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PLANCK-SCALE PHYSICS AND NEUTRINO MASSES

arXiv:hep-ph/9205230v2 5 Jun 1992IC/92/79SISSA-83/92/EPLMU-04/92May 1992PLANCK-SCALE PHYSICS AND NEUTRINO MASSESEugeni Kh. Akhmedov(a,b,c) ∗, Zurab G. Berezhiani(d,e) † ‡,Goran Senjanovi´c(a) §(a)International Centre for Theoretical Physics, I-34100 Trieste, Italy(b)Scuola Internazionale Superiore di Studi Avanzati, I-34014 Trieste, Italy(c)Kurchatov Institute of Atomic Energy, Moscow 123182, Russia(d)Sektion Physik der Universit¨at M¨unchen, D-8000 Munich-2, Germany(e)Institute of Physics, Georgian Academy of Sciences, Tbilisi 380077, GeorgiaAbstractWe discuss gravitationally induced masses and mass splittings of Majorana, Zeldo-vich-Konopinski-Mahmoud and Dirac neutrinos.Among other implications, theseeffects can provide a solution of the solar neutrino puzzle.

In particular, we show howthis may work in the 17 keV neutrino picture.∗E-mail: akhmedov@tsmi19.sissa.it, akhm@jbivn.kiae.su†Alexander von Humboldt Fellow‡E-mail: zurab@hep.physik.uni-muenchen.de, vaxfe::berezhiani§E-mail: goran@itsictp.bitnet, vxicp1::gorans

A. IntroductionIt is commonly accepted, although not proven, that the higher dimensional operatorsinduced through the quantum gravity effects are likely not to respect global symmetries.This is, at least in part, a product of one’s experience with black holes and wormholes. Ifso, it becomes important to study the impact of such effects on various global symmetriesof physical interest.

Recently, an attention has been drawn to the issues of Peccei-Quinnsymmetry [1] and global non-abelian symmetry relevant for the textures [2]. Here, instead,we study the possible impact of gravity on the breaking of lepton flavor and lepton number,more precisely its impact on neutrino (Majorana) masses.

It is clear that all such effects,being cut offby the Planck scale, are very small, but on the other hand even small neutrinomass can be of profound cosmological and astrophysical interest.We start with a brief review of a situation in the standard model where such effectscan induce, as pointed out by Barbieri, Ellis and Gaillard [3] (in the language of SU(5)GUT), a large enough neutrino masses to explain the solar neutrino puzzle (SNP) throughthe vacuum neutrino oscillations. We also derive the resulting neutrino mass spectrum andthe mixing pattern, which have important implications for the nature of the solar neutrinooscillations.

From there on we center our discussion on the impact of the mass splits betweenthe components of Dirac and Zeldovich-Konopinski-Mahmoud (ZKM) [4] neutrinos. Ourmain motivation is the issue of SNP, but we will discuss the cosmological implications aswell.

The important point resulting from our work is a possibility to incorporate the solutionof the SNP in the 17 keV neutrino picture in a simple and natural manner. Finally, we makesome remarks on the see-saw mechanism and also mirror fermions in this context.B.

Neutrino masses in the standard modelSuppose for a moment that no right-handed neutrinos exist, i.e. the neutrinos are onlyin left-handed doublets.Barring accident cancellations (or some higher symmetry), the1

lowest-order neutrino mass effective operators are expected to be of dimension five:αij lTi Cτ2⃗τljHTτ2⃗τHMPl(1)where li = (νiL eiL)T, H is the usual SU(2)L × U(1) Higgs doublet, MPl is the Planckmass ≈1019 GeV and αij are unknown dimensionless constants. The operator (1) was firstwritten down by Barbieri et al.

[3] who, as we have mentioned before, based their discussionon SU(5) GUT although strictly speaking it only involves the particle states of the standardmodel. If gravity truly breaks the lepton number and induces terms in (1), neutrino maybe massive even in the minimal standard model.

This important result of ref. [3] seems notto be sufficiently appreciated in the literature.

As was estimated in [3], for αij ∼1 one getsfor the neutrino masses mν ∼10−5 eV, which is exactly of the required order of magnitudefor the solution of the SNP through the vacuum neutrino oscillations.We would like to add the following comment here. The universality of gravitationalinteractions makes it very plausible that all the αij constants in (1) should be equal to eachother: αij = α0.

If so, the neutrino mass matrix must take the ”democratic” form withall its matrix elements being equal to m0 ∼α0 · 10−5 eV. For three neutrino generationsthis pattern implies two massless neutrinos and a massive neutrino with mν = 3m0.

Thesurvival probability of νe due to the νe →νµ, ντ oscillations isP(νe →νe; t) = 1 −89 sin2 m2ν4E t! (2)For mν ∼10−5 eV, the oscillations length l = 4πE/m2ν for the neutrinos with the energyE ∼10 MeV is of the order of the distance between the sun and the earth.

Since eq. (2)describes the large-amplitude oscillations, one can in principle get a strong suppression ofthe solar neutrino flux.

This, so-called ”just so”, oscillation scenario leads to well definedand testable consequences [5].We should add that if the relevant cut-offscale in (1) would be one or two orders ofmagnitude smaller than MPl (as can happen in the string theory, where the relevant scalemay be the compactification scale), mν could be as large as 10−4 −10−3 eV. One can2

distinguish two cases then. For 10−10 < m2ν < 10−8 eV2 we are faced with the conventionalvacuum oscillations for which eq.

(2) gives the averaged νe survival probability ¯P(νe →νe) ≃5/9. For m2ν > 10−8 eV2 the MSW effect [6] comes into the game.

Although all threeneutrino flavors are involved, one can readily make sure that the resonant oscillation patterncan be reduced to an effective two-flavor one. If the adiabaticity condition is satisfied, theneutrino emerging from the sun is the massive eigenstate ν3 = (νe + νµ + ντ)/√3.

Thus,P(νe →νe) = 1/3 in this case.C. ZKM and Dirac neutrinosThe major point of the result (1) is that the emerging neutrino masses are on theborderline of the range needed for the solution of the SNP.

For αij ≪1, the mechanismwould not work. The situation changes drastically if there is an additional mechanism ofgenerating neutrino mass.This is particularly interesting in the case of ZKM or Diracneutrinos.

By ZKM we generically denote any situation with degenerate active neutrinoflavors νiL and νjL when the lepton charge Li−Lj is conserved. In other words, the resultingstate is a four-component neutrino νZKM = νiL + (νjL)c. In the conventional Dirac pictureone has νD = νL + nR, where nR is sterile and the conserved charge is just the particularlepton flavor defined through νL.In any case, if one desires to have the oscillations between the components of νZKM (νD),one has to break the degeneracy, i.e.

induce Majorana masses which violate the conservedcharge in question. This, as before, can be achieved through the gravity induced operatorsof (1) and also in the same manner by(nTRCnR)"αnH†HMPl+ βnS2MPl#,(3)where S is any SU(2)L×U(1) singlet scalar field, which may or may not be present.

Here, asthroughout our analysis, we assume no direct right-handed neutrino mass such as nTRCnR Swhich is implicit in our assumption of having a Dirac state 1.1The nTRCnR S terms could be forbidden by a global charge.3

Let us focus first on the ZKM case. In general, the terms in (1) will induce the split∆m ≤10−5 eV2.

The oscillation probability depends on ∆m2 ∼m∆m. For this to berelevant for the SNP, one of the components νi or νj must be νe and therefore m ≤10 eV[8], or ∆m2 ≤10−4 eV2.

Clearly, for any value of m ≥10−5 eV, this can provide a solutionto the SNP through the vacuum oscillations 3.We would like to mention that ∆m2 in the above range can be also relevant for anotherpossible explanation of the SNP, namely, resonant spin-flavor precession of neutrinos due tothe transition magnetic moment between νi and νj [9].The same qualitative analysis holds true for the Dirac neutrino, the only difference beingthe additional contribution of (3) to the Majorana masses. The possible presence of the S2term (if < S ≯= 0) could modify drastically the predictions for ∆m.

Strictly speaking, in ageneral case no statement is possible at all since < S > could be in principle as large as MPl.Of course, in the most conservative scenario of no new Higgs fields above the weak scale, theanalysis gives the same result as for the ZKM situation. Oscillations (or resonant spin-flavorprecession) between the components of a Dirac neutrino can also provide a solution to theSNP; however, the experimental consequences for experiments such as SNO or Borex willbe different.

Namely, the detection rates in the neutral current mediated reactions will bereduced since the resulting neutrino is sterile.Another important consequence of the induced mass splitting between the componentsof a Dirac neutrino is a possibility to have a sterile neutrino brought into the equilibriumthrough the neutrino oscillations at the time of nucleosynthesis [10]. This has been analysedat length in ref.

[11], and can be used to place limits on ∆m2ij and neutrino mixings.2A ZKM neutrino with a small energy split between its components is usually called pseudo-Diracneutrino [7].3Although ∆m2 could easily be in the range ∆m2 ≃10−8 −10−4 eV2, the MSW effect is irrelevant forthe SNP in this case since the vacuum mixing angle is practically equal to 45◦.4

D. 17 keV neutrinoA particularly interesting application of the above effects finds its place in the problem of17 keV neutrino [12]. Although the very existence of this neutrino is not yet established, it istempting and theoretically challenging to incorporate such a particle into our understandingof neutrino physics.Many theoretical scenarios on the subject were proposed; however, it is only recentlythat the profound issue of the SNP in this picture has been addressed.

The problem isthat the conventional scenario of three neutrino flavors νeL, νµL and ντL cannot reconcilelaboratory constraints with the solar neutrino deficit. Namely, the combined restrictionfrom the neutrinoless double beta decay and νe ↔νµ oscillations leads to a conserved (orat most very weakly broken) generalization of the ZKM symmetry: Le −Lµ + Lτ [13] 4.This in turn implies the 17 keV neutrino mainly to consist of ντ and (νµ)c, mixed with theSimpson angle θS ∼0.1 with the massless νe.

Clearly, in this picture there is no room forthe solution of the SNP due to neutrino properties.It is well known by now that the LEP limit on Z0 decay width excludes the existenceof yet another light active neutrino. However, the same in general is not true for a sterileneutrino n. Of course, once introduced, n (instead of νµ) can combine with, say, ντ to formν17 or just provide a missing light partner to νe needed for the neutrino-oscillations solutionto the SNP.

The latter possibility has been recently addressed by the authors of ref. [15].The introduction of a new sterile state n allows for a variety of generalizations of aconserved lepton charge Le −Lµ + Lτ.

This will be analyzed in detail in the forthcomingpublication [11]. Here we concentrate on the simplest extension ˆL = Le −Lµ + Lτ −Lnc 5and assume the following physical states:ν17 ≃ντ + (νµ)c,νlight ≃νe + n(4)4This lepton charge was first introduced in another context in ref.

[14].5The phenomenology of the system with the conserved lepton charge ˆL (with n being an active neutrinoof the fourth generation) was analysed in [16].5

mixed through θS. In the limit of the conserved charge ˆL, the light state is a Dirac particleand no oscillations are possible which would be relevant for the SNP.

Furthermore, the onlyallowed oscillations are νe ↔ντ and νµ ↔n with ∆m2 ≃(17 keV)2. The situation changesdrastically even with a tiny breaking of ˆL and we show here how the potential gravitationaleffects in (1) and (3) may naturally allow for the solution of the SNP without any additionalassumptions.Clearly, the main impact of the above effects is to induce the mass splittings betweenthe ντ and νµ on one hand, and νe and n on the other hand.

Recall that we expect thesecontributions to be of the order of 10−5 eV or so, if no new scale below MW is introduced.This tells us that∆m2ντ νµ ≤10−1 eV2,∆m2νen ≤10−4 eV2(5)As we can see, the scenario naturally allows for the solution of the SNP due to the vacuumoscillations 3 and furthermore predicts the νµ ↔ντ oscillations potentially observable inthe near future. Notice that although we discussed the case of only one sterile neutrino,any number would do.Also, we should emphasize that the result is completely modelindependent, as long as one deals with (almost) conserved charge ˆL.

However, a simplemodel can easily be constructed and will be presented elsewhere [11].E. DiscussionWe have seen how gravitation may play a major role in providing neutrino masses andmass splittings relevant for the SNP.

Purely on dimensional grounds, at least in the caseof only left-handed neutrinos, the Planck-scale physics induced masses and splittings are≤10−5 eV. The smaller they are, the larger the neutrino masses generated by some othermechanism should be, in order to obtain large enough ∆m2.

For this reason a ZKM neutrinois rather interesting, especially with cosmologically relevant mνe close to its experimentalupper limit ∼10 eV.The situation is less clear in the case of a Dirac neutrino, since the gravitationally6

induced mass term mn nTCn could in principle give mn as large as MPl. In fact, in this caseit is hard to decide whether one is dealing with a Dirac neutrino or actually with a see-sawphenomenon [17].

The situation depends on the unknown aspects at very high energies, i.e.whether or not the scale of the n physics is much above MW.The see-saw effects may be even more important if one is willing to promote the wholeSU(2)L × SU(2)R × U(1) electroweak symmetry or, in other words, if one is studying aparity conserving theory. This is a natural issue in many GUTs, such as SO(10) or E6.Normally, in order for the see-saw mechanism to work, one introduces a Higgs field whichgives directly a mass to n. In the spirit of our discussion it is clear that gravitation mayalso do the job and, if so, one would expect mn ∼M2R/MPl where MR is the scale of theSU(2)R breaking, i.e.

the scale of parity restoration. In other words, even with large MR,mn could be quite small allowing far more freedom for light neutrino masses.

Of course, thedetails are model-dependent (i.e. MR scale dependent) and we do not pursue them here.As we have seen in section D, all the cases relevant to the 17 keV neutrino lead to thelarge mixing angle solution of the SNP.

Before concluding this paper, we would like to offersome brief remarks regarding an interesting possibility of mirror fermions picture providingthe desirable MSW solution of this problem.Imagine a world which mimics ours completely in a sense that these mirror states havetheir own, independent weak interactions. In other words, let gravitation be the only bridgebetween the leptonic sectors of the two worlds.

One would then have the following newmass operators in addition to those in eq. (1):α0 lTMiCτ2⃗τlMjHTMτ2⃗τHMMPl,α0 ¯lMiljH†HMMPl(6)where M stands for mirror particles.

Therefore, in addition to the 17 keV neutrino describedbefore there should be an analogous (ν17)M neutrino in the mirror world, and anothermassless state much as like as the massless state in the standard generalized ZKM picture.It turns out that, due to the mixing in eq. (6), one of these two massless states picks up amass ∼α0M2WM−1Pl θ−2 with θ ≃< H >/< HM > whereas the other still remains massless.7

Therefore they can oscillate into each other with the mixing angle θ, and so in principlethis can provide a solution of the SNP through the appealing MSW effect. Namely, for< HM >≃1 −10 TeV the mass difference ∆m2 could easily be in the required MSW range.The above range of < HM > makes this scenario in principle accessible to the SSC physics.AcknowledgementsWe would like to thank I. Antoniadis, J. Harvey and G. Raffelt for discussions and M.Lusignoli for bringing ref.

[16] to our attention. E.A.

is grateful to SISSA for its kindhospitality during the initial stage of this work.Note added. After this work was completed we received a paper by Grasso, Lusignoli andRoncadelli [18] who also discuss gravitationally induced effects in the 17 keV neutrino pic-ture.References[1] R. Holman, S.D.H.

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