The neutrino oscillation data are well explained by the tri-bimaximal
pattern. Recently it has been shown that also the bimaximal pattern could be a
very good starting point in order to describe the lepton mixing. In this paper
I review both the flavour structures and then I present an explicit example.Comment: 4 pages; Talk given at IFAE 2009: Incontri di Fisica delle Alte
  Energie, Bari, Italy, 15-17 April 200

Merlo, Luca

English

arXiv

The neutrino oscillation data are well explained by the tri-bimaximal pattern. Recently it has been shown that also the bimaximal pattern could be a very good starting point in order to describe the lepton mixing. In this paper I review both the flavour structures and then I present an explicit example

Merlo, L.

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2009-10425-5Colloquia: IFAE 2009IL NUOVO CIMENTO Vol. 32 C, N. 3-4 Maggio-Agosto 2009Tri-bimaximal and bimaximal neutrino mixing fromdiscrete symmetriesL. Merlo(∗)Dipartimento di Fisica “G. Galilei”, Universita` di Padova and INFN, Sezione di PadovaVia Marzolo 8, I-35131 Padova, Italy(ricevuto il 19 Settembre 2009; pubblicato online il 14 Ottobre 2009)Summary. — The neutrino oscillation data are well explained by the tri-bimaximalpattern. Recently it has been shown that also the bimaximal pattern could be avery good starting point in order to describe the lepton mixing. In this paper Ireview both the flavour structures and then I present an explicit example.PACS 11.30.Hv – Flavour symmetries.PACS 14.60.Pq – Neutrino mass and mixing.1. – The lepton mixing matrixThe solar and the atmospheric neutrino anomalies are well explained by the oscillationof three massive neutrinos: in table I there are the results of two independent global fitsfrom [1] and [2] of the experimental data.The pattern of the mixings is characterized by two large angles and a small one: theatmospheric angle θ23 is compatible with a maximal value, but the accuracy admits rel-atively large deviations; the solar angle θ12 is large, but about 5σ errors far from themaximal value; the reactor angle θ13 only has an upper bound, which can be parame-terized at 3σ as sin2 θ13 ≤ λ2C , where λC  0.23 is the Cabibbo angle. Measuring thereactor angle in future experiments represents a fundamental task in order to understandthe pattern of the neutrino mixing [3].From the theoretical point of view, for long time, people tried to reproduce modelswith a maximal angle and a vanishing one. The most general mass matrix which providesθ13 = 0 and θ23 maximal owns the μ − τ symmetry, for which (mν)2,2 = (mν)3,3 and(∗) E-mail: merlo@pd.infn.itc© Societa` Italiana di Fisica 333334 L. MERLOTable I. – Neutrino oscillation parameters from independent global fits [1, 2].Ref. [1] Ref. [2]Parameter Best fit (1σ) 3σ-interval Best fit (1σ) 3σ-intervalΔm221 (10−5 eV2) 7.67+0.16−0.19 7.14–8.19 7.65+0.23−0.20 7.05–8.34|Δm231| (10−3 eV2) 2.39+0.11−0.8 2.06–2.81 2.40+0.12−0.11 2.07–2.75sin2 θ12 0.312+0.019−0.018 0.26–0.37 0.304+0.022−0.016 0.25–0.37sin2 θ23 0.466+0.073−0.058 0.331–0.644 0.50+0.07−0.06 0.36–0.67sin2 θ13 0.016+0.010−0.010 ≤ 0.046 0.010+0.016−0.011 ≤ 0.056(mν)1,2 = (mν)1,3, and it is given bymν =⎛⎝ x y yy z wy w z⎞⎠ .(1)It depends on four real parameters: the three masses and the solar angle, which can bewritten in terms of the parameters of the matrix assin2 2θ12 =8y2(x− y − z)2 + 8y2 .(2)Many models have been constructed introducing a flavour symmetry in addition to thegauge group of the Standard Model in order to provide the μ− τ symmetric pattern forthe neutrino mass matrix, but in all of these the solar angle remains undetermined. Somenew ingredient other than the μ− τ symmetry is therefore needed to describe correctlythe neutrino mixings from the theoretical point of view.A second relevant aspect concerns the maximality of the atmospheric angle: it is avery well-known result [4] that a maximal atmospheric angle can be achieved only whenthe flavour symmetry is broken. As a consequence, the new flavour symmetry needed tocorrectly reproduce the three mixing angles has to be broken, spontaneously or softly.1.1. The tri-bimaximal mixing . – In what follows I review two relevant flavour pat-terns, which can be considered an upgrade of the μ − τ symmetry. The first one is thetri-bimaximal (TB) pattern [5] which assumes a vanishing reactor angle, a maximal at-mospheric angle and sin2 θ12 = 1/3. From eq. (2) it results w = x + y − z and thereforethe most generic mass matrix of the TB-type is given bymTBν =⎛⎝ x y yy z x + y − zy x + y − z z⎞⎠ .(3)This matrix satisfies the μ − τ symmetry and the so-called magic symmetry, forwhich (mTBν )1,1 = (mTBν )2,2 + (mTBν )2,3 − (mTBν )1,3. It is diagonalized by a unitaryTRI-BIMAXIMAL AND BIMAXIMAL NEUTRINO MIXING FROM DISCRETE SYMMETRIES 335transformation in such a way that (mTBν )diag = (UTB)T mTBν UTB, where the unitarymatrix is given byUTB =⎛⎝√2/3 1/√3 0−1/√6 1/√3 −1/√2−1/√6 1/√3 +1/√2⎞⎠ .(4)Note that UTB does not depend on the mass eigenvalues, in contrast with the quarksector, where the entries of the CKM matrix can be written in terms of the ratio of thequark masses. Moreover it is a completely real matrix, since the factors with the Diracphase vanish (the Majorana phases can be factorized outside).Many flavour symmetries have been used to recover the TB pattern: the most knownare the discrete groups A4 [4, 6], T ′ [7], S4 [8] and some continuous groups [9]. In allthese models, when the symmetry is broken, some corrections to the mixing angles areintroduced: in general all of them are of the order of λ2C and therefore these modelssuggest a value for the reactor angle which is very small, at most of the order of 0.05.1.2. The bimaximal mixing . – In agreement with the data in table I, the reactor angleis suggested to be non-vanishing and, if a value close to the present upper bound is foundin the future experiments, this could be interpreted as an indication that the agreementwith the TB mixing is only accidental. Looking for an alternative leading principle, itis interesting to note that the data suggest a numerical relationship between the leptonand the quark sectors, known as the complementarity relation, for which θ12+λC  π/4.However, there is no compelling model which manages to get this nice feature, withoutparameter fixing. Our proposal is to relax this relationship. Noting that√mμ/mτ  λC ,we can write the following expression, which we call weak complementarity relation:θ12  π4 −O(√mμmτ).(5)The idea is first to get a maximal value both for the solar and the atmospheric anglesand then to correct θ12 with relatively large terms. To reach this task, the bimaximal(BM) pattern can be extremely useful: it corresponds to the requirement that θ13 = 0and θ23 = θ12 = π/4. For the solar angle we can alternatively write that sin2 θ12 = 1/2,which brings to w = x− z from eq. (2). The most general mass matrix of the BM-typeand its diagonalizing unitary matrix are then given bymBMν =⎛⎝ x y yy z x− zy x− z z⎞⎠ , UBM =⎛⎝ 1/√2 −1/√2 01/2 1/2 −1/√21/2 1/2 +1/√2⎞⎠ ,(6)where mBMν satisfies to the μ − τ symmetry and to an additional symmetry for which(mBMν )1,1 = (mBMν )2,2 + (mBMν )2,3. Note that UBM does not depend on the mass eigen-values and is completely real, like in the TB pattern.Starting from the BM scheme, the corrections introduced from the symmetry breakingmust have a precise pattern: δ sin2 θ12  λC , while δ sin2 θ23 ≤ λ2C and δ sin θ13 ≤ λCin order to be in agreement with the experimental data. This feature is not triviallyachievable.336 L. MERLO2. – The model buildingIn this part I present a flavour model in which the neutrino mixing matrix at theleading order (LO) is the BM scheme, while the charged lepton mass matrix is diagonalwith hierarchical entries; moreover the model allows for corrections that bring the mixingangles in agreement with the data (for details see [10]). The strategy is to use the flavourgroup Gf = S4×Z4×U(1)FN , where S4 is the group of the permutations of four objects,and let the SM fields transform non-trivially under Gf ; moreover some new fields, theflavons, are introduced which are scalars under the SM gauge symmetry, but transformunder Gf ; these flavons, getting non-vanishing vacuum expectation values (VEVs), spon-taneously break the symmetry in such a way that two subgroups are preserved, G = Z4in the charged lepton sector and Gν = Z2×Z2 in the neutrino sector. It is this breakingchain which assures that the LO neutrino mixing matrix is the BM pattern and thatthe charged lepton mass matrix is diagonal and hierarchical. As a consequence the LOlepton mixing matrix U ≡ (U)†Uν , the only which can be observable, coincides withUBM.Thanks to the particular VEV alignment, the next-to-leading order (NLO) correc-tions, coming from the higher order terms, are not democratic and the corrected mixingsaresin2 θ12 =12− 1√2(a + b), sin2 θ23 =12, sin θ13 =1√2(a− b),(7)where a and b parameterize the VEVs of the flavons. When a and b are of the order of theCabibbo angle, then the θ12 is brought in agreement with the experimental data; in themeantime the reactor angle is corrected of the same amount, suggesting a value for θ13close to its present upper bound. Note that the atmospheric angle remains uncorrectedat this order.It is then interesting to verify the agreement of the model with other sectors ofthe neutrino physics, such as the 0ν2β-decay and the leptogenesis. The result ofthe analysis is that the model presents a normal-ordered—moderate hierarchical orquasi-degenerate—spectrum with a suggested lower bound for the lightest neutrino massand for the effective 0ν2β-mass parameter |mee| of about 0.1meV. On the other hand,it is compatible with the constraints from leptogenesis as an explanation of the baryonasymmetry in the Universe.∗ ∗ ∗I thank the organizers of IFAE 2009 – Incontri di Fisica delle Alte Energie for givingthe opportunity to present my talk and for the kind hospitality in Bari. Alike I thankG. Altarelli and F. Feruglio for the pleasant and advantageous collaboration.REFERENCES[1] Fofli G. L., Lisi E., Marrone A., Palazzo A. and Rotunno A. M., arXiv:0809.2936[hep-ph].[2] Schwetz T., Tortola M. and Valle J. W. F., New J. Phys., 10 (2008) 113011.[3] For more details see the talks of Mauro Mezzetto and Maurizio Bonesini, theseproceedings.[4] Altarelli G. and Feruglio F., Nucl. Phys. B, 720 (2005) 64.TRI-BIMAXIMAL AND BIMAXIMAL NEUTRINO MIXING FROM DISCRETE SYMMETRIES 337[5] Harrison P. F., Perkins D. H. and Scott W. G., Phys. Lett. B, 530 (2002) 167; XingZ. Z., Phys. Lett. B, 533 (2002) 85.[6] Ma E. and Rajasekaran G., Phys. Rev. D, 64 (2001) 113012; Babu K. S., Ma E. andValle J. W. F., Phys. Lett. B, 552 (2003) 207; Altarelli G. and Feruglio F., Nucl.Phys. B, 741 (2006) 215.[7] Feruglio F., Hagedorn C., Lin Y. and Merlo L., Nucl. Phys. B, 775 (2007) 120;Chen M. C. and Mahanthappa K. T., Phys. Lett. B, 652 (2007) 34.[8] Bazzocchi F. and Morisi S., arXiv:0811.0345 [hep-ph]; Bazzocchi F., Merlo L. andMorisi S., Nucl. Phys. B, 816 (2009) 204, arXiv:0902.2849 [hep-ph].[9] King S. F., JHEP, 0508 (2005) 105; de Medeiros Varzielas I. and Ross G. G., Nucl.Phys. B, 733 (31) 2006.[10] Altarelli G., Feruglio F. and Merlo L., JHEP, 0905 (2009) 020.

Tri-bimaximal and bimaximal neutrino mixing from discrete symmetries

Abstract

Similar works

Full text

Available Versions

Scientific Open-access Literature Archive and Repository