Embedding the Zee-Wolfenstein neutrino mass matrix in an SO(10) x A4 GUT scenario
aa r X i v : . [ h e p - ph ] F e b UWThPh-2007-21
Embedding the Zee–Wolfenstein Neutrino MassMatrix in an
S O (10) × A GUT Scenario
Walter Grimus ∗ and Helmut K¨uhb¨ock § Fakult¨at f¨ur Physik, Universit¨at WienBoltzmanngasse 5, A–1090 Wien, Austria
January 22, 2008
Abstract
We consider renormalizable SO (10) Yukawa interactions and put the three fermi-onic 16-plets into the 3-dimensional irreducible A representation. Scanning thepossible A representation assignments to the scalars, we find a unique case which al-lows to accommodate the down-quark and charged-lepton masses. Assuming type IIseesaw dominance, we obtain a viable scenario with the Zee–Wolfenstein neutrinomass matrix, i.e., the Majorana mass matrix with a vanishing diagonal. Contribu-tions from the charged-lepton mass matrix resolve the well-known problems withlepton mixing arising from the vanishing diagonal. In our scenario, fermion massesand mixings are well reproduced for both normal and inverted neutrino mass spec-tra, and b – τ Yukawa unification and definite predictions for the effective mass inneutrinoless double- β decay are obtained. ∗ E-mail: [email protected] § E-mail: [email protected] Introduction
Grand Unified Theories (GUTs) based on the gauge group SO (10) [1] are a frameworkfor attempts to understand the observed fermion masses and mixings. These theories fea-ture a 16-dimensional irreducible representation (irrep), the spinor representation, whichnaturally accommodates all chiral fermions of one Standard Model (SM) generation plusa right-handed neutrino. Furthermore, SO (10) GUTs allow for type I [2] and type II [3]seesaw mechanisms (see also [4]) for explaining the smallness of the light neutrino masses.By employing only one scalar in the and one in the irrep of SO (10) inrenormalizable Yukawa couplings, the so-called “Minimal Supersymmetric SO (10) GUT”(MSGUT) [5] has been successful in accounting for all fermion masses and mixings, ifone focuses solely on its fermion mass matrices. Moreover, this model has built-in thegauge-coupling unification of the Minimal Supersymmetric Standard Model (MSSM). In-depth studies of the MSGUT have been performed [6, 7], also with inclusion of small [8]and prominent [9] effects of the irrep. Despite its success in reproducing the knownfermion masses and mixings, it should be stressed, however, that the MSGUT consideredas a whole is too constrained, as its scalar sector does not admit the vacuum expectationvalues (VEVs) required for the fit in the fermionic sector—see [10] and [7].However, one is not really satisfied by just reproducing known fermion masses andmixings, one would also like to explain, for instance, the threefold replication of fermiongenerations, or the peculiar mixing properties of the lepton sector [11], that is, maxi-mal atmospheric and large but non-maximal solar mixing and a small mixing angle θ . SO (10) models have not yet been successful in explaining such features, but one may tryto meet such challenges by considering the possibility of an underlying flavor symmetrygroup G . In GUT models based on the product group SO (10) × G , the three knownfermion generations can be assigned to representations of the flavor group G . Our choiceof G is guided by the following considerations. Since all irreps of abelian groups are one-dimensional, only non-abelian groups are suited to explain the existence of more thanone fermion generation. Furthermore, unlike continuous symmetries, the break-down ofdiscrete global symmetries does, in general, not give rise to undesired Goldstone bosons.This suggests to stick to non-abelian, discrete flavor groups. Only a few SO (10) × G mod-els, with G being non-abelian and discrete, have been studied so far. For instance, modelswith SO (10) × S [12] and SO (10) × A [13] symmetries have already been investigated.In particular, models employing an A [14, 15] flavor symmetry may give tri-bimaximalleptonic mixing [16], but, in general, in such models right- and left-handed fermion fieldstransform differently under A . However, in SO (10) GUTs right- and left-handed fermionfields have to transform in the same way under A [17], since there all chiral matterfields of one generation belong to the same SO (10) irrep. In [13] a non-supersymmetric SO (10) × A model with type I seesaw dominance has been analyzed, which successfullypreserves tri-bimaximal leptonic mixing and can accommodate all known fermion masses.The quark mixing angles, however, are assumed to be zero.In this paper, we investigate the fermionic sector of renormalizable SO (10) × A GUTscenarios, with the three fermion families in the three-dimensional irrep of A , while forthe SO (10) scalar irreps occurring in Yukawa couplings we allow all possible A irreps.We do not discuss the difficult problem of vacuum alignment, but rather assume that we2an dispose of the VEVs according to our needs. With this assumption, we will see thatthe SO (10) × A structure enforces the building blocks of the fermion mass matrices toconsist of diagonal and off-diagonal matrices. A crucial role will play the mass matricesof the down quarks and the charged leptons. Requiring solely that the scenario is ableto reproduce down-quark masses and charged-lepton masses, singles out a unique casewith respect to the transformation of the scalars under SO (10) × A . In that uniqueviable scenario, under the assumption of type II seesaw dominance, we will find theZee–Wolfenstein form [18, 19] of the mass matrix of light neutrinos. The well-knownphenomenological problems [20] of this mass matrix turn out to be completely resolvableby contributions to lepton mixing from the charged-lepton sector. We want to stress,however, that our usage of the A flavor symmetry does not enforce tri-bimaximal mixingin the lepton sector.Though we have in mind a supersymmetrized scenario, supersymmetry enters our con-siderations only via the fermion masses. For the numerics we use masses at the GUT scale,which have been obtained through the renormalization group equations of the MSSM.Our paper is organized as follows. In Section 2 we summarize the properties of theZee–Wolfenstein neutrino mass matrix. The SO (10) × A GUT scenario is developed inSection 3. The methods and results of our numerical analysis are discussed in Section 4.Section 5 is devoted to the conclusions.
The Zee model generates Majorana neutrino masses at the one-loop level [18, 21]. Itsneutrino mass matrix has, in general, non-zero elements on the diagonal. However, witha suitable Z symmetry one can enforce a vanishing diagonal in M ν at the one-looplevel [19]. This Zee–Wolfenstein neutrino mass matrix is a symmetric 3 × M ν = a ba cb c . (1)From now on we will discuss only this case. The restricted Zee model has the propertythat one can make a basis transformation such that the charged-lepton mass matrix isdiagonal but the form of M ν given by (1) persists. Thus, without loss of generality, wewill assume in this section that the charged-lepton mass matrix is diagonal.In diagonalizing the matrix (1), one can first remove the phases of M ν . These phasescan be absorbed into the charged-lepton fields. Thus we take the matrix entries a, b, c tobe real. Since the Zee–Wolfenstein mass matrix is traceless and symmetric one has [22] λ + λ + λ = 0 , (2)where the λ i denote the real eigenvalues of M ν . Writing m i for the masses of the lightneutrinos, we have m i = | λ i | .In the case of inverted ordering m < m < m of the neutrino masses (∆ m ⊙ = m − m , ∆ m = m − m ), it has been pointed out in Ref. [20] that the mass matrix (1)together with ∆ m ⊙ ≪ ∆ m leads, for all practical purposes, to maximal solar mixing3 = π/ θ = 0. Furthermore, the atmospheric mixing angle θ can be chosen tobe maximal. While the latter two properties are most welcome, maximal solar mixing isexcluded by more than 5 σ by experimental data [23]. In [24] it is has been shown thatdeviations from maximal solar mixing are severely constrained through | cos 2 θ | < ∼
14 ∆ m ⊙ ∆ m . (3)The neutrino masses are approximately given by m ≃
12 ∆ m ⊙ p ∆ m , m ≃ m ≃ p ∆ m . (4)Thus one obtains m ≪ m ≃ m and the resulting neutrino mass spectrum exhibits aninverted hierarchy.In the case of the normal ordering m < m < m of the neutrino masses (∆ m ⊙ = m − m , ∆ m = m − m ) things are even worse. Although it is now possible to havemaximal atmospheric mixing and, at the same time, allowing the solar mixing angle to bein perfect agreement with experimental data, the mixing angle θ turns out to be muchtoo large [20]: sin θ ≃ . (5)The neutrino mass spectrum can be estimated by m ≃ m ≃ m ≃ r ∆ m . (6)Therefore, all neutrino masses will be of the same order of magnitude, but the massspectrum cannot be quasi-degenerate.Concerning neutrinoless double- β decay, the relevant observable is the effective Majo-rana neutrino mass |h m ββ i| ≡ | P i U ei m i | , where U denotes the unitary leptonic mixing(PMNS) matrix. The mass |h m ββ i| is equal to the modulus of the ( e, e ) matrix elementof M ν , which is exactly zero in the Zee–Wolfenstein case. Thus the model preventsneutrinoless double- β decay.In summary, the Zee–Wolfenstein model is not viable because it does not give a con-sistent explanation of all current experimental data of the neutrino sector ( i.e. two mass-squared differences plus three mixings angles). It is the purpose of this paper to embedthe Zee–Wolfenstein neutrino mass matrix in an SO (10) GUT. In such an environment,the zeros in the diagonal of M ν are not stable under a basis change such that the charged-lepton mass matrix becomes diagonal. Therefore, as we will show, contributions from thecharged-lepton sector can provide the necessary remedy for correcting the too large mixingangle θ in the case of inverted hierarchy and θ in the case of normal hierarchy [25]. Asan additional bonus, a non-vanishing |h m ββ i| and, therefore, neutrinoless double- β decaybecomes possible. 4 The SO (10) × A model The tensor product of the SO (10) spinor representation of the fermions is given by [26, 27] ⊗ = ( ⊕ ) S ⊕ AS , (7)where the subscripts S and AS refer to symmetric and antisymmetric Yukawa couplingmatrices, respectively. Renormalizable SO (10) GUTs can generate fermion masses at thetree level only by the scalar irreps , and .The 12-element group A is popular as a family symmetry in model building— see [14]for a selection of the vast A literature and [15] for a review on the group A and models.It has three one-dimensional irreps and one three-dimensional irrep. The tensor product ⊗ contains all one-dimensional irreps exactly once, but the three-dimensional irrepis contained twice. While the Yukawa couplings corresponding to the one-dimensionalirreps are diagonal and, therefore, symmetric, the couplings of the ⊕ ∈ ⊗ areoff-diagonal, but no special symmetry property is fixed. However, Eq. (7) suggests tochoose one three-dimensional irrep with symmetric and the other one with antisymmetrictensor indices: ⊗ = ( ⊕ ′ ⊕ ′′ ⊕ ) S ⊕ AS . (8)Now we consider SO (10) × A and investigate possible Yukawa couplings and fermionmass matrices under the assumption that the fermions transform as ⊗ , which isis clearly the only reasonable choice if we want to take advantage of the non-abeliancharacter of A . Equations (7) and (8) dictate that the can only transform as a under A , while for the for and singlet and triplet irreps of A are possible. Letus consider the case where the scalars responsible for Yukawa couplings transform as ⊗ ( ⊕ ′ ⊕ ′′ ) and ⊗ . (9)Then, in a symbolic way, writing down only the A part, the Yukawa couplings are givenby X i =1 h i X a =1 ω ( i − a − a a i + (10)( + ) + ( + ) + ( + ) , (11)where a is a family index and ω = ( − i √ /
2. Furthermore, we make two assumptions:i) All VEVs which occur in the scalars can have independent values.ii) Type II seesaw dominates in the neutrino mass matrix.These assumptions together with Eq. (9) define the scenario we will investigate in thefollowing.We furthermore assume that our models can be extended in a suitable way to solve thedoublet-triplet splitting problem. Moreover, since we have in mind the MSSM, with only For instance, the Dimopoulous–Wilczek mechanism [29] and the missing partner mechanism [30]provide viable solutions of the doublet-triplet splitting problem in SO (10) GUTs. SO (10) models we must assume a suit-able doublet-doublet splitting as well, which is usually achieved by finetuning [31]. Theseassumptions are not innate to the models presented here but are well-known problems inGUTs.Let us now derive some consequences of our scenario. Because of assumption i), theYukawa couplings (10) produce diagonal mass terms with three independent entries, onefor the up-quark mass matrix ( q = u ) and another one for the down quark mass matrix( q = d ):diag (cid:0) h v q + h v q + h v q , h v q + ωh v q + ω h v q , h v q + ω h v q + ωh v q (cid:1) , (12)where the v qi are the VEVs appearing in the scalar 10-plets. Next we consider the Yukawacouplings (11). Again because of assumption i), this Yukawa interaction generates twoindependent off-diagonal contributions to the mass matrices of up and down quarks.Studying the system of mass matrices, we find that assumptions i) and ii) lead toa decoupling of the up-quark mass matrix M u from the rest of the system. This is sobecause the quark mass matrices are given by M u = H ′ + F ′ and M d = H + F , where H ′ and H are independent diagonal matrices, while F ′ and F are independent off-diagonalmatrices. Therefore, M u would only be related to the system of mass matrices throughthe neutrino Dirac-mass matrix M D = H ′ − F ′ , but this relationship is irrelevant dueto assumption ii). Since M u is a general symmetric matrix independent of the rest of thesystem of mass matrices, the CKM matrix can always be reproduced. The other side ofthe coin is that our scenario loses predictivity because it is neither restricted by the valuesof the up-quark masses nor by the experimental information on the CKM matrix.The remaining system of mass matrices which we want to study consists of the massmatrices of down-type quarks and charged-leptons, given by M d = H + F and M ℓ = H − F, (13)respectively, where H is diagonal, while F is off-diagonal, and of the neutrino mass matrix M ν of Eq. (1). Without loss of generality, H can be assumed to be real, but F and M ν have complex entries. Note that in view of assumption i) the entries in M ν areindependent of F , but M ℓ and M ν are coupled via the PMNS matrix U = U † ℓ U ν with U Tℓ M ℓ U ℓ = diag ( m e , m µ , m τ ) , U Tν M ν U ν = diag ( m , m , m ) . (14)Counting the number of parameters, we find nine absolute values and five phases, whilethe number of observables to be fitted is 11: three charged-lepton masses, three down-quark masses, two neutrino mass-squared differences and three lepton mixing angles. Thefitting procedure and predictions of our scenario will be exposed in the next section.We note that the family symmetry A has the effect of generating independent diagonaland off-diagonal contributions to the quark and lepton mass matrices. Adhering to thetwo assumptions presented above but using other A representations than those of Eq. (9),we can find several other scenarios. E.g., with 120-plets, antisymmetric off-diagonal mass Now we cannot absorb the phases of M ν into the charged-lepton fields since M ℓ is not diagonal. Of the three phases in M ν one can be removed. m e . +0 . − . m µ . +0 . − . m τ . +1 . − . ∆ m ⊙ (7 . ± . × − ∆ m (cid:16) . +0 . − . (cid:17) × − s . ± . s . ± . s < . m d . ± . m s . ± . m b . +141 . − . Table 1: Input data (central values and 1 σ errors) at the GUT scale of M GUT = 2 × GeV for tan β = 10. The charged-fermion masses are taken from [32], except for thevalues of m d and m s ; these were obtained by taking their low-energy values from [33] andscaling them to M GUT . As for ∆ m , we use the value obtained in [23]. We have copiedthe remaining input from Table I in [7]. Charged-fermion masses are in units of MeV,neutrino mass-squared differences in eV . We have used the abbreviations s ≡ sin θ , etc . The angles in the left table refer to the PMNS matrix.matrix contributions are generated. A list of such cases is presented in Table 2. Therewe confine ourselves to a maximum of three scalars per SO (10) irrep, the 126-plet mustalways be present to allow a viable type II seesaw neutrino mass matrix and the and are not present at the same time; the latter condition is for avoiding a proliferationof parameters. However, it will turn out that the only viable scenario is the one definedvia Eq. (9). We perform a global χ analysis of the SO (10) × A scenario defined by Eq. (9) andassumptions i) and ii) by employing the downhill simplex method [28]. In Table 1 theobservable quantities O i are specified in the form O i = ¯ O i ± σ i , (15)where ¯ O i and σ i denote central values and 1 σ deviations, respectively. The index i =1 , . . . ,
11 labels the different observables given in Table 1. The masses in that table referto the mass values at a GUT scale of 2 × GeV, obtained via the renormalizationgroup equations of the MSSM, for the ratio of Higgs doublet VEVs tan β = 10. Writing Since we do not have quasi-degenerate neutrino mass spectra, the effect of the renormalization grouprunning on the lepton mixing angles is negligible [34].
10 120 126 χ dℓ A − B − − ⊕ ′ ⊕ ′′ − ⊕ ′ ⊕ ′′ − × E ⊕ ′ ⊕ ′′ − . SO (10) × A . Each linecorresponds to a distinctive model scenario. Columns 2–4 specify the transformationproperties of the SO (10) scalar multiplets under the flavor symmetry group A . The lastcolumn gives the best-fit values χ dℓ when fitting charged-lepton and down-type quarkmasses. x for the set the 14 model parameters and P i ( x ) for the resulting model predictions, onecan define a χ function by χ ( x ) = X i =1 (cid:18) P i ( x ) − ¯ O i σ i (cid:19) . (16)The global minimum of χ will represent the best possible agreement of theoretical pre-dictions and experimental data. This minimization task is performed using the downhillsimplex method.For investigating the variation of χ as a function of the value b O of an observable O , we add the “pinning term” ( P ( x ) − b O ) / (0 . b O ) to χ , where P ( x ) represents thetheoretical prediction for O . Note that if O agrees with one of observables O i occurringin χ of Eq. (16), the O i term has to be removed from Eq. (16). The small error in thedenominator of the “pinning term” guarantees to pin the observable O down to the value b O . The pinning procedure performs as desired when the contribution of the pinning termto χ is negligible.As mentioned at the end of Section 3, we have not only investigated the scenario definedby Eq. (9) but also a variety of others which are characterized by the A transformationproperties of their scalar SO (10) multiplets in columns 2–4 of Table 2 (models A–E’). Wehave found that all these scenarios fail already to reproduce the down-quark and charged-lepton masses—see the value of the corresponding χ dℓ in the last column of Table 2. Forcomparison we have also presented the χ dℓ of our successful scenario in the line labeledby E, which will be investigated in the rest of this paper. For case A this failure is trivial: M d and M ℓ are symmetric with a vanishing diagonal, therefore,Eq. (2) holds, which is in contradiction to the strong hierarchy in the down-quark and charged-leptonmasses. .1 Predictions for the case of normal neutrino mass ordering We search for the best-fit solution for the normal neutrino mass spectrum m < m < m .In this case we find an excellent fit with the following properties: χ = 0 . ,m = 2 . × − eV , m = 2 . × − eV , m = 5 . × − eV . (17)The corresponding values of the matrix elements of H , F , M ν are given by H = . . . ,F = . e i . π . e − i . π . e i . π . e i . π . e − i . π . e i . π , (18) M ν = . e i . π . e i . π . e i . π . e i . π . e i . π . e i . π × − , where the numerical values in H and F are in units of MeV, while the entries in M ν arein units of eV.The non-zero value of χ stems from the deviation of the bottom-quark mass m b fromits central value by +0 . σ . The remaining observables of Table 1 are fitted perfectly.Thus the model succeeds in correcting the too large value for the mixing angle θ ofthe Zee–Wolfenstein model, despite the close relationship between M d and M ℓ given byEq. (13) which, on the other hand, leads to the desired unification of m b and m τ , as willbe discussed in Section 4.3.As explained in Section 2, the three light neutrino masses cannot be independentof each other. The sum of the eigenvalues of M ν must be zero, which translates into m + m − m = 0. This can easily be verified for the neutrino masses of the best-fit (17).The sum of the neutrino masses is Σ ≡ P i m i = 2 m = 0 .
11 eV, which lies safely belowthe cosmological bound Σ < ∼ m from Table 1 into Eq. (6) gives m ≃ m ≃ . × − eV and m ≃ . × − eV, which is in good agreement with the above best-fit results.The quantity R ≡ m / p ∆ m ⊙ measures how hierarchical a neutrino mass spectrumis. χ as a function of R is depicted in the right panel of Figure 1. We read off that R ∼ . . < ∼ R < ∼ . χ < ∼
15. Thusthe mass spectrum is neither hierarchical nor quasi-degenerate, but is located betweenthese extrema. The narrow range of allowed values for R reflects the clear-cut predictionof the Zee–Wolfenstein mass matrix for the neutrino mass spectrum.Figure 1 (left panel) shows the constraints on the atmospheric mixing angle θ . Onecan see that values of sin θ smaller than 0.38 ( − σ ) are strongly disfavored and thus Typically, quasi-degenerate neutrino spectra would correspond to
R > ∼ q ∆ m / ∆ m ⊙ ≃ . θ is established. However, very good fits are also possible forvalues of sin θ significantly larger than the best-fit value of 0.5.Concerning the solar mixing angle, Figure 1 (middle panel) shows that the wholephysically allowed range for sin θ gives excellent fits and therefore no prediction can beobtained.Regarding the mixing angle θ , the best-fit solution gives a value of sin θ = 2 × − .However, also significantly smaller (down to 10 − ) and larger values (up to 0.1) for sin θ are equally allowed. Thus the severe problem of the original Zee–Wolfenstein mass matrix(sin θ ≃ /
3) can be resolved completely by contributions from the charged-leptonsector.The best-fit gives δ PMNS = 31 ◦ for the leptonic CP phase. However, varying δ PMNS shows that the whole [0 ◦ , ◦ ] range allows for very good fits and therefore no predictioncan be made.The effective Majorana mass of neutrinoless double- β decay |h m ββ i| for the normalspectrum is given by |h m ββ i| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) m c + q m + ∆ m ⊙ s e iβ (cid:19) c + q m + ∆ m s e iβ (cid:12)(cid:12)(cid:12)(cid:12) , (19)where β and β are Majorana phases. Here and in the following we use the abbreviations c ≡ cos θ , s ≡ sin θ , etc . After inserting into Eq. (19) m from the best-fit (17),employing for the other parameters the corresponding central values from Table 1, andvarying the two phases β and β freely between 0 ◦ and 360 ◦ , we obtain the bounds10 . ≤ |h m ββ i| ≤ . . (20)On the other hand, the phases β and β are actually functions of the parameters ofour scenario and are determined by the fit. Using the best-fit parameters (18) for thecalculation of the effective Majorana mass, we obtain |h m ββ i| = 28 . χ when |h m ββ i| is varied. We can read off that the range for the effective mass is much more restrictedthan (20) would suggest. Obviously, the increase of χ for larger values of |h m ββ i| is causedby exceeding the upper bound of (20). The strong increase of χ for smaller values of |h m ββ i| , however, is a clear-cut model prediction. For instance, allowing for only moderategood fits with χ < ∼ < ∼ |h m ββ i| < ∼
31 meV,which could be tested by future neutrinoless double- β decay experiments sensitive to |h m ββ i| > ∼
10 meV.
The best-fit solution for the inverted neutrino mass spectrum m < m < m turns outto be excellent as well. It is characterized by the following properties: χ = 0 . ,m = 4 . × − eV , m = 5 . × − eV , m = 7 . × − eV , (21)10 .3 0.4 0.5 0.6 0.7 sin θ χ sin θ R Figure 1: χ as a function of sin θ (left panel), sin θ (middle panel) and of R ≡ m min / p ∆ m ⊙ (right panel), where m min = m for the normal and m for the invertedneutrino mass spectrum. The solid lines correspond to normal neutrino mass ordering,while the dashed lines refer to the inverted neutrino mass spectrum.with the matrices H = . . . ,F = . e iπ . . e iπ . . .
271 0 , (22) M ν = . . e iπ . . e iπ . e iπ . e iπ × − , where the numerical values in H and F are in units of MeV, while the entries in M ν arein units of eV.The χ analysis reveals that the removal of the non-trivial complex phases from F and M ν does not affect the goodness of the fit. Thus we specified here the fitting parametersfor the CP conserving case. However, the subsequent numerical analysis is performedwith the inclusion of the five phase parameters (CP non-conservation). As in the case ofnormal neutrino mass ordering, the main contribution to χ is caused by the bottom-quarkmass m b , being too large by 0 . σ . All the other observables are fitted very accurately. For the normal neutrino spectrum, however, the CP conserving case results in a worse, but stillvery good fit with χ = 1 .
94. Here, the main contributions to χ stem from m b (+1 . σ ) and sin θ ( − . σ ) . | 〈 m ββ 〉 | [meV] χ Figure 2: χ as a function of the effective Majorana mass |h m ββ i| probed in neutrinolessdouble- β decay experiments. The solid line corresponds to normal neutrino mass ordering,while the dashed line refers to the inverted neutrino mass spectrum.Thus the GUT model allows for a considerably reduction of the maximal solar mixingangle θ , which spoiled the Zee–Wolfenstein model. M ν being of the Zee–Wolfenstein form implies m − m + m = 0 for the three lightneutrino masses. Taking the neutrino masses from the best-fit (21), we get Σ ≡ P i m i =2 m = 0 .
10 eV, which is safely below the cosmological limit [35]. Inserting the centralvalues for the mass-squared differences from Table 1 into Eqs. (4) gives m ≃ m ≃ × − eV and m ≃ . × − eV, which is in good agreement with the numericallyobtained best-fit values (21). χ as a function of R ≡ m / p ∆ m ⊙ is shown in Figure 1 (right panel). We canread off that R ∼ .
09 is preferred and for the values 0 . < ∼ R < ∼ .
11 one gets fitswith χ < ∼
15. As in the case of normal neutrino mass ordering, the range for R is veryrestricted. Hierarchy is strongly preferred, however, too small values for m are strictlyforbidden.Figure 1 (middle panel) depicts the constraints on the solar mixing angle θ . We canread off that values for sin θ smaller than 0.3 become increasingly disfavored. However,very good fits can also be found for values of sin θ larger than the best-fit value, andmaximal solar mixing also represents a very good fit.Regarding the atmospheric mixing angle θ , Figure 1 (left panel) reveals that thewhole physically allowed range for sin θ gives very good fits and therefore no predictioncan be obtained. This property is seemingly a legacy of the original Zee–Wolfensteinmodel, where the atmospheric mixing angle for inverted neutrino mass ordering is uncon-strained [20].As for the mixing angle θ , we find sin θ = 2 . × − for the best fit. However,12inning sin θ in χ shows that also smaller (down to 10 − ) and larger values (up to 0.1)are possible. For instance, enforcing sin θ = 10 − still allows χ = 3 . δ PMNS , the specified best-fit, which employs onlytrivial complex phases, gives δ PMNS = 180 ◦ . However, varying δ PMNS in the range(90 ◦ , ◦ ) allows for fits of equally good quality. Only the neighborhood of δ PMNS ≃ ◦ seems to be slightly disfavored by χ ≃ . β decay is given by |h m ββ i| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:18)q m + ∆ m − ∆ m ⊙ c e iβ + q m + ∆ m s e iβ (cid:19) c + m s (cid:12)(cid:12)(cid:12)(cid:12) . (23)With m from the best-fit and taking for the other parameters in (23) the correspondingcentral values in Table 1, free variation of the two complex phases results in the followingbounds on the effective Majorana neutrino mass:18 . ≤ |h m ββ i| ≤ . . (24)On the other hand, employing the best-fit parameters (22), we obtain |h m ββ i| = 18 . χ under variations of |h m ββ i| . We can see that therange for the effective mass is less restricted than in the case of normal neutrino spectrum.Clearly, the strong increase of χ for smaller values of |h m ββ i| comes from falling belowthe lower bound of (24). The rise of χ is less dramatic when moving to larger valuesof |h m ββ i| . However, there is a clear bias towards values of |h m ββ i| in the lower half ofthe range spanned by (24). We can also read off from Figure 2 that allowing moderatelygood fits with χ < ∼ < ∼ |h m ββ i| < ∼
35 meV. Moreover, we can see that the |h m ββ i| regions where χ > ∼ |h m ββ i| between normal and inverted mass spectrum in theoverlap region. However, |h m ββ i| < ∼
20 meV (which is preferred) or |h m ββ i| > ∼
33 meV isonly possible for an inverted hierarchy in our scenario. b − τ unification As has already been noticed in Sections 4.1 and 4.2, both best-fit values of m b are locatednear the upper 1 σ bound of its input value from Table 1. The ratio m b /m τ , using themean values from Table 1, is 0 .
82. Employing the best-fit values for m b and m τ , this ratiois higher, namely m b /m τ = 0 .
93 for both normal and inverted neutrino mass ordering.Figure 3 depicts χ as a function of m b for both neutrino mass orderings. This figureclearly reflects the feature mentioned above since for values of m b below 1190 MeV, χ increases dramatically and lower values of m b become strictly ruled out.There also exists an upper bound on m b in Figure 3 at about 1250 MeV, which islocated below the central input value of m τ at 1292 MeV. In contrast to the normalneutrino mass spectrum, however, the inverted spectrum seems to prefer values for m b near its lower bound, as can be read off from Figure 3.13 .17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 m b [GeV] χ Figure 3: χ as a function of the bottom-quark mass m b . The solid line corresponds tonormal neutrino mass ordering, while the dashed line refers to the inverted neutrino massspectrum. The shaded region indicates the 1 σ interval for m b from Table 1.In summary, our scenario imposes rather rigid constraints on m b and favors b − τ unification. This feature is apparently caused by the SO (10) relation (13) between themass matrices of charged-leptons and down-quarks, which differ only by a factor of − In this paper we have presented an attempt to combine an SO (10) GUT with the familysymmetry A . We have considered renormalizable Yukawa interactions, therefore, thechoice of scalar SO (10)-plets for fermion mass matrices is confined to , and .The three fermion families are accommodated in an A triplet. For fitting purposes weuse the fermion masses at the GUT scale evolved by the renormalization group equationsof the MSSM. As a further important prerequisite we assume that the VEVs occurring inthe scalar SO (10) multiplets can be freely chosen for the purpose of fitting fermion massesand mixings. Our investigation consists of two steps—for the details see Section 3.In the first step we have considered only the down-quark and charged-lepton massmatrices. We have assigned all possible A representations to the , and andchecked, if the down-quark and charged-lepton masses can correctly be reproduced. Inthis way we have identified a unique successful scenario given by the in ⊕ ′ ⊕ ′′ of A and the in the of A . This is a non-trivial result, because we use only sixmasses for probing mass matrices constructed with more than six parameters. The massmatrices (13) of the successful scenario (9) reflect the SO (10) × A structure: The 10-plets14ontribute a general diagonal and the 126-plets a general off-diagonal matrix to the massmatrices.In the second step we have assumed type II dominance in the seesaw mechanismgenerating light neutrino masses. Since only the 126-plets contribute to the neutrinomass matrix, we obtain the Zee–Wolfenstein mass matrix.The scenario gives an excellent fit to all known data on fermion masses and mixingsand shows, therefore, the compatibility of the family group A with SO (10) GUTs. Insummary, we have found the following features: • All mass matrices are symmetric. • In the charged-fermion sector, as an effect of SO (10) × A , the building blocks ofthe mass matrices are general diagonal and off-diagonal matrices, generated by theVEVs of scalar 10-plets and 126-plets, respectively. • M ν is given by the Zee–Wolfenstein matrix with its definite predictions for theneutrino masses derived from Eq. (2). • The scenario can equally well accommodate normal and inverted neutrino massspectra. • The lepton mixing angles of the Zee–Wolfenstein mass matrix which are in dis-agreement with the data are corrected by contributions to the PMNS matrix from M ℓ . • There are definite predictions for |h m ββ i| for both spectra. • Our scenario gives b – τ Yukawa unification.On the negative side we note that our scenario does not feature tri-bimaximal lepton mix-ing, however, some minor constraints on the lepton mixing angles exist—see Sections 4.1and 4.2. Moreover, up-quark masses and the CKM matrix are completely free and canthus be adapted to the data without imposing any restrictions on the parameters of themass matrices M d , M ℓ and M ν . Acknowledgments:
We thank L. Lavoura for valuable suggestions and reading themanuscript. 15 eferences [1] H. Fritzsch, P. Minkowski, Ann. Phys. 93 (1975) 193.[2] P. Minkowski, Phys. Lett. B 67 (1977) 421;T.Yanagida, in
Proceedings of the Workshop on Unified Theory and Baryon Numberin the Universe , O. Sawata and A. Sugamoto eds., KEK report 79-18, Tsukuba,Japan, 1979;S.L. Glashow, in
Quarks and Leptons, Proceedings of the Advanced Study Institute(Carg`ese, Corsica, 1979) , J.-L. Basdevant et al. eds., Plenum, New York, 1981;M. Gell-Mann, P. Ramond, and R. Slansky, in
Supergravity , D.Z. Freedman and F.van Nieuwenhuizen eds., North Holland, Amsterdam, 1979;R.N. Mohapatra, G. Senjanovi´c, Phys. Rev. Lett. 44 (1980) 912.[3] G. Lazarides, Q. Shafi, C. Wetterich, Nucl. Phys. B 181 (1981) 287;R.N. Mohapatra, G. Senjanovi´c, Phys. Rev. D 23 (1981) 165;R.N. Mohapatra, P. Pal,
Massive Neutrinos in Physics and Astrophysics , WorldScientific, Singapore, 1991, p. 127.[4] J. Schechter, J.W.F. Valle, Phys. Rev. D 22 (1980) 2227;S.M. Bilenky, J. Hoˇsek, S.T. Petcov, Phys. Lett. B 94 (1980) 495;I.Yu. Kobzarev, B.V. Martemyanov, L.B. Okun, M.G. Shchepkin, Yad. Phys. 32(1980) 1590 [Sov. J. Nucl. Phys. 32 (1981) 823];J. Schechter, J.W.F. Valle, Phys. Rev. D 25 (1982) 774.[5] C.S. Aulakh, R.N. Mohapatra, Phys. Rev. D 28 (1983) 217;T.E. Clark, T.K. Kuo, N. Nakagawa, Phys. Lett 115B (1982) 26;K.S. Babu, R.N. Mohapatra, Phys. Rev. Lett. 70 (1993) 2845 [hep-ph/9209215];C.S. Aulakh, B. Bajc, A. Melfo, G. Senjanovi´c, F. Vissani, Phys. Lett. B 588 (2004)196 [hep-ph/0306242].[6] K. Matsuda, Y. Koide, T. Fukuyama, Phys. Rev. D 64 (2001) 053015[hep-ph/0010026];K. Matsuda, Y. Koide, T. Fukuyama, H. Nishiura, Phys. Rev. D 65 (2002) 033008(Err. ibid.
D 65 (2002) 079904) [hep-ph/0108202];T. Fukuyama, N. Okada, JHEP 11 (2002) 011 [hep-ph/0205066];B. Bajc, G. Senjanovi´c, F. Vissani, Phys. Rev. Lett. 90 (2003) 051802[hep-ph/0210207];H.S. Goh, R.N. Mohapatra, S.P. Ng, Phys. Lett. B 570 (2003) 215 [hep-ph/0303055];H.S. Goh, R.N. Mohapatra, S.P. Ng, Phys. Rev. D 68 (2003) 115008[hep-ph/0308197];B. Bajc, G. Senjanovi´c, F. Vissani, Phys. Rev. D 70 (2004) 093002 [hep-ph/0402140];K.S. Babu, C. Macesanu, Phys. Rev. D 72 (2005) 115003 [hep-ph/0505200].[7] S. Bertolini, T. Schwetz, M. Malinsk´y, Phys. Rev. D 73 (2006) 115012[hep-ph/0605006]. 168] K. Matsuda, T. Fukuyama, H. Nishiura, Phys. Rev. D 61 (2000) 053001[hep-ph/9906433];S. Bertolini, M. Frigerio, M. Malinsk´y, Phys. Rev. D 70 (2004) 095002[hep-ph/0406117];S. Bertolini, M. Malinsk´y, Phys. Rev. D 72 (2005) 055021 [hep-ph/0504241].[9] C.S. Aulakh, hep-ph/0602132;W. Grimus, H. K¨uhb¨ock, Phys. Lett. B 643 (2006) 182 [hep-ph/0607197];W. Grimus, H. K¨uhb¨ock, Eur. Phys. J. C 51 (2007) 721 [hep-ph/0612132];C.S. Aulakh, hep-ph/0607252;C.S. Aulakh, talk presented at , Moscow, Russia, July 26–August 2, 2006, hep-ph/0610097;C.S. Aulakh, S.K. Garg, hep-ph/0612021.[10] C.S. Aulakh, expanded version of the plenary talks at the
Workshop Series on The-oretical High Energy Physics , IIT Roorkee, Uttaranchal, India, March 16–20, 2005,and at the , ICTP, Trieste, Italy, May 23–28, 2005, hep-ph/0506291;B. Bajc, A. Melfo, G. Senjanovi´c, F. Vissani, Phys. Lett. B 634 (2006) 272[hep-ph/0511352];C.S. Aulakh, S.K. Garg, Nucl. Phys. B 757 (2006) 47 [hep-ph/0512224].[11] M. Maltoni, T. Schwetz, M.A. T´ortola, J.W.F. Valle, New. J. Phys. 6 (2004) 122[hep-ph/0405172];G.L. Fogli, E. Lisi, A. Marrone, A. Palazzo, Prog. Part. Nucl. Phys. 57 (2006) 742[hep-ph/0506083].[12] D.G. Lee, R.N. Mohapatra, Phys. Lett. B 329 (1994) 463 [hep-ph/9403201];C. Hagedorn, M. Lindner, R.N. Mohapatra, JHEP 0606 (2006) 042 [hep-ph/0602244];Yi Cai, Hai-Bo Yu, Phys. Rev. D 74 (2006) 115005 [hep-ph/0608022].[13] S. Morisi, M. Picariello, E. Torrente-Lujan, Phys. Rev. D 75 (2007) 075017[hep-ph/0702034].[14] E. Ma, G. Rajasekaran, Phys. Rev. D 64 (2001) 113012 [hep-ph/0106291];K.S. Babu, E. Ma, J.W.F. Valle, Phys. Lett. B 552 (2003) 207 [hep-ph/0206292];M. Hirsch, E. Ma, A. Villanova del Moral, J.W.F. Valle, Phys. Rev. D 72 (2005)091301 (Err. ibid.
D 72 (2005) 119904) [hep-ph/0507148];G. Altarelli and F. Feruglio, Nucl.Phys. B 720 (2005) 64 [hep-ph/0504165;G. Altarelli and F. Feruglio, Nucl.Phys. B 741 (2005) 215 [hep-ph/0512103];E. Ma, Phys. Rev. D 73, 057304 (2006) [hep-ph/0511133];E. Ma, H. Sawanaka, M. Tanimoto, Phys. Lett. B 641 (2006) 301 [hep-ph/0606103];L. Lavoura, H. K¨uhb¨ock, Mod. Phys. Lett. A 22 (2007) 181 [hep-ph/0610050];and references therein.[15] G. Altarelli, Lectures given at the 61st Scottish Universities Summer School inPhysics, St. Andrews, Scottland, 8–23 August 2006, hep-ph/0611117.1716] P.F. Harrison, D.H. Perkins, W.G. Scott, Phys. Lett. B 530 (2002) 167[hep-ph/0202074].[17] E. Ma, Mod. Phys. Lett. A 21 (2006) 2931 [hep-ph/0607190].[18] A. Zee, Phys. Lett. 93B (1980) 389; ibid.
Numerical recipes in C:The art of scientific computing , Cambridge University Press, 1992.[29] S. Dimopoulos, F. Wilczek, in:
The Unity of the Fundamental Interactions , Pro-ceedings of the 19th Course of the International School of Subnuclear Physics, Erice,Italy, 1981, edited by A. Zichini (Plenum Press, New York, 1983) 237-249;K.S. Babu, S.M. Barr, Phys. Rev. D 48 (1993) 5354 [hep-ph/9306242].[30] K.S. Babu, I. Gogoladze, Z. Tavartkiladze, Phys. Lett. B 650 (2007) 49[hep-ph/0612315].[31] C.S. Aulakh, A. Girdar, Int. J. Mod. Phys. A 20 (2005) 865 [hep-ph/0204097];B. Bajc, A. Melfo, G. Senjanovi´c, F. Vissani, Phys. Rev. D 70 (2004) 035007[hep-ph/0402122]. 1832] C.R. Das, M.K. Parida, Eur. Phys. J. C 20 (2001) 121 [hep-ph/0010004].[33] W.-M. Yao et al.,