After an Introduction briefly describing the rise and fall of the three-zero
texture of the Zee model, we discuss still-allowed two-zero textures for the
Majorana three-neutrino mass matrix. Finally, a model with two right-handed
neutrinos and two Dirac texture zeros is described (FGY model) which can relate
CP violation in leptogenesis to CP violation in long-baseline neutrino
oscillations.Comment: 9 pages latex. Talk at Coral Gables 2003. Added reference

Frampton P. H.

Fukugita M.

P. H. FRAMPTON

Sakharov A. D.

Zee A.

English

arXiv

Crossref

ASPECTS OF NEUTRINO MASS MATRICES

After an Introduction briefly describing the rise and fall of the three-zero texture of the Zee model, we discuss still-allowed two-zero textures for the Majorana three-neutrino mass matrix. Finally, a model with two right-handed neutrinos and two Dirac texture zeros is described (FGY model) which can relate CP violation in leptogenesis to CP violation in long-baseline neutrino oscillations

Frampton, P.H.

Carolina Digital Repository

arXiv:hep-ph/0403164v2  21 May 2004July 29, 2021 10:49 WSPC/Trim Size: 9in x 6in for Proceedings PF0403cgASPECTS OF NEUTRINO MASS MATRICESP.H. FRAMPTONDepartment of Physics and Astronomy,University of North Carolina at Chapel Hill,Chapel Hill, NC 27599-3255.E-mail: frampton@physics.unc.eduAfter an Introduction briefly describing the rise and fall of the three-zero textureof the Zee model, we discuss still-allowed two-zero textures for the Majorana three-neutrino mass matrix. Finally, a model with two right-handed neutrinos and twoDirac texture zeros is described (FGY model) which can relate CP violation inleptogenesis to CP violation in long-baseline neutrino oscillations.1. IntroductionThe minimal standard model involves three chiral neutrino states, butit does not admit renormalizable interactions that can generate neutrinomasses. Nevertheless, experimental evidence show that both solar and at-mospheric neutrinos display flavor oscillations, and hence that neutrinos dohave mass.Among the many renormalizable and gauge-invariant extensions of thestandard model that can provide neutrino masses are:• The introduction of a complex triplet of mesons (T++, T+, T 0)coupled bilinearly to pairs of lepton doublets1. They must alsocouple bilinearly to the Higgs doublet(s) so as to avoid spontaneousB − L violation and the appearance of a massless and experimen-tally excluded majoron. This mechanism can generate an arbitrarycomplex symmetric Majorana mass matrix for neutrinos.• The introduction of singlet counterparts to the neutrinos with verylarge Majorana masses. The interplay between these mass termsand those generated by the Higgs boson—the so-called see-sawmechanism—yields an arbitrary but naturally small Majorana neu-trino mass matrix.• The introduction of a charged singlet meson f+ coupled antisym-1July 29, 2021 10:49 WSPC/Trim Size: 9in x 6in for Proceedings PF0403cg2metrically to pairs of lepton doublets, and a doubly-charged singletmeson g++ coupled bilinearly both to pairs of lepton singlets andto pairs of f -mesons. An arbitrary Majorana neutrino mass matrixis generated in two loops.• The introduction of a charged singlet meson f+ coupled antisym-metrically to pairs of lepton doublets and (also antisymmetrically)to a pair of Higgs doublets. This simple mechanism was first pro-posed by Zee 2, and results (at one loop) in a Majorana mass matrixin the flavor basis (e, µ, τ) of a special form:M =0 meµ meτmeµ 0 mµτmeτ mµτ 0 (1)Because the diagonal entries of M are zero, the amplitude for no-neutrino double beta decay vanishes at lowest order and this pro-cess cannot proceed at an observable rate. Furthermore, the pa-rameters meµ, meτ and mµτ may be taken real and non-negativewithout loss of generality, whence M becomes real as well as trace-less and symmetric. With this convention, the analog to theKobayashi-Maskawa matrix becomes orthogonal: M is explicitlyCP invariant and δ = 0.The sum of the neutrino masses (the eigenvalues of M) vanishes: m1 +m2+m3 = 0. When the squared-mass hierarchy is taken into account, thatthe solar squared mass difference is very much smaller than the atmosphericsquared mass difference, there are two possibilities. In case A, we havem1+m2 = 0 and m3 = 0. This case arises iff at least one of the three parametersin M vanishes. In case B, we have m1 = m2 and m3 = −2m2 > 0. It hasbeen shown3,4 that neither case is consistent with the SuperKamiokandeand SNO data so the the Zee model is ruled out by these experiments.2. Two-Zero TexturesHaving excluded one three-zero texture we now study two-zero possibilities.Quark masses and mixings are described by two 3×3 matrices which in-volve a total of ten convention-independent real quantities: four Kobayashi-Maskawa parameters and six positive quark masses. The lepton sector re-quires only three additional parameters (the charged lepton masses) in theJuly 29, 2021 10:49 WSPC/Trim Size: 9in x 6in for Proceedings PF0403cg3minimal—and manifestly incomplete—standard model wherein neutrinosare massless.We assume that neutrino phenomenology can be formulated in termsof three left-handed neutrino states with a complex symmetric Majoranamass matrix M. We do not commit ourselves to any particular mechanismby which neutrino masses are generated. Of the twelve real parameterscharacterizing M, three are arbitrary phases of the three flavor eigenstates.Thus the neutrino sector involves nine convention-independent parameters.The two squared-mass differences and the four Kobayashi-Maskawa analogsare likely to be measureable in practice. So also is the quantity |Mee|through the search for neutrinoless double beta decay. There remain twoparameters that are measureable in principle but apparently not easily inpractice.It will be useful to define the ratio of squared-mass differences, whoseestimated value is: Rν ≡ ∆s∆a ≈ 2× 10−2.We turn to the question of which two independent entries of M canvanish in the basis wherein the charged lepton mass matrix is diagonal.Of the fifteen logical possibilities, we find just seven to be in accord withour empirical hypotheses. We discuss them individually, with the non-vanishing entries in each case denoted by X ’s. Our results are presented toleading order in the small parameter s2. We begin with a texture in whichMee = Meµ = 0:Case A1:0 0 X0 X XX X XFor this texture, Mee = 0 so that the amplitude for no-neutrino double betadecay vanishes to lowest order in neutrino masses. If Case A1 is realized innature, the neutrinoless process simply cannot be detected. There is evenmore to say. Two of the squared neutrino masses are suppressed relativeto the third by a factor of s22. As a result we find that s2 can lie closeto its present experimental upper limit. This prediction will become moreprecise when θ3 is better measured. However, the CP-violating parameterδ is entirely unconstrained. For case A1 the subdominant angle θ2 is likelyto be measureable and neutrinos may display observable CP violation.Case A2:0 X 0X X X0 X XThis texture has Mee = Meτ = 0. Its phenomenological consequences areJuly 29, 2021 10:49 WSPC/Trim Size: 9in x 6in for Proceedings PF0403cg4nearly the same as those of Case A1.Case B1:X X 0X 0 X0 X XWith Mµµ = Meτ = 0, we find an acceptable solution if and only if|s2 cos δ tan 2θ1| ≪ 1, in which case the three neutrinos are nearly de-generate in magnitude because t21 ≃ 1. The texture is also promisingin regard to the search for neutrinoless double beta decay. We obtainMee ≃ −t21√∆a/|1− t41| likely to exceed 100 meV. The rate of neutrinolessdouble beta decay may approach its current experimental upper limit.Case B2:X 0 X0 X XX X 0This texture, with Mττ = Meµ = 0, is as in B1 with t1 replaced by−1/t1. Its phenomenological consequences are nearly the same as those ofCase B1.Case B3:X 0 X0 0 XX X X and Case B4:X X 0X X X0 X 0The phenomenological implications of all four cases Bi (i = 1, 2, 3, 4) aresubstantially the same.Case C:X X XX 0 XX X 0Our seventh and last allowed texture has Mµµ = Mττ = 0.No other two-zero texture of the neutrino mass matrix is compatiblewith our empirical hypotheses. It is easily verified that no two of our allowedtwo-zero textures can be simultaneously satisfied while remaining consistentwith our empirical hypotheses. It follows that there is no tolerable three-zero texture.The seven allowed two-zero neutrino textures fall into three classes: A(with two members), B (with four members), and C. The textures withineach class are difficult or impossible to distinguish experimentally, but eachof the three classes has radically different implications. For class A, theJuly 29, 2021 10:49 WSPC/Trim Size: 9in x 6in for Proceedings PF0403cg5subdominant angle θ2 is expected to be relatively large, but no-neutrino ββdecay is forbidden. For class B, the latter process should be measureableby the next generation of double beta decay experiments, whilst s2 may ormay not be large enough to be detected. However, if s2 is comparable to itsexperimental upper limit, CP violation must be nearly maximal and shouldbe readily detectable by proposed experiments. For class C, no-neutrino ββdecay is likely to be observable and θ2 ought to be large enough to measureand to permit a search for CP violation. It is surprising that such a greatvariety of textures of the neutrino mass matrix can fit what is presentlyknown about neutrino masses and oscillations. Future data should revealwhich, if any, of these textures should serve as a guide to the model builder.3. FGY ModelOne of the most profound ideas is6 that baryon number asymmetry arisesin the early universe because of processes which violate CP symmetry andthat terrestrial experiments on CP violation could therefore inform us ofthe details of such cosmological baryogenesis.The early discussions of baryogenesis focused on the violation of baryonnumber and its possible relation to proton decay. In the light of presentevidence for neutrino masses and oscillations it is more fruitful to associatethe baryon number of the universe with violation of lepton number7. In thepresent Letter we shall show how, in one class of models, the sign of thebaryon number of the universe correlates with the results of CP violation inneutrino oscillation experiments which will be performed in the forseeablefuture.Present data on atmospheric and solar neutrinos suggest that there arerespective squared mass differences ∆a ≃ 3 × 10−3eV 2 and ∆s ≃ 5 ×10−5eV 2. The corresponding mixing angles θ1 and θ3 satisfy tan2θ1 ≃ 1and 0.6 ≤ sin22θ3 ≤ 0.96 with sin2θ3 = 0.8 as the best fit. The thirdmixing angle is much smaller than the other two, since the data requiresin22θ2 ≤ 0.1.A first requirement is that our model5 accommodate these experimentalfacts at low energy.In the minimal standard model, neutrinos are massless. The most eco-nomical addition to the standard model which accommodates both neutrinomasses and allows the violation of lepton number to underly the cosmolog-ical baryon asymmetry is two right-handed neutrinos N1,2.These lead to new terms in the lagrangian:July 29, 2021 10:49 WSPC/Trim Size: 9in x 6in for Proceedings PF0403cg6L =12(N1, N2)(M1 00 M2)(N1N2)++ (N1, N2)(a a′00 b b′)l1l2l3H + h.c. (2)where we shall denote the rectangular Dirac mass matrix by Dij . We haveassumed a texture for Dij in which the upper right and lower left entriesvanish. The remaining parameters in our model are both necessary andsufficient to account for the data.For the light neutrinos, the see-sawmechanism leads to the mass matrix8L̂ = DTM−1D=a2M1aa′M10aa′M1(a′)2M1+ b2M2bb′M20 bb′M2(b′)2M2(3)We take a basis where a, b, b′are real and where a′is complex a′ ≡|a′ |eiδ. To check consistency with low-energy phenomenology we temporar-ily take the specific values (these will be loosened later) b′= b and a′=√2aand all parameters real. In that case:L̂ =a2M1√2a2M10√2a2M12a2M1+ b2M2b2M20 b2M2b2M2(4)We now diagonalize to the mass basis by writing:L =12νT L̂ν =12ν′TUT L̂Uν′(5)whereU =1/√2 1/√2 0−1/2 1/2 1/√21/2 −1/2 1/√2××1 0 00 cosθ sinθ0 −sinθ cosθ (6)We deduce that the mass eigenvalues and θ are given bym(ν′3) ≃ 2b2/M2; m(ν′2) ≃ 2a2/M1; m(ν′1) = 0 (7)July 29, 2021 10:49 WSPC/Trim Size: 9in x 6in for Proceedings PF0403cg7andθ ≃ m(ν′2)/(√2m(ν′3)) (8)in which it was assumed that a2/M1 ≪ b2/M2.By examining the relation between the three mass eigenstates and thecorresponding flavor eigenstates we find that for the unitary matrix relevantto neutrino oscillations thatUe3 ≃ sinθ/√2 ≃ m(ν2)/(2m(ν3)) (9)Thus the assumptions a′=√2a, b′= b adequately fit the experimentaldata, but a′and b′could be varied around√2a and b respectively to achievebetter fits.But we may conclude that2b2/M2 ≃ 0.05eV =√∆a2a2/M1 ≃ 7× 10−3eV =√∆s (10)It follows from these values thatN1 decay satisfies the out-of-equilibriumcondition for leptogenesis (the absolute requirement ism < 10−2eV 9) whileN2 decay does not. This fact enables us to predict the sign of CP violationin neutrino oscillations without ambiguity.Let us now come to the main result. Having a model consistent with alllow-energy data and with adequate texture zeros10 in L̂ and equivalently Dwe can compute the sign both of the high-energy CP violating parameter(ξH) appearing in leptogenesis and of the CP violation parameter whichwill be measured in low-energy ν oscillations (ξL).We find the baryon number B of the universe produced by N1 decayproportional to11B ∝ ξH = (ImDD†)212 = Im(a′b)2= +Y 2a2b2sin2δ (11)in which B is positive by observation of the universe. Here we have loosenedour assumption about a′to a′= Y aeiδ.At low energy the CP violation in neutrino oscillations is governed bythe quantityξL = Im(h12h23h31) (12)where h = L̂L̂†.July 29, 2021 10:49 WSPC/Trim Size: 9in x 6in for Proceedings PF0403cg8Using Eq.(3) we find:h12 =(a3a′∗M21+a|a′ |2a′∗M21)+aa′b2M1M2h23 =(bb′a′2M1M2)+(b3b′M22+bb′3M22)h31 =(aa′∗bb′M1M2)(13)from which it follows thatξL = −a6b6M31M32sin2δ[Y 2(2 + Y 2)] (14)Here we have taken b = b′because the mixing for the atmospheric neutrinosis almost maximal.Neutrinoless double beta decay (ββ)0ν is predicted at a rate correspond-ing to L̂ee ≃ 3× 10−3eV .The comparison between Eq.(11) and Eq.(14) now gives a unique rela-tion between the signs of ξL and ξH .This fulfils in such a class of models the idea of 6 with only the smallchange that baryon number violation is replaced by lepton number viola-tion.AcknowledgementI wish to thank Ruth Daly, Tom Kephart, the late Behram Kursunoglu,Steve Mintz, Arnold Perlmutter and Ina Sarcevic for organizing the verymemorable Fest. This work was supported in part by the Department ofEnergy under Grant Number DE-FG02-97ER-410236.References1. G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B181, 287 (1981); R.N.Mohapatra and G. Senjanović, Phys. Rev. D23, 165 (1981); C. Wetterich,Nucl. Phys. B187, 343 (1981).2. A. Zee, Phys. Lett, 93B 389 (1980).3. P.H. Frampton and S.L. Glashow, Phys. Lett. B461, 95 (1999).hep-ph/9906375.4. P.H. Frampton, M.C. Oh and T. Yoshikawa, Phys. Rev. B65, 073014 (2002).hep-ph/0110300.July 29, 2021 10:49 WSPC/Trim Size: 9in x 6in for Proceedings PF0403cg95. P.H. Frampton, S.L. Glashow and T. Yanagida, Phys. Lett. B548, 119(2002). hep-ph/02081576. A.D. Sakharov, Pis’ma Zh. Eksp. Teor. Fiz. 5, 32 (1967) [JETP Lett. 5, 24(1967)]7. M. Fukugita and Y. Yanagida, Phys. Lett. B174, 45 (1986).8. T. Yanagida, in Proceedings of the Workshop on Unified Theories and BaryonNumber of the Universe Editors:O.Sawada and A. Sugamoto, (Tsukuba,Japan, 1979), KEK Report KEK-79-18, page 95.S.L. Glashow, Cargese lectures (1979).M. Gell-Mann, P. Ramond and R. Slansky, in Supergravity Editors: D.Z.Freedman and P. Van Nieuwenhuizen, (North-Holland, Amsterdam, 1979).9. W. Buchmüller, in Proceedings of the Eighth Symposium on Particles,Strings and Cosmology. Editors: P.H. Frampton and Y.J. Ng. Rinton Press,New Jersey (2001), page 97.10. P.H. Frampton, S.L. Glashow and D. Marfatia, Phys. Lett. B536, 79 (2002).11. See e.g. W. Buchmüller and M. Plumacher, Phys. Reps. 320, 329 (1999),and references therein.

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Aspects of Neutrino Mass Matrices

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