We show in a model independent manner that, if the magnitudes of Majorana
masses of neutrinos are exactly equal at some high scale, the radiative
corrections cannot reproduce the observed masses and mixing spectrum at the low
scale, irrespective of the Majorana phases or the mixing angles at the high
scale.Comment: 12 pages ReVTeX, A few typos corrected in the 2nd versio

Dighe, Amol S.

Joshipura, Anjan S.

English

arXiv

We show in a model independent manner that, if the magnitudes of Majorana masses of neutrinos are exactly equal at some high scale, the radiative corrections cannot reproduce the observed masses and mixing spectrum at the low scale, irrespective of the Majorana phases or the mixing angles at the high scale.We show in a model independent manner that, if the magnitudes of Majorana masses of neutrinos are exactly equal at some high scale, the radiative corrections cannot reproduce the observed masses and mixing spectrum at the low scale, irrespective of the Majorana phases or the mixing angles at the high scale

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hep-ph/0010079CERN-TH/2000-301No-go for exactly degenerate neutrinos at high scale ?Amol S. Dighe1 and Anjan S. Joshipura1,2y1Theory Division, CERN, CH-1211 Geneva 23, Switzerland.2Theoretical Physics Group, Physical Research Laboratory,Navarangpura, Ahmedabad, 380 009, India.AbstractWe show in a model independent manner that, if the magnitudes of Majo-rana masses of neutrinos are exactly equal at some high scale, the radiativecorrections cannot reproduce the observed masses and mixing spectrum atthe low scale, irrespective of the Majorana phases or the mixing angles at thehigh scale.The data from the solar and atmospheric neutrino experiments can be explained throughthe mixing of three active neutrinos with nonzero masses. The atmospheric neutrino solutionneeds νµ−ντ mixing with the corresponding mass squared dierence of m2atm  10−3−10−2eV2, and a large mixing angle, sin2 2θatm > 0.8 [1]. The solar neutrino solution needs themixing of νe with a combination of νµ and ντ with a corresponding mass squared dierencem2 < 10−4 eV2, the mixing angle may be small or large, depending on the particularsolution [2].Since m2atm >> m2, there are three distinct patterns for the neutrino masses fromthe point of view of mass hierarchy. We dene the mass eigenstate ν3 as the one such thatjm231j  jm232j  m2atm, and the states ν1 and ν2 as the ones separated by jm221j m2, with m231 and m221 having the same sign (here m2ij  jmij2 − jmjj2). With thisconvention, the possible patterns for the neutrino masses are (i) completely hierarchical:jm1j  jm2j  jm3j, (ii) partially degenerate: jm1j  jm2j  jm3j, also called as invertedhierarchy or (iii) completely degenerate: jm1j  jm2j  jm3j. In contrast to the rst twocases, the common mass in the third case can be near the direct limit on the electron neutrinomass as obtained from the Kurie plot [3]. Such a mass can have its own signature in theneutrinoless double beta decay [4]. In order for the neutrinos to contribute even a smallfraction of the dark matter [5] of the universe, degenerate neutrino masses are essential.amol.dighe@cern.chyanjan.joshipura@cern.ch1The pattern of mixing in the lepton sector (which denitely has at least one large mixingangle, as suggested by the atmospheric neutrino data) is very dierent from that in thequark sector (where all the mixing angles are small). The presence of large mixing anglesis another indication that the neutrino masses may be partially or completely degenerate(cases ii and iii above). The origin of this degeneracy, as well as its breaking (which leadsto the mass splittings among neutrinos) needs to be understood theoretically. Here, weconcentrate on the latter.The SU(2)U(1) interactions in the standard model (and its supersymmetric generalisa-tions) break the symmetry among generations explicitly through dierent Yukawa couplings.The resulting radiative corrections modify the neutrino mass matrices while evolving from ahigh scale to a low scale. The eects of radiative evolutions on the masses and mixings havebeen studied in [6{8]. The most economical possibility would be to have exactly degenerateneutrinos at the high scale (the seesaw scale, or the GUT scale, for example), and let theradiative corrections lead to the required mass dierences and mixing angles. The exactdegeneracy of neutrino masses at the high scale may be obtained by a symmetry [9], whichis broken later in the charged lepton sector.The models with exact degeneracy at the high scale are constrained by the masses andmixings observed at the low scale (at the experiments). In the context of specic models,some constraints have been found. Refs. [10,11] have examined the consequences of thebimaximal mixing pattern at the high scale which was motivated to suppress the eect oflarge degenerate masses of neutrinos in the neutrinoless double beta decay. Taking the modelof Georgi and Glashow [12] as an example, they show that the bimaximal pattern is unstableunder radiative corrections. Moreover, one obtains the inverted hierarchy (m1 > m2 > m3)for the renormalized masses in case of the MSSM, which rules out the MSW solution forthe solar neutrinos. Ref. [13] assumes Ue3 = 0 at the high scale, and claims that the mixingpattern is unstable under radiative corrections irrespective of the initial values of the angles.The above arguments are not complete in ruling out the case of degenerate neutrinosfor the following reasons. In the standard model, one does indeed obtain the normal masshierarchy (m1 < m2 < m3) for the renormalized neutrino masses1. Even in the case ofthe inverted hierarchy (that one gets with MSSM), matter can play an important role inthe solution to the solar neutrino problem, and this possibility is not experimentally ruledout [14]. Moreover, one need not insist on the bimaximal mixing pattern if the commondegenerate mass is not much larger than the experimental limit coming from the neutrinolessdouble beta decay. Finally, while the arguments of [13] hinge on the assumption of Ue3 = 0,nonzero values of Ue3 are allowed by the experiments [15].In this paper, we reinvestigate the viability of the exact degenerate spectrum in a modelindependent way. We show that (I) in any model with the magnitudes of the neutrino massesexactly equal at some high scale, the mixing at the high scale can always be dened in sucha way that its structure is preserved in the process of evolution, so that the issue of thestability of mixing angles does not arise. (II) However, irrespective of the structure of theoriginal mass matrix, the degenerate spectrum at a high scale cannot lead to the observed1Ref. [13] claim inverted hierarchy in case of the SM rather than the MSSM, but this can betraced to their having the wrong sign for the radiative correction parameter  to be defined later.2masses and mixings at the low scale, as long as the charged lepton masses are generated byonly one Higgs particle. The results are valid irrespective of the model (in particular, theyhold for the SM and the MSSM), the Majorana phases of the degenerate neutrinos or theirmixing angles at the high scale.The most general neutrino mass matrix Mν in the Majorana basis is a symmetric complexmatrix and it can always be diagonalized by a unitary matrix U0 in the standard way:UT0 Mν U0 = D , (1)where D refers to a diagonal matrix with real and positive masses. In particular, D isproportional to the identity matrix for the exactly degenerate spectrum. It then followsthat for degenerate neutrinos, the Majorana mass matrix MF (X) at the high scale X in theflavour basis (in which the charged lepton mass matrix is diagonal) can always be writtenasMF (X) = mU0 U y0  m V , (2)where m is a positive real number denoting the common mass of the degenerate system andV is a symmetric unitary matrix. The freedom in dening the phases of the charged leptonscan be used to replace V ! KV K without loss of generality, where K is a diagonal phasematrix. We use this freedom to make the third row and the third column of V (and hence ofMF (X)) real and positive. The matrix V can be parametrised in terms of only two anglesand a phase [16].The radiative corrections modify MF as one evolves down to the low scale x. Withonly one Higgs giving masses to the charged leptons, the Yukawa couplings of the chargedleptons are hierarchical (he : hµ : hτ = me : mµ : mτ  3  10−4 : 6  10−2 : 1). In the limit ofneglecting the electron and muon Yukawa couplings, the radiative modication of the massmatrix is given byMF (x) = IτMF (X)Iτ , (3)where Iτ  Diag(1, 1,pIτ ). We dene the radiative correction parameter  throughpIτ 1 + . The value of  is determined by the model. In particular, for the SM and the MSSM, can be written in the form  C h2τ(4pi)2lnXx, (4)where hτ  mτ/v is the tau Yukawa coupling in the standard model. The value of theconstant C is (1/2) and (−1/ cos2 β) in the case of SM and MSSM respectively. Note thatthe sign of  is opposite in these two cases, and hence the mass shifts in these two cases arein opposite directions.Using (2), (3) and the unitarity of V , we getMF (x)MF (x)y = m2I + (Iτ − 1)V 213 V13V23 V13V33pIτV13V23 V223 V23V33pIτV13V33pIτ V23V33pIτ 1 + V233Iτ , (5)3where I is an identity matrix. All the entries in the above matrix are real and positivedue to the phase convention we have chosen (the third column of V has real and positiveelements). The matrix appearing in the second term has one zero eigenvalue, indicatingthat one of the originally degenerate eigenvalues remains unchanged in magnitude after theradiative corrections. The other two masses are aected by the radiative corrections so thatthe eigenvalues of (5) are given to leading order in  byjmaj2 = m2jmbj2 = m2 + 4 sin2 ~Ψ + O(2)jmcj2 = m2 + 4 cos2 ~Ψ + O(2) , (6)where the angle ~Ψ is dened such thatV33 = − cos 2 ~Ψ . (7)The corresponding eigenvectors are given by the columns of the following matrix:Ux = R12(Ω)R23(Ψ) =cΩ sΩcΨ sΩsΨ−sΩ cΩcΨ cΩsΨ0 −sΨ cΨ , (8)where cθ  cos θ, sθ  sin θ, and the matrices Rij are the rotation matrices in the corre-sponding planes. The angles Ω and Ψ are given bytan Ω =V13V23; tanΨ = (1 + 2) tan ~Ψ + O(2) (9)The matrix Ux (8) is the mixing matrix at the low scale x. The corresponding matrix UXat the high scale is dened only up to UX ! UXOK due to the exact degeneracy in masses(here O is an orthogonal matrix, and K is a diagonal phase matrix), however a \natural"choice of UX can be made. Indeed, following the arguments in [16], one can always write Vin the formV =cΩ′ sΩ′ 0−sΩ′ cΩ′ 00 0 1e2iα 0 00 c2Ψ′ −s2Ψ′0 −s2Ψ′ −c2Ψ′cΩ′ −sΩ′ 0sΩ′ cΩ′ 00 0 1 , (10)= R12(Ω0)Diag(eiα, 1, 1)R23(Ψ0)Diag(1, 1,−1)RT23(Ψ0)Diag(eiα, 1, 1)RT12(Ω0) , (11)which conforms to our phase convention of a real third row and third column. The matrixUX that diagonalizes this V (and hence, MF (X)) can be chosen to beUX = R12(Ω0)R23(Ψ0) =cΩ′ sΩ′cΨ′ sΩ′sΨ′−sΩ′ cΩ′cΨ′ cΩ′sΨ′0 −sΨ′ cΨ′ . (12)This mixing matrix UX at the high scale has the same form as the matrix Ux at the lowscale (8) resulting after the radiative corrections. Expanding the right hand side of (11),we get V33 = − cos 2Ψ0, so that using (7), we can identify ~Ψ = Ψ0. This implies that4tan Ψ = tanΨ0[1 + O()]. Also, tan Ω0 = V13V23, so that from (9), we get Ω = Ω0. Thus,the radiative corrections leave Ω0 unchanged2, and modify the angle Ψ0 only at O(). Thespecic choice for UX made in (12) is thus a \natural" choice, such that the mixing is notperturbed in the process of evolution.The specic structure of the mixing matrix Ux (eq. (8)) is imposed in a model indepen-dent way just by the requirement that the neutrinos be exactly degenerate at some highscale X. This structure does not allow large angle solutions to the solar neutrino problemwithout conflicting either with the atmospheric neutrino data [1] or the CHOOZ constraints[15]. The small angle solution is still allowed (see [17] for the detailed allowed region inangles Ω, Ψ in this case). However, even in that case it is not possible to obtain the requiredmass squared dierences, as we show below.Note that the rows of Ux in (8) are labelled by the flavour indices e, µ, τ , while itscolumns are labelled by the mass eigenstates a, b, c. Reordering of the mass eigenvaluesamounts to the interchange of columns of Ux. We have denoted by a the eigenvalue whichremains unchanged and the corresponding eigenvector at low scale (which does not haveany τ flavour component) is given by the rst column of Ux. The other columns correspondto the eigenvalues jmbj2 and jmcj2 respectively. The neutrino mass squared dierences aregiven from eq.(6) bym2cb  4 cos 2 ~Ψ , m2ca  4 cos2 ~Ψ , m2ba  4 sin2 ~Ψ . (13)It is seen from (13) that two hierarchical mass squared dierences are possible if (i) ~Ψ  pi/4,(ii) ~Ψ  pi/2, (iii) ~Ψ  0. These three cases correspond to the identication of the masseigenstate ν3 with a, b, c respectively. We shall show below that all the three identicationslead to phenomenological problems. Note that since the arguments below depend only onthe ratios of the mass squared dierences, they are independent of the magnitude or sign of, and hence are valid for all the models.In case (i), the rst column of Ux should be identied with the third column of theleptonic mixing matrix U , leading to the predictionjUe3j2 + jUµ3j2 = 1 .This is clearly in contradiction with the required values for the atmospheric neutrino mixingand the bound on jUe3j from CHOOZ [15].In case (ii), one hasm2m2atm cot2 ~Ψ , (14)sin2 2θatm = 4c2Ωc2Ψ(1− c2Ωc2Ψ) . (15)Since m2/m2atm < 0.1, from (14) we have cos2 Ψ  cos2 ~Ψ < cot2 ~Ψ < 0.1. Then eq. (15)gives sin2 2θatm < 0.4, which is clearly inconsistent with the atmospheric neutrino data.2This general result has been proved in [17] in the case of the mixing matrix of the form U =R12 R23. Here we have shown that U can always be brought in this form in the present context.5In case (iii),m2m2atm tan2 ~Ψ , (16)sin2 2θatm = 4c2Ωs2Ψ(1− c2Ωs2Ψ) . (17)Since m2/m2atm < 0.1, from (16) we have sin2 Ψ  sin2 ~Ψ < tan2 ~Ψ < 0.1. Then eq. (17)gives sin2 2θatm < 0.4, which is clearly inconsistent with the atmospheric neutrino data,similar to the case (ii) above.The above reasoning holds both in case of the small angle as well as the large anglesolution to the solar neutrino problem. As already remarked, the latter can be independentlyruled out from the structure of the mixing matrix Ux. Thus, it is not possible to generatethe observed pattern of neutrino masses and mixings solely through radiative corrections,starting from exactly degenerate Majorana neutrinos at some high scale X. This conclusionis valid irrespective of the Majorana phases or the mixing angles at the high scale, or thevalue of the high scale itself as long as there is no new physics between the high and the lowscale.When the masses at the high scale are not exactly equal, but the mass splittings aresmall compared to the magnitude of radiative corrections (m2ij(X)  jmij2), the radiativecorrections determine the mixing matrix at the low scale, and hence the mixing matrix is Uxas given by eq.(8). The mass squared dierences at the low scale are still dominantly givenby eq.(13). (In the case where the original Majorana masses of the neutrinos are mi = m,this is shown explicitly in [18].) This implies that the required masses and mixings at thelow scale still cannot be reproduced even with the introduction of small mass splittings(m2ij(X)  jmij2) at the high scale. On the other hand, when m2ij(X)  jmij2, theradiative corrections clearly fail to have any signicant impact on the values of m2ij andthe mixing angles. The net result is that at least one of the mass splittings at the high scale(m2ij(X)) needs to be approximately equal to the one observed at the low scale (m2ij(x)),and hence needs to be generated through some other mechanism. The radiative correctionscan help only in generating the other (most likely smaller) mass splitting. Such masses andmixing patterns at the high scale have been shown to lead to correct masses and mixings atthe low scale in several models [19].The other possibility is that the degeneracy is completely broken at the high scale explic-itly by a source other than the Yukawa couplings. Specic models have been investigated in[13,20]. These may give rise to quasi-degenerate neutrino masses at the high scale. One pos-sible origin of such splitting is the running of the degenerate right handed neutrino massesfrom the Planck scale to the GUT or the scale of the right handed neutrino masses [11].The radiative corrections can then modify the mass splittings as well as the mixing angles.The change in the mass splittings will be proportional to the magnitude of the radiativecorrections, but the mixing angles can get modied drastically { they can get magnied orcan be driven to zero, depending on the magnitude and sign of the radiative corrections ascompared to the original mass splittings at the high scale [6,7,18]. It may even be possibleto obtain large angles at the low scale irrespective of the mixing angles at the high scale [8].Throughout this analysis, we have assumed that the charged lepton Yukawa couplingsinvolve only one Higgs. 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No-go for exactly degenerate neutrinos at high scale?

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