Generalised Geometrical CP Violation in a T ′ Lepton Flavour Model
SSISSA 56/2013/FISIRM3-TH/13-10TTP13-04
Generalised Geometrical CP Violationin a T (cid:48) Lepton Flavour Model
Ivan Girardi a, , Aurora Meroni a,b, , S. T. Petcov a,c, , Martin Spinrath a,d, a SISSA/INFN, Via Bonomea 265, I-34136 Trieste, Italy b Dipartimento di Matematica e Fisica, Universit`a di Roma Tre,Via della Vasca Navale 84, I-00146 Rome, andINFN, Laboratori Nazionali di Frascati, Via E. Fermi 40, I-00044 Frascati, Italy c Kavli IPMU (WPI), University of Tokyo, Tokyo, Japan d Institut f¨ur Theoretische Teilchenphysik, Karlsruhe Institute of Technology,Engesserstraße 7, D-76131 Karlsruhe, Germany
Abstract
We analyse the interplay of generalised CP transformations and the non-Abelian discretegroup T (cid:48) and use the semi-direct product G f = T (cid:48) (cid:111) H CP , as family symmetry actingin the lepton sector. The family symmetry is shown to be spontaneously broken in ageometrical manner. In the resulting flavour model, naturally small Majorana neutrinomasses for the light active neutrinos are obtained through the type I see-saw mechanism.The known masses of the charged leptons, lepton mixing angles and the two neutrino masssquared differences are reproduced by the model with a good accuracy. The model allowsfor two neutrino mass spectra with normal ordering (NO) and one with inverted ordering(IO). For each of the three spectra the absolute scale of neutrino masses is predicted withrelatively small uncertainty. The value of the Dirac CP violation (CPV) phase δ in thelepton mixing matrix is predicted to be δ ∼ = π/ π/
2. Thus, the CP violating effects inneutrino oscillations are predicted to be maximal (given the values of the neutrino mixingangles) and experimentally observable. We present also predictions for the sum of theneutrino masses, for the Majorana CPV phases and for the effective Majorana mass inneutrinoless double beta decay. The predictions of the model can be tested in a varietyof ongoing and future planned neutrino experiments. E-mail: [email protected] E-mail: [email protected] Also at: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia,Bulgaria. E-mail: [email protected] a r X i v : . [ h e p - ph ] M a y Introduction
Understanding the origin of the patterns of neutrino masses and mixing, emerging fromthe neutrino oscillation, H β − decay, cosmological, etc. data is one of the most challengingproblems in neutrino physics. It is part of the more general fundamental problem in particlephysics of understanding the origins of flavour, i.e., of the patterns of the quark, chargedlepton and neutrino masses and of the quark and lepton mixing.At present we have compelling evidence for the existence of mixing of three light massiveneutrinos ν i , i = 1 , ,
3, in the weak charged lepton current (see, e.g., [1]). The masses m i ofthe three light neutrinos ν i do not exceed approximately 1 eV, m i ∼ < × U PMNS . In the widely used standard parametrisation [1], U PMNS is expressed in terms of the solar, atmospheric and reactor neutrino mixing angles θ , θ and θ , respectively, and one Dirac - δ , and two Majorana [2] - β and β , CP violation phases: U PMNS ≡ U = V ( θ , θ , θ , δ ) Q ( β , β ) , (1.1)where V = c s − s c c s e − i δ − s e i δ c c s − s c
00 0 1 , (1.2)and we have used the standard notation c ij ≡ cos θ ij , s ij ≡ sin θ ij , 0 ≤ θ ij ≤ π/ , ≤ δ ≤ π . The matrix Q contains the two physical Majorana CP violation (CPV) phases: Q = Diag(e − i β / , e − i β / , . (1.3)The parametrization of the phase matrix Q in Eq. (1.3) differs from the standard one [1] Q = Diag(1 , e i α / , e i α / ). Obviously, one has α = ( β − β ) and α = β . In the caseof the seesaw mechanism of neutrino mass generation, which we are going to employ, theMajorana phases β and β (or α and α ) vary in the interval [3] 0 ≤ β , ≤ π . If CPinvariance holds, we have δ = 0 , π, π , and [5] β = k ( (cid:48) ) π , k ( (cid:48) ) = 0 , , , , θ , θ and θ and thetwo neutrino mass squared differences ∆ m and ∆ m (or ∆ m ), which drive the observedoscillations involving the three active flavour neutrinos and antineutrinos, ν l and ¯ ν l , l = e, µ, τ ,with a relatively high precision [6, 7]. In Table 1 we give the values of the 3-flavour neutrinooscillation parameters as determined in the global analysis performed in [6].An inspection of Table 1 shows that although θ (cid:54) = 0, θ (cid:54) = π/ θ (cid:54) = π/
4, thedeviations from these values are small, in fact we have sin θ ∼ = 0 . (cid:28) π/ − θ ∼ = 0 . π/ − θ ∼ = 0 .
20, where we have used the relevant best fit values in Table 1. The value of θ and the magnitude of deviations of θ and θ from π/ θ , θ and θ might originate from certain “symmetry” values which undergo relativelysmall (perturbative) corrections as a result of the corresponding symmetry breaking. Thisidea was and continues to be widely explored in attempts to understand the pattern of mixing The interval beyond 2 π , 2 π ≤ β , ≤ π , is relevant, e.g., in the calculations of the baryon asymmetrywithin the leptogenesis scenario [3], in the calculation of the neutrinoless double beta decay effective Majoranamass in the TeV scale version of the type I seesaw model of neutrino mass generation [4], etc. ± σ ) 3 σ ∆ m [10 − eV ] 7.54 +0 . − . | ∆ m | (NO) [10 − eV ] 2.47 +0 . − . | ∆ m | (IO) [10 − eV ] 2.46 +0 . − . θ (NO or IO) 0.307 +0 . − . θ (NO) 0.386 +0 . − . +0 . − . θ (NO) 0.0241 +0 . − . +0 . − . Table 1:
The best-fit values and 3 σ allowed ranges of the 3-flavour neutrino oscillationparameters derived from a global fit of the current neutrino oscillation data (from [6]). Iftwo values are given, the upper one corresponds to neutrino mass spectrum with normalhierarchy (NO) and the lower one - to spectrum with inverted hierarchy (IO) (see text forfurther details). in the lepton sector (see, e.g., [8–16]). Given the fact that the PMNS matrix is a product oftwo unitary matrices, U = U † e U ν , (1.4)where U e and U ν result respectively from the diagonalisation of the charged lepton and neu-trino mass matrices, it is usually assumed that U ν has a specific form dictated by a symmetrywhich fixes the values of the three mixing angles in U ν that would differ, in general, by per-turbative corrections from those measured in the PMNS matrix, while U e (and symmetrybreaking effects that we assume to be subleading) provide the requisite corrections. A varietyof potential “symmetry” forms of U ν , have been explored in the literature on the subject(see, e.g., [17]). Many of the phenomenologically acceptable “symmetry” forms of U ν , as thetribimaximal (TBM) [18] and bimaximal (BM) [19,20] mixing, can be obtained using discreteflavour symmetries (see, e.g., the reviews [21–23] and the references quoted there in). Discretesymmetries combined with GUT symmetries have been used also in attempts to constructrealistic unified models of flavour (see, e.g., [21]).In the present article we will exploit the approximate flavour symmetry based on the group T (cid:48) , which is the double covering of the better known group A (see, e.g., [23]), with the aim toexplain the observed pattern of lepton (neutrino) mixing and to obtain predictions for the CPviolating phases in the PMNS matrix and possibly for the absolute neutrino mass scale andthe type of the neutrino mass spectrum. Flavour models based on the discrete symmetry T (cid:48) have been proposed by a number of authors [24–26] before the angle θ was determined witha high precision in the Day Bay [27] and RENO [28] experiments (see also [29–31]). All thesemodels predicted values of θ which turned out to be much smaller than the experimentallydetermined value.In [25, 26], in particular, an attempt was made to construct a realistic unified supersym-metric model of flavour, based on the group SU (5) × T (cid:48) , which describes the quark masses,the quark mixing and CP violation in the quark sector, the charged lepton masses and theknown mixing angles in the lepton sector, and predicts the angle θ and possibly the neutrinomasses and the type of the neutrino mass spectrum as well as the values of the CPV phasesin the PMNS matrix. The light neutrino masses are generated in the model by the type I2eesaw mechanism [32] and are naturally small. It was suggested in [25, 26] that the complexClebsch-Gordan (CG) coefficients of T (cid:48) [33] might be a source of CP violation and hence thatthe CP symmetry might be broken geometrically [34] in models with approximate T (cid:48) symme-try. Since the phases of the CG coefficients of T (cid:48) are fixed, this leads to specific predictionsfor the CPV phases in the quark and lepton mixing matrices. Apart from the incorrect pre-diction for θ , the authors of [25,26] did not address the problem of vacuum alignment of theflavon vevs, i.e., of demonstrating that the flavon vevs, needed for the correct description ofthe quark and lepton masses and of the the mixing in both the quark and lepton sectors, canbe derived from a flavon potential and that the latter does not lead to additional arbitraryflavon vev phases which would destroy the predictivity, e.g., of the leptonic CP violation ofthe model.A SUSY SU (5) × T (cid:48) model of flavour, which reproduces the correct value of the leptonmixing angle θ was proposed in [35], where the problem of vacuum alignment of the flavonvevs was also successfully addressed . In [35] it was assumed that the CP violation in thequark and lepton sectors originates from the complexity of the CG coefficients of T (cid:48) . Thiswas possible by fixing the phases of the flavon vevs using the method of the so-called “discretevacuum alignment”, which was advocated in [37] and used in a variety of other models withdiscrete flavour symmetries [38]. The value of the angle θ was generated by charged leptoncorrections to the TBM mixing using non-standard GUT relations [13, 39, 40].After the publication of [35] it was realised in [41,42] that the requirement of CP invariancein the context of theories with discrete flavour symmetries, imposed before the breakingof the discrete symmetry leading to CP nonconservation and generation of the masses ofthe matter fields of the theory, requires the introduction of the so-called “generalised CPtransformations” of the matter fields charged under the discrete symmetry. The explicit formof the generalised CP transformations is dictated by the type of the discrete symmetry. It wasnoticed in [41], in particular, that due to a subtle intimate relation between CP symmetry andcertain discrete family symmetries, like the one associated with the group T (cid:48) , it can happenthat the CP symmetry does not enforce the Yukawa type couplings, which generate the matterfield mass matrices after the symmetry breaking, to be real but to have certain discrete phasespredicted by the family symmetry in combination with the generalised CP transformations.In the SU (5) × T (cid:48) model proposed in [35], these phases, in principle, can change or modifycompletely the pattern of CP violation obtained by exploiting the complexity of some of the T (cid:48) CG coefficients.In the present article we address the problem of the relation between the T (cid:48) symmetryand the CP symmetry in models of lepton flavour. After some general remarks about theconnection between the T (cid:48) and CP symmetries in Section 2, we present in Section 3 a fullyconsistent and explicit model of lepton flavour with a T (cid:48) family symmetry and geometricalCP violation. We show that the model reproduces correctly the charged lepton masses, allleptonic mixing angles and neutrino mass squared differences and predicts the values of theleptonic CP violating phases and the neutrino mass spectrum. We show also that this modelindeed exhibits geometrical CP violation. We clarify how the CP symmetry is broken in themodel by using the explicit form of the constructed flavon vacuum alignment sector; withoutthe knowledge of the flavon potential it is impossible to make conclusions about the origin ofCP symmetry breaking in flavour models with T (cid:48) symmetry. In the Appendix we give some A modified version of the model published in [25, 26], which predicts a correct value of the angle θ , wasconstructed in [36], but the authors of [36] left open the issue of the vacuum alignment of the flavon vevs. T (cid:48) and present a “UV completion” of the model, whichis necessary in order to to select correctly certain T (cid:48) contractions in the relevant effectiveoperators. T (cid:48) Symmetry and Generalised CP Transformations
In this Section we would like to clarify the role of a generalised CP transformation combinedwith the non-Abelian discrete symmetry group T (cid:48) . Let G f = T (cid:48) (cid:111) H CP be the symmetrygroup acting in the lepton sector such that both T (cid:48) and H CP act on the lepton flavour space.Motivated by this study we will present in the next section a model where G f is broken suchthat all lepton mixing angles and physical CP phases of the PMNS mixing matrix can bepredicted in terms of two mixing angles and two phases. The breaking of G f will be achievedthrough non zero vacuum expectation values (vevs) of some scalar fields, the so-called flavons. The discrete non-Abelian family symmetry group T (cid:48) is the double covering of the tetrahedralgroup A and its complete description in terms of generators, elements and representationsis given in Appendix A. An interesting feature of this group is the fact that it is the smallestgroup that admits 1-, 2-, and 3-dimensional representations and for which the three repre-sentations can be related by the multiplication rule ⊗ = ⊕ . T (cid:48) has seven differentirreducible representations: the 1- and 3-dimensional representations , (cid:48) , (cid:48)(cid:48) , are notfaithful, i.e., not injective, while the doublet representations , (cid:48) and (cid:48)(cid:48) are faithful. One in-teresting feature of the T (cid:48) group is related to the tensor products involving the 2-dimensionalrepresentation since the CG coefficients are complex.We define now the transformation of a field φ ( x ) under the group T (cid:48) and H CP respectivelyas: φ ( x ) → ρ r ( g ) φ ( x ) , φ ( x ) → X r φ ∗ ( x (cid:48) ) , (2.1)where ρ r ( g ) is an irreducible representation r of the group element g ∈ T (cid:48) , x (cid:48) ≡ ( x , − (cid:126)x )and X r is the unitary matrix representing the generalised CP transformation. In order tointroduce consistently the CP transformation for the family symmetry group T (cid:48) , the matrix X r should satisfy the consistency conditions [41, 42, 44]: X r ρ ∗ r ( g ) X − r = ρ r ( g (cid:48) ) , g, g (cid:48) ∈ T (cid:48) . (2.2)Following the discussion given in [41,42,44] it is important to remark that the consistencycondition corresponds to a similarity transformation between the representation ρ ∗ r and ρ ◦ CP.Since the structure of the group is preserved and an element g ∈ T (cid:48) is always mapped into anelement g (cid:48) ∈ T (cid:48) , this map defines an automorphism of the group. In general g and g (cid:48) mightbelong to different conjugacy classes: in this case the map defines an outer automorphism .It is worth noticing that the matrices X r are defined up to an arbitrary global phase.Indeed, without loss of generality, for each matrix X r , one can define different phases θ r for The only other 24-element group that has representation of the same dimensions is the octahedral group O (which is isomorphic to S ). In this case, however, the product of two doublet reps does not contain atriplet [43]. For details concerning the group of outer and inner automorphisms, Out( G ) and Inn( G ), see [41, 44]. X r up to a group transfor-mation (change of basis): in fact the consistency conditions in Eq. (2.2) are invariant under X r → e i θ r X r and X r → ρ r (˜ g ) X r with ˜ g ∈ T (cid:48) .It proves convenient to use the freedom associated with the arbitrary phases θ r to definethe generalised CP transformation for which the vev alignments of the flavon fields can bechosen to be all real. We will show later on that the phases θ r are not physical and thereforethe results we present are independent from the specific values we assume. In the contextof the T (cid:48) group this choice however helps us to extract a real flavon vev structure whichis a distinctive feature of some models proposed in the literature where the origin of thephysical CP violation arising in the lepton sector is tightly related to the combination of realvevs, complex CGs and eventual phases arising from the requirement of invariance of thesuperpotential under the generalised CP transformation.Before going into details of the computations, let us comment that in the analysis pre-sented in [41] related to the group T (cid:48) , the CP transformations are defined as elements of theouter automorphism group and are derived up to inner automorphisms of T (cid:48) (up to conjugacytransformations). In the present work we will consider instead all the possible transformationsincluding the inner automorphism group and we will discuss all the convenient CP transfor-mations which can be used to clarify the role of a generalised CP symmetry in the context ofthe group T (cid:48) . We give now all the possible equivalent choices of generalised CP transformations for anyirreducible representation of T (cid:48) .The group T (cid:48) is defined by the group generators T and S , then from the consistencyconditions in Eq. (2.2) it is sufficient to require that X r ρ ∗ r ( S ) X − r = ρ r ( ˆ S ) , X r ρ ∗ r ( T ) X − r = ρ r ( ˆ T ) . (2.3)It is easy to show that the CP transformation leaves invariant the order of the elementof the group g meaning that denoting n ( g ) the order of g , we have n ( g ) = n ( g (cid:48) ). Sincethe element S has order four and the element T has order three we have ˆ S ∈ C andˆ T ∈ (cid:48) C [45]. The latter result is derived using the action of CP on the one-dimensionalrepresentations, i.e. ρ , (cid:48) , (cid:48)(cid:48) ( ˆ T ) = ρ ∗ , (cid:48) , (cid:48)(cid:48) ( T ) which can be satisfied only if ˆ T ∈ (cid:48) C .The conjugacy classes 6 C and 4 (cid:48) C contain the group elementsˆ S ∈ C = (cid:8) S, S , T ST , T ST, S T ST , S T ST (cid:9) , ˆ T ∈ (cid:48) C = (cid:8) T , S T ST, S T S, S T (cid:9) . (2.4)We recall that we have the freedom to choose arbitrary phases θ r , so for instance in thecase of X , X (cid:48) and X (cid:48)(cid:48) we are allowed to write the most general CP transformations for thethree inequivalent singlets of T (cid:48) as → e i θ ∗ , (cid:48) → e i θ (cid:48) (cid:48)∗ , (cid:48)(cid:48) → e i θ (cid:48)(cid:48) (cid:48)(cid:48)∗ . (2.5)Differently from the case of the A family symmetry discussed in [44] in which one canshow that the generalised CP transformation can be represented as a group transformation, in This idea was pioneered in [25]. T (cid:48) we will show that this is true only for the singlet and the triplet representations.For the doublets the action of the CP transformation cannot be written as an action of a groupelement (i.e. (cid:64) g ∈ T (cid:48) such that X r = ρ r ( g ) for r = , (cid:48) , (cid:48)(cid:48) ).We give a list of all the possible forms of X r , which can be in general different for eachrepresentation: the CP transformations on the singlets, X , (cid:48) , (cid:48)(cid:48) , are complex phases, asmentioned above while the CP transformations on the doublets, X , (cid:48) , (cid:48)(cid:48) , and the triplets X , are given respectively in Table 2 and 3. We stress that all the possible forms of X r aredefined up to a phase, which can be in general different for each representation. Each CPtransformation we found generates a Z symmetry.The generalised CP transformation H CP , acting on the lepton flavour space is given by,see also [41], u : (cid:40) T → T ,S → S T ST . (2.6)This definition of the CP symmetry is particularly convenient because it acts on the 3- and1- dimensional representations trivially. This particular transformation however is related toany other possible CP transformation by a group transformation.In other words, different choices of CP are related to each other by inner automorphisms ofthe group i.e. the CP transformations listed in Tables 2 and 3 are related to each other througha conjugation with a group element. For example, another possible CP transformation wouldbe v : (cid:40) T → T ,S → S , (2.7)which is related to u via u = conj ( T ) ◦ v . Indeed S v (cid:55)−→ S conj ( T ) (cid:55)−−−−−−→ T S ( T ) − = S T ST ,T v (cid:55)−→ T conj ( T ) (cid:55)−−−−−−→ T T ( T ) − = T . (2.8)Without loss of generality we choose as CP transformation the one defined throughEq. (2.6) and from Eq. (2.1) using the results of Table 2 and Table 3 we can write therepresentation of the CP transformation acting on the fields as → e i θ ∗ , (cid:48) → e i θ (cid:48) (cid:48)∗ , (cid:48)(cid:48) → e i θ (cid:48)(cid:48) (cid:48)(cid:48)∗ , → e i θ ∗ , (2.9) → e i θ (cid:18) ω ¯ p
00 ¯ ωp (cid:19) ∗ , (cid:48) → e i θ (cid:48) (cid:18) ω ¯ p
00 ¯ ωp (cid:19) (cid:48)∗ , (cid:48)(cid:48) → e i θ (cid:48)(cid:48) (cid:18) ω ¯ p
00 ¯ ωp (cid:19) (cid:48)(cid:48)∗ , where ω = e i 2 π/ , p = e i π/ and ω ¯ p = e i π/ . Notice that we did not specify the values ofthe phases θ r . Further we can check that the CP symmetry transformation chosen generatesa Z symmetry group. Indeed it is easy to show that u = E , therefore the multiplicationtable of the group H CP = { E, u } is obviously equal to the multiplication table of a Z group,from which we can write H CP ∼ = Z .Since we want to have real flavon vevs – following the setup given in [35] – it turnsout to be convenient to select the CP transformations with θ = θ (cid:48) = θ (cid:48)(cid:48) = θ = 0 and θ (cid:48)(cid:48) = − θ (cid:48) = π/ . With this choice the phases of the couplings of renormalisable operators Since in our model later on we do not have fields in a representation of T (cid:48) the phase θ is irrelevant inour further discussion and we do not fix its value. A possible convenient choice might be θ = 0 which makesthe mass term of a two-dimensional representation real. , g (cid:48) X = ρ ( g ) = ρ ( g (cid:48) ) T → ˆ T S → ˆ ST , S T ω
00 0 ω T S T , S T ω
00 0 ω T S T ST E , S T S T STT S , S T S − / / ω / ω / − / ω / ω / / ω − / ω S T S ST ST , S T ST − / / / / − / / / / − / S T S T STS T ST , T ST − / / ω / ω / − / ω / ω / / ω − / ω S T S S T ST ST , S T − / / ω / ω / ω − / ω / / ω / − / ω S T ST S S T , ST − / / / / ω − / ω / ω / ω / ω − / ω S T ST T ST S , S − / / ω / ω / ω − / / ω / ω / ω − / S T ST T STS T S , ST S ω + 4 ω + 1 − ω − ω + 4 − ω − ω + 4 − ω + 4 ω − ω + ω + 4 − ω + 4 ω − ω − ω − ω − ω − ω + 4 ω + 4 S T SS T S , T S − / / ω / ω / ω − / ω / / ω / − / ω S T T ST T ST , S T ST − / / ω / ω / ω − / / ω / ω / ω − / S T S T ST Table 2:
The generalised CP transformation for the triplet representation of the group T (cid:48) derived using the consistency conditions. We have defined ω = e i 2 π/ . , X (cid:48) , X (cid:48)(cid:48) T → ˆ T S → ˆ S X , X (cid:48) , X (cid:48)(cid:48) T → ˆ T S → ˆ S (cid:18) ¯ p p (cid:19) T S (cid:113) (cid:18) ¯ p / √ p / √ (cid:19) S T ST S (cid:18) ¯ p p (cid:19) T S T ST √ (cid:18) p √ q √ q ¯ p (cid:19) S T ST T ST (cid:18) e i π/
00 e − iπ/ (cid:19) T S T ST (cid:18) e i π/ √ q √ q e − iπ/ (cid:19) S T ST T ST (cid:18) p √ q √ q ¯ p (cid:19) S T S S √ (cid:18) p √ q √ q ¯ p (cid:19) S T S √ (cid:18) e − iπ/ − i √ − i √ i π/ (cid:19) S T S T ST √ (cid:18) ¯ p √ √ p (cid:19) S T T ST (cid:113) (cid:18) q / √ pp ¯ q/ √ (cid:19) S T S S T ST (cid:113) (cid:18) e i 5 π/ / √ − iπ/ e − i π/ e − i π/ / √ (cid:19) S T S T ST Table 3:
The generalised CP transformation for the doublet representation of the group T (cid:48) derived using the consistency conditions. We have defined ω = e i 2 π/ , p = e i π/ , q = e i π/ and note that ω ¯ p = e i π/ . is fixed up to a sign by the CP symmetry. In fact, supposing one has a renormalisable operatorof the form λ O = λ ( A × B × C ) where λ is the coupling constant and A , B , C represent thefields, then the generalised CP phase of the operator is defined as β ≡ CP[ O ] / O ∗ . The phaseof λ is hence given by the equation λ = βλ ∗ which is solved by (cid:40) arg( λ ) = arg( β ) / β ) / − π if arg( β ) > , arg( λ ) = arg( β ) / β ) / π if arg( β ) ≤ . (2.10)In Table 4 we give a list of the phases of λ for all renormalisable operators without fixing the θ r and with the above choice for θ r in Table 5.Under the choice we made, the CP transformation acting on the fields using the abovechoice for the θ r reads → ∗ , (cid:48) → (cid:48)∗ , (cid:48)(cid:48) → (cid:48)(cid:48)∗ , → ∗ , (cid:48) → (cid:18) − i (cid:19) (cid:48)∗ , (cid:48)(cid:48) → (cid:18) i 00 1 (cid:19) (cid:48)(cid:48)∗ , (2.11)where we have again skipped the representation because we will not need it later on. In this section we try to clarify the origin of the phases entering the Lagrangian after T (cid:48) breaking which are then responsible for physical CP violation. We will use the choice of the θ r discussed in the previous section, i.e. θ = θ (cid:48) = θ (cid:48)(cid:48) = θ = 0 and θ (cid:48)(cid:48) = − θ (cid:48) = π/ ψ i to an operator O r containing matter fields and transforming in the representation r of T (cid:48) .8 O = λ ( A × B × C ) β ≡ CP[ O ] / O ∗ ( × ) × e i( θ +2 θ ) ( (cid:48) × (cid:48)(cid:48) ) × e i( θ + θ (cid:48) + θ (cid:48)(cid:48) ) ( (cid:48) × (cid:48) ) (cid:48)(cid:48) × (cid:48) e i( θ (cid:48) +2 θ (cid:48) ) ( × (cid:48)(cid:48) ) (cid:48)(cid:48) × (cid:48) e i( θ (cid:48) + θ + θ (cid:48)(cid:48) ) ( (cid:48)(cid:48) × (cid:48)(cid:48) ) (cid:48) × (cid:48)(cid:48) e i( θ (cid:48)(cid:48) +2 θ (cid:48)(cid:48) ) ( × (cid:48) ) (cid:48) × (cid:48)(cid:48) e i( θ (cid:48)(cid:48) + θ + θ (cid:48) ) [( × ) × ] − i e i(2 θ + θ ) [( (cid:48) × (cid:48)(cid:48) ) × ] − i e i( θ (cid:48) + θ (cid:48)(cid:48) + θ ) [( (cid:48) × (cid:48) ) × ] − i e i(2 θ (cid:48) + θ ) [( × (cid:48)(cid:48) ) × ] − i e i( θ + θ (cid:48)(cid:48) + θ ) [( (cid:48)(cid:48) × (cid:48)(cid:48) ) × ] − i e i(2 θ (cid:48)(cid:48) + θ ) [( × (cid:48) ) × ] − i e i( θ + θ (cid:48) + θ ) [( × ) × ] e i( θ +2 θ ) [( × ) (cid:48) × (cid:48)(cid:48) ] e i( θ (cid:48)(cid:48) +2 θ ) [( × ) (cid:48)(cid:48) × (cid:48) ] e i( θ (cid:48) +2 θ ) [( × ) S , A × ] e iθ [( × ) × ] e i(3 θ ) [( (cid:48) × (cid:48) ) (cid:48)(cid:48) × (cid:48) ] e i(3 θ (cid:48) ) [( (cid:48)(cid:48) × (cid:48)(cid:48) ) (cid:48) × (cid:48)(cid:48) ] e i(3 θ (cid:48)(cid:48) ) [( (cid:48) × (cid:48)(cid:48) ) × ] e i( θ + θ (cid:48) + θ (cid:48)(cid:48) ) Table 4:
List of operators, which form a singlet and constraints on the phase of the coupling λ from the invariance under the generalised CP transformations Eq. (2.9). O = λ ( A × B × C ) β ≡ CP[ O ] / O ∗ arg( λ )( (cid:48) × (cid:48)(cid:48) ) × π ( (cid:48) × (cid:48) ) (cid:48)(cid:48) × (cid:48) − i ± ¯Ω( (cid:48)(cid:48) × (cid:48)(cid:48) ) (cid:48) × (cid:48)(cid:48) i ± Ω[ (cid:48) × ( (cid:48)(cid:48) × ) (cid:48)(cid:48) ] π [ (cid:48)(cid:48) × ( (cid:48) × ) (cid:48) ] π [( (cid:48) × (cid:48)(cid:48) ) × ] − i ± ¯Ω[( (cid:48) × (cid:48) ) × ] − ± π/ (cid:48)(cid:48) × (cid:48)(cid:48) ) × ] π [( × ) × ] π [( × ) (cid:48) × (cid:48)(cid:48) ] π [( × ) (cid:48)(cid:48) × (cid:48) ] π [( × ) S , A × ] π [( × ) × ] π [( (cid:48) × (cid:48) ) (cid:48)(cid:48) × (cid:48) ] π [( (cid:48)(cid:48) × (cid:48)(cid:48) ) (cid:48) × (cid:48)(cid:48) ] π [( (cid:48) × (cid:48)(cid:48) ) × ] π Table 5:
List of some operators, for which it is possible construct a singlet, and constraintson the phase of the coupling λ from invariance under the generalised CP transformationsEq. (2.9) with the choice θ = θ (cid:48) = θ (cid:48)(cid:48) = θ = 0 and θ (cid:48)(cid:48) = − θ (cid:48) = π/
4. We have omitted thetransformations which include the representation of T (cid:48) because they do not appear in ourmodel, but they can be read off from Table 4 after choosing θ . W ⊃ O r Φ ¯ r , where Φ ¯ r = (cid:32)(cid:89) i ψ i (cid:33) ¯ r . (2.12)In order to obtain a singlet, the flavons (the doublets) have to be contracted to the represen-tation ¯ r which is the complex conjugate representation of r .If the operator O r by itself conserves physical CP – by which we mean that it does notintroduce any complex phases into the Lagrangian including the associated coupling constant– the only possible source of CP violation is coming from the doublet vevs and the complexCG factors appearing in the contraction with the operator and the doublets. For illustrativepurpose we want to discuss this explicitly if we have two doublets ψ (cid:48) ∼ (cid:48) and ψ (cid:48)(cid:48) ∼ (cid:48)(cid:48) .For r = there is only one possible combination using only ψ (cid:48) and ψ (cid:48)(cid:48) which is ( ψ (cid:48) ⊗ ψ (cid:48)(cid:48) ) .Using the tensor products of T (cid:48) —see for example [35]— we find that the combination is realif the vevs fulfill the following conditions ψ (cid:48) = (cid:18) X e i α X e i β (cid:19) , ψ (cid:48)(cid:48) = (cid:18) Y e − i β Y e − i α (cid:19) , (2.13)with X , X , Y , Y , α and β real parameters. For r = (cid:48) and r = (cid:48)(cid:48) the only possiblecontractions ( ψ (cid:48) ⊗ ψ (cid:48) ) (cid:48)(cid:48) and ( ψ (cid:48)(cid:48) ⊗ ψ (cid:48)(cid:48) ) (cid:48) vanish due to the antisymmetry of the contraction.For r = there are three possible contractions. Either a flavon with itself or both flavonstogether. • For the selfcontractions ( ψ (cid:48) ⊗ ψ (cid:48) ) and ( ψ (cid:48)(cid:48) ⊗ ψ (cid:48)(cid:48) ) the Lagrangian will not contain aphase if the flavon fields ψ (cid:48) and ψ (cid:48)(cid:48) have the following structure ψ (cid:48) = (cid:18) X e i α X e − i( α + π/ (cid:19) , ψ (cid:48)(cid:48) = (cid:18) Y e − i( β − π/ Y e i β (cid:19) , (2.14)with X , X , Y , Y being real and α , β = 0 , ± π/ , π . • The contraction ( ψ (cid:48) ⊗ ψ (cid:48)(cid:48) ) does not introduce phases if ψ (cid:48) , ψ (cid:48)(cid:48) have the following struc-ture: ψ (cid:48) ∼ (cid:18) X e i( β − π/ X e i( α + π/ (cid:19) , ψ (cid:48)(cid:48) ∼ (cid:18) Y e − iβ Y e − iα (cid:19) , (2.15)with X , X , Y , Y real and β − α = − π/ ψ (cid:48) ⊗ ψ (cid:48)(cid:48) ) and another entry by ( ψ (cid:48) ⊗ ψ (cid:48)(cid:48) ) we see that we cannot fulfill both conditionssimultaneously if the doublet vevs do not vanish. That means we would expect CP violationif both of these contractions are present in a given model.Later on in our model we will have real doublet alignments with ψ (cid:48) ∼ (cid:18) (cid:19) and ψ (cid:48)(cid:48) ∼ (cid:18) (cid:19) . (2.16)These alignments would conserve CP for sure only if the model contains only the contractions( ψ (cid:48) ⊗ ψ (cid:48) ) , ( ψ (cid:48)(cid:48) ⊗ ψ (cid:48)(cid:48) ) and ( ψ (cid:48) ⊗ ψ (cid:48)(cid:48) ) . Adding the contraction ( ψ (cid:48) ⊗ ψ (cid:48)(cid:48) ) would add a phaseto the Yukawa matrix resulting possibly in physical CP violation.11 The Model
In this section we discuss a supersymmetric model of lepton flavour based on T (cid:48) as a familysymmetry. Because it considers only the lepton sector we can consider it as a toy model. Thegeneralised CP symmetry will be broken in a geometrical way as we will discuss later on andwe can fit all the available data of masses and mixing in the lepton sector.The gauge symmetry of the model is the Standard Model gauge group G SM = SU (3) c × SU (2) L × U (1) Y . The discrete symmetries of the model are T (cid:48) (cid:111) H CP × Z × Z × Z × Z ,where the Z n factors are the shaping symmetries of the superpotential required to forbidunwanted operators.There are a few comments about this symmetry in order. First of all, the symmetryseems to be rather large but in fact compared to the first works on T (cid:48) with geometricalCP violation [25, 26] we have only added a factor of Z × Z but included the full flavonvacuum alignment and messenger sector. This symmetry is also much smaller than theshaping symmetry we have used before in [35].One might wonder where this symmetry originates from and it might be embedded into(gauged) continuous symmetries or might be a remnant of the compactification of extra-dimensions. But a discussion of such an embedding goes clearly beyond the scope of thiswork where we just want to discuss the connection of a T (cid:48) family symmetry with CP andillustrate it by a toy model which is nevertheless in full agreement with experimental data.In this section we will only discuss the effective operators generated after integrating outthe heavy messenger fields. The full renormalisable superpotential including the messengerfields is given in Appendix B. We will start the discussion of the model with the flavon sector which is self-contained. Howthe flavons couple to the matter sector will be discussed afterwards.The model contains 14 flavon fields in 1-, 2- and 3-dimensional representations of T (cid:48) and 5auxiliary flavons in 1-dimensional representations. Before we will discuss the superpotentialwhich fixes the directions and phases of the flavon vevs we will first define them. We havefour flavons in the 3-dimensional representation of T (cid:48) pointing in the directions (cid:104) φ (cid:105) = φ , (cid:104) ˜ φ (cid:105) = ˜ φ , (cid:104) ˆ φ (cid:105) = ˆ φ , (cid:104) ξ (cid:105) = ξ . (3.1)The first three flavons will be used in the charged lepton sector and the fourth one couplesonly to the neutrino sector. These flavon vevs, like all the other flavon vevs, are real.Further we introduce three doublets of T (cid:48) : ψ (cid:48) ∼ (cid:48) , ψ (cid:48)(cid:48) ∼ (cid:48)(cid:48) and ˜ ψ (cid:48)(cid:48) ∼ (cid:48)(cid:48) . We recallthat the doublets are the only representations of the family group T (cid:48) which introduce phases,due to the complexity of the Clebsh-Gordan coefficients. For the doublets we will find thealignments (cid:104) ψ (cid:48) (cid:105) = (cid:18) (cid:19) ψ (cid:48) , (cid:104) ψ (cid:48)(cid:48) (cid:105) = (cid:18) (cid:19) ψ (cid:48)(cid:48) , (cid:104) ˜ ψ (cid:48)(cid:48) (cid:105) = (cid:18) (cid:19) ˜ ψ (cid:48)(cid:48) . (3.2)And finally, we introduce 7 flavon fields in one-dimensional representations of the familygroup. In particular, we have (the primes indicate the types of singlet) (cid:104) ζ (cid:48) (cid:105) = ζ (cid:48) , (cid:104) ζ (cid:48)(cid:48) (cid:105) = ζ (cid:48)(cid:48) , (cid:104) ˜ ζ (cid:48) (cid:105) = ˜ ζ (cid:48) , (cid:104) ˜ ζ (cid:48)(cid:48) (cid:105) = ˜ ζ (cid:48)(cid:48) , (cid:104) ζ (cid:105) = ζ , (cid:104) ρ (cid:105) = ρ , (cid:104) ˜ ρ (cid:105) = ˜ ρ . (3.3)12 SM T (cid:48) U (1) R Z Z Z Z Z Z φ ( , , φ ( , , φ ( , , ξ ( , , ψ (cid:48) ( , , (cid:48) ψ (cid:48)(cid:48) ( , , (cid:48)(cid:48) ψ (cid:48)(cid:48) ( , , (cid:48)(cid:48) ζ ( , , ζ (cid:48) ( , , (cid:48) ζ (cid:48) ( , , (cid:48) ζ (cid:48)(cid:48) ( , , (cid:48)(cid:48) ζ (cid:48)(cid:48) ( , , (cid:48)(cid:48) ρ ( , , ρ ( , , (cid:15) ( , , (cid:15) ( , , (cid:15) ( , , (cid:15) ( , , (cid:15) ( , , Table 6:
List of the flavon fields and their transformation properties. We also list here theauxiliary flavon fields (cid:15) i , i = 1 , . . . ,
5, which are needed to fix the phases of the vevs of theother flavon fields. SM T (cid:48) U (1) R Z Z Z Z Z Z D φ ( , , D φ ( , , D φ ( , , D ψ ( , , D ψ ( , , D ψ ( , , D ξ ( , , S ζ ( , , S (cid:48) ζ ( , , (cid:48) S (cid:48) ζ ( , , (cid:48) P ( , , Table 7:
List of the driving fields and their T (cid:48) transformation properties. The field P standsfor the fields ˜ S ζ , S ξ , S ρ and S ε i , with i = 1 , . . . ,
5, which all have the same quantum numbers.
The ρ and ˜ ρ couple only to the neutrino sector while the other one-dimensional flavons coupleonly to the charged lepton sector. Also the five auxiliary flavons (cid:15) i , i = 1 , . . . , (cid:15) i which are only needed to fix the phases of the otherflavon vevs and all acquire real vevs by themselves.We discuss now the superpotential in the flavon sector which “aligns” the flavon vevs. Wewill use so-called F -term alignment where the vevs are determined from the F -term conditionsof the driving fields. The driving fields are listed with their quantum numbers in Table 7,where we have indicated for simplicity P = ˜ S ζ , S ξ , S ρ and S ε i , with i = 1 , . . . ,
5, becausethey have all the same quantum numbers under the whole symmetry group.The fields labeled as P play a crucial role in fixing the phases of the flavon vevs. Theyare fixed by the discrete vacuum alignment method as it was first proposed in [37]. Having aflavon (cid:15) (for the moment we assume it is a singlet under the family symmetry) charged undera Z n symmetry the superpotential will contain a term W ⊃ P (cid:18) (cid:15) n Λ n − ∓ M (cid:19) . (3.4)Remember that the P fields are total singlets. Due to CP symmetry in this simple exampleall parameters and couplings are real. The F -term equation for P reads | F P | = (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) n Λ n − ∓ M (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (3.5)which gives for the phase of the flavon vevarg( (cid:104) (cid:15) (cid:105) ) = (cid:40) πn q , q = 1 , . . . , n for “ − ” in Eq. (3.5), πn q + πn , q = 1 , . . . , n for “+” in Eq. (3.5). (3.6)14his method will be used to fix the phases of the singlet and triplet flavon vevs (including the (cid:15) i ). Note that we have to introduce for every phase we fix in this way a P field and only aftera suitable choice of basis for this fields we end up with the simple structure we show later,see also the appendix of [37]. For the directions of the triplets we use standard expressions,cf. also the previous paper [35].For the doublets, nevertheless, we use here a different method. Take for example the term D ψ (cid:104) ( ψ (cid:48)(cid:48) ) − φ ζ (cid:48) (cid:105) . The F -term equations read | F D ψ | = ( ψ (cid:48)(cid:48) ) − φ ζ (cid:48) = 0 , (3.7) | F D ψ | = i( ψ (cid:48)(cid:48) ) − φ ζ (cid:48) = 0 , (3.8) | F D ψ | = (1 − i) ψ (cid:48)(cid:48) ψ (cid:48)(cid:48) − φ ζ (cid:48) = 0 . (3.9)Note the phases coming from the complex CG coefficients of T (cid:48) . Plugging in the (real) vevsof φ and ζ (cid:48) it turns out that only the second component of ψ (cid:48)(cid:48) does not vanish and is indeedreal as well.The full superpotential for the flavon vacuum alignment reads W f = D φ ε Λ (cid:2) φ − φ ζ (cid:48)(cid:48) (cid:3) + ˜ D φ ε Λ (cid:104) ˜ φ − ˜ φ ˜ ζ (cid:48) (cid:105) + ˆ D φ Λ (cid:34) ε ˆ φ ˆ φ + ε ˜ φ ˜ ζ (cid:48)(cid:48) + ε ˜ ζ (cid:48)(cid:48) ˜ φ Λ (cid:35) + D ψ (cid:104)(cid:0) ψ (cid:48)(cid:48) (cid:1) − φ ζ (cid:48) (cid:105) + ¯ D ψ (cid:0) i ψ (cid:48) ψ (cid:48) + φ ζ (cid:48) (cid:1) + ˜ D ψ (cid:34)(cid:16) ˜ ψ (cid:48)(cid:48) (cid:17) − ˜ φ ˜ ζ (cid:48)(cid:48) − ε ˆ φ ˆ φ Λ (cid:35) + S (cid:48) ζ (cid:18) ζ (cid:48) ζ (cid:48) − ξ ξ − ε ξ Λ (cid:19) + ˜ S (cid:48) ζ (cid:16) ˜ ζ (cid:48) ˜ ζ (cid:48) − ˜ φ ˜ φ (cid:17) + ˜ S ζ Λ (cid:20)(cid:16) ˜ ζ (cid:48)(cid:48) (cid:17) − M ζ (cid:48)(cid:48) (cid:21) + S ζ (cid:16) ζ ζ + ˆ φ ˆ φ (cid:17) + S ε Λ (cid:0) ε − M ε (cid:1) + S ε (cid:18) ε Λ − M ε (cid:19) + S ε (cid:0) ε − M ε (cid:1) + S ε (cid:0) ε − M ε (cid:1) + S ε (cid:0) ε ε − M ε (cid:1) + D ξ ε Λ (cid:0) ξ + ξρ + ξ ˜ ρ (cid:1) + S ξ (cid:0) ξ − M ξ (cid:1) + S ρ (cid:0) ρ + ˜ ρ − M ρ (cid:1) . (3.10)We will not go through all the details and discuss each F -term condition but this potentialis minimized by the vacuum structure as in Eqs. (3.1), (3.2) and (3.3). Finally, we wantto remark that the F -term equations do not fix the phase of the field ζ . However, thephase of this field will turn out to be unphysical because it can be canceled out through anunphysical unitary transformation of the right-handed charged lepton fields as we will showlater explicitly. Since we have discussed now the symmetry breaking flavon fields we will now proceed with thediscussion on how these fields couple to the matter sector and generate the Yukawa couplingsand right-handed Majorana neutrino masses.The model contains three generations of lepton fields, the left-handed SU (2) L doubletsare organized in a triplet representation of T (cid:48) , the first two families of right-handed chargedlepton fields are organized in a two dimensional representation, (cid:48)(cid:48) , and the third family sitsin a (cid:48)(cid:48) . There are two Higgs doublets as usual in supersymmetric models. They are both15 ¯ E ¯ E N R H d H u SU (2) L U (1) Y -1 2 2 0 -1 1 T (cid:48) (cid:48)(cid:48) (cid:48)(cid:48) U (1) R Z Z Z Z Z Z Table 8:
List of the matter and Higgs fields of the model and their transformation propertiesunder T (cid:48) , U (1) R and the shaping symmetries. We also give the quantum numbers under SU (2) L × U (1) Y . All fields are singlets of SU (3) C . singlets, under T (cid:48) . The model includes three heavy right-handed Majorana neutrino fields N , which are organized in a triplet. The light active neutrino masses are generated throughthe type I seesaw mechanism [32]. At leading order tri-bimaximal mixing (TBM) is predictedin the the neutrino sector which is corrected by the charged lepton sector allowing a realisticfit of the measured parameters of the PMNS mixing matrix. The quantum numbers of thematter fields are summarized in Table 8.In this work we use the right-left convention for the Yukawa matrices − L ⊃ (Y e ) ij ¯ e R i e L j H d + H.c. , (3.11)i.e. there exists a unitary matrix U e which diagonalizes the product Y † e Y e and contributes tothe physical PMNS mixing matrix. The Yukawa matrix Y e is generated after the flavons acquire their vevs and T (cid:48) is broken. Theeffective superpotential describing the couplings of the matter sector to the flavon sector isgiven by W Y e = y ( e )33 Λ (cid:0) ¯ E H d (cid:1) (cid:48)(cid:48) ( L φ ) (cid:48) + y ( e )32 Λ (cid:0) ¯ E H d (cid:1) (cid:48)(cid:48) (cid:16) L ˆ φ (cid:17) (cid:48) ζ + ˆ y ( e )32 Λ ¯Ω (cid:0) ¯ E H d (cid:1) (cid:48)(cid:48) (cid:2) L (cid:0) ψ (cid:48) ψ (cid:48)(cid:48) (cid:1) (cid:3) (cid:48) + y ( e )22 Λ (cid:0) ¯ E ψ (cid:48) (cid:1) H d ( L φ ) + y ( e )21 Λ ¯Ω (cid:0) ¯ E ψ (cid:48) (cid:1) H d (cid:2)(cid:0) ψ (cid:48) ψ (cid:48)(cid:48) (cid:1) L (cid:3) + y ( e )11 Λ Ω (cid:16) ¯ E ˜ ψ (cid:48)(cid:48) (cid:17) (cid:48) H d ˜ ζ (cid:48) (cid:16) L ˜ φ (cid:17) (cid:48) , (3.12)where Λ denotes a generic messenger scale. Note the explicit phase factors Ω = (1+i) / √ − i) / √ T (cid:48) as indices at the brackets. These contractions aredetermined by the messenger sector which will be discussed in Appendix B.16fter plugging in the flavon vevs from Eqs. (3.1)-(3.3) we find for the structure of theYukawa matrix Y e Y e = Ω a b c d + i k e ≡ Ω a b c ρ e i η e , (3.13)where we define ρ = √ d + k and η = arg ( d + i k ).The parameters a , b , c , d , e , k depend on the unfixed phase of the vev of ζ , ζ , which canbe explicitly factorized asY e = Ω ¯ a ( ζ ) 0 0i ζ b ζ c ζ ( d + i k ) ζ e = e i arg(Ω ¯ a ( ζ )) ζ
00 0 ζ | Ω ¯ a ( ζ ) | b c d + i k e , (3.14)from which it is clear that an eventual phase of ζ drops out in the physical combinationY † e Y e and we can choose the parameters in the Yukawa matrix to be real.We remind that there are in principle three possible sources of complex phases which canlead to physical CP violation: complex vevs, complex couplings whose phases are determinedby the invariance under the generalised CP symmetry and complex CG coefficients. In ourmodel all vevs are real due to our flavon alignment and the convenient choice of the θ r phases.Then the (physical) phases in Y e are completely induced by the complex couplings andcomplex CG coefficients. In fact the insights we have gained before in Section 2.3 can be usedhere. The phase in the 1-1 element is unphysical (it drops out in the combination Y † e Y e . Sothe physical CP violation is to leading order given by the phases of the ratios (Y e ) / (Y e ) and (Y e ) / (Y e ) . Let us study for illustration the second ratio which has two components,one with a non-trivial relative phase and one without. The real ratio d/e is coming fromthe operators with the coefficients y ( e )32 and y ( e )33 and from the viewpoint of T (cid:48) (cid:111) H CP there isnot really any difference between the two because we have only added a singlet which cannotbreak CP in our setup as we said before.For the second ratio i k/e this is different. Using the notation from Section 2.3 we have O Φ = (cid:0) ¯ E H d (cid:1) (cid:48)(cid:48) ( L Φ ) (cid:48) . If Φ = φ (the operator with y ( e )32 ) we cannot have a phasebecause φ is a triplet flavon. For Φ = ( ψ (cid:48) ψ (cid:48)(cid:48) ) (the operator with ˆ y ( e )32 ) we can checkif condition (2.15) is fulfilled which is not the case because both vevs are real, while thecondition demands a relative phase difference between the vevs of π/
4. This demonstratesthe usefulness of the conditions given in Section 2.3 in understanding the origin of physicalCP violation in this setup.
The neutrino sector is constructed using a superpotential similar to that used in [35]: thelight neutrino masses are generated through the type I see-saw mechanism, i.e. introducingright-handed heavy Majorana states which are accommodated in a triplet under T (cid:48) . We havethe effective superpotential W Y ν = λ N N ξ + N N ( λ ρ + λ ˜ ρ ) + y ν Λ (
N L ) ( H u ρ ) + ˜ y ν Λ (
N L ) ( H u ˜ ρ ) . (3.15)17he Dirac and the Majorana mass matrices obtained from this superpotential are identicalto those described in [35] and we quote them here for completeness M R = Z + X − Z − Z − Z Z − Z + X − Z − Z + X Z , M D = ρ (cid:48) Λ , (3.16)where X, Z and ρ (cid:48) are real parameters which can be written explicitly as X = λ √ ρ + λ √ ρ , Z = λ √ ξ and ρ (cid:48) = y ν √ ρ v u + ˜ y ν √ ρ v u . (3.17)The right-handed neutrino mass matrix M R is diagonalised by the TBM matrix [18] U TBM = (cid:112) / (cid:112) / − (cid:112) / (cid:112) / − (cid:112) / − (cid:112) / (cid:112) / (cid:112) / , (3.18)such that the heavy RH neutrino masses read: U T TBM M R U TBM = D N = Diag(3 Z + X, X, Z − X )= Diag( M e i φ , M e i φ , M e i φ ) , M , , > . (3.19)Since X and Z are real parameters, the phases φ , φ and φ take values 0 or π . A lightneutrino Majorana mass term is generated after electroweak symmetry breaking via the typeI see-saw mechanism: M ν = − M TD M − R M D = U ∗ ν Diag ( m , m , m ) U † ν , (3.20)where U ν = i U TBM
Diag (cid:16) e i φ / , e i φ / , e i φ / (cid:17) ≡ i U TBM Φ ν , Φ ν ≡ Diag (cid:16) e i φ / , e i φ / , e i φ / (cid:17) , (3.21)and m , , > m i = (cid:18) ρ (cid:48) Λ (cid:19) M i , i = 1 , , . (3.22)The phase factor i in Eq. (3.21) corresponds to an unphysical phase and we will drop it inwhat follows. Note also that one of the phases φ k , say φ , is physically irrelevant since it canbe considered as a common phase of the neutrino mixing matrix. In the following we willalways set φ = 0. This corresponds to the choice ( X + 3 Z ) > θ r At this point we want to comment on the role of the phases θ r appearing in the definitionof the CP transformation in Eq. (2.11). These phases are arbitrary and hence they shouldnot contribute to physical observables. This means, for instance, that these arbitrary phasesmust not appear in the Yukawa matrices after T (cid:48) is broken. However it is not enough to lookat the Yukawa couplings alone but one also has to study the flavon vacuum alignment sector.18e want to show next a simple example for which, as expected, these phases turn out to beunphysical.In order to show this we consider as example (Y e ) and (Y e ) respectively generated bythe following operators:(Y e ) ∼ (cid:0) ¯ E ψ (cid:48) (cid:1) ( L φ ) ζ H d , (Y e ) ∼ (cid:0) ¯ E ψ (cid:48) (cid:1) ( L φ ) (cid:48)(cid:48) ζ (cid:48) H d . (3.23)The fields together with their charges have been defined before in Table 6. We will now bemore explicit and consider all the possible phases arising in each of the given operators underthe CP transformation of Eq. (2.9) where the θ r were included explicitly. For each flavon vevin the operators we will denote the arising phase with a bar correspondingly, i.e. for the vevof the flavon φ we will have φ → e i ¯ φ φ where φ is the modulus of the vev. Then using thetransformations in Eq. (2.9) and Table 2.9 we getarg ((Y e ) − (Y e ) ) = ¯ ζ − ¯ ζ (cid:48) + ( θ − θ (cid:48) ) / . (3.24)The vevs of the flavons ζ and ζ (cid:48) are determined at leading order by S ζ (cid:104) ζ − ( ˆ φ ˆ φ ) (cid:105) and S ζ (cid:48) (cid:2) ( ζ (cid:48) ) − ( ξ ξ ) (cid:48)(cid:48) (cid:3) . (3.25)where S ζ,ζ (cid:48) are two of the so-called “driving fields” which in this case are singlet of type and (cid:48) under T (cid:48) . From the F -term equations one gets that ¯ ζ = ¯ˆ φ + ( θ − θ ) / ζ (cid:48) = ¯ ξ + ( θ − θ (cid:48) ) / e ) − (Y e ) ) = ¯ˆ φ − ¯ ξ . (3.26)the phases θ , θ (cid:48) and θ cancel out.This shows how the θ r cancel out in a complete model and become unphysical. Includingthem only in one sector, for instance, in the Yukawa sector they might appear to be physicaland only after considering also the flavon alignment sector it can be shown that they areunphysical which is nevertheless quite cumbersome in a realistic model due to the many fieldsand couplings involved. In this section we want to provide a better understanding of the quality of symmetry break-ing our model exhibits. To be more precise we will argue that our model breaks CP in ageometrical fashion and then we will discuss the residual symmetries of the mass matrices.Geometrical CP violation was first defined in [34] and there it is tightly related to theso-called “calculable phases” which are phases of flavon vevs which do not depend on theparameters of the potential but only on the geometry of the potential. This applies also toour model. All complex phases are determined in the end by the (discrete) symmetry group ofour model. In particular the symmetries T (cid:48) (cid:111) H CP and the Z n factors play a crucial role here.For the singlets and triplets in fact the Z n symmetries (in combination with CP) make thephases calculable using the discrete vacuum alignment technique [37]. For the doublets thenthe symmetry T (cid:48) (cid:111) H CP enters via fixing the phases of the couplings and fixing relative phasesbetween different components of the multiplets. In particular, all flavon vevs are left invariantunder the generalised CP symmetry and hence protected by it. However the calculable phases19re necessary but not sufficient for geometrical CP violation. For this we have to see if CP isbroken or not.For this we will have a look at the residual symmetries of the mass matrices after T (cid:48) (cid:111) H CP is broken. First of all, we observe that the vev structure mentioned in Section 3 gives abreaking pattern which is different in the neutrino and in the charged lepton sector, i.e. theresidual groups G ν and G e are different.In the charged lepton sector the group T (cid:48) is fully broken by the singlet, doublet andtriplet vevs. If it exists, the residual group in the charged lepton sector is defined throughthe elements which leave invariant the flavon vevs and satisfy ρ † ( g e i ) Y † e Y e ρ ( g e i ) = Y † e Y e with g e i ∈ G e < T (cid:48) (cid:111) H CP ,X † e (cid:16) Y † e Y e (cid:17) X e = (cid:16) Y † e Y e (cid:17) ∗ with X e ∈ G e < T (cid:48) (cid:111) H CP . (3.27)The first condition is the ordinary condition to study residual symmetries while the secondone is relevant only for models with spontaneous CP violation.In our model this conditions are not satisfied for any ρ ( g ) or X e . Hence there is no residualsymmetry group in the charged lepton sector and even more CP is broken spontaneously.Together with the fact that all our phases are determined by symmetries (up to signs anddiscrete choices) we have demonstrated now that our model exhibits geometrical CP violation.In the neutrino sector we can write similar relations that take into account the symmetricalstructure of the Majorana mass matrix, and in particular as before the residual symmetry isdefined through the elements which leave invariant the flavon vevs and satisfy ρ T ( g ν i ) M ν ρ ( g ν i ) = M ν with g ν i ∈ G ν < T (cid:48) (cid:111) H CP ,X Tν M ν X ν = M ∗ ν with X ν ∈ G ν < T (cid:48) (cid:111) H CP . (3.28)In our model M ν is a real matrix and therefore ρ ( g ν i ) and X ν are defined through the sameconditions. Defining O as the orthogonal matrix which diagonalizes the real symmetric matrix M ν we find from Eq. (3.28) ( O X Tν O T ) M diag ν ( O X ν O T ) = M diag ν (3.29)and hence the matrix D = O X ν O T has to be of the form D = ( − p − q
00 0 ( − p + q . (3.30)The same argument can be applied to the matrix ρ ( g ν i ), because the matrices X ν , ρ ( g ν i )and M ν are simultaneously diagonalisable by the same orthogonal matrix O . It is easy tofind that O = / √ / √ / √ − / √ / √ − (cid:112) / / √ / √ , (3.31)which is expected since M ν is diagonalized by U TBM and O is just a permutation of U T TBM which corresponds to a permutation of the eigenvalues.20he residual symmetry coming from T (cid:48) is generated only by T S T = O T · − − · O , (3.32)which also leaves invariant the vev structure. This symmetry is a Z symmetry. In summarythe residual symmetry in the neutrino sector is a Klein group K ∼ = Z × Z , in which one Z comes from H CP and the other one from T (cid:48) . H CP is conserved because in the neutrino sector X ν can be chosen as the identity matrix and M ν is real.Combining the two we find G f ≡ T (cid:48) (cid:111) H CP ≡ T (cid:48) (cid:111) Z −→ (cid:40) G e = ∅ ,G ν = K , (3.33)so that T (cid:48) (cid:111) H CP is completely broken and there is no residual symmetry left. Before we consider the mixing angles and phases in the PMNS matrix we first will discussthe neutrino spectra predicted by the model. We get the same results as in [35] because ourneutrino mass matrix has exactly the same structure. The forms of the Dirac and Majoranamass terms given in Eq. (3.16) imply that in the model considered by us both light neutrinomass spectra with normal ordering (NO) and with inverted ordering (IO) are allowed (seealso [35]). In total three different spectra for the light active neutrinos are possible. Theycorrespond to the different choices of the values of the phases φ i in Eq. (3.21)). More specif-ically, the cases φ = φ = φ = 0 and φ = φ = 0 and φ = π correspond to NO spectra ofthe type A and B, respectively. For φ = φ = 0 and φ = π also IO spectrum is possible.The neutrino masses in cases of the three spectra are given by:NO spectrum A : ( m , m , m ) = (4 . , . , . · − eV , (3.34)NO spectrum B : ( m , m , m ) = (5 . , . , . · − eV , (3.35)IO spectrum : ( m , m , m ) = (51 . , . , . · − eV , (3.36)where we have used the best fit values of ∆ m and | ∆ m | given in Table 1. Employingthe 3 σ allowed ranges of values of the two neutrino mass squared differences quoted in Table1, we find the intervals in which m , , can vary: • NO spectrum A: m ∈ [4 . , . · − eV, m ∈ [9 . , . · − eV, m ∈ [4 . , . · − eV; • NO spectrum B: m ∈ [5 . , . · − eV, m ∈ [9 . , . · − eV, m ∈ [4 . , . · − eV; • IO spectrum: m ∈ [4 . , . · − eV, m ∈ [4 . , . · − eV, m ∈ [1 . , . · − eV.21orrespondingly, we get for the sum of the neutrino masses:NO A : (cid:88) j =1 m j = 6 . × − eV , . × − ≤ (cid:88) j =1 m j ≤ . × − eV , (3.37)NO B : (cid:88) j =1 m j = 6 . × − eV , . × − ≤ (cid:88) j =1 m j ≤ . × − eV , (3.38)IO : (cid:88) j =1 m j = 12 . × − eV , . × − ≤ (cid:88) j =1 m j ≤ . × − eV , (3.39)where we have given the predictions using the best fit values and the 3 σ intervals of theallowed values of m , m and m quoted above. We will derive next expressions for the mixing angles and the CPV phases in the standardparametrisation of the PMNS matrix in terms of the parameters of the model. The expres-sion for the charged lepton mass matrix Y e given in Eq. (3.13) contains altogether sevenparameters: five real parameters and two phases, one of which is equal to π/
2. Three (com-binations of) parameters are determined by the three charged lepton masses. The remainingtwo real parameters and two phases are related to two angles and two phases in the matrix U e which diagonalises the product Y † e Y e and enters into the expression of the PMNS matrix: U PMNS = U † e U ν , where U ν is of TBM form (see Eq. (3.19)), while U e ∝ R R , R and R are orthogonal matrices describing rotations in the 2-3 and 1-2 planes, respectively. It provesconvenient to adopt for the matrices U e and U ν the notation used in [14]: (cid:40) U e = Ψ e R − ( θ e ) R − ( θ e ) U ν = R ( θ ν ) R ( θ ν ) Φ ν (3.40)where Ψ e = diag (cid:0) , e i ψ e , e i ω e (cid:1) , θ ν = − π/ θ ν = sin − (1 / √ ν is a diagonal phase matrixdefined in Eq. (3.21), and R ( θ e ) = cos θ e sin θ e − sin θ e cos θ e
00 0 1 , R ( θ e ) = θ e sin θ e − sin θ e cos θ e . (3.41)Using the expression for the charged lepton mass matrix Y e given in Eq. (3.13) andcomparing the right and the left sides of the equationY † e Y e = U e diag (cid:0) m e , m µ , m τ (cid:1) U † e , (3.42)we find that m e = a , m µ = c and m τ = e . For U e given in Eq. (3.40) this equality holdsonly under the condition that sin θ e and sin θ e are sufficiently small. Using the leading termsin powers of the small parameters sin θ e and sin θ e we get the approximate relations:sin θ e e i ψ e (cid:39) ± i (cid:12)(cid:12)(cid:12)(cid:12) bc (cid:12)(cid:12)(cid:12)(cid:12) , sin θ e e i( ψ e − ω e ) = e ρc − e e i η (cid:39) (cid:12)(cid:12)(cid:12) ρe (cid:12)(cid:12)(cid:12) e i ξ e , (3.43)22here ψ e = ± π/ ξ e = ψ e − ω e , ξ e ∈ [0 , π ], θ e (cid:39) | b/c | and θ e (cid:39) | ρ/e | .In the discussion that follows θ e , θ e , ψ e and ω e are treated as arbitrary angles andphases, i.e., no assumption about their magnitude is made.The lepton mixing we obtain in the model we have constructed, including the Dirac CPVphase but not the Majorana CPV phases, was investigated in detail on general phenomeno-logical grounds in ref. [14] and we will use the results obtained in [14]. The three angles θ , θ and θ and the Dirac and Majorana CPV phases δ and β and β (see Eqs. (1.1) - (1.3)),of the PMNS mixing matrix U PMNS = U † e U ν = R ( θ e ) R ( θ e )Ψ ∗ e R ( θ ν ) R ( θ ν ) Φ ν , canbe expressed as functions of the two real angles, θ e and θ e , and the two phases, ψ e and ω e present in U e . However, as was shown in [14], the three angles θ , θ and θ and the Diracphase δ are expressed in terms of the angle θ e , an angle ˆ θ and just one phase φ , wheresin ˆ θ = 12 (1 − θ e cos θ e cos( ω e − ψ e )) , (3.44)and the phase φ = φ ( θ e , ω e , ψ e ). Indeed, it is not difficult to show that (see the Appendixin [14]) R ( θ e ) Ψ ∗ e R ( θ ν ) = P Φ R (ˆ θ ) ˜ Q . (3.45)Here P = diag(1 , , e − i α ), Φ = diag(1 , e i φ ,
1) and ˜ Q = diag (cid:0) , , e i β (cid:1) , where α = γ + ψ e + ω e , β = γ − φ , (3.46)and γ = arg (cid:16) − e − i ψ e cos θ e + e − i ω e sin θ e (cid:17) , φ = arg (cid:16) e − i ψ e cos θ e + e − i ω e sin θ e (cid:17) . (3.47)The phase α is unphysical (it can be absorbed in the τ lepton field). The phase β contributesto the matrix of physical Majorana phases, which now is equal to Q = ˜ Q Φ ν . The PMNSmatrix takes the form: U PMNS = R ( θ e ) Φ( φ ) R (ˆ θ ) R ( θ ν ) Q , (3.48)where θ ν = sin − (1 / √ θ , θ , θ and δ are functions of threeparameters θ e , ˆ θ and φ . As a consequence, the Dirac phase δ can be expressed as a functionof the three PMNS angles θ , θ and θ , leading to a new “sum rule” relating δ and θ , θ and θ [14]. Using the measured values of θ , θ and θ , the authors of [14] obtainedpredictions for the values of δ and of the rephasing invariant J CP = Im( U ∗ e U ∗ µ U e U µ ), whichcontrols the magnitude of CP violating effects in neutrino oscillations [46], as well as for the2 σ and 3 σ ranges of allowed values of sin θ , sin θ and sin θ . These predictions are validalso in the model under discussion.To be more specific, using Eq. (3.48) we get for the angles θ , θ and θ of the standardparametrisation of U PMNS [14]:sin θ = | U e | = sin θ e sin ˆ θ , sin θ = | U µ | − | U e | = sin ˆ θ − sin θ − sin θ , cos θ = cos ˆ θ − sin θ , sin θ = | U e | − | U e | = 13 (cid:32) √ θ sin θ cos φ − sin θ − cos θ cos θ (cid:33) , (3.49)23here the first relation sin θ = sin θ e sin ˆ θ was used in order to obtain the expressionsfor sin θ and sin θ . Clearly, the angle ˆ θ differs little from the atmospheric neutrinomixing angle θ . For sin θ = 0 .
024 and sin θ ∼ = 0 .
39 we have sin θ e ∼ = 0 .
2. Comparingthe imaginary and real parts of U ∗ e U ∗ µ U e U µ , obtained using Eq. (3.48) and the standardparametrisation of U PMNS , one gets the following relation between the phase φ and the Diracphase δ [14]: sin δ = − √
23 sin φ sin 2 θ , (3.50)cos δ = 2 √
23 sin 2 θ cos φ (cid:18) − θ sin θ cos θ + sin θ (cid:19) + 13 sin 2 θ sin 2 θ sin θ sin θ cos θ + sin θ . (3.51)The results quoted above, including those for sin δ and cos δ , are exact. As can be shown, inparticular, we have: sin δ + cos δ = 1.Equation (3.49) allows to express cos φ in terms of θ , θ and θ , and substituting theresult thus obtained for cos φ in Eqs. (3.50) and (3.51), one can get expressions for sin δ andcos δ in terms of θ , θ and θ . We give below the result for cos δ [14]:cos δ = tan θ θ sin θ (cid:2) (cid:0) θ − (cid:1) (cid:0) − cot θ sin θ (cid:1)(cid:3) . (3.52)For the best fit values of sin θ , sin θ and sin θ , one finds in the case of NO and IOspectra (see also [14]): cos δ ∼ = − . , sin δ = ± . . (3.53)These values correspond to δ = 93 . ◦ or δ = 266 . ◦ . (3.54)Thus, our model predicts δ (cid:39) π/ π/
2. The fact that the value of the Dirac CPVphase δ is determined (up to an ambiguity of the sign of sin δ ) by the values of the threemixing angles θ , θ and θ of the PMNS matrix, (3.52), is the most striking predictionof the model considered. For the best fit values of θ , θ and θ we get δ ∼ = π/ π/ J CP factor, which determinesthe magnitude of CP violation in neutrino oscillations, is also a function of the three angles θ , θ and θ of the PMNS matrix: J CP = J CP ( θ , θ , θ , δ ( θ , θ , θ )) = J CP ( θ , θ , θ ) . (3.55)This allows to obtain predictions for the range of possible values of J CP using the currentdata on sin θ , sin θ and sin θ . For the best fit values of these parameters (see Table1) we find: J CP (cid:39) ± . δ and J CP were obtained first on the basis of a phenomenologicalanalysis in [14]. Here they are obtained for the first time within a selfconsistent model oflepton flavour based on the T (cid:48) family symmetry. Due to the slight difference between the best fit values of sin θ and sin θ in the cases of NO and IOspectra (see Table 1), the values we obtain for cos δ in the two cases differ somewhat. However, this differenceis equal to 10 − in absolute value and we will neglect it in what follows.
24n [14] the authors performed a detailed statistical analysis which permitted to determinethe ranges of allowed values of sin θ , sin θ , sin θ , δ and J CP at a given confidencelevel. We quote below some of the results obtained in [14], which are valid also in the modelconstructed by us.Most importantly, the CP conserving values of δ = 0; π ; 2 π are excluded with respect tothe best fit CP violating values δ ∼ = π/
2; 3 π/ σ . Correspondingly, J CP = 0is also excluded with respect to the best-fit values J CP (cid:39) ( − . J CP (cid:39) .
034 at morethan 4 σ . Further, the 3 σ allowed ranges of values of both δ and J CP form rather narrowintervals. In the case of the best fit value δ ∼ = 3 π/
2, for instance, we have in the cases of NOand IO spectra: NO : J CP ∼ = − . , . ∼ < J CP ∼ < . , or (3.56) − . ∼ < J CP ∼ < − . , (3.57)IO : J CP ∼ = − . , . ∼ < J CP ∼ < . , or (3.58) − . ∼ < J CP ∼ < − . , (3.59)where we have quoted the best fit value of J CP as well. The positive values are related to the χ minimum at δ = π/ δ and J CP for the 1 σ and 2 σ ranges of allowed valuesof sin θ , sin θ and sin θ , which were taken from Table 1. The figure was producedassuming flat distribution of the values of sin θ , sin θ and sin θ in the quoted intervalsaround the corresponding best fit values. As can be seen from Fig. 1, the predicted values ofboth sin δ and J CP thus obtained form rather narrow intervals .As it follows from Table 1, the angle θ is determined using the current neutrino oscillationdata with largest uncertainty. We give next the values of the Dirac phase δ for two valuesof sin θ from its 3 σ allowed range, sin θ = 0 .
50 and 0 .
60, and for the best fit values ofsin θ and sin θ :sin θ = 0 .
50 : cos δ = − . , δ = 97 . ◦ or 262 . ◦ ; (3.60)sin θ = 0 .
60 : cos δ = − . , δ = 100 . ◦ or 259 . ◦ . (3.61)These results show that | sin δ | , which determines the magnitude of the CP violation effectsin neutrino oscillations, exhibits very weak dependence on the value of sin θ : for any valueof sin θ from the interval 0 . ≤ sin θ ≤ .
60 we get | sin δ | ≥ . δ and J CP will be tested in the experiments searchingfor CP violation in neutrino oscillations, which will provide information on the value of theDirac phase δ . Using the expressions for the angles θ e and ˆ θ and for cos φ in terms of sin θ , sin θ andsin θ and the best fit values of sin θ , sin θ and sin θ , we can calculate the numerical The 2 σ ranges of allowed values of sin δ and J CP shown in Fig. 1 match approximately the 3 σ ranges ofallowed values of sin δ and J CP obtained in [14] by performing a more rigorous statistical ( χ ) analysis. + - - - - - - - - - - - - - - ∆ J C P ++ - - - - - - - - - - - - - ∆ J C P ++ ∆ J C P ++ ∆ J C P Figure 1:
Possible values of sin δ and J CP , obtained by using the 1 σ (light brown areas)and 2 σ ranges (blue + light brown areas) of allowed values of the mixing angles θ , θ and θ for NO spectrum (left panels) and IO spectrum (right panels), and for sin δ < δ > θ , θ and θ ,corresponding to δ = 266 . ◦ (sin δ <
0) and δ = 93 . ◦ (sin δ > U PMNS from which we can extract the values of the physical CPV Majorana phases.We follow the procedure described in [47]. Obviously, there are two such forms of U PMNS corresponding to the two possible values of δ . In the case of δ = 266 . ◦ and φ (cid:39) . ◦ wefind: U PMNS = .
822 e − i 7 . ◦ .
547 e i 16 . ◦ .
155 e i 102 . ◦ .
436 e − i 104 . ◦ .
658 e i 114 . ◦ .
614 e i 102 . ◦ . − .
517 0 . Q . (3.62)Recasting this expression in the form of the standard parametrisation of U PMNS we get: U PMNS = P .
822 0 .
547 0 .
155 e i 93 . ◦ .
436 e i 169 . ◦ .
658 e i 4 . ◦ . .
365 e i 16 . .
517 e i 172 . . Q Q , (3.63)where P = diag(e i(16 . − . ◦ , e i 102 . ◦ , Q = diag(e − i 16 . ◦ , e i 7 . ◦ ,
1) and e i 93 . ◦ =e − i(360 − . ◦ = e − i 266 . ◦ .Similarly, in the case of δ = 93 .
98 and φ (cid:39) . ◦ we obtain: U PMNS = .
822 e i 7 . ◦ .
547 e − i 16 . ◦ .
155 e − i 102 . ◦ .
436 e i 104 . ◦ .
658 e − i 114 . ◦ .
614 e − i 102 . ◦ . − .
517 0 . Q . (3.64)Extracting again phases in diagonal matrices on the right hand and left hand sides to get thestandard parametrisation of U PMNS we find: U PMNS = ˜ P .
822 0 .
547 0 .
155 e − i 93 . ◦ .
436 e − i 169 . ◦ .
658 e − i 4 . ◦ . .
365 e − i 16 . .
517 e − i 172 . . ˜ Q Q , (3.65)where P = diag(e i( − . . ◦ , e − i 102 . ◦ ,
1) and ˜ Q = diag(e i 16 . ◦ , e − i 7 . ◦ , P and ˜ P can be absorbed by the charged lepton fields and are unphysical. Incontrast, the phases in the matrices Q and ˜ Q contribute to the physical Majorana phases.We can finally write the Majorana phase matrix in the parametrization given in (1.1) ( φ = 0): − β ∓ . ◦ − β − φ , sin δ = ∓ . , (3.66) − β ± . ◦ − β − φ − φ , sin δ = ∓ . . (3.67)In order to calculate the phase β = γ − φ we have to find the value of γ . It follows fromEqs. (3.44) and (3.47) thatcos γ = sin θ e cos ω e √ θ , sin γ = ± cos θ e − sin θ e sin ω e √ θ , (3.68)cos φ = sin θ e cos ω e √ θ , sin φ = ∓ cos θ e − sin θ e sin ω e √ θ , (3.69)where we used the fact that ψ e = ± π/
2. These equations imply the following relations:cos γ = cos φ cos ˆ θ sin ˆ θ , (3.70)sin φ cos ˆ θ + sin γ sin ˆ θ = − √ θ e sin ω e . (3.71)27t is clear from Eq. (3.70) that the value of cos γ can be determined knowing the values ofcos φ and sin ˆ θ , independently of the values of θ e and ω e . This, obviously, allows to find also | sin γ | , but not the sign of sin γ . In the case of sin θ e sin ω e (cid:28) γ with the sign of sin φ and thus to determine γ for a given φ : wehave sin γ < φ >
0, and sin γ > φ <
0. Thus, for φ = 102 . ◦ (correspondingto δ = 266 . ◦ ) we find γ = − . β = γ − φ = − . ◦ = − (180 + 27 . ◦ ,while for φ = − . ◦ (corresponding to δ = 93 . ◦ ) we obtain γ = + 105 . β = +207 . ◦ = + (180 + 27 . ◦ .The results thus derived allow us to calculate numerically the Majorana CPV phases. Forthe best fit values of the neutrino mixing angles we get: β = (23 .
84 + 360 + φ ) ◦ , β = (70 .
88 + 360 − φ + φ ) ◦ for φ = − . ◦ ( δ = 93 . ◦ ) ;(3.72) β = ( − . −
360 + φ ) ◦ = ( − .
84 + 360 + φ ) ◦ ,β = ( − . − − φ + φ ) ◦ = ( − .
88 + 360 − φ + φ ) ◦ for φ = 102 . ◦ ( δ = 266 . ◦ ) , (3.73)where we have used the fact that β and β + 4 π lead to the same physical results.In the cases of the three types of neutrino mass spectrum allowed by the model, which arecharacterised, in particular, by specific values of the φ and φ we find: • NO A spectrum, i.e., φ = φ = 0: β = (23 .
84 + 360) ◦ , β = (70 .
88 + 360) ◦ for φ = − . ◦ ( δ = 93 . ◦ ) ,β = ( − .
84 + 360) ◦ , β = ( − .
88 + 360) ◦ for φ = 102 . ◦ ( δ = 266 . ◦ ) ; (3.74) • NO B spectrum, i.e., φ = 0 and φ = π : β = (23 .
84 + 540) ◦ , β = (70 .
88 + 540) ◦ for φ = − . ◦ ( δ = 93 . ◦ ) ,β = ( − .
84 + 540) ◦ , β = ( − .
88 + 540) ◦ for φ = 102 . ◦ ( δ = 266 . ◦ ) ; (3.75) • IO spectrum, φ = 0 and φ = π : β = (23 .
84 + 360) ◦ , β = (70 .
88 + 180) ◦ for φ = − . ◦ ( δ = 93 . ◦ ) ,β = ( − .
84 + 360) ◦ , β = ( − .
88 + 180) ◦ for φ = 102 . ◦ ( δ = 266 . ◦ ) , (3.76)where again we have used the fact that β and β + 4 π are physically indistinguishable. Knowing the values of the neutrino masses and the Majorana and Dirac CPV phases we canderive predictions for the neutrinoless double beta (( ββ ) ν -) decay effective Majorana mass |(cid:104) m (cid:105)| (see, e.g., [48]). Since |(cid:104) m (cid:105)| depends only on the cosines of the CPV phases, we get thesame result for φ = + 102 . ◦ ( δ = 266 . ◦ ) and φ = − . ◦ ( δ = 93 . ◦ ).Thus, for φ = ± . ◦ , using the best fit values of the neutrino mixing angles, we obtain: |(cid:104) m (cid:105)| = 4 . × − eV , NO A spectrum ; (3.77) |(cid:104) m (cid:105)| = 7 . × − eV , NO B spectrum ; (3.78) |(cid:104) m (cid:105)| = 26 . × − eV , IO spectrum . (3.79)28 A T R I N GERDA (cid:43)
IGEX (cid:43)
HdM (cid:43)(cid:43) (cid:43) (cid:45) (cid:45) m min (cid:64) eV (cid:68) (cid:200) (cid:88) m (cid:92) (cid:200) (cid:64) e V (cid:68) Figure 2:
The 3 σ allowed regions of values of the effective Majorana mass |(cid:104) m (cid:105)| as func-tions of the lightest neutrino mass m min for the NO (blue area) and IO (red area) neutrinomass spectra. The regions are obtained by using the experimentally determined values of theneutrino oscillation parameters (including the 1 σ uncertainties) quoted in Table 1. The blackcrosses correspond to the predictions of the model constructed in the present article, Eqs.(3.77) - (3.79). The horizontal band indicates the upper bound |(cid:104) m (cid:105)| ∼ . − . Ge reported in [49]. The dotted line representsthe prospective upper limit from the β -decay experiment KATRIN [50].
29n Fig. 2 we show the general phenomenologically allowed 3 σ range of values of |(cid:104) m (cid:105)| for the NO (blue area) and IO (red area) neutrino mass spectra as a function of the lightestneutrino mass. The values of |(cid:104) m (cid:105)| quoted above and corresponding to the three types ofneutrino mass spectrum (NO A, NO B and IO), predicted by the model constructed in thepresent article, are indicated with black crosses. The vertical lines in Fig. 2 correspond to m min = 8 . × − eV and 1 . × − eV; for a given value of m min from the interval determinedby these two values, [8 . × − , . × − ] eV, one can have |(cid:104) m (cid:105)| = 0 for specific values ofthe Majorana CPV phases. Finally, there are two interesting limiting forms of the charged lepton Yukawa coupling (mass)matrix Y e : they correspond to i) k = 0, i.e., η = 0 or π , and ii) d = 0, i.e., η = ± π/
2. In thecase of k = 0, the TBM prediction for θ does not depend on θ e anymore; if d = 0, even θ itself does not depend on θ e anymore. Up to next-to-leading order we find:i) sin θ = 13 + 13 sin θ ≈
13 for k = 0 , η = 0 , π , ii) sin θ = 12 −
12 sin θ ≈
12 for d = 0 , η = ± π/ , (3.80)where we have written the corrections in terms of θ . Both cases could be realised by choosinga certain set of messengers. If we remove the messenger pair Σ A (cid:48) , ¯Σ A (cid:48)(cid:48) , our model wouldcorrespond to the case i), while if we remove the messenger pair Σ C (cid:48)(cid:48) , ¯Σ C (cid:48) , the model wouldcorrespond to the case ii). The model we have constructed, which includes both messengerpairs, gives a somewhat better description of the current data on the neutrino mixing angles.This brief discussion shows how important the messenger sector can be for getting meaningfulpredictions. In this work we have analyzed the presence of a generalised CP symmetry, H CP , combinedwith the non-Abelian discrete group T (cid:48) in the lepton flavour space, i.e. the possibility ofthe existence of a symmetry group G f = T (cid:48) (cid:111) H CP acting among the three generations ofcharged leptons and neutrinos. The phenomenological implications of the breaking of such asymmetry group both in the charged lepton and neutrino sectors are thus explored especiallyin connection with the CP violation appearing in the leptonic mixing matrix, U PMNS .First of all we have derived in Section 2 all the possible generalised CP transformationsfor all the representations of the T (cid:48) group i.e. we found all possible outer automorphismsof the group T (cid:48) following the consistency conditions given in [41, 42, 44]. We have chosen asgeneralised CP symmetry the transformation u : ( T, S ) → ( T , S T S T ) which correspondsto a Z symmetry and it is defined up to an inner automorphism. The transformation u isparticularly convenient since, in the basis chosen for the generators S and T , for the 1 and3-dimensional representations it is trivially defined as the identity up to a global unphysicalphase θ r where the index r refers to the representation. More importantly we found that,given this specific generalised CP symmetry combined with T (cid:48) , it is possible to fix the vevs30f the flavon fields to real values in such a way that no complex phases, and thus no physicalCP violation, stem from the vevs themselves.Moreover, for a list of possible renormalisable operators, namely λ O = λ ( A × B × C ) where λ is the coupling constant and A , B , C are fields, we derived the constraints on the phase λ under the assumption of invariance under the generalised CP transformation. This list ofpossible operators can be used to construct a CP-conserving renormalisable superpotentialfor the flavon sector and therefore can be used in order to show that real vev structures canbe achieved.Motivated by this preliminary study we constructed in Section 3 a supersymmetric flavourmodel able to describe the observed patterns and mixing for three generations of chargedlepton fields and the three light active neutrinos.We have constructed an effective superpotential with operators up to mass dimensionsix giving the charged lepton and neutrino Yukawa couplings and the Majorana mass termfor the RH neutrinos. Naturally small neutrino masses are generated by the type I see-saw mechanism. At leading order, the mixing in the neutrino sector is described by thetri-bimaximal mixing, which is then perturbed by additional contributions coming from thecharged lepton sector. The latter are responsible for the compatibility of the predictions onthe mixing angles with the experimental values and, in particular, with the non-zero value ofthe reactor mixing angle θ .Similarly to what was found in [35], we find that both types of neutrino mass spectrum- with normal ordering (NO) and inverted ordering (IO) - are possible within the model andthat the NO spectrum can be of two varieties, A and B. They differ by the value of the lightestneutrino mass. Only one spectrum of the IO type is compatible with the model. For each ofthe three neutrino mass spectra, NO A, NO B and IO, the absolute scale of neutrino masses ispredicted with relatively small uncertainty. This allows us to predict the value of the sum ofthe neutrino masses for the three spectra. The Dirac phase δ is predicted to be approximately δ ∼ = π/ π/
2. More concretely, for the best fit values of the neutrino mixing angles quotedin Table 1 we get δ = 93 . ◦ or δ = 266 . ◦ . The deviations of δ from the values 90 ◦ and270 ◦ are correlated with the deviation of atmospheric neutrino mixing angle θ from π/ ν A, etc.), ii) experiments aiming to determine the absoluteneutrino mass scale, and iii) experiments searching for neutrinoless double beta decay.It is important to comment that in this model the physical CP violation emerging in thePMNS mixing matrix stems only from the charged lepton sector. Indeed, in the neutrinosector the Majorana mass matrix and the Dirac Yukawa couplings are real and the CP viola-tion is caused by the complex CP violating phases arising in the charged lepton sector. Thepresence of the latter is a consequence of the requirement of invariance of the theory underthe generalised CP symmetry at the fundamental level and of the complex CGs of the T (cid:48) group.We also found that the residual group in the charged lepton sector is trivial i.e. G e = ∅ andsince the phases of the flavon vevs are completely independent of the coupling constants of theflavon superpotential, the CP symmetry is broken geometrically (according to the definition31f “geometrical CP violation” given in [34]). In the neutrino sector, the residual subgroupis instead a Klein group, G ν = K = Z × Z with one Z coming from the generalised CPsymmetry H CP .Concluding, we have shown that the spontaneous breaking of a symmetry group G f = T (cid:48) (cid:111) H CP in the leptonic sector through a real flavon vev structure is possible and, at thesame time, CP violation in the leptonic sector can take place. In this scenario the appearanceof the CP violating phases in the PMNS mixing matrix can be traced to two factors: i) therequirement of invariance of the Lagrangian of the theory under H CP at the fundamentallevel, and ii) the complex CGs of the T (cid:48) group. The model we have constructed allowsfor two neutrino mass spectra with normal ordering (NO) and one with inverted ordering(IO). For each of the three spectra the absolute scale of neutrino masses is predicted withrelatively small uncertainty. The value of the Dirac CP violation (CPV) phase δ in the leptonmixing matrix is predicted to be δ ∼ = π/ π/
2. Thus, the CP violating effects in neutrinooscillations are predicted to be nearly maximal and experimentally observable. We presentalso predictions for the sum of the neutrino masses, for the Majorana CPV phases and forthe effective Majorana mass in neutrinoless double beta decay. The predictions of the modelcan be tested in a variety of ongoing and future planned neutrino experiments.
Note added
After the submission of our article to the arXiv, an update of the global fits to the neutrinooscillation data appeared [51]. The results reported in [51] are in agreement with the predic-tions of our model. More specifically, the authors of [51] find that the best fit value of δ is δ ≈ π/
2, which is one of the two possible values predicted by in our model. Similar resultson δ were obtained in the global analysis of the neutrino oscillation data performed in [52]. Acknowledgements
The work of M. Spinrath was partially supported by the ERC Advanced Grant no. 267985“DaMESyFla”, by the EU Marie Curie ITN “UNILHC” (PITN-GA-2009-237920). A. Meroniacknowledges MIUR (Italy) for financial support under the program Futuro in Ricerca 2010(RBFR10O36O). This work was supported in part also by the European Union FP7 ITNINVISIBLES (Marie Curie Actions, PITN-GA-2011-289442-INVISIBLES), by the INFN pro-gram on “Astroparticle Physics”(A.M., S.T.P.) and by the World Premier International Re-search Center Initiative (WPI Initiative), MEXT, Japan (S.T.P.).
A Technicalities about T (cid:48) The group T (cid:48) is the double covering group of A and it is defined through the algebraicrelations: S = R R = T = ( ST ) = E RT = T R . (A.1)The number of the unitary irreducible representations of a discrete group is equal to thenumber of the conjugacy classes. For T (cid:48) they are seven, which are classified given the elements32 , S , because R ≡ S , we summarize them as1 C : { E } , (cid:48) C : (cid:8) S (cid:9) C : (cid:8) T, S T S, ST, T S (cid:9) , (cid:48) C : (cid:8) T , S T ST, S T S, S T (cid:9) (cid:48)(cid:48) C : (cid:8) S T, ST S, S T, S T S (cid:9) , (cid:48)(cid:48)(cid:48) C : (cid:8) S T , T ST, T S, ST (cid:9) C : (cid:8) S, S , T ST , T ST, S T ST , S T ST (cid:9) (A.2)The representations of T (cid:48) can be expressed as : T = 1 , R = 1 , S = 1 ; (cid:48) : T = ω , R = 1 , S = 1 ; (cid:48)(cid:48) : T = ω , R = 1 , S = 1 ; : T = (cid:18) ω ω (cid:19) , R = (cid:18) − − (cid:19) , S = − i √ − (cid:113) p (cid:113) ¯ p i √ ; (cid:48) : T = (cid:18) ω ω (cid:19) , R = (cid:18) − − (cid:19) , S = − i √ − (cid:113) p (cid:113) ¯ p i √ ; (cid:48)(cid:48) : T = (cid:18) ω
00 1 (cid:19) , R = (cid:18) − − (cid:19) , S = − i √ − (cid:113) p (cid:113) ¯ p i √ ; : T = ω
00 0 ω , R = , S = −
13 2 ω ω ω −
13 2 ω ω ω − . We use the definition of the representation of T (cid:48) given in [24] in which ω and p are fixed tobe respectively ω = e π and p = e i π . Finally T (cid:48) has n = 13 subgroups excluding the wholegroup: • Trivial subgroup E = { E } ; • Z subgroup Z S = (cid:8) E, S (cid:9) ; • Z subgroups Z T = (cid:8) E, T, T (cid:9) , Z S T S = (cid:8) E, S T S, S T ST (cid:9) , Z ST = (cid:8) E, ST, S T S (cid:9) , Z T S = (cid:8) E, T S, S T (cid:9) ; • Z subgroups Z S = (cid:8) E, S, S , S (cid:9) , Z T ST = (cid:8) E, T ST , S , S T ST (cid:9) , Z T ST = (cid:8) E, T ST, S , S T ST (cid:9) ; • Z subgroups Z S T = (cid:8) E, S T, T , S , T, S T (cid:9) , Z ST S = (cid:8) E, ST S, S T ST, S , S T S, T ST (cid:9) , Z S T = (cid:8) E, S T, S T S, S , ST, T S (cid:9) , Z S T S = (cid:8) E, S T S, S T , S , T S, ST (cid:9) .A complete table of the CGs coefficients can be found in [35].33 U (2) U (1) Y T (cid:48) U (1) R Z Z Z Z Z Z Ξ , ¯Ξ , , A , ¯Σ A , , B , ¯Σ B , , A (cid:48) , ¯Σ A (cid:48)(cid:48) , (cid:48) , (cid:48)(cid:48) B (cid:48) , ¯Σ B (cid:48)(cid:48) , (cid:48) , (cid:48)(cid:48) A (cid:48)(cid:48) , ¯Σ A (cid:48) , (cid:48)(cid:48) , (cid:48) B (cid:48)(cid:48) , ¯Σ B (cid:48) , (cid:48)(cid:48) , (cid:48) C (cid:48)(cid:48) , ¯Σ C (cid:48) , (cid:48)(cid:48) , (cid:48) A (cid:48) , ¯Σ A (cid:48)(cid:48) , (cid:48) , (cid:48)(cid:48) A (cid:48)(cid:48) , ¯Σ A (cid:48) , (cid:48)(cid:48) , (cid:48) A , ¯∆ A , , B , ¯∆ B , , A (cid:48) , ¯∆ A (cid:48)(cid:48) , (cid:48) , (cid:48)(cid:48) A (cid:48) , ¯∆ A (cid:48)(cid:48) , (cid:48) , (cid:48)(cid:48) A , ¯∆ A , , B , ¯∆ B , , C , ¯∆ C , , D , ¯∆ D , , E , ¯∆ E , , Table 9:
List of the messengers fields and their transformation properties.
B Messenger Sector
The effective model we have considered so far contains only non-renormalisable operatorsallowed by the symmetry group G f × Z × Z × Z × Z × U (1) R . But in fact using onlythis symmetry there would be more effective operators allowed which might spoil our modelpredictions.Therefore we discuss in in this section we a so-called ultraviolet completion defining arenormalisable theory which gives the effective model described in the previous sections afterintegrating out the heavy messenger superfields. In this way we can justify why we have chosenonly a certain subset of the effective operators allowed by the symmetries. The quantumnumbers of the messenger fields are given in Table 9. We label them with Σ, Ξ and ∆ for thecharged lepton, neutrino and flavon sector respectively.For the charged lepton sector we find the renormalisable superpotential W ren e W ren e = L φ Σ A (cid:48)(cid:48) + L φ Σ A + ¯ E H d ¯Σ A (cid:48) + L ˆ φ Σ C (cid:48)(cid:48) + ζ ¯Σ C (cid:48) Σ A (cid:48)(cid:48) + H d ¯Σ A Σ B + ¯ E ψ (cid:48) ¯Σ B + L ψ (cid:48) Σ A (cid:48)(cid:48) + ψ (cid:48)(cid:48) ¯Σ A (cid:48) Σ A + L ˜ φ Σ B (cid:48) + ˜ ζ (cid:48) ¯Σ B (cid:48) Σ B (cid:48) + H d ¯Σ B (cid:48)(cid:48) Σ C (cid:48) + ¯ E ˜ ψ (cid:48)(cid:48) ¯Σ C (cid:48)(cid:48) + L ψ (cid:48) Σ A (cid:48) + ψ (cid:48)(cid:48) ¯Σ A (cid:48)(cid:48) Σ A (cid:48)(cid:48) , (B.1)which through the diagrams of Fig. 3 generates at low energy the non-renormalisable super-potential W Y e of Eq. (3.12). 34 φ ¯ E H d L ˆ φ ¯ E H d LφLψ ′ L ˜ φ ¯ Eψ ′ ¯ Eψ ′ ¯ E ˜ ψ ′′ ζH d ψ ′′ H d ˜ ζ ′ H d ¯Σ A ′ Σ A ′′ Σ C ′′ Σ A ′′ ¯Σ C ′ ¯Σ A ′ Σ A ¯Σ A Σ B ¯Σ B Σ A ′′ ¯Σ A ′ Σ A ¯Σ A Σ B ¯Σ B Σ B ′′ ¯Σ B ′ Σ B ′ ¯Σ B ′′ Σ C ′ ¯Σ C ′′ L ψ ′ ψ ′′ ¯ E H d Σ A ′′ ¯Σ A ′ Σ A ′ ¯Σ A ′′ Figure 3:
The supergraphs before integrating out the messengers for the charged leptonsector.
For the neutrino and the flavon sector we obtained similarly to the previous case W ren ν = N ξ + N ρ + N ˜ ρ + LN Ξ + H u ¯Ξ ρ + H u ¯Ξ ˜ ρ , (B.2) W renflavon = D φ φ ∆ B + ε φ ¯∆ B + D φ ζ (cid:48)(cid:48) ∆ B + ˜ D φ ˜ φ ∆ C + ε ˜ φ ¯∆ C + ˜ D φ ˜ ζ (cid:48) ∆ C + ˆ D φ ˆ φ ∆ D + ε ˆ φ ¯∆ D + ˆ D φ ˜ ζ (cid:48)(cid:48) ∆ E + ε ˜ φ ¯∆ E + ˜ S ζ ˜ ζ (cid:48)(cid:48) ∆ A (cid:48) + ˜ ζ (cid:48)(cid:48) ˜ ζ (cid:48)(cid:48) ¯∆ A (cid:48)(cid:48) + S ε ε ∆ B + ε ε ¯∆ B + ε ε ¯∆ A + S ε ∆ A ∆ A . (B.3)The corresponding diagrams that generate the effective operators in the neutrino and flavonsector in our model are given in Figs. 4 and 5. References [1] K. Nakamura and S. T. Petcov, in J. Beringer et al. (Particle Data Group), Phys. Rev.D (2012) 010001.[2] S.M. Bilenky, J. Hosek and S.T. Petcov, Phys. Lett. B (1980) 495.[3] E. Molinaro and S. T. Petcov, Eur. Phys. J. C (2009) 93.[4] A. Ibarra, E. Molinaro and S. T. Petcov, Phys. Rev. D (2011) 013005.[5] L. Wolfenstein, Phys. Lett. B (1981) 77; S.M. Bilenky, N.P. Nedelcheva and S.T.Petcov, Nucl. Phys. B (1984) 61; B. Kayser, Phys. Rev. D (1984) 1023.35 N ξ NN ρ , ˜ ρNL ρ , ˜ ρH u Ξ ¯Ξ Figure 4:
The supergraphs before integrating out the messengers for the neutrino sector. D φ φ , ζ ′′ ∆ B ¯∆ B ε φ ∆ C ¯∆ C ˜ D φ ˜ φ , ˜ ζ ′ ε ˜ φ ˆ D φ ˆ φ ∆ D ¯∆ D ε ˜ φ ˆ D φ ∆ E ¯∆ E ε ˜ φ ˜ ζ ′′ ˜ S ζ ˜ ζ ′′ ∆ A ′ ¯∆ A ′′ ˜ ζ ′′ ˜ ζ ′′ S ε ε ∆ B ¯∆ B ε ε ε ε ε ε S ε ¯∆ A ¯∆ A ∆ A ∆ A Figure 5:
The supergraphs before integrating out the messengers for the flavon sector. Wehave omitted for simplicity the supergraphs of the higher order corrections in the flavon su-perpotential. (2012) 013012.[7] M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado and T. Schwetz, JHEP (2012) 123.[8] C. Giunti and M. Tanimoto, Phys. Rev. D (2002) 053013, Phys. Rev. D (2002)113006.[9] P.H. Frampton, S.T. Petcov and W. Rodejohann, Nucl. Phys. B (2004) 31.[10] S.T. Petcov and W. Rodejohann, Phys. Rev. D71 (2005) 073002.[11] A. Romanino, Phys. Rev. D (2004) 013003.[12] K.A. Hochmuth, S.T. Petcov and W. Rodejohann, Phys. Lett B (2007) 177.[13] D. Marzocca, S. T. Petcov, A. Romanino, M. Spinrath, JHEP (2011) 009.[14] D. Marzocca, S. T. Petcov, A. Romanino and M. C. Sevilla, JHEP (2013) 073.[15] G. Altarelli, F. Feruglio and I. Masina, Nucl. Phys. B (2004) 157; S. F. King, JHEP (2005) 105; I. Masina, Phys. Lett. B (2006) 134; S. Antusch and S. F. King,Phys. Lett. B (2005) 42; S. Dev, S. Gupta and R. R. Gautam, Phys. Lett. B (2011) 527; S. Antusch and V. Maurer, Phys. Rev. D (2011) 117301; A. Meroni,S.T. Petcov and M. Spinrath, Phys. Rev. D (2012) 113003; C. Duarah, A. Das andN. N. Singh, [arXiv:1210.8265].[16] W. Chao and Y. -j. Zheng, JHEP (2013) 044; D. Meloni, JHEP (2012) 090;S. Antusch, C. Gross, V. Maurer and C. Sluka, Nucl. Phys. B (2013) 255; G. Altarelli,F. Feruglio, L. Merlo and E. Stamou, JHEP (2012) 021; G. Altarelli, F. Feruglioand L. Merlo, [arXiv:1205.5133]; F. Bazzocchi and L. Merlo, [arXiv:1205.5135]; S. Gollu,K. N. Deepthi and R. Mohanta, [arXiv:1303.3393].[17] C. H. Albright, A. Dueck and W. Rodejohann, Eur. Phys. J. C (2010) 1099.[18] P. F. Harrison, D. H. Perkins and W. G. Scott, Phys. Lett. B (2002) 167; Phys.Lett. B (2002) 163; Z. Z. Xing, Phys. Lett. B (2002) 85; X. G. He and A. Zee,Phys. Lett. B (2003) 87; see also L. Wolfenstein, Phys. Rev. D (1978) 958.[19] S.T. Petcov, Phys. Lett. B (1982) 245.[20] F. Vissani, [arXiv:hep-ph/9708483]; V. D. Barger, S. Pakvasa, T. J. Weiler and K. Whis-nant, Phys. Lett. B (1998) 107; A. J. Baltz, A. S. Goldhaber and M. Goldhaber,Phys. Rev. Lett. (1998) 5730.[21] S. F. King and C. Luhn, Rept. Prog. Phys. (2013) 056201.[22] G. Altarelli and F. Feruglio, Rev. Mod. Phys. (2010) 2701.[23] H. Ishimori et al. , Prog. Theor. Phys. Suppl. (2010) 1.3724] P. H. Frampton and T. W. Kephart, Int. J. Mod. Phys. A (1995) 4689; F. Feruglio,C. Hagedorn, Y. Lin and L. Merlo, Nucl. Phys. B (2007) 120 [Erratum-ibid. (2010) 127]; G. J. Ding, Phys. Rev. D (2008) 036011; P. H. Frampton, T. W. Kephartand S. Matsuzaki, Phys. Rev. D (2008) 073004; D. A. Eby, P. H. Frampton andS. Matsuzaki, Phys. Lett. B (2009) 386; P. H. Frampton and S. Matsuzaki, Phys.Lett. B (2009) 347.[25] M.-C. Chen and K. T. Mahanthappa, Phys. Lett. B (2007) 34.[26] M.-C. Chen and K. T. Mahanthappa, Phys. Lett. B (2009) 444.[27] F. P. An et al. [DAYA-BAY Collaboration], Phys. Rev. Lett. (2012) 171803.[28] J. K. Ahn et al. [RENO Collaboration], Phys. Rev. Lett. (2012) 191802.[29] K. Abe et al. [T2K Collaboration], Phys. Rev. Lett. (2011) 041801.[30] Y. Abe et al. [DOUBLE-CHOOZ Collaboration], Phys. Rev. Lett. (2012) 131801;Y. Abe et al. [DOUBLE-CHOOZ Collaboration], Phys. Rev. D (2012) 052008.[31] P. Adamson et al. [MINOS Collaboration], Phys. Rev. Lett. (2011) 181802.[32] P. Minkowski, Phys. Lett. B (1977) 421; M. Gell-Mann, P. Ramond and R. Slansky inSanibel Talk, CALT-68-709, Feb 1979, and in Supergravity (North Holland, Amsterdam1979); T. Yanagida in
Proc. of the Workshop on Unified Theory and Baryon Number ofthe Universe , KEK, Japan, 1979; S.L.Glashow, Cargese Lectures (1979); R. N. Mohap-atra and G. Senjanovic, Phys. Rev. Lett. (1980) 912.[33] J.-Q. Chen and P.-D. Fan, J. Math. Phys. (1998) 5519.[34] G. C. Branco, J. M. Gerard and W. Grimus, Phys. Lett. B (1984) 383.[35] A. Meroni, S. T. Petcov and M. Spinrath, Phys. Rev. D (2012) 113003.[36] M. -C. Chen, J. Huang, K. T. Mahanthappa and A. M. Wijangco, arXiv:1307.7711.[37] S. Antusch, S. F. King, C. Luhn and M. Spinrath, Nucl. Phys. B (2011) 477.[38] S. Antusch, S. F. King and M. Spinrath, Phys. Rev. D (2013) 096018; S. F. King,JHEP (2013) 137 and arXiv:1305.4846; S. Antusch, C. Gross, V. Maurer andC. Sluka, arXiv:1305.6612 and arXiv:1306.3984.[39] S. Antusch and V. Maurer, Phys. Rev. D (2011) 117301.[40] S. Antusch and M. Spinrath, Phys. Rev. D (2009) 095004; M. Spinrath,arXiv:1009.2511; S. Antusch, S. F. King and M. Spinrath, arXiv:1311.0877.[41] M. Holthausen, M. Lindner and M. A. Schmidt, JHEP (2013) 122.[42] F. Feruglio, C. Hagedorn and R. Ziegler, JHEP (2013) 027.[43] A. Aranda, C. D. Carone and R. F. Lebed, Phys. Rev. D (2000) 016009.[44] G.J. Ding, S. F. King and A. J. Stuart, arXiv:1307.4212.3845] W. Grimus and P. O. Ludl, J. Phys. A (2012) 233001.[46] P.I. Krastev and S. T. Petcov, Phys. Lett. B (1988) 84.[47] A. Meroni, E. Molinaro and S.T. Petcov, Phys. Lett. B (2012) 435;[48] S. M. Bilenky and S. T. Petcov, Rev. Mod. Phys. (1987) 671;W. Rodejohann, Int. J.Mod. Phys. E20 (2011) 1833.[49] M. Agostini et al. , arXiv:1307.4720.[50] K. Eitel et al. , Nucl. Phys. Proc. Suppl.143