Present neutrino data are consistent with neutrino masses arising from a
common seed at some ``neutrino unification'' scale $M_X$. Such a simple
theoretical ansatz naturally leads to quasi-degenerate neutrinos that could lie
in the electron-volt range with neutrino mass splittings induced by
renormalization effects associated with supersymmetric thresholds. In such a
scheme the leptonic analogue of the Cabibbo angle $\theta_{\odot}$ describing
solar neutrino oscillations is nearly maximal. Its exact value is correlated
with the smallness of $\theta_{reactor}$. These features agree both with latest
data on the solar neutrino spectra and with the reactor neutrino data. The two
leading mass-eigenstate neutrinos present in \ne form a pseudo-Dirac neutrino,
avoiding conflict with neutrinoless double beta decay.Comment: RevTex format, 2 figures, 4 pages, a few new references, no other
  important change, figures unchanged, version to be published in PR

A. G. Cohen

A. N. Ioannisian

B. T. Cleveland

C. Wetterich

D. E. Groom

E. Dudas

E. J. Chun

G. M. Asatrian

H. Georgi

H. S. Hirata

J. A. Casas

J. Ellis

J. N. Abdurashitov

J. Schechter

J. W. F. Valle

K. R. Balaji

K. S. Babu

L. Ibanez

L. Wolfenstein

M. Apollonio

M. C. Gonzalez-Garcia

P. Binetruy

P. H. Chankowski

R. Becker-Szendy

R. Davis

S. M. Bilenki

S. Pokorski

S. Weinberg

T. Takeuchi

W. Hampel

W. W. M. Allison

Y. Fukuda

Y. Grossman

Y. Nir

Y. Suzuki

English

arXiv

Crossref

Physical Review Letters

Neutrino Unification

Present neutrino data are consistent with neutrino masses arising from a common seed at some 'neutrino unification' scale MX. Such a simple theoretical ansatz naturally leads to quasi-degenerate neutrinos that could lie in the electron-volt range with neutrino mass splittings induced by renormalization effects associated with supersymmetric thresholds. In such a scheme the leptonic analogue of the Cabibbo angle θ⊙ describing solar neutrino oscillations is nearly maximal. Its exact value is correlated with the smallness of θreactor. These features agree both with latest data on the solar neutrino spectra and with the reactor neutrino data. The two leading mass-eigenstate neutrinos present in \ne form a pseudo-Dirac neutrino, avoiding conflict with neutrinoless double beta decay

Chankowski, Piotr H.

Ioannisian, Ara

Pokorski, Stefan

Furtado Valle, José Wagner

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:hep-ph/0011150v2  14 Feb 2001hep-ph/0011150FERMILAB-Pub-00/250-TIFIC/00-68, IFT-00/26Neutrino UnificationP. H. Chankowski1, A. Ioannisian2,3, S. Pokorski1,4 and J. W. F. Valle21 Univ. of Warsaw Inst. Physics Ul. Hoz˙a 69, 00-681, Warsaw, Poland2 Instituto de F´ısica Corpuscular – C.S.I.C. – Univ. of Vale`nciaEdificio Institutos de Paterna – Apartado de Correos 2085 - 46071 Vale`ncia, Spain3 On leave from Yerevan Physics Institute, Alikhanyan Br.2, Yerevan, 375036, Armenia4 Frontier Fellow at Fermilab, P.O. Box 500, Batavia IL 60510, USA(August 6, 2013)Present neutrino data are consistent with neutrino masses arising from a common seed at some “neu-trino uniﬁcation” scale MX . Such a simple theoretical ansatz naturally leads to quasi-degenerateneutrinos that could lie in the electron-volt range with neutrino mass splittings induced by renormal-ization eﬀects associated with supersymmetric thresholds. In such a scheme the leptonic analogueof the Cabibbo angle θ⊙ describing solar neutrino oscillations is nearly maximal. Its exact value iscorrelated with the smallness of θreactor. These features agree both with latest data on the solarneutrino spectra and with the reactor neutrino data. The two leading mass-eigenstate neutrinospresent in νe form a pseudo-Dirac neutrino, avoiding conﬂict with neutrinoless double beta decay.The standard model (SM) and its minimal supersym-metric extension (MSSM) fail only in accounting for thesolar and atmospheric neutrino data [1–3] which stronglyindicate the need for neutrino conversions, as would arisefrom neutrino-mass-induced oscillations [4]. It is thusreasonable to investigate simple theoretically motivatedansatz that can account for these observations.Inspired by the idea of unification of fundamental in-teractions [5] here we propose the idea of neutrino unifi-cation: namely that the neutrino mass and mixings ob-served at low energies take a very simple form at somehigh energy scale MX . Thus, we add to the basic La-grangean the dimension–five operator [6]λ0δabMX(φℓa)(ℓbφ) + H. c. (1)where ℓa denote the three lepton doublets and φ is thestandard Higgs doublet. As a working hypothesis, we as-sume this operator to be characterized at the scale MX ,by a single real dimension–less parameter λ0. The break-ing of the electroweak symmetry due to a non-zero vac-uum expectation value (VEV) 〈φ〉 will generate, in addi-tion to the known SM masses, the seesaw-type neutrinomass operator MνMν =〈φ〉2MXλ0 (2)For MX ∼ MPlanck neutrino masses would be too smallto account for the atmospheric neutrino anomaly. Thuswe will adopt an intermediate value for the MX scale. Inthe basis in which the charged lepton Yukawa couplingsare diagonal (weak basis), λ0δij gets transformed intoΛ = λ0ΩTΩ (3)where Ω is an arbitrary unitary matrix. After the elec-troweak symmetry breaking we can rotate back the neu-trino fields, ν = Ω−1ν′, and in this basis (the chargedlepton mass matrix is still diagonal) we getL ⊃ λ0v2MXν′ν′ − g√2eLσ¯µW−µ Uν′ + H. c. (4)where U = Ω† and the fields ν′ are the neutrino masseigenstates (in this paper we number the eigenvalues sothat ∆m221= m22−m21≡ ∆m2⊙ corresponds to the solarneutrino oscillations and ∆m232 = m23 − m22 ≡ ∆m2atmcorresponds to the atmospheric neutrino oscillations). Ingeneral, the mixing matrix U is characterized by threemixing angles and three CP violating phases, one Diracplus two Majorana-type phases [7–9] and can be writtenas the product of VCKM by 1 0 00 eiα1 00 0 eiα2 (5)where we have chosen to parametrize VCKM exactlyas the Cabibbo-Kobayashi-Maskawa matrix describingquarks, with the familiar three angles and one phaseδ. It is clear that with our ansatz (1) we have an ad-ditional freedom of performing any arbitrary real 3 × 3rotation of the neutrinos. The general form of such arestricted matrix U can be derived by applying a con-venient parametrization [7] of a general unitary matrixto the matrix M ≡ ΩTΩ and by imposing the symmetrycondition M = MT . Using the freedom of re-phasingthe charged leptons, one can show that the U matrixcannot be eliminated by redefining the neutrinos. Thus,despite the mass degeneracy amongst the neutrinos, theyexhibit non-trivial mixing effects [10]. In contrast to thecase investigated in ref. [10] where neutrinos were mass-less but non-orthonormal, here it is the fact that the masseigenvalues have different phases which prevents one fromrotating away the mixing amongst the neutrinos. Never-theless, the structure of the mixing in our case is morerestricted than the general case, since only two anglesand one phase remain physical. The above constructionhas the virtue of explicitly listing the relevant mixingparameters for exactly degenerate Majorana neutrinos.Once the neutrino mass degeneracy is lifted by quan-tum corrections the mixing effects require the full matrixU with, however, a non-trivial relationship amongst thethree mixing angles.Non-zero values of the Dirac phases mean CP non-conservation, while Majorana phases of π/2 indicate dif-ferent CP parities of the neutrino mass eigenstates anddo not imply CP violation [11]. Thus in our case CP isconserved if the phase δ = 0 mod π and α1, α2 = 0 modπ/2. Our main conclusions do not depend on whether ornot CP is conserved in the neutrino sector and from nowon we will assume it is conserved. Different CP paritiesof neutrino mass eigenstates can then be accounted forby different signs of neutrino masses (which for α1 = π/2and/or α2 = π/2 can be achieved by simply absorbingthese phases in the neutrino fields).Now we turn to the renormalization effects. One-looprenormalization group equations (RGEs) for the Λ coeffi-cients characterizing the dimension-5 non-renormalizableterms can easily be written down both for the SM andMSSM [12]. The RGEs take a particularly useful formwhen written directly for mass eigenvalues and for theelements of the mixing matrix [13]. The flavour indepen-dent corrections to the Λ = O (1) coefficients are irrel-evant for our discussion. The flavour-dependent correc-tions are due to lepton Yukawa couplings and, to a goodapproximation, determined by the τ Yukawa coupling.Using e.g. the results of ref. [13] one can easily verifythat, with the ansatz (1), such effects cannot explain|∆m232| ≫ |∆m221| with phenomenologically acceptablemixing angles.It is remarkable that supersymmetry can induceflavour-dependent threshold corrections associated withslepton mass splittling [14] which can dominate over theτ Yukawa corrections. We show in this paper that suchcorrections can lead to the desirable low energy patternwith the ansatz (1). This is possible provided the softsupersymmetry breaking scalar mass terms deviate suffi-ciently from universality.The quantum corrections to our ansatz (1) in the basisdefined by eq. (4) are given by [14]mabν = maδab +ma(UT IU∗)ab +mb(U†IU)ab (6)where in our case all |ma| = m up to a common flavour-independent renormalization O (1) factor. Here the in-dices a, b refer to our starting basis used in eq. (1). Thecorrection I (calculated in the electroweak charged leptonmass eigenstate basis) consists of two partsI = IRG + ITH (7)where IRG is the renormalization group correction [15]and ITH are the electroweak scale threshold corrections[14]. Assuming no lepton flavour violation in other sec-tors of the theory (e.g. in the case of supersymmetry) thematrix I is diagonal, IAB = IAδAB (the index A refers tothe electroweak basis). Without loss of generality in ourdiscussion we take Iµ = 0. We adopt the general formfor the CP conserving mixing matrix U [7,16] c12c13 s12c13 s13−s12c23 − c12s23s13 c12c23 − s12s23s13 s23c13s12s23 − c12c23s13 −c12s23 − s12c23s13 c23c13 (8)in which θ13 ≡ θreactor, and allow for different CP paritiesof neutrino eigenstates. For degenerate masses |ma| = mthe matrix (8) contains one redundant angle and this de-gree of freedom is used to re-diagonalize the mass matrix(6). As long as we assume CP conservation we can usethe standard perturbation theory for degenerate zerothorder eigenvalues in order to determine the non-trivial re-lation amongst the three mixing angles in eq. (8). Taking|Iτ | ≫ |Ie| we easily recover the conclusions of ref. [13].In this letter we investigate the alternative possibilitythat |Ie| ≫ |Iτ |. We begin with m1 = −m2 = m3 ≡ min which casemabν = m 1 + 2U2A1IA 0 2UA1UA3IA0 −1− 2U2A2IA 02UA1UA3IA 0 1 + 2U2A3IA (9)Since m1 = m3, the ordinary perturbation calculus tellsus that the neutrino mass basis should be chosen so thatthe off-diagonal 13 entry of the perturbation is zero. Wetherefore require that UA1UA3IA = 0 which fixes the re-dundant angle in the matrix (8). With our assumptionsabout IA’s (|Ie| ≫ |Iτ |) this givess13 = −s12c12s23c23r +O(r2) (10)where r ≡ Iτ/Ie. It is then trivial to compute the cor-rected mass eigenvalues and one finds∆m2atm ≈ −4m2Ies212 , (11)∆m2⊙ ≈ −4m2Ie[C12(1− s223r)+ (1 + 2c212)s213](12)where C12 ≡ c212 − s212. For maximal θ12 = π/4 oneobtains that ∆m2atm ∼ Ie and ∆m2⊙ ∼ Ier2, in agreementwith the experimental requirement ∆m2⊙ ≪ ∆m2atm.Eqs. (11) and (12) lead to the following relation be-tween the neutrino oscillation parameters∆m2⊙ ≈ 2∆m2atm[C12(1− s223r)+ 2s213](13)which is the prediction of our model, independent of thevalue of Ie. Since we need c212 ≈ s212 we conclude thatsmall mixing solution to the solar neutrino problem isruled out in our neutrino unification picture.It is easy to check that with our assumptions aboutIA’s the same mechanism works in the case −m1 = m2 =m3. However, in the case m1 = m2 = −m3 the condi-tion UA1UA2IA = 0 means that s12c12c213 = O(r) whichmeans that one cannot have simultaneously large solarneutrino mixing and remain consistent with the reac-tor experiments [17] (this is also incompatible with theSMA solution to the solar neutrino problem because fors12c12 ≈ 0 one gets ∆m2⊙ ∼ ∆m2atm).Since ∆m2atm ≈ 3 × 10−3eV2 is rather fixed by theanalysis of the data [4] and since, as we shall see, typi-cally |Ie|<∼10−3, it follows that m>∼ O (1) eV. Thus thispicture leads to neutrino masses in the range accessibleto β decay and hot dark matter searches. Different CPparities of the neutrinos are however sufficient to ensure adestructive interference in the amplitude for neutrinolessdouble beta decay which is suppressed by the fact thatthe two leading mass eigenstate neutrinos in the νe forma pseudo-Dirac neutrino [18].How can one realize our neutrino unification scenario?First of all, it requires the existence of new states givinga dominant contribution to Ie. Rather than extendingthe gauge and/or multiplet structure of the SM, we pre-fer to ask whether in supersymmetry one can arrange thecorrections IA to satisfy our requirements. Taking a di-agonal slepton mass matrix (in the same basis in whichthe charged leptons mass matrix is diagonal) and takinginto account only the contribution of pure Wino of massm˜ one finds for IA ≡ I(yA) the following formIA =g232π2{− 1yA+y2A − 1y2Aln(1 − yA)− (A→ µ)}(14)where yA = 1−(MA/m˜)2,MA is the A-th charged sleptonmass. It is easy to see that for Ie ∼ 10−3 and positive weneed Me ≈ 1.7Mµ,τ for the charged slepton masses.Such slepton mass patterns can arise in models with in-verted hierarchy [19], although the proposed mechanismmay have other realizations. Being an operator of dimen-sion five, the nontrivial neutrino mass matrix does not af-fect the evolution of the lepton Yukawa and slepton masssquared matrices. Hence, if there exists a basis at MXin which both lepton Yukawa and slepton mass squaredmatrices are simultaneously diagonal (fermion-sfermionmass alignment [20] present e.g. in models with abelianU(1) gauge symmetry [21]), this basis remains unchangedduring the RG evolution to the electroweak scale andhence the slepton masses remain diagonal. Thus in thiscase lepton flavor-violating decays such as µ → eγ arenot induced.In Fig. 1 we illustrate the neutrino unification ideaby displaying the supersymmetric evolution of solar andatmospheric neutrino mass splittings as a function ofenergy, from the high energy scale MX where neutrinomasses unify, down to the weak scale for tanβ = 2 anda starting neutrino mass at MX of 1.2 eV. Above thesupersymmetry breaking scale the neutrino mass-squaredifferences are ∆m221= 13∆m231= 12∆m232≃ −m2Iτ . Af-ter threshold corrections they change according to Eqs.(11,12,13). Depending on the chosen value of MX thiscan fit large mixing angle solutions to the solar neutrinoproblem. For example for MX = 1013 GeV and Ie ≈ 2×10−3, r = IτIe= −0.029 we obtain quasi-degenerate neu-trino masses |mi| = 0.9 eV split by ∆m221 = −1.52×10−5eV2 and ∆m232 = −3.13 × 10−3 eV2. Neutrino mixingis bi-maximal s12 ≈ s23 ≈ 0.707 with s13 ≈ 0.0143,nicely matching the LMA solution. In order to achievethe LOW solution one needs a smaller Iτ value. Bar-ring a possible cancellation between the terms in ther.h.s of Eq. (13) this requires a low neutrino unificationscale MX = 105 GeV. In this case we find that quasi-degenerate masses |mi| = 1.1 eV for r = IτIe = −0.0103and Ie ≈ 1.3 × 10−3 with ∆m221 = −2.74 × 10−7 eV2and ∆m232= −3.16 × 10−3 eV2. Mixing is nearly bi-maximal with s13 ≈ 0.0051. Due to the observed flatnessof the latest recoil electron energy spectrum, present so-lar neutrino data globally prefer large mixing angle MSWsolutions over the small mixing solution (SMA) [4]. Thisfeature fits well in our neutrino unification scheme, wherethe SMA solution can not be realized. From ref. [4] wefind that in our scheme the LMA solution is present atthe 97% C.L. while the LOW solution appears already at90% C.L.Conclusions.We have proposed a simple theoretical ansatz whereneutrino masses arise from a common dimension-5 oper-ator at some “neutrino unification” scaleMX . This scalemust be lower than that which characterizes the unifica-tion of gauge and Yukawa couplings. The ansatz natu-rally implies quasi-degenerate neutrinos at the electron-volt range, thus accessible to future β decay and hot darkmatter searches. However neutrinoless double beta de-cay is naturally suppressed due to the intrinsic Majorananeutrino CP parities. We have shown that neutrino split-tings can be induced by supersymmetric thresholds aris-ing from non-universal soft breaking terms, without nec-essarily conflicting with limits on lepton flavour violatingprocesses such as µ → eγ. A non-trivial consequence ofthe neutrino unification idea is that the solar and atmo-spheric data can only be accounted for in terms of largemixing angle-type MSW oscillations. Note also that theconsistency of our scheme correlates the smallness of θ13indicated by reactor data and, to a lesser extent, also byFIG. 1. Schematic evolution of solar and atmospheric splittings corresponding to LMA (left) and LOW solutions (right)the atmospheric data [4], to the largeness of θ12 requiredby the solar neutrino data.Acknowledgements: Work supported by Spanish DG-ICYT under grants PB98-0693 and SAB/1998-0136(A.I.), by the European Commission TMR networkHPRN-CT-2000-00148, by the European Science Foun-dation network grant N. 86 and by the Polish State Com-mittee for Scientific Research grant KBN 2 P03B 060 18.P.H.Ch. thanks the CERN Theory group for hospitalityduring the completion of this paper.[1] Y. Suzuki, talk at XIX International Conference onNeutrino Physics and Astrophysics, Sudbury, Canada,http://nu2000.sno.laurentian.ca; T. Takeuchi, talk atthe XXXth International Conference on High En-ergy Physics, ICHEP 2000 http://www.ichep2000.rl.-ac.uk. Homestake Collaboration, B.T. Cleveland etal., Astrophys. J. 496, 505 (1998); R. Davis, Prog.Part. Nucl. Phys. 32, 13 (1994); K. 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Weinberg, Opening Address - Neutrinos ’78, In WestLafayette 1978, Proceedings, Neutrinos ’78.[7] J. Schechter and J. W. F. Valle, Phys. Rev. D22, 2227(1980).[8] As shown in J. Schechter and J. W. F. Valle, Phys. Rev.D23 (1981) 1666 Majorana phases are physical, thoughtheir eﬀects are small due to the small neutrino masses.[9] For a recent discussion see S.M. Bilenki, C. Giunti andW. Grimus, Prog. Part. Nucl. Phys. 48, 1 (1999).[10] Degenerate states can mix even if massless, seeJ. W. F. Valle, Phys. Lett. B199, 432 (1987) and Prog.Part. Nucl. Phys. 26 (1991) 91.[11] J. Schechter and J. W. F. Valle, Phys. Rev. D24, 1883(1981) & D25, 283 (1982)[12] C. Wetterich, Nucl. Phys. B187, 343 (1981); G. M. Asa-trian and A. N. Ioannisian, Sov. J. Nucl. Phys. 51, 115(1990); P. H. Chankowski and Z. P luciennik, Phys. Lett.B316, 312 (1993); K. S. Babu, C. N. Leung and J. Pan-taleone, Phys. Lett. B319, 191 (1993).[13] P. H. Chankowski, W. Kro´likowski and S. Pokorski, Phys.Lett. B473 (2000) 109; J.A. 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Neutrino unification

http://dx.doi.org/10.1103/PhysRevLett.86.3488

Neutrino Unification

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Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura