Physics Letters B 288 (1992) 140-144 North-Holland PHYSICS LETTERS B

Global continuous symmetry and the 17 keV neutrino

D. G r a s s o a M. Lus igno l i a a n d M. R o n c a d e l l i b a Dipartimento di Fisica, Universit& "'La Sapienza". 1-00185 Rome, Italy

and INFN, Sezione di Roma L Rome, Italy b INFN, Sezione di Pavia, Pavia, Italy

Received 27 February 1992

The effects of a very small explicit breaking of a continuous global symmetry advocated in elegant and phenomenologically successful models for the 17 keV neutrino are discussed. It is shown that these effects are substantial, even if the explicit breaking terms are suppressed by powers of a scale as large as the Planck mass.

It has recently been cla imed that a massive com- ponent at 17 keV of the electron neutr ino exists [ 1- 3 ]. This has p rompted a large number of theoret ical suggestions on possible ways to include such a strange object in a coherent descr ipt ion of electroweak interactions.

One of the most economic models, in that it con- tains a min imal number of fermions, was suggested by Petcov [4] several years ago ~ . After the first in- d icat ion in favour of the 17 keV neutr ino [ 1 ], the model was reconsidered by Dugan, Gelmini , Georgi and Hall [ 6] who suggested that a global U ( 1 )e-u+ symmetry generated by the combina t ion of individ- ual lepton numbers L e - L , + L ~ could provide a ra- t ionale for a structure o f the neutr ino mass matr ix still compat ib le with other phenomenological con- straints. Quite recently, the model has been revisi ted by Barbieri and Hall [ 7 ], who payed considerable at- tent ion to the scalar sector. They have given a de- tai led descr ipt ion o f the fast decay of the heavy neu- tr ino into a final state consist ing in a light neutr ino and one of the Golds tone bosons - d u b b e d f l a v o n s -

present in their model, along with a thorough discus- sion of the consistency with cosmological con- straints. The interesting novel ty of ref. [7] is that cosmological bounds are obeyed and neutr inos are naturally lighter than charged fermions, in spite of the fact that only the Fermi scale V - (x /2GF) - 1/2 = 246 GeV appears in the model.

~ See also ref. [5].

One may now take the a t t i tude to consider the s tandard model as the ul t imate theory, giving up all a t tempts of further un i fca t ion , which is a perfectly sensible possibility. In this case, the 17 keV neutr ino ( i f conf i rmed) could well be inserted in the theory in the way suggested by ref. [ 7 ].

Many physicists however believe that the s tandard model is the low-energy manifes ta t ion of a more fun- damenta l theory - relevant at a very large scale A - and that the only true symmetr ies are local ("gauge

dogma" ) . Then one is led to assume that cont inuous global symmetr ies will not be exact at A, and this en- tails in turn an explicit breaking of the symmetry in the low-energy effective lagrangian. It is the purpose of this let ter to investigate the consequences of such an expl ic i t b reak ing for the model [ 7 ].

Two different types of effects are expected, with the common feature of disappear ing as A goes to infinity. On one hand, the breakdown of the "exact" symme- try U( 1 )e-~+~ will generate new terms in the neu- tr ino mass matrix. On the other hand, the explicit breaking of cont inuous symmetr ies will give nonzero masses to the flavons. One would naively think that the impor t of both effects is negligible since the scale A is supposed to be much larger than the Fermi scale V. In addi t ion, this seems technically na tura l [ 8 ], for as A--. oo an exact symmetry is recovered ~2. Surpris- ingly enough, it turns out that the stringent experi-

,2 This is true also for nonlinearly realized symmetries (i.e. for a spontaneously broken symmetry).

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Volume 288, number 1,2 PHYSICS LETTERS B 20 August 1992

mental bounds on neutrino oscillations and the need to keep the pseudo-Goldstone bosons lighter than 17 keV (to allow for the heavy neutrino decay) can barely be fulfilled, and only for the very largest values of A, close to the Planck m a s s M p I '~ 1019 GeV. More- over, the massive flavons are essentially stable (i.e. they have a lifetime larger than the age of the uni- verse) so that they will overclose the universe if their mass is larger than ~ 100 eV, as it appears to be in our scenario. Although arguments of this kind [9 ] are only able to provide order-of-magnitude estimates, it should be stressed that the effects considered here should not be ignored and that they are potentially dangerous for model building.

In ref. [7] an exact U( 1 )~®U( 1 ) , ® U ( 1 ), global symmetry is spontaneously broken to U ( 1 )e-~+~ by nonzero vacuum expectation values (VEVs) o f neu- tral singlet Higgs fields. The neutrinos get nonzero masses and the (Majorana) neutrino mass matrix has the form [ 6 ]

M = 0 , (1)

P~

2 2 with [3] ~ = m = 1 7 keV and /~//z~= tan 0= 0.1. The eigenvalues of the above mass matrix are (0, + m) and the two degenerate massive Major- ana eigenstates can be combined in a ZKM-Di rac neutrino [ 10 ].

The effect of a small explicit breakdown of the global remaining symmetry U( 1 )~_,+~ was already discussed by one of us [ 11 ] from a different and now obsolete point of view. In the present approach, we expect this breaking to induce in the effective lagran- gian, among other terms, dimension-five operators of the form [ 9 ]

C o - - c "~ c ~-Ltz~L)O r~O + h . c . , (2)

where ~ and L, are the usual Higgs and lepton dou- blets, i and j are generation indices and the super- script c denotes charge conjugation (repeated indices are summed over, unless explicitly stated, in all our formula). The operators in (2) will generate new contributions to neutrino masses, replacing the zeros in ( 1 ) by terms that are generally of order VZ/A. For A equal to the Planck mass, this corresponds to ~ 10-s eV [12].

The previously degenerate eigenvalues are now separated by a quantity 8rn of this order, which looks a very small correction indeed. However, the impor- tant parameter for neutrino oscillations is the differ- ence o f squared masses Am 2 = 2 m 8m ~ 0.3 eV 2, and this is comparable to (in fact, slightly higher than) the present bound for maximal mixing ~3 in v~-dis- appearance experiments [13]. The situation be- comes of course even more difficult for A < Mm.

We consider now the effect of a tiny explicit global symmetry breaking on the flavon masses. In ref. [ 7 ] two different options for the Higgs sector have been proposed. In one o f them, which will be considered in more detail below, besides two Higgs doublets H1 and H2, three neutral (S e~, S ~, Set) and three charged Higgs singlets are present. The doublets carry no lep- tonic charge and one o f their VEVs can be put equal to zero without loss of generality. An exact U( 1 )e®U( 1 )g®U( 1 ), global symmetry is sponta- neously broken to U (1)e-~+* by the nonzero VEVs of the neutral singlets S e~ and S ~. Omitting terms containing charged singlet fields (that are irrelevant to the following discussion) the most general form of the Higgs potential is:

4 / = l~ab S ab S ab

+/&/ /~H, + U2/4~H2 + (/z3n~H2 +h .c . )

+2abcaS~bSaeS~dS cd+2 ~ (H~ H1 )2

+~,2 (HIH2) 2 +23H~H2H~H,

" r ~ 1 ab H~ H l Sat, S ab + 22ab H~ U 2 Sab S ab

+ (23ahH~ H2 SabSa~+ h.c. )

+ (24H~HIH~H2 +h.c. )

+ (25H~H2H~H1 + h . c . ) . (3)

In (3) the indices a, b .... run over the leptonic fla- vours and S ab- * Sab.

The explicit breaking of the global U ( 1 ) symme- tries at the large scale A gives rise to new terms in the Higgs potential of the low-energy theory, that also break these symmetries. They are nonrenormalizable (hard) terms as well as soft breaking terms, that are present since the global symmetry is not "auto-

#3 T h e m i x i n g angle for v . osc i l la t ions is in fact m a x i m a l in the

m o d e l u n d e r d iscuss ion , due to the s t ruc tu re o f the mass m a -

t r ix ( 1 ). For a detailed analysis, see ref. [ 11 ].

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matic" ~4. The coefficients of all these terms should go like some inverse power of A, and we will consider for simplicity a soft breaking term of the form

1 all' = -~ ( PlabH~Hl sab + VzabH~H2S ab

+ V3abH~H2Sab+h.c.) , (4)

where Ilia b are new dimension-two parameters of or- der V z. Similar results would be obtained starting from a dimension-five, nonrenormalizable breaking term like

~[" = ~ - ( sabSab ) 2S cd . (5)

As usual, we parametrize the neutral fields in terms of their VEVs Vg

1 H° = ~7~ ( Vl +p , + ir/l ) ,

1 H ° = ~7~ (P2 + iq2),

Sab = Vab + Pab + i~lab • (6)

We recall that V2 has been chosen zero and that be- fore explicit breaking V¢r= 0 (this was necessary to keep U ( 1 ) ~_ ~ + ~ unbroken ). Moreover V~ = V at tree level.

In presence of the breaking terms (4) the extre- mum conditions are

l~l ~-2 I V2 ~-2 lab Vab Vab

Re Vlab Vab=0 + 2 ~ , (7)

Re 2 3~b Vab vab+ Re P3 + ½ Re 24 V~

V~b + R e V3ab--~-- = 0 , (8)

Im 23ab Vab v~b+ Im/~3 + ½ Im 24 V 2

v ab + I m v3~b ~ - - = 0 , (9)

~4 As usual, by "automatic" symmetry we mean any global sym- metry which is present in the most general renormalizable la- grangian invariant under the gauge group.

Vab []Aab "~- (2 abcd Ved v cd) ' "~- 22 ~bab Vab V ab + ½2 lab V 21 ]

Re Vlab V ~ = 0 , (10) +½ A

Im Vlab=0. (1 1)

In (10) no summation over the indices (a, b) should be made and the prime ( ' ) denotes a sum with (c, d) ~ (a, b).

Computing the quadratic terms in the potential ~ + ~//' in the extremum, we obtain the (mass) 2 ma- trix for the pseudoscalar bosons q~. For simplicity, we first restrict our attention to the representative case u3ab = 0, where the matrix is diagonal.

The results are

m 2, = 0 , (12)

m22 =/~2 + ½23 V~ + ½22a b Vab V ab

+ 1 Re v2ab V ab , (13)

2 Re vlab V 2 mab= (14)

A Vab"

As (12) shows, q~ is the Goldstone boson that is "ea- ten" by the Z gauge bonson, q2 is a Higgs boson which in the symmetric limit has a mass ~ V, only slightly changed by the last term in (13) .

In (14) no summation over the indices (a, b) should be made. This equation gives the masses of the neutral singlets in a form valid for any (a, b). We stress however that for (a, b) = (e, T) the resulting mass is still of order V. This is so, because in the sym- metric limit Ve, = 0 and only as an effect o f the ex- plicit symmetry breaking Se, takes a small nonzero vacuum expectation value. As (10) shows, for a non- zero V~e,~ V 2 the VEV becomes Ve~~ VZ/A.

Our main result is obtained from (14) for (a, b) = (e, ~t) and (p, z). The particles rle~ and rl~ which were the massless flavons of the model acquire now masses

V 3 2 mn . . . . > ~1 (keV) 2, (15)

mpl

where we assumed all dimensionful terms to be about equal to the Fermi scale V, with the exception of A<~Mpt.

The limit in ( 15 ), although true only as an order

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of magnitude result, leads to a cosmological disaster if taken literally. In fact, the usual limit for the sum of the masses of light and stable neutrinos (m < 100 eV) applies equally well to the two flavons of the model we are considering. This follows from the fact that the flavon interactions are very weak, so that the primordial flavons decouple well before the QCD phase transition and their relic number density is considerably diluted [7]. Later on, neutrino decays generate one flavon for each heavy neutrino degree of freedom, whence the same limit for their masses, if the flavons are stable.

In this model the flavon couplings to fermions are off-diagonal, always involving at least one ~ 17 keV mass eigenstate and therefore lighter flavons cannot decay. Only in presence of new terms in the lagran- gian due to the explicit breaking of lepton numbers the flavons could be allowed to decay into two light neutrinos: however, we expect these terms to be sup- pressed by the very small factor V/A ~ 2 × 10 - t y, giv- ing rise to flavon lifetimes at least as long as

16tch ( A ) z (M~l) 5/2 "(fl . . . . - - - - ~ 4 . 5 X 1 0 1 6 S . (16)

mflavon

This should be compared with the limit following from the request not to overclose the Universe [ 14 ]. Using (15), it can be written as rn . . . . < 2 × 1015 (A/ M p l ) S. An even stronger, but less certain limit, Tfl . . . . < 1010 (A /MpI ) S, can be derived requiring matter dominance of the Universe to allow for galaxy formation [ 15 ]. Shorter lifetimes could result for a scale A lower than the Planck scale, however the fla- von mass m would correspondingly increase, and for m ~> 17 keV heavy neutrinos would become in turn stable and the model would be ruled out again.

The inclusion of terms with l]3ab~ 0 in (4) does not change the results. Indeed, if no accidental degener- acies are present, the further corrections would be of second order in the small parameter V/A.

Furthermore, the second Higgs structure consid- ered in ref. [7 ] and consisting of the usual doublet, three singlets and three triplets and also having two flavons in the limit of exact U( 1 )~_~+~ can be ana- lyzed along similar lines. Defining a two-by-two ma- trix for every Higgs triplet

- A J~b (17)

and its hermitian conjugate

(j_ z jab~(~ab) t= d - - , (18)

the globally symmetric Higgs potential, analogous to (3) , is

~x = l~abSabSab + pH HT Hl + It~ab Tr( A~b A~b )

_]_ 2 abcdS ab SabS cdSCd..i_ ~ n ( H~ H 1 )2

+ 2 ~abcdTr ( A~b A~b ) Tr ( ACd Acd )

+TabH~Hl SabS~b + fiobH~HI Tr( A~bAab )

+ p~bcdSabS~Tr( ACdAcd)

+aabH]A~bAabHI

+ (v~bS~b~I[A~bHl +h.c.) , (19)

where we used the definition I~ = iz2H*. The triplet fields A°b have vanishing VEVs at tree level in this model, to avoid the presence of a new scale much smaller than the Fermi scale. The nonzero VEVs and the other dimensionful parameters are generally of order V.

To illustrate the effect of an explicit breakdown of the global symmetry, we choose a term

~/gr= ~ (H~H,)S,b+h.c. . (20)

In the presence of this term, the (mass): matrix in the neutral pseudoscalar sector is still diagonal, and the eigenvalues corresponding to the pseudo-flavons q~ and q~ are

2 Re ~ab V~ (21) mab =

A Vab'

so that the conclusion remains the same, see (15). Again we note that the particular choice of the break- ing term (20) is representative of the general situa- tion, barring of course accidental cancellations.

As we have seen, in spite of the fact that the explicit symmetry breaking terms have coefficients of order V~/A ~ 10 -5 eV, the resulting mass for the pseudo- flavons is larger than this number by about eight or- ders of magnitude. An analogous result was recently obtained for the triplet majoron model [ 16 ]. In fact, the high sensitivity of the pseudo-Goldstone boson masses to small explicit symmetry breaking was first

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no ted th i r ty years ago by Jona -Las in io and N a m b u

[ 17 ] in the context o f dynamica l symmet ry breaking.

T h e m o d e l o f refs. [4 ,6 ,7] is an elegant and phe-

nomeno log ica l ly successful scheme ~5 to inc lude the

17 keV neu t r ino in the theory. We h a v e d iscussed in

this let ter what happens i f the exact U ( 1 ) e - ~ + , global

s y m m e t r y charac te r iz ing this m o d e l is expl ic i t ly bro- ken by a very tiny a m o u n t , as a low-energy effect o f

new physics at a scale A m u c h larger t han the F e r m i

scale. We have shown that the b reak ing t e rms gener-

ally induce i m p o r t a n t neu t r ino osc i l la t ion effects and

unaccep tab ly large masses for the f lavons. As a f inal caveat , i t shou ld be m e n t i o n e d tha t one

canno t o f course exc lude that the s y m m e t r y b reak ing terms, (2 ) and (4 ) , are actual ly mu l t i p l i ed by coef- f ic ients O ( 1 0 - 2 ) , so that the resul t ing masses wou ld

b e c o m e (bare ly ) accep tab le for cosmology and neu-

t r ino osci l la t ions p h e n o m e n o l o g y for A--Mp~. The

po in t we want to m a k e is that effects o f physics at the

P lanck scale m a y be d ramat i ca l ly i m p o r t a n t in the

m o d e l we have cons idered : to calcula te t h e m explic-

i t ly one wou ld need to k n o w what the actual theory

at the scale A is.

We thank the referee for cons t ruc t ive cr i t ic ism.

~5 Although somebody could criticize it on the grounds of not predicting a reduction in the solar neutrino flux. We do not.

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