Realization of the minimal extended seesaw mechanism and the T M 2 type neutrino mixing
PPreprint version
Realization of the minimal extended seesawmechanism and the
T M type neutrino mixing R. Krishnan, a Ananya Mukherjee b and Srubabati Goswami b a Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, India b Physical Research Laboratory, Ahmedabad- 380009, India
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We construct a neutrino mass model based on the flavour symmetry group A × C × C × C which accommodates a light sterile neutrino in the minimal extendedseesaw (MES) scheme. Besides the flavour symmetry, we introduce a U (1) gauge symmetryin the sterile sector and also impose CP symmetry. The vacuum alignments of the scalarfields in the model spontaneously break these symmetries and lead to the constructionof the fermion mass matrices. With the help of the MES formulas, we extract the lightneutrino masses and the mixing observables. In the active neutrino sector, we obtainthe TM mixing pattern with non-zero reactor angle and broken µ - τ reflection symmetry.We express all the active and the sterile oscillation observables in terms of only four realmodel parameters. Using this highly constrained scenario we predict sin θ = 0 . +0 . − . , sin δ = − . +0 . − . , | U e | = 0 . +0 . − . , | U µ | = 0 . +0 . − . and | U τ | = 0 . +0 . − . which are consistent with the current data. a r X i v : . [ h e p - ph ] O c t ontents Observations made in the neutrino oscillation experiments have confirmed that neutrinoshave mass, albeit tiny. The Standard Model (SM) of particle physics can not accommodatethe neutrino mass due to the absence of right-handed neutrinos, unlike the case for thecharged leptons and the quarks. The inclusion of additional right-handed neutrino fieldsalong with the seesaw mechanism [1–4] plays a vital role in modeling properties of massiveneutrinos. The well known PMNS matrix encodes the mixing between the neutrino flavoureigenstates and their mass eigenstates. This matrix is parametrised in terms of three mixingangles and three CP phases (in a three flavoured paradigm), U PMNS = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c c s e iδ − c s − s c s e iδ c c . U Maj , (1.1)where c ij = cos θ ij , s ij = sin θ ij . The diagonal matrix, U Maj = diag (1 , e iα , e i ( β + δ ) ) , containsthe Majorana CP phases α, β which become observable if the neutrinos behave as Majoranaparticles.Although the last two decades of neutrino oscillation experiments made tremendousprogress in determining the three flavour mixing angles, efforts are underway to measurethese parameters more precisely. We do not yet know whether the atmospheric mixingis maximal or not. If it is not, the octant of the atmospheric mixing angle, θ , is tobe determined. Measurement of the Dirac CP phase, δ , will confirm CP violation in theleptonic sector and may explain the observed baryon asymmetry via leptogenesis. The– 1 –ature of the neutrinos, i.e. whether they are Dirac or Majorana, is still an open questionwhich can not be settled with the help of the oscillation experiments. On the other hand, theobservation of neutrino-less double-beta decays ( νββ ) will establish the Majorana nature.Such decays are yet to be observed. The oscillation experiments have determined the mass-squared differences (solar: ∆ m and atmospheric: ∆ m ), but they are not sensitive tothe absolute neutrino mass scale. Data from the Planck satellite provides an upper boundon the sum of neutrino masses, (cid:80) i m i ≤ . eV [5]. Experimental searches are also beingmade to directly measure the electron neutrino mass using the kinematics of beta decays.Recently, the KATRIN collaboration has announced its first result on the effective electronantineutrino mass using the tritium beta decay, H → He + e − + ¯ ν e , and reported theupper bound for the effective antineutrino mass [6], m ¯ ν e < . eV at the confidencelevel (CL).Although the three flavour paradigm of neutrino oscillation is well established, thereare some experimental results that motivate us to go beyond this and postulate the exis-tence of one or more sterile neutrinos. This possibility has gained considerable attentionin recent years. In principle, the presence of a fourth neutrino can impressively explainseveral sets of experimental anomalies. The first indication came from the LSND exper-iment which showed evidence of oscillation with mass scale ∼ eV [7] in ¯ ν µ - ¯ ν e channel.Later MiniBooNE experiment also confirmed it [8, 9]. The Reactor Anomaly involves adeficit of reactor antineutrinos detected in short-baseline (<500 m) experiments with re-calculated neutrino fluxes [10–12]. The short-baseline neutrino oscillations can also explainthe so-called
Gallium Anomaly observed during the calibrations runs of the radiochemicalexperiments, GALLEX and SAGE. The ratio of the experimental flux to the theoreticalestimate was found to be . ± . . The resolution of both the Reactor and the Galliumanomalies with the help of the active-sterile oscillations point towards a common region ofthe parameter space with the sterile neutrino having mass in the ∼ eV scale [13–16].The proposed sterile neutrino is an SM singlet which does not participate in the weakinteractions, but they can mix with the active neutrinos enabling them to be probed inthe oscillation experiments. The addition of a single sterile neutrino field leads to anoscillation parameter space consisting of a × unitary mixing matrix along with threeindependent mass-squared-differences. Among them, the preferred scenario, often calledthe 3+1 scheme [17–20], has three active neutrinos and one sterile neutrino in the sub-eVand eV scale respectively. The 2+2 scheme, in which two pairs of neutrino mass states differby O ( eV ) , is not consistent with the solar and the atmospheric data [21]. The 1+3 schemein which the three active neutrinos are in eV scale and the sterile neutrino is lighter thanthe active neutrinos is disfavored by cosmology. Therefore, in this paper, we assume the3+1 scenario. The recently proposed Minimal Extended Seesaw (MES) [22, 23] has manyappealing features. The active-sterile mixing obtained in MES is suppressed by the ratioof masses of the active and sterile sectors. With the active neutrino mass of the order of ∼ . eV and the sterile neutrino mass of the order of eV, this suppression is consistentwith the active-sterile mixing as observed in LSND and MiniBooNe The data from solar and atmospheric neutrino oscillations as well as oscillations observed in accelerator – 2 – large number of neutrino mass models based on discrete flavour symmetry groupshave been proposed [24–27] in the last decade. These models generate various mixingpatterns such as the well known tribimaximal mixing (TBM) [28–34]. Since the non-zerovalue of the reactor mixing angle [35–37] has ruled out TBM, one of the popular ways toachieve realistic mixings is through either its modifications or extensions [38–44]. Unlikethe active-only mixing scenarios, realising the minimal extended type-I seesaw with thehelp of discrete groups is somewhat recent and limited [45–48]. It is in this context that wepropose a model to implement the MES and obtain oscillation observables consistent withthe latest experiments. Our model produces an extension of the TBM called the TM [49–57] in which the second column of the TBM is preserved. We use A × C as the flavourgroup for our model. We propose several scalar fields, often called the flavons, which couplewith the charged-lepton fields as well as the various neutrino fields. The inherent propertiesof A and C as well as the residual symmetries of the vacuum alignments of the flavons,determine the structure and the symmetries of the mass matrices.The content of this paper is organised as follows. The features of the MES schemeare outlined in Section 2. In Section 3, we briefly explain the representation theory of theflavour group and move on to construct the Yukawa Lagrangian based on the proposedflavon content of the model. We also assign Vacuum Expectation Values (VEVs) for theseflavons. We justify our assignments of VEVs with the help of symmetries in Appendix A. InSection 4, the mass matrices are constructed in terms of the VEVs. We provide the formulaefor various experimental observables as functions of the model parameters. In Section 5, wecompare these formulae with the experimental results and make predictions. We provide arepresentative set of model parameters in Appendix B and numerically extract the values ofthe observables so as to verify the validity of the various approximations used in the paper.Finally, we conclude in Section 6. In the Standard Model, the left-handed charged-lepton fields, l L = ( e L , µ L , τ L ) T , and theneutrino fields, ν L = ( ν e , ν µ , ν τ ) T , transform as the SU (2) doublet, L = ( ν L , l L ) T . Theycouple with the right-handed charged-lepton fields, l R = ( e R , µ R , τ R ) T , to form the charged-lepton mass term, ¯ Ly l l R H, (2.1) experiments like T2K, MINOS, NOvA and reactor experiments KamLAND and Daya-Bay, RENO, Double-Chooz etc. can be explained in terms of the three neutrino framework. Because dominant oscillations tosterile neutrinos is disfavored as a solution to solar and atmospheric neutrino anomalies, the 2+2 picture isdisfavored. In the 3+1 picture, the oscillations to sterile neutrinos is a sub-leading effect to the dominant 3flavour oscillations and the 3 generation global fit results are not altered. The short baseline eV oscillationscan be explained using the One-mass-scale-Dominance approximation governed by ∆ m . However, there isa tension between observance of non-oscillation in disappearance experiments and observation of oscillationin LSND and MiniBOONE which makes the goodness of fit in the 3+1 picture worse. – 3 –here y l are the Yukawa couplings. In general, y l is a × complex matrix. The electroweaksymmetry is spontaneously broken when the Higgs acquires the VEV, (cid:104) H (cid:105) = (0 , v ) T . (2.2)Subsequently, the mass term, Eq. (2.1), becomes ¯ l L M l l R , (2.3)where M l = vy l is the charged-lepton mass matrix.In the type-I seesaw framework, we add extra right-handed neutrino fields, ν R , to theSM. We may assume that three families of such fields exist, i.e. ν R = ( ν R , ν R , ν R ) T . Theycouple with the left-handed fields, L , forming the Dirac neutrino mass term, ¯ Ly ν ν R ˜ H, (2.4)where ˜ H = iσ H ∗ . As a result of the Spontaneous Symmetry Breaking (SSB), this termbecomes ¯ ν L M D ν R , (2.5)where M D = vy ν is the Dirac neutrino mass matrix. The right-handed neutrino fields cancouple with themselves resulting in the Majorana mass term,
12 ¯ ν cR M R ν R , (2.6)where M R is the × Majorana neutrino mass matrix which is assumed to be at a very highscale in order to cause the seesaw suppression of the light neutrino masses. The canonicaltype-I seesaw can be extended to accommodate an eV-scale sterile neutrino at the cost ofno fine-tuning of the Yukawa coupling. To implement this MES scheme we need to includean SM gauge singlet field, ν s , which couples with the heavy neutrino fields, ν R , leading to ¯ ν cs M s ν R , (2.7)where M s is a × mass matrix. We assume that the coupling of the sterile field ( ν s ) withitself as well as with the left-handed fields ( L ) is forbidden.Combining Eqs. (2.5, 2.6, 2.7), we obtain the Lagrangian containing the neutrino massmatrices relevant to the MES: L ν = ¯ ν L M D ν R + ¯ ν cs M s ν R + 12 ¯ ν cR M R ν R + h.c. (2.8)The Lagrangian, Eq. (2.8), leads to the following × neutrino mass matrix in the( ν L , ν cs , ν cR ) basis: M × ν = M D M s M TD M Ts M R . (2.9)Being analogous to the canonical type I seesaw, the MES scheme allows us to have thehierarchical mass spectrum assuming M R >> M s > M D . The right-handed neutrinos are– 4 –uch heavier compared to the electroweak scale enabling them to be decoupled at the lowscale. As a result, Eq.(2.9) can be block diagonalized to obtain the effective neutrino massmatrix in the ( ν L , ν cs ) basis, M × ν = − (cid:32) M D M − R M TD M D M − R M Ts M s ( M − R ) T M TD M s M − R M Ts (cid:33) . (2.10)This particular type of model is a minimal extension of the type I seesaw in the sense thatonly an extra sterile field is added whose mass is also suppressed along with that of thethree active neutrinos. Since M × ν has rank 6 and subsequently M × ν has rank three, thelightest neutrino state becomes massless .Assuming M s > M D , we may apply a further seesaw approximation on Eq.(2.10) toget the active neutrino mass matrix, M × ν (cid:39) M D M − R M Ts ( M s M − R M Ts ) − M s M − R M TD − M D M − R M TD . (2.11)It is worth mentioning that the RHS of Eq. (2.11) remains non-vanishing since M s is a rowvector × rather than a square matrix. Under the approximation M s > M D , we alsoobtain the mass of the th mass eigenstate , m (cid:39) M s M − R M Ts . (2.12)The charged-lepton mass matrix, M l , Eq. (2.3), is a × complex matrix in general.Its diagonalisation leads to the charged-lepton masses, U L M l U † R = diag ( m e , m µ , m τ ) , (2.13)where U L and U R are unitary matrices. The low energy effective × neutrino mass matrix, M × ν , Eq. (2.11), is complex symmetric. Its diagonalisation is given by U † ν M × ν U ∗ ν = diag ( m , m , m ) , (2.14)where U ν is a unitary matrix and m , m and m are the light neutrino masses . Using U L and U ν , we obtain the × light neutrino mixing matrix, U (cid:39) (cid:32) U L (1 − RR † ) U ν U L R − R † U ν − R † R (cid:33) , (2.15)where the three-component column vector R is given by R = M D M − R M Ts ( M s M − R M Ts ) − . (2.16) If we want to accommodate more than one sterile neutrino at the eV scale, we need to increase thenumber of heavy neutrinos as well. Otherwise, more than one active neutrino becomes massless which isruled out experimentally. Since the active-sterile mixing is small, the th mass eigenstate ( ν ) more or less corresponds to thesterile state ( ν s ). In the MES framework, we have m = 0 . – 5 – , Eq. (2.15), relates the neutrino mass eigenstates with the neutrino flavour eigenstates, U ( ν , ν , ν , ν ) T = ( ν e , ν µ , ν τ , ν s ) T , (2.17)in the basis where the charged-lepton mass matrix is diagonal. From Eq. (2.15), it is evidentthat the strength of the active-sterile mixing is governed by U L R = ( U e , U µ , U τ ) T . (2.18)Note that R is suppressed by the ratio O ( M D /M s ) . The × mixing matrix involving thethree active neutrinos, ( ν e , ν µ , ν τ ) , and the three lightest mass eigenstates, ( ν , ν , ν ) , isoften called the PMNS mixing matrix, U PMNS . In MES models with active-sterile mixing, U PMNS will not be unitary. It is given by the upper-left block of Eq. (2.15), U PMNS = U L U ν − U L RR † U ν . (2.19)The deviation of U PMNS from unitarity, i.e. − U L RR † U ν , is suppressed by O ( M D /M s ) . We construct the model in the framework of the discrete group A × C . A , which isthe smallest group with a triplet irreducible representation, has been studied extensivelyin the literature [25, 27, 31, 58–62]. Here we briefly mention the essential features of thisgroup in the context of model building. A is the rotational symmetry group of the regulartetrahedron. It has the group presentation, (cid:104) S, T | S = T = ( ST ) = I (cid:105) . (3.1) A has 12 elements which fall under four conjugacy classes. Its conjugacy classes andirreducible representations are listed in Table 1. (1) (12)(34) (123) (132) ω ω ¯ ω ¯ ω ω ω − Table 1 . The character table of the A group. A denotes the even permutations of four objects.The conjugacy class (12)(34) represents two inversions carried out in two separate pairs of objects.(123) and (132) represent two inversions carried out in a set of three objects in the forward sense andthe backward sense respectively. is the trivial representation. ω and ¯ ω are singlets transformingas ω = e i π and ¯ ω = e − i π under (123) and (132) . represents the three-dimensional rotationalsymmetries of a regular tetrahedron. For the triplet representation, , we choose the following basis, S = − − , T = . (3.2)– 6 – e R µ R τ R ν R ν s φ l η φ φ s η ν H H s A C − i i i i − C − ω − ω ¯ ω − ω − ω − ω ω ω ωC × U (1) s e iqθ − − × e − iqθ Table 2 . The particle content and their charges under the flavour group of the model
The representations ω and ¯ ω transform as ω and ¯ ω respectively under the generator T and trivially under the generator S . The tensor product of two triplets, ( x , x , x ) and ( y , y , y ) , leads to ≡ x y + x y + x y , (3.3) ω ≡ x y + ¯ ωx y + ωx y , (3.4) ¯ ω ≡ x y + ωx y + ¯ ωx y , (3.5) s ≡ ( x y + x y , x y + x y , x y + x y ) T , (3.6) a ≡ ( x y − x y , x y − x y , x y − x y ) T . (3.7)The triplets s and a , both of which transforming as under A , are constructed as thesymmetric and the antisymmetric products respectively of x and y .We extend the SM particle sector by the inclusion of three right-handed neutrinos, ν R = ( ν R , ν R , ν R ) T , a sterile neutrino ( ν s ), several flavon multiplets, φ l , η , φ , φ s , η ν anda sterile sector Higgs, H s . Along with the A group, the model includes several Abeliandiscrete groups ( C , C and C ) acting variously on these fields. We also introduce a gaugegroup, U (1) s , in the sterile sector. The field content of the model, along with the irreduciblerepresentations they belong to, are given in Table 2.From these field assignments, we obtain the following Yukawa Lagrangian: L Y = Y τ ¯ L φ l Λ τ R H + Y µ ¯ L φ ∗ l Λ µ R H + ¯ L Q Λ e R H + Y η ¯ Lν R η Λ ˜ H + Y φs (cid:0) ¯ Lν R (cid:1) T s φ Λ ˜ H + Y φa (cid:0) ¯ Lν R (cid:1) T a φ Λ ˜ H + Y s φ Ts Λ ¯ ν cs ν R H s + Y ν η ν ¯ ν cR ν R (3.8)where Y τ , Y µ , Y η , Y φs , Y φa , Y s , and Y ν are the Yukawa-like dimensionless coupling con-stants, Λ is the cut-off scale of the theory. () s and () a represent the symmetric and theantisymmetric tensor products given in Eq. (3.6) and Eq. (3.7) respectively. Q representsall the quadratic flavon terms forming a triplet under A and an invariant under the restof the flavour group. It consists of ( φ ∗ l φ l ) s , ( φ ∗ l φ l ) a , ( φ ∗ s φ s ) s , ( φ ∗ s φ s ) a , ( φ ∗ φ ) s , ( φ ∗ φ ) a , η ∗ φ , ηφ ∗ along with the corresponding Yukawa-like coupling constants. Besides the flavoursymmetries, we also impose the CP symmetry at high energy scales where the Higgsesand the flavons have not acquired their VEVs, i.e. all the Yukawa-like couplings in theLagrangian are real. We also assume that these couplings are of the order of one.– 7 –n the above Lagrangian, the first three terms are responsible for the charged-leptonmass generation. Since φ l , φ ∗ l and Q transform as − ω , − ¯ ω and under C , they couplewith ¯ Lτ R , ¯ Lµ R and ¯ Le R (which transform as − ¯ ω , − ω and ) respectively. The rest ofthe terms involve the right-handed neutrino triplet ν R and they contribute to the neutrinomass generation. The terms in the second line of Eq. (3.8) contain ¯ Lν R which transformsas − i under C and remains invariant C . The flavons φ and η , which transform as i under C and remain invariant C , couple with ¯ Lν R . These terms result in the Dirac massmatrix for the neutrinos. Under A × C × C × C × U (1) s , the term ¯ ν cs ν R transforms as × × ¯ ω × × e iqθ . This term couples with φ s and H s which transform as × × ¯ ω × − × and × × ¯ ω × − × e − iqθ respectively and forms the sterile mass term. Finally, we have theMajorana mass term consisting of ¯ ν cR ν R coupled with the flavon η ν . We may also constructthe neutrino mass terms ( ¯ L ˜ H )( ˜ H T L c ) Q Λ , ¯ Lν s Q Λ ˜ HH s and ¯ ν cs ν s Q Λ H s where Q , Q and Q represent the quadratic flavon terms transforming as × ¯ ω × , − i × ω × − and − × × respectively under C × C × C . Since these mass terms are heavily suppressed, we havenot included them in the Lagrangian. In fact, a part of the reason for assigning the variousAbelian charges to the flavons, Table 2, is to ensure that this suppression occurs so thatour model leads to the standard MES framework where these terms are assumed to vanish(the block of zeros in Eq. (2.9)).Like the SM Higgs, the other scalar fields in the model also acquire VEVs through SSB.We assign them the following values: (cid:104) φ l (cid:105) = v l (1 , ¯ ω, ω ) T , (3.9) (cid:104) η (cid:105) = v η , (3.10) (cid:104) φ (cid:105) = v φ (0 , − i, T , (3.11) (cid:104) φ s (cid:105) = v s (1 , , T , (3.12) (cid:104) η ν (cid:105) = v ν , (3.13) (cid:104) H s (cid:105) = v (cid:48) . (3.14)The VEVs of the various flavons in the model break the discrete flavour group, A × C × C × C in specific ways. In Appendix A, we study the residual symmetries of these VEVsand describe how their alignments can be uniquely defined.The VEV of the sterile Higgs ( H s ) breaks the U (1) s gauge group and leads to amassive gauge boson. This particle can mediate the so-called secret interactions of thesterile neutrinos proposed in the cosmological context [63–65]. The VEV of the sterileHiggs is assumed to be an order of magnitude higher than the VEV of the SM Higgs,i.e. v = 176 GeV, v (cid:48) ≈ GeV. We also assume that the flavon VEVs are at a very highenergy scale v x ≈ GeV where v x denotes v l , v η , v φ , v s and v ν and that the cut-offscale Λ ≈ GeV. Under these assumptions, we may calculate the scales of our massterms (after SSB); ¯ l L τ R , ¯ l L µ R : v v l Λ ≈ − , ¯ l L e : v O ( v x )Λ ≈ − , ¯ ν L ν R : v O ( v x )Λ ≈ − , ¯ ν cs ν R : v (cid:48) v s Λ ≈ , ¯ ν cR ν R : v ν ≈ , ¯ ν L ν cL : v O ( v x )Λ ≈ − , ¯ ν L ν s : vv (cid:48) O ( v x )Λ ≈ − and ¯ ν cs ν s : v (cid:48) O ( v x )Λ ≈ − where all the units are in GeV.– 8 – Mass Matrices and Observables
Substituting the Higgs VEV and the flavon VEVs in the Lagrangian for the charged-leptonsector (first line of Eq. (3.8)), we obtain the charged-lepton mass matrix, Eq. (2.3), M l = v O ( v x )Λ O (1) 0 0 O (1) 0 0 O (1) 0 0 + v v l Λ Y µ Y τ ωY µ ¯ ωY τ ωY µ ωY τ . (4.1)This mass matrix is diagonalised using the transformation, U L M l U † R = diag ( m e , m µ , m τ ) , (4.2)where U L (cid:39) √ ω ω ω ¯ ω , (4.3) U R is an unobservable unitary matrix and m e = v O ( v x )Λ , m µ (cid:39) √ Y µ v v l Λ , m τ (cid:39) √ Y τ v v l Λ (4.4)are the masses of the charged leptons. Here, the electron mass is suppressed by a factorof v x Λ ≈ − compared to the muon and the tau masses. This is similar to the Froggatt-Nielsen mechanism which was proposed to explain the hierarchy of the fermion masses. Itcan be shown that the error in the expression of U L given in Eq. (4.3) is of the order of v x Λ . The relative errors in the expressions of m µ and m τ given in Eqs. (4.4) are also of thesame order. These errors are very small and hence can safely be ignored. For a numericalverification, please refer to Appendix B.The terms in the second line of the Lagrangian, Eq. (3.8), generate the Dirac massmatrix for the neutrinos. Substituting Higgs VEV and the VEVs of the flavons η and φ ,Eqs. (3.10, 3.11), in these terms, we obtain the Dirac neutrino mass matrix, Eq. (2.5), M D = v v η Y η − iv φ ( Y φs − Y φa )0 v η Y η − iv φ ( Y φs + Y φa ) 0 v η Y η . (4.5)Substituting the VEV of φ s , Eq. (3.12), in the mass term for the sterile neutrino, Y s ¯ ν cR ν s φ s Λ H s , we obtain the mass matrix representing the couplings between ν s and ν R ,Eq. (2.7), M s = v (cid:48) v s Λ Y s (1 , , . (4.6)Finally, from the term Y ν η ν ¯ ν cR ν R , we obtain the mass matrix for the heavy right-handedneutrinos, Eq. (2.6), M R = Y ν v ν I, (4.7)where I is the × identity matrix. – 9 –e implement the MES scheme, Eq. (2.10), using the neutrino mass matrices, M D , M s , M R , Eqs. (4.5, 4.6, 4.7), and obtain the following effective neutrino mass matrix: M × ν = m − ( κ s − κ a ) − iκ s − iκ s − ( κ s + κ a ) √ mm s √ − i ( κ s − κ a )01 − i ( κ s + κ a ) √ mm s √ (cid:16) − i ( κ s − κ a ) 0 1 − i ( κ s + κ a ) (cid:17) m s , (4.8)where m = v v η Y η Y ν v ν Λ , m s = 2 v (cid:48) v s Y s Y ν v ν Λ , κ s = v φ Y φs v η Y η , κ a = v φ Y φa v η Y η . (4.9)Here, the mass m is suppressed by the very high value of v ν ( ≈ GeV) and also bythe ratio Λ v η ( ≈ ). Hence, we obtain m at around . eV. The ratio mm s relates thescale of the active sector of the neutrinos to that of the sterile sector and it is given by mm s = v v η Y η v (cid:48) v s Y s . Since v (cid:48) is assumed to be an order of magnitude higher than v , the ratio mm s becomes small ( ≈ . ). As a result, we can use Eq. 2.11 to obtain the effective × neutrino mass matrix, M × ν = m ( κ s − κ a − i ) κ a − ( κ s − i ) − κ a − ( κ s − i ) κ s + κ a − i ) . (4.10)Using the unitary matrix, U ν = 1 √ κ i + κ s + κ a − i + κ s − κ a i √ κ i + κ s − κ a i − κ s − κ a with κ = (cid:112) (1 + κ s + κ a ) , (4.11)we diagonalise M × ν , Eq. (4.10), U † ν M × ν U ∗ ν = m diag (cid:0) , , κ s + κ a (cid:1) , (4.12)to obtain the light neutrino masses, m = 0 , m = m, m = m (1 + κ s + κ a ) . (4.13)Using the expressions of U L and U ν , Eqs. (4.3, 4.11), we obtain the PMNS mixingmatrix ( U PMNS (cid:39) U L U ν , Eq. (2.19)) in terms of the parameters κ s and κ a , U PMNS (cid:39) √ κ i + κ s ) i √ κ − κ a ( i + κ s )(1 + ω ) + κ a (1 − ω ) i √ κ ¯ ω ( − i + κ s )(1 − ω ) − κ a (1 + ω )( i + κ s )(1 + ¯ ω ) + κ a (1 − ¯ ω ) i √ κω ( − i + κ s )(1 − ¯ ω ) − κ a (1 + ¯ ω ) . (4.14) U PMNS obtained in this approximation is unitary. For a numerical analysis without this approximationwhich leads to non-unitarity, please refer to Appendix B. – 10 –he absolute values of the elements of the middle column of this mixing matrix are equalto √ , i.e. the mixing has the TM form. Note that the eigenvalue m should correspondto the second neutrino eigenstate because this eigenstate should remain unmixed with theothers in order to obtain the TM mixing, Eq. (4.14). Therefore, the mass ordering shouldbe either (0 , m, m (1 + κ s + κ a )) or ( m (1 + κ s + κ a ) , m, . The second case is inconsistentwith the experimental observation of m < m given that κ s and κ a are real parameters.Hence, the model predicts the normal ordering (the first case) of the light neutrino masses.From Eq. (4.14), we extract the three mixing angles in the active sector, sin θ = 2 κ a κ s + κ a ) , (4.15) sin θ = 1 + κ s + κ a κ s + κ a , (4.16) sin θ = 3 + 3 κ s + 2 √ κ a + κ a κ s + κ a ) . (4.17)Eq. (4.15) and Eq. (4.16) are consistent with the TM constraint, sin θ cos θ = .Using Eqs. (4.13, 4.15-4.17), we obtain another constraint among the observables, sin θ = 12 + 3 √ θ sin θ (cid:18) ∆ m ∆ m (cid:19) , (4.18)which shows the deviation from the maximal atmospheric mixing, i.e. sin θ = . We alsocalculate the Jalskog’s CP-violation parameter [66] in the active sector, J = Im ( U e U µ U ∗ e U ∗ µ ) = − κ s κ a √ κ s + κ a ) . (4.19)Eliminating κ s and κ a from Eq. (4.19) using Eqs. (4.13, 4.15), we can express J in termsof the reactor angle and the light neutrino masses, J = − √ θ (cid:115) −
32 sin θ − (cid:112) ∆ m (cid:112) ∆ m . (4.20)Given the three mixing angles and J in terms of the model parameters, we can obtain sin δ using the following expression: sin δ = J/ (sin θ sin θ sin θ cos θ cos θ cos θ ) . (4.21)Note that the × mixing matrix is parametrised using six mixing angles ( θ , θ , θ , θ , θ , θ ) and three Dirac CP phases ( δ , δ , δ ) with the help of the parametrizationmentioned in Ref. [22]. However, we used the parametrisation for the × mixing matrix,Eq. (1.1), to extract the mixing angles ( θ , θ , θ ) and the CP phase ( δ = δ ) given inEqs. (4.15, 4.16, 4.17, 4.21). This approximation is valid since the active-sterile mixing isquite small.Comparing Eqs. (2.10, 2.12) with Eq. (4.8), it is clear that the model parameter m s corresponds to the mass of the th mass eigenstate, m = m s . (4.22)– 11 –sing Eq. (2.16), we obtain the three-component column vector R , R = (cid:114) m m s − i ( κ s − κ a )01 − i ( κ s + κ a ) . (4.23)Substituting Eqs. (4.3, 4.23) in Eq. (2.18) we obtain U e = √ m √ m s (1 − iκ s ) , (4.24) U µ = − ¯ ω √ m √ m s (1 − iκ s + √ κ a ) , (4.25) U τ = − ω √ m √ m s (1 − iκ s − √ κ a ) . (4.26)Using Eqs. (4.24, 4.25, 4.26), we can write the three active-sterile mixing angles in termsof the model parameters, sin θ = 23 mm s (1 + κ s ) , (4.27)sin θ = 16 mm s (1 + κ s + 2 √ κ a + 3 κ a )1 − mm s (1 + κ s ) , (4.28)sin θ = 16 mm s (1 + κ s − √ κ a + 3 κ a )1 − mm s (1 + κ s ) − mm s (1 + κ s + 2 √ κ a + 3 κ a ) . (4.29)For the extraction of δ and δ , we need to calculate the Jarlskog-like rephasinginvariants from the active-sterile sector. In this context, we refer the readers to a recentwork [67], in which nine independent rephasing invariants in terms of the six mixing anglesand the three Dirac phases have been evaluated in the context of the × mixing matrix.With the help of these invariants, we may extract δ and δ .The effective neutrino mass applicable to the neutrinoless double-beta decay [68, 69] isgiven by m ββ = (cid:12)(cid:12) m U e + m U e + m U e + m s U e (cid:12)(cid:12) . (4.30)Substituting the values of the neutrinos masses, Eq. (4.13), the elements of the first row ofthe mixing matrix, Eq. (4.14), and the expression for U e , Eq. (4.24), in the above equation,we obtain m ββ = (cid:12)(cid:12)(cid:12) m − κ s + 2 κ a − iκ s ) (cid:12)(cid:12)(cid:12) = m (cid:112) (1 − κ s + 2 κ a ) + 16 κ s . (4.31) Our model allows only four degrees of freedom in the neutrino Yukawa sector, denoted bythe free parameters κ s , κ a , m , m s . There are eleven independent experimentally measuredquantities, sin θ , sin θ , sin θ , sin δ , ∆ m , ∆ m , m ββ , ∆ m , | U e | , | U µ | , | U τ | all of which can be expressed in terms of the above mentioned four model parameters. So,– 12 –t is clear that the model is extremely constrained. In this section, we calculate the modelparameters using the experimental data and also make predictions. To calculate the allowedranges of the model parameters, κ s , κ a , m and m s , we utilize the observables sin θ , ∆ m , ∆ m and ∆ m whose experimental values are given in Table 3. These values are obtainedfrom the global fit data published by the nufit group [70] and the active-sterile mixing datafrom Ref. [71].Using the expression of the reactor mixing angle given in Eq. (4.15) and its range givenin Table 3, we obtain . ≤ κ a κ s + κ a ) ≤ . . (5.1)Using Eq. (4.13), we calculate the ratio of the mass-squared differences of the activeneutrinos, ∆ m ∆ m = (1 + κ s + κ a ) . (5.2)Given the ranges of ∆ m and ∆ m (Table 3), Eq. (5.2) leads to . ≤ (1 + κ s + κ a ) ≤ . . (5.3)We use Eqs. (5.1, 5.3) to constrain the parameters κ s and κ a . In this analysis, we chose sin θ and ∆ m ∆ m because these are the most precisely measured observables that can beused for constraining κ s and κ a . The results are shown in Figure 1 where the blue andthe red regions represent the constraints Eq. (5.1) and Eq. (5.3) respectively. The allowedrange of κ s and κ a is given by the intersection of the red and the blue regions. Note thatthese parameters can be directly expressed in terms of the observables: κ s = (cid:18) −
32 sin θ (cid:19) (cid:112) ∆ m (cid:112) ∆ m − , κ a = 32 sin θ (cid:112) ∆ m (cid:112) ∆ m . (5.4)The best fit values, i.e. sin θ = 0 . , ∆ m = 7 . × − eV and ∆ m = 2 . × − eV , leads to κ s = 2 . and κ a = 0 . consistent with Figure 1.Substituting this range of values in the expression of the solar mixing angle, Eq. (4.16),we predict . ≤ sin θ ≤ . . (5.5) σ range sin θ . → . m . × − eV → . × − eV ∆ m . × − eV → . × − eV ∆ m . eV → . eV Table 3 . The mixing observables which are used to evaluate the model parameters κ s , κ a , m and m s . The σ ranges of sin θ , ∆ m and ∆ m are taken from Ref. [70] and that of ∆ m is takenfrom Ref. []. – 13 – .0 0.5 1.0 1.5 2.0 2.50.00.51.01.52.02.5 κ s κ a Figure 1 . The parameters κ s and κ a constrained using the reactor mixing angle and the ratio ofthe mass-squared differences of the active neutrinos. Prediction Experimental range sin θ . → .
342 0 . → . θ . → .
548 0 . → . δ − . → − . − → . | U e | . → .
038 0 . → . | U µ | . → .
013 0 . → . | U τ | . → . < . m ββ . eV → . eV < . eV Table 4 . The values of the observables predicted by the model in comparison to their exper-imental ranges [70–72]. m ββ < . eV is the most stringent bound from the KamLAND-Zenexperiment [72]. TM mixing fixes | U e | to be . We also have | U e | = sin θ cos θ . Therefore, TM scheme strongly constrains θ given the precise experimental determination of θ . The re-sulting prediction, Eq. (5.5), is consistent with the σ experimental range . ≤ sin θ ≤ . , Table 4. However, a more precise determination of the solar mixing angle, for instancefrom reactor experiments [73, 74] can test this prediction.Substituting the allowed range of κ s and κ a in the expressions of the atmospheric– 14 – ����� ������ ������ ������ ������ ������ �������������������������������������� ��� � θ �� � � � � θ �� ����� ����� ����� ����������������������������������������������� ��� � θ �� � � � � θ �� ���� ���� ���� ������������������������������������ ��� � θ �� � � � � θ �� ����� ����� ����� ����������������������������������������������� ��� � θ �� � � � � θ �� ������ ������ ������ ������ ������ ������ ���������������������� ��� � θ �� � � � � θ �� ����� ����� ����� ������������������������� ��� � θ �� � � � � θ �� Figure 2 . Correlations among the mixing angles found using equations 4.15, 4.16 and 4.17. Foran explanation please see the text. In the left panel, the predicted ranges of the observables areplotted against their entire σ ranges. The right panel shows the correlations among the angleswhich is not clearly visible in the left panel. mixing angle, the Jarlskog invariant and the Dirac CP phase, Eqs. (4.17-4.21), we predict . ≤ sin θ ≤ . , (5.6) − . ≤ sin δ ≤ − . with − . ≤ J ≤ − . . (5.7)These predictions are also consistent with the experimental ranges, Table 4. Note thatthe determination of the octant of θ is still an open problem experimentally. If the µ - τ reflection symmetry [75–80] is broken, we have θ either in the first or the second octant.The model predicts it to be in the second octant.In Figure 2, we show the correlations among the mixing angles resulting from themodel. The top panel of the figure shows the TM constraint between the solar and reactormixing angles, sin θ cos θ = . For small θ , we obtain a linear relationship, sin θ (cid:39) (1 + sin θ ) . The mild positive correlations of the atmospheric angle with the solar andthe reactor angles shown in the middle and the bottom panels respectively are nothingbut the result of the constraint among these quantities given in Eq. (4.18). We also note– 15 – .535 0.540 0.545 0.550 0.555 - - - - - - θ s i n δ Figure 3 . The predicted ranges of sin θ and sin δ as constrained by the parameters κ s and κ a . that the deviation from maximal atmospheric mixing shown in these plots ( . - . ) isconsistent with the deviation obtained in Eq. (4.18).The global fit [70] of oscillation data gives hints for CP violation. Even though themeasurement is not precise, ◦ ≤ δ ≤ ◦ , it favours a relatively large negative valuefor sin δ . Our prediction, Eq. (5.7), supports this scenario. In Figure 3, we have shown thepredictions for sin θ and sin δ .Under the MES scheme, the mass of the lightest neutrino, m , vanishes . Therefore,the experimental ranges of ∆ m and ∆ m , Table 3, leads to the prediction, . × − eV ≤ m ≤ . × − eV , . × − eV ≤ m ≤ . × − eV . (5.8)The model parameter m corresponds to the neutrino mass m , so its allowed range is thesame as that of m given above. Using the range of ∆ m from Table 3, we obtain, . eV ≤ m ≤ . eV , (5.9)which also corresponds to the range of the model parameter, m s , Eq. (4.22).The active-sterile mixing observables, Eqs. 4.24-4.26, depend on all the four modelparameters, κ s , κ a , m , and m s . By varying these parameters within their respective rangeswe predict the values of these observables, i.e. | U e | , | U µ | and | U τ | , Table 4. Thesepredictions are well within their corresponding experimental ranges. In Figure 4, we haveplotted them against the parameter, m s .Substituting the allowed ranges of κ s , κ a , m , and m s in Eq. (4.31), we predict thevalue of the neutrinoless double-beta decay mass, Table 4, which is also shown in Fig. 5.This range is quite narrow because the model strongly constrains the first row of the mixing The higher-order corrections in the MES framework will generate non-zero mass for m , albeit tiny.Here we ignore this mass. In Appendix B, we estimate it to be of the order of − eV. – 16 – .6 0.8 1.0 1.2 1.4 1.6 1.80.010.020.030.040.05 m s | U e m s | U μ m s | U τ (a) | U e | vs m s (b) | U µ | vs m s (c) | U τ | vs m s Figure 4 . The active-sterile mixing observables predicted by the model plotted against m s . Thesecurves show the inverse relationship between the moduli-squared values of the active-sterile mixingelements and the sterile neutrino mass as can be inferred from Eqs. (4.24-4.26). Figure 5 . The prediction of the effective neutrino mass, m ββ , in relation to the active-sterilemixing strength. matrix through the parameters κ s and κ a , which effectively constrains the Majorana phasesas well.Cosmological observations set upper bounds to the sum of the neutrino masses forthree generations of neutrinos, Σ m i = m + m + m . However, in the presence of thesterile neutrino, the bound gets affected. At the same time, some recent cosmologicalmodels offer an explanation in favour of the existence of the sterile neutrino via the so-called secret interactions. The broken U (1) s gauge symmetry in our model will lead toa massive gauge boson which can mediate such an interaction. For a detailed discussionabout the cosmological implications of sterile neutrinos having secret interactions, pleasesee the references [63–65]. – 17 – Discussion and Conclusion
In this paper, we construct the leptonic mass matrices in terms of the VEVs of a set offlavon fields transforming under the discrete symmetry group A × C × C × C and theVEVs of the SM Higgs and a sterile sector Higgs. In the charged-lepton sector, we obtaina non-diagonal mass matrix. In the neutrino sector, we use the MES formula, Eq. (2.11),to construct the effective × seesaw mass matrix. The unitary matrices U L and U ν diagonalise the charged-lepton and the neutrino mass matrices respectively. Their productdetermines the mixing in the active sector, i.e. U PMNS (cid:39) U L U ν , Eq. (2.19). In our model,the unitary contribution from the charged-lepton sector ( U L ) has a × trimaximal form,Eq. (4.3). On the other hand, the contribution from the active neutrino sector ( U ν ) hasthe form which corresponds to the second flavour eigenstate being equal to the second masseigenstate as evident from the off-diagonal zeros in U ν , Eq. (4.11). Consequently, the secondcolumn of U L is preserved in the product U L U ν and as a result, we obtain the TM mixing. U ν obtained in the model contains two parameters κ s and κ a . These parameters cor-respond to the symmetric and the antisymmetric parts of the neutrino mass matrix, M D ,which in turn originate from the symmetric and the antisymmetric parts of the tensorproduct of triplets of A . If κ a vanishes, U ν becomes bimaximal, i.e. κ a → ⇒ U ν → √ − √ √ √ , (6.1)which will lead to tribimaximal (TBM) mixing. The observation of non-zero reactor anglehas ruled out TBM. Hence, the parameter κ a plays the vital role of generating the non-zeroreactor angle in the model. This role has been emphasized in Ref. [81 ? ].We obtain CP violation even though all the free parameters in the model are real. Thecharged-lepton mass matrix, M l , Eq. (4.1), and the neutrino mass matrix, M D , Eq. (4.5),turn out to be complex on account of the complex VEVs (cid:104) φ l (cid:105) and (cid:104) φ (cid:105) respectively. Hence CPis broken spontaneously in the model. Since M l and M D are complex, the correspondingdiagonalising matrices U L and U ν also become complex and they generate the complexmixing matrix, U PMNS (cid:39) U L U ν . It can be shown that if U ν were real, the resulting mixingmatrix U L U ν would be symmetric under µ - τ reflection implying θ = π . In such a scenario,despite U ν being real, CP would be maximally broken ( δ = ± π ) because of the complexcontribution from the charged-lepton sector ( U L ) alone. Our model, with U ν also beingcomplex, breaks µ - τ reflection symmetry and we obtain θ (cid:54) = π . The complex U ν alsoshifts δ away from its maximal value, i.e. δ (cid:54) = ± π . Therefore, the origin of the non-maximalvalues of the atmospheric mixing as well as the CP phase is the complex VEV, (cid:104) φ (cid:105) .LSND and MiniBooNE observations suggest the existence of sterile neutrinos. Theobserved active-sterile mixing ( | U e | , | U µ | ) is found to be of the order of √ ∆ m √ ∆ m . TheMinimal Extended Seesaw provides a natural framework to achieve this relationship. It isin this context that we built the model to explain both the active and the sterile mixingobservables. In the model, these observables are given in terms of four parameters, κ s , κ a ,– 18 – and m s . We use the experimental ranges of the reactor mixing angle, sin θ , and themass-squared differences, ∆ m and ∆ m , to extract the allowed values of κ s , κ a and m ,as we obtain m < m corresponding to normal hierarchy. The extracted values of κ s and κ a are used to predict θ and δ . These predictions can be tested when these observablesare measured more precisely in future oscillation experiments. The model parameter, m s ,corresponds to the sterile neutrino mass and is determined by the active-sterile mass-squareddifference, ∆ m . The three model parameters, κ s , κ a and m (constrained using sin θ , ∆ m and ∆ m ), as well as the fourth parameter, m s (constrained using ∆ m ), are usedto evaluate the active-sterile mixing. We find that these values are consistent with theexperimental results. We also obtain strong constraints on the range of the effective massgoverning the neutrinoless double-beta decay. A Uniquely defining the flavon VEVs
The model contains the flavon multiplets, φ l , η , φ , φ s and η ν . The alignments of theirVEVs in the flavour space play a crucial role in determining the model’s phenomenology.In this Appendix, we provide justifications for these alignments by uniquely defining themusing symmetries.The flavon φ l couples in the charged-lepton sector. Consider the alignment (cid:104) φ l (cid:105) ∝ (1 , ¯ ω, ω ) T , Eq. (3.9). This VEV is invariant under the following group action: ωT (cid:104) φ l (cid:105) = (cid:104) φ l (cid:105) , (A.1)where ω and T are elements of C and A respectively under which φ l transforms. In otherwords, ωT generates the residual symmetry of (cid:104) φ l (cid:105) and this symmetry uniquely defines (cid:104) φ l (cid:105) (up to multiplication with a constant ). A triplet flavon whose VEV is defined by theresidual symmetry, Eq. (A.1), was recently utilised in the construction of the charged-leptonmass matrix in Ref. [82].The singlet η and the triplet φ couple in the neutrino Dirac sector. The VEV of thesinglet, Eq. (3.10), is assumed to be real. This can be ensured by assuming the residualsymmetry under conjugation, (cid:104) η (cid:105) ∗ = (cid:104) η (cid:105) . (A.2)The VEV (cid:104) φ (cid:105) ∝ (0 , − i, T , Eq. (3.11), is uniquely defined using the residual symmetries, T ST (cid:104) φ (cid:105) = (cid:104) φ (cid:105) , − (cid:104) φ (cid:105) ∗ = (cid:104) φ (cid:105) . (A.3)These symmetries ensure that the first and the third components of (cid:104) φ (cid:105) vanish and thephase of the second component is − i . Note that the constant of proportionality in theVEV, i.e. v φ in Eq. (3.11), is real.The triplet φ s couples in the sterile sector. Its VEV, (cid:104) φ s (cid:105) ∝ (1 , , T , Eq. (3.12), hasthe following residual symmetries: − T ST (cid:104) φ s (cid:105) = (cid:104) φ s (cid:105) , (cid:104) φ s (cid:105) ∗ = (cid:104) φ s (cid:105) . (A.4) This constant, i.e. v l in Eq. (3.9), can be complex in general. Note that v l being complex does not alterthe phenomenology of the charged-lepton sector. – 19 –he VEV of the A triplet in the form ∝ (1 , , T has been widely used in the literature [83–85]. This alignment can be uniquely obtained from the symmetric tensor product involvingthe alignments (0 , , T and (1 , , T , (cid:0) (0 , , T , (1 , , T (cid:1) s ∝ (1 , , T . (A.5)The VEV of the singlet η ν can be trivially assigned any complex constant since itsphase has no effect on the model’s phenomenology.Here we have assumed that various flavon VEVs have distinct residual symmetries. It isinteresting to note that if all the VEVs are taken together, none of these symmetries survive.A comprehensive study of the origin of the VEVs requires the construction of the flavonpotential. If different irreducible multiplets are decoupled in the potential, then it can leadto the corresponding VEVs having separate residual symmetries. The construction of apotential with an inbuilt mechanism to ensure the decoupling of the irreducible multipletsis beyond the scope of this work. However, we assume that our VEVs are generated fromsuch a potential. B Numerical verification of approximations
In this Appendix, we construct the charged-lepton and the neutrino mass matrices usinga representative set of model parameters and numerically extract the masses and the mix-ing observables without employing approximations. Thus we verify the correctness of thevarious results obtained in the main body of the paper. In our calculations, we use
Λ = 10 , v x = 10 , (B.1)where v x represents v l , v η , v φ , v s and v ν . All mass units are given in GeV, unless otherwisespecified. The SM Higgs and the sterile Higgs are given the following VEVs, v = 176 , (B.2) v (cid:48) = 2000 . (B.3)Substituting Eqs. (B.1, B.2) in Eq. (4.1), we obtain the charged-lepton mass matrix in theform, M l = 176 × − (1 . − i . Y e . − i . Y e . − i . Y e + 176 × − Y µ Y τ ωY µ ¯ ωY τ ωY µ ωY τ . (B.4)Since several higher-order flavon triplets contribute in the construction of the electron massterm, there will be a corresponding set of Yukawa-like free parameters. We assign randomreal numbers to these parameters resulting in the first column of M l , Eq. (B.4) . Forconvenience, we have introduced a parameter Y e in this column. Diagonalising M l using The alignment (1 , , T can be uniquely defined by the residual symmetry, T (1 , , T = (1 , , T . The exact values of these numbers are irrelevant in an order-of-magnitude calculation. – 20 –he unitary matrices U L and U R , Eq. (4.2), produces the charged-lepton masses. Theirexperimental values ( m τ = 1777 MeV, m µ = 106 MeV and m e = 0 . MeV) are obtainedwith the substitution, Y τ = 1 √ . , Y µ = 1 √ . , Y e = 0 . . (B.5)These coupling constants are of the order of one as we expect. The corresponding diago-nalising matrices are U L = .
577 0 .
577 0 . . − . − i . − .
289 + i . . − .
289 + i . − . − i . , (B.6) U R = . − i .
618 0 .
001 + i .
001 0 . i .
001 1 .
000 0 . .
000 0 .
000 1 . . (B.7)We find that the above calculation of U L is consistent with the expression given in Eq. (4.3)within a deviation of the order of v x Λ .To construct the neutrino mass matrices, we make the following assignments: Y η = 1 . , Y φs = 3 . , Y φa = 0 . , Y s = 1 . , Y ν = 1 . . (B.8)Substituting Eqs. (B.1-B.3, B.8) in Eqs. (4.5-4.7), we obtain M D = .
294 0 − i . .
294 0 − i .
773 0 0 . , M s = (2 . , , . , M R = diag (1 , , . (B.9)We also introduce mass matrices involving ¯ ν L ν cL , ¯ ν L ν s and ¯ ν cs ν s , which are highly suppressedas described at the end of Section 3, ¯ ν L ν cL : M = . − . − i . − .
39 + i . − . − i .
02 1 .
24 4 . − .
39 + i .
02 4 .
65 1 . − , (B.10) ¯ ν L ν s : M = . − i . .
11 + i . − , ¯ ν cs ν s : M = (cid:16) . (cid:17) − . (B.11)Here also, the matrices have been populated with a random choice of Yukawa-like couplingscorresponding to the higher-order terms. In terms of Eqs. (B.9-B.11), we construct theneutrino mass matrix in the ‘ ν = ( ν L , ν cs , ν cR ) ’-basis: ¯ νν c : M × ν = M M M D M T M M s M TD M Ts M R . (B.12)– 21 –his matrix is diagonalised to obtain the neutrino masses, U × † ν M × ν U × ∗ ν = diag (cid:0) . × -14 , . × -12 , . × -11 , . × -9 , . × , . × , . × (cid:1) , (B.13)where U × ν = (cid:32) U × ν O (10 − ) O (10 − ) U heavy ν (cid:33) . (B.14)The structure of U heavy ν depends on the higher-order corrections to M R which break thedegeneracy of the heavy neutrino masses. These corrections are of no significance to thepresent work. We note that the mass acquired by the lightest neutrino is of the order of M = O (10 − ) .The × light-neutrino mixing matrix is obtained as U = (cid:32) U L
00 1 (cid:33) U × ν = − .
332 + i .
713 0 .
571 + i . − .
059 + i .
137 0 . − i . − .
323 + i . − . − i . − . − i .
581 0 .
096 + i . .
157 + i . − .
359 + i .
452 0 . − i . − . − i . .
185 0 .
000 0 .
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