ≡ v/ √ Y D ) iα are the Dirac Yukawa couplings and M N is the Majorana mass matrix for heavy singlet fermions. Without loss of generality, one canconsider a basis in which the 3 × Y l and M N are diagonalwith real and positive elements. We denote such diagonal elements in M N as M N α . If M N α (cid:29) (cid:104) φ (cid:105) then after the electroweak symmetry breaking, the SM neutrinos get the masseswhich can be expressed in terms of the fundamental couplings in eq. (2) as M ν = (cid:104) φ (cid:105) Y D M − N Y TD . (3)The symmetric matrix M ν is diagonalized by a unitary matrix such that U † M ν U ∗ = Diag . ( m ν , m ν , m ν ) , (4)4here m νi are the masses of light neutrinos and U is the leptonic mixing matrix, also knownas the PMNS matrix.The Yukawa interaction of N α with the light neutrinos and Higgs induces finite correctionto the µ parameter at one-loop [27–30]. Such a correction is estimated as | δµ | ≈ π (cid:88) i,α | ( Y D ) iα | M N α . (5)The electroweak naturalness criteria therefore imposes a constraint | ( Y D ) iα | M N α ≤ O (TeV) . (6)It requires the singlet fermions at TeV scale if the neutrino Yukawa couplings are of O (1)or small couplings if the mass scale of singlet fermion is heavier than TeV. The latterpossibility is actually further constrained by the observed neutrino masses and one cannotconsider arbitrarily small ( Y D ) iα and large M N α . In a simplified case of single generation oflight neutrino and fermion singlet, eq. (3) implies Y D ≈ M N m ν / (cid:104) φ (cid:105) leading to a genericbound from the criteria of naturalness: M N m ν π (cid:104) φ (cid:105) (cid:46) (TeV) ⇒ M N (cid:46) . × × (cid:32) (cid:112) m m ν (cid:33) / GeV . (7)The above bound on the mass of singlet fermion does not get drastically modified if threegenerations of neutrinos and fermion singlets are considered. A numerical investigationperformed in [31] shows that all the three fermion singlets are generically required to be ≤ GeV in order to produce viable neutrino masses and to maintain the electroweaknaturalness. In a special case when the lightest neutrino is massless, one of the threesinglets can have arbitrarily large mass unconstrained by the electroweak naturalness. Thiscan also be understood from the fact that in such a case, a linear combination of the states N α decouples completely from the SM and it has only the self interactions giving no contributionin the Higgs mass correction.The electroweak naturalness demands the scale of fermion singlets below 10 GeV. Atypical observation from eq. (3) then implies that the Dirac type Yukawa couplings arerequired to be small to account for light neutrino masses. If m i ≤ . | ( Y D ) iα | (cid:46) O (10 − ). Further, if M N α ≈ | y iα | are typically required to be (cid:46) − . Such smallcouplings can arise from a more fundamental theory in which an underlying mechanismensures the smallness of effective couplings. An example of such framework is Froggatt-Neilsen based models in which an extra global U(1) symmetry and its spontaneous breakingis utilized to produce tiny effective couplings from the fundamental couplings of O (1) [42].Another example is the theories based on extra spatial dimension in which the fermionsinglets are localized far away from the SM brane (on which the Higgs is localized) leading tothe small effective Dirac neutrino Yukawa couplings in four spacetime dimensions [43, 44]. Analternative way to generate small neutrino masses with light fermion singlets and Yukawas5f order unity is to have specific structures in Y D and M N such that M ν vanishes in eq.(3). This is termed as “seesaw cancellation” and its phenomenology is studied in [45, 46].The structures of Y D and M N leading to the seesaw cancellations remain very fine-tunedand unstable with respect to higher order corrections if they are not consequences of somesymmetry or a dynamical mechanism. One such framework is proposed in [45] where theseesaw cancellation is shown to arise due to a global U(1) symmetry equivalent to the leptonnumber conservation. We offer an alternative framework in which seesaw cancellation arisesfrom the residual discrete symmetries of the SM leptons and fermion singlets. III. DISCRETE SYMMETRIES AND SEESAW NATURALNESS
We provide a symmetry based origin of natural type-I seesaw in this section. When allthe three SM neutrinos are strictly massless, the low energy effective theory obtained afterintegrating out the singlet fermions from eq. (2) possesses a maximal accidental U(1) ≡ U(1) e × U(1) µ × U(1) τ global symmetry. The lepton doublets L e,µ,τ transform non-triviallyunder U(1) e,µ,τ and hence such symmetry leads to the same consequences as the leptonnumber conservation in the SM. Such a symmetry therefore can be used as guiding principleto forbid neutrino masses at the leading order in seesaw mechanism. The small perturbationare then induced in order to generate tiny neutrino masses. One such framework using aglobal U(1) symmetry and assuming three generations of fermion singlets is constructed in[45]. The lepton doublets have +1 charge under this global symmetry while the charges ofthree singlet fermions are chosen appropriately such that all the Dirac Yukawa couplingsdo not vanish. It is shown that such a choice would always leads to one of the fermionsinglets completely decoupled from the SM and the other two degenerate in masses whichhas non-vanishing Yukawa couplings with SM leptons and Higgs.We adopt an alternate approach in which the suppression in neutrino masses originatesfrom a discrete symmetry. The discrete symmetries are extensively used in order to predictthe structure of leptonic mixing matrix. A different class of such symmetries can be utilizedto predict particular mass patterns for Majorana neutrinos as it is recently shown in [37].Appropriately chosen residual symmetry of neutrino mass matrix can lead to one masslessneutrino [37, 38] or two degenerate and one massive or massless neutrinos [39]. A similarspectrum can be obtained by means of flavour antisymmetry [40, 41]. Here, we extend thisnovel idea to all the three neutrinos and demand that such symmetry leads to masslessneutrinos by arranging appropriate cancellations in the seesaw formula eq. (3).Consider a discrete flavour group G f as a symmetry of the leptonic part of Lagrangian ineq. (2). Under G f , the three generations of lepton doublets and fermion singlets transformrespectively as L i → ( S L ) ij L j and N i → ( S N ) ij N j , (8)where i, j = 1 , , S L and S N are 3 × S † L Y D S N = Y D and S TN M N S N = M N . (9)If there exists three massive fermion singlets then det. S N = ±
1. We choose det. S N = 1 andhence S N as an element of DSG of SU(3) which is also a subgroup of underlying flavourgroup G f . The most general such S N in arbitrary basis is S N = V N Diag . ( η , η , η ∗ η ∗ ) V † N , (10)where V N is unitary matrix representing arbitrary basis and η , are arbitrary phase factors.An invariance in eq. (9) then implies D N (cid:102) M N D N = (cid:102) M N , (11)where (cid:102) M N = V TN M N V N and D N = Diag . ( η , η , η ∗ η ∗ ). A requirement of three massivefermion singlets leads to two possibilities: (a) η , = ± η = η ∗ (cid:54) = ±
1. The choice(a) leads to three massive fermion singlets with no restrictions on their masses while (b)implies that two of the three fermion singlets are degenerate in masses. Such a symmetry isdiscussed earlier in [39] in the context degenerate solar neutrino pair.It is easily seen from eq. (3) and eq. (9) that the effective neutrino mass matrix M ν possesses residual symmetry such that S † L M ν S ∗ L = M ν . (12)We assume that S L is an element of a group Z m × Z n × Z p with m, n, p ≥ S L can be written as S L = V L Diag . ( ζ , ζ , ζ ) V † L , (13)where V L is a unitary matrix and ζ , , are phase factors. The M ν = 0 requires ζ , , (cid:54) = ± ζ i ζ j (cid:54) = 1 for all i (cid:54) = j . Using this S L and S N from eq. (10), the symmetry constraints onthe structure of Dirac Yukawa couplings can be determined from an invariance condition ineq. (9). One obtains D ∗ L (cid:101) Y D D N = (cid:101) Y D , (14)where (cid:101) Y D = V † L Y D V N , D L = Diag . ( ζ , ζ , ζ ) and D N as specified earlier. The matrix (cid:101) Y D completely vanishes if η , = ±
1. Therefore the non-vanishing Dirac Yukawa couplingsnecessarily requires η = η ∗ ≡ η (cid:54) = ±
1. The symmetry allowed by all three massive fermionsinglets and non-vanishing Dirac Yukawa couplings therefore corresponds to S N = V N D N V † N with D N = Diag . ( η, η ∗ ,
1) and η (cid:54) = ± . (15)The (cid:102) M N invariant under the above symmetry is (cid:102) M N = M M M . (16)7t leads to a degenerate pair of Majorana fermions forming a psedo-Dirac state with mass M . It is also straightforward to see that the third column of (cid:101) Y D vanishes and therefore thefermion singlet with mass M decouples completely from the SM.The structure of the first two columns of (cid:101) Y D depends on the choice of phase factors. Onegets non-vanishing element in the first (second) column and j th row for ηζ ∗ j = 1 ( ηζ j = 1).As it is discussed earlier, ζ i ζ j (cid:54) = 1 for any i (cid:54) = j is required for M ν = 0 which implies thateither the first or second column of (cid:101) Y D must entirely vanish. Therefore, if all the threeSM neutrinos are arranged to couple with one fermion singlet then the following choices areallowed for S L . S L = V L D L V † L with D L = Diag . ( η, η, η ) or Diag . ( η ∗ , η ∗ , η ∗ ) . (17)They respectively lead to (cid:101) Y D = ˜ y y y or y
00 ˜ y
00 ˜ y . (18)In the mass basis of fermion singlets, one gets Y D = y ± iy y ± iy y ± iy and M N = Diag . ( M, M, M ) , (19)where y i = ˜ y i / √
2. Note that one can also choose D L = ( η, ζ , ζ ) with ζ (cid:54) = ζ ∗ and ζ , (cid:54) = ± y = ˜ y = 0 in the above (cid:101) Y D . These two cases are physically inseparable asboth lead to M ν = 0.At this point we would like to compare our results with those obtained in [45]. The sameresults have been obtained by the authors of [45] enforcing the lepton number conservationwithout using discrete symmetries. We emphasize that the residual symmetry we use for theleptons, characterized by a generator S L given in eq. (17), can be seen as a DSG of U(1) L global symmetry which corresponds to the lepton number conservation. Therefore while thebasic mechanism to obtain the massless neutrinos is same, our approach offers an alternativeway to realize seesaw cancellations through class of discrete symmetries. It therefore opensup a platform for discrete symmetry based model building for the electroweak natural seesaw. IV. MODELS OF NATURAL SEESAW BASED ON DISCRETE SYMMETRIES
We now provide some specific examples of G f which lead to massless neutrinos despite O (1) Yukawa couplings and low seesaw scale. As it is discussed in the previous section,such a G f must contain both S N and S L , given in eq. (15) and eq. (17), as a symmetry ofthree generations of fermion singlets and lepton doublets respectively. If S N and S L both8re simultaneously chosen to be diagonal, then it is sufficient to work with G f that has one-dimensional irreducible representations. Such a G f can be abelian group, Z n (with n ≥ S L and S N as its representations. In the simplest case, G f = Z is sufficient to generate the structure of Y D and M N given in eqs. (16,18) if each of thethree generations of lepton doublets and two of the three generations of fermion singletstransform non-trivially as one dimensional irreducible representations of Z . For example,if L , , → ωL , , , N → ωN and N → ω N where ω = e πi/ then one obtains Y D and M N as shown in eqs. (16,18). Hence Z is the smallest group which can be the symmetryof leptons leading to the electroweak natural seesaw.We however discuss more interesting class of symmetries under which either the three gen-erations of N i or both N i and L i transform as three dimensional irreducible representationsof an underlying group G f . Let us first find out a suitable G f in which the three generationsof fermion singlets can be assigned to a three dimensional irreducible representation. Sucha group must contain S N = V N Diag . ( η, η ∗ , V † N as one of its elements with η (cid:54) = ± Z n with n ≥
3. The smallest such group is A which possesses two3-dimensional and three 1-dimensional irreducible representations. The lepton doublets canbe assigned to suitable 1-dimensional representations. We outline a complete model basedon A in the next subsection.If both N i and L i are chosen as 3-dimensional irreducible representations under a discretegroup G f then such a group must contain both S N and S L given in eqs. (15) and (17)respectively. Since det. S L (cid:54) = 1, such a group must be DSG of U(3) which is not a subgroupof SU(3). The DSG of U(3) containing at least one faithful three dimensional irreduciblerepresentation and of order upto 512 are listed in [47]. We look for the groups which containdesired S L and S N as their elements. The smallest such group is found to be of order 81 andknown as Σ(81) in literature [36, 48, 49]. We also construct a model based on this groupand discuss it in the second subsection below. A. An A Model
The group A is the smallest DSG of SU(3) possessing 3-dimensional irreducible repre-sentation. This group has been widely used as a flavour symmetry for the leptons becauseof its ability to predict tri-bimaximal flavour mixing pattern in the lepton sector [32–36]. Ithas one 3-dimensional ( ) and three 1-dimensional ( , (cid:48) and (cid:48)(cid:48) ) irreducible representations.The tensor products and their decomposition rules are given in [36]. We assume that thethree flavours of fermion singlets transform as and each of the three flavours of leptondoublets transforms as (cid:48) . The non-zero Dirac Yukawa couplings then require an existenceof SM singlet scalar field χ = ( χ , χ , χ ) T which transform as under A . The gauge and A invariant Lagrangian involving the leading order interactions of fermion singlets can begiven as: − L N = 1Λ y i L i ( N χ ) (cid:48) ˜ φ + 12 M ( N c N ) + 12 λ ( N c N ) χ + h . c . , (20)9here ( ... ) r denotes the component of the tensor product of the fields inside the bracket thattransform as r -dimensional irreducible representation.Let’s now discuss the breaking of A symmetry induced by non-trivial VEV of flavon field χ . In order to ensure that the mass matrix of fermion singlets remains invariant under Z symmetry characterized by a generator similar to the one given in eq. (10), one needs tofind such a generator in the representation of flavon field χ and demand that the vacuumof χ is invariant under the transformation induced by this generator. The generator of Z subgroup of A in the triplet representation is given by [36] S N = . (21)The constraint S N (cid:104) χ (cid:105) = (cid:104) χ (cid:105) implies the VEV structure (cid:104) χ (cid:105) = (cid:104) χ (cid:105) = (cid:104) χ (cid:105) ≡ v χ . It isdiscussed in detail in the Appendix B that such a VEV structure is naturally favoured forsome range of parameters in the scalar potential. After the A symmetry is broken by theVEV, eq. (20) leads to Y D = v χ Λ y y ω y ω y y ω y ω y y ω y ω and M N = M λv χ λv χ λv χ M λv χ λv χ λv χ M . (22)The Y D and M N obtained in the above respect the constraints given in eq. (9) and leadto massless neutrinos at the leading order. They can be brought into the form given in eq.(19) through a basis transformation M N → U T M N U , Y D → Y D U, with U = (cid:113) √ − √ − √ √ − √ − √ √ . (23) B. A
Σ(81)
Model
We now discuss the model in which both the fermion singlets and lepton doublets can beassigned 3-dimensional irreducible representations. The group Σ(81) has eight triplets ( A , B , C , D , ¯3 A , ¯3 B , ¯3 C and ¯3 D ) and nine singlets ( kl with k, l = 0 , ,
2) [36]. The generatorsrepresented on each of the triplets are listed in [36] which we reproduce in the Appendix Afor a convenience of reader. A complete set of tensor product decomposition rules are alsolisted in the Appendix A. We assign D ( C ) representation to the three flavours of fermionsinglets (lepton doublets). We require four flavon fields to reproduce completely the ansatzgiven in eq. (19). They are denoted as ϕ ∼ D , ψ A ∼ A , ψ B ∼ B and ψ C ∼ C . The10elevant part of the Lagrangian of model at the leading order is −L N = 1Λ y ( LN ) A ψ A ˜ φ + 1Λ y (cid:48) ( LN ) B ψ B ˜ φ + 1Λ y (cid:48)(cid:48) ( LN ) C ψ C ˜ φ + 12 λ ( N c N ) D ϕ + 12 λ (cid:48) ( N c N ) D ϕ + h . c . , (24)where ( N c N ) D in the last two terms represent two different invariant combinations of theproduct N c N as listed in the tensor decomposition rule eq. (A24) given in Appendix A.The Σ(81) symmetry is to be broken down to the Z symmetry corresponding to thegenerator a (cid:48) , represented on D as given in eq. (A5), in the fermion singlet sector. TheVEV of ϕ therefore must respect a (cid:48) (cid:104) ϕ (cid:105) = (cid:104) ϕ (cid:105) . This implies (cid:104) ϕ (cid:105) = (cid:104) ϕ (cid:105) = 0 , (cid:104) ϕ (cid:105) ≡ v ϕ (cid:54) = 0 . (25)We further require the Dirac Yukawa couplings to be invariant under the symmetry trans-formation given in eq. (9) with S L = aa (cid:48) a (cid:48)(cid:48) = Diag . ( ω, ω, ω ). Using the tensor product de-composition rules we have derived for ¯3 C ⊗ D given in eq. (A23), we find that the VEVs offlavons must follow ( aa (cid:48) ) A (cid:104) ψ A (cid:105) = (cid:104) ψ A (cid:105) , ( a a (cid:48)(cid:48) ) B (cid:104) ψ B (cid:105) = (cid:104) ψ B (cid:105) and ( a (cid:48) a (cid:48)(cid:48) ) C (cid:104) ψ C (cid:105) = (cid:104) ψ C (cid:105) ,where the ( ... ) r implies the generators in the parenthesis to be chosen in r th representation.These constraints lead to (cid:104) ψ A (cid:105) = v ψ A , (cid:104) ψ B (cid:105) = v ψ B , (cid:104) ψ C (cid:105) = v ψ C . (26)The resulting structures of Y D and M N when compared to those in eq. (2) are Y D = 1Λ yv ψ A y (cid:48)(cid:48) v ψ C y (cid:48) v ψ B and M N = v ϕ λ λ λ (cid:48) . (27)Again, they can be brought into the form given in eq. (19) by the basis transformation M N → U T M N U , Y D → Y D U, with U = √ − i √ √ i √
00 0 1 , (28)resulting into y = y √ v ψA Λ , y = y (cid:48)(cid:48) √ v ψC Λ , y = y (cid:48) √ v ψB Λ , M = λv ϕ and M = λ (cid:48) v ϕ . Clearly, thismodel requires more flavons and therefore is less economical than the model based on A symmetry discussed earlier. We also discuss the viability of VEV structures in Appendix B.It is shown that the above vacuum structure is not natural in the given minimal model. Toobtain the required vacuum alignment without any fine tunning in the scalar potential, oneneeds to suitably modify the model. We have disccussed one such possibility in AppendixB in which an additional U (1) symmetry is imposed under which all the flavon fields posses11ifferent charges. However a set of new flavons, charged under the U (1) but singlet underΣ(81), is required to maintain the form of interactions in Eq. (24). We refer reader toAppendix B for more details.One of the motivations to assign 3-dimensional irreducible representations to the leptonsis to predict their flavour mixing pattern through underlying symmetry. However, in thepresent case all the mixing angles are not physical as the neutrino masses are vanishing atthe leading order. So the presence of discrete symmetry here does not correspond to anyprediction for the mixing angles. Once the symmetry is broken to generate tiny neutrinomasses, it gives rise to the leptonic mixing parameters. We quantitatively discuss somegeneric cases of the symmetry breaking in the next section.Before ending this section, we comment on the viability of the above models in thecontext of electroweak naturalness. Both the proposed models involve non-renormalizableinteractions as well as new SM singlet flavon fields. For example, such a flavon field ξ cancouple to the SM Higgs φ with a coupling like L φ − ξ = κ | φ | | ξ | , (29)which is not forbidden by the SM gauge or flavour symmetry. This interaction contributesin the µ correction at one-loop level which is estimated to be [26] δµ ∼ κ π M ξ , (30)where M ξ represents the mass of flavon field. The elecroweak naturalness then requireseither κ (cid:28) M ξ ≤ O (1) then electroweaknaturalness again dictates the mass scale of such new fields to be O (TeV) [26]. Henceone finds the scale of discrete symmetry breaking and the cutoff scale Λ to be close and ∼ O (TeV) in natural theories. If the criteria of naturalness is given up and low scale type-Iseesaw mechanism is still considered then the scale of discrete symmetry breaking and Λcan be arbitrarily large keeping the ratio (cid:104) ξ (cid:105) / Λ to be of O (1). V. BREAKING OF RESIDUAL SYMMETRIES & NONZERO NEUTRINO MASSES
We now discuss generic perturbations in Y D and M N given in eq. (19) derived by de-manding invariance under the residual symmetries S L and S N . Such perturbations mayarise from different sources depending on the exact model under consideration. The mostcommon source of perturbation is the next-to-leading order corrections in Y D and/or M N which do not respect the invariance conditions eq. (9). Another source of perturbation is12 small deviation from the exact vacuum alignments in the flavon fields which may ariseagain from the next-to-leading order corrections in the flavon potential. We however do notdiscuss here the origin of such model specific perturbations and only analyze phenomenolog-ical consequences of generic perturbations. In the mass basis of fermion singlets, the mostgeneral deviations form Y D and M N in eq. (19) can be parametrized as Y (cid:48) D = y (1 − (cid:15) ) iy (1 + (cid:15) ) (cid:15) y (1 − (cid:15) ) iy (1 + (cid:15) ) (cid:15) y (1 − (cid:15) ) iy (1 + (cid:15) ) (cid:15) , M (cid:48) N = Diag . ( M (1 − (cid:15) M ) , M (1 + (cid:15) M ) , M ) . (31)We discuss two phenomenologically interesting cases. In the first case, we assume that themass matrix of fermion singlets still possesses suitable residual Z n symmetry characterized by S N given in eq. (15) while the Dirac Yukawa interactions do not respect such symmetry i.e. S † L Y D S N (cid:54) = Y D . This case is characterized by (cid:15) M = 0 in eq. (31). Such scenario may arise, forexample in case of Σ(81) model when the vacuum of flavons ψ A,B,C has small deviations fromtheir structures given in eq. (26) but the VEV alignment of ϕ remains intact. To analyzethis case, we fix M = 2 M = 2 TeV and randomly vary all y i in the range: | y i | ∈ [0 . , . y i ) ∈ [0 , π ]. For each of these point, we optimize the values of (cid:15) i such that theyreproduce the solar and atmospheric squared mass differences within the 3 σ ranges of theirglobal fit values. These ranges are taken from the recent global fit of the neutrino oscillationdata given in [50]. Here, we do not impose any restrictions on (cid:15) i from the neutrino mixingangles since the mixing angles depend also on the parameters in the charged lepton massmatrix which, in general case, is also perturbed by symmetry breaking effects. We first set (cid:15) , , = 0 in eq. (31) which corresponds to a case with perturbations but still with decoupledfermion singlet corresponding to the mass M . The results of this case are displayed inFig. 2. It can be seen that one requires very small perturbations < ∼ O (10 − − − ) inorder to produce viable neutrino mass spectrum. The resulting neutrino mass spectrumhas a massless neutrino as one of the fermion singlets completely decouples from the SM.In Fig. 3, we show the similar results but with (cid:15) , , (cid:54) = 0. As it can be seen, in this casethe magnitude of perturbation can be as large as O (10 − ) and the lightest neutrino can beheavier compared to the previous case.In the second case, we assume that perturbation breaks both the residual symmetriesand both Y D and M N do not satisfy the invariance conditions given in eq. (9). In this case, (cid:15) M can also be nonzero together with all the (cid:15) i in Y D . We find that the magnitude of (cid:15) , , dominate over all the other perturbations. The largest magnitude of perturbations requiredin this case is similar to the one shown in the left panel of Fig. 3. The other results of thiscase are displayed in Fig. 4. We find that the allowed splitting between the fermion singletsof first two generations varies in the range 10 − -10 eV for M = 1 TeV. Although we assumenormal ordering for the neutrino masses in the above numerical analysis of perturbations,we get similar results in the case of inverted ordering.13 - - - - Max {| ϵ i |} F r e q u e n c y FIG. 2. The largest magnitude of perturbation required in eq. (31), with (cid:15) , , = (cid:15) M = 0, inorder to generate ∆ m and ∆ m within the 3 σ of their global fit values [50]. The values of y i are taken from the random and flat distribution corresponding to the range | y i | ∈ [0 . , .
5] andarg.( y i ) ∈ [0 , π ] and M = 2 M is fixed at 2 TeV. - - - - Max {| ϵ i |} F r e q u e n c y - - - - m ν [ eV ] F r e q u e n c y FIG. 3. Left: Same as the caption of Fig. 2 but with (cid:15) , , (cid:54) = 0. Right: The correspondingpredictions for the mass of the lightest neutrino.. VI. SUMMARY AND DISCUSSION
In the models of neutrino masses based on type-I seesaw mechanism, the naturalnessof electroweak scale restricts the masses of fermion singlets to be ≤ GeV and theirYukawa couplings to the SM fermion to be of O (10 − ). If these couplings are assumed tobe of the order of unity then fermion singlets are required to be as light as few TeV. In thiscase, the seesaw mechanism cannot be considered as the dominant mechanism responsiblefor small neutrino masses. In order to produce phenomenologically viable neutrino mass14 - - - - m ν [ eV ] F r e q u e n c y - - - - - - | ϵ M | F r e q u e n c y FIG. 4. The predictions for the mass of the lightest neutrino (left) and splittings between themasses of fermion singlets of the first and second generations (right) allowed by the most generalperturbation in eq. (31). The other details are same as mentioned in the caption of Fig. 2. spectrum in this setup, one needs specific finely tuned correlations among the O (1) Yukawacouplings and masses of singlet fermions. We motivate such fine-tuning through the presenceof finite discrete symmetry under which all the SM leptons and fermion singlets transformnon-trivially. The three generations of lepton doublets are assumed to possess Z m × Z n × Z p symmetry with m, n, p ≥ G f depending on the representations to beassigned to the leptons and fermion singlets. The G f can be abelian symmetry if the onedimensional representations are chosen for the leptons and fermion singlets. If 3-dimensionalirreducible representation is assigned to the three generations of fermion singlets, then onecan have G f as a DSG of SU(3). The smallest such group is A and we have provided amodel realization of it. If the leptons are also to be chosen as 3-dimensional irreducible rep-resentation then the G f is necessarily DSG of U(3). We find the group Σ(81) as the smallestDSG of U(3) which qualifies to be a symmetry of massless neutrinos and we have outlineda model based on this group. It is found that A symmetry provides more economical andnatural option for model compared to the Σ(81). In all the cases, the underlying symmetryleads to massless neutrinos at the leading order and the tiny neutrino masses arise throughsmall perturbations to the symmetry. We also study the phenomenology of generic pertur-bations and find the magnitude of such perturbations required to generate viable neutrinomasses. We find that small deviations from degeneracy in the masses of fermion singlets iscompatible with the data and therefore the resonant leptogenesis mechanism may naturally15merge in this class of models as an alternative of the standard thermal leptogenesis. It isalso possible to have the similar symmetry based realization for seesaw cancellations in caseof only two generations of fermion singlets. One still gets the degenerate pair of fermionsinglets at the leading order by the symmetry conditions. This setup naturally leads to theminimal low scale type-I seesaw model with (quasi)degenerate RH neutrinos and its updatedphenomenology is recently studied in [51, 52]. It is shown there that such a framework cansuccessfully account for the baryon asymmetry of the universe through resonant leptogenesis. ACKNOWLEDGMENTS
We thank Anjan S. Joshipura for reading the manuscript and useful suggestions. KMPthanks the Department of Science and Technology, Government of India for research grantsupport under INSPIRE Faculty Award (DST/INSPIRE/04/2015/000508). We acknowl-edge the use of JaxoDraw [53].
Appendix A: The group
Σ(81)
Here we outline some important features of the group Σ(81). The reader is advised tosee [36] for more details. The Σ(81) group is a finite discrete subgroup of U(3) which has81 elements. The elements can be written as g = b k a l a (cid:48) m a (cid:48)(cid:48) n with k, l, m, n = 0 , ,
2. Thegenerators b, a, a (cid:48) and a (cid:48)(cid:48) satisfy b = a = a (cid:48) = a (cid:48)(cid:48) = 1, b − ab = a (cid:48)(cid:48) , b − a (cid:48) b = a and b − a (cid:48)(cid:48) b = a (cid:48) . The generators a , a (cid:48) and a (cid:48)(cid:48) commute with each others. The elements areclassified into seventeen conjugacy classes. There are nine singlets represented by kl where k, l = 0 , , A , B , C , D , A , B , C and D . Below welist the set of four generators for each of these 3-dimensional representations.On all of the triplets, the generator b is represented as b = . (A1)The representation of the generators a , a (cid:48) and a (cid:48)(cid:48) on each of the triplets are: a = ω , a (cid:48) = ω
00 0 1 , a (cid:48)(cid:48) = ω , on A (A2) a = ω
00 0 ω , a (cid:48) = ω ω , a (cid:48)(cid:48) = ω ω
00 0 1 , on B (A3)16 = ω ω
00 0 ω , a (cid:48) = ω ω
00 0 ω , a (cid:48)(cid:48) = ω ω
00 0 ω , on C (A4) a = ω ω , a (cid:48) = ω ω
00 0 1 , a (cid:48)(cid:48) = ω
00 0 ω , on D . (A5)The representations on A , B , C , D are complex conjugate of the representations of A , B , C , D respectively. The generators are represented on the singlets kl as b = ω l , a = a (cid:48) = a (cid:48)(cid:48) = ω k .We now list the complete set of tensor product decompositions for the triplets. a a a A ⊗ b b b A = a b a b a b A ⊕ a b a b a b B ⊕ a b a b a b B (A6) a a a A ⊗ b b b A = (cid:16) (cid:88) l =0 , , ( a b + ω l a b + ω l a b ) l (cid:17) ⊕ a b a b a b D ⊕ a b a b a b D (A7) a a a A ⊗ b b b B = a b a b a b C ⊕ a b a b a b A ⊕ a b a b a b A (A8)17 a a a A ⊗ b b b B = (cid:16) (cid:88) l =0 , , ( a b + ω l a b + ω l a b ) l (cid:17) ⊕ a b a b a b D ⊕ a b a b a b D (A9) a a a A ⊗ b b b C = a b a b a b B ⊕ a b a b a b C ⊕ a b a b a b C (A10) a a a A ⊗ b b b C = (cid:16) (cid:88) l =0 , , ( a b + ω l a b + ω l a b ) l (cid:17) ⊕ a b a b a b D ⊕ a b a b a b D (A11) a a a A ⊗ b b b D = a b a b a b A ⊕ a b a b a b B ⊕ a b a b a b C (A12) a a a A ⊗ b b b D = a b a b a b A ⊕ a b a b a b B ⊕ a b a b a b C (A13), 18 a a a B ⊗ b b b B = a b a b a b B ⊕ a b a b a b C ⊕ a b a b a b C (A14) a a a B ⊗ b b b B = (cid:16) (cid:88) l =0 , , ( a b + ω l a b + ω l a b ) l (cid:17) ⊕ a b a b a b D ⊕ a b a b a b D (A15) a a a B ⊗ b b b C = a b a b a b A ⊕ a b a b a b B ⊕ a b a b a b B (A16) a a a B ⊗ b b b C = (cid:16) (cid:88) l =0 , , ( a b + ω l a b + ω l a b ) l (cid:17) ⊕ a b a b a b D ⊕ a b a b a b D (A17) a a a B ⊗ b b b D = a b a b a b A ⊕ a b a b a b B ⊕ a b a b a b C (A18)19 a a a B ⊗ b b b D = a b a b a b A ⊕ a b a b a b B ⊕ a b a b a b C (A19) a a a C ⊗ b b b C = a b a b a b C ⊕ a b a b a b A ⊕ a b a b a b A (A20) a a a C ⊗ b b b C = (cid:16) (cid:88) l =0 , , ( a b + ω l a b + ω l a b ) l (cid:17) ⊕ a b a b a b D ⊕ a b a b a b D (A21) a a a C ⊗ b b b D = a b a b a b A ⊕ a b a b a b B ⊕ a b a b a b C (A22) a a a C ⊗ b b b D = a b a b a b A ⊕ a b a b a b B ⊕ a b a b a b C (A23) a a a D ⊗ b b b D = a b a b a b D ⊕ a b a b a b D ⊕ a b a b a b D (A24)20 a a a D ⊗ b b b D = (cid:16) (cid:88) l =0 , , ( a b + ω l a b + ω l a b ) l (cid:17) ⊕ (cid:16) (cid:88) l =0 , , ( a b + ω l a b + ω l a b ) l (cid:17) ⊕ (cid:16) (cid:88) l =0 , , ( a b + ω l a + ω l a b ) l (cid:17) (A25)The tensor products of singlets are given by kl ⊗ k (cid:48) l (cid:48) = k + k (cid:48) ( mod l + l (cid:48) ( mod (A26) Appendix B: The scalar potential and vacuum alignment1. The A model This model contains two scalars: an SM singlet and A triplet real scalar field χ and theSM Higgs which is singlet under A . The most general renormalizable potential invariantunder the SM gauge symmetry and A flavour symmetry is written as V = V ( φ ) + V ( χ ) + V ( φ, χ ) (B1)where V ( φ ) = µ φ † φ + λ ( φ † φ ) ,V ( χ ) = µ χ ( χχ ) + σ ( χχχ ) + λ ( χχ ) ( χχ ) + λ ( χχ ) (cid:48) ( χχ ) (cid:48)(cid:48) + λ ( χχ ) ( χχ ) ,V ( χ, φ ) = κ ( φ † φ )( χχ ) . (B2)where ( ... ) r denotes the component of the tensor product of the fields inside the bracket thattransform as r -dimensional irreducible representation.The minimization consditions of the complete potential evaluated at the required mini-mum (cid:104) χ (cid:105) = (cid:104) χ (cid:105) = (cid:104) χ (cid:105) ≡ v χ lead to0 = (cid:20) ∂V∂χ i (cid:21) χ j = v χ = 2 v χ ( µ χ + κv + 3 σv χ + 2(3 λ + 4 λ ) v χ ) . (B3)The nontrivial vacuum is obtained as v χ = − σ ± (cid:113) σ − λ + 4 λ )( µ χ + κv )4(3 λ + 4 λ ) . (B4)The above minima is global for λ , > µ χ + κv ) <
0. Therefore in this model thedesired vacuum alignment can be obtained without any fine-tunning in the potential. It iseasy to see that the required vacuum alignment can be global minimum of V ( χ ). The SMHiggs φ , being A singlet, does not change this result.21 . The Σ(81) model
The model contains several flavon fields: ψ A , ψ B , ψ C and ϕ transforming as A , B , C and D respectively. The complete flavon potential can be decomposed into the followingpieces for simplicity. V = V ( ϕ ) + (cid:88) i ( V ( ψ i ) + V ( φ, ψ i )) + (cid:88) i (cid:54) = j ( V ( ψ i , ψ j ) + V ( φ, ψ i , ψ j ))+ V ( ψ A , ψ B , ψ C ) + V ( φ, ψ A , ψ B , ψ C ) , (B5)where i, j = A, B, C . We find that the desired vacuum of ϕ given in Eq. (25) can be aglobal minima of V ( ϕ ). Similar results are found for all the other flavons. However somecross-coupling terms in V ( φ, ψ i ), V ( ψ i , ψ j ), V ( φ, ψ i , ψ j ), V ( ψ A , ψ B , ψ C ) and V ( φ, ψ A , ψ B , ψ C )destroy the alignment and require special conditions on the parameters of the most generalpotential. We therefore find that the desired vacuum of all the flavon fields require severalunnatural conditions.One of the well-known solutions to the vacuum alignment problem is to extend the flavourgroup as discussed in detail in [54]. The flavour group in this case can be extended in sucha way that it preserves the flavour structure and leads to some accidental symmetry inthe flavon potential which ensures the desired vacuum structures. Investigation of suchpossibility is however beyond the scope of the studies presented in this paper and shouldbe taken up elsewhere. We provide a less radical solution to this problem. Some of theunwanted cross-coupling can be avoided by imposing additional symmetries on the flavonpotential. Consider a U (1) symmetry under which ψ A → e iα ψ A , ψ B → e iβ ψ B , ψ C → e iγ ψ C and φ → e iδ φ . Alternatively, U (1) can also be replaced by Z n symmetry with sufficientlylarge value of n . It is straightforward to see that the invariance of V under U (1) implies V ( φ, ψ i , ψ j ) = V ( ψ A , ψ B , ψ C ) = V ( φ, ψ A , ψ B , ψ C ) = 0 (B6)in Eq. (B5). The remaining terms can be obtained using the tensor product rules given inthe previous Appendix. They are V ( ϕ ) = µ ϕ φ † φ + (cid:88) a =1 λ ϕa ( φ † φ ) a ( φ † φ ) a + (cid:88) a =1 κ ϕa ( φφ ) a ( φ † φ † ) a ,V ( ψ i ) = µ i ψ † i ψ i + (cid:88) a =1 λ ia ( ψ † i ψ i ) a ( ψ † i ψ i ) a + (cid:88) a =1 κ ia ( ψ i ψ i ) a ( ψ † i ψ † i ) a ,V ( φ, ψ i ) = (cid:88) a =1 λ ϕia ( ψ † i ψ i ) a ( φ † φ ) a + (cid:88) a =1 κ ϕia ( ψ i φ ) a ( ψ † i φ † ) a ,V ( ψ i , ψ j ) = (cid:88) a =1 λ ija ( ψ † i ψ i ) a ( ψ † j ψ j ) a + (cid:88) a =1 κ ija ( ψ i ψ j ) a ( ψ † i ψ † j ) a , (B7)where a = 1 , , ... denotes various possible ways to contract the flavon. For example, inthe last term of V ( φ ), a = 1 , .., φφ ) and ( φ † φ † )22an be contracted to get singlets. They arise from the fact that ( φφ ) transform as threedifferent D representations under Σ(81). Since ( φφ ) and ( φ † φ † ) are symmetric, it leads tosix independent way to make ( φφ ) a ( φ † φ † ) a as singlet.Using Eqs. (B5,B6,B7) and the vacuum structures in Eqs. (25,26), one obtains (cid:20) ∂V∂ϕ (cid:21) min . = (cid:20) ∂V∂ϕ (cid:21) min . = 0 , (cid:20) ∂V∂ψ A (cid:21) min . = (cid:20) ∂V∂ψ A (cid:21) min . = 0 , (cid:20) ∂V∂ψ B (cid:21) min . = (cid:20) ∂V∂ψ B (cid:21) min . = 0 , (cid:20) ∂V∂ψ C (cid:21) min . = (cid:20) ∂V∂ψ C (cid:21) min . = 0 , (B8)and (cid:20) ∂V∂ϕ (cid:21) min . = 2 v ϕ ( µ ϕ + c v ψ A + c v ψ B + c v ψ C + c v ϕ ) , (cid:20) ∂V∂ψ A (cid:21) min . = 2 v ψ A ( µ A + c A v ψ A + c A v ψ B + c A v ψ C + c v ϕ ) , (cid:20) ∂V∂ψ B (cid:21) min . = 2 v ψ B ( µ B + c B v ψ A + c B v ψ B + c B v ψ C + c v ϕ ) , (cid:20) ∂V∂ψ C (cid:21) min . = 2 v ψ C ( µ C + c A v ψ A + c B v ψ B + c C v ψ C + c v ϕ ) , (B9)where c = κ ϕA + λ ϕA + λ ϕA + λ ϕA ,c = κ ϕB + λ ϕB + ωλ ϕB − ω λ ϕB ,c = κ ϕC + λ ϕC − ω λ ϕC + ωλ ϕC ,c = 2( κ ϕ + (cid:88) a =1 λ ϕa ) ,c A = 2( κ A + (cid:88) a =1 λ Aa ) ,c A = κ AB + λ AB − ω λ AB + ωλ AB ,c A = κ AC + λ AC + ωλ AC − ω λ AC ,c B = 2( κ B + (cid:88) a =1 λ Ba ) ,c B = κ BC + λ BC − ω λ BC + ωλ BC ,c C = 2( κ C + (cid:88) a =1 λ Ca ) , (B10)23he four equations in (B9) when equated to zero determine the four VEVs. As it can beseen, there are large number of parameters in potential which lead to the desired values ofVEVs. Clearly, no fine-tuning is required to achieve required vacuum alignment. It is tobe noted that the U (1) symmetry forbids the terms in Yukawa Langrangian, Eq. (24). Toavoid this problem, one can introduce a scalar singlet for each flavon triplet with opposite U (1) charge. Each flavon triplet in Eq. (24) then can be replaced by the a combination ofthat flavon triplet and its singlet partner. Since the new scalars are singlets under the fullflavour group, they do not modify the vacuum alignment conditions, Eqs. (B8,B9). Thesefields only shift the mass term of flavon triplets in a similar way the VEV of Higgs field φ changes the µ χ to µ χ + κv as shown in the case of A model. The SM Higgs also give rise tothe same effects without perturbing the vacuum structure when its interactions are includedin V given in Eq. (B5). [1] P. Minkowski, Phys.Lett. B67 , 421 (1977).[2] T. Yanagida, Conf.Proc.
C7902131 , 95 (1979).[3] S. Glashow, NATO Adv.Study Inst.Ser.B Phys. , 687 (1980).[4] R. N. Mohapatra and G. Senjanovic, Phys.Rev.Lett. , 912 (1980).[5] M. Gell-Mann, P. Ramond, and R. Slansky, Conf.Proc. C790927 , 315 (1979), arXiv:1306.4669[hep-th].[6] J. Schechter and J. Valle, Phys.Rev.
D22 , 2227 (1980).[7] J. Schechter and J. W. F. Valle, Phys. Rev.
D25 , 774 (1982).[8] F. del Aguila, J. A. Aguilar-Saavedra, A. Martinez de la Ossa, and D. Meloni, Phys. Lett.
B613 , 170 (2005), arXiv:hep-ph/0502189 [hep-ph].[9] S. Bray, J. S. Lee, and A. Pilaftsis, Phys. Lett.
B628 , 250 (2005), arXiv:hep-ph/0508077[hep-ph].[10] T. Han and B. Zhang, Phys. Rev. Lett. , 171804 (2006), arXiv:hep-ph/0604064 [hep-ph].[11] F. del Aguila, J. A. Aguilar-Saavedra, and R. Pittau, Elementary particle physics. Proceedings,Corfu Summer Institute, CORFU2005, Corfu, Greece, September 4-26, 2005 , J. Phys. Conf.Ser. , 506 (2006), arXiv:hep-ph/0606198 [hep-ph].[12] D. Atwood, S. Bar-Shalom, and A. Soni, Phys. Rev. D76 , 033004 (2007), arXiv:hep-ph/0701005 [hep-ph].[13] S. Bray, J. S. Lee, and A. Pilaftsis, Nucl. Phys.
B786 , 95 (2007), arXiv:hep-ph/0702294[HEP-PH].[14] F. M. L. de Almeida, Jr., Y. do Amaral Coutinho, J. A. Martins Simoes, A. J. Ramalho,S. Wulck, and M. A. B. do Vale, Phys. Rev.
D75 , 075002 (2007), arXiv:hep-ph/0703094[HEP-PH].[15] F. del Aguila, J. A. Aguilar-Saavedra, and R. Pittau, JHEP , 047 (2007), arXiv:hep-ph/0703261 [hep-ph].
16] B. Bajc, M. Nemevsek, and G. Senjanovic, Phys. Rev.
D76 , 055011 (2007), arXiv:hep-ph/0703080 [hep-ph].[17] F. F. Deppisch, P. S. Bhupal Dev, and A. Pilaftsis, New J. Phys. , 075019 (2015),arXiv:1502.06541 [hep-ph].[18] A. Das and N. Okada, (2017), arXiv:1702.04668 [hep-ph].[19] W. Buchmuller and C. Greub, Nucl. Phys. B363 , 345 (1991).[20] G. Ingelman and J. Rathsman, Z. Phys.
C60 , 243 (1993).[21] C. A. Heusch and P. Minkowski, Nucl. Phys.
B416 , 3 (1994).[22] G. Aad et al. (ATLAS), Phys. Lett.
B716 , 1 (2012), arXiv:1207.7214 [hep-ex].[23] S. Chatrchyan et al. (CMS), Phys. Lett.
B716 , 30 (2012), arXiv:1207.7235 [hep-ex].[24] V. Khachatryan et al. (CMS), Eur. Phys. J.
C75 , 212 (2015), arXiv:1412.8662 [hep-ex].[25] M. Farina, D. Pappadopulo, and A. Strumia, JHEP , 022 (2013), arXiv:1303.7244 [hep-ph].[26] A. de Gouvea, D. Hernandez, and T. M. P. Tait, Phys. Rev. D89 , 115005 (2014),arXiv:1402.2658 [hep-ph].[27] F. Vissani, Phys. Rev.
D57 , 7027 (1998), arXiv:hep-ph/9709409 [hep-ph].[28] J. A. Casas, J. R. Espinosa, and I. Hidalgo, JHEP , 057 (2004), arXiv:hep-ph/0410298[hep-ph].[29] Z.-z. Xing, Particle physics beyond the standard model. Proceedings, 16th Yukawa InternationalSeminar, YKIS2008, Kyoto, Japan, January 26-March 25, 2009 , Prog. Theor. Phys. Suppl. , 112 (2009), arXiv:0905.3903 [hep-ph].[30] H. Davoudiasl and I. M. Lewis, Phys. Rev.
D90 , 033003 (2014), arXiv:1404.6260 [hep-ph].[31] J. D. Clarke, R. Foot, and R. R. Volkas, Phys. Rev.
D91 , 073009 (2015), arXiv:1502.01352[hep-ph].[32] G. Altarelli and F. Feruglio, Rev.Mod.Phys. , 2701 (2010), arXiv:1002.0211 [hep-ph].[33] G. Altarelli, F. Feruglio, and L. Merlo, Fortsch.Phys. , 507 (2013), arXiv:1205.5133 [hep-ph].[34] A. Y. Smirnov, J.Phys.Conf.Ser. , 012006 (2011), arXiv:1103.3461 [hep-ph].[35] S. F. King and C. Luhn, Rept.Prog.Phys. , 056201 (2013), arXiv:1301.1340 [hep-ph].[36] H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada, et al. , Prog.Theor.Phys.Suppl. , 1 (2010), arXiv:1003.3552 [hep-th].[37] A. S. Joshipura and K. M. Patel, Phys.Lett. B727 , 480 (2013), arXiv:1306.1890 [hep-ph].[38] A. S. Joshipura and K. M. Patel, JHEP , 009 (2014), arXiv:1401.6397 [hep-ph].[39] A. S. Joshipura and K. M. Patel, Phys.Rev.
D90 , 036005 (2014), arXiv:1405.6106 [hep-ph].[40] A. S. Joshipura, JHEP , 186 (2015), arXiv:1506.00455 [hep-ph].[41] A. S. Joshipura and N. Nath, Phys. Rev. D94 , 036008 (2016), arXiv:1606.01697 [hep-ph].[42] G. Altarelli, F. Feruglio, and I. Masina, JHEP , 035 (2003), arXiv:hep-ph/0210342[hep-ph].[43] R. Kitano and T.-j. Li, Phys.Rev.
D67 , 116004 (2003), arXiv:hep-ph/0302073 [hep-ph].[44] F. Feruglio, K. M. Patel, and D. Vicino, JHEP , 095 (2014), arXiv:1407.2913 [hep-ph].
45] J. Kersten and A. Yu. Smirnov, Phys. Rev.
D76 , 073005 (2007), arXiv:0705.3221 [hep-ph].[46] C.-H. Lee, P. S. Bhupal Dev, and R. N. Mohapatra, Phys. Rev.
D88 , 093010 (2013),arXiv:1309.0774 [hep-ph].[47] P. O. Ludl, J.Phys.
A43 , 395204 (2010), arXiv:1006.1479 [math-ph].[48] C. Hagedorn, M. A. Schmidt, and A. Y. Smirnov, Phys.Rev.
D79 , 036002 (2009),arXiv:0811.2955 [hep-ph].[49] Y. BenTov and A. Zee, Nucl. Phys.
B871 , 452 (2013), arXiv:1202.4234 [hep-ph].[50] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, and T. Schwetz, (2016),arXiv:1611.01514 [hep-ph].[51] G. Bambhaniya, P. S. B. Dev, S. Goswami, S. Khan, and W. Rodejohann, (2016),arXiv:1611.03827 [hep-ph].[52] T. Rink, K. Schmitz, and T. T. Yanagida, (2016), arXiv:1612.08878 [hep-ph].[53] D. Binosi, J. Collins, C. Kaufhold, and L. Theussl, Comput. Phys. Commun. , 1709(2009), arXiv:0811.4113 [hep-ph].[54] M. Holthausen and M. A. Schmidt, JHEP , 126 (2012), arXiv:1111.1730 [hep-ph]., 126 (2012), arXiv:1111.1730 [hep-ph].