Quasidegeneracy of Majorana Neutrinos and the Origin of Large Leptonic Mixing
aa r X i v : . [ h e p - ph ] F e b Quasidegeneracy of Majorana Neutrinosand the Origin of Large Leptonic Mixing
G.C. Branco a , b , , M.N. Rebelo a , J.I. Silva-Marcos a , Daniel Wegman a , a Centro de F´ısica Te´orica de Part´ıculas - CFTP, b Departamento de F´ısica,Instituto Superior T´ecnico - IST, Universidade de Lisboa - UL,Avenida Rovisco Pais, 1049-001 Lisboa, Portugal
Abstract
We propose that the observed large leptonic mixing may just reflect a quasidegeneracy of three Ma-jorana neutrinos. The limit of exact degeneracy of Majorana neutrinos is not trivial, as leptonic mixingand even CP violation may occur. We conjecture that the smallness of | U | , when compared to the otherelements of U PMNS , may be related to the fact that, in the limit of exact mass degeneracy, the leptonicmixing matrix necessarily has a vanishing element. We show that the lifting of the mass degeneracycan lead to the measured value of | U | while at the same time accommodating the observed solar andatmospheric mixing angles. In the scenario we consider for the breaking of the mass degeneracy there isonly one CP violating phase, already present in the limit of exact degeneracy, which upon the lifting ofthe degeneracy generates both Majorana and Dirac-type CP violation in the leptonic sector. We analysesome of the correlations among physical observables and point out that in most of the cases considered,the implied strength of leptonic Dirac-type CP violation is large enough to be detected in the next roundof experiments. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]
Introduction
The observed pattern of fermion masses and mixing continues being a major puzzle in particle physicsand the discovery of large leptonic mixing rendered the question even more intriguing. A large number ofmodels have been suggested in the literature for providing an understanding of neutrino masses and mixing.These models cover a large number of possibilities, going from models with discrete abelian or non-abeliansymmetries [1] to the suggestion that in the neutrino sector, anarchy prevails [2].In this paper we conjecture that the observed large mixing in the lepton sector may just reflect a Majoranacharacter of neutrinos and quasidegeneracy of neutrino masses. It is well known that, for Dirac neutrinos,leptonic mixing can be rotated away in the limit of exact neutrino mass degeneracy. For Majorana neutrinos,it has been pointed out that leptonic mixing and even CP violation can occur in the limit of exact neutrinomass degeneracy and in this limit, leptonic mixing is characterized by two angles and one CP violatingphase [3]. In this limit, the leptonic unitarity triangles are collapsed in a line, since one of the entries ofthe leptonic mixing vanishes, thus implying no Dirac-type CP violation. However, the Majorana trianglesdo not all collapse into the real or imaginary axis, thus implying [4] CP violation of Majorana type. Weidentify the zero entry of the leptonic mixing matrix with U and show that a small perturbation around thedegenerate limit generates the observed neutrino mass differences as well as leptonic mixing in agreementwith experiment, including the recent measurements of the smallest mixing angle, q , at reactor [5], andaccelerator [6] neutrino experiments. In this framework, one also finds a possible explanation for the small-ness of | U | , compared to the other entries of the U PMNS . This may just reflect the fact that, in the exactdegenerate limit of Majorana neutrinos, one of the entries of U PMNS necessarily vanishes.As soon as it became clear that the experimental evidence favoured a nonvanishing U , many proposals[7] were put forward in the literature analysing how small perturbations around various textures obtainedfrom symmetries, could accommodate a non-vanishing U while also correctly reproducing the data on thesolar and atmospheric mixing angles. The distinctive feature of our proposal is the fact that we start fromthe non-trivial limit of exactly degenerate Majorana neutrinos.This paper is organised as follows. In the next section, we study the limit of exact degeneracy of threeMajorana neutrinos, pointing out that in this limit the Majorana mass matrix is proportional to a unitarymatrix and describing the implications for leptonic mixing and CP violation. In section 3, we study thelifting of the mass degeneracy with the generation of neutrino mass differences and a non-vanishing U .We analyse in detail the case where the unperturbed leptonic mixing is given by some of the most popularAns¨atze, allowing for Majorana-type CP violation, with special emphasis on the tribimaximal case [8]. Weconsider a scenario for the breaking of the degeneracy, where there is only one CP violating phase which,upon the lifting of the degeneracy, generates both Majorana and Dirac-type CP violation. Finally, in section4 we present our conclusions. 1 The Limit of Exact Degeneracy
Without loss of generality, we choose to work in a weak basis (WB) where the charged lepton mass matrixis diagonal, real and positive. We assume three left-handed neutrinos and consider a Majorana mass termwith the general form: L mass = − ( n L a ) T C − ( M o ) ab n L b + h . c . (1)where n L a stand for the left-handed weak eigenstates and M o is a 3 × M o is diagonalized by a unitary matrix U o through U To M o U o = diag ( m n , m n , m n ) , itfollows that in the limit of exact neutrino mass degeneracy, M o can be written: M o = µ S o (2)where µ is the common neutrino mass and S o = U ∗ o U † o . In the limit of exact degeneracy, a novel featurearises, namely M o is proportional to the symmetric unitary matrix S o . Under a WB transformation cor-responding to a rephasing of both n L and the charged lepton fields, the neutrino mass matrix transformsas: M o → L M o L (3)with L ≡ diag ( e i j , e i j , e i j ) . As a result, the individual phases of M o have no physical meaning, butone can construct polynomials in ( M o ) i j which are rephasing invariant [9] such as ( M ∗ o ) ( M ∗ o ) ( M ∗ o ) or ( M o ) ( M ∗ o ) ( M o ) . The fact that S o is symmetric and unitary implies that in general S o can beparametrized by two angles and one phase. In Ref. [3], the limit of exact degeneracy for Majorana neu-trinos was analysed in some detail and it was shown that leptonic mixing and even CP violation can occurin that limit. Leptonic mixing can be rotated away if and only if there is CP invariance and all neutrinoshave the same CP parity [10], [11]. Furthermore, it was also shown in Ref [3] that in the case of differentCP parities, the most general matrix S o can be parametrized in terms of two rotations with three-by-threeorthogonal matrices having only a two-by-two non diagonal block each, corresponding to a single mixingangle, together with one diagonal matrix with one phase: S o = c f s f s f − c f · c q s q s q − c q
00 0 e i a · c f s f s f − c f (4)this equation is of the form: S o = O ( f ) O ( q ) e i a O ( f ) (5)with each orthogonal matrix O i j chosen to be symmetric. Using the fact that S o = U ∗ o U † o one concludes that, in this limit, the leptonic mixing matrix is given by: U o = O ( f ) O (cid:18) q (cid:19) i
00 0 e − i a (6)2p to an orthogonal rotation of the three degenerate neutrinos.Given the Majorana character of neutrino masses, it is clear that even in the limit of exact degeneracy withCP conservation, but with different CP-parities, one cannot rotate U o away through a redefinition of theneutrino fields. It should be emphasized that the leptonic mixing matrix is only defined up to an orthogonalrotation of the three degenerate neutrinos. Indeed if U o diagonalizes M o so does U o O , as it is evident fromEq. (2) and the fact that S o = U ∗ o U † o . Without loss of generality one can eliminate the matrix O. It isimportant to notice that U o always has one zero entry which in the above parametrization appears in the ( ) position. This may be a hint that the limit of exact degeneracy is a good starting point to perform asmall perturbation around it, leading to the lifting of the degeneracy and the generation of a non-zero U e .At this stage, it should be noted that although the limit of exact degeneracy necessarily implies a zero entryin U o the location of the zero is not fixed. If we had interchanged the rˆoles of O and O in Eq. (4),the zero entry would appear in the ( ) position. Our choice of Eq. (4) was dictated by the experimentalfact that the leptonic mixing matrix has a small entry in the ( ) position. It should be stressed that theidentification of U PMNS with U o O can only be done after the lifting of the degeneracy, which will be done inthe sequel. The matrix O will be fixed by the perturbation of M o leading to the lifting of the degeneracy. Inthe exact degenerate limit the individual elements of the matrix U o O have no physical meaning. But thereare physical quantities which do have physical meaning even in the exact degenerate limit. These quantitiesare independent of the matrix O , depending only on combinations of the angles q , f and the phase a ,entering in the parametrisation of U o given in Eq. (4). An example of such a physical quantity, will be givenin the next subsection, where we evaluate in the exact degenerate limit the strength of Majorana type CPviolation, expressed in terms of the mixing angles q , f and the phase a . Of course, this quantity does notdepend on the matrix O .It is easy to understand why a symmetric unitary 3 × S o , can be parametrized by only twoangles and one phase. On one hand, there is the freedom of choice of WB given by Eq. (3) on the otherhand for a general 3 × U , one can define an asymmetry parameter, given by [12]: A s ≡ | U | − | U | = | U | − | U | = | U | − | U | (7)In the case of a unitary symmetric matrix, one has A s =
0, which leads to the loss of one parameter.The parametrization of S o in terms of two rotations and one phase is the most general one (apart from theunphysical complex phases which can be rotated away as in Eq. (3)). This can be seen by recalling thatthe parametrization of a general unitary matrix through Euler angles involves three orthogonal rotations,usually denoted by O , O , O . The fact that S o is symmetric implies the loss of one parameter and, as aresult, only two orthogonal matrices are needed. The rotation matrix O rs that is left out, dictates the entryof U o that is zero to be, ( r , s ) or ( s , r ) depending on the order chosen for the other two orthogonal matrices. S o unitarity triangles, leptonic mixing and CP violation Since S o is a unitary matrix, one can consider S o unitarity triangles, which are analogous to the ones [4]encountered in the leptonic mixing matrix U PMNS , but with a different physical meaning. A unitarity tri-angle corresponding to orthogonality of the two first columns of S o is displayed in Figure 1. Note that theorientation of the S o unitarity triangles rotates under the rephasing of Eq.(3) and therefore it has no physical3 S L H S * L H S L H S * L H S L H S * L Figure 1: Unitarity triangle built from the first two columns of S , for a generic unitary matrix, assumingCP violationmeaning. However, the area of the S o triangles has got physical meaning, giving a measure of the strengthof Majorana-type CP violation in leptonic mixing in the case of exact degeneracy. All S o unitarity triangleshave the same area A , which equals twice the absolute value of any of the rephasing invariant quartets Q s of S o : A = | Im Q s | = (cid:12)(cid:12) cos ( q ) sin ( q ) sin ( f ) sin ( a ) (cid:12)(cid:12) (8)with | Q s | ≡ | ( S o ) i j ( S o ) ∗ ik ( S o ) ∗ l j ( S o ) lk | with i = l , j = k . In the limit of exact degeneracy, we have seen thatthe leptonic mixing matrix U o has a zero entry, which implies that there is no Dirac-type CP violation andall the U o unitarity triangles collapse to lines. However, there is CP violation of the Majorana-type, since theMajorana unitarity triangles for U o are in general not collapsed along the real and imaginary axis [4]. Oncethe degeneracy is lifted the leptonic unitarity triangles open up and Dirac-type CP violation is generated. InRef. [13], it was shown how to express, in this case, the full PMNS matrix, including the strength of Dirac-type CP violation in terms of arguments of the six independent rephasing invariant bilinears correspondingto the orientation of the sides of Majorana-type unitarity triangles, thus showing that Dirac-type CP violationin the leptonic sector with Majorana neutrinos, necessarily implies Majorana-type CP violation. For definiteness and without loss of generality, we work in the weak basis where the charged lepton massmatrix is diagonal real. As emphasized in the previous section, in the case of exactly degenerate Majorananeutrinos, mixing is meaningful and it can be parametrized by two angles and one phase.Several textures for the leptonic mixing matrix have been studied in the literature, often in the context offamily symmetries [1]. In most of the proposed schemes, the pattern of leptonic mixing is predicted butthe spectrum of masses is not constrained by the symmetries. It is therefore consistent to consider theseschemes, together with the hypothesis of quasidegeneracy of Majorana neutrinos. A different approachconnecting the leptonic mixing parameters with certain kinds of degeneracy of the neutrino mass spectrumwas followed in [14]. 4ntil recently one of the most favoured Ans¨atze, from the experimental point of view, seemed to be thetribimaximal mixing [8] which has a zero in the ( ) entry. Other interesting textures which also have azero entry in this location [15] include the democratic mixing [16], bimaximal mixing [17], golden ratiomixings [18], [19], hexagonal mixing [20] and bidodeca mixing [21], [22]. Recent measurements of q ,the smallest of the mixing angles of U PMNS as given by the standard parametrization [23], have establisheda non-zero value for this angle [24]. In the standard parametrization, U PMNS is given by: U PMNS = c c s c s e − i d − s c − c s s e i d c c − s s s e i d s c s s − c c s e i d − c s − s c s e i d c c · P , (9)where c i j ≡ cos (cid:0) q i j (cid:1) , s i j ≡ sin (cid:0) q i j (cid:1) , with all q i j in the first quadrant, d is a Dirac-type phase and P = diag ( , e i a , e i b ) with a and b denoting the phases associated with the Majorana character of neutrinos.The clear experimental evidence for a non-zero q has motivated a series of studies on how to generate anon-vanishing q through a small perturbation of the tribimaximal and other schemes which predict q = The distinctive feature of our analysis , is the fact that we start from a non-trivial limit ofthree exactly degenerate Majorana neutrinos. In the previous section, we presented the most general mixingmatrix U o in this limit and explained that it can be parametrized by two angles and one CP violating phase: U o = O ( q , f ) · K (10)with K a diagonal matrix such as the one written in Eq. (6). This choice for the matrix K implies that inthe CP conserving limit corresponding to a = p , one neutrino has a CP parity different from the othertwo. Otherwise, in the limit of exact degeneracy, with CP conservation and all neutrinos having the sameCP parity, the two angles f and q could be rotated away.Lifting the degeneracy corresponds to adding a small perturbation to S o M = µ ( S o + e Q o ) (11)the matrix Q o is fixed in such a way that the correct neutrino masses are obtained. It will be a function ofthe neutrino mass differences given in terms of D m and D m defined by: D m = m − m D m = | m − m | (12)as well as the overall mass scale µ . The parameter e is chosen as: e ≡ D m µ (13)Quasidegeneracy forces the overall mass scale to be much larger than the neutrino mass differences andguarantees the smallness of the perturbation parameter e . See Table 1 and subsequent comments.Our strategy for confronting the data on neutrino masses and mixing is the following:5 i ) We assume that the physics responsible for the lifting of the degeneracy, does not introduce new sourcesof CP violation beyond the phase a , already present in the limit of exact degeneracy. As a result, after thelifting of the degeneracy, the leptonic mixing matrix is given by: U PMNS = U o · O (14)where O is an orthogonal matrix, parametrized by small angles. The fact that O is orthogonal, rather thana general unitary matrix, implies that U PMNS still diagonalizes S o , thus establishing a strong connection be-tween the degenerate and quasidegenerate case. This is particularly relevant since we shall take as startingpoint for U o some of the most interesting examples considered in the literature based on symmetries andwith a zero in the ( ) entry of U o . ( ii ) After the lifting of the degeneracy, the single phase a will generate both Dirac and Majorana-type CPviolations. This is a distinctive feature of our framework.With the notation of Eq.(14), Q o introduced in Eq. (11) is determined by: e Q o = U ∗ o · O (cid:18) µ D n − (cid:19) O T · U † o , D n = diag ( m n , m n , m n ) (15)In the limit of exact degeneracy the matrix O has no physical meaning, it only acquires meaning with thelifting of the degeneracy. A striking feature is the fact that new sources of CP violation are not introduced.However, once the matrix O is included, the CP violating phase present in K ceases to be a factorizablephase and in general gives rise to Dirac-type CP violation.The matrix O will be parametrized by three mixing angles which we denote by: O = O O O = c f s f − s f c f
00 0 1 c f s f − s f c f c f s f − s f c f (16)Our choice of U o ’s is based on the fact that q is known to be a small angle. Furthermore, in each case, theresulting O matrices represent small perturbations around U o matrices. Once the matrix O is fixed and thescale µ of neutrino masses is specified, Q o can be computed from Eq. (15)In our analysis, we use data from the global fit of neutrino oscillations provided in Ref. [24] requiringagreement within 1 s range. Table 1 summarizes the data obtained from Ref. [24]. From Table 1, assuming µ ∼ . e of the order 5 × − .In what follows we discuss separately several different cases of interest.6able 1: Neutrino oscillation parameter summary. For D m , sin q , sin q , and d the upper (lower) rowcorresponds to normal (inverted) neutrino mass hierarchy.Parameter Best fit 1 s range D m [ − eV ] D m [ − eV ] D m [ − eV ] q q ( . ) q q q d p pd -0.03 p p In this case our starting point is U o = U T BM · K with: U T BM = √ √ √ − √ √ √ − √ − √ and K = diag ( , i , e − i a / ) (17)In the notation of Eq. (6), this ansatz corresponds to f = ◦ and cos (cid:0) q (cid:1) = √ i.e., q = . ◦ . We allowthe angle a to vary, together with the three angles of the matrix O . In this example, agreement with theglobal fit for the experimental values requires lowering the values for the mixing angles q and q of U o and at the same time generating a q different from zero.Denoting the entries of U PMNS by U i j we have: | U | = | √ O + i √ O | = c c | U | = | √ O + i √ O | = s c | U | = | √ O + i √ O | = s | U | = | √ O − i √ O + √ e − i a / O | = s c (18)The first three equations allow to determine f , f and f , the fourth one puts bounds on the phase a thusconstraining the strength of leptonic CP violation [25]. At this stage it is worth emphasizing that there isstrong experimental evidence that in the quark sector the V CKM matrix is complex even if one assumes thepossible presence of physics beyond the Standard Model [26]. As a result, it is natural to assume that theleptonic sector also violates CP.This scenario allows for a particularly simple solution since, one can reach agreement with the experimentaldata by choosing a matrix O with only one parameter different from zero, namely the angle f . In this case7he relevant O i j simplify significantly and one can express sin ( q ) , sin ( q ) and sin ( q ) simply interms of f , and the phase a , or else. equivalently, in terms of | U | and the phase a :sin ( q ) ≡ | U | = sin ( f ) ( q ) ≡ sin ( q solar ) = − sin f − sin f = − | U | − | U | (20)sin ( q ) ≡ sin ( q atm ) = − √ ( a ) sin f cos f − sin f = − √ ( a ) | U | p − | U | − | U | (21)Clearly, | U | fixes the allowed range for the angle f and in this limit only sin ( q atm ) depends on the phase a . From Eq. (19) and taking the best fit value from Table 1 we obtain sin ( f ) = .
27. It is instructive todetermine Q o for this value of sin ( f ) . Making use of Eq. (15), and keeping only the dominant terms, bymaking the following approximations: µ s + D m µ ≃ µ ; µ s + D m µ ≃ µ (cid:18) + D m µ (cid:19) = µ ( + e ) (22)the explicit expression for Q o simplifies to: Q o = − . . − . e i a . + . e i a . − . e i a ( . i + . e i a ) − . − . e i a . + . e i a − . − . e i a ( . i − . e i a ) (23)It should be noticed that, even after factoring out e , most entries of the matrix Q o have modulus muchsmaller than one, thus confirming that we are doing a very small perturbation around the degeneracy limit.We find that the angle f cannot deviate significantly from the value of the Cabibbo angle. The constraintson the phase a obtained from Eq. (21), translate into bounds for the Dirac CP violating phase d . Thestrength of Dirac-type CP violation is often given in terms of the modulus of the parameter I CP defined asthe imaginary part of a quartet of the mixing matrix U PMNS , i.e., I CP ≡ Im | U i j U ∗ ik U ∗ l j U lk | with i = l , j = k .Due to the unitariry of U PMNS all quartets have the same modulus. For the standard parametrization, givenin Eq. (9), we have: I CP ≡ | sin ( q ) sin ( q ) sin ( q ) cos ( q ) sin ( d ) | (24)In our framework, with only f and a different from zero, I CP is given by: I CP = (cid:12)(cid:12)(cid:12)(cid:12) cos ( a / ) sin ( f ) cos ( f ) √ (cid:12)(cid:12)(cid:12)(cid:12) (25)and is predicted to be of order 10 − , meaning that it could be within reach of future neutrino experiments.This is a special prediction for this framework since from the values of Table 1 we can conclude that theexperimental bounds at 1 s level allow for the leptonic strength of Dirac-type CP violation to range from8 æææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò òò ò ò ò ò ò ò òò ò ò ò ò ò òò ò ò ò ò ò òò ò ò ò ò òò ò ò ò òò ò ò òò ò òò òà à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à àà à à à à à à àà à à à à à àà à à à à à àà à à à à àà à à à àà à à àà à àà à È U
13 2 s i n Θ Α=- Π Α=- Π Α= Π Α= Π ò Φ ¹ Α=- Π à Φ ¹ = Π Figure 2: sin q versus | U | obtained by perturbing tribimaximal mixing with f =
0. Each curvecorresponds to a fixed a and to f = f is the only variable. The points drifting away fromeach curve were obtained by varying also f .0 to about 4 × − . In Figure 2 we present sin ( q atm ) versus | U | . The dotted vertical lines delimit theallowed experimental values for | U | . The dotted horizontal lines delimit the two allowed experimentalregions for sin ( q atm ) according to Table 1. The authors of Ref. [24] consider the region of lower sin ( q atm ) to be experimentally favoured, therefore in our analysis we require that this region can be reached eventhough we also indicate the above region. The different solid lines correspond to our framework with onlyone parameter different from zero, the angle f , and for different values of the phase a as indicated in thefigure. The values for this phase are chosen in such a way as to give an indication of the intervals that arecompatible with the experimental data. Points represented by squares and triangles where obtained withone additional mixing angle, f , different from zero. Squares and triangles correspond to different values ofthe phase a respectively, as indicated in the figure. In Figure 3 we plot I CP versus | U | . Again the dottedvertical lines delimit the allowed experimental values for | U | and the different solid lines correspond toour framework with only one mixing angle different from zero and for different values of the phase a asindicated in the figure. The values chosen for this phase are based on the information contained in Figure2. Points represented by squares and triangles where obtained with one additional mixing angle differentfrom zero, which in this case was chosen to be f . Squares and triangles correspond to different values of9 æææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò ò òò ò ò ò ò ò ò òò ò ò ò ò ò ò òò ò ò ò ò ò òò ò ò ò ò òò ò ò ò òò ò ò òò ò òò òà à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à à àà à à à à à à àà à à à à à à àà à à à à à àà à à à à à àà à à à à àà à à à àà à à àà à àà à È U
13 2 I C P Φ ¹ Α=- Π Φ ¹ Α=- Π Φ ¹ Α= Π Φ ¹ Α= Π ò Φ ¹ Φ ¹ Α=- Π à Φ ¹ Φ ¹ Α= Π Figure 3: I CP versus | U | obtained by perturbing tribimaximal mixing with f =
0. Each curve corre-sponds to a fixed a and to f = f is the only variable. The points drifting away from eachcurve were obtained by varying also f .the phase a respectively, as indicated in the figure.Concerning the neutrinoless double beta decay, this process depends on the effective Majorana mass, m ee ,defined by; m ee = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:229) k = U k m k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (26)In the above framework, the dominant terms, ignoring in particular corrections of order D m / µ are: m ee = µ (cid:16) − D m µ sin ( f ) (cid:17) = µ (cid:16) − D m µ | U | (cid:17) (27)Agreement with the present experimental bounds, [27] taking into account nuclear physics uncertainties[29] requires | m ee | to be smaller than 0 . eV . The Heidelberg-Moscow experiment [28] claimed to have10btained a non-zero result close to 0 . eV which would imply all three neutrino masses close to 1 eV .These masses are somewhat above the bound favoured by cosmology, however the cosmological bounddepends on model assumptions and on the data set that is taken into consideration [30].In this framework, the angle f cannot deviate significantly from the value of the Cabibbo angle evenwhen we extend it to include other non-zero mixing angles. In fact, the range of the allowed experimentalparameters given in Table 1 can accommodate non zero values for the two other angles in the matrix O requiring them to be smaller than the Cabibbo angle. In this case the simple expressions given above mustbe replaced by somewhat more cumbersome and less transparent ones. The solar angle obtained in theunperturbed tribimaximal mixing case is larger than the allowed experimental values. The angle f is theonly one in O capable of lowering its value. The effect of the other two mixing angles is the opposite. As stated before, we analysed perturbations around some of the well known mixing textures considered inthe literature with a zero in the (13) entry. Examples of such textures include the democratic mixing, U DM [16], bimaximal mixing U BM [17], golden ratio mixings U GRM [18], U GRM [19], hexagonal mixing U HM [20] and bidodeca mixing U BDM [21], [22]. The democratic mixing and the bimaximal mixing are of theform: U DM = √ √ − √ √ q √ − √ √ ; U BM = √ √ −
12 12 1 √ −
12 1 √ These two cases are very constrained in our framework, since they correspond to sin ( q sol ) = . f , agreement with the experimental values given in Table 1 ishardly possible at 1 s level. Therefore, we do not further analyse these two cases.The other textures mentioned above are U GRM , U GRM and the hexagonal mixing U HM which coincideswith the bidodeca mixing U BDM : U GRM = r (cid:16) + √ (cid:17) q + √ − √ + √ q + √ √ √ + √ − q + √ √ ! ; U GRM = ( + √ ) q ( −√ ) − p −√ + √ √ √ p −√ − + √ √ √ ! U HM = U BDM = √
32 12 − √ q
32 1 √ √ − q
32 1 √ The case of the golden ratio mixing 2 is less favourable than the golden ratio mixing 1, due to the fact thatthe corresponding solar angle is larger. We analysed in more detail only the cases starting with U GRM and U HM . We have scanned the allowed region of parameter space for the angles f , f , f of our perturbation,and for the phase a . Both examples have very similar features. The exact analytic expressions are obtained11 òòòòò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òòòòòòòò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òòòòòòòò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òòòòòòò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òòòòòòò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òòòòòòò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òòòòòòò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òàààààà à à à à à à à à à à à à à à à à à à à à à à à àààààààà à à à à à à à à à à à à à à à à à à à à à à à àààààààà à à à à à à à à à à à à à à à à à à à à à à à àààààààà à à à à à à à à à à à à à à à à à à à à à à à ààààààà à à à à à à à à à à à à à à à à à à à à à à à à ààààààà à à à à à à à à à à à à à à à à à à à à à à à à ààààààà à à à à à à à à à à à à à à à à à à à à à à à à àææææææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææææææææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææææææææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææææææææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææææææææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææææææææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æææææææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ È U
13 2 s i n Θ – Φ = Φ = Α =- Π – Φ = Φ = Α = – Φ = Φ = Α = Π Figure 4: sin q versus | U | obtained by perturbing golden ratio 1 with f =
0. Each curve correspondsto a fixed a and fixed value of f . The points drifting away from each curve were obtained by varying f .from Eqs. (14) and (16). A novel feature of these examples is the fact that agreement with experimentcannot be obtained with the matrix O parametrized by one mixing angle only. Furthermore, unlike thetribimaximal mixing case, it is f that is required to differ from zero and on the other hand either f or f can be zero, although not simultaneously. These new features are related to the fact that in both cases thecorresponding solar angle lies below the experimental range unlike in the tribimaximal case. As pointedout, in the tribimaximal case the angle f played a fundamental rˆole in lowering this angle. In the case of f equal to zero, U is then given by: U = ( U o ) sin ( f ) sin ( f ) + ( U o ) cos ( f ) sin ( f ) (28)it is the second term that gives the dominant contribution. The fact that there are two independent parametersin the matrix O does not allow to express sin ( q solar ) , sin ( q atm ) and I CP in terms of | U | only. Howeverit is still instructive to plot these quantities as a function of | U | for certain choices of the parameters ofthe matrix O . For illustration, we present in Figure 4 sin ( q ) versus | U | with sin ( f ) =
0. This plotis done for golden ratio 1. The hexagonal mixing case presents similar features. Each curve in the figurecorresponds to a fixed value of the parameter f and of the phase a and is therefore obtained by varying12 . The points drifting away from each curve were obtained by varying in turn f still keeping a fixed and f = a , respectively, asindicated in the figure. The figure shows that it is possible to accommodate a =
0, corresponding to the CPconserving case, however agreement with experiment in this case is only possible for a small range of theparameter space. On the other hand, fixing f = f and f to be close to the Cabibbo angle. In this paper, we present a novel proposal for the understanding of the observed pattern of leptonic mixing,which relies on the assumption that neutrinos are Majorana particles. It is argued that the observed largeleptonic mixing may arise from a quasidegeneracy of three Majorana neutrinos. The essential point is thefact that the limit of exact mass degeneracy of three Majorana neutrinos is non-trivial as lepton mixing andeven CP violation can arise. This limit is particularly interesting since in this case leptonic mixing canbe parametrized by only two mixing angles and one phase, implying that without loss of generality theleptonic mixing matrix can be written with one zero entry. We have then conjectured that the smallnessof | U | when compared to the other elements of U PMNS may result from this fact. We show that theobserved pattern of mixing and neutrino mass differences can be generated through a small perturbationof the exact degenerate case, without the introduction of additional CP violating sources. A key point inour work is the assumption that the physics responsible for the lifting of the degeneracy does not introducenew sources of CP violation. Our perturbation requires the multiplication on the right by an orthogonalmatrix. The resulting unitary matrix U o O which can be identified as the U PMNS matrix, also diagonalizesthe neutrino mass matrix in the fully degenerate case. This allows to establish a strong connection betweenthe degenerate and quasidegenerate cases and at the same time reducing the number of free parameters.Upon the lifting of degeneracy, this single phase generates both Majorana and Dirac-type CP violation inthe leptonic sector. For definiteness, we have used as the starting point for the perturbation around the limitof exact degeneracy, some of the most interesting Ans¨atze considered in the literature, which were proposedin the past assuming q =
0. We analyse correlations among physical observables, and point out that inmost of the cases considered, the implied strength of leptonic Dirac-type CP violation is large enough to bedetectable in the next round of experiments.
Acknowledgments
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