Nuclear Physics B193 (1981) 297-316 © North-Holland Publishing Company

U N C O N V E N T I O N A L MODEL OF NEUTRINO MASSES*

Howard M. GEORGI

Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA

Sheldon Lee GLASHOW 1 and Shmuel NUSSINOV 2

Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received 25 June 1981

Gelmini and Roncadelli have proposed a model of neutrino masses in which B - L symmetry is spontaneously broken by a small vacuum expectation value of a Higgs triplet. We give an exegesis of this model. We show that the massive neutrinos in this model cannot be cosmologically relevant today and that conflicting analyses of double beta decay experiments can be reconciled.

1. Introduction

The advent of g rand unified theories, which predict observable violat ions of

b a r y o n - n u m b e r conservat ion, has directed theoret ical a t t en t ion towards other

tacitly assumed but u n p r o v e n symmetr ies : the several lepton numbers , ba ryon

n u m b e r minus lep ton number , vanishing neu t r ino masses, color SU(3), electric

charge, and even Lorentz invar iance. In this paper , we address the ques t ion of

neu t r ino masses and their phenomeno log ica l import . We see m a n y acceptable ways

to modify the s tandard theory so as to genera te neu t r ino masses. We pursue in

depth only one of these approaches , the scheme recent ly proposed by Ge lmin i and

Roncade l l i [1] (which we encoun t e r ed i ndependen t ly but subsequent ly) . Here ,

neu t r ino masses appear by vir tue of the spon taneous b r e a kdow n of global B - L

symmet ry and are accompanied by the necessary appearance of a truly massless

Go lds tone particle. In this respect, the model resembles an earl ier scheme of

Chikashige, Mohapa t ra , and Peccei [2], who in t roduced the term m a j o r o n to

describe the Go lds tone boson. The G R model differs f rom the CMP model in that

it does not involve the existence of heavy unobse rved neut r inos , wh i l e ' i t does

involve a much richer Higgs configurat ion. We follow the G R model with no

compel l ing a rgumen t that this is Na ture ' s way to neu t r ino mass, save that the

phenomeno log ica l sequelae are most bizarre and fascinating.

* This research is supported in part by the National Science Foundation under contract no. PHY77- 22864 and the Department of Energy under DE-AC02-76-ERO-3069.

1 On leave from Harvard University. 2 On leave from Tel-Aviv University; supported in part by the Israel Academy of Sciences.

297

298 H.M. Georgi et al. / Neutrino masses

To set the stage, we present in sect. 2 a brief sampling of the experimental data which suggest, but do not prove, the existence of non-vanishing neutrino masses. In sect. 3 we discuss the special role of B - L symmetry in the understanding of neutrino masses. In sect. 4 we present the model of Gelmini and Roncadelli in order to fix a notation and make this work self-contained.

The remaining sections are devoted to a discussion of the phenomenological implications of the model. In sect. 5 we discuss neutrino cosmology in terms of the

GR model, and show the impossibility of a neutrino-dominated universe. In sect. 6, severe constraints on the model are deduced from our knowledge of stellar structure. In sect. 7 we derive bounds on the coupling of the majoron to fermions.

In sect. 8 we discuss implications on the physics of e÷e - and e e collisions, and upon decay of the Z °.

2. Hints of neutrino masses

Recent papers address the possibility that neutrinos have measurable masses, and indeed, that effects due to these masses have already been seen. Here is a sampling of current literature.

Endpoint spectra. A finite neutrino mass will affect the shape of the electron spectrum from beta decay near its endpoint. In a recent study of tritium decay [3], evidence for a non-zero neutrino mass has been claimed:

46 e V > m~ > 14 eV. (1)

Interpretation of this careful experiment is clouded by uncertainty due to atomic corrections of the same magnitude as the alleged neutrino mass. Among experiments which could verify this result is the study of the endpoint of inner bremsstrahlung photons in radiative electron capture, a process now under consideration [4].

Neutrinoless double beta decay. Double beta decay is a rare but conventional second-order weak process, which has been established to take place for at least three different nuclides, 82Se, 128Te and 13°Te. If the electron neutrino has a

non-vanishing Majorana mass, their double beta decay can proceed in a neutrinoless mode. In a recent reanalysis of the experimental data on Te double beta decay, Doi et al. [5] conclude that the neutrinoless mode does take place, and indeed dominates the double beta decay of 128Te. In terms of a Majorana mass of the electron neutrino, they conclude

My = 3 0 + 1 eV, (2)

in agreement with (1). Other studies of double beta decay [6] yield results much like (2), but in the form of upper limits to the neutrino mass.

Neutrino oscillations. If neutrinos have mass, they may display the phenomenon of neutrino oscillations. These have been and are being searched for. There are two positive indications. One emerges from the apparent deficit of solar neutrinos,

H.M. Georgi et al. / Neutrino masses 299

an effect which could be exp la ined by neu t r ino osci l la t ions ove r an a s t ronomica l

unit*. P resen t d a t a can admi t of a l t e rna t ive exp lana t ions , and e x p e r i m e n t s sensi t ive

to l o w - e n e r g y neu t r inos (e.g., the gal l ium expe r imen t ) must be done . If neu t r ino

osc i l la t ions are the exp lana t ion of the solar neu t r ino p r o b l e m , smal l neu t r ino masses

would suffice (at leas t microvol ts ) , but large mixing angles are requ i red . The second

posi t ive ind ica t ion ar ises f rom the r eac to r e x p e r i m e n t s of Re ines and Sobe l [8].

Di f fe rences of neu t r ino squa red masses of o r d e r 1 (eV) 2 are ind ica ted , but the

resul t is not decisive.

D a r k m a s s in the universe. A t var ious levels, t he re are indica t ions of the exis tence

of subs tan t ia l quant i t i es of n o n - l u m i n o u s mass in the universe . Neu t r i nos with

masses - 3 0 eV which are relics of the big bang could compr i se the mass of galact ic

ha loes [9]. L igh te r neu t r inos , with masses of 6 + 3 eV, may p e r m e a t e galact ic clusters

and expla in the i r a p p a r e n t missing mass [10]. In e i ther case, these exp lana t ions

would imply that the mass of the universe is d o m i n a t e d by neut r inos . Indeed , a

s tab le rel ic neu t r ino with a mass - 1 0 0 eV would p rov ide sufficient mass to close

the universe . A l t h o u g h the case for subs tan t ia l a moun t s of da rk mass in the universe

is firm, it is by no means clear tha i this mass must res ide in the fo rm of mass ive

neut r inos .

W e have m e n t i o n e d four dis t inct p h e n o m e n o l o g i c a l man i fes t a t ions of non-

vanish ing neu t r ino masses. In a s t r a igh t fo rward m o d e l with M a j o r a n a neu t r ino

masses , all of these effects can be p re sen t s imul taneous ly . But the gene ra l s i tua t ion

is much more complex . F o r example , if n o n - z e r o masses a re conclus ively o b s e r v e d

in e n d p o i n t spec t roscopy , it does not fol low that any of the o the r effects must be

obse rvab le . Osci l la t ions may be supp re s sed by small or zero angles. The mass m a y

be a l e p t o n - n u m b e r conserv ing Di rac mass, in which case neu t r ino less /3/3 decay

is fo rb idden . The mass ive neu t r inos m a y decay or annih i la te and thus not survive as da rk mass in the p re sen t universe .

The f inancial a u t o n o m y of var ious po r t ions of H a r v a r d Unive r s i ty is of ten pu t

as an apho r i sm; " E a c h tub on its own b o t t o m " . It seems equa l ly apt to the cur ren t

s i tua t ion in neu t r ino physics. Le t the cosmologis t , the e n d p o i n t spec t roscopis t , the

neu t r ino osci l la tor , the doub le be ta and the solar en thus ias t s each do his (or her) thing.

I t is p r e m a t u r e to a t t e m p t to confine onese l f to one specific theo re t i ca l f r a me w ork ,

for this m a y easi ly lead to confus ion or con t rad ic t ion . In the next sect ion, we

c o m m e n t on the wide var ie ty of theore t i ca l op t ions tha t a re ava i lab le . Of course ,

in the r e m a i n d e r of this p a p e r we shall commi t ourse lves to an expl ici t and

pa r t i cu la r ly unconven t iona l theore t i ca l model . It is not so much fai th in this m o d e l

tha t mo t iva t e s us, as it is the unconven t iona l na tu re of the p h e n o m e n o l o g y . F o r

example , in this t heo ry the neu t r ino mass as o b s e r v e d in e n d p o i n t expe r imen t s

needs not agree with the neu t r ino mass as d e d u c e d f rom neu t r ino less /3/3 decay.

* For a recent review of neutrino oscillations and solar neutrino flux measurements in the Davis experiment, see ref. [7].

300 H.M. Georgi et al. / Neutrino masses

And, the oft-quoted cosmologist limit upon the sum of the neutrino masses is entirely

irrelevant in this theory. Kilovolt neutrinos are perfectly acceptable, for they would not have survived as relics of the big bang.

3. The importance of B - L

Take as a starting point the standard gauge theory of strong and electroweak

interactions based on G = SU(3)× SU(2)x U(1). Three (or more) fermion families are each 15-dimensional chirat representations of G, and there are no right-handed counterparts of neutrinos. A Higgs doublet implements the spontaneous breaking of G to SU(3) x U(1). This system admits a unique current which is free of gauge- gauge-current triangle anomalies, which is not a current of G, which commutes with the elements of G, and which treats the families alike. A linear combination

of its generator with the U(1) gauge generator yields the generator of B - L

symmetry, baryon number minus lepton number. Neither B nor L is a candidate for an exact symmetry. Both have triangle anomalies with two weak SU(2) gen- erators. The combination is not merely anomaly free, but it is, in fact, an exact

global symmetry of the standard theory. It remains an exact symmetry in the extension of G to the unifying group SU(5). This simplest version of a grand unified

theory predicts observable violations of B and of L, but conserves the combination B - L as an exact global symmetry.

We wish to ring changes upon the standard theory in order to produce non-zero neutrino masses. The various possibilities may be classified in terms of what has become of B - L symmetry. In O(10), for example, the generator of B - L lies within the gauge group and has become a local symmetry of the lagrangian. It

cannot be an exact symmetry, for there does not exist a second photon with a sensible coupling to B - L . It is spontaneously broken. In any model where B - L is a spontaneously broken local symmetry, the fermion families must be extended in order to assure that there are no (gauge) [3] triangle anomalies. A minimal

extension involves the introduction of one right-handed neutrino state into each fermion family, and so it is in O(10). In the simplest O(10) theories [9], the masses of observed neutrinos are Majorana masses, and neutrinoless double beta processes

are allowed. Perhaps B - L is a global symmetry of the lagrangian and not a local symmetry.

This is the case for conventional SU(5) unification, wherein neutrino masses are zero. In this scenario, while right-handed neutrinos need not be introduced, they may be. Couplings may be set to make small Dirac neutrino masses, and to maintain global B - L conservation. In such a scheme, neutrinoless /3fl processes are pro- hibited. The adjustments required to make Dirac neutrino masses small compared to quark and lepton masses do not seem particularly attractive to us. However, unless neutrinoless tiff decay or some other B - L violating process is confirmed experimentally, this scenario cannot be excluded on any but aesthetic grounds.

H . M . Georgi et al. / Neutr ino masses 301

More interesting is the possibility that global B - L symmetry is spontaneously broken, producing a true Goldstone boson. Two recent papers [1, 2] have introduced models of this kind. Even more interesting is the fact that the existence of this Goldstone boson cannot be excluded by available experimental data. In the next section, we exhibit the Gelmini-Roncadel l i model [1] in detail.

The final logical possibility is that B - L is not a symmetry of the lagrangian at all. The phenomenological implications of such schemes are sparse.

4. The G R m o d e l

In the standard theory, the Higgs doublet is almost unique. In order to be responsible for a quark or lepton mass term, a Higgs multiplet must have Yukawa couplings to the fermion fields allowed by the gauge symmetry and one of its components must be electrically neutral so that it may develop a vacuum expectation value (VEV) without breaking electromagnetic gauge invariance. Besides the Higgs doublet, there is only one other possible Higgs multiplet with both these properties - the complex triplet which couples to pairs of fermion doublets and can give Majorana neutrino masses.

If such a triplet exists, it carries lepton number ±2. Thus B - L can be a symmetry of the lagrangian only if the scalar Higgs meson self-interactions are invariant with respect to independent phase rotations on the doublet and triplet Higgs fields. If the triplet field has a non-zero VEV, the B - L symmetry is spontaneously broken. This is the G R model.

We denote the usual Higgs doublet by

and the complex triplet Higgs by the 2 × 2 matrix field

( x = x ""

The covariant derivative is

e e D " = 0 " + i T • W " + i S V ~' , (4.3)

sin 0 cos 0

where T (S) are the SU(2) (U(1)) generators. On the scalar fields, the generators act as follows:

TX = ~ ~'~ + ~g~" (4.4)

s 6 = - ½ 4 0 , s x = - x .

302 H.M. Georgi et al. / Neutrino masses

W e can now wri te down the most gene ra l l agrangian which is invar ian t u n d e r

SU(2) x U(1) and i n d e p e n d e n t phase ro ta t ions on ~b and X in the fo l lowing form:

~LP(&, X) = (D"& )*D,& + t r [(D"x)+D,x]- V(ob, X) , (4.5)

whe re

V(6 , X) = h i (& +o - l u 2 ) 2 + A2(tr (X+X) - ½v2)z + A3(&+& - l u2 ) ( t r 0 ( + X ) - i v 2 )

-1-/~4(~/~+(/~ tr (X+X)-cb+XX+Cb)+as((tr (X+X) )2 - t r (X+XX+X)). (4.6)

In this form, it is t r ivial to find the m i n i m u m of the po ten t i a l . If A1, A2, A4 and A5

are pos i t ive and ]A31<2 a-,/a~a~, then V(~b,x) is pos i t ive semidef in i te . But

V((~b), (x)) vanishes for

W e can t ake these to be the V E V ' s .

The V E V ' s of (4.7) give the fo l lowing W ± and Z ° masses:

2 M 2 w e ( u 2 + 2 v 2 )

- 4 sin 2 0

2 M2 e ( u 2 + 4 v 2 ) " (4.8)

z - 4 s i n 2 0 cos 2 0

If v # 0, the s t a n d a r d re la t ion M w = M z cos 0 is not satisfied, and the neu t ra l

cur ren t s do not have the canonica l s t rength. Thus f rom the success of the s t anda rd

pred ic t ions , we know tha t v << u -~ 250 GeV.

Of the ten real fields in (4.1), (4.2) th ree are ea t en by the Higgs mechan i sm.

They are

[u (~b ° - 4) °*) + 2v0 ( ° - x°*)]/242u 2 + 8v 2 ,

(u~b + ~ / 2 v X ) / ~ + 2 v 2 , (4.9)

and he rmi t i an conjuga te . One field is the G o l d s t o n e boson

M ° = [u (A "o - X °*) - 2v (oh o _ ~ o , ) ] / 2 # 2 u 2 + 8 v 2. (4.10)

The o the r fields descr ibe mass ive scalars. The neu t r a l Higgs fields

1 /Ti 0 O~x ½~/~(<b°+~b°*), ~'e~kX ,X ), (4.11)

have a mass squa red mat r ix

2Alu AaUV] A3uv 2A2v2j . (4.12)

Thus for u >> v, one l inear c o m b i n a t i o n (pr imar i ly 4)) has mass of o r d e r ,fA u and

the o the r (p r imar i ly X) is a l ight Higgs par t ic le with mass of o r d e r ~/A v.

H.M. Georgi et al. / Neutrino masses

There is a singly charged field

(u1" - 4 2 v4) ) / 4 u - ~ v 2

with mass squared

303

(4.13)

1)t4(U2 + 2V2) . (4.14)

And there is the doubly charged 1" with mass squared

A4U2+2AsU 2 (4.15)

Note that for u >> v, the t" is heavier than the singly charged field (primarily X-) by a factor of ,]2.

Notice that none of the spinless mesons in the GR model can have a mass which is large compared to GF 1/2. This result depends crucially on the assumption that

B - L symmetry is not explicitly broken, not just on the existence of the t" field. If we were to add a t" field with a large positive mass squared term

M 2 tr (1"~1") (4.16)

to the standard SU(2) x U(1) model and include the B - L breaking term

m& Tx~ d) + b.c., (4.17)

the 1"o field would develop of VEV of order mu2/M 2. Thus, this t" could be used

to give Majorana masses to the neutrinos, but we would encounter no other new physics at momenta small compared to M. In the GR model, with (4.17) ruled out by symmetry, the new physics is unavoidable.

The extra light charged bosons do not appear in the CMP model. The crucial difference is that the B - L symmetry in the CMP model is spontaneously broken at a very large mass scale. Except for the majoron, no trace of the B - L symmetry remains in the effective low-energy theory (indeed the majoron is not, properly

speaking, a part of the low energy theory because all its couplings to light fields are suppressed by inverse powers of a large mass).

We next discuss the couplings of the 1 ̀to leptons. Let

t/.,[ = (~ 'e ) (4.18) ( L

be the standard charged lepton doublets where ( runs over e, tx and r. Then the most general coupling of the triplet field X to the leptons is

g e e " e R X 'XVL +h.c . , (4.19)

where ~ec is the charge conjugate field. In a Majorana basis for the y matrices,

gt~. c = g t [ * . (4.20)

304 H.M. Georgi et al. / Neutrino masses

The coupling matrix gee' = ge'e is symmetric. The Majorana mass matrix of the

neutrinos is

ga~,v. (4.21)

We have no reason to assume that the coupling matrix g is diagonal. For that

matter , we have no a priori reason to assume anything about it at all. For lack of

anything better , we will sometimes assume that the ga.' are not wildly different for

different ( and ( ' , and we may refer to the components generically as g.

5. Neutrino cosmology

In this section, we will discuss the bounds on the parameters in the G R model

which derive from cosmological considerations. We will discuss the range of para-

meters in the G R model for which the neutrinos masses are cosmologically relevant.

Our discussion starts with the of t -quoted bound on neutrino masses*

Z m v ~ < 1 0 0 e V , (5.1)

where the sum extends over the stable light neutrino species. This remarkable

bound is many orders of magnitude bet ter than any which can be obtained in direct

particle physics experiments. The argument for the bound is part icularly simple and convincing. Just as there

are photons which are relics of the big bang in the famous 2.7 ° b lackbody radiation,

there should be relic neutrinos of each of the stable species. The neutrino tem-

perature ( T v - 1 . 9 K) is slightly lower than that of the photons (because e+e

annihilation added entropy to the photons after the neutrino decoupled. We expect

a neutrino number density

n~ ~ 100/cm 3 (5.2)

for each type. Then the total energy density due to the rest masses is

p , = n~ 2 rn,, . (5.3)

We know from rough observations of the decelerat ion parameter and from

bounds on the age of the universe to (from radioactive dating) that p, cannot be

much bigger than the critical density required to close the universe,

pc = 3 H ~ / ( 8 r r G ) , (5.4)

where Ho is the Hubble constant

Ho-=-- t~ /R ~ 100 km/s mpc , (5.5)

and G is Newton's constant. This comparison yields the bound (5.1).

* This bound was first pointed out in ref. [11].

H.M. Georgi et al. / Neutrino masses 305

Converse ly , if n ~m~, is less than a few pe rcen t of pc, the ene rgy dens i ty in neu t r inos

of t ype i is less than tha t in the o b s e r v e d luminous mass in the universe and the

cosmolog ica l i m p o r t a n c e of neu t r ino mass m,~ is small . Thus we d e m a n d as a

necessa ry c r i te r ion for cosmolog ica l r e levance of a neu t r ino species,

m,~ ~> 1 e V . (5.6)

In the G R mode l , the re a re severa l p rocesses which are not p re sen t in more

s t a n d a r d mode l s and which cause neu t r inos to d i s a p p e a r f rom the universe . The

most i m p o r t a n t are: decay of a neu t r ino UH into a l ighter neu t r ino uL plus a m a j o r o n ,

UH~ u L + M ; (5.7)

i n t e rconver s ion of heavy into light neu t r inos in pairs due to s -channel M exchange ,

PH 4- P H ' ~ M - ' ~ PL + PL ;

and ann ih i l a t ion in pai rs into m a j o r o n s ,

t , + u - ~ M + M .

(5.8)

(5.9)

The cosmolog ica l effect of neu t r ino decay has been discussed by C M P [12]. But

the decay is on ly r e l evan t if the l i fe t ime is shor t c o m p a r e d to to ( - 1 0 l° years). In

a m o d e l of the G R or C M P type, the neu t r inos are long- l ived . The reason is that

the m a j o r o n is a G o l d s t o n e boson. Its coupl ing to neu t r inos in the t ree a p p r o x i m a -

t ion is p r o p o r t i o n a l to the neu t r ino mass mat r ix and the re fo re p re se rve neu t r ino

ident i t ies . The l ead ing con t r ibu t ion to the decay (5.7) comes f rom the W exchange

d i ag ram in fig. 1. The resul t for the decay ra te is

F(I,H ~ u L + M ) = ~sml • 2 2c~ uFm~,,rnLv8rr5 In 2 , (5.10)

whe re mL is the mass of the cha rged l ep ton in fig. 1 and c~ is a mixing angle. If we

÷

w

~'H L - 1 I L - u L I I I I M o

Fig. 1. The decay of a massive neutrino into a lighter neutrino plus majoron proceeds, in lowest order, through this Feynman diagram.

306 l l .M. Georgi et al. / Neutrino masses

t ake L to be the r and assume max ima l mixing, we find

5 X 1027 S

r - g3(v/1 keV) s . (5.11)

The in te rconvers ion and ann ih i l a t ion processes (5.8), (5.9) a re m o r e in teres t ing .

The cross sect ions are

2 2 O'2VH+2VL = g H g L / 3 2 r r / 3 s , (5.12)

O'2u~2M = g 4 / 3 / 4 8 r r s , ( 5 . 1 3 )

where ~/s is the to ta l c.m. energy and /3 is the veloci ty of the init ial neu t r inos in

the cen te r of mass. The ann ih i l a t ion cross sect ion (5.13) is given in the a p p r o x i m a t i o n

of smal l /3 (at / 3 - 1 the re are i m p o r t a n t logar i thmic correc t ions , but we will not

need them).

W e will focus our a t t en t ion on the ann ih i l a t ion process . I t o p e r a t e s for any

neu t r ino species. The in t e rconvers ion process al lows heavy neu t r inos to d i s a p p e a r

even fas ter but as we will see the ann ih i l a t ion process a lone is enough to signif icantly

dec rease the dens i ty of neut r inos .

W h e n the neu t r ino t e m p e r a t u r e is large c o m p a r e d to the neu t r ino mass, the

inverse r eac t ion

M + M - ~ v + u (5.14)

is i m p o r t a n t and the m a j o r o n s and neu t r inos are in equ i l ib r ium. W h e n the t em-

p e r a t u r e d rops to the o r d e r of rn~, (at a t ime which we will call ti) the inverse

r eac t ion phases out. F r o m that t ime on we will ignore it. Le t n be the neu t r ino

dens i ty as a funct ion t ime, then

N = R 3 n (5.15)

is the n u m b e r of neu t r inos in a comoving cube of side R. Then for ti < t < to, N ( t )

satisfies

d N N 2 g4/32

dt rtN/3o'2v+2M- R3 192rrm 2 . (5.16) u

A t t = ti, the energy dens i ty p is d o m i n a t e d by the neu t r inos which are b e c o m i n g

non-re la t iv is t ic . F o r t > t~, the universe will r e m a i n m a t t e r d o m i n a t e d unless so

m a n y v 's ann ih i la te that the i r con t r ibu t ion to p b e c o m e s negl igible . But we are

t ry ing to d e t e r m i n e the cond i t ions u n d e r which the neu t r inos still con t r ibu te

signif icantly to p. Thus , we can assume tha t the un iverse is m a t t e r d o m i n a t e d for

t i < t < to. Then R is p r o p o r t i o n a l t o t 2/3, SO we can wri te

• t . 2 / 3

H . M . Georgi et al. / Neutr ino masses 3 0 7

where Ri = R (ti),

Then (5.16) becomes

where

Integrating, we find

t ,81

• t." 1o/3 1 d N . A ( t ) (5.19)

N 2 d t -

A _ 4 g

2 3 • 1927rm ~R i (5.20)

7 / 3

1 1 = 3 A t , [ 1 - ( ~ ) ] (5.21) N N i

If A t i N i >> 1, the final neutrino number approaches an asymptotic value independent of Ni:

1 9 2 ~-rn . R i No~ = 7 / ( 3 A t i ) 7 2 3 z - - _, 3 g4t i (5.22)

The contribution of the surviving neutrinos to the energy density is

7 192, m3o/RiV to / - - | (5.23)

P" - 3 g4 to k R o ] t~

W e can use (5.17) and the relation between R and temperature:

Ri kT~ (5.24) g o m, '

and write

1400m 3 / \ 'kTd 3/2 p , - - g4t ° ~ - ) . (5.25)

Requiring that pv exceed 1% of Pc, we obtain

( m J 1 keV)3/g ~ t> 1038 . (5.26)

Note that we do not have to worry about the contribution to the energy density from the majorons produced by the annihilation. Eq. (5.21) shows that most of the annihilation takes place for t = ti so the energy of the majorons is comparable to the photon tempera ture at the time. Thus, there is a majoron background somewhat larger than we would expect if we ignored annihilation. But like photons,

308 H.M. Georgi et al. / Neutrino masses

v 8 - - ~- 10 3e KeY:

1 G e V

I 1 M e V v

1 K e Y r v = t 0

1 e V m u ~- leV

I , , , I I I ] e V 1 K e Y 1 M e V 1 G e V

m~

Fig. 2. This graph establishes the impossibility of a neutrino-dominated universe in the GR model. If a given neutrino species is to contribute at least 1% to the critical mass density of the universe (.Q~ > 0.01 ), then its mass and the value of v must lie in the unshaded domain. The vertical constraint ensures m~ > 1 eV, while the skew constraint ensures that these neutrinos survive annihilation into massless Goldstone bosons. In the subsequent section, we show that v must lie below the dotted line, so that

the unshaded region is physically unacceptable.

t h e m a j o r o n s a r e m a s s l e s s , so t h e i r c o n t r i b u t i o n to t h e e n e r g y d e n s i t y is s i m i l a r to

t h a t of t h e m i c r o w a v e b a c k g r o u n d , a n d t h e r e f o r e n e g l i g i b l e .

W e s u m m a r i z e t h e d i s c u s s i o n in th i s s e c t i o n in fig. 2 w h i c h s h o w s t h e r a n g e of

v a n d rnv fo r w h i c h a g i v e n n e u t r i n o s p e c i e s c a n b e c o s m o l o g i c a l l y r e l e v a n t . T h e

r e l e v a n t r e g i o n sa t i s f ies t w o b o u n d s ,

m y / > 1 e V , vS/mSv ~ 1038 k e V 3 , (5 .27)

w h i c h c o m e f r o m (5.6) a n d (5 .26 ) r e s p e c t i v e l y . W e a l so s h o w t h e l ine c o r r e s p o n d i n g

to a n e u t r i n o l i f e t i m e e q u a l to t h e a g e of t h e u n i v e r s e ( for m a x i m a l m i x i n g ) . F i n a l l y ,

l e t us n o t e t h a t t h e t w o v e r y l i gh t s c a l a r s c o u n t as o n e n e u t r i n o . spec i e s in t h e

a n a l y s i s of p r i m o r d i a l h e l i u m n u c l e o s y n t h e s i s .

H.M. Georgi et al. / Neutrino masses 309

6. Astrophysics

The discussion of the previous section does not put any useful bounds on the parameters of the GR model. The "allowed" region in fig. 1 is the region in parameter space for which a given neutrino is cosmologically relevant (in fact, there

is a small part of the cosmologically relevant region which is cosmologically ruled out, but as we will see it is not interesting). However, there is an astrophysical

consideration which we can use to put a very strong bound on the VEV v, v < 100 keV. This bound, together with fig. 2, shows that neutrinos in the GR

model cannot be cosmologically relevant. The bound on v derives from consideration of majoron emission from the cores

of red supergiant stars. A similar process, the emission of light axions, has been studied by Dicus, Kolb, Teplitz and Wagoner [13]. The coupling of the GR majoron to fermions (except neutrinos) is similar in form to that of the axion, but suppressed by a factor of 2 v / u . It is a pseudoscalar coupling to mass eigenstates with coupling

2v m (6.1) b/ b/ '

where m is the quark or lepton mass. This follows from (4.10) which shows that the GR majoron has a component (with amplitude - 2 v / u ) which is the neutral

Goldstone boson of the standard SU(2)× U(1) model. Furthermore, Dicus et al. extended their calculations to very low mass axions ( - 1 0 eV) which for stellar core temperatures much greater than 105 K are just like massless majorons.

When axion reabsorption corrections are neglected (which will certainly be a good approximation for the much more weakly interacting majorons), Dicus et al. find that light axion emission from the helium core of a red supergiant (M = 95 solar masses, T = 10 s K, p = 104 g/cm 3) should cause energy loss of 1015 erg/g s.

This is thirteen orders of magnitude larger than the energy production due to the 3c~ process in the core. It would lead to the complete exhaustion of the thermonu- clear energy available in times much shorter than the red supergiant core lifetimes.

To avoid this effect in the GR model, we must decrease the coupling by taking 2 v / u <~ 10 -6 or v ~< 100 keV. In the remainder of this section, we will give a simplified

analysis of majoron emission. At the relevant core temperatures - 1 0 s K - 1 0 4 eV, we can estimate the mean

free path of photons in the core, Lv, as follows:

LS~ 1 =o'c" ne, (6.2)

where o-c is the cross section for Compton scattering (which dominates at these

energies), 2 2

cr c '~ Tro~ / m e , ( 6 . 3 )

and ne is the electron number density,

n~ = 1028/cm 3 , (6.4)

310 H.M. Georgi et al. / Neutrino masses

corresponding to a mass density

Thus

The radius of the helium core is

P = 10 4 g / c m 3 • (6.5)

Lv = 4 x 10 4 cm. (6.6)

R = (3M/4zrp) 1Is ---3 x 109 cm. (6.7)

The process of photon diffusion can be approximated as a random walk of step Lr. On the average ( R / L , ) 2 steps are required in order to diffuse a distance R. Thus the number of collisions Nc suffered by the photon in the process of its escape is

Nc ~-- ( R / L ~ ) 2 . (6.8t

In each such collision there is a small probability P that a majoron will be

produced instead of a photon, with

P = O r V + e ~ M + e / O V c . (6.9)

When this happens, the majoron will escape, carrying off the typical photon energy

E v ~ - - k T = 2 x l O 2 m e . (6.10)

Thus, the fraction f of the produced energy carried off by majorons is of the order of the probability that a given photon will convert to a majoron before it escapes,

roughly

f = P N e . (6.11)

We demand that f <~ 1, so that majoron emission will not determine photon emission. The majoron photoproduction cross section can be inferred from the axion

calculation [14]. Expanding to lowest order in E . J m e [see (6.10)] we find

t2~2 c~g /z v

O'v+e~M+e 3m 4 , (6.12)

where g' is the electron-majoron coupling constant given by (6.1) with rn = mo. Putting all this together, we find that PNc ~< 1 implies

v ~<75 keV, (6.13)

in agreement with the Dicus et al. result.

7. D o u b l e beta decay and bounds on g

It is important to set limits on this process, for neutrinoless double beta decay is logically separate from the question of neutrino masses. We may have large

H.M. Georgi et al. / Neutrino masses 311

Dirac neutrino masses, and exact B - L conservation. In this case, neutrinoless

double beta decay is fobidden*. We may (in the context of theories we do not discuss) have vanishing neutrino masses and yet allow neutrinoless double beta decay. This tub is on its own bottom.

Let us consider the question of neutrinoless double beta decay in the context of the GR theory of neutrino masses. Three processes are relevant to double beta decay. They correspond to the decay schemes

( Z , A ) ~ ( Z + 2 , A ) + e + e - + ~ + ~ , (7.1)

( Z , A ) ~ ( Z + 2 , A ) + e + e - , (7.2)

( Z , A ) ~ ( Z + 2 , A ) + e - + e +X °. (7.3)

In (7.3), X ° signifies the majoron or its very light scalar counterpart. The novelty

of the GR scheme in the occurrence of the third process, wherein no neutrinos are produced but a very light boson is emitted. If double beta decay is observed in the laboratory, it is difficult but conceivable to distinguish reaction (7.1) from reaction (7.2) by the observed shape of the electron spectrum. It is even more difficult to distinguish reaction (7.3) from reactions (7.1) and (7.2) in this fashion.

It may be possible to infer the occurrence of reactions (7.2) and (7.3) from the observation of the rates of double beta decays together with a certain confidence

in the underlying nuclear physics. Doi et al. [5] have recently analyzed the Hennecke et al. [16] data concerning the double beta decay of two isotopes of tellurium. For 13aTe, they conclude that the decay rate for the neutrinoless reaction (7.2) is about

thrice the rate for reaction (7.1). [Needless to say, they do not consider reaction

(7.3).] In terms of a Majorana mass of the electron neutrino, they conclude that my =41 eV.

It is straightforward to adapt the Doi calculation to deal with the GR model in which reaction (7.3) can effectively compete with reaction (7.2). Both of these processes depend upon the identical nuclear matrix elements. Their respective decay rates are related in the following manner:

F3/F2 = g2(847r2) l (Q/mv)2R(x ) , (7.4)

where Fi is the partial rate for reaction (7.1) and g is the Yukawa coupling of the majoron to the electron neutrino, rn~ is the mass of the electron neutrino, Q is the

available energy of the decay, and x = Q/me. We have omitted inessential complica- tions due to neutrino mixing. The factor R(x ) is a ratio of phase-space integrals,

Ix 4 + 14x 3 + 84x 2 + 210x + 210] R(x ) = [ x ~ - + ~ O ~ - ~ ~ 3. (7.5)

* Early theoretical work on the subject was done by Primakoff and Rosen [15].

312 H.M. Georgi et al. / Neutrino masses

In t e rms of the G R mode l , the t e l lu r ium analysis yields the s t a t emen t that F2 + F3 -~

3F~, and gives the cons t ra in t

g~e[v 2 + mZxZR (x)/(84~-2)] = (30 eV) 2 , (7.6)

whe re x is eva lua t ed at x = 1.7, so that x2R(x) = 8.4.

Sensi t ive searches have been p e r f o r m e d for the n o - n e u t r i n o doub le be t a decays

of 7 6 G e E17] and 82Se E18], resul t ing in u p p e r l imits for reac t ion (7.2) in which the

e lec t rons car ry off all of the decay energy. These expe r imen t s are insensi t ive to

reac t ions (7.1) and (7.3). Hax ton , S tephenson , and S t ro t tman [6] have ana lyzed

these expe r imen t s in te rms of the M a j o r a n a mass of the e lec t ron neu t r ino , and conc lude that its mass must be less than 15 eV. In our no ta t ion , this means

geov < 15 e V . (7.7)

The cons t ra in ts on gee and v f rom (7.6) and (7.7) are d i sp layed in fig. 3. It is to be

s t ressed that in the contex t of the G r mode l , the inequa l i ty (7.7) is not in conflict

with the equa l i ty (7.6).

I°%v 1 2 " / ~ ~

lO%V -

103eV -

my :15eV \ IOO eV

lOeV -

I eV - -

Doi Estimate

I I I lO-5 lO-4 lO-3 lO-Z lO-I

gee

Fig. 3. T h e Do i et al. ana lys i s of Te /3/3 d e c a y yie lds the sol id cu rve as a c o n s t r a i n t o n g ~ a n d v. A l so

s h o w n in the H a x t o n et al. u p p e r l imit on g~ev a n d the Dicus et al. u p p e r l imit on u. T h e G R m o d e l is c o m p a t i b l e wi th all t h ree cons t r a in t s .

H.M. Georgi et al. / Neutrino masses 313

We turn now to the process of double K capture, for which there are three

relevant reactions,

e - + e + ( Z , A ) - ~ ( Z - 2 , A ) + v + v , (7.8)

e +e + ( Z , A ) ~ ( Z - 2 , A ) , (7.9)

e +e + ( Z , A ) ~ ( Z - 2 , A ) + x ° . (7.10)

Reaction (7.8) is "conventional", though rarely discussed. Reaction (7.9) is not a reaction at all, and will command separate comment. Reaction (7.10) is characteris-

tic of the GR model, and will be of concern to us. Process (7.9) corresponds to an off-diagonal mass term which mixes two distinct

atomic species: the normal (Z, A) atom and an excited ( Z - 2 , A) atom with two holes in the atomic S-shell and, possibly, in an excited nuclear state. Conservation of angular momentum requires that the initial nuclear spin coincides with the final nuclear spin. The magnitude of the off diagonal mass term, dimensionally, is given

by 3 2

= (Zc~rne)-Gvm~m,~ (7.11)

and cannot reasonably be expected to exceed 10 -2o eV. Atomic mixing such as we have just described is analogous to KK mixing and

to neutron-antineutron mixing. It can lead to observable decay processes in the following fashion. The (Z - 2, A) atom is in an unstable configuration, due to the possibility of X-ray emission, the Auger process, or -/-ray emission from the excited nuclear state. It follows that the original (Z, A) atom will also be unstable with a

decay rate given by [ ' (Z , A ) = e2F(A 2 + ( 1 / . ) 2 ) 1 (7.12)

where F is the decay rate of the excited (Z = 2, A) atom and A is the difference

in mass between the two atoms. We know of no instance where this process can

realistically be detected. Our discussion of double K-capture may therefore be

limited to the conventional reaction (7.8) and the GR process (7.10). A favorable candidate for double K-capture is the decay of SSNi (the principal

isotopic species) to the relatively rare isotope SSFe, where the available energy is ~3 MeV, and reaction (7.10) may possibly dominate reaction (7.8). Although the

lifetime is certainly long, the signal of double K-capture is unique. It consists of two characteristic Fe X-rays, with slightly different energies, in time coincidence. A careful calculation of the rates of reactions (7.8) and (7.10), coupled with a measurement of double K-capture in ~SNi may provide a unique test of the GR

model of neutrino masses. Another rare decay mode that is generally permitted in the GR model is the

unobserved process

/x- -~e-e e + . (7.13)

314 H . M . Georgi et al. / Neutr ino masses

A n effective four - fe rmion coupling generat ing this decay is p roduced by X exchange, and its s trength is given by

= g e e g , e / M 2 , (7.14)

where M is the mass of the doubly charged X meson. The exper imental upper limit on the branching ratio for react ion (7.13) is 10 9. It follows that the G R model is const ra ined to satisfy

10 ~<3 x 10 -5 GF. (7.15)

We know f rom the preceding analysis that gee < 10 -3. If it is t rue that g ,e - gee, we

may conclude that M > 50 GeV. However , we cannot be sure that gue is not larger than ge~, or contrariwise that it is not zero. In ei ther case, there is a large range of parameters in which the G R model is compat ible with the observed suppression of/.t -+ 3e. Fur ther searches for this decay mode are mandatory .

8. Lepton-lepton collisions and Z ° decay

Lepton- l ep ton collisions present the cleanest labora tory in which to discover the existence of the complex triplet of Higgs bosons, the essential new c o m p o n e n t of the G R model . This is clearly true for X ++ and X +, whose strongest couplings are their e lec t romagnet ic couplings. Well above threshold X+X - pairs will be p roduced

1 + - + + + - - i n e+e collisions with a cross section ao'(e e + / z /x ), while X X pairs will

have a cross section equal to the tz+/x- product ion cross section. The masses of these particles satisfy the relations

15 G e V ~ < M ++ = x/2M+ ~< 250 G e V , (8.1)

where the lower limit is experimental , the equali ty is an essential p roper ty of the G R model , and the upper limit is p robably necessary if the theory is to be self-consistent. Whe the r these particles will be accessible to the next genera t ion of e+e - machines is a mat ter of luck, as it is for the top quark and the convent ional neutral Higgs boson.

H o w do these particles decay? They can decay into two leptons:

4 + + + )¢ ~ # g , (8.2)

+ + _

X --> ( v, (8.3)

or, they can decay weakly:

+ + + J + X ~ A " + ,

+ 0 + j + (8.4) X -+X

where J+ denotes the decay products associated with a charged weak current : ud, + + +

cg, e v, b~ v, r v, etc. The weak decays are complete ly predictable in terms of

H.M. Georgi et al. / Neutrino masses

the mass M of the X ++. The partial rate for these decays is of order

0.01 G~M 5

if M <~ Mw, and of order

a 2M/sin4 0

if M ~> Mw. The partial width for the leptonic decay (8.2) is

[ga,,12M/64rr ,

and that for the decay (8.3) is

315

(8.5)

(8.6)

(8.7)

5Z [gee,12M/167r~/2. (8.8) ~ ,

+ + + +

It is clear from the double beta decay constraint that the • ~ e e partial width is small compared to (8.5) or (8.6). Apart from that, we can say little about the

direct leptonic decays. If the weak decays of the X ++ dominate, the decay products which result from

X++X production in e+e + annihilation consist of J+J+J J X°X °. The neutral mesons are unobservable light Higgs or Goldstone bosons. Thus, these events will be highly spherical events involving four quark (or lepton) pairs with more than 20% missing energy. They should be readily distinguished from the larger jetlike

background. If the leptonic decays dominate, we can anticipate the more spectacular + + + +

final statestx tz tz /x o r z z z r . If gee is not too small, it may be possible to produce X singly in e - e - collisions.

Here the only hadronic background comes from two photon exchange and the purely leptonic background (e /~ , /z tz , etc.) is nil. For M - 1 0 0 G e V and gee--10 3 (saturating the double beta decay bound (7.6)) the e e width of the X is about 1 keV. The full width is much larger, at least 10 MeV. The X may well be the most exciting potential discovery to be found in an e e facility. A reasonable criterion that such a machine should satisfy is that it will reveal the X

if it exists. It may be possible to observe X ++ and X + as decay products of Z ° and W + which

are produced in PP or PP colliding beam machines. The partial widths for various decay modes are given in table 1.

TABLE 1

Decay mode Partial width Mass range

o + + Z ~ X ~ 0 . 1 1 G e V M < < 4 7 G e V

W + ~ X++)C- 0.24 G e V M << 49 G e V

Z ° ~ x+X 0.02 G e V M << 66 G e V

W+~)C+X° 0.24 G e V M<< 117 G e V

Z°-+ X°X ° 0.34 G e V for any mass

316 H.M. Georgi et al. / Neutrino masses

T h e Z ° decay modes in table 1 can be conven ien t ly s tudied on the Z ° r e sonance

which will be abundant ly p roduced and will decay in a known fashion into all

k inemat ica l ly accessible channels. The charged X's can be seen if they are light

enough to be produced . H o w e v e r , the Z ° can cer ta inly decay into X°X °, or m o r e

accurately, into our new light Higgs boson plus a ma jo ron , i n d e p e n d e n t of the

value of M. The part ial decay width for Z ° to decay into this dark channel is

precisely twice its decay width into each neu t r ino -an t ineu t r ino mode . It is env isaged

that L E P will succeed in compar ing the actual Z ° width with the "v i s ib le" and

smal ler width deduced f rom the leptonic width and branching ratio. Thus, L E P

will a t t empt to count the n u m b e r of fe rmion famil ies by count ing the n u m b e r of

neu t r ino decay modes . In the G R model , L E P ' s count ing must reveal at least five

" n e u t r i n o s " . If the measu red n u m b e r is less than five, the G R mode l is shown to

be false. If it is five or more , we shall have a lovely ambigui ty for fu ture expe r imen t s

to resolve. A winner e i ther way.

We would like to thank D u a n e Dicus, Dan F r e e d m a n , Alan Guth , Ken Lane,

Dav id Schramm, and Vigdor Tepl i tz for useful conversa t ions .

References

[1] G.B. Gelmini and M. Roncadelli, Phys. Lett. 99B (1981) 411 [2] Y. Chikashige, R.N. Mohapatra and R.D. Peccei, Phys. Lett. 98B (1981) 265 [3] V.A. Lubimov, V.Z. Nozsik, E.G. Novikov, E.F. Tertyakov and V.S. Kosik, Phys. Lett. 94B

(1980) 266 [4] A. De Rtijula, Nucl. Phys. B188 (1981) 414 [5] M. Doi, T. Kotani, H. Nishiura, K. Okuda and E. Takasugi, Osaka University preprint number

OS-GE 80-27 (1980), revised version [6] W.C. Haxton, G.J. Stephenson, Jr. and D. Strottman, Phys. Rev. Lett. 46 (1981) 698 [7] J.N. Bahcall et al., Phys. Rev. Lett. 45 (1981) 945 [8] F. Reines, H.W. Sobel and E. Pasierb, Phys. Rev. Lett. 45 (1980) 1307 [9] E. Witten, Proc. 1st Workshop on Grand unification, Durham, New Hampshire, April, 1980;

H. Sato, Kyoto University preprint RIFP-423 (1981) [10] D.N. Schramm and G. Steigman, Gen. Rel. Gray., to be published [11] R. Cowsik and J. McLelland, Phys. Rev. Lett. 29 (1972) 669; Astrophys. J. 180 (1973) 2 [12] Y. Chicashige, R.N. Mohapatra and R.D. Peccei, Phys. Rev. Lett. 45 (1980) 1927 [13] D.A. Dicus, E.W. Kolb, V.L. Teplitz and R. Wagoner, Phys. Rev. D18 (1978) 1829;

M.I. Vysotsskii, M. Yu. Khlopov and V.M. Chechetkin, JETP Lett. 27 (1978) 502; K. Sato and H. Sato, Prog. Theor. Phys., 51 (1975) 1564

[14] T.W. Donnelly, S.J. Freedman, R.S. Lytel, R.D. Peccei, and M. Schwartz, Phys. Rev. D18 (1978) 1607

[15] H. Primakoff and S.P. Rosen, Phys. Rev. 184 (1969) 1925 [16] E.W. Hennecke, O.K. Manuel and D.D. Sabu, Phys. Rev. C l l (1975) 1378 [17] E. Fiorini et al., Nuovo Cim. A13 (1973) 247 [18] B.T. Cleveland et al., Phys. Rev. Lett. 35 (1975) 757