Neutrino Mass Textures and Partial μ - τ Symmetry
aa r X i v : . [ h e p - ph ] A p r Neutrino Mass Textures and Partial µ – τ Symmetry
E. I. Lashin , , ∗ , N. Chamoun , † , C. Hamzaoui ‡ , and S. Nasri , § Ain Shams University, Faculty of Science, Cairo 11566, Egypt. Centre for Theoretical Physics, Zewail City of Science and Technology,Sheikh Zayed, 6 October City, 12588, Giza, Egypt. The Abdus Salam ICTP, P.O. Box 586, 34100 Trieste, Italy. Physics Department, HIAST, P.O.Box 31983, Damascus, Syria. Physikalisches Institut der Universit¨ a t Bonn, Nußalle 12, D-53115 Bonn, Germany. Groupe de Physique Th´eorique des Particules,D´epartement des Sciences de la Terre et de L’Atmosph`ere,Universit´e du Qu´ebec `a Montr´eal, Case Postale 8888, Succ. Centre-Ville,Montr´eal, Qu´ebec, Canada, H3C 3P8. Department of Physics, UAE University, P.O.Box 17551, Al-Ain, United Arab Emirates. Laboratoire de Physique Th´eorique, ES-SENIA University, DZ-31000 Oran, Algeria.
October 8, 2018
Abstract
We discuss the viability of the µ – τ interchange symmetry imposed on the neutrino mass matrix inthe flavor space. Whereas the exact symmetry is shown to lead to textures of completely degeneratespectrum which is incompatible with the neutrino oscillation data, introducing small perturbations intothe preceding textures, inserted in a minimal way, lead however to four deformed textures representingan approximate µ – τ symmetry. We motivate the form of these ‘minimal’ textures, which disentangle theeffects of the perturbations, and present some concrete realizations assuming exact µ – τ at the Lagrangianlevel but at the expense of adding new symmetries and matter fields. We find that all these deformedtextures are capable to accommodate the experimental data, and in all types of neutrino mass hierarchies,in particular the non-vanishing value for the smallest mixing angle. Keywords : Neutrino masses,
PACS numbers : 14.60.Pq; 11.30.Hv; 14.60.St
The elusive neutrino particles proved, so far, to be the only feasible window for the physics beyond theStandard Model (SM) of particle physics. The observed solar and atmospheric neutrino oscillations in theSuper-Kamiokande [1] experiment constitute a compelling evidence for the massive nature of neutrinoswhich is a clear departure from the SM particle physics. In the flavor basis where the charged lepton massmatrix is diagonal, the mixing can be solely attributed to the effective neutrino mass matrix M ν . In such ∗ [email protected], [email protected] † [email protected] ‡ [email protected] § [email protected] M ν can be parameterized by nine free parameters: three masses ( m , m and m ), three mixing angles ( θ x , θ y and θ z ) and three phases (two Majorana-type ρ , σ and one Dirac-type δ ). The culmination of experimental data [2, 3, 4, 5] amounts to constraining the masses and themixing angles, while for the phases there is no, so far, a feasible experimental set for their determination.The recent results from the T2K[6], MINOS[7], and Double Chooz[8] experiments reveal a nonzero valueof θ z . The more recent Daya Bay [9] and RENO[10] experiments confirm a sizable value with relativelyhigh precision. The discovery of relatively large mixing angle θ z has a tremondous impact on searchingfor a sizable CP-viloation effect in neutrino oscillations that enables measuring the Dirac phase δ . Theimpact could also extend to our understanding of matter-antimatter asymmetry that shaped our universe.In order to cope with a relatively large mixing angle θ z , one might be compelled to introduce newideas in model building that may enrich our theoretical understanding of the neutrino flavor problem orthe flavor problem in general in case we are fortunate enough. One of the common ideas, often discussedin the literature[11], is using flavor symmetries, and one of the most attractive ideas in this regard is the µ – τ symmetry [12, 13]. This symmetry is enjoyed by many popular mixing patterns such as tri-bimaximalmixing (TBM) [14], bimaximal mixing (BM) [15], hexagonal mixing (HM) [16] and scenarios of A mixing[17], and it was largely studied in the literature [18]. Actually, many sorts of these symmetries happento be ‘accidental’ - just a numerical coincidence of parameters without underlying symmetry, but rathera symmetry resulting from a mutual influence of different and independent factors. The authors of [19]showed that the TBM symmetry falls under this category in that large deviations from its predictions areallowed experimentally. Nonetheless, one can adopt a more ‘fundamental’ approach and construct modelsincorporating the symmetry in question at the Lagrangian level. In this context, recent, particularlysimple, choices for discrete and continuous flavor symmetry addressing the non-vanishing θ z questionwere respectively worked out in [20] and [21].For the µ – τ symmetry, it is well known that the exact form often requires vanishing θ z and, thus, therecent results on non-vanishing θ z force us to abandon the idea of exact µ – τ symmetry and to invokesmall perturbation violating it. The idea of introducing perturbations over a µ – τ symmetric massmatrix was recently introduced in [22, 23, 24], where the authors analyzed the effect of perturbations andthe correlation of their sizes with those corresponding to the deviation of θ z and θ y − π from zero. In[22], the perturbations are introduced into the µ - τ symmetric neutrino mass matrix at all entries, whilein [23] the perturbations are introduced only at the mass matrix entries which are related through µ – τ symmetry. The perturbations in [24] were imposed on four and three zero neutrino Yukawa textures.In fact, approximate interchange symmetry between second and third generation fields goes back to [25]where µ – τ symmetry was extended to all fermions with a concrete realization in a two-doublets Higgsmodel.In this present work, we follow a similar procedure as in [23], and insert the perturbations only at massmatrix entries related by µ – τ symmetry. In our approach, however, the deformed relations are thought ofas defining textures, and this way of thinking provides deep insight about the µ – τ symmetry itself and itsbreaking. The two relations defining the approximately µ – τ symmetric texture contain two parameters,generally complex, controlling the strength of the symmetry breaking. For the sake of simplicity andclarity, we disentangle each parameter to be kept alone in the relations defining the texture. The ‘minimal’textures obtained in this way (minimal in the sense of containing just one symmetry breaking parameter)may be considered as a ‘basis’ for all perturbations. Moreover, the numerical study of [23] with normalhierarchy spectrum required one of the two symmetry breaking parameters to be small with respect tothe other, and this motivated us to consider the extreme case where one of the two symmetry breakingparameters is absent.As we shall see, the exact µ — τ symmetry can be realized in two different ways as equating to zerotwo linear combinations of the mass matrix entries. Thus, upon deforming these two defining linearcombinations, in each of the possible two ways of realizing µ — τ symmetry, by two parameters (eachparameter affecting one linear combination) and separating the two parameters effects, we end up withfour possible textures. The two equations defining each textures provide us with four real equations,which are used to reduce the independent parameters of the neutrino mass matrix in this specific texturefrom nine to five. We choose the five input parameters to be the mixing angles ( θ x , θ y , θ z ), the Dirac phase δ and the solar mass square difference δm , and we vary them within their experimentally acceptable2egions. Moreover, we vary also the complex parameter defining the deformation. Therefore, in this waywe can reconstruct the neutrino mass matrix out of 7-dimensional parameter space, and compute theunknown mass spectrum ( m , m , m ) and the two Majorana phases ρ and σ . We perform consistencycheck with the other experimental results, and find that all possible four textures could accommodatethe data. However, no singular models, where one of the masses equals zero, could be viable.In contrast to the analysis of [23] which stated that normal type hierarchy is not compatible withsmall perturbations ( ǫ < χ = 2 ǫ < m /m < m /m < . m ≪ m ≪ m ). Thus we believe our analysis ismore thorough and our conclusions are more solid.As to the origin of the perturbations, there are few strategies to follow. First, one can add termsviolating explicitly the µ — τ symmetry in the Lagrangian, as was done in [26]. Second, one may assumeexact symmetry, leading to θ z = 0, at high scale. Then renormalization group (RG) running of theneutrino mass matrix elements creates a term which breaks the µ — τ symmetry at the electroweak scale.However, many studies showed that the RG effects are negligible. In [27], this process of symmetrybreaking via RG running within multiple Higgs doublets model was only valid, for a sizable θ z , in aquasi-degenrate spectrum. In [28], the same conclusion, about the inability of radiative breaking togenerate relatively large θ z , was reached in minimal supersymmetric standard model (MSSM) schemes.Thus, we shall not consider RG effects, but impose approximate µ — τ symmetry at high scale (seesawscale, say) which would remain valid at measurable electroweak scale. Third, as was done in [29], the µ – τ symmetry is replaced by another symmetry including the former as a subgroup. In this spirit and inline with [23, 25], we address in detail the question of the perturbations root and present some concreteexamples at the Lagrangian level for the ‘minimal’ texture form having only one breaking parameter bymeans of adding extra Higgs fields and symmetries, in both types I and II of seesaw mechanisms. In typeII seesaw, we achieve the desired perturbed form by adding a new Z -symmetry to the one characterizingthe µ - τ symmetry (which we denote henceforth by S ) and three Higgs triplets responsible for givingmasses to the left-handed (LH) neutrinos and by substituting three Higgs doublets for the SM Higgsfield for the charged lepton masses. On the other hand, we achieve the desired form in type I seesaw byconsidering a flavor symmetry of the form S × Z and by having three SM-like Higgs doublets for thecharged leptons masses, four other Higgs doublets for the Dirac neutrino mass matrix and additional twoHiggs singlets for the Majorana right-handed (RH) neutrino mass matrix.The plan of the paper is as follows: in section 2, we review the standard notation for the neutrinomass matrix and its relation to the experimental constraints. In section 3, we present the µ – τ symmetryand its implications. The realization of µ – τ symmetry as textures and its consequences for non-singluarand singular cases are respectively worked out in section 4 and 5. In section 6, we present the minimalpossible ways for breaking the µ – τ symmetry leading to four cases being interpreted as four possibletextures, and we classify all the hierarchy patterns regarding the mass spectra. The detailed relevantformulae and the results of the phenomenological analysis of each texture are presented in Section 7 (fornonsingular cases) and Section 8 (for singular ones). In section 9, we present a possible Lagrangian forthe approximate µ – τ leading to the ‘minimal’ textures we adopted. The last section 10 is devoted fordiscussions and conclusions. In the flavor basis, where the charged lepton mass matrix is diagonal, we diagonalize the symmetricneutrino mass matrix M ν by a unitary transformation, V † M ν V ∗ = m m
00 0 m , (1)3ith m i (for i = 1 , ,
3) real and positive. We introduce the mixing angles ( θ x , θ y , θ z ) and the phases( δ, ρ, σ ) such that [30]: V = U PP = diag( e iρ , e iσ , U = c x c z s x c z s z − c x s y s z − s x c y e − iδ − s x s y s z + c x c y e − iδ s y c z − c x c y s z + s x s y e − iδ − s x c y s z − c x s y e − iδ c y c z , (2)(with s x ≡ sin θ x . . . ) to have M ν = U λ λ
00 0 λ U T . (3)with λ = m e iρ , λ = m e iσ , λ = m . (4)In this parametrization, the mass matrix elements are given by: M ν = m c x c z e i ρ + m s x c z e i σ + m s z ,M ν = m (cid:16) − c z s z c x s y e i ρ − c z c x s x c y e i (2 ρ − δ ) (cid:17) + m (cid:16) − c z s z s x s y e i σ + c z c x s x c y e i (2 σ − δ ) (cid:17) + m c z s z s y ,M ν = m (cid:16) − c z s z c x c y e i ρ + c z c x s x s y e i (2 ρ − δ ) (cid:17) + m (cid:16) − c z s z s x c y e i σ − c z c x s x s y e i (2 σ − δ ) (cid:17) + m c z s z c y ,M ν = m (cid:16) c x s z s y e i ρ + c y s x e i ( ρ − δ ) (cid:17) + m (cid:16) s x s z s y e i σ − c y c x e i ( σ − δ ) (cid:17) + m c z s y ,M ν = m (cid:16) c x s z c y e i ρ − c y s x e i ( ρ − δ ) (cid:17) + m (cid:16) s x s z c y e i σ + s y c x e i ( σ − δ ) (cid:17) + m c z c y ,M ν = m (cid:16) c x c y s y s z e i ρ + s z c x s x ( c y − s y ) e i (2 ρ − δ ) − c y s y s x e i ( ρ − δ ) (cid:17) + m (cid:16) s x c y s y s z e i σ + s z c x s x ( s y − c y ) e i (2 σ − δ ) − c y s y c x e i ( σ − δ ) (cid:17) + m s y c y c z . (5)Note that under the transformation given by T : θ y → π − θ y and δ → δ ± π, (6)the mass matrix elements are transformed amongst themselves by swapping the indices 2 and 3 andkeeping the index 1 intact: M ν ↔ M ν , M ν ↔ M ν M ν ↔ M ν , M ν ↔ M ν . (7)On the other hand, the mass matrix is transformed into its complex conjugate i.e M νij ( T ( δ, ρ, σ )) = M ∗ νij (( δ, ρ, σ )) (8)under the mapping given by: T : ρ → π − ρ, σ → π − σ, δ → π − δ, (9)The above two symmetries T , are quite useful in classifying the models and in connecting thephenomenological analysis of patterns related by them.It is straightforward to relate our parametrization convention Eq. (2) to the more familiar one usedin the recent data analysis of [31]. In fact, the mixing angles in the two parameterizations are equal θ x ≡ θ , θ y ≡ θ , θ z ≡ θ . (10)whereas there is a simple linear relation, discussed in [20, 32], between the phases defined in ourparametrization and those corresponding to the standard one.4he solar and atmospheric neutrino mass-squared differences are characterized by two independentneutrino mass-squared differences[31]: δm ≡ m − m , (cid:12)(cid:12) ∆ m (cid:12)(cid:12) ≡ (cid:12)(cid:12)(cid:12)(cid:12) m − (cid:0) m + m (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) , (11)whereas the parameter R ν ≡ δm | ∆ m | . (12)characterizes the hierarchy of these two quantities.The neutrino mass scales are constrained in the reactor nuclear experiments on beta-decay kinematicsand neutrinoless double-beta decay by two parameters which are the effective electron-neutrino mass: h m i e = vuut X i =1 (cid:0) | V ei | m i (cid:1) , (13)and the effective Majorana mass term h m i ee : h m i ee = (cid:12)(cid:12) m V e + m V e + m V e (cid:12)(cid:12) = | M ν | . (14)Another parameter with an upper bound coming from cosmological observations is the ‘sum’ parameterΣ: Σ = X i =1 m i . (15)Moreover, the Jarlskog rephasing invariant quantity is given by[33]: J = s x c x s y c y s z c z sin δ (16)There are no experimental bounds on the phase angles, and we take the principal value range for δ, ρ and 2 σ to be [0 , π ]. As to the other oscillation parameters, the experimental constraints give the valuesstated in Table (1) with 1, 2, and 3- σ errors [31, 34]. Actually, the fits of oscillation data found in [31]and [34] are consistent with each other except that the latter fits are stricter for θ z . In our numericalanalysis, we prefer to use the former fit having a wider range for θ z in order to easily catch the patternof variation depending on θ z . Other groups [35, 36] have also carried out global fits for the oscillationdata and their findings are in line with those of the group of [31]. Parameter Best fit 1 σ range 2 σ range 3 σ range δm (10 − eV ) 7 .
58 [7 . , .
80] [7 . , .
99] [6 . , . (cid:12)(cid:12) ∆ m (cid:12)(cid:12) (10 − eV ) 2 .
35 [2 . , .
47] [2 . , .
57] [2 . , . θ x . o [32 . o , . o ] [31 . o , . o ] [30 . o , . o ] θ y . o [38 . o , . o ] [36 . o , . o ] [35 . o , . o ] θ z . o [7 . o , . o ] [6 . o , . o ] [4 . o , . o ]8 . o (8 . o , . o ) (7 . o , . o ) (7 . o , . o ) R ν . . , . . , . . , . Table 1:
The global-fit results of three neutrino mixing angles ( θ x , θ y , θ z ) and two neutrino mass-squared differ-ences δm and ∆ m as defined in Eq. (11). The results [ · · · ] and ( · · · ) as respectively extracted from [31] and [34].In [31], it is assumed that cos δ = ± δ is not restrictedand the old reactor flux is used. We adopt the less conservative 2- σ range as reported in [37] for the non oscillation parameters h m i e ,Σ, whereas for the other non-oscillation parameter h m i ee we use values found in [38]: h m i e < . , Σ < .
19 eV , h m i ee < . − .
78 eV . (17)5 The µ – τ symmetry and neutrino mass matrix The µ – τ symmetry can be described by the following general set of conditions[22], | V µ i | = | V τ i | , for i = 1 , , . (18)According to our adopted parameterizations for V in Eq.(2) these conditions imply two classes of solutions.The first class, hereafter labeled by class I, is characterized by, θ y = π , s x c x s z c δ = 0 , (19)while the second class, hereafter labeled by class II, is determined by, θ z = π , s x s y c δ = c y c x , (20)The two classes, I and II, are distinguished by the possible allowed values for mixing angles θ y and θ z .In class I, the mixing angle θ y is fixed to be π , while for class II the mixing angle θ z is fixed to be π .These restrictions are the only nontrivial consequence of the µ – τ symmetry. Regarding the other mixingangles and phases for each class, the restriction imposed through the symmetry is rather loose. However,according to the allowed values for mixing angles and phases, the class II cannot be divided into a finitenumber of sub-classes in contrast to the class I which can be divided into four sub-classes as follows, (a) θ y = π and θ x = 0 while θ z , δ, ρ and σ are free, (b) θ y = π and θ x = π while θ z , δ, ρ and σ are free, (c) θ y = π and θ z = 0 while θ x , δ, ρ and σ are free, (d) θ y = π and δ = ± π while θ x , θ z , ρ and σ are free.The sub-classes (a) and (b) seem unsatisfactory because the predicted θ x is far from the experimentallypreferred value. The remedy for this defect is to introduce a small perturbation having a large effecton θ x as was done in [22]. As to the sub-class (c), it seems to be the most interesting class, from aphenomenological point of view, when joint by fixing θ x near the experimentally preferred value. In asense, it can contain models with tri-bimaximal, bimaximal, hexagonal, and A symmetries. The lastremaining sub-class (d), predicting maximal CP violation, can include the tetramaximal symmetry [39].The class II is phenomenologically disfavored since θ z = π is far from the experimentally preferred value,which might justify dropping this whole class in the analysis carried out in[22].We can get more insight into the µ – τ symmetry by writting its implications on the neutrino massmatrix entries. The class I and its sub-classes are found to imply (a) M ν = M ν and M ν = M ν , (b) M ν = M ν and M ν = M ν , (c) M ν = − M ν and M ν = M ν , (d) M ν = M ∗ ν and M ν = M ∗ ν for vanishing Majorana phases, otherwise no simple algebraicrelation between the mass entries is found.In the second class II, the implied mass relations are, M ν = M ν = 0 , and | M ν | = | M ν | . (21)The above mentioned considerations motivate us to take as a starting point one of the following massrelations as defining the µ – τ symmetry. The first relation is taken to be M ν = M ν , and M ν = M ν . (22)while the second one is M ν = − M ν , and M ν = M ν . (23)6hese two alternative ways for imposing µ – τ symmetry in Eq.(22) and Eq.(23) are respectively designatedby S + and S − in order to ease the corresponding referral. The other possible relations like ( M ν = M ∗ ν and M ν = M ∗ ν ) or ( M ν = M ν = 0 and | M ν | = | M ν | ) are disfavored because theyinvolve non analytical algebraic relation between mass entries that cannot be generated by usual discreteflavor symmetries. There is still a further motivation for imposing µ – τ symmetry via S + or S − whichcan be easily inferred from the symmetry properties enjoyed by the neutrino mass matrix as explainedin section 2. In fact, the transformation rule in Eq.(6) singles out θ = π as a fixed point for thetransformation and the mass relations in Eq.(7) already links the mass matrix entries relevant for the µ – τ symmetry. The difference in sign between the two alternative realizations, M ν = ± M ν , can beattributed to the different phases assigned to the third neutrino filed ν τ . µ – τ symmetry as a texture for non singular neu-trino mass matrix The exact exact µ – τ symmetry can be treated as a texture defined by, M ν ∓ M ν = 0 , (24) M ν − M ν = 0 , where the minus and plus sign correspond respectively to the cases of Eq.(22) and Eq.(23).Using Eqs. (2-4), the relation defining the texture can be expressed as M ν ∓ M ν = 0 , ⇒ X j =1 ( U j U j ∓ U j U j ) λ j = 0 ⇒ A ∓ λ + A ∓ λ + A ∓ λ = 0 M ν − M ν = 0 , ⇒ X j =1 ( U j U j − U j U j ) λ j = 0 , ⇒ B λ + B λ + B λ = 0 (25)where A ∓ j = U j ( U j ∓ U j ) , and B j = U j − U j , (no sum over j ) . (26)The coefficients A ∓ and B can be written explicitly in terms of mixing angles and Dirac phase as, A ∓ = − c x c z (cid:2) c x s z ( s y ∓ c y ) + s x ( c y ± s y ) e − i δ (cid:3) ,B = (cid:0) c x s y s z + s x c y e − i δ (cid:1) − (cid:0) − c x c y s z + s x s y e − i δ (cid:1) ,A ∓ = − s x c z (cid:2) s x s z ( s y ∓ c y ) ∓ c x ( s y ± c y ) e − i δ (cid:3) ,B = (cid:0) − s x s y s z + c x c y e − i δ (cid:1) − (cid:0) s x c y s z + c x s y e − i δ (cid:1) ,A ∓ = s z c z ( s y ∓ c y ) ,B = c z (cid:0) s y − c y (cid:1) . (27)Provided λ is non-vanishing, the equations (25) can be treated as two inhomogeneous linear equationsof the ratios λ λ and λ λ which can be solved to get, λ λ = A ∓ B − A ∓ B A ∓ B − A ∓ B ,λ λ = A ∓ B − A ∓ B A ∓ B − A ∓ B . (28)Computing the mass spectrum, we find that it is always a degenerate one ( m = m = m ) leadingto vanishing mass-squared differences, which is unacceptable phenomenologically. Explicitly, for thecases (a) to (c) mentioned in the previous section and respecting exact µ – τ symmetry, we have all thecoefficients A ∓ ’s and B ’s vanishing except: A +3 = − A +1 = s z c z (case a), A +3 = − A +2 = s z c z (case b) and A − = − A − = −√ s x c x e − iδ (case c). 7 The exact µ – τ symmetry as a texture for singular neutrinomass matrix One may wonder that our analysis might lead to non trivial results for singular neutrino mass matrix.Thus, it is crucial to carry the same study for singular case, and keep in mind that the viable singularneutrino mass matrices have to be characterized by vanishing m or m . The vanishing of m leading tothe simultaneous vanishing of m and m is not at all phenomenologically consistent. m = 0Realization m m S − (cid:12)(cid:12)(cid:12) A − A − (cid:12)(cid:12)(cid:12) ≈ q − s y s y s z s x c x + O ( s z ) (cid:12)(cid:12)(cid:12) B B (cid:12)(cid:12)(cid:12) ≈ c x (1 + 2 t x t y c δ s z ) + O ( s z ) S + (cid:12)(cid:12)(cid:12) A +3 A +2 (cid:12)(cid:12)(cid:12) ≈ q s y − s y s z s x c x + O ( s z ) (cid:12)(cid:12)(cid:12) B B (cid:12)(cid:12)(cid:12) ≈ c x (1 + 2 t x t y c δ s z ) + O ( s z ) m = 0Realization m m S − (cid:12)(cid:12)(cid:12) A − A − (cid:12)(cid:12)(cid:12) ≈ − (1 − s y ) c δ s z c y s x c x + O ( s z ) (cid:12)(cid:12)(cid:12) B B (cid:12)(cid:12)(cid:12) ≈ t x (cid:16) t y c δ s z s x c x (cid:17) + O ( s z ) S + (cid:12)(cid:12)(cid:12) A +1 A +2 (cid:12)(cid:12)(cid:12) ≈ (1+ s y ) c δ s z c y s x c x + O ( s z ) (cid:12)(cid:12)(cid:12) B B (cid:12)(cid:12)(cid:12) ≈ t x (cid:16) t y c δ s z s x c x (cid:17) + O ( s z )Table 2: The approximate mass ratio formulae for the singular light neutrino mass realizing exact µ – τ symmetry.The forumlae are calculated in terms of A’s or B’s coefficients m singular neutrino mass matrix having exact µ – τ symmetry The mass spectrum in this case turns out to be, m = 0 , m = √ δm , m = r ∆ m + δm ≈ √ ∆ m , (29)which puts the mass ratio m m in the form m ≡ m m = s R ν R ν ≈ p R ν , (30)where the phenomenologically acceptable value for R ν is given in Table (1). The vanishing of m togetherwith imposing the exact µ – τ symmetry as stated in Eqs.(25) leads to, A ∓ λ + A ∓ λ = 0 ,B λ + B λ = 0 , (31)which gives non trivial solutions, provided A ∓ B − A ∓ B = 0, i.e. m = (cid:12)(cid:12)(cid:12)(cid:12) A ∓ A ∓ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) B B (cid:12)(cid:12)(cid:12)(cid:12) , σ = 12 Arg (cid:18) − A ∓ m A ∓ m (cid:19) = 12 Arg (cid:18) − B m B m (cid:19) . (32)The Majorana phase ρ becomes unphysical, since m vanishes, in this case, and can be dropped out.8hese patterns can be easily shown to be unviable just by comparing the two approximate expressionsobtained for m m . As an example we consider the case S − where we have, as reported in Table (2), m ≈ q − s y s y s z s x c x + O ( s z ) , from A − ’s , c x (1 + 2 t x t y c δ s z ) + O ( s z ) , from B’s , (33)This mass ratio, m m should be consistent with the constraint of Eq. (30), which means that it shouldbe much less than one. It is hard to satisfy this constraint because the first expression, obtained from A − ’s, starts from O ( s z ) and can be tuned to a small value, while the second one, obtained from B ’s,has a leading contribution ( c x ) which is greater than one for the admissible range of θ x . To properlytune the second expression, one needs large negative higher order corrections which can be achieved bychoosing negative c δ and letting θ y approach π , but this tends in its turn to diminish the first expressionof the mass ratio more than required. Thus, the two expression cannot be made compatible. A similarreasoning can be applied to the case S + to show the incompatibility of the two derived expressions for themass ratio. Our numerical study confirms this conclusion where all the phenomenologically acceptableranges for mixing angles and Dirac phase are scanned, but no solutions could be found satisfying themass constraint expressed in Eq. (30) m singular neutrino mass matrix having exact µ – τ symmetry Along the same lines of the previous subsection, we can treat the case of vanishing m . This time, themass spectrum is found to be, m = r ∆ m − δm , m = q ∆ m + δm , m = 0 , (34)forcing the mass ratio m m to be m ≡ m m = s R ν − R ν ≈ R ν & . (35)The vanishing of m together with imposing exact µ – τ symmetry as stated in Eqs.(25) result in thefollowing equations, A ∓ λ + A ∓ λ = 0 ,B λ + B λ = 0 , (36)which have non trivial solutions as, m = (cid:12)(cid:12)(cid:12)(cid:12) A ∓ A ∓ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) B B (cid:12)(cid:12)(cid:12)(cid:12) , ρ − σ = 12 Arg (cid:18) − A ∓ m A ∓ m (cid:19) = 12 Arg (cid:18) − B m B m (cid:19) , (37)provided A ∓ B − A ∓ B = 0. It is clear that the only relevant physical combination of Majorana phasesin such a case is the difference ρ − σ . One can use the same reasoning explained in the case of vanishing m , based on approximate formulae for mass ratios, as reported in Table (2), to show that the constraintof Eq. 35 cannot be satisfied, which makes the patterns unviable. Again, our numerical study based onscanning all phenomenologically acceptable ranges for mixing angles and Dirac phase reveals no solutionsfound satisfying the constraint of Eq. (35).Our investigations, which are so far model independent, point out that imposing exact µ – τ symmetryalways produces phenomenologically unsatisfactory results. Thus one might find the solace by demandingviolation of the exact µ – τ symmetry. In breaking the symmetry, we are going to try the simplest andminimal ways of breaking. 9 Deviation from exact µ – τ symmetry We consider the simplest minimal possible deviation from the exact µ – τ symmetry that can be param-eterized by only one parameter. The relations characterizing these deviations can assume the followingtwo forms, M ν (1 + χ ) = ± M ν , and M ν = M ν , (38)and M ν = ± M ν , and M ν (1 + χ ) = M ν , (39)where χ = | χ | e i θ is a complex parameter measuring the deviation from exact µ – τ symmetry. Theabsolute value | χ | is restricted to fall in the range [0 , . θ is totally free. The chosenrange for χ is made to ensure a small deviation that can be treated as a perturbation.The deviation from exact µ – τ symmetry can be treated in an illuminating way by considering therelations in Eqs.(38,39) as defining the following textures M ν (1 + χ ) ∓ M ν = 0 , and M ν − M ν = 0 , (40)and M ν ∓ M ν = 0 , and M ν (1 + χ ) − M ν = 0 . (41)Following the same procedure as described in section 4, we find that the coefficients A ’s and B ’s corre-sponding to the textures defined in Eq.(40) and Eq.(41) are respectively, A ∓ j = U j [ U j (1 + χ ) ∓ U j ] , and B j = U j − U j , (no sum over j ) . (42)and A ∓ j = U j ( U j ∓ U j ) , and B j = U j (1 + χ ) − U j , (no sum over j ) . (43)Assuming λ = 0, the resulting λ ’s ratio are found to be, λ λ = A B − A B A B − A B ,λ λ = A B − A B A B − A B , (44)From these λ -ratios, the mass ratios (cid:16) m m , m m (cid:17) and Majorana phases ( ρ, σ ) can be determined in termsof the mixing angles (( θ x , θ y , θ z ), the Dirac phase δ and the complex parameter χ . Thus, we can vary( θ x , θ y , θ z , δm ) over their experimentally allowed regions and ( δ, | χ | , θ ) in their full range to determinethe unknown mass spectra and Majorana phases. We can then confront the whole predictions with theexperimental constraints given in Table (1) and Eq. (17) to find out the admissible 7-dim parameterspace region. For a proper survey of the allowed parameter space, one can illustrate graphically all thepossible correlations, at the three levels of σ -error, between any two physical neutrino parameters. Wechose to plot for each pattern and for each type of hierarchy thirty four correlations at the 3- σ errorlevel involving the parameters ( m , m , m , θ x , θ y , θ z , ρ, σ, δ, J, m ee , | χ | , θ ) and the lowest neutrino mass( LNM ). Moreover, for each parameter, one can determine the extremum values it can take according tothe considered precision level, and we listed in tables these predictions for all the patterns and for thethree σ -error levels.The resulting mass patterns are found to be classifiable into three categories: • Normal hierarchy: characterized by m < m < m and is denoted by N satisfying numerically thebound: m m < m m < . • Inverted hierarchy: characterized by m < m < m and is denoted by I satisfying the bound: m m > m m > . Degenerate hierarchy (meaning quasi- degeneracy): characterized by m ≈ m ≈ m and is denotedby D . The corresponding numeric bound is taken to be:0 . < m m < m m < . m , and m ) being equal to zero (the data prohibits the simultaneousvanishing of two masses and thus m can not vanish). µ – τ symmetry We present now the results of our numerical analysis for the four simplest possible patterns violatingexact µ – τ as described in the previous section and quantified in Eq.(40) and Eq.(41). The coefficients A ′ s and B ’s are expressed in Eq.(42) and Eq.(43) according to the pattern under study. Moreover,analytical expressions of the relevant parameters up to leading order in s z are provided in order to getan “understanding” of the numerical results. The relevant parameters include mass ratios, Majoranaphases, R ν parameter, effective Majorana mass term h m i ee and effective electron’s neutrino mass h m i e .We stress here that our numerical analysis is based on the exact formulae and not on the approximateones.The large number of correlation figures is organized in plots, at the 3- σ -error level, by dividing eachfigure into left and right panels (halves) denoted accordingly by the letters L and R. Additional labels (D,N and I) are attached to the plots to indicate the type of hierarchy (Degenerate, Normal and Inverted,respectively). Any missing label D, N or I on the figures of certain pattern means the absence of thecorresponding hierarchy type in this pattern.We list in tables (3) and (4), and for the three types of hierarchy and the three precision levels,the extremum values that the different parameters can take. It is noteworthy that our numericalstudy is based, as was the case in [32], on random scanning of the 7-dim parameter space composedof (cid:0) θ x , θ y , θ z , δ, δm , | χ | and θ (cid:1) . This kind of randomness implies that the reported values in the tablesare meant to give only a strong qualitative indication, in that they might change from one run to another,providing thus a way to check for the stability of the results. M ν (1 + χ ) − M ν = 0 , and M ν − M ν = 0 . In this pattern, C1,the relevant expressions for A ’s and B ’s are A = − c x c z (cid:0) c x s y s z + s x c y e − i δ (cid:1) (1 + χ ) − c x c z (cid:0) − c x c y s z + s x s y e − i δ (cid:1) ,A = s x c z (cid:0) − s x s y s z + c x c y e − i δ (cid:1) (1 + χ ) + s x c z (cid:0) s x c y s z + c x s y e − i δ (cid:1) ,A = s z s y c z (1 + χ ) − s z c y c z ,B = (cid:0) c x s y s z + s x c y e − i δ (cid:1) − (cid:0) − c x c y s z + s x s y e − i δ (cid:1) ,B = (cid:0) − s x s y s z + c x c y e − i δ (cid:1) − (cid:0) s x c y s z + c x s y e − i δ (cid:1) ,B = s y c z − c y c z , (48)leading to mass ratios, up to leading order in s z , as m ≡ m m ≈ s δ s θ | χ | s z t x T ,m ≡ m m ≈ − t x s δ s θ | χ | s z T , (49)where T is defined as, T = | χ | c y + 2 | χ | c θ c y ( c y + s y ) + 1 + s y . (50)11hile the Majorana phases as, ρ ≈ δ + s δ s z (cid:16) − s y c y | χ | + | χ | c θ ( c y − s y ) + c y (cid:17) t x T ,σ ≈ δ − s δ t x s z (cid:16) − s y c y | χ | + | χ | c θ ( c y − s y ) + c y (cid:17) T . (51)The parameters R ν , mass ratio square difference m − m , h m i e and h m i ee can be deduced to be, R ν ≈ − s δ s θ | χ | s z s x T ,m − m ≈ − s δ s θ | χ | s z s x T , h m i e ≈ m (cid:20) s θ s δ | χ | s z t x T (cid:21) , h m i ee ≈ m (cid:20) s θ s δ | χ | s z t x T (cid:21) . (52)Our expansion in terms of s z is justified since s z is typically small for phenomenologically acceptablevalues where the best fit for s z ≈ . m as, m = 1 + ∞ X i =1 c i ( θ x , θ y , δ, | χ | , θ ) s iz , (53)where c i is the i th -Taylor expansion coefficient depending on θ x , θ y , δ, | χ | and θ . In this pattern, putting θ y equal to π makes the spectrum degenerate ( m = m = 1) irrespective of the values for θ x , δ, | χ | and θ . There are two possible alternatives to match this finding: in the first one, all the c i (cid:0) θ y = π (cid:1) ’sare vanishing, whereas in the second one some of the c i (cid:0) θ y = π (cid:1) ’s are finite and non-vanishing providedthat an infinite number of c i (cid:0) θ y = π (cid:1) ’s are divergent such that the coefficients recombine in a delicateway to make the sum P ∞ i =1 c i (cid:0) θ x , θ y = π , δ, | χ | , θ (cid:1) s iz equaling zero for any s z ∗ . Explicit calculationreveals that c is finite and non vanishing at θ y = π as is evident from Eq.(49), while c i is divergentat θ y = π for all i ≥
2. A similar consideration applies also to the mass ratio m . These divergences,at θ y = π , appearing in the expansion coefficients c i for mass ratios resurface again in the expansioncoefficients corresponding to h m i e and h m i ee but surprisingly enough the divergences associated with R ν and m − m start only from the third order coefficients. All these subtleties are an artifact of theexpansion, whereas no such problems arise if we use exact formulae. Thus, the formulae due to expansionmust be dealt with caution.All the possible fifteen pair correlations related to the three mixing angles and the three Majoranaand Dirac phases ( θ x , θ y , θ z , δ, ρ, σ ) are presented in the left and right panels of Figure 1, while the lastplot in the right panel is reserved for the correlation of m against θ y .In Fig. 2, left panel, we present five correlations of J against ( θ z , δ, σ, ρ and LNM ) and the correlationof ρ versus LNM . As to the right panel, we include presentation for the correlations of h m i ee against θ x , θ z , ρ , σ , LNM, and J .As to Fig. 3, and in a similar way, we present correlations for θ against θ y and δ and for | χ | versus θ y and θ z . The correlation of m against m and m are also included. All correlations are exhibited forall three types of hierarchy and for each type we have thirty four depicted correlations. ∗ One can see this simply by noting that in case all the c i ’s are bounded then the analyticity of the series forces themto vanish. On the other hand, one can not have a finite number of ‘unbounded’ expansion coefficients, otherwise we could,assuming without loss of generality two coefficients ( c i , c i , i < i ) whose limits at y = y = π are divergent, write c i ( y ) t i + c i ( y ) t i = g ( y, t ) where g is a well behaved function if the infinite sum of ‘bounded’ terms converge. It sufficesthen to let y , for t = t , approach y in the relation c i ( y ) = g ( y,t ti − g ( y,t ti t i − i − t i − i to reach a contradiction. Pattern having M ν (1 + χ ) − M ν = 0 , and M ν − M ν = 0: The left panel (the left three columns)presents correlations of δ against mixing angles and Majorana phases ( ρ and σ ) and those of θ x against θ y , ρ and σ . Theright panel (the right three columns) shows the correlations of θ z against θ y , ρ , σ , and θ x and those of ρ against σ and θ y ,and also the correlation of θ y versus σ and m . Figure 2:
Pattern having M ν (1 + χ ) − M ν = 0 , and M ν − M ν = 0: Left panel presents correlations of J against θ z , δ , σ , ρ , and lowest neutrino mass ( LNM ), while the last one depicts the correlation of LNM against ρ . The right panelshows correlations of m ee against θ x , θ z , ρ , σ , LNM and J . Pattern having M ν (1 + χ ) − M ν = 0 , and M ν − M ν = 0: The first two rows presents the correlationsof θ against θ y and δ , while the second two rows depict those of | χ | versus θ y and θ z . The last two rows shows the correlationsof mass ratios m and m against m . Before dwelling into examining the correlations provided by the various figures we can infer somerestrictions concernning mixing angles and phases in each pattern just by considering the expression for R ν as given in Eq.(52). The parameter R ν must be positive, nonvanishing and at the 3 − σ level isrestricted to be in the interval [0 . , . s z , s δ , s θ and | χ | . The nonvanishing of s z means θ z = 0 which is phenomenologically favorable, while vanishingof s δ , s θ implies excluding 0, π and 2 π for both δ and θ . The nonvanishing of | χ | is naturally expectedotherwise there would not be a deviation from exact µ – τ symmetry. The other required restriction,namely, s θ s δ < δ falls in the first and second quadrants then θ falls in third and fourthquadrants and vice versa. These conclusions remain valid if one used the exact expression for R ν insteadof the first order expression. Explicit computations of R ν using its exact expression tell us that θ y cannotbe exactly equal to π otherwise R ν would be zero, but nevertheless θ y can possibly stay very close to π .We see in Fig. 1 (plots: a-L → c-L, as examples) that all the experimentally allowed ranges of mixingangles, at 3 σ error levels, can be covered in this pattern except for normal and inverted hierarchy typeswhere θ y is restricted to be around 45 , by at most, plus or minus 1 . . This restriction on θ y is acharacteristic of the normal and inverted hierarchy type in this pattern. This characteristic behaviour of14 y can be understood by expressing the mass ratios, using Eqs. (49, 50 and 52), as m = 1 − c x R ν + O (cid:0) s z (cid:1) ,m = 1 + 12 s x R ν + O (cid:0) s z (cid:1) , (54)where the first order correction is identified consistently with R ν expressed up to this order. All theremaining higher order corrections to the mass ratios contribute significantly and in a spiky way in thevicinity of θ y = π leading to mass ratios considerably greater or smaller than unity. Therefore, to producethe various hierarchy types as marked in Eqs.(45–47), θ y can take in the degenerate hierarchy type valuesfar from π corresponding to small higher order corrections in Eq. (54) which would keep m and m near the value one. However, in order to get normal or inverted hierarchies, the higher order correctionsin Eq. (54) should contribute in a noticeably large amount, which could not be happened unless θ y staysclose to π , and this is what the corresponding ranges for θ y reported in Table 3 confirm. As to the DiracCP-phase δ , the whole range is allowed except the regions around 0 and π whose extensions depend onthe type of hierarchy and the precision level as evident from the same plots and the reported values inTable 3. Likewise, the plots (g-L, h-L), in Figure 1 and the values reported in Table 3 show that theMajorana phases ( ρ, σ ) are covering their ranges excluding regions around 0 and π .The plots in Figure 1 can reveal many obvious clear correlations. For example, the plots (a-R) showsthat as θ z decreases θ y tends to be very close to 45 . The plots (d-L, e-L) show a sort of distorted linearcorrelation of δ versus ( ρ, σ ) in all hierarchy types which confirms the relations presented in Eq. (51)which give linear relations at zeroth order of s z , while the found distortion can be attributed to thehigher order corrections. We may see also also in (plot e-R), a very clear linear correlation between theMajorona phases ( ρ, σ ) in all hierarchy types which again confirms the relations presented in Eq. (51)which at zeroth order produces the linear relation ρ ≈ σ .The Figure 2 (plots: a-L ,b-L) shows that the correlations ( J, θ z ) and ( J, δ ) have each a specificgeometrical shape irrespective of the hierarchy type. In fact, Eq. (16) indicates that the correlation (
J, δ )can be seen as a superposition of many sinusoidal graphs in δ , the ‘positive’ amplitudes of which aredetermined by the acceptable mixing angles, whereas the ( J, θ z ) correlation is a superposition of straight-lines in s z ∼ θ z , for small θ z , the slopes of which are positive or negative according to the sign of s δ .The resulting shape for ( J, θ z ) correlation being trapezoidal rather than isosceles is due to the exclusionof zero and its vicinity for θ z considering the latest oscillation data. The unfilled region in the plotsoriginates from the disallowed region of δ around 0 and π , which would have led, if allowed, to zero J .The left panel of Figure 2 (plots: c-L, d-L), unveils a correlation of J versus ( ρ, σ ) which is a directconsequence of the ‘linear’ correlations of δ against ( ρ, σ ) and of the ‘geometrical’ correlation of ( J, δ ). Thetwo correlations concerning the
LNM (plots: e-L, f-L) reveals that as the
LNM increases the parameterspace becomes more restricted. This seems to be a general tendency in all the patterns, where the
LNM can reach in the degenerate case values higher than in the normal and inverted hierarchies.To gain more insight about the correlattions involving h m i ee as defined in Eq. (14), we work outapproximate formulae for h m i ee corresponding to different hierarchy types. It is helpful in deriving theseapproximate formulae to realize that ρ ≈ σ and m ≈ m in all hierarchy types as is evident respectivelyfrom Fig. 1 (plots: e-R) and Fig. 3 (plots: f), and also to realize that the normal hierarchy is moderate(meaning m is of the same order as m ) while the inverted one is acute as can be inferred from Fig. 3(plots: e-N, e-I). Thus, the resulting formulae are, h m i ee ≈ m (cid:0) − s z c z s σ (cid:1) For normal and degenerate cases , h m i ee ≈ m (cid:0) − s z (cid:1) For inverted case . (55)The correlations of h m i ee against ( θ x , θ z , ρ, σ ) as depicted in the right panel of Fig. 2 (plots: a-R –d-R) canbe understood by exploiting the approximate expression for h m i ee in conjunction with the correlationsfound between θ z and ( θ x , ρ, σ ). The totality of correlations of h m i ee presented in the right panel of Fig. 2indicate that the increase of h m i ee would on the whole constrain the allowed parameter space. We notealso a general trend of increasing h m i ee with increasing LNM in all cases of hierarchy (plots e-R). Thevalues of h m i ee can not reach the zero-limit in all types of hierarchy, as is evident from the graphs or15xplicitly from the corresponding covered range in Table 3. Another point concerning h m i ee is that itsscale is triggered by the scale of m ( ≈ m ) as is evident from both the approximate formula in Eq. (55)and the corresponding covered range in Table 3.The plots in Fig. 3 (plots: b) disclose a clear correlation between θ and δ which is in accordance withwhat was derived before in that ( s θ s δ < θ and δ , which must definitely contain domains around 0 and π besides other possible additionalareas. The disallowed regions can be also checked with the help of Tables (3–4) where one additionallyfinds that the regions around 0 and π tend to be shrunk for the degenerate case. The plots (c) in Fig. 3show that as θ y deviates slightly from π then | χ | tends to increase.For the mass spectrum, we see from Fig. 3 (plots: e, f) that the normal hierarchy is mild in thatthe mass ratios do not reach extreme values. In contrast, the inverted hierarchy can be acute in thatthe mass ratio m can reach values up to O (10 ). The values of m and m are nearly equal in allhierarchy types. We also see that if m is large enough then only the degenerate case with m ∼ m canbe phenomenologically acceptable. M ν (1 + χ ) + M ν = 0 , and M ν − M ν = 0 . In this pattern, C2,the relevant expressions for A ’s and B ’s are A = − c x c z (cid:0) c x s y s z + s x c y e − i δ (cid:1) (1 + χ ) + c x c z (cid:0) − c x c y s z + s x s y e − i δ (cid:1) ,A = s x c z (cid:0) − s x s y s z + c x c y e − i δ (cid:1) (1 + χ ) − s x c z (cid:0) s x c y s z + c x s y e − i δ (cid:1) ,A = s z s y c z (1 + χ ) + s z c y c z ,B = (cid:0) c x s y s z + s x c y e − i δ (cid:1) − (cid:0) − c x c y s z + s x s y e − i δ (cid:1) ,B = (cid:0) − s x s y s z + c x c y e − i δ (cid:1) − (cid:0) s x c y s z + c x s y e − i δ (cid:1) ,B = s y c z − c y c z , (56)leading to mass ratios, up to leading order in s z , as m ≈ − s δ s θ | χ | s z t x T ,m ≈ t x s δ s θ | χ | s z T , (57)where T is defined as, T = | χ | c y + 2 | χ | c θ c y ( c y − s y ) + 1 − s y . (58)The Majorana phases are given by ρ ≈ δ − s δ s z (cid:16) s y c y | χ | + | χ | c θ ( c y + s y ) + c y (cid:17) t x T ,σ ≈ δ + s δ t x s z (cid:16) s y c y | χ | + | χ | c θ ( c y + s y ) + c y (cid:17) T . (59)The parameters R ν , mass ratio square difference m − m , h m i e and h m i ee can be deduced to be, R ν ≈ s δ s θ | χ | s z s x T ,m − m ≈ s δ s θ | χ | s z s x T , h m i e ≈ m (cid:20) − s θ s δ | χ | s z t x T (cid:21) , h m i ee ≈ m (cid:20) − s θ s δ | χ | s z t x T (cid:21) . (60)16ne can notice that the all results concerning this pattern, C2, can be derived from those of the previousone, C1, by simply making the substitutions s y → − s y and δ → δ + π . Unfortunately, the found relationcannot be used in practice to derive the predictions of one pattern from the other because the mapping s y → − s y takes θ y from a physically admissible region to a forbidden one. However, one can also verifythat the two patterns have the same properties regarding divergences for the expansion coefficients of themass ratios.The approximate expression for R ν in Eq. (60) provides us with similar restrictions like those of theprevious pattern C1, except that both δ and θ should now fall in same upper or lower semicircles. Onceagain the derived restriction remains unchanged when using the exact expression for R ν .We plot the corresponding correlations in Figures (4, 5 and 6) with the same conventions as before.In contrast to the C1 case, we see here that the mixing angle ( θ y ) can cover a wider range in the normaland inverted hierarchy cases instead of being confined around θ y = π . In the normal hierarchy case θ y falls in the interval [41 o − o ], while it almost covers all the admissible range in the inverted case. In thedegenerate case, however, there is no restriction on θ y , as it was in the C1 pattern. Another contrastingfeature is the range of θ z in the normal hierarchy type, where it is now restricted to be less than 10 o ,whereas it can, similarly to the C1 pattern, cover all its allowed range in the inverted and degeneratecases.We can understand the behaviour of θ y , compared to that of the previous pattern C1, by expressingthe mass ratios, from Eqs. (57,58) and (60), as m = 1 − c x R ν + O (cid:0) s z (cid:1) ,m = 1 + 12 s x R ν + O (cid:0) s z (cid:1) , (61)where the first order correction is identified consistently with R ν expressed up to this order, and thusrepresenting a small quantity. In contrast to the situation in the pattern C1, the remaining higher-order corrections in the mass ratios can be tuned to have a significant contribution in the vicinity ofany θ y depending on the other combinations of mixing angles and phases, which would lead to massratios considerably greater or smaller than unity. Therefore the various hierarchy types as marked inEqs.(45–47) can be generated for almost all θ y in its allowed range, and the values of θ y reported inTable 3 confirm this. As to the Dirac CP-phase δ , the whole range is allowed except the regions around0 and π whose extensions depend on the type of hierarchy and the precision level as is evident from thecorresponding plots and from the reported values in Table 3.The plots in Figure 4 can disclose many obvious clear correlations. For example, the plots (a-R) show,in normal and inverted hierarchy cases, that as θ z decreases θ y tends to be spread over its admissiblerange while the contrary occurs when θ z increases. The plots (d-L, e-L) do not show a simple correlationof δ versus ( ρ, σ ) in the various hierarchy types which would have been consistent with the zeroth orderlinear relation given in Eq. (59). In fact, the higher order corrections bring a severe distortion thatinvalidate the zeroth order linear relation even at the approximate level. These higher order correctionsdo not work in the same manner for both ρ and σ , so they do not cancel out upon subtraction producingambiguous correlation between ρ and σ , as depicted in the (plot e-R), contrasted with the simple linearityin the previous pattern C1. The absence of linear relations among the phases ( δ, ρ, σ ) forbids the allowedregion of Majorana phases to be straightforwardly determined from that of the Dirac phase ( δ ), as canbe figured out looking at the corresponding allowed values in Table 3.The special ‘sinusoidal’ and ‘trapezoidal’ shapes of J versus δ and θ z remain intact (Fig. 5, plots:a-L, b-L), and as before the unfilled region in the trapezoidal shaped plots is attributed to the disallowedregion for δ around 0 and π . The usual correlations of J versus ρ and σ (Fig. 5 plots: c-L, d-L) emergefrom those of δ versus ρ and σ . The two correlations concerning the LNM (plots: e-L, f-L) indicate thatas the
LNM increases (say, larger than 0 . LNM can reach in the degenerate case valueshigher than the other hierarchies.The correlations involving h m i ee can be made more transparent by deriving an approximate formulafor h m i ee capturing the essential observed features for all kinds of hierarchies in this specific pattern C217hich are: first, the equality of m and m as is clear in Fig. 6 (plots: f); second, the mild hierarchy inboth normal and inverted cases as is evident from Fig. 6 (plots: e-N, e-I). Thus, one can deduce fromEq. (14) that h m i ee is approximated by h m i ee ≈ m c z q(cid:2) − s x sin ( ρ − σ ) (cid:3) . (62)Now, the correlations of h m i ee against ( θ x , θ z , ρ, σ ) as displayed in the right panel of Figure 5 (plots: a-R–d-R) can be comprehended by invoking the approximate expression for h m i ee in conjunction with thepair correlations found amidst θ x , θ z , ρ and σ . The whole correlations of h m i ee presented in the rightpanel of Figure 5 point out that the increase of h m i ee would generally constrain the allowed parameterspace. We note also a general tendency of increasing h m i ee with increasing LNM in all cases of hierarchy(plots e-R). The values of h m i ee can not attain the zero-limit in all types of hierarchy, as is evident fromthe graphs or explicitly from the corresponding covered range in Table 3. Another point concerning h m i ee is that its scale is triggered by the scale of m ( ≈ m ) as is evident from both the approximate formulain Eq. (62) and the corresponding covered range stated in Table 3.The plots in Fig. 6 (plots: b) shows both that θ and δ must lie in the same upper or lower semicirclewhich confirms our inference based on the approximate formula for R ν in Eq. (60). The plots also revealthat there are disallowed regions for both θ and δ , which definitely should contain regions around 0 and π besides other possible additional regions. The disallowed regions can be also checked with the helpof Tables (3–4) where one can additionally find that the forbidden regions around 0 and π tend to beshrunk for the degenerate case and that the allowed range for θ is very limited in normal and invertedhierarchy. The Figure 6 (plots: c,d) shows that | χ | tends to increase in normal and inverted heirarchiesas θ y deviates from π or as θ z increases.For the mass spectrum, we see from Fig. 6 (plots: e) that all hierarchy types are characterized bynearly equal values of m and m . Moreover, Fig. 6 (plots: f) reveals that both normal and invertedhierarchies are of moderate type in that the mass ratios m does not reach extremely low nor high values.We also see that if m is large enough then only the degenerate case with m ∼ m can be compatiblewith data. 18igure 4: Pattern having M ν (1 + χ ) + M ν = 0 , and M ν − M ν = 0: The left panel (the left three columns)presents correlations of δ against mixing angles and Majorana phases ( ρ and σ ) and those of θ x against θ y , ρ and σ . Theright panel (the right three columns) shows the correlations of θ z against θ y , ρ , σ , and θ x and those of ρ against σ and θ y ,and also the correlation of θ y versus σ and m . Figure 5:
Pattern having M ν (1 + χ )+ M ν = 0 , and M ν − M ν = 0: Left panel presents correlations of J against θ z , δ , σ , ρ , and lowest neutrino mass ( LNM ), while the last one depicts the correlation of LNM against ρ . The right panelshows correlations of m ee against θ x , θ z , ρ , σ , LNM and J . Pattern having M ν (1 + χ ) + M ν = 0 , and M ν − M ν = 0: The first two rows presents the correlationsof θ against θ y and δ , while the second two rows depict those of | χ | versus θ y and θ z . The last two rows shows the correlationsof mass ratios m and m against m . M ν − M ν = 0 , and M ν (1 + χ ) − M ν = 0 . In this pattern, the relevant expressions for A ’s and B ’s are A = − c x c z (cid:0) c x s y s z + s x c y e − i δ (cid:1) − c x c z (cid:0) − c x c y s z + s x s y e − i δ (cid:1) ,A = s x c z (cid:0) − s x s y s z + c x c y e − i δ (cid:1) + s x c z (cid:0) s x c y s z + c x s y e − i δ (cid:1) ,A = s z c z ( s y − c y ) ,B = (cid:0) c x s y s z + s x c y e − i δ (cid:1) (1 + χ ) − (cid:0) − c x c y s z + s x s y e − i δ (cid:1) ,B = (cid:0) − s x s y s z + c x c y e − i δ (cid:1) (1 + χ ) − (cid:0) s x c y s z + c x s y e − i δ (cid:1) ,B = s y c z (1 + χ ) − c y c z , (63)leading to mass ratios, up to leading order in s z , as m ≈ r T T " − | χ | c y (cid:0) − c δ s y | χ | + c y c δ − θ (cid:1) s z t x (1 + s y ) T + O ( s z ) , ≈ r T T " | χ | c y t x (cid:0) − c δ s y | χ | + c y c δ − θ (cid:1) s z (1 + s y ) T + O ( s z ) , (64)where T and T are defined as, T = | χ | s y − | χ | c θ s y c y + c y ,T = | χ | c y + 2 | χ | c θ c y c y + c y , (65)While the Majorana phases as, ρ ≈
12 arctan " | χ | c y s y s δ − | χ | c y (cid:0) c y s δ c θ − s δ + θ (cid:1) − s δ c y | χ | c y s y c δ − | χ | c y (cid:0) c y c δ c θ − c δ + θ (cid:1) − c δ c y + O ( s z ) , ≈ δ for small enough | χ | ; | χ | ≤ . ,σ ≈
12 arctan " | χ | c y s y s δ − | χ | c y (cid:0) c y s δ c θ − s δ + θ (cid:1) − s δ c y | χ | c y s y c δ − | χ | c y (cid:0) c y c δ c θ − c δ + θ (cid:1) − c δ c y + O ( s z ) , ≈ δ for small enough | χ | ; | χ | ≤ . . (66)The parameters R ν , mass ratio square difference m − m , h m i e and h m i ee can be deduced to be, R ν ≈ | χ | c y (cid:0) − c δ s y | χ | + c y c δ − θ (cid:1) s z s x c x (1 + s y ) T + O (cid:0) s z (cid:1) ,m − m ≈ | χ | c y (cid:0) − c δ s y | χ | + c y c δ − θ (cid:1) s z s x c x (1 + s y ) T + O (cid:0) s z (cid:1) , h m i e ≈ m r T T " s z | χ | c y (cid:0) | χ | s y c δ − c y c δ − θ (cid:1) t x (1 + s y ) T + O (cid:0) s z (cid:1) , h m i ee ≈ m r T T " s z | χ | c y (cid:0) | χ | s y c δ − c y c δ − θ (cid:1) t x (1 + s y ) T + O (cid:0) s z (cid:1) . (67)It is worthy to mention that the expansions in terms of s z for this pattern are well behaved in the sensethat the expansion coefficients appearing in the mass ratio expressions are not divergent for certain valuesof the mixing angles as it is the case in the C1 and C2 patterns. Therefore, the expansion can be reliablyused as a perturbative expansion in which higher order terms have negligible contribution compared tothe lower ones. In this pattern, it remains forbidden for θ z or the difference ( θ y − π ) to vanish otherwise,as exact computations show, we would have degeneracy for m and m leading to vanishing R ν . Incontrast, the phases δ (Dirac phase) and θ can attain the values zero or π without implying vanishing R ν . These findings can be easily deduced using the approximate formula for R ν as given in Eq. (67).The complete degeneracy ( m = m = m ) is achieved when θ y = π and δ = π which can only bechecked using the exact complicated formulae for m and m . At this particular value, ( θ y = π , δ = π ),the zeroth order expansion coefficient, of say m p T /T
3, assumes the value of one, while the otherremaining coefficients are checked to be vanishing. The positivity of R ν and the constraint to lie withinthe interval [0 . , . − σ level) imposes a complicated relation between δ and θ rather thanthe simple constraint of belonging to alternate (identical) semicircles in the cases C1 (C2).The phenomenology of this pattern has many features in common with that of the pattern C1 in termsof correlations and allowed values for the parameters as can checked from the corresponding Figs.-(7–9)versus (1–3)- and Tables (3–4). Thus, we shall not repeat the same discussions and descriptions. Rather,we mention few dissimilarities: first, the mixing angle θ y is allowed to cover all of its admissible rangeeven in the cases of inverted and normal hierarchies; second, the correlation between δ and θ is not assimple as that of belonging to opposite semicircles in the pattern C1, where the R ν ’s expression allowsinterpreting it. 21igure 7: Pattern having M ν − M ν = 0 , and M ν (1 + χ ) − M ν = 0: The left panel (the left three columns)presents correlations of δ against mixing angles and Majorana phases ( ρ and σ ) and those of θ x against θ y , ρ and σ . Theright panel (the right three columns) shows the correlations of θ z against θ y , ρ , σ , and θ x and those of ρ against σ and θ y ,and also the correlation of θ y versus σ and m . Figure 8:
Pattern having M ν − M ν = 0 , and M ν (1 + χ ) − M ν = 0: Left panel presents correlations of J against θ z , δ , σ , ρ , and lowest neutrino mass ( LNM ), while the last one depicts the correlation of LNM against ρ . The right panelshows correlations of m ee against θ x , θ z , ρ , σ , LNM and J . Pattern having M ν − M ν = 0 , and M ν (1 + χ ) − M ν = 0: The first two rows presents the correlationsof θ against θ y and δ , while the second two rows depict those of | χ | versus θ y and θ z . The last two rows shows the correlationsof mass ratios m and m against m . M ν + M ν = 0 , and M ν (1 + χ ) − M ν = 0 . In this pattern, the relevant expressions for A ’s and B ’s are A = − c x c z (cid:0) c x s y s z + s x c y e − i δ (cid:1) + c x c z (cid:0) − c x c y s z + s x s y e − i δ (cid:1) ,A = s x c z (cid:0) − s x s y s z + c x c y e − i δ (cid:1) − s x c z (cid:0) s x c y s z + c x s y e − i δ (cid:1) ,A = s z c z ( s y + c y ) ,B = (cid:0) c x s y s z + s x c y e − i δ (cid:1) (1 + χ ) − (cid:0) − c x c y s z + s x s y e − i δ (cid:1) ,B = (cid:0) − s x s y s z + c x c y e − i δ (cid:1) (1 + χ ) − (cid:0) s x c y s z + c x s y e − i δ (cid:1) ,B = s y c z (1 + χ ) − c y c z , (68)leading to mass ratios, up to leading order in s z , as m ≈ r T T " | χ | c y (cid:0) − c δ s y | χ | + c y c δ − θ (cid:1) s z t x (1 − s y ) T + O ( s z ) , ≈ r T T " − | χ | c y t x (cid:0) − c δ s y | χ | + c y c δ − θ (cid:1) s z (1 − s y ) T + O ( s z ) . (69)While the Majorana phases as, ρ ≈
12 arctan " | χ | c y s y s δ − | χ | c y (cid:0) c y s δ c θ − s δ + θ (cid:1) − s δ c y | χ | c y s y c δ − | χ | c y (cid:0) c y c δ c θ − c δ + θ (cid:1) − c δ c y + O ( s z ) ,σ ≈
12 arctan " | χ | c y s y s δ − | χ | c y (cid:0) c y s δ c θ − s δ + θ (cid:1) − s δ c y | χ | c y s y c δ − | χ | c y (cid:0) c y c δ c θ − c δ + θ (cid:1) − c δ c y + O ( s z ) . (70)The parameters R ν , mass ratio square difference m − m , h m i e and h m i ee can be deduced to be, R ν ≈ | χ | c y (cid:0) + c δ s y | χ | − c y c δ − θ (cid:1) s z s x c x (1 − s y ) T + O (cid:0) s z (cid:1) ,m − m ≈ | χ | c y (cid:0) + c δ s y | χ | − c y c δ − θ (cid:1) s z s x c x (1 − s y ) T + O (cid:0) s z (cid:1) , h m i e ≈ m r T T " − s z | χ | c y (cid:0) | χ | s y c δ − c y c δ − θ (cid:1) t x (1 − s y ) T + O (cid:0) s z (cid:1) , h m i ee ≈ m r T T " − s z | χ | c y (cid:0) | χ | s y c δ − c y c δ − θ (cid:1) t x (1 − s y ) T + O (cid:0) s z (cid:1) . (71)Once again, and as it was for the two patterns C1 and C2, one can find the same interrelations betweenC3 and C4 where the results (formulae) of C4 can be derived from those of C3, by simply making thesubstitutions s y → − s y and δ → δ + π . Another time, the found relations cannot be used in a useful wayto derive the predictions of one pattern from the other because the mapping s y → − s y does not keep thephysically admissible region of θ y invariant. Furthermore, we are ill-fated that the properties regardingboundedness of the expansion coefficients of the mass ratios are mapped so that the bounded coefficientat ( θ y = π , δ = π ) in the pattern C3 may become divergent in the case of C4. This becomes clear bylooking at the expressions in Eq. (69), where the zeroth order expansion coefficient, for say m p T /T ,assumes the value one, and the first order coefficient is convergent at ( θ y = π , δ = π ) , whereas all higherorder expansion coefficients are divergent at this point while they were vanishing in the C3 pattern. Thisfinding is consistent with the infinite number of divergent terms summing up to a smooth function as wasdiscussed in Section (7.1). The divergence for R ν expansion is starting from the second order coefficientin harmony with the corresponding behaviour in the patterns C1 and C2. Using the exact expressionof R ν corresponding to this pattern shows that the mixing angle θ y is allowed to be exactly π withoutforcing R ν to vanish. The phases δ and θ can assume also any arbitrary values, but we should note thatthe point ( θ y = π , δ = π ) causes the exact form of R ν to be null. It is obvious that vanishing θ z leadsalso to vanishing R ν , but this choice is already excluded by data. As was the case in the C3 pattern,the correlation between δ and θ that emerges from the positivity of R ν and its allowed range cannot,due to the complicated expression of R ν that involves complicated dependence on phases even at theapproximate level, be described in a simple manner. We stress again that the expansion should be dealtand interpreted with caution in case of divergent coefficients and cannot be reliably used as perturbativeexpansion. Thus to avoid these kinds of problems, our numerical results are based on exact expressionsthat do not suffer from divergences.We checked when we spanned the parameter space that the normal hierarchy could accommodatethe data only at the 3 − σ error level, whereas the inverted hierarchy could do it at the 2 − σ errorlevels, and the degenerate hierarchy could survive at all error levels. The figures (10, 11 and 12) show thecorresponding correlation plots, with the same conventions as in the previous patterns. The appearanceof the normal hierarchy only at the 3 − σ error level makes it so special, and it turns out to be quiterestrictive in the sense that the mixing angle θ y is severely bounded to be around two possible values,namely, 36 or 52 , whereas θ z has only one narrow band close to 4 , while the Dirac phase δ coversalmost all its range excluding the region ]158 − . [. Moreover, in this normal hierarchy case the24arameter χ , parameterizing the deviation from exact µ – τ symmetry, cannot assume an arbitrary valuein its prescribed range: | χ | must be in the range [0 . − . θ can cover all its allowablerange excluding the region ]19 . − . [ S ]217 . − . [.Once again, there is a close resemblance between the pattern C4 and C2 in terms of correlationsand allowed values for the parameters, as can be checked respectively from the corresponding Figs.-(10–12) versus (4–6)- and Tables (3–4). Therefore it is not necessary to repeat the same discussions anddescriptions but rather we focus on the few dissimilarities: First, the mixing angle θ y is allowed to coverall of its admissible range in the inverted hierarchy type, and in particular the value π which is excludedwith its small neighborhood in the pattern C2; second, the Dirac phase δ is allowed to cover all of itsranges in the inverted and degenerate hierarchy types without any exclusion as was the case in the patternC2 concerning the values (0 , and π ) together with their neighborhoods; third, the mixing angle θ z tendsto have a far more restrictive range in case of the pattern C4 compared to that of C2; fourth, the normalhierarchy case for the pattern C4, as explained above, represents an exceptional situation, which was notthe case in the pattern C2. The figures depicting the correlations for the two patterns C2 and C4 look,more or less, similar provided the loose restrictions on θ y and δ associated with the pattern C4 are takeninto consideration.Figure 10: Pattern having M ν + M ν = 0 , and M ν (1 + χ ) − M ν = 0: The left panel (the left three columns)presents correlations of δ against mixing angles and Majorana phases ( ρ and σ ) and those of θ x against θ y , ρ and σ . Theright panel (the right three columns) shows the correlations of θ z against θ y , ρ , σ , and θ x and those of ρ against σ and θ y ,and also the correlation of θ y versus σ and m . Pattern having M ν + M ν = 0 , and M ν (1 + χ ) − M ν = 0: Left panel presents correlations of J against θ z , δ , σ , ρ , and lowest neutrino mass ( LNM ), while the last one depicts the correlation of LNM against ρ . Theright panel shows correlations of m ee against θ x , θ z , ρ , σ , LNM and J . Pattern having M ν + M ν = 0 , and M ν (1 + χ ) − M ν = 0: The first two rows presents the correlationsof θ against θ y and δ , while the second two rows depict those of | χ | versus θ y and θ z . The last two rows shows the correlationsof mass ratios m and m against m . attern: Mν
12 (1 + χ ) − Mν
13 = 0 , and Mν − Mν
33 = 0quantity θx θy θz m m m ρ σ δ h m i e h m i ee J Degenerate Hierarchy1 σ . − .
00 38 . − .
99 7 . − .
30 0 . − . . − . . − . . − .
00 0 . − .
96 [0 . − . S [180 . − .
27] 0 . − . . − . − . − . σ . − .
09 36 . − .
77 6 . − .
68 0 . − . . − . . − . . − .
17 0 . − .
16 [0 . − . S [180 . − .
99] 0 . − . . − . − . − . σ . − .
11 36 . − .
01 4 . − .
92 0 . − . . − . . − . . − .
55 0 . − .
46 [0 . − . S [180 . − .
79] 0 . − . . − . − . − . σ . − .
00 44 . − .
96 7 . − .
30 0 . − . . − . . − . . − .
30 9 . − . . − . S [188 − .
00] 0 . − . . − . − . − − . S [0 . − . σ . − .
09 [44 . − . S [45 . − .
07] 6 . − .
68 0 . − . . − . . − . . − .
71 7 . − .
56 [3 . − . S [188 . − .
46] 0 . − . . − . − . − − . S [0 . − . σ . − .
10 [43 . − . S [45 . − .
30] 4 . − .
92 0 . − . . − . . − . . − .
92 4 . − .
83 [8 . − . S [190 . − .
69] 0 . − .
050 0 . − . − . − − . S [0 . − . σ . − .
00 43 . − .
97 7 . − .
30 0 . − . . − . . × − − . . − .
84 0 . − .
49 [2 . − . S [235 . − .
6] 0 . − . . − . − . − − . S [0 . − . σ . − .
08 [43 . − . S [45 . − .
13] 6 . − .
68 0 . − . . − . . × − − . . − .
40 0 . − .
81 [7 . − . S [185 . − .
34] 0 . − . . − . − . − − . S [0 . − . σ . − .
11 [43 . − . S [45 . − .
35] 4 . − .
92 0 . − . . − . . × − − . . − .
39 0 . − .
73 [6 . − . S [188 . − .
83] 0 . − . . − . − . − − . S [0 . − . Mν
12 (1 + χ ) + Mν
13 = 0 , and Mν − Mν
33 = 0quantity θx θy θz m m m ρ σ δ h m i e h m i ee J Degenerate Hierarchy1 σ . − .
00 38 . − .
91 7 . − .
30 0 . − . . − . . − . . − . S [136 . − .
95] 0 . − .
29 [2 . − . S [199 . − .
94] 0 . − . . − . − . − . σ . − .
10 36 . − .
77 6 . − .
68 0 . − . . − . . − . . − . S [111 . − .
95] 0 . − .
45 [1 . − . S [189 . − .
1] 0 . − . . − . − . − − . S [0 . − . σ . − .
11 35 . − .
10 4 . − .
92 0 . − . . − . . − . . − . S [99 . − .
90] 0 . − .
36 [4 . − . S [188 − . . − . . − . − . − − . S [0 . − . σ . − .
99 40 . − .
05 7 . − .
16 0 . − . . − . . − . . − . S [156 . − .
81] [41 . − . S [100 . − . . − . S [279 − .
81] 0 . − . . − . − . − − . S [0 . − . σ . − .
09 [40 . − . S [46 . − .
31] 6 . − .
89 0 . − . . − . . − . . − . S [121 . − .
89] 18 . − .
63 [12 . − . S [185 . − .
12] 0 . − . . − . − . − − . S [0 . − . σ . − .
11 [40 . − . S [45 . − .
43] 4 . − .
87 0 . − . . − . . − . . − . S [112 . − .
44] 4 . − .
68 [10 . − . S [187 . − .
32] 0 . − . . − . − . − − . S [0 . − . σ . − .
00 38 . − .
46 7 . − .
30 0 . − . . − . . − . . − . S [160 . − .
77] [14 . − . S [109 . .
11] [18 . − . S [236 . − .
59] 0 . − . . − . − . − − . S [0 . − . σ . − .
09 [36 . − . S [46 . − .
77] 6 . − .
67 0 . − . . − . . − . . − . S [153 . − .
64] 9 . − .
19 [6 . − . S [196 . − . . − . . − . − . − − . S [0 . − . σ . − .
11 [35 . − . S [45 . − .
13] 4 . − .
84 0 . − . . − . . − . . − . S [137 . − .
93] 4 . − .
68 [4 . − . S [190 . − .
24] 0 . − . . − . − . − − . S [0 . − . Mν − Mν
13 = 0 , and Mν
22 (1 + χ ) − Mν
33 = 0quantity θx θy θz m m m ρ σ δ h m i e h m i ee J Degenerate Hierarchy1 σ . −
35 38 . − .
848 7 . − .
30 0 . − . . − . . − . . − .
30 0 . − .
29 0 . − .
94 0 . − . . − . − . − . σ . − .
09 [36 . − . S [45 . − .
77] 6 . − .
68 0 . − . . − . . − . . − .
84 0 . − .
84 0 . − .
88 0 . − . . − . − . − . σ . − .
11 [35 . − . S [45 . − . . − .
92 0 . − . . − . . − . . −
180 0 . −
180 0 . − .
86 0 . − . . − . − . − . σ . −
35 38 . − .
72 7 . − .
30 0 . − . . − . . − . . − .
98 0 . − .
97 [0 . − . S [193 . − . . − . . − . − . − . σ . − .
09 [36 . − . S [46 . − .
77] 6 . − .
68 0 . − . . − . . − . . − .
98 0 . − .
95 [0 . − . S [181 . − .
94] 0 . − . . − . − . − . σ . − .
11 [35 . − . S [46 . − .
12] 4 . − .
92 0 . − . . − . . − . . −
180 0 . − .
99 0 . − .
92 0 . − . . − . − . − . σ . − .
00 38 . − .
36 7 . − .
30 0 . − . . − . . − . . − .
77 0 . − .
73 59 . − .
52 0 . − . . − . − . − . σ . − .
09 [36 . − . S [45 . − .
77] 6 . − .
68 0 . − . . − . . − . . − .
93 0 . − .
96 0 . − .
95 0 . − . . − . − . − . σ . − .
10 [35 . − . S [45 . − .
13] 4 . − .
92 0 . − . . − . . − . . − .
99 0 . − .
84 0 . − .
91 0 . − . . − . − . − . Mν
12 + Mν
13 = 0 , and Mν
22 (1 + χ ) − Mν
33 = 0quantity θx θy θz m m m ρ σ δ h m i e h m i ee J Degenerate Hierarchy1 σ . −
35 38 . − .
98 7 . − .
30 0 . − . . − . . − . . − . S [135 . − .
67] 1 . − .
65 0 . − .
73 0 . − . . − . − . − . σ . − .
09 36 . − .
77 6 . − .
68 0 . − . . − . . − . . − . S [113 − .
59] 1 . − .
99 0 . − .
73 0 . − . . − . − . − . σ . − .
10 35 . − .
13 4 . − .
90 0 . − . . − . . − . . − .
71 0 . − .
30 0 . − .
90 0 . − . . − . − . − . σ × × × × × × × × × × × × σ × × × × × × × × × × × × σ . − .
11 [35 . − . S [50 . − .
13] 4 . − .
67 0 . − . . − . . − . . − .
35 0 . − .
58 [0 . − S [188 . − .
43] 0 . − . . − . − . − . σ × × × × × × × × × × × × σ . − .
09 36 . − .
77 6 . − .
07 0 . − . . − . . − . . − . S [94 . − .
27] 0 . − .
63 0 . − .
39 0 . − . . − . − . − . σ . − .
11 35 . − .
12 4 . − .
73 0 . − . . − . . − . . − .
52 0 . − .
96 0 . − .
92 0 . − . . − . − . − . Table 3:
The various prediction for the patterns of violating exact µ – τ symmetry. All the angles (masses) are evaluated in degrees ( eV ). attern: Mν
12 (1 + χ ) − Mν
13 = 0 , and Mν − Mν
33 = 0 | χ | θ σ σ σ σ σ σ Degenerate Hierarchy0 . − . . − . . − . . − . . − .
12 0 . − . . − . . − . . − . . − ∪ [269 . − .
77] [11 . − . ∪ [101 − . ∪ [8 . − . ∪ [104 − ∪ [188 . − . ∪ [277 . − .
13] [186 − ∪ [273 . − . . − . . − . . − . . − . ∪ [188 . − .
7] [8 . − . ∪ [188 − . − . ∪ [185 . − . Mν
12 (1 + χ ) + Mν
13 = 0 , and Mν − Mν
33 = 0 | χ | θ σ σ σ σ σ σ Degenerate Hierarchy0 . − . . − . . − . . − . ∪ [59 . − . ∪ [0 . − . ∪ [108 . − . ∪ [0 . − . ∪ [85 . − . ∪ [129 . − . ∪ [288 . − .
9] [300 . − .
78] [276 . − . . − . . − . . − . . − . ∪ [357 . − .
8] [0 . − . ∪ [176 . − . ∪ [0 . − . ∪ [173 . − ∪ [356 . − .
54] [355 . − . . − . . − . . − . . − . ∪ [180 . − .
12] [0 . − . ∪ [175 . − . ∪ [0 . − . ∪ [174 . − . ∪ [355 . − .
65] [353 . − . Mν − Mν
13 = 0 , and Mν
22 (1 + χ ) − Mν
33 = 0 | χ | θ σ σ σ σ σ σ Degenerate Hierarchy0 . − . . − . . − . . − .
75 0 . − .
84 0 . − . . − . . − . . − . . − . ∪ [285 . − .
12] [0 . − . ∪ [114 . − . ∪ [0 . − . ∪ [112 . − . ∪ [286 . − .
68] [285 . − . . − . . − . . − . . − .
33 [0 . − . ∪ [112 . − . ∪ [0 . − . ∪ [108 . − . ∪ [283 . − .
87] [283 . − . Mν
12 + Mν
13 = 0 , and Mν
22 (1 + χ ) − Mν
33 = 0 | χ | θ σ σ σ σ σ σ Degenerate Hierarchy0 . − . . − . . − . . − .
94 0 . − .
42 0 . − . × × . − . × × [0 . − . ∪ [139 . − . ∪ [340 . − . × . − . . − . × . − .
22 0 . − Table 4:
The allowed values for | χ | (pure number) and θ for the patterns of violating exact µ – τ symmetry. Allthe angles are evaluated in degrees. µ – τ symmetry As was the case in the exact symmetry, the violation of exact µ – τ symmetry does not allow for singularneutrino mass matrix. The same analysis and arguments against the viability of the singular patternshaving exact µ – τ symmetry in section (5) can be carried out here to show the inviability of the varioussingular deformed patterns. The numerical study based on scanning all acceptable ranges for the mixingangles and the Dirac phase δ assures the absence of any solution satisfying the mass ratio constraints asexpressed in Eq. (30) and Eq. (35). All the relevant formulae for mass ratios are collected in Table (5)in order to ease judging the inviability of patterns. The T and T present in the formulae are the onesdefined before in Eq. (65), while T is introduced as T = | χ | c y c δ + | χ | (cid:2) c δ c θ (cid:0) c y − (cid:1) + s θ s δ (cid:3) + 2 c δ c y . (72)29 = 0Pattern m m C1 (cid:12)(cid:12)(cid:12) A A (cid:12)(cid:12)(cid:12) ≈ r | χ | s y +2 | χ | c θ s y ( s y − c y )+1 − s y | χ | c y +2 | χ | c θ s y ( s y + c y )+1+ s y s z s x c x + O ( s z ) (cid:12)(cid:12)(cid:12) B B (cid:12)(cid:12)(cid:12) ≈ c x (1 + 2 t x t y c δ s z ) + O ( s z )C2 (cid:12)(cid:12)(cid:12) A A (cid:12)(cid:12)(cid:12) ≈ r | χ | s y +2 | χ | c θ s y ( s y + c y )+1+ s y | χ | c y +2 | χ | c θ s y ( c y − s y )+1 − s y s z s x c x + O ( s z ) (cid:12)(cid:12)(cid:12) B B (cid:12)(cid:12)(cid:12) ≈ c x (1 + 2 t x t y c δ s z ) + O ( s z )C3 (cid:12)(cid:12)(cid:12) A A (cid:12)(cid:12)(cid:12) ≈ q − s y s y s z s x c x + O ( s z ) (cid:12)(cid:12)(cid:12) B B (cid:12)(cid:12)(cid:12) ≈ c x q T T (cid:16) t x s y T s z T (cid:17) + O ( s z )C4 (cid:12)(cid:12)(cid:12) A A (cid:12)(cid:12)(cid:12) ≈ q s y − s y s z s x c x + O ( s z ) (cid:12)(cid:12)(cid:12) B B (cid:12)(cid:12)(cid:12) ≈ c x q T T (cid:16) t x s y T s z T (cid:17) + O ( s z ) m = 0Pattern m m C1 (cid:12)(cid:12)(cid:12) A A (cid:12)(cid:12)(cid:12) ≈ | χ | s y c y c δ + | χ | [ c δ c θ ( s y − c y ) − s θ s δ ] − c δ c y | χ | c y +2 | χ | c θ c y ( s y + c y )+1+ s y s z s x c x + O ( s z ) (cid:12)(cid:12)(cid:12) B B (cid:12)(cid:12)(cid:12) ≈ t x (cid:16) t y c δ s z s x c x (cid:17) + O ( s z )C2 (cid:12)(cid:12)(cid:12) A A (cid:12)(cid:12)(cid:12) ≈ | χ | s y c y c δ + | χ | [ c δ c θ ( s y + c y )+ s θ s δ ]+ c δ c y | χ | c y +2 | χ | c θ c y ( c y − s y )+1 − s y s z s x c x + O ( s z ) (cid:12)(cid:12)(cid:12) B B (cid:12)(cid:12)(cid:12) ≈ t x (cid:16) t y c δ s z s x c x (cid:17) + O ( s z )C3 (cid:12)(cid:12)(cid:12) A A (cid:12)(cid:12)(cid:12) ≈ − (1 − s y ) c δ s z c y s x c x + O ( s z ) (cid:12)(cid:12)(cid:12) B B (cid:12)(cid:12)(cid:12) ≈ t x (cid:16) T s y s z T s x c x (cid:17) + O ( s z )C4 (cid:12)(cid:12)(cid:12) A A (cid:12)(cid:12)(cid:12) ≈ (1+ s y ) c δ s z c y s x c x + O ( s z ) (cid:12)(cid:12)(cid:12) B B (cid:12)(cid:12)(cid:12) ≈ t x (cid:16) T s y s z T s x c x (cid:17) + O ( s z ) Table 5:
The approximate mass ratio formulae for the singular light neutrino mass violating exact µ – τ symmetry.The forumlae are calculated in terms of A’s and B’s coefficients µ - τ symmetry and realizations of the perturbed tex-tures We study now in detail how the perturbed textures can arise assuming an exact µ - τ symmetry at theLagrangian level but at the expense of introducing new matter fields and symmetries. To fix the ideas,let’s take the C M ν = A B B (1 + χ ) B C DB (1 + χ ) D C . (73)The exact µ - τ symmetry (the S symmetry) corresponding to this pattern is given by the matrix S = (74)30n that we have S = 1 and n(cid:16) M = M t (cid:17) ∧ h S t · M · S = M io ⇔ ∃ A, B, C, D : M = A B BB C DB D C , (75)We shall need also the following relations: n(cid:16) M = M t (cid:17) ∧ h S t · M · S = − M io ⇔ ∃ B, C : M = B − BB C − B − C , (76) h S t · M · S = M i ⇔ ∃ A, B, C, D : M = A B BE C DE D C , (77) h S t · M · S = − M i ⇔ ∃ B, C, D : M = B − BE C D − E − D − C , (78)[ S · M = M ] ⇔ ∃ A, B, C, D, E, F : M = A B CD E FD E F (79)We shall achieve the texture of Eq. 73 using both types II and I of the seesaw mechanism.
In the type II seesaw [40] mechanism, we show now how one can reach the desired form by assuming aflavor symmetry of the form S × Z and by having three Higgs triplets for the neutrino mass matrix andthree Higgs doublets for the charged lepton mass matrix. First, we extend the SM by introducing three SU (2) L scalar triplets H a , ( a = 1 , , H a ≡ (cid:2) H ++ a , H + a , H a (cid:3) . (80)In addition to the S symmetry, we introduce another Z symmetry, and we assume the following trans-formations: L S −→ SL , L Z −→ diag(1 , − , − L (81) H S −→ diag(1 , , − H , H Z −→ diag(1 , − , − H (82)where the L t = ( L , L , L ) , H t = ( H , H , H ) with L i ’s,( i = 1 , ,
3) are the components of the i th -family LH lepton doublets (we shall adopt this notation of ‘vectors’ in flavor space even for other fields,like l c , ν R and φ, . . . ). Note that the assignments of L , L should be the same under Z as the S symmetry interchanges them, otherwise the factor subgroups S and Z do not commute. For this reason,the S -charges of H , H are allowed to be different because Z acts on H diagonally. There will be alsothe RH charged lepton singlets and the Higgs fields responsible for the charged lepton mass matrix.31 .1.2 Neutrino mass matrix The Yukawa interaction relevant for neutrino mass has the form, L H,L = X i,j =1 3 X a =1 G aij (cid:2) H a ν TLi C ν Lj + H + a (cid:0) ν TLi C l Lj + l TLj C ν Li (cid:1) + H ++ a l TLi C l Lj (cid:3) , (83)where G aij are Yaukawa coupling constants, the indices i, j are flavor ones, and C is the charge conjugationmatrix.The field H a can get a small vacuum expectation value (vev), h H a i = v a leading to a Majorananeutrino mass matrix, M ν ij = X a =1 G aij h H a i . (84)The smallness of the vev h H a i is due to the largeness of the triplet scalar mass scale[40].The bilinear of ν Li ν Lj relevant for Majorana mass matrix transforms, via Eq. 81, under Z as: ν Li ν Lj Z ∼ B = − − − − , meaning: ν Li ν Lj Z −→ Z ( ν Li ν Lj ) = B ij ν Li ν Lj (no sum) (85)Thus we have: S t G S = G , G t = G G ij Z ( H ) Z ( ν Li ν Lj ) = G ij H ν Li ν Lj (no sum) ) Eqs. , , = ⇒ G = A C D D C (86) S t G S = G , G t = G G ij Z ( H ) Z ( ν Li ν Lj ) = G ij H ν Li ν Lj (no sum) ) Eqs. , , = ⇒ G = B B B B (87)The two Higgs fields H , H generate the unperturbed texture, whereas the perturbation is generated bythe field H : S t G S = − G , G t = G G ij Z ( H ) Z ( ν Li ν Lj ) = G ij H ν Li ν Lj (no sum) ) Eqs. , , = ⇒ G = B − B B − B (88)The mass matrix we get is of the form: M ν = v A v B + v B v B − v B v B + v B v C v D v B − v B v D v C . (89)Thus if the Yukawa couplings are all of the same order while the vevs satisfy v ≫ v we get the desiredform of the pattern C χ = − v B v B + v B . We need here to extend the symmetry to the charged lepton sector and arrange the couplings in orderto be in the ‘flavor basis’ where the charged lepton mass matrix is diagonal. For this we present threepossible options.1.
Just the SM Higgs
We have the usual Yukawa coupling term L = Y ij L i Φ l cj (90)32e assume the SM Higgs Φ is singlet under the flavor symmetry.Φ S −→ Φ , Φ Z −→ Φ (91)and present two scenarios for the RH charged lepton singlets l cj transformation under S × Z asfollows. • l cj transforms similarly as L We assume: l c S −→ Sl c , l c Z −→ diag(1 , − , − l c (92)We get via Eqs. (81,91 and 92) then: S t Y S = Y , L i l cj Z ∼ − − − − (93)which would lead, upon acquiring a vev v for the SM Higgs, to a charged lepton mass matrixof the form (see Eqs. 77, 93): M l = v A C D D C ⇒ M l M † l = v | A | | C | + | D | ℜ ( CD ∗ )0 2 ℜ ( CD ∗ ) | C | + | D | . (94)Thus we need to perform a rotation across the 1 st -axis by an angle θ y = π/ • l cj is singlet under flavor symmetry We assume: l c S −→ l c , l c Z −→ l c (95)We get via Eqs. (81,91 and 95) then: SY = Y , L i l cj Z ∼ − − − − − − (96)which would lead, upon acquiring a vev v for the SM Higgs, to a charged lepton mass matrixof the form (see Eqs. 79, 96): M l = v A B C ⇒ M l M † l = v | A | + | B | + | C | . (97)The squared mass matrix is diagonal, but it predicts two vanishing eigen masses for the 2ndand 3rd families which is not acceptable experimentally.2. Three SM-like Higgs doublets
We extend the SM to include three scalar doublets φ k playing the role of the ordinary SM-Higgsfield. The Lagrangian reponsible for the charged lepton mass is given by: L = f jik L i φ k l cj (98)We assume the Higgs fields φ k , k = 1 , , L i under S × Z : φ S −→ Sφ , φ Z −→ diag(1 , − , − φ (99)33qually, the RH charged leptons are supposed to transform as singlets under S : l c S −→ l c (100)whereas we present two scenarios for their transformations under Z as follows. • l cj transforms similarly as L under Z We assume l c Z −→ diag(1 , − , − l c (101)We get via Eqs. (81,99, 100 and 101) then: S t f ( j ) S = f ( j ) , L i φ k Z ∼ − − − − (102)where f ( j ) is the matrix whose ( i, k ) th -entry is the Yukawa coupling f jik . Then, Eqs. (77, 101and 102) lead to the following forms of the Yukawa coupling matrices: f (1) = A C D D C , f (2) = B B E E , f (3) = B B E E (103)If there is cute hierarchy in the vevs: v ≫ v , v , say, we get, for real entries, a charged leptonmass matrix of the form M l = v B B D C (104)We see that this choice of Z -charge assignments for the RH lepton singlets leads to onevanishing mass, which is excluded by experiment. Thus we turn to the other choice whichwould prove capable of producing the charged lepton mass spectrum. • l cj transforms differently from L under Z We assume l c Z −→ diag(1 , , − l c (105)We get the same Eq. (102), but Eq. (105) leads now to: f (1) = A C D D C , f (2) = A C D D C , f (3) = B B E E (106)The hierarchy ( v ≫ v , v ) would now lead to the following form for the charged lepton massmatrix: M l = v B D D C C ⇒ M l M † l = v | B | | D | D · C C · D | C | , (107)where B = (0 , , B ) T , D = ( D , D , T and C = ( C , C , T , and where the dot product isdefined as D · C = P i =3 i =1 D i C i ∗ . Now, one can adjust the Yukawa couplings so that to requirean infinitesimal rotation in order to diagonalize the squared charged lepton mass matrix and34e in the flavor basis. In fact, let us just assume the magnitudes of the three vectors comingin ratios comparable to the lepton mass ratios: | B || C | ≡ λ e ∼ m e m τ = 2 . × − , | D || C | ≡ λ µ ∼ m µ m τ = 5 . × − , (108)Then it is easy to see that the matrix: U ( θ, α, β ) = c θ e − iα s θ e − iβ − s θ e − iα c θ e − iβ : (109) α − β = arg( D · C ) , tan 2 θ = 2 D · C | D | − | C | ≃ | D || C | cos ψ (110)where ψ is the angle between the two complex vectors D and C , defined by cos ψ = D · C / ( | D |·| C | ), does diagonalize M l M † l . Note that one can absorb the individual phases α, β , using thefreedom of multiplying the unitary diagonalizing matrix by a diagonal phase matrix, whichwould leave us with only one ‘physical’ phase α − β : U ( θ, α, β ) = c θ s θ e − i ( β − α ) − s θ e i ( β − α ) c θ (111)Thus, we are in the flavor basis, as required, up to an infinitesimal rotation of angle less than10 − (See Eqs. 108 and 110).3. SM plus three Higgs singlets
One might keep the SM Higgs doublet Φ, with the same flavor transformations of Eq. (91) butadd three Higgs singlets ∆ k so that to contribute to the charged lepton mass through dimension-5operators. The Lagrangian responsible for the charged lepton mass is given by: L = L + L = Y ij L i Φ l cj + g jik Λ L i Φ∆ k l cj (112)where Λ is a mass high scale characterizing the Higgs singlets. We assume the Higgs singlet fields∆ k , k = 1 , , L i under S × Z :∆ S −→ S ∆ , ∆ Z −→ diag(1 , − , − S (Eq.100), whereas for Z we have the following options: • l cj transforms similarly as L under Z We have thus Eq.(101). The invariance of L implies SY = Y , L i l cj Z ∼ − − − − (114)This leads, when Φ acquires a vev, to a contribution to the mass matrix (see Eqs. 79, 93): M = a e f e f (115)Eq. (113) would lead, exactly as the three Higgs doublets did in the previous enumeration, toa mass contribution M of the form of Eq. 104 when the Higgs singlets acquire vevs ( δ k ), withthe hierarchy δ ≫ δ , δ . Thus we get the charged lepton mass matrix in the form: M l = M + M = a B B D e fC e f (116)35ith the condition that D = C in order not to make the determinant of the matrix equal tozero implying a vanishing mass. • l cj transforms differently from L under Z We have thus Eq.(105). The invariance of L implies: SY = Y , L i l cj Z ∼ − − − − − (117)so when Φ acquires a vev we get a contribution to the mass matrix (see Eqs. 79, 117): M = a b
00 0 f f (118)Eq. (113) would lead, exactly as the three Higgs doublets did in the previous case, to a masscontribution M of the form of Eq. 107 when the Higgs singlets acquire vevs ( δ k ), with thehierarchy δ ≫ δ , δ . Thus we get the charged lepton mass matrix in the form: M l = M + M = a b B D D fC C f (119) • In both previous items we get a charged lepton mass matrix of the form M l = A T B T C T (120)adjustable so that the three vectors are linearly independent making the mass matrix invert-ible. The discussion in [41] on the charged lepton mass matrix of the same form showed thepossibility to adjust Yukawa couplings in order to get the charged lepton mass hierarchy, andthen automatically the working basis will become the flavor basis up to order λ µ . We shall notrepeat the same analysis here, but just note that in case the parameters a, b, f (correspondingto L ) are negligible compared to B, C, D (related to L ) then the last item (Eq. 119) is similarto the last item of the past enumeration (Eq. 107), where we showed explicitly the chargedlepton mass diagonalizing matrix being an infinitesimal rotation, which allows to consider thematrices as being those in the flavor basis, with a good approximation.Before we finish this subsection, we note that there is an advantage for using the type-II seesawmechanism in that the flavor changing neutral current due to the triplet is highly suppressed because ofthe heaviness of the triplet mass scale, or equivalently the smallness of the neutrino masses. We proceed now to find a realization of the perturbed texture of pattern C1 (Eq. 73) in type-I seesawmechanism where the effective neutrino mass matrix ( M ν ) is expressed in terms of the Dirac neutrinomass matrix ( M D ) and the RH Majorana neutrino mass matrix ( M R ) through: M ν = M D M − R M TD (121)For the flavor symmetry, we start by adding a new Z symmetry (called Z ′ ) to the flavor symmetry of thetype II case, but we shall see that it is not enough to achieve the desired form, and needs to be expandedto a larger group (say to S × Z ) for this. 36 .2.1 S × Z × Z ′ -flavor symmetry We consider here a minimal extension to the flavor group of the type II seesaw by adding a new Z -symmetry so that to get the group ( Z ) .1. Matter content and symmetry transformations
We have three SM-like Higgs doublets ( φ i , i = 1 , ,
3) which would give mass to the charged leptonsand another three Higgs doublets ( φ ′ i , i = 1 , ,
3) for the Dirac neutrino mass matrix. The RHneutrinos are denoted by ( ν Ri , i = 1 , , ν R Z ′ −→ − ν R , φ ′ Z ′ −→ − φ ′ (122) L Z ′ −→ L, l c Z ′ −→ l c , φ Z ′ −→ φ, (123) ν R Z −→ diag(1 , − , − ν R , φ ′ Z −→ diag(1 , − , − φ ′ (124) L Z −→ diag(1 , − , − L, l c Z −→ diag(1 , , − l c , φ Z −→ diag(1 , − , − φ, (125) ν R S −→ Sν R , φ ′ S −→ diag(1 , , − φ ′ (126) L S −→ SL, l c S −→ l c , φ S −→ Sφ, (127)2.
Charged lepton mass matrix-flavor basis
As was the case of type-II seesaw with three SM-like Higgs doublets and where the RH chargedlepton singlets transform differently from L under Z , the Lagrangian responsible for the chargedlepton mass is given by Eq. (98). The Z ′ does not play a role here, since all the fields involved aresinglets under it, except for the fact that it does forbid the trilinear coupling between φ ′ , L and l c .Again, assuming a hierarchy in the Higgs φ ’s fields vevs ( v ≫ v , v ) we end up with a chargedlepton mass matrix of the form (Eq. 107) which can be adjusted to be in the flavor basis to a goodapproximation.3. Dirac neutrino mass matrix
The Lagrangian responsible for the neutrino mass matrix is L D = g kij L i ˜ φ ′ k ν Rj , where ˜ φ ′ = iσ φ ′∗ (128)This lagrangian is clearly invariant under Z ′ (see Eq. 122) which forces the existence of φ ′ ratherthan φ in L D . For the S × Z factor, we get via Eqs. (124,125, 126 and 127) then: S t g ( k =1 , S = g ( k =1 , , S t g ( k =3) S = − g ( k =3) , L i ν Rj Z ∼ − − − − (129)where g ( k ) is the matrix whose ( i, j ) th -entry is the Yukawa coupling g kij . Then, Eqs. (77, 78, 124and 129) lead to the following forms of the Yukawa coupling matrices: g (1) = A C D D C , g (2) = B B E E , g (3) = B − B E − E (130)Upon acquiring vevs ( v ′ i , i = 1 , ,
3) for the Higgs fields ( φ ′ i ), we get the following Dirac neutrinomass matrix: M D = Σ k =3 k =1 v ′ k g ( k ) = A D B D B D (1 + α ) E D C D D D E D (1 + β ) D D C D (131)37ith α = − v ′ B v ′ B + v ′ B , β = − v ′ E v ′ E + v ′ E (132)If the vevs satisfy v ′ ≪ v ′ and the Yukawa couplings are of the same order then we get perturbativeparameters α, β ≪ Majorana neutrino mass matrix
The mass term is directly present in the Lagrangian L R = M Rij ν Ri ν Rj (133)It is invariant under Z ′ . Then Eqs. (126,124) lead to: S t M R S = M R , ν Ri ν Rj Z ∼ − − − − Eq. = ⇒ M R = A R C R D R D R C R (134)5. Effective neutrino mass matrix
One can see by direct computation that plugging Eqs. (131,134) in the seesaw formula (Eq. 121)would result in an effective neutrino mass matrix of the form: M ν = M ν M ν M ν (1 + χ ) M ν M ν M ν M ν (1 + χ ) M ν M ν (1 + ξ ) (135)where ( Y = A, B, C, D, E ) χ = χ ( α, β, Y D , Y R ) , ξ = ξ ( β, Y D , Y R ) : β = 0 ⇒ ξ = 0 (136)Thus, in general, we do not get the desired C ξ = 0.However, for some choices of the Yukawa couplings satisfying E = 0 we get this form (see Eq.132), with χ , as α , is a small parameter for moderate values of Yukawa couplings. S × Z -flavor symmetry In order to get a realization of the C S × Z .1. Matter content and symmetry transformations
The matter spectrum consists of three SM-like Higgs doublets ( φ i , i = 1 , ,
3) responsible for thecharged lepton masses, and of four Higgs doublets ( φ ′ j , j = 1 , , ,
4) giving rise when acquiring avev to Dirac neutrino mass matrix, and, as before, of left doublets ( L i , i = 1 , , l cj , j = 1 , ,
3) and RH neutrinos ( ν Rj , j = 1 , , k , k = 1 ,
2) related to Majorana neutrino mass matrix. We denote the octic root of theunity by w = e iπ . The fields transform under the flavor symmetry as follows. L S −→ SL, l c S −→ l c , φ S −→ Sφ, (137) ν R S −→ Sν R , φ ′ S −→ diag(1 , , , − φ ′ , ∆ S −→ ∆ (138) L Z −→ diag(1 , − , − L, l c Z −→ diag(1 , , − l c , φ Z −→ diag(1 , − , − φ, (139) ν R Z −→ diag( w, w , w ) ν R , φ ′ Z −→ diag( w, w , w , w ) φ ′ , ∆ Z −→ diag( w , w )∆ (140)Note here that we have the following transformation rule for ˜ φ ′ ≡ iσ φ ′∗ :˜ φ ′ S −→ diag(1 , , , −
1) ˜ φ ′ , ˜ φ ′ Z −→ diag( w , w , w, w ) ˜ φ ′ (141)38. Charged lepton mass matrix-flavor basis
As in the previous case of S × Z × Z ′ -flavor symmetry, the charged lepton mass Lagrangian is givenagain by Eq. (98). Since the transformations of the involved fields ( L, l c , φ ) are identical under S in both flavor symmetry groups and are equally the same under Z (in S × Z ) compared to Z (in S × Z × Z ′ ), we end up, assuming again a hierarchy in the Higgs φ ’s fields vevs ( v ≫ v , v ), witha charged lepton mass matrix of the form (Eq. 107) adjustable to be approximately in the flavorbasis. Note also here that no terms of the form f k ′ ij L i φ ′ k l cj can exist since we have: L i l cj Z ∼ − − − − − Eq. = ⇒ ∄ i, j, k : L i φ ′ k l cj = Z ( L i φ ′ k l cj ) (142)3. Dirac neutrino mass matrix
The Lagrangian responsible for the neutrino mass matrix is again given by Eq. (128). By meansof Eqs. (137,138, 139 , 140 and 141) we have: S t g ( k =1 , S = g ( k =1 , , , S t g ( k =4) S = − g ( k =4) , L i ν Rj Z ∼ w w w w w w w w w (143)where, as before, g ( k ) is the matrix whose ( i, j ) th -entry is the Yukawa coupling g kij . Then, Eqs. (77,78 and 141 and 143) impose the following forms on the Yukawa coupling matrices: g (1) = A , g (2) = B B , g (3) = C D D C , g (4) = B − B (144)When the Higgs fields ( φ ′ i ) get vevs ( v ′ i , i = 1 , , , M D = Σ k =4 k =1 v ′ k g ( k ) = A D B D B D (1 + α )0 C D D D D D C D (145)with α = − v ′ B v ′ B + v ′ B (146)If the vevs satisfy v ′ ≪ v ′ and the Yukawa couplings are of the same order then we get a perturbativeparameter α ≪ Majorana neutrino mass matrix
The mass term is generated from the Lagrangian L R = h kij ∆ k ν Ri ν Rj (147)Under Z we have the bilinear: ν Ri ν Rj Z ∼ w w w w w w w w w Eq. = ⇒L R = h ∆ ν R ν R + h ∆ ν R ν R + h ∆ ν R ν R + h ∆ ν R ν R + h ∆ ν R ν R (148)If we call h ( k ) the matrix whose ( i, j ) th -entry is the coupling h kij then we have (the cross sign denotea non-vanishing entry): h (1) = × , h (2) = × × × × (149)39hen Eq. (138) leads to: S t h ( k ) S = h ( k ) , Eqs. , = ⇒ h (1) = a R , h (2) = c R d R d R c R (150)Thus when the Higgs singlets ∆ acquires vevs ( δ , δ ) we get the Majorana neutrino mass matrix: M R = X k =1 δ k h ( k ) = A R C R D R D R C R (151)5. Effective neutrino mass matrix
By direct computation, plugging Eqs. (145,151) into the seesaw formula (Eq. 121) results in aneffective neutrino mass matrix of the desired C M ν = M ν M ν M ν (1 + χ ) M ν M ν M ν M ν (1 + χ ) M ν M ν (152)where the perturbation parameter χ is given by χ = α ( C D − D D )( C R + D R )(1 + α )( C R D D − D R C D ) + C R C D − D R D D (153)Before ending this section, we would mention that introducing multiple Higgs doublets as we didin our constructions might display flavor-changing neutral currents. However, the effects are calculablein the models and in principle one can adjust the Yukawa couplings so that processes like µ → eγ aresuppressed [42]. Moreover, and as was discussed in the introduction, the RG running effects are expectedto be small when multiple Higgs doublets are present, so that not to spoil the predictions of the symmetryat low scale.
10 Summary and Discussion
We have carried out a thorough phenomenological analysis for the patterns of the neutrino mass matrixmeeting the µ – τ symmetry. We found that exact symmetry leads to a totally degenerate spectrum andso is excluded on phenomenological grounds.We thus introduced and in a minimal way perturbations such that the neutrino mass matrix satisfiesan approximate µ – τ symmetry. We got four such patterns and carried out a complete phenomenologicalanalysis of them. We found that all these ‘deformed’ patterns can accommodate the current data withoutneed to adjust the input parameters. However, no singular such patterns could meet the experimentalconstraints.All the four patterns can produce all types of hierarchy and all have complex entries able to showCP-violation effects. The mixing angle θ x can cover all its admissible range in all four patterns. As to theangle θ y , it is unconstrained in the patterns C3 except that it should not equal the value 45 , whereas itis restricted to be around 45 , without taking this value, in the C1 pattern for the normal and invertedhierarchies, and around 36 or 52 in the C4 pattern of normal hierarchy type. Again, θ y can not takethe value 45 in the C2 pattern of normal or inverted hierarchy types, where it is just mildly constrainedin the normal type to be around 45 . However, for this latter pattern C2, the mixing angle θ z can notbe larger than 10 . Actually, there is a narrow interval ]4 , . [ for θ z in the C4 pattern of normal type,whereas this mixing angle is bounded by 8 in the inverted type.The phases are not constrained in the C3 or C4 patterns, except that in the C4 pattern of normaltype the Dirac phase δ can not be in the interval ]160 , [ and the Majorana phase ρ ( mod π ) cannot belong to ] − , [. As to the C1 pattern of normal type, the phases σ, ρ ( mod π ) can not takevalues in the interval ] − , [ around the origin, whereas the Dirac phase δ in all hierarchy types is40xcluded from a narrow band ]177 , . [ around π . For the C2 pattern, the phase ρ is excluded fromthe interval ]94 , [ in the degenerate case, and from broader intervals in the normal (]90 , [) andinverted (]48 , [) types. The phase σ ( mod π ) is bound not to be around zero in the normal andinverted types, whereas the Dirac phase δ in all hierarchy types is excluded from narrow bands aroundzero (] − , [) and around π (]178 , [).There exist linear correlations between δ, ρ, σ for the patterns C1 and C3 in all types of hierarchy,and a linear correlation between < m ee > and the LNM in the degenerate type for these two patterns.The strength of the hierarchies is characterized by the ratio m , and the normal type hierarchy isusually mild taking values of order 1 in all patterns. However, the inverted hierarchy type in the patternsC1 and C3 can be very acute taking values of order O (10 ).All these features might help in distinguishing between the independent patterns. For example, if bymeasuring the mass ratios we find a very pronounced hierarchy, then we know that we have either C1 orC3 pattern, of an inverted hierarchy type. Consequently, if by measuring the angle θ y we find a value farfrom 45 then we know we have a C3 pattern. Also if δ gives a value around π then again we have a C3pattern. On the other hand, if by measuring the masses we get a mild hierarchy then we do not actuallyhave enough signatures to determine the pattern. Rather, we have exclusion rules which help to drop asmuch patterns as possible. For example, if ρ ( mod π ) ∈ ] − , [ or θ z > or θ y = 36 , then wecan drop the C4 pattern of normal type, whereas if θ z > we exclude the C4 of inverted type possibility.If | ρ ( mod π ) | < then no C1 pattern of normal type, while if ρ ∈ ]94 , [ then we drop the possibilityof a C2 pattern. Also if θ z ≥ then we conclude that we do not have a C2 pattern of normal type.Moreover, the knowledge of all the phase angles and other mass parameters jointly and referring to the‘narrow’ bands of the correlation plots can help in deciding which texture does fit the data.We note finally that the deformation parameter | χ | can cover all its ‘perturbative’ range ( ≤ | χ | ≥ µ − τ symmetry augmented by new matterfields and abelian symmetries at the Lagrangian level, and we have presented some concrete examplesusing both types I and II of seesaw mechanism.Our analysis follows a bottom-up approach and, in view of the full parameter space we adopted for theobservables, can be considered as new. In particular, it shows in a very transparent way the correlationbetween the perturbation χ and the non-vanishing θ z . We can summarize the mainly new results in ourwork as follows. First, we presented the complete analytical expressions (full or expanded) for all theobservables and in all patterns. Second, we raised the question of convergence of the expansion series (Eq.53) and analyzed it. Third, we presented an exhaustive analysis plotting all the possible correlations.Fourth, we disentangled the effects of the two perturbation parameters and presented detailed theoreticalrealizations of the resulting perturbed patterns. Fifth, we treated also the case of singular neutrino massmatrix. Sixth, we reached different conclusions compared to some other works with far more restrictedparameter space. Acknowledgements
Part of the work was done within the associate scheme and short visits program of ICTP.N.C. acknowledges funding provided by the Alexander von Humboldt Foundation.
References [1] Y. Fukuda et al. , Phys. Lett. B , 33 (1998); Phys. Rev. Lett. , 1562 (1998). For a review, see:C.K. Jung, C. McGrew, T. Kajita, and T. Mann, Ann. Rev. Nucl. Part. Sci. , 451 (2001).[2] SNO Collaboration, Q.R. Ahmad et al. , Phys. Rev. Lett. , 011301 (2002); Phys. Rev. Lett. ,011302 (2002).[3] KamLAND Collaboration, K. Eguchi et al. , Phys. Rev. Lett. , 021802 (2003).414] K2K Collaboration, M.H. Ahn et al. , Phys. Rev. Lett. , 041801 (2003).[5] CHOOZ Collaboration, M. Apollonio et al. , Phys. Lett. B , 397 (1998); Palo Verde Collaboration,F. Boehm et al. , Phys. Rev. Lett. , 3764 (2000).[6] T2K Collaboration, K. Abe et al. , Phys. Rev. Lett , 041801 (2011).[7] MINOs Collaboration, P. Adamson et al. , Phys. Rev. Lett , 181802 (2011).[8] DOUBLE-CHOOZ Collaboration, Y. Abe et al. , Phys. Rev. Lett , 131801 (2012).[9] F. P. An et al. , [DAYA-BAY Collaboration], Phys. Rev. Lett , 171803 (2012).[10] J. K. Ahn et al. , [RENO Collaboration], Phys. Rev. Lett. , 191802 (2012).[11] See e.g., M. Hirsch, D. Meloni, S. Morisi, S. Pastor, E. Peinado, J. W. F. Valle, A. Adulpravitchaiand D. Aristizabal Sierra et al. , arXiv:1201.5525 [hep-ph].[12] R. N. Mohapatra and S. Nussinov, Phys. Rev. D , 013002 (1999); C. S. Lam, Phys. Lett. B ,214 (2001); P. F. Harrison and W. G. Scott, Phys. Lett. B , 219 (2002). T. Kitabayashi andM. Yasue, Phys. Rev. D , 015006 (2003).[13] W. Grimus and L. Lavoura, JHEP , 045 (2001); W. Grimus and L. Lavoura, Phys. Lett. B ,189 (2003); Y. Koide, Phys. Rev. D , 093001 (2004); R. N. Mohapatra, JHEP , 027 (2004).[14] P. F. Harrison, D. H. Perkins and W. G. Scott, Phys. Lett. B 530 , 167 (2002).[15] V. D. Barger, S. Pakvasa, T. J. Weiler and K. Whisnant, Phys. Lett.
B 437 , 107 (1998); A. J. Baltz,A. S. Goldhaber and M. Goldhaber, Phys. Rev. Lett. , 5730 (1998).[16] C. H. Albright, A. Dueck and W. Rodejohann, Eur. Phys. J. C , 1099 (2010).[17] Y. Kajiyama, M. Raidal and A. Strumia, Phys. Rev. D , 117301 (2007); L. L. Everett and A. J.Stuart, Phys. Rev. D — , 085005 (2009).[18] E. Ma and M. Raidal, Phys. Rev. Lett. , 011802 (2001); W. Grimus and L. Lavoura, JHEP ,045 (2001); E. Ma, Phys. Rev. D , 117301 (2002); R. N. Mohapatra and S. Nasri, Phys. Rev.D , 033001 (2005); R. N. Mohapatra, S. Nasri and H. -B. Yu, Phys. Lett. B , 231 (2005);S. Nasri, Int. J. Mod. Phys. A , 6258 (2005); T. Kitabayashi and M. Yasue, Phys. Lett. B ,133 (2005); S. Choubey and W. Rodejohann, Eur. Phys. J. C , 259 (2005); R. N. Mohapatra,S. Nasri and H. B. Yu, Phys. Lett. B , 114 (2006); R. N. Mohapatra, S. Nasri and H. B. Yu,Phys. Lett. B , 318 (2006); Z. -z. Xing, H. Zhang and S. Zhou, Phys. Lett. B , 189 (2006);T. Ota and W. Rodejohann, Phys. Lett. B , 322 (2006); Y. H. Ahn, S. K. Kang, C. S. Kim andJ. Lee, Phys. Rev. D , 093005 (2006); I. Aizawa and M. Yasue, Phys. Rev. D , 015002 (2006);K. Fuki and M. Yasue, Phys. Rev. D , 055014 (2006); K. Fuki and M. Yasue, R. Jora, S. Nasriand J. Schechter, Int. J. Mod. Phys. A , 5875 (2006), Nucl. Phys. B , 31 (2007); B. Adhikary,A. Ghosal and P. Roy, JHEP (2009) 040; Z. z. Xing and Y. L. Zhou, Phys. Lett. B , 584(2010); R. Jora, J. Schechter and M. Naeem Shahid, Phys. Rev. D , 093007 (2009) [Erratum-ibid.D , 079902 (2010)]; S. -F. Ge, H. -J. He and F. -R. Yin, JCAP , 017 (2010); I. de MedeirosVarzielas, R. Gonz´alez Felipe and H. Serodio, Phys. Rev. D , 033007 (2011); H. -J. He and F. -R. Yin, Phys. Rev. D , 033009 (2011); Y. H. Ahn, H. Y. Cheng, S. Oh, Phys. Lett. B , 203(2012); H. -J. He and X. -J. Xu, Phys. Rev. D , 111301 (2012); S. Gupta, A. S. Joshipura and K.M. Patel, JHEP , 035 (2013); B. Adhikary, M. Chakraborty and A. Ghosal, JHEP , 043(2013); C. Hamzaoui, S. Nasri and M. Toharia: arXiv:1311.2188 [hep-ph] (2013).[19] M. Abbas and A. Y. Smirnov, Phys. Rev. D , 013003 (2013).[23] S. Gupta, A. S. Joshipura, and K. M. Patel, JHEP , 035 (2013), arXiv:1301.7130 [hep-ph].[24] B. Adhikary, A. Ghosal and P. Roy, Int. J. Mod. Phys. A 28 (2013) 24, 1350118[25] A. S. Joshipura, Eur. Phys. J.
C 53 (2008) 77,[26] R.N.Mohapatra and W. Rodejohann, Phys. Rev. D 72, 053001 (2005) [hep-ph/0507312]; N. Habaand W. Rodejohann, Phys. Rev. D 74, 017701 (2006) [hep-ph/0603206]; S. Luo and Z.-Z. Xing, Phys.Lett. B 646, 242 (2007) [hep-ph/0611360]; Y. Koide and H. Nishiura, Int. J. Mod. Phys. A 25, 3661(2010) [arXiv:0911.2279].[27] W. Grimus and L. Lavoura, Eur. Phys. J. C 39, 219 (2005) [hep-ph/0409231][28] A. Dighe, S. Goswami and P. Roy, Phys. Rev. D , 096005 (2007); S. Luo and Z. -z. Xing, Phys.Rev. D , 073003 (2012).[29] W. Grimus and L. Lavoura, Fortsch. Phys. , 535 (2013)[30] Z.Z. Xing; Phys. Lett. B (2002), 159-166.[31] G. L. Fogli et al. , Phys. Rev. D , 053007 (2011).[32] E. Lashin and N. Chamoun, Phys. Rev. D , (2012) 113011.[33] C. Jarlskog, Phys. Rev. Lett. , 1039 (1985); Z. Phys. C 29, 491 (1985); Phys. Rev. D 35, 1685(1987).[34] G. L. Fogli, E. Lisi, A. Marrone, D. Montanino, A. Palazzo and A. M. Rotunno, Phys. Rev. D ,013012 (2012).[35] T. Schwetz, M. Tortola and J. W. F. Valle, New J. Phys. , 063004 (2011); T. Schwetz, M. Tortolaand J. W. F. Valle, New J. Phys. , 109401 (2011); D. V. Forero, M. Tortola and J. W. F. Valle,Phys. Rev. D , 073012 (2012).[36] M. C. Gonzalez-Garcia, M. Maltoni and J. Salvado, JHEP , 056 (2010); M. C. Gonzalez-Garcia,M. Maltoni, J. Salvado and T. Schwetz, JHEP , 123 (2012).[37] G. L. Fogli et al. , Phys. Rev. D , 033010 (2008).[38] E. Andreotti et al. , Astropart. Phys. , 822 (2011).[39] Z.Z. Xing; Phys. Rev. D , 011301(R) (2008).[40] T. P. Cheng and L. F. Li, Phys. Rev. D , 2860 (1980); R. N. Mohapatra and G. Senjanovic, Phys.Rev. D , 165 (1981).[41] E. I. Lashin, N. Chamoun, E. Malkawi and S. Nasri, Phys. Rev. D , 013002 (2011)[42] C. Hagedorn, J. Kersten and M. Lindner, Phys. Lett. B597