Masses of neutrinos in unification gauge theories and neutrino-antineutrino oscillations

P H Y S I C A L R E V I E W D V O L U M E 2 3 , N U M B E R 9 1 M A Y 1 9 8 1

Masses of neutrinos in unification gauge theories and neutrino-antineutrino oscillations

Dan-di Wu* Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305

and Lyman Laboratory of Physics, Haruard University, Cambridge, Massachusetts 02138 (Received 18 August 1980)

We discuss two types of neutrino oscillation: (I) the v,-v, type of oscillations and (11) the Y-+$ type of neutnno- antineutrino oscillations. To connect the oscillation phenomenology with mass differences of the neutrinos, we discuss possible neutrino mass patterns in a general form including the Dirac mass and the Majorana mass, when there are both left- and right-handed neutrinos. Then we extract four extreme and interesting cases of the mass patterns. We connect these mass patterns with the Weinberg-Salam model, grand unification models such as SU(5), S0(10), and E,, and some constituent models. At the end of this paper we deduce from one of the interesting neutrino-mass patterns that one more neutral intermediate boson with mass at the same order as the mass of the W boson may exist.

I. INTRODUCTION

Three species of neutrino have been found ex- perimentally. All three a r e consistent with being left-handed neutrinos. We do not lcnow whether these neutrinos a r e mass less neutrinos. We do not know whether they have right-handed partners. Whether they have family -number-unconserved and/or lepton-number-unconserved interactions, we also do not know. We can imagine how diffi- cult it is to answer these questions, if we remem- ber that the neutrion was invented by Pauli' in the beginning of the 1930's, some 35 years later than the discovery of the P-type radioactivity (1896), and i t s existence was "verified" by Cowan and Reines et ~ 2 . ~ some 20 yea r s after i t s inven- tion. Of course, the point is that the interactions of the neutrinos a r e so weak, especially fo r right- handed neutrinos if they exist. Also, the masses of the left-handed neutrinos a r e extremely s n ~ a l l . ~ In this situation the neutrino-oscillation experi- ments415 may play a la rge role. A naive estimate says the neutrino- oscillation experiments may be sensitive to the mass differences Am2 in the following region:

A given experiment may cover a part of this region' depending upon the character of the neutrino source and the distance between the source and the detector. But most of this region cannot be reached by spectrometer experiment^.^

The neutrino-oscillation experiments may also help to answer whether there a r e right-handed neutrinos if the relative mass difference fal ls in the 4?nz region in Eq. (1). Actually we may have two types of neutrino oscillation: The f i r s t type of oscillation i s among the left-handed neu- trinos4 (or among their antineutrinos)

These oscillations conserve the lepton number but break the family numbers, whereas the second type of oscillation7 i s between the neutrino and the antineutrino with the same helicity

type 11: vc-t v C . (3)

These oscillations change the lepton number by AL =i 2. We shall re fer to type I1 a s the "neutrino-antineutrino oscillations" and to type I a s "neutrino-neutrino oscillations."

11. NEUTRINO OSCILLATIONS IN C E N E R A L ~ ~

The left-handed neutrinos have the SU(2) x U(1) weak gauge interactionsg which can be shown by putting the neutrinos in doublets:

Here we have three families (or generations) of leptons. Because the gauge bosons meet only the particles in the same representation of the gauge group in an interaction vertex, the family quantum number, which differentiates different representations of the same dimension, is crucial for specifying the weak gauge interactions of the neutrinos. Here the family quantum number i s e , p , or T . Every neutrino in the table of Eq. (4) has fixed quantum numbers T, T,, and family number. Thus we call these neutrinos the eigenstates of interactions. Because the SU(2) x ~ ( 1 ) gauge interactions a r e the dominant interaction of neutrinos, in a reaction a neutrino in a pure eigenstate of interactions i s produced o r anni- hilated. The right-handed neutrinos (if they exist) a r e in an SU(2) singlet with the hypercharge of

2038 O 198 1 The American Physical Society

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U(1) being zero. They belong to the other kind of eigenstates of interactions in the sense that they have fixed quantum numbers of the SU(2) x U(1) gauge group, though they do not have any gauge interactions of the SU(2) x ~ ( 1 ) type.

However, the Yukawa interaction between fermions and Higgs part icles [we will call them "superweakJ' here; the typical strength of the effective four-fermion superweak interactions is (mFZ/nzHZ)GF] violates family numbers, and some grand unification interactions mediated by extremely heavy-boson transform leptons t o anti- leptons. These interactions will produce non- diagonal masses among different neutrino eigenstates of interactions. The contribution of these interactions t o the mass matrix i s much la rger than to the reactions which violate the conservation laws of the main SU(2) x ~ ( 1 ) gauge interactions because, roughly speaking, the former i s an S matrix but the lat ter i s the absolute value squared of the S matrix. In this way, the eigenstates of interactions become different s tates from the eigenstates of masses .

Let v, be the eigenstates of interactions and vi be the eigenstates of masses

where Vmi a r e the elements of the unitary matrix V, VV' = 1. By a physical reaction we produce an eigenstate of interaction / v, ) which is a special combination of 1 vi ) , a s shown in Eq. (5). [v i ) with different i's a r e different eigenstates of mas- s e s , i.e., different eigenstates of propagation. So there will be an evolution of the neutrino state

1 v,) in i t s propagation process. Let ( v,(x, t)) be the evolution state, / v,(O, 0)) = 1 v,); then

/ v,(x, t ) ) = x Vai / vi)ei'Pi'-Eit' i

where we use vni << E, and p i = E i - m,2/2Ei. To measure the neutrino flux is to pick up an eigenstate of interaction if the corresponding spectrometer cannot differentiate the different neutrino m a ~ s e s . ~ From the number of the events

we know the flux of v,, in the neutrino beam. We have

where At is the time resolution of the detector (we have assumed that A t i s large enough so that E , and E 5 have to be almost the same, E i =E5 =El ,

and N,+&x, E) means the transition probability of a neutrino v, with energy E at x = O to be a neutrino v, a t x. We notice that

This i s the expression of the conservation of the flux of probability.

In an ordinary case, only left-handed neutrinos a r e produced and only left-handed neutrinos can be detected. Suppose v, i s a left- handed neutrino and in Eq. (9) we sum over only left-handed neutr inos, then we get the number of the left-handed neutrinos in total

Of course we have

The equality happens only when there 'are no right- handed neutrinos in the set labeled by /3 o r mixings between left-handed neutrinos and right-handed antineutrinos a r e zero. Equation (10) i s also for the number of neutral-current (NC) events because the neutral-current interactions a r e diagonal and proportional t o the flux of left-handed neutrinos in total.

A typical function N(x,E) normed by the spec- t r a l function f(E) i s shown in Fig. 1 when only one m a s s difference i s concerned. The curve i s a triangle function of x / 2 E . This curve will be measurable if the following conditions hold.

(1) A ( x / ~ E ) satisfies

where A ( x / ~ E ) i s the uncertainty of the measurement of x / 2 ~ , which i s caused mainly by the size of the source and the uncertainty of the energy measurement. If this condition i s not satisfied, i .e., the hm2 i s too large for given x and E , the oscillation part of the flux will be wiped out and the flux becomes a constant relative to the mixing angles a s follows:

and the diagonal transition probability has a low bound

FIG. 1. A typical function A'(x ,E) ; see Eqs. (7) and (10).

where N i s the number of the species of neutrinos. Fo r instance, if there a r e three species of left-handed neutrinos, then N = 3; if for each left-handed neutrino we have a l so a rightlhanded neutrino, then 1V=6. If the oscillation part of the number of the neutral-current events in Eq. (10) i s wiped out, then the flux of the neutral current becomes constant and has a low bound

Of course, if the oscillation part of the flux i s wiped out, we will not be able to get any information about the m a s s of neutrinos f rom the flux measurement, except a low bound of m a s s if we know the intensity and the size of the source be- forehand.

(2) The relative mixing is not too smal l a s the oscillation t e r m s escape from the sensitivity of the detector. By increasing the stat is t ics of the experiment, we can measure smal le r and smaller mixing angles.

(3) x/2E satisfies

If x/2E i s too small , then the oscillation cannot be seen.

If a l l v, in Eq. (5) a r e left-handed neutrinos, then we get oscillations of type I only. The uni- ta ry transformation between se t v, and set vi is s imi la r to the Kobayashi-Maskawa matrixlo between (df , s r , b') and (d, s , 6 ) . We cannot see oscillations but mixing angles (or the KM ma- t r ~ ' ) in the quark case because the mass differences a r e too large here and the condition of Eq. (12) i s not satisfied.

In the case of only one left-handed and one right- handed neutrino, we have pure type-II oscillation. This oscillation i s quite s imi la r to the well-known p-xo oscillation1' and the recently discussed neutron-antineutron o~c i l l a t i on . "~ If there i s more

than one family, we get generally mixed oscillations of types I and 11.

111. POSSIBLE MASS PATTERNS OF NEUTRINOS','~~'~

The situation of oscillations and i t s measur- ability depend on the mass pattern of the neutrinos. Let u s discuss the case wherein there i s only one left-handed neutrino and one right- handed neutrino.

Suppose v and vC a r e Dirac neutrinos

and

where (2 and @ a r e charge-conjugation and space- inverse operator, respectively. We take normal definition of these operators a s follows:

(2+ee1 =c$T, C = Q y Z y O (19)

and

s$('x, t )N1 =yo$(-%, t ) . (20)

We a lso take the definition of chirality s tates a s follows:

We notice that at the limit m - 0 , a particle with left-handed chirality v, has left-handed helicity whereas an antiparticle with left-handed chirality (v,)' has right-handed helicity:

We have the most general m a s s t e r m s of one- family neutrinos in the Lagrangian a s follows:

Here a is the Dirac mass and the t e r m s b and c the left-handed and the right-handed Majorana masses , respectively. The Majorana mass t e rms violate lepton-number conservation with 4L=+2. Defining

a s the eigenstates of interactions, we can rewrite Eq. (23) a s

$ $ R ~ ~ + L +H.c., (24)

where

23 M A S S E S O F N E U T R I N O S I N U N I F I C A T I O N G A U G E T H E O R I E S ...

is the m a s s matrix. The diagonal elements a r e the same because of the CPT theorem. In order to make ILf diagonalized, let u s define two unitary matr ices U, and U, (m- and 772, a r e positive numbers ) such that

We also define eigenstates of masses FIG. 2 . The mass spectrum of the neutrinos. Pat tern

(a) : Three l ines, if there a r e only left-handed neutrinos. Pattern (b) : Three doubly degenerate lines in the Weinberg-Salam model. Pat tern ( c ) : Neutrino masses in the SO(10) model with ter r ib le heavy right-handed Leptons. Pattern (d): An extreme case with double-line fine structures. If 4 =0 , the mass t e rm Eq. (22) becomes

Pat te rn (c) B =0 , C >>A. Here we compare two matr ices by comparing al l nonzero elements. In this case we have

~ - E ( A ~ ) ~ / C ~ , X _ = V ~ + ( V ~ ) ~ , (33)

where

a r e Majorana neutrinos: and

X+ = c o s ~ [ v , + (vL)C] +sinQ[vR + (vR)'1, (30')

7 7 2 , ~ cb, X + = vR+ ( v , ) ~ . (34)

The mass sepctrum is shown in Fig. 2(c). Pa t te rn (d) All A , B , and C a r e smal l a s t o

match the present bounds3 on neutrino mass . The shape of the spectrum is ugly. There i s a much more interesting special case of this pattern when each doubly degenerate line in Fig. 2(b) spl i ts into two lines with very small space [Fig. 2(d)] as the fine structure in optics.

Only in pattern (d ) may there exist observable oscillations of type 11.

IV. MORE PHENOMENOLOGY OF THE O S C I L L A T I O N S ~ ~ ~ , ~

X - = s i n ~ [ v L - (v,)C]+cos~[v, - (v,)C].

If there a r e three families of neutrinos, our mass matrix will be a 6 x 6 matrix

where A is the Dirac mass matrix and B and C a r e the Majorana m a s s matrices. When diagonal- izing Eq. (31) to

with m- and m+ 3 X 3 diagonal matrices, the gene- r a l solutions a r e very complicated, but there a r e four very interesting extreme cases. The discussion of these four m a s s patterns will give u s a good insight into the physics.

Pa t te rn (a) A = C = 0, if there a r e no right- handed neutrinos. Then X - is a linear combina-

he left-handed neutrino and i t s antineu- tr ino

Let u s return to our discussion with one family, see Eq. (22). To simplify the discussion, we con- centrate on the case when b =c<< a . From Eq. (30), we get

The mass spectrum of neutrinos has only three lines [Fig. 2(a)]. Of course, in th is case only oscillations of type I a r e possible.

Pa t te rn (b) B =C =O, if there a r e no lepton- number-violating interactions. The mass spectrum of neutrinos i s shown in Fig. 2(b), which is s imi lar to case (a), but every line is doubly degenerate.

PX-6-' = X + . According to Eq. (7), we have

Because the right-handed neutrino escapes from our detection, we do not write it here.

2042 D A N - D I W U 23 -

Going back t o the three-family case , choosing spectruni Fig. 2(d) and the simplifying form Eq. (35) a s an example, we get

where cu and P a r e left-handed neutrinos, V,, a r e t ransformation mat r ices between the eigen- s t a t e s of interaction and the eignestates of m a s s e s when omitting the fine s t ruc tures , 4m2, = (??z:)~

~ ? " n ~ ~ , = n ? ~ ~ - in;, nli=+(?nt+n7;), and ~ 7 1 1 ~ , < < l m z Z , f o r any i # j and any k is the charac- t e r i s t i c of the spectrum in Fig. 2d. According to Eq. ( l o ) , the number of the neutral-current events is

It is distance dependent because there a r e a l s o type-I1 oscillations. However, if A7nzi = O (i.e., the re a r e no type-I1 oscillations), Eq. (39) becomes identical to Eq. (9), the measurable flux conservation. The dependence of the number of the neutral-current events on the distance is the character is t ic of the neutrino-antineutrino oscillations. This charac te r i s t i c is preserved in general , a s was shown in Eq. (10).

The observation of neutrino oscillations of both types I and I1 will be simplified if the neutrino- m a s s pat tern is that shown i n Fig. 2d. In this c a s e we could use the advantage Anz2,<< 4?nZif f o r any i # j and any k. F o r instance, f i r s t we should use a s m a l l source (such a s a reac tor o r acce le ra tor ) and short distance to m e a s u r e 4mZi, and find [compare the condition of Eq. (16)]

and

Then we use a l a rge source (such a s the Sun) and la rge distance to m e a s u r e and find [compare the condition of Eq. (12)] in the s implest c a s e Eqs. (35) and (36)

and

Will nature repeat the s a m e pat tern that occur red

in the spectrum of an atom, then in the spectra of nuclei? We do not know.

V. MODELS WHICH GIVE DIFFERENT NEUTRINO-MASS PATTERNS

Now we a r e going t o give a brief discussion about the neutrino-mass pat tern in some grand unification models of weak, electromagnetic, and s trong interactions. The constituent models of leptons and quarks a r e proliferating r e ~ e n t l y , ' ~ " ~ We will d i scuss the neutrino-mass pattern in some of these models also.

In the original SU(5) model of Georgi-Glashow.lg where the fe rmions a r e in 5 and lo* representations, only left-handed neutrinos, which a r e in 5-plets, and Higgs 24- and 5-plets a r e in- volved. Incidentally, the re is a globalB - L c o n ~ e r v a t i o n ~ ~ (B is the baryon number, L is the lepton number) which makes neutrino mass - l ess . However, i f we have a Higgs 10- o r 15- plet, B - L will be brokenz0 and a Majorana neu- t r ino f o r the left-handed neutrinos will exist. In Fig. 3 we show a two-loop diagram which in- volves the Higgs 10-plet and contributes a Maj- orana m a s s to the neutrino

where Mlo i s the m a s s of the 10-plet. h and P a r e the dimensional and dimensionless coupling con- s tants , respectively. G i (i = 1 , 2 , 3 ) a r e the Yuka- wa couplings between 10-plet Higgs bosons and 5- plet fermions. There is no information about the o r d e r of magnitude of those parameters . If we take a plausible position, and put MI,- V , X - a2V, p - a2, GI-G2-G3- sCtnF/v where cr =$/4a, m, is the typical m a s s of fermions (about 1 GeV), and s, i s the Cabibbo angle, we get m u - eV. We notice h e r e that a Majorana m a s s t e r m comes f r o m the violation of B - L conserva t~on .

The Weinberg-Salam model may suit m a s s pat tern (b) with doubly degenerate l ines if the re a r e Yukawa couplings between the right-handed singlet neutrinos and the left-handed doublets and if the Lagrangian respec t s lepton-number conservation. The SU(2) instanton may break

/ I \

vi GI b c ) ~ G 2 TL G 3 lvcIR

FIG. 3 . A two-loop diagram which contributes a Maj- orana mass to the neutrino in the SU(5) model with 10-plet Higgs particles. See caption of Fig. 4(b).

23 - M A S S E S O F N E U T R I N O S I N U N I F I C A T I O N G A U G E T H E O R I E S . . .

lepton-number conservation and baryon-number conservation simultaneously, so it cannot split the double line by a very small space.21

The SO(10) model,z2 where al l fermions a r e in 16 spinoral representations can, in principle suit any mass pattern one wants, especially mass patterns (c) and (d). Most authors doing S q 1 0 ) model favor pattern (c). Giving to one neutral color-singlet component of a Higgs 126- plet [which transforms a s (1,3, lo*) under the subgroup SU(2), x SU(2), x SU(4) and i s responsible for the Majorana mass of the right-handed neutrino], a huge vacuum expectation value (VEV) of 1014 GeV o r so, whereas the other neutral color- singlet component of 126 [which transforms a s (3,1,10) under the subgroup SU(2), x SU(2), x SU(4) and i s responsible fo r the Majorana mass of the left-handed neutrino] a zero vacuum expectation value, we break the SO(10) down to SU(3) x SU(2) x U(1). Then we use a Higgs 10- plet to break SU(2) x U(l) down to U(1) a t 300 GeV and give the fermions Dirac masses . This i s the simplest way to make m a s s pattern (c) in the SO(10) model. We notice that the components which may get VEV in (1,3, lo*) and (3,1,10) of 126 have B - L =*2, respectively. Thus giving any one of them a nonzero VEV means to break B - L gauge symmetry.

However, we have two methods to arrange mass pattern (d) in the S0(10) model.14 Method A. We make the model by the following three steps: (1) Give the neutral color-singlet components of the Higgs 45-plet [which transform a s (1,1,15) and (1,3,1) under the subgroup] different VEV's to break SO(10) down to SU(3) x SU(2),x U(1), x U(1). (2) Give the neutral components of the Higgs 10- plet and 126-plet [which transforms a s (2 ,2 ,1) and (2,2,15), respectively, under the subgroup S U ( ~ ) , x SU(2), x SU(4)] VEV's close t o 300 GeV and make the Dirac mass of neutrino zeroa t t r ee level. Here we take the extreme smallness of the neutrino m a s s a s an ansatz for the assignment of the VEV's and the Yukawa couplings. Because there a r e four VEV's (two of them a r e relevant t o neutrino masses) and two Yukawa coupling constants, we can definitely make such arrangement. (3) Give the neutral components in (2,2,10) and (2,2, lo*) of the Higgs 210-plet, which have nonzero B - L , VEV's close t o 300 GeV.

In such a model, the Dirac m a s s of the neutrino is given a t one-loop level in Fig. 4(a) and i s nearly equal

where m, i s the m a s s of the right-handed W boson. The Majorana massz3 of the neutrino i s given a t

FIG. 4. The neutrino mass in the SO(10) model. (a) The one-loop diagram that gives neutrino a Dirac mass through the mixing between right- and left-handed gauge bosons. (b) Two-loop diagram that gives neutrino a Majorana mass . The waved lines a re gauge bosons. The dotted lines a r e Higgs particles with the dimensions of their SO(10) representations along the lines. The nota- tion x means VEV. The numbers in parentheses show the transformation properties of the components under the subgroup SU(2), x SU(2), x SO(6) which develop VEV. Q, i s the linear combination of Higgs 10- and 126-plet which gives neutrino zero mass at tree level. Both dia- grams (a) and @) a r e invergent.

two-loop level by Fig. 4(b); the result i s extremely small:

where 8, and pz a r e dimensionless coupling constants in the Higgs potential, m, i s a typical up- quark mass. For the T neutrino, Am,,- lo-' eV.

Method B. The f i r s t two steps a r e the same a s those in Method A, but the third becomes (3) give the two neutral components of Higgs 16-plet, which have nonzero B - L also, VEV's close to 300 GeV. The Dirac mass and the Majorana mass of the neutrino a r e shown in Figs. 5(a) and 5(b), respectively, and we have

The neutrino-mass pattern in the E, modelz4 i s quite s imi lar to that in the SO(10) model. Es- pecially, we can also get mass pattern (d) in the simplest E, model. In the simplest E, model, al l fermions a r e in 27 and Higgs part icles a r e in 27 x 27=(27* + 351), +351, only. To make m a s s pattern (d), the principles of the model

D A N - D I W U 23 -

110 (or 126) I

FIG. 5. Neutrino mass in the SO(10) model. (a) Dirac mass; (b) Majorana mass. See the caption of Fig. 4(b).

building a r e the following: (1) The neutrino m a s s at the t r e e level equals ze ro , whether it is the Dirac m a s s o r the Majorana m a s s . (2) All the neutral components which t rans form under the subgroup SU(2), x SU(2), x SU(3), a s the representat ion (2 ,2 , I ) , ( 1 , 2 , I ) , o r (2,1,1) get the vacuum expectation values n e a r 300 GeV. The neutrino will get a Dirac m a s s a t the one-loop level s i m i l a r t o Eq. (42) and a Majorana m a s s a t the two-loop level s i m i l a r to Eq. (44).

We notice that in both the SO(10) model and the E, model, a Majorana m a s s of the neutrino (whether left- o r right-handed) a l so comes from the violation of B - L gauge symmetry.

M a s s and mixing pat terns among neutrinos in different famil ies could be discussed using the horizontal-symmetry (discretez5 o r continuous26 groups) o r cer tain grand unification models.27 Some resu l t s about neutrino m a s s e s and mixings have been found in th i s area.28 We a r e not going t o give detai ls here . What we want t o say is that in any huge grand unification models, the interaction among different fami l ies is gauged and unified with the interact ions within a family. At the s a m e t ime , the original family s t ruc ture , which is our f i r s t insight to the nature, is de- s t royed and reorganized according to the way it f i l l s in the representat ion of the huge groups.

What do neutrino m a s s e s look like in the constituent models? This may be another point the constituent r n ~ d e l s ' ~ ~ ~ ~ can touch on a t the i r pr i - m a r y stage of development. Let u s take the

Terazawa modelL6 and the H a r a r i modelL7as two examples.

In the Terazawa model, l6 the constituents a r e th ree kinds of fermions, a l l with spin $. They a r e the c a r r i e r s of weak-SU(2) quantum number w i (i = 1 , 2 ) , the c a r r i e r s of color-SU(4) quantum number C, (a = 0 , 1 , 2 , 3 ) , and the c a r r i e r s of generation numbers h, @ = 1 , 2 , 3 , ... ). Leptons and quarks a r e made up by picking up one f rom each kind and combining them together a s

In the H a r a r i model,17 there a r e only two spinor constituents, T and V , with the e lec t r ic charges -+ and 0, respectively. The construction of the leptons and quarks is given in Eq. (46):

The next generations a r e some excitation of the f i r s t generation. Both of these models give very interest ing resu l t s in a very s imple way. F o r instance, in the H a r a r i model the proton decay is just a rearrangement of the constituents

Although these models suffer f rom some diffi- cul t ies (e.g., the H a r a r i model has a s tat is t ical problem), the constituent models a r e at t ract ive and interesting both f rom the viewpoint of physics and philosophy. If we take them seriously, and go ahead a s f a r a s possible, we would say that a l l these model^'^-^* give neutrinos m a s s pat tern (d) in Sec. 111, probably s i m i l a r to what is shown in Fig. 2(d) with fine s t ructures . The reasons a r e the following: F i r s t , they have right-handed neutrinos in the i r models because the neutral constituents have t o have two chiral i t ies t o get cor rec t quantum numbers f o r the quarks; second, i f we assume that the main contribution t o the m a s s of the composite part ic le is f rom i t s inner interactions, then the Di rac m a s s of a neutrino must be much l a r g e r than the Majorana m a s s of the neutrino.

Some constituent model^^'*^^ s e e par t i c les grouped in famil ies a s a basic fact and s e e the interaction among famil ies a s leaking interactions. So in the zeroth o r d e r of the m a s s e s of

23 - M A S S E S O F N E U T R I N O S I N U N I F I C A T I O N G A U G E T H E O R I E S . . . 2045

the fe rmions (leptons and quarks ) , they have a t l eas t SO(10) symmetry? ' + CROSSED

VI. THE IMPLICATIONS OF NEUTRINO OSCILLATIONS

A complete neutrino-oscillation experiment may give us the information about (1) the m a s s pattern of the neutrinos, (2) the mixing m a t r i x among dif - ferent fami l ies , and (3) the mixing m a t r i x between the left-handed neutrinos and the right-handed anti-

>, neutr inos if the right-handed neutrinos exist. FIG. 6. P- ey process when masses of neutrinos a re

However, i f we find only neutrino-neutrino zero. oscillations (type I) , we can say nothing definite about the right-handed neutrinos because there a r e two possibilities: (i) There a r e no r ight- handed neutrinos, o r (ii) the re a r e right -handed neutrinos, but the AL = 2 interactions a r e too weak a s to give almost m a s s pat tern (b) o r the right-handed neutrinos a r e too heavy to give m a s s pattern (c). If we a l so find neutrino-antineutrino oscillations (both type I and type 11), then we can say definitely that t h e r e a r e right-handed neutr i - nos and there a r e AL = 2 interactions.

To have an insight on the m a s s pattern of neu- t r inos and the interact ions among neutrinos, we would like t o cite 't Hooft's wordsz9 here : "At any energy sca le p , a physical parameter o r s e t of physical p a r a m e t e r s a i ( p ) is allowed t o be very s m a l l only i f the replacement a i ( p ) = O would increase the symmetry of the system." We notice that if the Majorana m a s s of the neu- t r ino equals ze ro , we will have B - L conservation and may be a m a s s l e s s B - L gauge boson. If a l l the m a s s e s of the neutrinos equal ze ro , o r the mixing mat r ix is tr ivial , then we will have the family-lepton-number conservation to a very high extent, for instance, the decay r a t e of p - e y is ( see Fig. 6)

where 8, is the Cabibbo-type angle in quark-lepton interaction^,^' m, i s the typical m a s s of f e r - mions, and M is the m a s s sca le of grand unification. The e x t r a (mF/M)4 factor comes f rom the Glashow -1liopoulos-Maiani m e ~ h a n i s m . ~ ~ Ex- perimentally, the m a s s e s of the left-handed neu- t r inos a r e very smal l , s o we do not need a mixing mat r ix of left-handed neutrinos to mee t the demands of a very good family -lepton-number -

conservation law. However, if we want to have a n extremely good

B -L conservation (much bet ter than B and L separa te conservations a t any energy sca les ) , we should give neutr inos (left- o r right-handed) very smal l Majorana m a s s e s , a s we did in the SU(5), S0(10) , and E, models ( s e e Figs. 4 and 5). F r o m the viewpoint of cosmology, t o get net s u r - vived baryon number in the universe, a B - LYL , where a, i s a constant, conservation law i s wanted a t the t empera ture 108-10' GeV.32 The models giving m a s s pattern in Fig. 2(d) with fine s t r u c - t u r e s satisfy th i s requirement automatically. The exciting low-energy physics of these models , which give neutr ino-mass pattern fine s t r u c t u r e s , is that we would expect two neutral gauge b o ~ o n s ~ ~ with m a s s e s a t nearly the s a m e energy level a s that of the w b o ~ o n . ~ In the near future, we wil l be able to answer whether o r not this i s t rue .

ACKNOWLEDGMENTS

The author wishes to thank S. Weinberg for giving inspirations to understand the problems ra i sed h e r e by his ta lks and discussions, and to thank N. Nakazawa, B. More l , G. Shore, E. Witten, N. Snyderman, Bam-bi Hu, Alan Guth, T. Veltman, and Y. S. T s a i for enjoyable and helpful discussions. This work w a s finished a t SLAC. The author a l so wishes t o thank S. Wein- berg and S. Drel l fo r making my stay a t SLAC possible and thank the SLAC people for the i r very w a r m hospitality. This work w a s supported in par t by the Department of Energy, contract No. DE-AC03-76SF00515, and by the National Science Foundation, Grant No. PHY77 -22864.

2046 D A N - D I W U 23 -

*On leave f rom the Institute of High Energy Phys ics , P.O. Box 918, Beijing, China.

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'cTL. Cowan, F . Reines, F . B . Har r i son , H. W. Kruse , and A. D. McGuire, Science 124, 103 (1956).

3 ~ h e upper l imits of neutrino m a s s obtained by measurement of decays of charged par t ic les ( spec t rometer experiments) a r e as follows: m,<60 e V in P decay of nucleus, m, < 0.5 MeV in p decay o r rr decay. nz, < 250 MeV in T decay. This means the m a s s e s of the neu- t r inos a r e s o s m a l l a s to escape f r o m detection by spec t rometers . However, a /3 spec t rometer using H~ decay a s the source got a positive resu l t

See: V. A. Lyubimov, E . G. Novikov, V. Z. Nozik, E . F . Tretyakov, and V. S. Kosik, Report No. ITEP-62 (unpublished). Of course , this i s a big progress . But a s an electron neutrino may not be a m a s s eigenstate, the resu l t suffers the ambiguity and the value may be some weighted average value of a few m a s s e s . See the text.

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2 5 ~ i s c r e t e s y m m e t r y among fami l ies , s e e Dan-di Wu, High Energy P h y s . Nucl. P h y s . (Beijing) 3, 681 (1979); Phys . Let t . E, 364 (1979); S. Pakvasa and H. Suga- w a r a , P h y s . Let t . E, 61 (1978); Zhong-qi M a , Pe i - you Xue, Zhong-wu Yue, and Xian-jian Zhou, i n P r o - ceedings o f the Guangzhou Conference on Theoretical Part icle Phys ics , Guangzhou, China, 1980 (Academics

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"see fo r example Zhong-qi M a , Tung-sheng T u , Pei-you Xue , and Xian-jian Zhou, Repor t s Nos. BIHEP-TH-1 to 3 Beijing 1980 (unpublished), H. Georg i , Nucl. P h y s . B156, 126 (1979); P . Frampton , Phys . Let t . E, 352 - (1980); E, 299 (1979); P . Frampton and S. Nandi, Phys . Rev. Let t . 43, 1460 (1980); R . Ramond, Repor t No. CALT-68-709, 1979 (unpublished).

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"G. 't Hooft, l ec ture given a t C a r g e s e Summer Inst i tute , Utrecht r e p o r t , 1979 (unpublished).

3 0 ~ h e n this paper was typed, we learned f r o m H a r a r i about this angle; s e e the Proceed ings of the SLAC Summer Institute.

3 ' ~ e e Ref. 9 and S. Glashow and S. Weinberg, P h y s . Rev. I3 15, 1958 (1977); B . L e e and R. Shrock, Phys . Rev. D 1 6 , 1444 (1977).

3 2 ~ . W e i n b e r g , ta lk a t the F i r s t Workshop on Grand Uni- fication, New Hampshi re , Harvard Report No. HUTP 80/A023, 1980 (unpublished).

3 3 ~ h o n g Shou Gao and Dan-di Wu, Phys. Rev . D (to b e published).

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Masses of neutrinos in unification gauge theories and neutrino-antineutrino oscillations