Flavon models for the 17 keV neutrino

Nuclear Physics B364 (1991) 27-42 North-Holland

FLAVON MODELS

FOR THE 17 keV NEUTRINO* R. BARBIERI

Physics

Department,

University

of Piss and INFN,

Piss, Italy

L.J. HALL Department

of Physics,

University

of California

and Theoretical

Physics Group, I Cyclotron

Physics Division, Road, Berkeley,

Lawrence Berkeley CA 94720, USA

Laboratory,

Received 10 April 1991

Models for a 17 keV neutrino based on the symmetry breaking U(l), X U(l),, X U(l), + are discussed. The models involve only the three usual neutrinos (which are naturally U(l),-,+, light), have no new symmetry breaking beneath the weak scale and have lifetimes for the 17 keV neutrino as short as 0.1 s. Unusual signatures are discussed for r decays, Higgs decays and supernova neutrinos.

1. Introduction In 1985 Simpson presented evidence in the p decay spectrum of 3H for a structure consistent with the hypothesis that the electron neutrino v, contains an admixture of a 17 keV mass eigenstate [l]. Other experiments, also using solid state detectors, have confirmed the effect, most recently in 14C [2] and 35S [3], giving a mixing probability near 1%. The experimental situation is controversial, however, since several experiments using magnetic spectrometers do not have any signal. It therefore seems likely that there are systematic effects which have yet to be understood. In this letter we assume the 17 keV neutrino does exist, and study the implications. We give a set of criteria for constructing models which incorporate Simpson’s neutrino, and we show that these criteria lead us to a class of models which involve a type of Goldstone boson that we call a flavon. *This work was supported in part by the U.S. Department of Energy under Contract DE-ACO376SF00098 and in part by the National Science Foundation under grant PI-W8515857. Elsevier Science Publishers B.V. All rights reserved

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2. General criteria In the standard model, without right-handed neutrinos and with any number of Higgs doublets, the three neutrinos are guaranteed to be exactly massless by gauge invariance. This is a major success of the standard model. There are innumerable ways of going beyond the standard model to get massive neutrinos. Our main guide in seeking a model which incorporates Simpson’s neutrino is that it should not give up the successes of the standard model, and it should not add to its shortcomings. Hence we require that the model provides an understanding of why the neutrinos are much lighter than the charged leptons. In particular the lightness of the neutrinos should follow from the structure of the renormalizable interactions of the theory and should not be the result of choosing extremely small values for some parameters. A major shortcoming of the standard model is the absence of any understanding of the pattern of the charged fermion masses. We will similarly not be able to provide a model where the numerical values of the neutrino masses and mixing angles are predicted. Another important shortcoming of the standard model is the absence of any understanding of the scale of electroweak symmetry breaking; in particular why the W boson is so much lighter than the Planck scale. In order not to exacerbate this problem, we require that our model does not introduce further scales of physics beneath the weak scale. Such scales would be very hard to generate naturally; certainly the ideas of supersymmetry would be of no use. The ideas of technicolor would then suggest the existence of a whole new gauge sector of the theory to explain the new scale. On the grounds of simplicity of the low-energy theory, we demand that no additional scales of symmetry breaking beneath the weak scale are introduced.

3. Implications

of criteria

One implication of this criterion is that the only bosonic elementary particles beneath the weak scale are either massless gauge particles or massless Goldstone bosons. Another consequence of our criteria is that we do not have to consider neutral fermions much lighter that the Z other than the three left-handed doublet neutrinos v.. It is an experimental fact from the Z width that any such light neutral fermions N must have zero TsL charge. For the structure of the theory to explain why V~ are light, a symmetry must forbid a Dirac mass coupling of N to v,. (We ignore the possibility that N is in a 5-plet or higher representation of SU(2),, and we ignore the possibilities of more elaborate schemes for understanding small Dirac masses.) To do its job this symmetry must be unbroken down to the scale of the light neutrino masses; our criteria then imply that it is never broken. Hence, if

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any light N exist they never mix with the neutrinos and can be ignored. We need only consider models with three light two-component neutrino states.

4. Double beta decay

Massive neutrinos with large ve components are constrained by neutrinoless double beta decay data. In the case of the 17 keV neutrino we are aware of three ways of satisfying this constraint: (a) The process may be forbidden by a symmetry [4]. (b) Two heavy neutrino states may have masses and mixings which are fine tuned in such a way as to provide a large cancellation in the /3p amplitude [5]. (c) A zero entry in the ee element of the neutrino mass matrix, together with light right-handed neutrinos, give a more natural interpretation of the cancellation in (b) [6]. In our view cases (b) and (c) add to the problems of the standard model. Certainly the fine tuning in (b) is very artificial. Although (c) is an improvement, having a light right-handed neutrino means that the lightness of the neutrinos is no longer guaranteed by the structure of the theory. Consequently we prefer that the stringent experimental limits on neutrinoless double p decay be understood in terms of an exact symmetry. We are led to models for neutrino masses which have a linear combination of lepton numbers unbroken, an idea with a long history [7]. For our case of three light neutrinos such a symmetry is uniquely chosen to be U(l),-,+,. In this case the spectrum of light neutrinos is

where m,7 z 17 keV and

(ij=(-‘,:)(::)

(2)

and s2 = sin* 0 = 0.01. The simplicity of this structure cannot be overemphasized. The neutrino mass matrix involves just two parameters, m,7 and 8, and these are fixed by the p decay data. The scheme is so constrained that it cannot be responsible for the observed deficit of solar neutrinos. The predictions for neutrino oscillation probabilities are P(v,,, t) v,) = 0 and, for any distance from the source much greater than the oscillation length, P(v, t) v,) = i sin* 28 = 0.02. The alternative choice for the unbroken symmetry is U(l)e+P-r, which is obtained by /.L f) r. This is excluded by ZJ@ --) ve oscillation data.

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5. Singlet admixtures

Having argued for the case of three light neutrinos with masses given by (1) and (2) we must now ask whether these states are pure SU(2) doublet, or whether they can have small admixtures of other SU(2) properties. Such small admixtures arise in the seesaw mechanism, which is a mechanism for understanding the lightness of neutrinos which are not massless. The simplest model which incorporates this mechanism is that of the singlet Majoron model (SMM) [8] 2i?SMM=im~~++~~~, L

(3)

where 1 and H are lepton and Higgs doublets, N are neutrino singlets, m and M are 3 x 3 mass matrices, and L’ is the doublet v.e.v. The three light states have Majorana masses which are, in leading m/M order, the eigenvalues of mTM-‘m; they are mainly doublet but have O(m/M) admixtures of singlet. Can we construct a SMM which is consistent with an unbroken U(l),-,+r symmetry? Since the singlet fields N,mP.7 have charges (- l,+ 1, - 1) under this symmetry, the matrix M has a zero eigenvalue. This means that the seesaw mechanism does not work for one of the light neutrinos. Imposing U(l),-,+, on the SMM destroys our understanding of why all three v0 are light. If the heaviest of the three ran is identified as v,, [9], which must be mainly v,, one has lost the understanding of why vr is much lighter than the r. The problem of imposing U(l),-,+7 on the SMM is that there are an odd number of N states which cannot all pair up to form very heavy Dirac states. This can be fixed up by spoiling the family structure of the theory and requiring an even number of N states. A model with four N states has been constructed by Grinstein et al. [lo]. We will briefly mention a model with two N states. In this letter we concentrate on models where all the fermions fit into three identical families. In this case with our three basic assumptions: (i) no new symmetry breaking beneath the weak scale, (ii) the need to understand why r~~,&,~are light, (iii) all fermions arranged in identical families, we are able to limit our study to models with just three light neutrinos, all of which are pure doublet, and whose mass matrix is given in (1) and (2) and is described by just two parameters. We believe that this framework is well motivated and that it leads to the simplest models. 6. Cosmology

In the standard big bang cosmology the three light neutrinos decouple from the hot electron-photon plasma when the temperature is a few MeV. Subsequently their comoving number density is unchanged but their energies are redshifted, so

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that today they have a number density and spectrum characteristic of a 1.9 K temperature. If this scheme is changed only by giving one of the three neutrinos a mass of 17 keV, the picture is disastrously altered. As the universe cools beneath a temperature of 17 keV the energy density of the universe becomes dominated by the rest mass energy of the 17 keV neutrinos. This would lead to a universe where the ratio of the total energy density to the photon energy density is orders of magnitude larger than is observed. There are at least two ways of recovering consistency: (al The 17 keV neutrino should have a lifetime T,, < 10” s, using a conservative bound. (b) The 17 keV neutrino should be cosmologically depleted by annihilation. A third alternative of relative depletion by dilution from entropy production does not work. Significant dilution after nucleosynthesis is in conflict with the nucleosynthesis constraint on the baryon density. Before nucleosynthesis the 17 keV neutrino is in thermal equilibrium, so that entropy dumping does not change its relative abundance. Even if the 17 keV neuhino is mainly, but not completely, sterile, thermal equilibrium would be maintained by neutrino oscillations [ll]. While depletion via cosmological annihilation is certainly possible, we now argue that it requires a new scale of physics well below the weak scale. For example, consider T,,v,, + VLvL where vL is some light neutrino. The rate mediated by Z exchange is much too small. The exchanged boson must be lighter than a GeV, and our criteria require that they be either light gauge or Goldstone particles. The Goldstone couplings are themselves of order l/V where V is the scale of symmetry breaking which produces the Goldstone. Hence Goldstone exchange gives sufficient depletion only for V < GeV. A new massless gauge particle (e.g. for U(l),-,) would give sufficient depletion via GTfiT+ Gefie for a gauge coupling (r > lo-“. However, such massless gauge particles are excfuded by fifth force experiments, and giving them a mass (,< GeV) would introduce a new physics scale. Identical arguments apply to other annihilation channels of vr7, for example to new light fermions, Goldstone or gauge bosons. Our criteria therefore imply that IJ,, must decay with r,, < 10” s. Physics at a scale V> c which produces this decay is most likely to give a decay rate of either r,, = Cm&/V

z,

(44

or r,, = C’m&/V4,

(4b)

where C and C’ are dimensionless constants containing coupling constants and phase space factors. The decay rate r,, = C”m,, would require new bosons lighter than 17 keV which are neither gauge or Goldstone bosons. For illustration of these

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rates we consider two examples, one of which we have already discarded for other reasons. Theories with right handed neutrinos where neutrino masses have both singlet and doublet components (the case of GIM violation) have a Z-mediated decay rate of the type (4b). In a second example v17 decays via the exchange of a heavy scalar triplet: cT --f 317,. This also produces decay rate (4b), where V is the mass of the heavy triplet, which must be taken to be near the weak scale. In this example the neutrino masses can be those of eq. (1). After including phase space and coupling constant factors in C’, both of these models can just satisfy T,~ < lOi s. However, the resulting universe has been radiation dominated ever since 1/,7 decay, and since perturbations in the energy density of the universe do not grow during the radiation dominated era, such a scheme conflicts with the conventional scenario for large scale structure formation. In fact for perturbations of size 5 Mpc today to have grown by a factor of 10’ since horizon crossing requires T,~ < 3 X lo5 s [12]. It is by no means clear that this conventional mechanism for the origin of large-scale structure is correct, nevertheless we will construct models which satisfy this more stringent lifetime constraint. We must therefore arrange for “fast” decays, by which we mean the decay rate formula (4a). With C = l/167? this gives T,~ = 2 x 10m3 s X (V/300 GeV12 so that the constraint on 7i7 becomes V< 3 x lo6 GeV. Within our framework there are only two decay modes which can lead to these fast decays: u,~ --f u,y and v,~ + Y~G, where G is a Goldstone boson. The decay to photons is constrained by supernova explosions. Providing 7i7 > 1 s the Y,~ escape from the exploding star, and their subsequent decay leads to an excessive cosmic gamma ray background, certainly for all T17 up to the cosmological limit of 1014 s. The very rapid decay, T,~ < 1 s, is also excluded because the energy dumped in the envelope of the star would increase the visible luminosity of the star by several orders of magnitude. We have reached an important conclusion: our prejudices against new lowenergy symmetry breaking scales and for fast vi7 decays implies that Goldstone bosons exist, and that the dominant decay mode of Y,~ is v,7 + v,G. 7. Low-energy effective lagrangians for Goldstone bosons We need to generate neutrino masses and appropriate couplings of the neutrinos to Goldstone bosons. It is most economical if the Goldstones bosons arise from the mass generation mechanism. We consider the case that above some scale V( 2 u) we have a global symmetry group G which acts on the lepton fields. The theory may be very complicated with many extra particles. At scale V the group G is broken to H, which contains U(l)e--p+r, by vevs of SU(2) singlet scalars, which can have a variety of G transformation properties: S”, Sab, S;, . . . (a = 1,2,3 for e, p, T; for the case that G is abelian a superscript is for charge - 1, a subscript for charge + 1). The interactions of the neutrinos and Goldstones can be described by

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an effective lagrangian valid for momenta less than Vz

L&f = v,v,{A,S”Sb + AJab + A,S,“Sb”+ . . .} + v+“i(Yb(8,Sa*Sb+ G,S,*,Sb’+ 6,S,b+ . . .} + h.c. + (4 four fermion operators) + . . . ,

(5)

where ua are the three light Weyl neutrino fields and the S’s have a non-linear dependence on the Goldstone fields, as dictated by the symmetry. The Ai and 6,. are couplings which depend on flavor. Their flavor dependence would be removed only if G were non-abelian. However, in this case G + H produces familons f which have direct off-diagonal couplings to charged leptons. The experimental limit on EL--f ef is in conflict with the requirement from r,, that V < 3 x lo6 GeV. G = SU(3) --) H = U(l),-,+, is excluded. There are other nonabelian possibilities however. The case of G = SU(2),, X U(l), has been considered in ref. [lo] and the case SU(3) + SU(2),, X U(l), may be worth considering. In this paper we limit ourselves to the case of abelian G; the Goldstones have no tree-level couplings to charged leptons and V may be as low as the weak scale. The most familiar example of this class of model is the singlet Majoron model [81: Sub = Pb ei.x/v where G = U(1) e+/.l+r and .& is the Majoron. To leading order in the ratio of mass matrix elements, m/M, with m and M defined in (31, the light neutrinos can be identified with the pure doublet components and L?$~” reduces to LZ’$~” = Aabv,vb ei.l/”

+ v+“i (I/ * + h.c *,

(6)

where A = mTM-‘m. This gives O((m/M>2> = O(m,/V> Majoron couplings diagonal in the mass basis. The off-diagonal couplings arise at O((m/M)4). This is because the light states, acquiring an O(m/M) singlet component, no longer have a definite lepton charge, which allows a Majoron independent mass term AovOvb to appear in (5). In fact, taking v, to be the mass eigenstates, A, and A, have equal and opposite O((m/Mj4) off-diagonal elements. These contribute to the O((m/M>‘) flavor changing Majoron couplings, which lead to the decay rate (4b). (In fact,, one power of m,7 should be replaced by the lighter neutrino mass.) From this example we see that if all contributions to the neutrino masses have the same transformation property under G then the leading tree-level Goldstone couplings are diagonal in the mass eigenstate basis so that neutrino decays are highly suppressed. To obtain “fast” neutrino decays requires at least two comparable sources to the neutrino mass which have different G transformation properties*. *The importance for fast neutrino decays of a spontaneously broken global symmetry distinguishing among different lepton families was emphasized in ref. [13].

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This very general result can be illustrated by two G = U(1) x U(1) models, the first with slow neutrino decays and the second with fast decays. Consider G = UWe+rX U(l), + H = U(l),-,+, via the vev of a singlet scalar carrying charge + 1 of each U(1): S = ,Y- = P” = eiFIV where F is the associated Goldstone boson field. For models where G acts differently on different generations we call the resulting Goldstone boson the flavon, F. There are two allowed mass terms:

It is because both terms have the same G property that S factors. Clearly F couples only to the heavy Dirac state ~~6~ and not to the massless state I;,. On the other hand a theory with G = U(l),-, X U(l), breaking to U(l),-,+, via the vev of S which carries charges (+ 1, - 1) under G has fast heavy neutrino decays. To see this notice that S = Y = S,i = eiFIV appears only in one of the two allowed mass terms:

As in the previous model this produces our desired minimal mass structure: with 6e massless, but now the flavon couplings are non-diagonal

m,,v,C7

where the Simpson mixing is s/c = A,/h,. r=-

The decay rate vi, + fieF is “fast”:

1 4 32~

-C2S2

V2

’

There are several simple ways of writing the full theory above scale V to generate (8). For example with a scalar doublet, H, and triplet T, the interactions I,l,T, H2T generate the vPve mass via heavy T exchange. The term uPv7 can be generated by the exchange of a heavy triplet which carries G charges: Tp7. The relevant interactions are l,l,Tp” and H2Tp7Sz,.

8. G= U(l),X

U(l),

X U(l),

We now consider models with the maximum abelian lepton family symmetry: G = U(l), X U(l), X U(l),, breaking to U(l),-,+,. In fact these models incorporate all of our requirements: an understanding of the lightness of neutrinos, no new low symmetry breaking scales, fast neutrino decays and fermions in identical families. The resulting absence of any heavy singlet neutrinos distinguishes our models from those of ref. [lo].

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Our models have two singlet scalars which acquire vevs giving rise to two Goldstone bosons: SPe = eiFc/b and SF’ = eiFT/c. The important low-energy interactions of the Goldstones with leptons 1, = (v,,e,) and ecOcan be described in a model independent way by

+ me7 e, ecTS~,Spc + mse e, eccS~,Spr

(10) All fields are two component left handed Weyl fields. The first line of (10) generates the neutrino masses of eq. (1) with m,, = dm and tan 8 = p,/pL,. Since the two terms have different G transformations, vi, has fast decays to Goldstones. The second line of (10) corrects the diagonal bare mass terms m, e, ecn of the charged leptons. The third line represents off-diagonal wave function mixing. The model dependence is described by the parameters pe, p,, m e77 m,,, 5, and 12. We now consider two explicit models which yield this low-energy theory. In a first model, called “T” for “triplet”, the relevant new interactions involve scalar triplets Tab and singlets Sab pT = f,,l,lbTab

+ f;bH2S”bT,*, .

(11)

Although only two S and T fields are needed (~7 and pe) it is more family-symmetric to add a third (re). The fact that neutrino physics is so non-family-symmetric is largely because S’” has zero vev unlike 9“ and SILe. The neutrino masses are generated by the exchange of the neutral components of the heavy triplets, as shown in fig. la:

Pa=fp”f:”

“2K (m~o)w .

The lightness of vi, is due to the large mass of the neutral component of the triplets, mT,. A second model*, called “I” for “loop”, involves the neutral singlets Sob, charge + 1 singlets Cab and one extra doublet H’, whose vev is taken to vanish without loss of generality. The relevant new interactions are

*This model was invented, for other purposes, by Z. Berezhiani.

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*. ’” H’

*. ” ” ” ‘.

” ‘\

‘.

‘1”

Fig. 1. Diagrams which generate the 17 keV neutrino mass in (a) T model, (b) L model.

In this L model the neutrinos are massless at tree-level, and acquire mass at one loop, as shown in fig. lb:

(14) where we have assumed the charged scalars of mass m, are the heaviest particles in the loop. Here m, and m, refer to the charged lepton masses. There are several reasons why v,, is light: the mass occurs at loop-level, it is proportional to four powers of coupling constants, and the charged scalars C may be heavy. Although there are too many parameters to make a prediction for tan 0 = kLe/pL7 it is interesting to note that tan 0 could naturally be proportional to m,/m, = 0.06. The effective lagrangian (10) describes the interactions of the flavons F, and F, with leptons. For example u,, decays to the massless ce and the flavon F, = c,F, + s,F, via the interaction

.m, _ I-UUF V pe

1,

where VW2= V;* + V,* and ci = cos 13,= V/V,. The corresponding 7

,, = 2 x 10-i s( V/300 GeV)*,

lifetime (15)

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which

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‘a

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4 mixing eq. (16).

between

I, and

I, with

coefficient

(,, given

in

is given in terms of just one model dependent parameter V, which is basically the lower of the two scales V, and V,. The largest couplings of the flavons to the charged leptons are given by the helicity conserving terms proportional to C, and l2 in (10). In both models 5, arises from the l-loop diagram of fig. 2, giving

(16) for the L model, whole 2 is the quartic coupling (for the T model g +f, C + T). The most important consequence of such a coupling is that it induces T + eF, with a branching ratio

B(r+eF,)

12rr2 = 5

{: G2V2m2 F z

2: 106[f(300

GeV/V)‘.

No definitive prediction is possible because the coupling 2 is not related to g’ which appears in the expression for m,7. Nevertheless our expectation is that there should be no very small couplings. Taking all unknown couplings equal, A, and all unknown masses Cm,, mT, VJ equal, M, we find Ii = A3/167r2, and B( T + eF) = 10’A6(300 GeV/M)2.

(18)

In the T model this is expected to be very small, since a natural understanding of the lightness of the neutrinos requires M to be very large. However, in the L model m,r = (h3/16rr2)(300 GeV/M) GeV so that A4 cannot be very large. In fact both B(T + eF) and m,7 depend on the same combination A3/M so we find an

3s

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expectation: B(T --) eF) = lo- 4. This is our central expectation; two orders of magnitude larger or smaller would not be unreasonable. Exotic decay modes of the T actually occur at tree level, by exchange of the heavy T or C fields. However, these amplitudes are l/M* rather than l/M and hence give smaller decay rates B(r + ep+p-) = h6(300 GeV/M14. For low h4 this becomes competitive with (181, and certainly has a much better signature. Also in the L model there will be non-standard tree level contributions to processes such as p + eui7 and AL = 2 neutrino-lepton scattering, since this model resembles the Zee model [14]. It is well known that the cooling of red giants provides a very stringent limit to the coupling of Goldstones to electrons, veeeCF: r], < 10-r*. In the L model there is no such tree-level coupling. In the T model there is a very small r], at tree-level because of an induced triplet vev (T) N nz,,. As a consequence the flavon acquires a component (T)*/cV of the Higgs doublet which couples with Yukawa h, to electrons, so that 7e N A,rnf,/~V. This is negligible even for V= 1’. None of the terms of the effective lagrangian (10) will have couplings eecF, or eecF, as can be seen by considering the non-linear U(l), and U(lj7 symmetry. The third line of eq. (10) induces wave function mixing. After rediagonalizing the kinetic energy terms it is found that the flavon interactions from these terms are still purely off-diagonal. Thus the coupling n, is generated only by the terms of the second line of eq. (101, which correct the charged lepton mass matrix. On diagonalization of the mass matrix a diagonal coupling of the flavon is induced, with ne having terms of magnitude m,,m,,/m,V and t,m,,/V. In both our models mre occurs at two-loops and these contributions to r], are very much smaller than lo-‘*. The L model has two weak-scale Higgs doublets which are not distinguished by any symmetry. This is usually thought to be disastrous: the Higgs with no vev could have large flavor changing Yukawa interactions giving much too large a contribution to the K,-KS mass difference. However, the Yukawa couplings that give quark masses break the SU(3)” chiral symmetry of the quarks with parameters that are hierarchical and generally small. It seems only natural that the second Higgs should break the SU(3)3 symmetry with similar symmetry breaking parameters, in which case its flavor changing interactions are not problematic. Alternatively one can have H and not I-I’ coupled to quarks by imposing a discrete symmetry which changes the sign of H’, ec and S fields. The spontaneous breaking of this symmetry by the S vevs will only induce finite and negligibly small Zf’qqC couplings. 9. Nucleosynthesis The primordial abundances of the light elements produced in the big bang are sensitive to the energy density of the universe during the MeV era. This constrains

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the number of particle species that can be in thermal equilibrium at this era, and is usually specified as a constraint on N,,, the effective number of light neutrino species. Many models for the 17 keV neutrino (those of refs. [6,8] for example) have light right-handed neutrinos and Goldstone bosons giving N, 2 4, which leads to an uncomfortably large primordial 4He abundance [15]. At first sight our models have the same difficulty: although there are only three light neutrinos the model has two flavon Goldstone bosons leading to the expectation N, = 3 + 2(4/7) = 4.14. We now show that our models actually have N, = 3. When the U(D3 symmetry spontaneously breaks at a temperature of order the scale V, the flavons will rapidly be brought into thermal equilibrium. However, for all V> 30 GeV, which includes all the values of V of interest to us, the reaction ~ir~i, t, FF is insufficient to maintain flavon equilibrium down to the QCD phase transition. Although we cannot calculate the entropy release from the QCD phase transition, it is clear just from the reduction in the number of particle species that the flavons will be diluted so that they contribute less than 0.1 to N,. It is interesting to note that a flavon background does re-appear later in the universe when the vi, decay. However, given 7i7 of eq. (151, and including the time dilation factor, these decays occur well after vi, have decoupled from the thermal plasma. Even if u,, decays turned on during nucleosynthesis, N, would be unchanged. In fact the decays and inverse decays (Y,, c, v,F,) create a flavon background and cool the background neutrinos slightly. As the temperature drops beneath 17 keV, the decays predominate so that today the ve background is larger than in the standard model. Our models both have N, = 3, giving predictions for primordial element abundances that are indistinguishable from the standard model. 10. Supernova neutrinos The observation of neutrinos from supernova 1987a implies limits on new physics which cools the supernova core more rapidly than neutrino emission. Volume emission of flavons produced by neutrino annihilations, u,,Y,, + FF, will cool the supernova too rapidly unless I/> 30 GeV [16]. Such values of V also give a sufficiently long ri, that flavon emission by v,, + LJ~F is not an important cooling mechanism. Similarly, the matter induced decays inside the core
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components of v,,, decay to v, and a flavon. The average energy of these decay ve will be comparable to the average energy of the thermally emitted ve. Hence we predict that the flux of v, from the supernova core will be approximately doubled compared to the standard model. The same argument applies to the anti-neutrinos. A more accurate calculation of the flux and spectrum will require numerical work; this model can clearly be tested if a supernova occurs in our galaxy. We expect not only a change in the magnitude of the ve and ZC signal, but also in the ratio of prompt neutronization ve to core v,, and most importantly in the ratio of neutral current to charge current events. It has been suggested that the ve interactions with the outer stellar remnants may be responsible for maintaining the outgoing shock-wave of the supernova explosion. It is of interest to note that in our models the vi, decay may be sufficiently fast that the magnitude of this effect is roughly doubled.

11. The three physical scalars In addition to the two massless flavons, our U(113 models have three massive neutral scalars which mix among each other. We call the mass eigenstates S,,2,3, which are linear combinations of Re Ho, Re Spe and Re S*‘. In the T model it is possible that V=* u so that we expect the singlets to be much heavier than the doublets. There will be little mixing between singlets and doublets so that the Higgs phenomenology to be discovered at collider experiments is that of the standard model. However, in either model with V= u it is likely that there is a great deal of mixing and we assume that Si each contain large doublet and singlet components. This is crucially important for Higgs physics because the singlet component has a fast decay to two flavon Goldstone bosons which are invisible. Any Si with mass mi < 2Mw will have a branching ratio to visible decay modes which may be very small

(19) where A, = 0.02 is the b quark Yukawa coupling and V is the smaller of V, and V,. Thus in Z decay the relevant searches are for events with a charged lepton pair recoiling against an invariant missing mass. Present data already rule out mi < 45 GeV for the case that Si is predominantly doublet. However, for our model it is important to collect more statistics since such a light Si might be mainly singlet. Similar searches at LEP2 will extend the range of accessible mi, at least for the case of Si with sizable doublet components. For heavy Si with mi > 2Mw, the visible branching ratio will be something like Bvis= (1 + u*/V*)-‘. This could be small if u > V, which would greatly reduce the statistics for the search for a very heavy Higgs at hadron colliders.

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It is important to stress that a large invisible width of the Higgs boson to a pair of Goldstone bosons is not a contrived signal which only occurs under very restricted circumstances. It is to be expected whenever any global symmetry is spontaneously broken at the weak scale by an SU(2) singlet.

12. Conclusions At first sight Simpson’s 17 keV neutrino is difficult to reconcile with both particle physics and cosmology. In fact there are very many ways of extending the standard model to successfully incorporate it. However, these models vary greatly in such things as: the size of new mass scales, the number of light neutrino states, the neutrino lifetime and simplicity. We have argued in favor of models which have no new symmetry breaking scales beneath the weak scale, give an understanding of the lightness of the neutrinos and have all fermions fitting into identical families. These criteria uniquely imply a low energy theory with just three light neutrinos with an unbroken U(l),-,+, which allows a single mass term, m,,v&v, + ev,). Thus the mass matrix involves no more than the two parameters which Simpson claimed to measure. Imposing a cosmological prejudice that u,, have fast decays further implies that the decays are to (v, - ev,) and a Goldstone boson. We have stated the general condition under which Goldstone models lead to fast decays. All such models for v,, will having striking consequences for supernova neutrino signatures; there will be no v* or v, flux reaching earth, but the core v, and pc signals will be roughly doubled. Furthermore, these models predict N, = 3 for big bang nucleosynthesis. The particular models which we prefer are based on breaking the family symmetric group U(l), x U(l), X U(l),. We give two examples: one in which neutrino masses are generated by the tree exchange of very heavy triplets (T model) and one in which they occur at l-loop (L model). These models have interesting particle physics consequences: (i) The decay T + eF (F is a Goldstone, which we call a flavon) could have a branching ratio as large as 10m2 in the L model; (ii) Mixing of doublet and singlet scalars leads to the expectation that the physical scalars will have dominant decay modes to invisible flavons, both for scalar masses below 2M, and also for very large scalar masses. This completely changes the Higgs search signatures. We thank ITP Santa Barbara for hospitality during part of this work. We thank Eric Carlson, Sasha Dolgov, Kim Griest, Luciano Maiani, Lisa Randall, Graham Ross and Uri Sarid for useful conversations. LH was supported in part by a Presidential Young Investigator award.

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