Predicting Lepton Flavor Mixing from Δ(48) and Generalized CP Symmetries
PPredicting Lepton Flavor Mixing from ∆(48) and Generalized CP Symmetries
Gui-Jun Ding ∗ and Ye-Ling Zhou † Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918, Beijing 100049, China
We propose to understand the mixing angles and CP-violating phases from the ∆(48) familysymmetry combined with the generalized CP symmetry. A model-independent analysis is performedby scanning all the possible symmetry breaking chains. We find a new mixing pattern with only onefree parameter, excellent agreement with the observed mixing angles can be achieved and all theCP-violating phases are predicted to take nontrivial values. This mixing pattern is testable in thenear future neutrino oscillation and neutrinoless double-beta decay experiments. Finally, a flavormodel is constructed to realize this mixing pattern.
PACS numbers: 14.60.Pq, 14.80.Cp, 11.30.Hv
Discrete family symmetry has been widely used to ex-plain the lepton flavor mixing [1] in the past years. Thediscovery of a sizable value of θ by reactor experi-ments [2] excludes many neutrino mixing models, andopens the possibility of measuring the Dirac CP-violatingphase in the next generation neutrino experiments. Theunderlying physics of flavor mixing and CP violation isstill an open question. The history of physics tells us thatsymmetry always plays a crucial role in understandingthe natural world. Inspired by the success of the familysymmetry paradigms, it is natural to extend the fam-ily symmetry to include a generalized CP (GCP) sym-metry H CP [3–6], to predict both flavor mixing anglesand CP phases. This idea has been implemented within S [4, 6, 7], A [8] and T (cid:48) [9] family symmetries, wherethe lepton mixing matrix is found to depend on one sin-gle parameter ϑ , which can be fixed by the measurementof the mixing angle θ .In this paper, we propose to impose the ∆(48) familysymmetry together with GCP symmetry on the theory.Compared with the well-known S and A family symme-tries, ∆(48) provides many candidates for GCP transfor-mations which could lead to new mixing patterns. Aftera brief introduction to the GCP symmetry and the grouptheory of ∆(48), we derive all the GCP transformationswhich are consistent with ∆(48). Then we present possi-ble lepton mixing patterns derived from different symme-try breaking chains in a model-independent way. Finallywe focus on phenomenological implications and modelbuilding aspect of a new pattern that has not been dis-cussed in the literature. A longer and more completeversion of this paper has been presented in [10].A field multiplet φ transforms under the family andGCP symmetries as φ g −→ ρ ( g ) φ and φ CP −→ Xφ ∗ , (1)respectively, where ρ ( g ) is a representation matrix of the ∗ Electronic address: [email protected] † Electronic address: [email protected] group element g ∈ G f , and X ∈ H CP is the GCP trans-formation matrix. Both of them are unitary matrices. Itis nontrivial to combine the family symmetry with theGCP symmetry. The so-called consistence equation hasto be satisfied: Xρ ∗ ( g ) X − = ρ ( g (cid:48) ) for g, g (cid:48) ∈ G f [3–5].Furthermore, X maps one group element g into anotherelement g (cid:48) , consequently X corresponds to an automor-phism of G f . It has been established that there is a one-to-one correspondence between the GCP transformationand the automorphism of G f [11].In the present work, the family symmetry is chosento be G f = ∆(48) ∼ = ( Z × Z ) (cid:111) Z , which is a finitesubgroup of SU (3) of order 48 with generators a , c and d satisfying a = c = d = 1 , cd = dc,aca − = c − d − , ada − = c, (2)where a generates Z and c , d are generators of Z × Z . Itbelongs to the ∆(3 n ) series [12] with n = 4. Any groupelement g ∈ ∆(48) can be expressed as g = a k c m d n with k = 0 , , m, n = 0 , , , • Three 1-dimensional (1d) representations , (cid:48) , (cid:48)(cid:48) ; • Five 3d representations , , (cid:48) , (cid:48) and (cid:101) , where ( (cid:48) ) is the complex conjugate of ( (cid:48) ). The formerfour are the faithful representations of ∆(48), whilethe last one is not.We shall work in the generator a diagonal basis. For therepresentation , we choose: a = (cid:32) ω
00 0 ω (cid:33) , c = 13 − √ √
31 + √ − √ − √ √ , (3)with ω = e i π/ , and the representation matrix of d isgiven by d = a − ca . Some Kronecker products that willbe used later are presented here: ⊗ = ⊕ (cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48) , ⊗ = S ⊕ A ⊕ (cid:101) , (cid:48) ⊗ (cid:48) = (cid:48) S ⊕ (cid:48) A ⊕ (cid:101) , ⊗ (cid:48) = ⊕ (cid:48) ⊕ (cid:101) , ⊗ (cid:48) = ⊕ (cid:48) ⊕ (cid:101) , (cid:48) ⊗ (cid:101) = ⊕ ⊕ (cid:48) . a r X i v : . [ h e p - ph ] O c t The basic paradigm is that the symmetry ∆(48) (cid:111) H CP is respected at high energy scales, and is then sponta-neously broken to different subgroups G ν (cid:111) H νCP and G l (cid:111) H lCP in the neutrino and charged lepton sectorsby flavon fields. This misalignment between the symme-try breaking patterns leads to particular predictions formixing angles and CP phases. Without loss of generality,three generations of the left-handed lepton doublets areassigned to ∆(48) triplet . The invariance of the La-grangian under residual family symmetries and residualGCP symmetries implies that the neutrino mass matrix m ν and the charged lepton mass matrix m l should satisfy ρ † ( g ν ) m ν ρ ∗ ( g ν ) = m ν , ρ † ( g l ) m l m † l ρ ( g l ) = m l m † l , (4a) X † ν m ν X ∗ ν = m ∗ ν , X † l m l m † l X l = ( m l m † l ) ∗ , (4b)where neutrinos are assumed to be Majorana particles, g ν , g l denote the group elements of the residual familysymmetries G ν , G l , and X ν , X l denote the elements ofthe remnant GCP symmetries H νCP , H lCP , respectively.By systematically scanning all the possible remnantfamily subgroups G ν and G l , we find that only the case G ν = Z and G l = Z can lead to viable phenomenology.One can choose G ν = { , c } and G l = { , a, a } withoutloss of generality, since all the possible choices are relatedby group conjugation. From the constraint of Eq. (4a),we find that the charged lepton mass matrix m l is diag-onal in the chosen basis, and the neutrino mass matrix m ν takes the form: m ν = α − − − − − − + β + γ + (cid:15) − − − , (5)where α , β , γ and (cid:15) are complex parameters, and they arefurther constrained by the neutrino residual GCP sym-metry H νCP , as shown in Eq. (4b).Each GCP transformation corresponds to an automor-phism of the family symmetry G f . The automorphismgroup of ∆(48) is Aut(∆(48)) ∼ = ∆(48) (cid:111) D with 384group elements. Its outer automorphism group is provento be a dihedral group Out(∆(48)) ∼ = D , with generators u and u defined as (cid:40) a u −→ a c u −→ cd , (cid:40) a u −→ ac u −→ cd . (6)The following multiplication rules are fulfilled u = u = ( u u ) = id . (7)Each group element in Out(∆(48)) can be expressed as u µ u ν for µ = 0 , , , ν = 0 ,
1. The generators u and u act on the irreducible representation of ∆(48) as (cid:48) u ←→ (cid:48)(cid:48) , u −→ (cid:48) u −→ u −→ (cid:48) u −→ , (cid:101) u −→ (cid:101) , (cid:48) u −→ (cid:48) , (cid:48)(cid:48) u −→ (cid:48)(cid:48) , u ←→ (cid:48) , u ←→ (cid:48) , (cid:101) u −→ (cid:101) . (8) The 8 outer automorphisms generated by u and u lead to different CP transformations and should havedistinct physical implications. In the present work, weminimally extend the ∆(48) family symmetry to includeonly those nontrivial CP transformations which map oneirreducible representation into its complex conjugate. Wefind that there are three outer automorphisms, u , u u ,and u u , satisfying this requirement. • The first automorphism u interchanges all 3d ir-reducible representations with their complex con-jugate representations. The corresponding GCPmatrix in each 3d irreducible representation is de-termined to be just a permutation X ( u ) = P .Thus, the GCP transformation acts on a 3d field φ = ( φ , φ , φ ) T as φ φ φ CP −→ P φ ∗ φ ∗ φ ∗ = φ ∗ φ ∗ φ ∗ . (9)This is the so-called µ − τ reflection symmetry [13]. • The second automorphism u u interchanges (cid:48) with (cid:48) but maps and into themselves. In (cid:48) and (cid:48) representation space, we find it to be X ( u u ) = , i.e., the conventional CP transfor-mation φ φ φ CP −→ φ ∗ φ ∗ φ ∗ = φ ∗ φ ∗ φ ∗ . (10) • The third one u u exchanges with while maps (cid:48) and (cid:48) into themselves. In and , the corre-sponding GCP matrix is X ( u u ) = as well.Taking account of the inner automorphisms, we find thatresidual GCP transformations compatible with the rem-nant family symmetry G ν = { , c } can be expressed as X ν = ρ ( c m d n ) P or ρ ( c m d n ) (11)in all 3d representations with m, n = 0 , , , m ν in Eq. (5) by applying the residualGCP symmetry and then derive the PMNS matrix. Inthe PDG convention [14], the PMNS matrix is cast in theform U P MNS = V diag(1 , e i α , e i α ) , (12)with V = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c c s e iδ − c s − s c s e iδ c c , (13) in which c ij = cos θ ij , s ij = sin θ ij , δ is the DiracCP-violating phase and α , α are the Majorana CP-violating phases. It is more convenient to redefine the pattern A pattern B pattern C pattern DGCP matrix X ρ ( c k + k d k ) P ρ ( c k + k d k ) ρ ( c k + k +1 d k ) P ρ ( c m d k +1 ) P sin θ
13 13 − cos ϑ − cos ϑ − √ cos ϑ − √ √ cos ϑ sin θ
12 12+cos ϑ ϑ √ ϑ √ √ ( √ ) cos ϑ sin θ
23 12 12 ∓ √ ϑ ϑ ∓ √ ϑ √ ϑ ∓ ( −√ ) cos ϑ √ ( √ ) cos ϑ J CP − sin ϑ √ ∓ sin ϑ √ ∓ sin ϑ √ | tan δ | + ∞ (cid:12)(cid:12) √ ϑ √ ϑ tan ϑ (cid:12)(cid:12) (cid:12)(cid:12) √ ( √ ) cos ϑ −√ −√ ϑ tan ϑ (cid:12)(cid:12) | tan α | or | cot α | (cid:12)(cid:12) √ ϑ sin ϑ (cid:12)(cid:12) (cid:12)(cid:12) √ √ ϑ + ( −√ ) sin ϑ √ √ ϑ − ( −√ ) sin ϑ (cid:12)(cid:12) | tan α (cid:48) | (cid:12)(cid:12) √ ϑ − ϑ (cid:12)(cid:12) (cid:12)(cid:12) ϑ − √ ( √ ) cos 2 ϑ (cid:12)(cid:12) TABLE I: The predictions for lepton mixing patterns and the associated mixing parameters for all possible choices of residualGCP symmetries in the neutrino sector, and all the mixing patterns are found to depend on only one parameter ϑ varying from0 to 2 π . The related GCP matrices hold for all faithful 3d representations , , (cid:48) and (cid:48) with k , k = 0 , m = 0 , , ,
3. Thesign “ + ∞ ” for | tan δ | implies that the corresponding Dirac CP-violating phase is ± π/ Majorana phase α (cid:48) ≡ α − δ during the analysis ofthe neutrinoless double-beta decay.With different choices of remnant GCP transforma-tions in Eq. (11) and abandoning the cases which predictdegenerate neutrino masses, we obtain 4 kinds of mix-ing patterns, denoted by patterns A, B, C and D. Eachmixing pattern depends on one free parameter ϑ and pre-dicts sin θ = 1 / (3 cos θ ) since the structure of m ν in Eq. (5) preserves the second column of the PMNSmatrix as (1 / √ , / √ , / √ T . As a consequence, mix-ing angles as well as CP phases are strongly correlated,as shown in Table I. For proper values of ϑ , all cases arecompatible with the present neutrino oscillation data [15]within 3 σ range, except θ in pattern C.In the following, we will focus on pattern D which iscompletely new as far as we know. In this case, the GCPsymmetry corresponding to the outer automorphism u should be implemented. All the mixing parameters, inparticular the CP phases are nontrivially dependent on ϑ ,and the correlations between the mixing parameters areplotted in Fig. 1. Excellent agreement with the presentglobal-fitting data of mixing angles can be achieved. It isinteresting to note that the relation between sin α (orcos α ) and sin α (cid:48) , shown in low right panel, lookslike the “compound eyes” of an insect. Taking the 3 σ ranges of mixing angles from [15], we obtain0 . (cid:54) | ϑ | (cid:54) . , (14)and the CP-violating phases are constrained to lie in thefollowing intervals:0 . (cid:54) sin δ (cid:54) . , . (cid:54) sin α (cid:48) (cid:54) , . (cid:54) sin α or cos α (cid:54) . . (15)The quadrants of CP-violating phases cannot be deter-mined in the present model-independent approach. No-tice that the Dirac phase δ is large but not maximal, andthis prediction could be tested in the next generationneutrino oscillation experiments LBNE and Hyper-K. σ σ ϑ =0 ϑ = π / ϑ = π / ϑ =2 π / ϑ = π ϑ bf sin θ s i n θ ϑ =0 ϑ = π / ϑ = π / ϑ =2 π / ϑ = π ϑ bf sin θ s i n θ ϑ =0 ϑ = π / ϑ = π / ϑ =2 π / ϑ = π ϑ bf sin θ s i n δ ϑ =0, π ϑ = π / ϑ = π / ϑ =2 π / ϑ bf sin α or cos α s i n α ′ FIG. 1: Correlations among the mixing angles and CP-violating phases in pattern D. We mark the best-fit value θ bf of the parameter ϑ with a red star, and also label ϑ =0 , π/ , π/ , π/ , π with a red cross on the curve. In the topright panel, the results of sin θ for the first octant and thesecond octant of θ , are shown in a solid line and dashed line,respectively. The 1 σ and 3 σ ranges of the mixing angles aretaken from Ref. [15]. This pattern is also testable in the future neutrino-less double-beta (0 νββ ) decay experiments. The rateof 0 νββ decay is determined by the nuclear matrix el-ement and the effective parameter (cid:104) m (cid:105) ee = | m c c + m s c e iα + m s e iα (cid:48) | . In Fig. 2, we show the pre-diction for (cid:104) m (cid:105) ee as a function of the lightest neutrinomass, where the constraint in Eq. (14) has been takeninto account. The upper bounds from cosmology (thesum of neutrino masses (cid:80) m i < .
23 eV) [16] and the - - - - - - - - m lightest @ eV D ¨ m ee ¨ @ e V D Disfavoured by 0 nbb D i s f avo u re db y C o s m o l ogy FIG. 2: The prediction of the effective neutrino mass (cid:104) m (cid:105) ee as a function of the lightest neutrino mass m in normal or-dering (NO) or m in inverted ordering (IO) in pattern D. Thesplits within each mass ordering come from the uncertaintiesof the quadrants of CP-violating phases. current 0 νββ bound ( (cid:104) m (cid:105) ee < .
32 eV) [17] are alsoincluded in the figure. The next generation 0 νββ de-cay experiments will reach the sensitivity of (cid:104) m (cid:105) ee (cid:39) (0 . − .
05) eV after 5 years of data taking [17]. As aconsequence, if the signal of 0 νββ would not be observed,the inverted mass ordering scenario of this pattern wouldbe excluded, since we have (cid:104) m (cid:105) ee > .
02 eV in this caseas shown in Fig. 2.Finally, we shall construct a simple flavor model inwhich pattern D is realized. The field arrangement islisted in Table II, where (cid:96) L , e R , µ R , τ R denote the left-handed and right-handed lepton fields, H represents theHiggs field, and φ l , ϕ l , ρ l , ϕ , ξ are the gauge-singletflavon fields. The additional Z × Z × Z symmetryis used to eliminate undesired dangerous operators andderive suitable vacuum alignments. Yukawa couplingsinvariant under ∆(48) × Z × Z × Z are − L (cid:96) = y τ Λ φ l (cid:96) L Hτ R + y µ Λ ( φ l ϕ l ) (cid:96) L Hµ R + y e Λ (cid:0) ρ l ( φ l φ l ) (cid:101) (cid:1) (cid:96) L He R + y e Λ (cid:0) φ l ( ϕ l ϕ l ) (cid:48) S (cid:1) (cid:96) L He R + y ϕ Λ ϕ ( (cid:96) L (cid:101) H (cid:101) H T (cid:96) cL ) S + y ξ Λ ξ ( (cid:96) L (cid:101) H (cid:101) H T (cid:96) cL ) (cid:101) + h.c., (16)in which (cid:101) H = iσ H ∗ , and Λ is the cut-off scale. More-over, all coupling coefficients are real since the GCP sym-metry is imposed. The flavon vacuum expectation valuescan be realized by using the supersymmetric driving fieldmethod. Here we directly list them as (cid:104) φ l (cid:105) = v φ l (1 , , T , (cid:104) ϕ l (cid:105) = v ϕ l (0 , , T , (cid:104) ρ l (cid:105) = v ρ l (0 , , T , (cid:104) ϕ (cid:105) = e i π v ϕ (1 , , T , (cid:104) ξ (cid:105) = v ξ (0 , − ω , ω ) T , (17)where v φ l , v ϕ l and v ρ l are generally complex, while v ϕ and v ξ are real. Notice that the vacuum of the neutrino Fields (cid:96) L e R µ R τ R H φ l ϕ l ρ l ϕ ξ ∆(48) (cid:48) (cid:48) (cid:101) Z ω ω ω ω ω ω ωZ − i i − i i i − − Z ω ω ω ω ω × Z × Z × Z , where ω = e i π/ and (cid:96) L =( (cid:96) τL , (cid:96) µL , (cid:96) eL ) T . flavons ϕ and ξ preserve Z (cid:111) H νCP symmetry, wherethe residual GCP matrix is X ν = ρ ( d ) P . As a result,pattern D is naturally produced.Leptons acquire masses after symmetry breaking. Thecharged lepton mass matrix is found to be diagonal with m e = (cid:12)(cid:12)(cid:12) y e v ρ l v φ l Λ v + 2 ωy e v φ l v ϕ l Λ v (cid:12)(cid:12)(cid:12) ,m µ = (cid:12)(cid:12)(cid:12) y µ v φ l v ϕ l Λ v (cid:12)(cid:12)(cid:12) , m τ = (cid:12)(cid:12)(cid:12) y τ v ϕ l Λ v (cid:12)(cid:12)(cid:12) , (18)in which v = (cid:104) H (cid:105) = 175 GeV. The neutrino mass matrixis of the form of Eq. (5) with α = e i π y ϕ v ϕ v Λ , β = − ω y ξ v ξ v Λ , γ = − (cid:15) = ω y ξ v ξ v . (19)The PMNS matrix is exactly pattern D, and the param-eter ϑ fulfills tan ϑ = y ξ v ξ / (2 √ y ϕ v ϕ ). For the neutrinomasses, we find that the neutrino mass spectrum can onlybe NO. A detailed calculation shows that∆ m ∆ m = 12 − ϑ −
14 sin ϑ . (20)To be compatible with data | ∆ m / ∆ m | (cid:39) .
03, wefind ϑ (cid:39) − . y ξ v ξ (cid:39) − . y ϕ v ϕ ,which leads to the predictions: θ (cid:39) . ◦ , θ (cid:39) . ◦ , θ (cid:39) . ◦ ,δ (cid:39) . ◦ , α (cid:39) . ◦ , α (cid:39) . ◦ . (21)The lightest neutrino mass m and the effective mass (cid:104) m (cid:105) ee is fixed in this model, m (cid:39) . , (cid:104) m (cid:105) ee (cid:39) . . (22)In summary, we have proposed the discrete group∆(48) to explain lepton mixing angles and predict CP-violating phases in the framework of generalized CP sym-metries. ∆(48) has a large automorphism group and thusprovides rich choices for GCP transformations. By sys-tematically scanning all the possible symmetry break-ing chains, we find 4 different mixing patterns compat-ible with experimental data. Among them, pattern Dis a completely new mixing pattern that has not beendiscussed in the literature. It predicts nontrivial CP-violating phases, which can be tested in the future neu-trino oscillation and neutrinoless double-beta decay ex-periments. We have realized this pattern in an effectiveflavor model, where all the neutrino flavor mixing param-eters, the absolute scale of neutrino masses and (cid:104) m (cid:105) ee arefixed.We would like to thank Z.Z. Xing for simulating dis-cussion, continuous support, and reading the manuscript. We are also grateful to Y.F. Li for his helpful discussion.This work was supported in part by the National NaturalScience Foundation of China under Grant Nos. 11275188,11179007, and 11135009. [1] for some reviews, see e.g., G. Altarelli and F. Feruglio,Rev. Mod. Phys. , 2701 (2010) [arXiv:1002.0211 [hep-ph]]; H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu,H. Okada and M. Tanimoto, Prog. Theor. Phys. Suppl. , 1 (2010) [arXiv:1003.3552 [hep-th]]; S. F. Kingand C. Luhn, Rept. Prog. Phys. , 056201 (2013)[arXiv:1301.1340 [hep-ph]].[2] F. P. An et al. [DAYA-BAY Collaboration], Phys. Rev.Lett. , 171803 (2012) [arXiv:1203.1669 [hep-ex]];J. K. Ahn et al. [RENO Collaboration], Phys. Rev. Lett. , 191802 (2012) [arXiv:1204.0626 [hep-ex]]; Y. Abe etal. [DOUBLE-CHOOZ Collaboration], Phys. Rev. Lett. , 131801 (2012) [arXiv:1112.6353 [hep-ex]].[3] G. Ecker, W. Grimus and W. Konetschny, Nucl. Phys. B , 465 (1981); G. Ecker, W. Grimus and H. Neufeld,Nucl. Phys. B , 70 (1984); J. Phys. A , L807 (1987);H. Neufeld, W. Grimus and G. Ecker, Int. J. Mod. Phys.A , 603 (1988).[4] F. Feruglio, C. Hagedorn and R. Ziegler, JHEP ,027 (2013) [arXiv:1211.5560 [hep-ph]].[5] M. Holthausen, M. Lindner and M. A. Schmidt, JHEP , 122 (2013). [arXiv:1211.6953 [hep-ph]].[6] G. -J. Ding, S. F. King, C. Luhn and A. J. Stuart, JHEP , 084 (2013) [arXiv:1303.6180 [hep-ph]].[7] C. -C. Li and G. -J. Ding, Nucl. Phys. B , 206 (2014)[arXiv:1312.4401 [hep-ph]].[8] G. -J. Ding, S. F. King and A. J. Stuart, JHEP ,006 (2013) [arXiv:1307.4212 [hep-ph]].[9] I. Girardi, A. Meroni, S. T. Petcov and M. Spinrath,JHEP , 050 (2014) [arXiv:1312.1966 [hep-ph]]. [10] G. J. Ding and Y. L. Zhou, JHEP , 023 (2014)[arXiv:1404.0592 [hep-ph]].[11] W. Grimus and M. N. Rebelo, Phys. Rept. , 239(1997) [hep-ph/9506272].[12] C. Luhn, S. Nasri and P. Ramond, J. Math. Phys. ,073501 (2007) [hep-th/0701188].[13] P. F. Harrison and W. G. Scott, Phys. Lett. B , 163 (2002) [hep-ph/0203209]; Phys. Lett. B , 219 (2002) [hep-ph/0210197]; W. Grimus andL. Lavoura, Phys. Lett. B , 113 (2004) [hep-ph/0305309]; P. F. Harrison and W. G. Scott, Phys.Lett. B , 324 (2004) [hep-ph/0403278]; Y. Farzanand A. Yu. Smirnov, JHEP , 059 (2007) [hep-ph/0610337]; P. M. Ferreira, W. Grimus, L. Lavoura andP. O. Ludl, JHEP , 128 (2012) [arXiv:1206.7072[hep-ph]]; W. Grimus and L. Lavoura, Fortsch. Phys. ,535 (2013) [arXiv:1207.1678 [hep-ph]].[14] J. Beringer et al. [Particle Data Group Collaboration],Phys. Rev. D , 010001 (2012).[15] M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado andT. Schwetz, JHEP , 123 (2012) [arXiv:1209.3023[hep-ph]].[16] P. A. R. Ade et al. [Planck Collaboration], Astron. As-trophys. (2014) [arXiv:1303.5076 [astro-ph.CO]].[17] W. Rodejohann, Int. J. Mod. Phys. E , 1833 (2011)[arXiv:1106.1334 [hep-ph]]; S. M. Bilenky and C. Giunti,Mod. Phys. Lett. A , 1230015 (2012) [arXiv:1203.5250[hep-ph]]; S. T. Petcov, Adv. High Energy Phys.2013