Neutrino masses and the scale of B-L violation

Nuclear Physics B187 (1981) 343-375 O North-Holland Publishing Company

NEUTRINO

MASSES AND THE SCALE OF B-L

VIOLATION

C. W E T T E R I C H

Fakultiit ffir Physik der Universitiit Freiburg, Hermann-Herder-Str. 3, 7800 Freiburg, B R D

Received 23 February 1981 A systematic study of neutrino masses in models with local B - L symmetry is presented. The observed SU(4)c violation in fermion masses, which is necessary to explain why me is not equal md, is related to the scale of B - L violation. An alternative approach uses renormalization group methods to determine this scale. The heaviest neutrino mass is predicted to be 0.1-50 eV in the case of four fermion generations. Two different generation patterns for neutrino masses are found, one predicting large mixing between ve and u, (and eventually v~) and the other predicting leptonic mixing angles of the same order as quark mixing angles.

1. Introduction It is k n o w n from /3-decay e x p e r i m e n t s that the mass of a n y n e u t r i n o with s u b s t a n t i a l weak c o u p l i n g to the e l e c t r o n is smaller t h a n a b o u t 45 eV. F o r n e u t r i n o s n o t c o u p l i n g to the e l e c t r o n (or with very small couplings) e x p e r i m e n t a l u p p e r b o u n d s o n the n e u t r i n o masses are less s t r i n g e n t , b u t a cosmological a r g u m e n t indicates that the s u m of light n e u t r i n o masses should n o t exceed - 5 0 eV. O t h e r w i s e the total mass of the u n i v e r s e w o u l d be too large. If the n e u t r i n o masses are n o t too far from this u p p e r b o u n d (my/> 0.5 eV) n e u t r i n o oscillations can in principle be o b s e r v e d by r e a c t o r or accelerator e x p e r i m e n t s [1]. W h y are n e u t r i n o masses m u c h smaller t h a n typical q u a r k or charged l e p t o n masses? In m o d e l s with r i g h t - h a n d e d n e u t r i n o s , D i r a c mass t e r m s --rn~D)~TRULarise n a t u r a l l y in the s a m e o r d e r as q u a r k mass t e r m s since n o s y m m e t r y forbids them. T h e r e f o r e , a n a t u r a l e x p l a n a t i o n of small n e u t r i n o masses has b e e n c o n s i d e r e d for some time as a p r o b l e m in unified gauge theories with r i g h t - h a n d e d n e u t r i n o s ( n e u t r i n o mass p r o b l e m [2]). A first step towards the s o l u t i o n of this p r o b l e m was the o b s e r v a t i o n [3] that the r i g h t - h a n d e d n e u t r i n o s have a s u p e r h e a v y M a j o r a n a 1 mass --~AvavR. This mass t e r m is a singlet u n d e r SU(3)cX SU(2)L × U ( l ) ~ H a n d it violates B - L s y m m e t r y . T h u s it m u s t be p r o p o r t i o n a l to a s u p e r h e a v y mass scale M a L w h e r e B - L s y m m e t r y is b r o k e n . T h e most g e n e r a l n e u t r i n o mass t e r m (neglecting g e n e r a t i o n s for a m o m e n t ) is given by ~L#~

=

1-

-¢

A 343

B

c

+h.c.,

(1.1)

344

C. Wetterich / Neutrino masses

with mass e i g e n v a l u e s ~ A a n d C - B Z / A . G e l l - M a n n et al. [3] p u t C = 0 a n d f o u n d that the light n e u t r i n o mass is supressed by a factor B / A - - M L / M B L compared with a c o r r e s p o n d i n g q u a r k mass. (ME is the mass of the WL boson.) E v e n if this a p p r o a c h has given the i m p o r t a n t insight that the large scale M B L m a y be r e s p o n s i b l e for the small n e u t r i n o masses, the p r o b l e m of n a t u r a l n e s s has r e m a i n e d unsolved. It has only b e e n r e m o v e d since C = 0 is as u n n a t u r a l as B = 0. A g a i n no s y m m e t r y forbids C = 0. Especially the v a c u u m e x p e c t a t i o n value (VEV) of a scalar field c o u p l i n g to ULUL a n d thus c o n t r i b u t i n g to C does not vanish*. T w o solutions for the p r o b l e m of n a t u r a l n e s s have b e e n p r o p o s e d : W i t t e n [4] p r o p o s e d a m o d e l w h e r e no scalars couple to URUR a n d ULUL. T h e M a j o r a n a mass A is t h e n i n d u c e d by radiative corrections. As we will see later, this possibility of radiatively i 0 d u c e d r i g h t - h a n d e d M a j o r a n a masses seems not to be realized. Lowest o r d e r c o n t r i b u t i o n s are always d o m i n a n t for the scale M B - L in the r a n g e of interest. In a m o r e g e n e r a l a p p r o a c h it has b e e n shown [5] that small l e f t - h a n d e d n e u t r i n o masses are an intrinsic f e a t u r e of unified gauge theories if the scale of B - L violation is m u c h larger t h a n ML. In fact, C is always of the o r d e r C <~M L2/ M B - L even in the p r e s e n c e of arbitrary scalar fields**. T h u s n e u t r i n o masses m u c h smaller t h a n q u a r k masses are an i m p o r t a n t prediction of unified gauge theories ( U G T ) . Inversely, n e u t r i n o masses in the eV region can only be n a t u r a l l y e x p l a i n e d within U G T s : A large mass ratio MB L/ML ~ 10 9 which is characteristic for U G T s (gauge hierarchy) is necessary for the s u p p r e s s i o n of n e u t r i n o masses c o m p a r e d with q u a r k masses. T h u s the d e t e c t i o n of n e u t r i n o masses b e t w e e n 10 2 a n d 5 0 eV w o u l d be an i m p o r t a n t success of U G T s which can be c o m p a r e d with the possible d e t e c t i o n of p r o t o n decay. W h e r e a s the p r o t o n decay e x p e r i m e n t s test the scale of b a r y o n n u m b e r violation, the n e u t r i n o masses indicate the scale of l e p t o n n u m b e r violation! T h e simplest U G T b a s e d on SU(5) [7] predicts MB L ~ > M x ~ 1 0 1 5 G e V a n d therefore n e u t r i n o masses*** smaller t h a n a b o u t 10 -1 eW. N e u t r i n o masses in the e V region f a v o u r U G T s with local B - L s y m m e t r y like SO(10) [8, 9] or E6 [10, 6]. In these m o d e l s M B - L can be smaller t h a n the unification scale M x a n d for L ~ 1013 G e V n e u t r i n o masses are expected to be in the e V region. C a n we predict the n e u t r i n o masses in these m o d e l s ? In this p a p e r we give a systematic discussion of n e u t r i n o masses in m o d e l s c o n t a i n i n g S U ( 4 L × SU(2)L × SU(2)R [11]

MB

* SU(2)L symmetry requires C ~
C. Wetterich / Neutrino masses

345

as a subgroup. (These models include SO(10) and E6.) Two independent low energy p h e n o m e n a are used to determine M B - L and to predict the neutrino masses: (i) The SU(4)~ relation m r = md (at the unification scale) is contradicted by experiment. It has been shown [9] that the SU(4)c violating contributions to charged fermion masses are of the order (MB L / M x ) 2 ML. This contribution has to be at least in the MeV range and a lower bound on MB L is derived. The successful relation m~ = mb (at the unification scale) suggests that M B - L is near this lower bound. (ii) Because of left-right (L-R) symmetry the gauge group SU(4)cxSU(2)L× SU(2)R has only two independent gauge couplings. This partial unification permits a renormalization group (RG) determination of MB L, starting from the low energy couplings O~em(ML), Ots(ML) and sin 2 tgw(mL). The two methods to determine M B - L are independent and we first discuss the constraints on neutrino masses from both observations separately. We find that both methods are in good agreement for sin 2 0 w ( M L ) ~ 0 . 2 3 and predict M B _ L ~ 1 0 1 3 - 1 0 1 4 G e V . Finally we combine the predictions of both methods. For four fermion generations we find that the heaviest neutrino mass is between 0.1 and 50 eV. We start in sect. 2 with a description of the relevant mass scales in UGTs containing SU(4)c × SU(2)L × SU(2)R. A general discussion of the possible contributions to left- and right-handed neutrino masses is given in sects. 3, 4. We classify the possible generation patterns for the light neutrino masses: either the ratio of neutrino masses of different generations is related to quark mass ratios and the leptonic mixing angles are of the same order as the quark mixing angles, or the neutrino masses show no strong generation dependence and the leptonic mixing angles are large. In sect. 5 the possible SU(4)c violating contributions to the masses of charged fermions (leading to me ~ rod) are investigated. This information is used in sect. 6 to classify different viable SO(10) models and to predict the neutrino masses within these models. In sect. 7 renormalization group methods are used to predict MB L and the scale of light neutrino masses. For given values of c~(ML) and sin 20w(ML) M B - L can be predicted very accurately. Even including uncertainties from scalar fields MB L can be determined within an error of at most 30%. A combination of the constraints from mr # md and from the R G method is used for a determination of the Yukawa couplings of the neutrinos and therefore to narrow down the uncertainty in the neutrino masses. We conclude in sect. 8 with a short discussion of the consequences of our prediction for ongoing and future neutrino oscillation experiments.

2. Symmetry breaking scales In the next sections we describe the possible contributions to neutrino masses in a general approach. Then information from SU(4)c × SU(2)L × SU(2)R ----K is incorporated. In order to establish our scenario we give in this section a brief description

346

C. Wetterich / Neutrino masses

of the possible scales of spontaneous symmetry breakdown within a unified gauge group G containing K as a subgroup. Not all of the scales discussed here need to be different. In the last sections of this p a p e r we will concentrate on models with three scales Mx, MB-L and ML. The unification scale (q~) is given by the vacuum expectation value (VEV) of some scalar field ¢G which breaks G spontaneously. This is the largest mass scale (except the Planck mass) in this model. Some of the gauge bosons aquire a mass MG ~ g(q~G) and generic scalar masses are all of the order M~. However, some scalar fields with mass smaller M e are needed for further symmetry breakdown to S U ( 3 ) c x U ( 1 ) .... No natural explanation for these small scalar masses has been found so far (gauge hierarchy problem [12]). Throughout this p a p e r we suppose that o n l y the scalars needed to induce spontaneous symmetry breaking at a scale smaller than (~0~) can have masses much smaller M c , whereas all the other scalars in the model have masses of the order M e . In general G could break to K in several steps. The only scale relevant for our discussion is (~px), where K gets unified in a simple subgroup of G. For the smallest possible group G = SO(10) (q~x) and ( ~ ) are identical, but a larger group like E6 ~'~) ~'px) may involve distinct scales (~Ox)< (qgG) (for example: E6 • S O ( 1 0 ) x U(1) , K). At this scale the gauge bosons mediating proton decay get a mass M x =

g<¢,,>. The breaking K--* SU(3)c x SU(2)L x U(1)--- H can be described by three mass scales [13]: The scale (q~c) where SU(4)~ breaks to SU(3)c, the scale (¢R) relevant for the breaking of left-right (L-R) symmetry and the scale (q~B-L) where local B - L symmetry is spontaneously broken. The masses of the corresponding gauge bosons are denoted by M~ (for the color triplet belonging to the SU(4)~ bosons), MR (for the charged W~ bosons of SU(2)R) and MB-L (for the remaining neutral boson not contained in H). We note that B - L breaking implies the breaking of SU(4)c and SU(2)R since it lowers the rank of the group and the remaining U(1) group in H is not contained in SU(4)c or SU(2)R. Thus one has

(~B-L) ~ ( ~ r ) ,

(2.1)

(q~n-L) ~<(~0c).

(2.2)

If SU(4)c is broken by a scalar field in a complex representation (by this we mean that the representation is not self-conjugate) one has (q~B-L) = (q~c). If the same field is also responsible for the breaking of L - R symmetry the three scales (q~), (~0r) and (q~B-L) are all equal. This interesting breaking pattern of K can be realized by a (4, 1, 2) scalar multiplet (belonging to 16 or 144 of SO(10)) or by a (1--0, 1, 3) multiplet (belonging to 126 of SO(10)). (The notation (A, B, C) denotes the transformation properties under SU(4)~, SU(2)L and SU(2)r respectively.) In a final step H is broken to SU(3)~ x U(1)em at the scale (q~L)= 174.1 GeV. The mass of the charged WL bosons is ML = x/~g(~PL). The ensemble of these mass scales describes the general scenario for our discussion of neutrino masses.

C. W e t t e r i c h / N e u t r i n o m a s s e s

347

3. Masses of left-handed neutrinos In this section we discuss the different contributions to the masses of the lefthanded neutrinos in the language of an effective low energy theory [14]. Physics at present energies (except gravitation) are described by a gauge theory based on SU(3)cx SU(2)LX U(1) with NG generations of 15 left-handed fermions and one scalar doublet CpL. All the other particles existing in a more complete theory have large masses M >>ML. Within the effective theory the interactions mediated by these heavy particles appear as nonrenormalizable interactions suppressed by one or more powers M L / M . In this effective theory neutrinos have only left-handed helicity states. Therefore the only possible mass term is a Majorana mass ~1m VL/,'Lb'L. The operator b ' L P L belongs to a S U ( 2 ) L triplet and a possible Majorana mass involves the doublet ~L twice. Since Majorana masses do not conserve B - L symmetry the natural mass scale suppressing this interaction is M s L. The VL mass matrix is then given by or ii

( M " L )'i = M B - L

(~L)2

"

(3.1)

In unified gauge theories, these effective interactions contributing to My1, are mediated by the exchange of heavy particles and the coefficients O'zi are calculable as functions of the parameters of the UGT. Two types of tree diagrams can contribute to an effective i n t e r a c t i o n b'Lb'L@ 2 " heavy fermion exchange (fig. la) and heavy scalar exchange (fig. lb). Since in some cases M S - L may be not very far from the Planck mass we also include gravitational interactions ("black hole exchange", fig. lc). (Predictions of unified gauge theories alone are only reliable if the contributions from gravity are small). We discuss the different contributions to M vL separately (MvL -- -- M ' i At(b) --(c) ~ ' '(Pa L) "ai-lVl VL + M v L ) "

3.1. HEAVY FERMION EXCHANGE

The heavy fermions in fig. l a are the right-handed neutrinos vr~ and the Yukawa coupling f for the tTRVL~Lvertex is related to the neutrino Dirac mass:

( M I D ) a- = ~/~fij(q~u)

(3.2)

The contribution of fig. l a corresponds to the Dirac mass contribution described in ref. [3]. It is given by M(a) ~

T

-1

"'' "L )ii = x / ~ f i k ( ~ L ) ( M . R )kl~/~fit(~e) = M~ D) ( M . . ) - ~ ( M ~ ) )v /~1- (D)/-~ t~- -- 1 F ) T ( M (D) ~T = -........ -- .......

(3.3)

where M y . is the mass matrix for the right-handed neutrinos. We choose a basis

348

C. Wetterich / Neutrino masses

\ ,x- q)L" /

\

/

\ \

/ VR

/

Vn

Y I

I~p 110.3.1) I

\

/

2< Fig. 1. C o n t r i b u t i o n s to l e f t - h a n d e d n e u t r i n o masses.

where M~ TM is diagonal. MvR an orthogonal transformation The Dirac mass M ~ ) can generations. If <~DL) belongs unification scale)

is a symmetric matrix which can be diagonalized by D, AT/v,~ = D T M ~ D = m iv~6ii. be fairly reliably estimated at least for the higher to an SU(4)c singlet we have the relation (at the M ~ ) = Mu,

(3.4)

where M , is the mass matrix for the u-type quarks. This relation has to be modified for the first generation, but it should hold at least for the third (and fourth) generation (see sect. 5). The mass of the top quark and the masses of a possible fourth fermion

349

C. Wetterich / Neutrino masses

generation have been estimated [9] in a model where (¢R)= finds (see sect. 7), including renormalization effects:

(~L)=

(~B-L).

rn~D) = 7 - 9 G e V , (D)

my, =68-139

One

(3.5)

GeV.

(3.6)

-(a)

The main uncertainty in the estimate of M ~ L c o m e s from M ~ . which may be parametrized by M ~ . = 2/~(~oB L).

(3.7)

The generation pattern and the mixings in the neutrino sector depend strongly on /~. If (¢B-L) belongs to a (10, 1, 3) representation of K the /~i are just the Yukawa couplings h~j between ¢B L and yrs. In more general cases/~ may depend on other Yukawa couplings and may even be generated by radiative corrections. For a discussion of the generation dependence of neutrino masses we distinguish three possible scenarios for the/~*. (1) First consider the case/~i = cf~i. This is realized if there is only one independent Yukawa coupling between scalars and fermions (see for example ref. [4]). Then we have D = 1 and M(a)= 1 M(y~ (~£~L) v~ 2~/2c (q~B L)"

(3.8)

In this case the generation pattern of neutrino masses is proportional to the u-type quark masses at least for the higher generations. Furthermore, the generation mixing matrix in the quark and in the lepton sector is the same (except for possible corrections to the relations M ~ ~ = Mu and ME = MD for the first two generations). (2) Next consider the case that/~ is generation independent: /~i =/~S~j. Then the VR masses are degenerate (M~,~ = m ~R6i/). One has D = 1 and

M(a) ~L =

1

}~,/(D) ( / ~ ( D )

...........

IT

(3.9)

mvR

The ratios of neutrino masses are now given by the squared ratios of u-type quark masses of the corresponding generation. Again, we have the same generation mixing in the quark and in the lepton sector. We may weaken the condition/~ii =/~8# and admit different eigenvalues/~ of/~ which are not separated by orders of magnitude. (Let us say a factor 10 between the lowest and highest/~/.) Then ~/ ~vt ( a )R is in general not diagonal in the basis where Mu is diagonal. The leptonic mixing matrix UL is given by UL = Uq Ua, where Uq is the quark mixing matrix (Kobayashi-Maskawa (a)

* A general condition for the eigenvalues of MIva~, MvR and M ~ t denoted by ( M . e )i, (MvR)i and (M~))i follows from eq. (3.3): NG

NG

NG

lq (M~!), II (Mo,), = II (M~'t,~ .

i--1

i--I

i--1

C. Wetterich/ Neutrino masses

350

matrix for three generations) and Ua diagonalizes M~a~). However, the non-diagonal -~f-(a) D D elements (....~)ii (i < j ) are suppressed by a factor (M~)ff(M~ )i compared with A4(a) D D ('""L)JJ" [(M~ )i are the eigenvalues of M , .] Thus the mixing angles in Ua are of the order 10 -~ or smaller and the generation mixing in the leptonic sector is still (a) (a) of the same order as in the quark sector. Ratios of eigenvalues (M~)i/(M~,_ )i are again of the order (m u) i 2/ ( m .i) .2 (3) Finally we consider a strong generation dependence of /~ (the highest and lowest /~i are separated by several orders of magnitude) which is independent of the generation pattern of f in the sense that the matrices diagonalizing f and/~ are different (D has large non-diagonal elements). Suppose mi, v,<
= (My(D) )iDil 1 _ _ Djl ( M ~ ) )i

(3.10)

/~VR

Here we suppose that there is a substantial mixing of the third (fourth) generation 1 to the eigenstate of m.~. Introducing "weighted masses" 5)I~= (M(~TM )iDil one has ("t~/. ' ~ " (La ) ) ij --~

11 m vR

~Qi2Qi"

(3.11)

~[(a) The eigenvalues of /.._~ are zero except one which is (a)

(M~L)~o=

1

1 EM~.

(3.12)

m VR i

(This corresponds to the approximation m i ~ ( j > 1 ) ~ , see previous footnote.) -(a) The non-diagonal elements in M,L and in Ua are suppressed by factors /I~/i/)IT/i compared with corresponding diagonal elements as long as the/I)/i are well separated. Thus we expect the mixing angles in the leptonic sector again to be of the same order as in the quark sector. For the lower neutrino masses we repeat the same ]~,/(a) 2 -1 procedure for the contributions to ---.L of order (m.R) and so on. Typical ratios of neutrino masses are given by - -(a) ~ M~L,i

(~)g-

{ ( M ~ ' ) i ) 2 m yi .

[(M~)i'~ z

~ __-7?-Kf<<\~} I~(M, )i ] m ~

.

(3.13)

[Kf involves ratios of mixing angles (Kf > 1).] The generation dependence of neutrino masses is even stronger than quadratic in the corresponding quark mass ratios. ]~[(a) To conclude we list the general features for the Dirac mass contribution -.-.L : The largest neutrino mass is proportional to the square of the largest Dirac mass: 2 " (M~a)~)No=( m(n) , )No/ho(~OB c),

where /~o is a typical Yukawa coupling which depends on the details of/~.

(3.14)

C. Wetterich/ Neutrino masses

351

Smaller neutrino masses are suppressed by at least one power of quark mass ratios. For /~ independent of f, typical suppression factors involve two powers of quark mass ratios and may even be stronger. The mixing angles in the teptonic sector are of the same order as in the quark sector. A situation [1] with large mixing between ve and vz but small mixing between ve and v,, cannot be explained if neutrino masses are induced by heavy fermion exchange. 3.2. HEAVY SCALAR EXCHANGE A second possible source for the VL masses is the exchange of a heavy neutral scalar field q~, belonging to the (10, 3, 1) representation of K (fig. lb). This corresponds to the induced V E V of SU(2)I_ triplet fields as discussed in ref. [5]. The Yukawa coupling in hiil.'Lil.'Li~t is related by L-R symmetry to the Yukawa coupling of (10, 1, 3). (Within SO(10) both (10, 3, 1) and (10, 1, 3) belong to the complex 126 dimensional representation.) The contribution to VL masses from heavy scalar exchange is M ( bv L) =

2h (q~L)a/3M~-2

)

(3.15)

where /3 is the effective trilinear scalar coupling and Mt the mass of Ct. Since q~t carries two units of the B - L charge, one has /3 = ) t l ( ¢ z L),.

(3.16)

Here )tl is the coupling constant of the quartic scalar coupling involving (10, 3, 1) x (10, 1, 3) × (1, 2, 2) × (1, 2, 2) which is expected to be of the order g2 and (¢s L)t is the B - L violating V E V of (10, 1, 3). We distinguish two possible scenarios: (A) Local B - L symmetry and L-R symmetry is broken by the VEV of the (10, 1, 3) multiplet ((¢B-L) = (¢B-L)~ = (¢R)). Then L-R symmetry requires M~ = )t 2(q~B_L)2, where )t2 is again typically of order g2. Introducing Kb = )t 1/)t2 (we expect Kb to be of order one) one has M(b) = 2hKb(~OL)2(~OB_L)-I vL

(3.17)

(B) B - L symmetry is broken by the VEV of the SU(2)R doublet in (~,, 1, 2). Then (~PB-L)t is induced by scalar self-couplings involving (q~B L) twice and is therefore of the order (¢'n L)Z/(q~G) (see sect. 4). In this case Mt is given by the largest mass scale MG(M2t = A'2(¢G)2). Thus the contribution to/~(b) is suppressed by a factor ((~B-L)/(~G)) 3 compared with eq. (3.17). If M(b] is the leading contribution to M~L, the generation pattern of neutrino masses and the leptonic mixing angles depend entirely on h. In general, h is not related to f and the symmetric matrix h involves ~2NG(NG+ 1) supplementary free parameters of the model. In principle these can be determined by experiments testing neutrino masses and oscillations. There is no reason why f and h should

352

C. Wetterich / Neutrino masses

be diagonal in the same basis. Thus large generation mixing in the leptonic sector is expected in this case. 3.3. CONTRIBUTIONS FROM GRAVITATIONAL INTERACTIONS Neutrino mass contributions from gravity have been discussed in ref. [15] in the context of SU(5) with global B - L symmetry. Typical neutrino masses of the order (qOL)2/mp--~ 10 -6 e g are found. In models with local B - L symmetry the contribution from gravity to M , L is further suppressed since local symmetries are conserved by gravity. One finds -(c) • m v L = CG(qOL)2(qOB-L)2Mp 3 . (3.18) Thus gravitational contributions to M~,. can be completely neglected. To conclude this section, we classify the possible generation patterns for M~L A/f(a) and ...~L hal'(b) are present. for the most general case where both contributions *.-~L -/~[ (a) (M(b) ] . C o m p a r e the largest eigenvalues (...... )max and . . . . . . . . . . . M(a) ]

2

......... M ( bvl_ ) lmax ] --'*

fmax

1

]~0hmax4KbF"

(3.19)

For (q~_/_) = (q~s L)t one has F = 1 and the above ratio is independent of (~0B-L), whereas for B - L breaking by a SU(2)R doublet this factor is given by F = O((~0s L)3/(~G)3). Possible generation patterns depend on ratios of Yukawa douplings and on F. P a t t e r n A : Suppose that (~pB L) belongs to a SU(2)R triplet and the Yukawa couplings f and h are independent and both of the same order of magnitude. Except for a possible contribution to the heaviest neutrino mass the Dirac mass contribution can be neglected since it is suppressed by factors (m iu/m ~)2 for the lower generations. The generation pattern is therefore determined by h and we expect large leptonic mixing angles. If h is smaller than f only the neutrino masses of the lower generations / . At(b) are essentially determined oy Jvi ~L whereas the heaviest neutrino mass comes from ~L" In this case leptonic mixing angles are large between the first two (or three) M(a) generations, but there are only small mixings with the heaviest neutrino. P a t t e r n B : For (~B L) beonging to a SU(2)R doublet and (q~B-L)<< (q~G) the main ) For/~ and f independent we expect a strong contribution to M ~ . comes from M -.-( a~L" suppression of neutrino masses of the lower generations by factors roughly of order ( m ui / m u i) 2 . For four generations the constraint ( A/f . . -(a) ~ ) 4 < 50 eV gives approximate upper bounds for lower generation neutrino masses: MCa) ... VL)3~ 0.5 eV, -

Ca)

(MvL)2~5" (/~r( a ) + ' ' V L )1 ~

10 3eV,

(3.20)

10-6 e V .

(For three generations these bounds are enhanced by a factor 102.) Only for the

C. Wetterich / Neutrino masses

353

first generation contributions from M~b] may be relevant. Leptonic mixing angles are of the same order as in the quark sector (at least for mixings involving the third and fourth generation). Pattern C: The Yukawa coulings/~ and f are not independent (/~ii = cfii). In this case My,. is proportional to the u-quark mass matrix Mu except possible modifications for the first generation.

4. Masses of right-handed neutrinos In order to predict the vL masses we need more information on (~B-L) and on the mechanism generating VR masses. These estimates of (q~B L) and M . R are interesting in their own right since they determine the strength of interactions leading to B - L violation in proton decay or n-fi oscillations. The decay of righthanded neutrinos may be responsible for the baryon asymmetry in the universe. For our discussion of the possible mechanisms responsible for vn masses we consider an effective theory for the particles with mass smaller than (~PB-L). These particles include: The gauge bosons of the group which is broken at (~0B L); NG generations of 16 left-handed fermions (the 15 "light" fermions and the left-handed antineutrino v~) and the corresponding right-handed fields; the scalar multiplet ~oL with zero B - L charge ( Y B - L ) which is responsible for the breaking of SU(2)L; the scalar multiplet ~0~-L responsible for the breaking of local B - L symmetry. If we restrict ourself to SU(2)R doublets or triplets this field has YB L = 1 or 2 (Oem = I 3 L + I 3 R + I y B - L ) . (A) B - L

BREAKING BY SU(2)R TRIPLETS (YB L =2)

In this case ¢B~C has direct Yukawa couplings to the right-handed neutrinos (¢B-L belongs to the (10, 1, 3) representation of K). As discussed before, the contribution to M , ~ is 2h(q~B t). (B) B - L

BREAKING BY SU(2)a DOUBLETS (YB L =1)

Similar as for the left-handed neutrinos the coupling VRVR ~OB-L is now forbidden by B - L symmetry and M~,~ involves (~B-L) twice. ( ¢ B - L belongs to the (7~, 1, 2) 2 representation of K). The effective coupling VRVR q~B-L is induced either by loop effects or by the exchange of particles with masses of the order M x or MG (see fig. 2). Two-loop contributions have been considered in the context of a SO(10) model [4]. Their contribution to M ~ is given by 2

(4.1)

C. Wetterich / Neutrino masses

354

<~PB_L> / / / / /

\ \ < kl)B.L)'

\ \

\ N

NO

NO

"~ x

-r

J

I

I Lp 110.1.3 ) I



\

< k~B-L> / /

,,,

%

/

Fig. 2. Contributions to right-handed neutrino masses.

where H(Mi) depends on the masses of the particles exchanged in the loops and is typically of the order (¢B_L) -1. There is an extreme suppression factor f(a/~r) 2 (for the first generation this is less than 10-8!) for the right-handed neutrino masses. Thus tree contributions from the exchange of heavy particles (including gravity) may be much more important. (a) Heavy fermion exchange. The contribution to Mv~ shown in fig. 2a occurs in models with K-singlet fermions (for example the SO(10) singlet in the 27 representation of E6) and is given by 1~/(a)

-1 T ~,~ = k(¢B L)MNo k (~B t ) ,

(4.2)

C. Wetterich / Neutrino masses

355

where k is a Y u k a w a matrix. For an SO(10) singlet one has

MN,, = / ~ ( ¢ c ) ,

(4.3)

and for h, k and k all in the same order of m a g n i t u d e M (a)v. is suppressed by a factor (¢B-L)/(~G) c o m p a r e d with (A). (b) Heavy scalar exchange. If the model contains H-singlet scalars with both Yn-L = 1 and YB L = 2, the scalars with YB-L = 2 acquire an induced V E V even if the main breaking of B - L s y m m e t r y is d o n e by the scalars with YB-L = 1. T h e contribution of fig. 2b is given by M(b) " / ~2/M2 vR = h OR~PB-L~ / 2,

(4.4)

where/3R is the trilinear coupling b e t w e e n two scalars with YB-L = 1 and the scalar with Yn L = 2 and M2 is the mass of the YB-L = 2 scalar. In general one expects /3R = A3(~00) and M 2 = A4(q~o)2. M(B2 is suppressed by a factor (¢B-L)/(q~O) c o m p a r e d with (A). W e note that for (¢B-L)/(q~G)> 10 3, which seems to be necessary for reasons shown in the next sections, the t w o - l o o p contributions can be completely neglected c o m p a r e d with --(a) M~R and t~(b) ... ,~. (c) Contributions from gravity. In models without appropriate heavy fermions or h e a v y scalars the only tree contribution can c o m e from gravity. Since q~n-L has the same q u a n t u m n u m b e r s as fiR, no local s y m m e t r y can forbid the interaction URUR ~ - L and the contribution from gravitational interactions is M(C) v.

=

cc(¢~-L)2/Mp.

(4.5)

This contribution dominates the t w o - l o o p contribution even for the third generation as long as >t33

~ 10 -5 .

(4.6)

T w o loop contributions could be relevant only for ( ~ B - L ) < 1014 G e V . H o w e v e r , such small values of (~0B-L) would imply neutrino masses not compatible with the cosmological u p p e r b o u n d of 50 eV. Thus tree contributions are d o m i n a n t and loop contributions can be neglected for the estimate of neutrino masses (except for a possible contribution to the neutrino mass of the fourth generation). W e can use the u p p e r b o u n d for the light neutrino mass to derive a lower b o u n d for (~B-L) for the different cases described above. F r o m eq. (3.14) one finds (M(D) ) 2~vo ( ¢ B - L ) > 50 e V ' /~"

(4.7)

For case (A) /~o is a c o m b i n a t i o n of Y u k a w a couplings h~j. Validity of p e r t u r b a t i o n t h e o r y requires/~0 < 1. For four generations, we find (~0B-L) > 101~ G e V .

(4.8)

356

C. Wetterich/ Neutrino masses

In the case of three generations this b o u n d is lowered by a factor 10 -2. For case (B) the b o u n d is even higher. It is e n h a n c e d by a factor (~OG)/(q~B L) and Mp/(~B L) for cases B(a), B(b) and B(c), respectively. If only gravity contributes to M ~ one has ( ~ B - L ) > 1015 G e V ( N o = 4).

5. Fermion mass relations and the scale of SU(4)¢ violation So far we have only used the scale of s p o n t a n e o u s b r e a k d o w n of local B - L s y m m e t r y for our discussion of neutrino masses. In models containing K as a s u b g r o u p there is a n o t h e r scale which reflects itself at low energies: the scale (~c) of s p o n t a n e o u s b r e a k d o w n of SU(4)c. Since (q~c)/> (q~B L), restrictions on (~'c) can also be used to constrain (~0B-L) and therefore give information on neutrino masses, especially if SU(4)c is b r o k e n by a complex representation ((~0c) -- (~0B-L)). T h e SU(4)~ s y m m e t r y relies quarks and leptons and implies the mass relations: ME = MD and M ~ ~ = Mu at the scale Me. A n y deviation from these mass relations must be p r o p o r t i o n a l to an SU(4)¢ violating V E V . T h e successful relation mb(Mc) = m~(M~) suggests that (~L) belongs to a singlet of SU(4)~*. In an effective theory at the scale Mc based on K all SU(4)c violating mass terms are induced by interactions involving (~OL) and one or m o r e powers of (~c). These mass terms are suppressed by a factor ((q~c)/Mo) N where M0 corresponds to (~Px), (¢G) or Mp d e p e n d i n g on the full theory. This can explain why SU(4)c violating mass terms are sufficiently small not to disturb the relation mb(Mc) = m,(M¢). O n the other hand the relation ma(Mc) = me(Mc) seems not to be realized in nature. The r e q u i r e m e n t that SU(4)c violating contributions to fermion masses are at least about 1 M e V gives a lower b o u n d on (q~c) which d e p e n d s on the specific m e c h a n i s m of SU(4)c breaking. If we believe that the relation ms(Me)= m~,(Mc) holds within 10%, (~p¢) has to be near this lower bound. The most general SU(4)c violating Dirac mass term is represented in fig. 3. H e r e the box represents the exchange of particles (either tree or loop contributions)

.f'~L ( ~. 2.1) Fig. 3. General structure of SU(4)c violation in fermion masses. * Dirac mass terms for the light charged fermions transform as (1, 2, 2) or (15, 2, 2) with respect to K.

357

C. Wetterich / Neutrino masses

<%OL>11,2,2 )

< ~ c ="

I

-%

(/.,1,2) or (10.1,3)

(Z-.1.2) or ( 1-0,1, 3)

I q) (15,2.2) I I

)

~L(L.2.1}

I,~)R ( /*.1, 2 )

<~L>(1,2.2 }

< ~Cm>(~,1.2) <~PL>(1,2,2 }

\



/I

/ \

\

/

<'-Pc > (4.1, 2)

(Z',l, 2 )

I

I

g

i

'

.,p ( ~ 2 , 1 ) i

I

\/

i

/ \

/

\

I

I I

I (4,2.1)

{ 2",1, 2 )

\

I I (

I

(6 1.1)

(/..1.2)

(4.2.11

[1,2.21

< ~pL> \

/ \

/

¥

{Z.,1,2 )



/ \

/ \

/

Fig. 4. SU(4)c violating contributions to fermion masses.

which transform as an effective (15, 2, 2) representation. A represents one or more SU(4)c breaking VEVs. Since (~L)XA must contain (15,2,2), A transforms as (15, 1, 1), (15, 3, 1), (15, 1, 3) or (15, 3, 3). Consider the case where (~c) belongs to a complex representation (~0c--=q~B-L=(24, 1,2) or (19, 1,3)). Since A is real it involves at least (~c) and (~*) and the leading SU(4)c violating mass term is given by

~M

-

~

2

= -~,R ~-~-<~L>I<~B-L>I ~L + h.c.

The possible tree contributions to AM are shown in fig. 4.

(5.1)

358

C. Wetterich / Neutrino masses

5.1. SCALAR EXCHANGE T h e e x c h a n g e of scalar fields c o n t a i n e d in (15, 2, 2) c o n t r i b u t e s : A M ~ ' = - 3 A M ~ ) =/7[{q~B_L)21(q0L)AsM152 = /~Kc{q~L)(qOB-L)2 (¢c,)2 ,

(5.2)

w h e r e the m a s s of the (15, 2, 2) scalar is M25 .)t6(q~G)2 a n d Kc = A s / a 6 is e x p e c t e d to b e of o r d e r one. A r o u g h e s t i m a t e , for A M D ~ > 0 . 5 M e V a n d k T < l gives (~0B-L)/(q~G) ~> 10 -3. T h e l o w e r limit on p r o t o n l i f e - t i m e r e q u i r e s ( ~ ) ~> 1015 G e V and therefore {qgB--L) ) 10 12 G e V . (5.3) =

This b o u n d is h i g h e r t h a n the b o u n d (4.8) d e r i v e d f r o m light n e u t r i n o masses. W i t h i n S O ( 1 0 ) t h e (15, 2, 2) field b e l o n g s to 126 o r 120. F o r (15, 2, 2) c 126 o n e has /7 = ~/rh a n d thus a r e l a t i o n b e t w e e n A M (a), M(b~) a n d M ~ since t h e y are all p r o p o r t i o n a l to h. This will allow us to m a k e p r e d i c t i o n s on n e u t r i n o m a s s e s which we discuss in sect. 6. F o r ( 1 5 , 2 , 2 ) c 1 2 0 t h e Y u k a w a couplings /7 a n d h a r e i n d e p e n d e n t . In this case /7 is an a n t i s y m m e t r i c m a t r i x a n d c o n t r i b u t e s o n l y to n o n - d i a g o n a l f e r m i o n m a s s m a t r i x e l e m e n t s . A s u b s t a n t i a l c o n t r i b u t i o n to t h e m a s s e i g e n v a l u e s of the first g e n e r a t i o n r e q u i r e s n o n - d i a g o n a l e l e m e n t s at least of the o r d e r of 10 M e V , thus r o u g h l y an o r d e r of m a g n i t u d e l a r g e r t h a n c o n t r i b u t i o n s f r o m (15, 2, 2 ) c 126. If b o t h 126 a n d 120 a r e p r e s e n t the l e a d i n g SU(4)c v i o l a t i n g c o n t r i b u t i o n to me a n d md is e x p e c t e d to c o m e f r o m 126. W e also n o t e t h a t if t h e r e a r e no s u p e r h e a v y K - s i n g l e t f e r m i o n s N O a c o n t r i b u t i o n f r o m (15, 2, 2) fields to A M suggests the e x i s t e n c e of a 126 scalar in the t h e o r y . O t h e r w i s e only g r a v i t a t i o n c o u l d c o n t r i b u t e to M,,~ a n d for (q~B-L) < 10 15 G e V (which is n e c e s s a r y for m b = rn,) t h e l e f t - h a n d e d n e u t r i n o m a s s e s w o u l d c o m e o u t t o o large. T h u s we can restrict o u r discussion to the case w h e r e (15, 2, 2) b e l o n g s to 126. 5.2. FERMION EXCHANGE S u p e r h e a v y charged f e r m i o n s can c o n t r i b u t e to A M (b). Such f e r m i o n s exist in the c o n t e x t of E6* w h e r e t h e f u n d a m e n t a l 27 r e p r e s e n t a t i o n c o n t a i n s the following fermions: (4, 2, 1) = (v, ui, e, d i ) , (3,, 1, 2) = (d~, e ~, u~, v~), (6, 1, 1) = (I)i, I)~),

(5.4)

(1, 2, 2) = (i¢,1,]~, l~c, l~C), (1, 1, 1 ) = I ~ . * Fermion masses in an E6 model have been discussed in ref. [6]. In this context it has been proposed [16] that the SU(4)c violating fermion mass contributions arise through loop effects involving heavy fermions. However, in general the tree diagrams (fig. 4a and 4b) contributing in this model are expected to dominate. For (¢c)= (q~x}= (~0o) no prediction on m b or mt is possible in this model without a very special choice of Yukawa couplings or quartic scalar couplings.

C. Wetterich / Neutrino masses

359

T h e first step of s y m m e t r y b r e a k i n g could be induced* by the V E V s of the K singlets c o n t a i n e d in the 351 representation of E6. A t this scale the neutral fermion N gets a M a j o r a n a mass (this is possible since it has zero B - L charge) and the fermions N, I~ and I)i (which have the same SU(3)c × U(1) q u a n t u m n u m b e r s as v, e and di) acquire a Dirac mass. T h e SU(4)c violating contributions to the light f e r m i o n masses f r o m the exchange of these h e a v y fermions are shown in fig. 4b. In contrary to fig. 4a these graphs are only possible for ¢c = (7[, 1, 2)**. T h e contribution to AME is a M ~ ) = k E M { ' I~Edc(~OL)](~OB_L)I2(~OX) -1 .

(5.5)

H e r e M ~ = ffE(q~X); kE, /~E and /~E are a p p r o p r i a t e Y u k a w a couplings and Y~ is the ratio of two quartic scalar couplings. T h e r e is a similar contribution to A M p and A M (D) but no contribution to AM~. For k, /~, /~ and h not very different A M (~) and A M (b) are roughly of the same o r d e r of magnitude. W e note that within SO(10) and for q~o= (7~, 1, 2) there is a relation b e t w e e n A M (a) and M(b~ since b o t h (]O, 1,3) and (15,2, 2) belong to the same 126 representation. A similar relation b e t w e e n A M (b) and .M(a) . . ~ exists in the context of E6: the neutral lepton in fig. 2a corresponds to N which belongs to the same representation as 1~ and I). Thus, for ¢o = (4, 1, 2) and both heavy fermions and scalars in the 126 representation of SO(10) present either the heavy fermion contribution is d o m i n a n t for b o t h rn~ # m d and M ~ , (large Y u k a w a coupling of ¢c, small M~,~,r3, small Y u k a w a coupling of 126 scalars) or inversely the scalar contribution is relevant for both p h e n o m e n a . 5.3. GRAVITATIONAL CONTRIBUTION TO rn,~ md In models without h e a v y charged fermions or (15, 2, 2) scalar fields (for example the SO(10) m o d e l of Witten [4]) only*** gravitation can contribute [17] to SU(4)c violation in fermion masses (fig. 4c). For ¢c belonging to a complex representation one has A M (c)= CG(CPL)(~PB-L)ZMp2. T h i s would lead to (¢B L ) > 1017 G e V and very small (10 -4 eV) neutrino masses. H o w e v e r the (15, 1, 1) field [belonging to 45 of SO(10)] in Wittens m o d e l also acquires a SU(4)c violating V E V of the o r d e r (Co). Thus the gravitational contribution is only suppressed by one p o w e r of (¢c)/Mp: A M (c) = CG(~PL)(qVc)Mp I ,

(5.6)

(~#c) > 10 is G e V .

(5.7)

and one has a lower b o u n d

This b o u n d coincides with the b o u n d for (q~B-L) given in sect. 4 for the same case. * For simplicity we suppose (~#x)= (~#G)for this discussion. ** The Yukawa couplings of (~0c) belonging to (10, 1, 3) can only contribute to Majorana masses. These are not possible for charged fermions. **'*In this model there are SU(4)c violating one-loop contributions to fermion masses. Since one has only one Yukawa coupling f one has AMc(a/rr)f(q~L). This is much too small to account for me ~ rod.

360

C. Wetterich / Neutrino masses

6. N e u t r i n o mass predictions from me ~ md

In this section we use the information on the SU(4)c violating fermion mass contributions for predictions on neutrino masses and leptonic mixing angles. T h r e e viable SO(10) models [or models based on groups containing SO(10)] are presented. Two of t h e m have an intermediate mass scale (~c) = (¢B-L) --~(3 • 10 3 _ 10-~). ( ¢ x ) . F o r the third m o d e l vr~ masses and SU(4)c violation in fermion masses are induced by gravitation. This m o d e l has only one superheavy mass scale (~0c) = (~0B L) = (¢x). All these models predict neutrino masses in the e V region.

6.1. MODEL A T h e first SO(10) m o d e l we discuss is based on the breaking chain [9] M x

SO(10)

54

MB--

126

, SU(4)~ x SU(2)L × SU(2)R L

, SU(3)~ × SU(2)I_ x U(1) y

M L 10

• SU(3)~ × U(1)em • m

T h e b r e a k d o w n of K is induced by the (10, 1, 3) contained in 126 and one has (q~B-L)=(~Oc)=(~OR). R i g h t - h a n d e d neutrinos get their masses f r o m the direct Y u k a w a coupling to (q~B-L). T h e induced V E V of (10, 3, 1) contained in 126 contributes to the VL masses (fig. lb). The SU(4)c violating fermion mass contributions arise f r o m the exchange of the (15, 2, 2) scalars also contained in 126 (fig. 4a). Thus MvL, Mv~ and A M are all p r o p o r t i o n a l to the Y u k a w a coupling h of the 126 scalar multiplet and to different powers of (q~B-L). I n f o r m a t i o n on A M can be used to restrict the neutrino masses. First we note that AMxl can be evaluated fairly reliably if the contributions of n o n - d i a g o n a l elements A M 0 to the electron and d - q u a r k mass can be neglected. This is the case if AMlj is not m u c h larger than AMll. W e can parametrize the electron and d - q u a r k mass: ma = y ( A + A M 1 1 ) , (6.1) me = O(A - 3AMll).

H e r e A is the SU(4)c invariant part of the mass and y arises f r o m the renormalization of md due to strong interactions. (We neglect small renormalization effects f r o m

361

C. Wetterich / Neutrino masses

S U ( 2 ) L × U ( 1 ) interactions in this section.) T h e factor 0 = +1 reflects the uncertainty in the relative sign of md and me. O n e has

1 3 A M n = ~me{zC + (¼C - 1)0},

(6.2)

w h e r e C = md/yme. For md = 7.5 M e V , me = 0.5 M e V and y ~ 3 . 5 - 4 , one has C ~ 4 and AM11 ~- 0.5 M e V . (6.3) Validity of p e r t u r b a t i o n t h e o r y requires h11 < 1 and we derive f r o m eq. (5.2): ./t~_g.

/

, (~OB-L) 2

(~B-L) 2

0.5 M e V = AM11 = ~/~rt11Kck~0L), . , - , - - ~ < 7 1 " 1 G e V . Kc (¢x)2

.

(6.4)

A s s u m i n g 1 < Kc < 5 (we note that in a m o r e perfect t h e o r y where the quartic scalar couplings are calculable Kc is typically a C l e b s c h - G o r d a n coefficient) and using MB-L = g(wB L), M x = ~/~g(~x), one finds

MB-L> 2 • 10 -3 .

(6.5)

Mx T h e Y u k a w a couplings h and f are independent. (The same generation pattern for h and f contradicts either rnb = m, or ma# me.) Thus the matrix transforming b e t w e e n eigenstates of f and h is expected to involve large mixing angles. In a basis w h e r e f is diagonal, the matrix elements hij are then expected to be all of the same o r d e r of m a g n i t u d e and the SU(4)c violating f e r m i o n mass contributions are all of the o r d e r of at m o s t a few M e V . T h e m o d e l predicts rob(Ms--L) = m-~(MB-L) and ms(MB-L) = m~,( M s - L ) within a few percent. T h e r e is no reason w h y h should be m u c h smaller than f. If we suppose 1 0 - 2 < hq < 1 we find, for M x = 1015 G e V 2 • 1012 G e V < M B - L < 1014 G e V .

(6.6)

A central value, for Kc = 1, hll = 0.1 is given by M s L = 1.3 • 1 0 - 2 M x = 1.3 • 1013 G e V .

(6.7)

W e also can use our estimate of A M 1 1 tO eliminate the scale (~0S-L) in the predictions of neutrino masses. O n e finds

6-1/4K K1/2 (~0L) ( ( ~ L ) ~1/2 3/2

(lar(b)'~

. . . . . )11 = 2 .

b

c

(~-~\~---M-7711]

h l l (q~t)

7 - /'1015 G e V \ 1.3/2 =8.9eV'KbVK~ ~/x ) ¢~11 '

(MvR)11= 2 "

(6.8)

~-1/2 --1/2~'1/21 ', \~----~11] ICe nl1 '~~Ox]

61/4 ((~0L)

[ Mx \ 1/2 = 1 . 4 . 1 0 1 3 G e V . K21/2 ~1015 G e v ) h n ,

(6.9)

362

C. Wetterich / Neutrino masses

( M u(a) L)44 =

K

:m(D)\21:" " a~, u~, ) / ( l V l u a ) l l

=0.73 eV.

(101' GeV~

Ka4--~\ -M-~ ] h~(/2"

(6.10)

Several comments are in order: The matrix elements VL(]~(b))ll, ,"(M,R)11 . . (. M . ( a ) , )44 are quoted in a basis where E and D fermion masses are diagonal except small SU(4)c violating contributions. The matrix elements ,--~LtM~b),,I ~ and (M,~)ii are expected to be in the same order of : ~ at(b) ~ magnitude as ~ 1 ~ ) 1 1 and (M~)11. Especially, upper bounds on (h,4(b) . . . . . . )11 and ( M ~ ) a l derived from the validity of perturbation theory ( h ~ ~< 1) are valid for all (M~b))ii and (M~)~j and therefore for the left- and right-handed neutrino mass eigenvalues. Eq. (6.10) is derived for the case of four fermion generations with the assumption M ( D ) = mu, m (D) ~, = 100 G e V (see sect. 7). The factor K~ reflects our lack of knowledge of hJh11. If the lightest right-handed neutrino mass is about ( M ~ ) 1 1 and if its eigenstate has a substantial admixture of u,, one has K ~ I . If there are eigenvalues of h much smaller than h11, Ka is enhanced (see sect. 3). -)~[(a) In the case of three generations (... ~L)33 is suppressed by a factor 10 -z c o m p a r e d - (a) with (MvL)44. For h11> 10-2(10 -1) the Dirac mass contribution to the neutrino masses of the first two (three) generations can be neglected. Thus we expect large leptonic mixing angles and no strong generation dependence of the neutrino masses for the lower generations (pattern A). ~ and M ~ can be derived from the assumption of validity U p p e r bounds on lvi , ,(b) of pe___rturbation theory (h~j
(6.11)

m v(b) L -<45

(6.12)

eV.

The upper bound on m (~t] coincides essentially with the cosmological upper bound. For four generations, we also can derive approximate lower bounds on the masses of the heaviest left- and right-handed neutrinos. For the left-handed neutrino mass - (a) ~ (Here we r e m e m b e r that for f44> 2~/2hal(m~)44 becomes larger than t:~r(b) ~ ~ )11. we assume Kb=K~; for K~>Kb the lower bound is even higher). Thus hll ~ f44/2",/2 = 0.2 corresponds to a minimal value of the heaviest neutrino mass m ~ : -

m,~"x ~>0.8 e V . Kb'fK'-~K~(101--5 GeV~ k Mx ].

(6.13)

A lower bound on hll and therefore on m ~ ~ is derived from the requirement (a) < (M~L)44 50 eV which yields h11 ~ 10 -4 ,

m , ~ x ~> 1011 G e V .

(6.14)

363

C. Wetterich / Neutrino masses

H o w e v e r , very small values of h 11 are not very appealing. For 10 -z < h ~1 < 1 one has 6 • 1011 G e V < (M,R)11 < 3 • 1013 G e V .

('6.15)

6.2. MODEL B T h e s y m m e t r y breaking of this m o d e l is the same as in m o d e l A, except that (q~B L) belongs to a (4, 1, 2) representation contained in 16 or 144 of SO(10). T h e SU(4)¢ violating fermion mass contributions and the right-handed neutrino masses are p r o d u c e d either by a s u p p l e m e n t a r y 126 scalar or by s u p e r h e a v y charged fermions. B o t h contributions are present in the context of E6 where a possible s y m m e t r y b r e a k i n g chain is Mx

E6

351(1,54) MB L 351'(16,144)

• SU(4)c x SU(2)L x SU(2)R

, SU(3)c x SU(2)L x U ( 1 ) y

ML 27(10)

, SU(3)~ x U(1)¢m.

First consider the contributions of the 126 scalars (contained in 351). For the SU(4)c violating f e r m i o n mass contributions the estimate of AM11 in m o d e l A applies also in this case. H o w e v e r , the r i g h t - h a n d e d fermion masses are now suppressed by a s u p p l e m e n t a r y factor (~B-L)/(~0X) (see fig. 2b): (My.)11

=

2h11(~Pt) = 2h11Ka(¢B-L)2/(~PX) = 4.7 " 10 a° G e V Ka 1015 Mx Kc GeV .

(6.16)

F u r t h e r m o r e M(~] is suppressed by a factor (q~-L)a/(q~X) 3 c o m p a r e d with m o d e l A. W e on n o. M~(b) -L < 10-6 e V which can be neglected except for a possible contribution to the neutrino mass of the first generation. T h e relevant contribution to light M ( a ) is e n h a n c e d by a factor (~x)/(q~s L) c o m p a r e d with m o d e l neutrino masses .._~L A. O n e finds .

- (a) KaK¢ 1015 G e V ( M ~ ) 4 4 = 213 e V . - Kd

(6.17)

MX

Thus for four generations the u,, neutrino is predicted to be near the cosmological u p p e r b o u n d since KaKc/Kd should not be m u c h smaller than one and M x cannot be m u c h larger than 1016 G e V (see sect. 7). T h e mass of the u, is a factor of a b o u t 10 -2 smaller than m~,. F o r three generations it is in the e V region. As discussed in sect. 3 we expect a strong generation d e p e n d e n c e of neutrino masses and leptonic mixing angles in the same o r d e r as q u a r k mixing angles (generation pattern B). Alternatively the heavy f e r m i o n contribution m a y d o m i n a t e A M and M~ R. W e first consider only the contributions f r o m (4, 1, 2) or (7~, 1, 2) scalar fields contained in the 351 representation of E6 (which has the Y u k a w a coupling h). W e note that

C. Wetterich / Neutrino masses

364

at the scale (~Ox) all (4, 1, 2) multiplets get strongly mixed. Thus (~0B L) has a substantial contribution from the (4, 1, 2) fields contained in 351 and )51 and one finds from eqs. (5.5) and (4.2):

A M = ~-61-ht~c(~L)

mo.

=

(~PB-L)2 (¢x) 2 ,

2hKd (q3B-L)2 "

(6.18)

(6.19)

Here Y~ involves Clebsch-Gordan coefficients and ratios of quartic scalar couplings and Yd denotes the SO(10) singlet contribution to (q~x). (Ya = ~(~ (35 l(1)))/(q~x) < ~.) Except for the difference between Y~, Yo and K~, Kd this is the same contribution as from scalar exchange. Next we include contributions from the (4, 1, 2) fields contained in 351', 351' and 27. If the Yukawa couplings of 351' (k) and 27 (f) are not much larger than h eqs. (6.18) and (6.19) still hold. For k >>h the Yukawa coupling h in these equations has to be replaced by kh-ak T. A somewhat smaller scale (q~a-L) is predicted, but the prediction of (Mv~) remains unaltered. Thus the predictions (6.16) and (6.17) for neutrino masses apply also for the contribution from heavy fermions. 6.3. MODEL C Finally we discuss a model where M ~ possible symmetry breaking chain is [4] SO(10) M~-L' SU(5) 16

Mx

and A M are induced by gravitation. A

• SU(3)c x SU(2)L x U(1) v

45

ML > SU(3)c

x U(1)em

•

10

A M involves the VEV of (1, 2, 2) in 10 and (15, 1, 1) in 45. Since 1 0 x 4 5 contains only 120 but not 126 the effective (15, 2, 2) field contained in (~OL)• (q~c) has only antisymmetric couplings to fermions. In order to account for me ¢ rna one needs AM12 ~- 1 7 - 2 0 MeV. Again, the mass relations mb(Mx)= rn,(Mx) and ms(Mx)= m , (Mx) are only slightly modified if all contributions are of this order of magnitude. This corresponds to (~0B-L)~ 1015 G e V [see eq. (5.6)] and suggests a two scale model with (~pB L) = (~Px)- For the right-handed neutrino masses one finds [eq. (4.5)] 2 Mv~ = 1011 G e V . CG/(CG)12,

(6.20)

where we expect all eigenvalues to be of the same order of magnitude. For the -(a) - (a) left-handed neutrinos only MVL contributes and (M~ L)44 is around 50 eV. The neutrino masses of the lower generations are strongly suppressed (m~. 0.5-1 eV; generation pattern B). Thus despite of the different scales for (¢S-L) models B and C have essentially the same predictions for neutrino masses.

C. Wetterich/ Neutrino masses

365

W e n o t e that for m o d e l C local B - L s y m m e t r y is n o t n e e d e d if we r e n o u n c e t h e usual " s u r v i v a l h y p o t h e s i s " for f e r m i o n s . In fact, we can i m a g i n e a h y b r i d SU(5) m o d e l with o n e 2 4 - p l e t a n d o n e 5 - p l e t of Higgs scalars which includes s u p p l e m e n t a r y r i g h t - h a n d e d n e u t r i n o s (SU(5) singlets). F o r (~Ox)~ 1015 G e V g r a v i t a t i o n a l c o n t r i b u t i o n s can a c c o u n t for me ¢ raa [17]. T h e M a j o r a n a m a s s of the r i g h t - h a n d e d n e u t r i n o s involves (q~x) twice a n d is of the o r d e r M , ~ ~(qgx)2/Mp. In this case a r - n e u t r i n o m a s s of a b o u t 1 e V a n d a p o s s i b l e r ' n e u t r i n o m a s s of s e v e r a l ten e V are expected.

7. Renormalization group determination of Mn-L and M x In this s e c t i o n w e use t h e r e n o r m a l i z a t i o n g r o u p to d e t e r m i n e the scales M B _ L a n d M x f r o m the l o w - e n e r g y p a r a m e t e r s aem(ML), as(ML) a n d sin 2 0 w ( M L ) . This a p p r o a c h is in s o m e sense c o m p l e m e n t a r y to the d e t e r m i n a t i o n of MB-L f r o m the SU(4)c v i o l a t i n g f e r m i o n m a s s c o n t r i b u t i o n s : in sect. 6 the r a t i o MB L/Mx has b e e n p r e d i c t e d as a f u n c t i o n of t h e k n o w n m a s s e s me a n d rod. U n c e r t a i n t i e s arise f r o m u n k n o w n Y u k a w a c o u p l i n g s a n d r a t i o s of q u a r t i c scalar couplings. T h e r e n o r m a l i z a t i o n g r o u p a p p r o a c h d e t e r m i n e s In (MB L/ME) as a f u n c t i o n of the l o w - e n e r g y couplings. N o u n k n o w n p a r a m e t e r s a r e involved, b u t MB L d e p e n d s e x p o n e n t i a l l y on sin 20w, OGrea n d as at the scale M r a n d is t h e r e f o r e v e r y sensitive to t h e i r p r e c i s e values. A t p r e s e n t , t h e u n c e r t a i n t y of the m e a s u r e d values of sin 2 0 w a n d as d o e s not allow a v e r y a c c u r a t e d e t e r m i n a t i o n of M B - - L . H o w e v e r , w h e n a m o r e p r e c i s e m e a s u r e m e n t of sin 2 0 w is a v a i l a b l e (for e x a m p l e f r o m t h e mass of t h e n e u t r a l w e a k b o s o n ) an a c c u r a t e d e t e r m i n a t i o n of M s L is possible. T h e n we can use M s L as an i n p u t to d e t e r m i n e s o m e of the free p a r a m e t e r s of sect. 6. In this s e c t i o n we assume (~B-L)= (~c)= (~R). This is the m i n i m a l * s c e n a r i o which covers m o d e l A , B a n d C of sect. 6. ( F o r m o d e l C o n e has ( ~ ; a - L ) = (~¢x) a n d the usual SU(5) p r e d i c t i o n s for M x apply.) W i t h i n SU(4)~ × SU(2)L × SU(2)R, t h e r e is a p a r t i a l unification of the g a u g e couplings: L - R s y m m e t r y p e r m i t s o n l y two i n d e p e n d e n t g a u g e couplings at the scale M s r. U s i n g t h e t h r e e low e n e r g y c o u p l i n g s for SU(3)c, SU(2)L a n d U ( 1 ) v as an input, the scale MB-L can be d e t e r m i n e d w i t h o u t any a s s u m p t i o n on w h a t h a p p e n s at e n e r g y scales b e y o n d

(,;~ L). A t t h e scale M B - L the e l e c t r o m a g n e t i c c o u p l i n g e is a l i n e a r c o m b i n a t i o n of the w e a k c o u p l i n g gw of SU(2)L x SU(2)R a n d t h e c o u p l i n g gB L of U(1)B-L (C~i= g2/47r): Olem(MB--L) = OLw(MB L) sin 2 0 w ( M B - L ) = ~C~w(MB L) sin 20v(MB--L) = 12aB-L(MB

L) COS2Ov(MB-L).

(7.1)

* For models with MB-L<
366

C. Wetterich/ Neutrinomasses

On the other hand U(1)B-L is a part of SU(4)~ and gB-c is related to the strong (SU(3)~) coupling constant gs:

OtB-L(MB-L)

=

3a~(MB L).

(7.2)

The coupling constants aem , ffw and c~ are partially unified: 2 6 adM~_L) + aw(MB-L)

3 aem(MB-t)= 0.

(7.3)

Below MB c the integration of the renormalization group equations yields in the one loop approximation:

1

1 _+flilnMB_L

C~i(MB L)

C~i(ML) 6rr

M~--L '

(7.4)

where /3s = 33 - 4 N G , flw = 22 - 4NG - 2~NH,

(7.5)

[~em 22 - 332NG- NH. =

H e r e NG denotes the n u m b e r of fermion generations and NH is the n u m b e r of Higgs doublets. Using eq. (7.3) one obtains MB-L = ME exp {22a

NIL) [1-2sin2Ow(ML)]

lla-fMLi"

(7.6)

Several comments are in order: MB-L depends strongly on sin 2 0w(ML), whereas the dependence on c~s(ML) is somewhat weaker. An accurate determination of MB-L requires also an accurate determination of C~em(ML) which we take to be 1/128.5. (A change of 1% in aem(ML) changes the prediction of MB-L by about 35%.) In the one-loop approximation M~-c is independent of the number of fermion generations and the n u m b e r of weak scalar doublets. We also have calculated two-loop corrections [18] to MB-c. These corrections turn out to be small. For three fermion generations they lower the prediction of MB-L by about 20%, whereas for four generations they lower MB-L only by 5%. (This is due to the cancellation of gauge boson and fermion contributions in the two-loop fl-function for four generations.) In table 1 we quote MB-L as a function of sin 2 0w(ML) and as(ML) including the two-loop effects (NG = 4). We also give aw(MB-L)/as(MB-L) as a measure how far M B - c is separated from the unification scale M x where both couplings are equal. Note that in our approach unification r e q u i r e s aw(MB-L) ~ as(MB-L) since ces evolves faster than Cewdue to the large colour group. This restricts the allowed range for sin e Ow(ML) and as(ML).

• 2 Ow(ML) sm

0.21 0.22 0.23 0.24

0.21 0.22 0.23 0.24

as(ML)

0.11

0.14

76.1 24.7 8.0 2.6

44.0 14.3 4.7 1.5

[1013 GeV]

MB L

TABLE 1

0.946 0.899 0.849 0.803

0.996 0.942 0.892 0.844

ot s ( M B - L )

C~w(MB L)

3.7' 10 l° 1.5 • 10 6 2.4. 1011 8 . 6 ' 1 0 -5 1.5.1012 5 . 1 0 -3 9.6.1012 0.28 6,6.1011 1.1 • 10 s 4.2.1012 7 . 4 . 1 0 4 2.7. 1013 3 . 5 . 1 0 .2 1 . 4 . 1 0 TM 1.6

2 . 4 . 1 0 .4 4.4.10 3 0.08 1.45

2.81 3.87 5.34 7.37

[eV]

(M~L(b))11

2.5" 10 s 4 . 5 ' 1 0 .`4 8.2.10 3 0.15

[GeV]

hI1(MB L) (M~R)11

0.5 0.68 0.95 1.31

[1015 GeV]

Mx

Model A

10 1.6 0.26 5.3' 10 2

251 40 6.5 1.1

[eV]

(Mfa) 'tvL/44

Mass scales and neutrino masses as functions of sin 20w(ML) and as(ML) (No = 4)

2.7' 10 -7 1.7' 10 7 1 . 0 ' 1 0 -7 6.6"10 8

2.1 . 101° 3.4' 101° 5.4' 101° 0 . 9 ' 1 0 ix

1.2" 1011 4.9" 10 s 1.9.1011 3.0" 10 8 3.1'1011 1.9' 10 8 4.9' lO ll 1.3' 10 s

[eV]

.L(M(b) ) 11 x"-

[GeV]

(MvR)11

Model B

54.5 34.9 22.5 13.8

435 281 181 117

[eV]

{a) (MvL)44

g

5.

3

368

C. Wetterich / Neutrino masses

We believe that in the context of an intermediate SU(4)c × SU(2)L X SU(2)R group broken by a complex scalar representation our prediction of MB-L is very reliable. There is no possible intermediate scale between ME and MB L where the number of gauge bosons contributing to the/3-functions could be increased. The number of light fermion generations is relevant only for the two-loop corrections which do not exceed a factor 2 for NG < 8. Besides the light fermions we only expect the right-handed neutrinos to have masses of the order MB-L or smaller. These right-handed neutrinos are singlets with respect to H and do not alter the renormalization group equations below MB L. Since the one-loop approximation is independent of NH, the question if the weak scalar doublet is a fundamental or a composite field (at energies between ME and MB-L) is not relevant for the determination of MB-L. The scalar fields responsible for the breakdown of K with a mass larger or equal MB-L do not contribute to the renormalization of coupling constants below MB-L. However, some of these scalars could have a mass somewhat below MB-L. 2 One-loop effects give a lower bound M~ >~aMB L. We have included such possible effects for scalar fields belonging to (10,3, 1)+(10, 1,3) (model A) and scalar fields belonging to (4, 2, 1)+ (4, 1, 2) (model B). In model B the only contributions can come from a weak doublet, color triplet field with hypercharge ½. If its mass is x/aMB L, this enhances MB-L by about 8%. In model A larger scalar multiplets could contribute. This contribution is maximal ( - 6 0 % - + 8 0 % change in MB-L) if only some of these scalars have a mass --~/-dMB--L whereas the others have a mass MB-L. However, this extreme situation is not very natural. If all remaining scalars ave a mass x/~MB L the correction is again + 8 % . In conclusion we believe that the main uncertainty in our prediction of MB L [eq. (7.6)] comes from the unknown number of fermion generations. For NG = 3 or 4 we estimate the total uncertainty to be smaller than 30%. The present world average for sin 2 0 w is sin 2 0 w = 0 . 2 2 8 + 0 . 0 1 0 . We note that sin 2 0 w > 0 . 2 1 implies (~0B-L)< 1015 GeV, thus excluding model C. However, we think that the present data on sin 2 0w are not sufficient to rule out this model definitely. A central value sin 2 0w = 0.23 predicts, for as(ML)= 0.11:

M°_L = 4 . 7 • 1013 G e V .

(7.7)

Interesting enough, this coincides rather well with the central value for MB-L in model A and B [see eq. (6.7)]. The renormalization group determination of MB L gives independent predictions for the neutrino masses in model A: MB

(Mvr~)11

=

()•,[(a) "'-UL )44

=

L

1.6. 10 TM G e V . hll M O L ,

(7.8)

0.06 e V . K,h11~ M ° - L

(7.9)

MB

L

'

369

C. Wetterich / Neutrino masses

o MB L (~ At(b) ~** .~ ~al = 0.76 eV • K b h l l - M B -L

(7.10)

For sin 2 0 w < 0 . 2 4 the contribution A/f(b) should not exceed a few eV. For small -(a) values of h and large values of K~(M~)44 can be as large as several ten eV. In model B M (a) is enhanced by a factor (¢X)/(¢B-L). Thus a heaviest neutrino mass VL near the cosmological upper bound is possible for h and f of the same order of magnitude. In any case, neutrino masses larger than 10 eV suggest the existence of a fourth fermion generation! Next we determine M x from aw(MB-L) and adMB-L). Since both couplings have to be equal at the scale M x one finds, in the one-loop approximation: 6nM x = MB L exp { 22 -- NH -- 2 N h

{

=MB-L exp 4 4 - 2 N I a - 4 N ~ I + ~NH(Sin 20w(ML) - 1))

1

(aw(MB-L)

1

Iaem-(-ML) 3 (6 sin 2 0w(ML) -- 1 10

as ( M L ~ (1

-r~0NH)] }

(7.11)

H e r e N H is the n u m b e r of complex (1, 2, 2) fields and we have N H = 1 if the scalar fields are not composite at energy scales below Mx. In model B the scalars belonging to (4, 2, 1) + (4, 1, 2) must have masses of the order MB-L. However, their contributions cancel out in the determination of Mx(N'H = 0). In model A the (10, 3, 1) + (10, 1, 3) fields give a contribution N ~ = 1. For three fermion generations the loop effects lower the predictions of M x by about 2 0 - 4 0 % depending o n a s ( M E ) and s i n 2 0 w ( M L ) . In the case of four fermion generations M x changes between - 5 % and + 1 0 % . In table 1 we give the dependence of M x o n s i n 2 0 w ( M E ) and a s ( M E ) for four fermion generations in model A (NH=N'H = 1). In model B ( N H = 1, N h = 0), M x is larger by about 30% (for sin 2 0 w = 0.23, as = 0.11)• We also studied the possible effects of scalars with mass M "2 = a M 2 in the context of SO(10). This gives an upper limit on the uncertainty in the estimate of Mx. Possible scalars with mass M's are (6, 1, 1) belonging to 10 or 126, (15, 2, 2) belonging to 126 and (1, 3, 3) and (20", 1, 1) belonging to 54. We find that the maximal contribution of scalars belonging to 10 or 126 is to enhance M x by 8%. For M ( 1 , 3, 3) M(20", 1, 1 ) ~ ~/aMx, we find that M x is enhanced by 16% whereas a factor 0.6 (2) arises in the (unnatural) extreme situation M ( 1 , 3 , 3 ) ~ x / ~ a M x ~/~M(20", 1, 1) (M(20", 1, 1 ) ~ x/~Mx ~-x/~M(1, 3, 3)). Summing up all uncertainties from Higgs scalars, including the question whether scalar fields are composite or not, we believe that our estimate of M x holds within a factor of two. Thus the uncertainty in M x in these models is not larger than the uncertainty in the simplest SU(5) model [19].

370

C. Wetterich / Neutrino masses

A central value of M x is (sin 20w(ML) = 0.23, as(ML)= 0, 11, NH = N ~ = 1, NG = 4 ) : M ° = 9.5 • 10 TM G e V .

(7.12)

This corresponds to a proton lifetime [20] r v = (8 - 160) x 103o yr.

(7.13)

We can use our estimate of M x to determine the Yukawa coupling hll from eq. (6.4):

1

2

h 1 1 = 0 . 0 0 6 _KCMx _~2.

02

MB-L M ~-L "

(7.14)

In table 1 we give the dependence of h l l o n sin 2 0 w ( M r ) and ~s(ML)(Kc = 1). If we restrict ourself to 0 . 0 1 < h 1 1 < 1 and 1 < K ~ < 5 , only a narrow range sin 20w(Me) = 0 . 2 4 + 0 . 0 1 (for as(ML)= 0.11) is allowed. The combination of the renormalization group predictions with the predictions from me # ma [eqs. (6.8)(6.10), (6.16), (6.17)] further constrains the neutrino masses. In table 1 we give numerical predictions for the neutrino masses in model A and B (for K a = K b = K c = Kd = 1). For these predictions we have included renormalization effects on the neutrino masses (the relevant R G equations are outlined in the appendix). In the case of ~/-(a) / ~ f (b) three generations renormalization effects enhance t---~L by 10% and -,-~L by 70%. For four generations the Yukawa coupling of the t' quark plays an important role. Using the relation [9] m t' (MB-L) = m T'(MB-L)

m~(Ms-L)

m. (MB-L)'

(7.15)

we can determine mc(2mt,) as a function of m , , ( 2 m , , ) . For the lowest possible mass mr, 150 G e V (this corresponds to m,, 17 GeV, m b ' 42 GeV, ~i~ 68 GeV; the numerical values are given for sin 20w(ML) = 0.23, as(ML)=0.11), (a) renormalization effects enhance ~r~b) by 20% and reduce M ~ by 6% For the largest possible value consistent with perturbation theory m t ' = 230 G e V (m,, = (a) 35 GeV, r o b ' = 118 GeV, m~TM = 139 GeV) renormalization effects reduce M ~ and (a) ~r(b) by a factor 1. Thus the renormalization effects reduce the dependence of M~L (a) on mr,. Over the range of r o t , discussed here M ~ varies only by a factor two and reaches its maximum for m t ' = 220 GeV. The values in table 1 are given for rnt, = 2 0 0 G e V (m,, = 25 GeV, rob, = 69 GeV, m~, ~D) = 99 GeV).

8. Conclusion We have determined the scale of B - L violation using the SU(4)¢ violation in fermion mass relations and renormalization group methods. Consistent with

C. Wetterich / Neutrino masses

371

m b ( M B L) = mT(MB--L), m d ( M B L) ~ m e ( M B L) and sin 20w(Mc) = 0.23 + 0.01, we found that M B L should be smaller than the unification scale Mx (4. 1 0 - 6 < (MB L / M x ) 2 < 4 " 10-2)- Within these constraints two models based on SO(10)

have been discussed. Model A predicts the neutrino mass of a possible fourth fermion generation to be several eV. Even a mass near the cosmological bound of 50 eV is not excluded. For sin 2 0 w = 0.23 the r neutrino has a mass around 10 1 eV whereas the neutrinos of the first two generations both have masses of order 10-3-10 -2 eV. For larger values of sin 20w(ML) the neutrinos of the lowest three generations may all have masses of several tenths of eV. The mixings between ~e and u~ (and uT for sin 20w(ML) > 0.23) are large, whereas the mixings of v~, [and t,~ for sin 20w(ML) < 0.23] are of the same order as in the quark sector for the corresponding generation. This model has some interesting features: The solar neutrino puzzle can be explained by large oscillations between ~'e and ~,, (and eventually t,~, depending on the precise value of sin 20w(Mc)). On the other hand the neutrino masses of the first three generations are too small to be observed by accelerator experiments, which are only sensitive for neutrino masses larger than 1 eV. Neutrino oscillations involving the fourth fermion generation are suppressed by the small leptonic mixing angles. This explains why transitions like ~,,, ~ ~'e have not been observed in accelerator experiments. Finally, a mass of u~, around 10 eV would imply many interesting cosmological consequences [21]. Model B predicts the neutrino mass of the fourth generation to be near the cosmological bound. The masses of the lower generations are suppressed by squared ratios of the quark masses of the corresponding generations. Leptonic mixing angles are predicted to be in the same order as quark mixing angles. Since these mixings are expected to be small it is difficult to explain the solar neutrino puzzle by neutrino oscillations within this model. We also discussed a model C where right-handed neutrino masses are induced by gravitation. This model requires M n - L = Mx(sin 20w(Mc) = 0 . 2 1 ) and predicts essentially the same neutrino masses as model B. No realization of a possible generation pattern where neutrino masses are proportional to quark masses has been found consistent with mb ----mr and me ~ md. In all of these models a neutrino mass between 10 and 50 eV favours strongly a fourth fermion generation! The neutrino of this generation dominates the neutrino oscillations which could be observed by accelerator experiments. Denoting PWa Ub) [1] the probability that Ua oscillates into ~'b we have for these experiments P(t,e ~ u e ) ~ l - P ( u e - ~ U~, ) ,

(8.1)

P(~% ~ u,) ~ 1 - P 0 ' , -~ ~'~') • The probability P(~,¢ ~ ~,~,) is essentially given by the squared mixing angle between

372

C. Wetterich / Neutrino masses

the first and the fourth generation and similar for P ( v , -+ v,,). We expect P(Ve-+ V,,)-----0.01--0.1 , P ( v , --) v., ) ~ 0 . 0 1 - 0 . 1 .

(8.2)

Therefore the deviation of the ratic~ P ( v e ~ V e ) / P ( v . ~ V.) from unity is expected between 1 and 10%. This is not very far from the sensitivity of present accelerator experiments. Neutrino oscillations between v. and v~ involve an intermediate v., and one has P ( v . ~ re) ~ , P ( v-, - ) V e ) ~ P1( v ~ , --> v , , ) P ( v ¢ - ) vT,) . -

(8.3)

We expect P(v~-)

Ve)~5 "

10 3 - 5 . 10 -s .

(8.4)

The error in present accelerator experiment measurements of P ( v , -) re) and P ( ~ , --) Pe) is around 10 3. Thus further improvement of these experiments could give important information on neutrino oscillations. Reactor experiments [22] are sensitive to smaller neutrino masses of about 0.5 eV. Our predictions depend strongly on the value of mv. and on the contribution of (b) M vc" For m ~. ~<0.1 eV the oscillation probabilities in reactor and accelerator experiments should be the same. However, for larger values of rn,. the r-neutrino gives an important contribution to neutrino oscillations in reactor experiments. If m.. is . (a) dominated by ~v/~L (this is the case for sin 20w(ML) < 0.23 or for model B), only p ( ~ -+(-) v.) is enhanced compared with accelerator experiments. P(ve-~ v.), P ( G -) ~), P(Ve -+ re) -- 1 and P ( G -~ G) - 1 are again suppressed by small generation mixing angles. However, for sin 20w(ML) ~ 0.23-0.24, M (b) can be in the order of 0.5 eV ~._~ (within model A). In this case, large oscillations can be seen by reactor experiments! We urge further improvement of both accelerator and reactor experiments to answer the question if neutrino oscillations exist in the predicted range! The author would like to thank J. Ellis for discussions on the subject. He thanks H. Kranz for typing the manuscript.

Appendix R E N O R M A L I Z A T I O N OF N E U T R I N O MASSES

Here we give the renormalization group equations for the Yukawa couplings relevant for the neutrino masses. For simplicity we restrict our discussion to model A. Below the scale M x we have to distinguish between the Yukawa couplings h (15) for the (15, 2, 2) scalars and h ¢1m for the (]-0, 1, 3 ) + ( 1 0 , 3, 1) scalars. Both scalar

C. Wetterich

/ Neutrino

masses

373

multiplets belong to the 126 multiplet of SO(10) which implies h(15)(Mx) = h(l°)(Mx).

(A.1)

Below M x these couplings evolve differently due to their different transformation properties with respect to K. The Yukawa coupling that is determined from me ¢ md in eq. (7.14) is h]]5)(2me). In order to predict the neutrino masses we integrate the renormalization group equations for h ~15) between me and Mx. Then the calculation of h (1°~(MB-L) permits the determination of the neutrino masses at the scale MB-L. The left-handed neutrino masses have finally to be rescaled from MB-L to m ~L. At the scale MB L the (15, 2, 2) scalars and the (1, 2, 2) scalars get mixed. The evolution of h (15) below MB-L is determined by the Yukawa coupling L defined by M E = L ( q 0 L ) : h(15)lM ij \ B

~ -/~!15)(2me) ' Lii(MB L)/Lii(2m~)

L ] - - ,~ tl

(A.2)

Below ML only small electromagnetic corrections contribute to the evolution of L. Between ME and MB-L we have the coupled equations [23] 2 dL 16zr ~ - =3LL+L+.Z • L - z g9 w2L - ~ g9i L2, dU 16772 ~ - = 3 ( U U + U - D D + U ) + X

2 rr • U - 8 g ~ t Jr r- ~ g9w U - 2 o g17~ t 2; r,r

(A.3)

2dD 167r ~ - =3(DD+D - U U + D ) + X • D - 8 g ~ D - ~9g w2 D - z 1g l 2D , where M y = U(~OL), MD = D(q~L), X = 3 Tr U ÷ U + 3 T r D + D + T r L + L 1/g2w + 5 / 3 g 2. At the scale MB-L the following relations [9] hold:

and 1/e 2=

U(MB L)= X/~f(MB--L)W +4k(15)(MB L)Y, D (MB L) = X/~2f(MB-L)X + x/~h (15)(MB_L)Z,

(A.4)

L(MB-L) = ~f~f(MB-L)X -- 3 ~ h (~5)(MB L)Z , with w

= (¢u(1, 2, 2))/(¢L),

x = (~pd(l, 2, 2))/(~pL),

(A.5) y = (¢u(15, 2,

2))/(¢L),

z = (¢a(15, 2, 2 ) ) / ( ~ e ) , and

lw?+Ixl

+ky?+Iz?= 1.

C. Wetterich/ Neutrino masses

374

Between MB L and M x the renormalization group equations for the relevant Yukawa couplings are given by 16~r 2 ~df= ~(fh(lO)+h(lO) + h~lO)h(lO)+f) +if+f+ 2 Tr (f+f)f_~_g~f_9g~f, 16zr2 dh(l°)dt - ~h~l°)h(l°)+h(l°) +½(h(l°)f+f+(f+f)Th(X°)

+Tr(h(lO)+h¢lO))h(lO) 16rr 2

d h (15)

63 2h(10) 15 2h(10) 7-g~ - Tgw ,

=~-(h(15)h~a°)+h ~1°) +h(10)h(lO)+h(15))

dt 1

+~(h

(15) +

f f+ff

+ (15)

h

9 2h(15)

)-~gs

(A.6)

Finally we need to know the renormalization of the four-point function involving two left-handed neutrinos and-two scalars eL below the scale MB L. The effective coupling H, defined by

M~ L (M) = H(M)(~pL)2/(~B_L) ,

(A.7)

H ( M B - L ) = N(MB-L)[h(10) (MB L)]-IN'r(MB-L) + 2Kbh(10)(MB-L),

(A.8)

N(MB-L) = "J~f(MB--L)W -- 3VC~gh(is)(MB-L)y,

(A.9)

with

evolves between ML and MB L:

H = I(LL+H + H(LL+)T)+ 2.~ . H - o gt-w 2r ~r r - ~9g l2rr rt . 167r 2 d-d--t-

(A.10)

Below ME the left-handed neutrinos decouple and there is no renormalization correction to H. References [1] S.M. Bilenky and B. Pontecorvo, Phys. Reports 41 (1978) 225; A. De Rtijula, M. Lusignoli, L. Maiani, S.T. Pectov and R. Pretronzio, Nucl. Phys. B168 (1980) 54 [2] G. Branco and G. Senjanovic, Phys. Rev. D18 (1978) 1621; R. Barbieri, D. Nanopoulos, G. Morchio and F. Strocchi, Phys. Lett. 90B (1980) 91 [3] M. Gell-Mann, P. Ramond and R. Slansky, unpublished [4] E. Witten, Phys. Lett. 91B (1980) 81 [5] M. Magg and C. Wetterich, Phys. Lett. 94B (1980) 61 [6] R. Barbieri and D. Nanopoulos, Phys. Lett. 91B (1980) 369 [7] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438; A. Buras, J. Ellis, M.K. Gaillard and D. Nanopoulos, Nucl. Phys. B135 (1978) 66 [8] H. Fritzsch and P. Minkowski, Ann. of Phys. 93 (1975) 193; H. Georgi, Particles and fields, ed. C.E. Carlson (AJP, 1975); M.S. Chanowitz, J. Ellis and M.K. Gaillard, Nucl. Phys. B128 (1977) 506 H. Georgi and D. Nanopoulos, Nucl. Phys. B155 (1979) 52

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[9] G. Lazarides, Q. Shaft and C. Wetterich, Nucl. Phys. B181 (1981) 287 [10] F. Gfirsey, P. Ramond and P. Sikivie, Phys. Lett. 60B (1976) 177; F. Gfirsey and M. Serdaroglu, Nuovo Cim. Lett. 21 (1978) 28; Y. Achiman and B. Stech, Phys. Lett. 77B (1978) 389; Q. Shaft, Phys. Lett. 79B (1978) 301 [11] J. Pati and A. Salam, Phys. Rev. D10 (1974) 275 [12] E. Gildener, Phys. Rev. D14 (1976) 1667; S. Weinberg, Phys. Lett. 82B (1979) 387 [13] Q. Shaft, M. Sondermann and C. Wetterich, Phys. Lett. 92B (1980) 304 [14] S. Weinberg, in Lectures on particles and field theory (Prentice Hall, Englewood Cliffs, N.J., 1964) vol. II, p. 439; E. Witten, Talk at First Workshop on Grand unification, New Hampshire, April, 1980 [15] R. Barbieri, J. Ellis and M.K. Gaillard, Phys. Lett. 90B (1980) 249 [16] R. Barbieri and D. Nanopoulos, CERN preprint TH 2870 (1980) [17] J. Ellis and M.K. Gaillard, Phys. Lett. 88B (1979) 315 [18] D. Jones, Nucl. Phys. B75 (1974) 531; W. Caswell Phys. Rev. Lett. 33 (1974) 244 [19] C. Cook, K. Mahanthappa and M. Sher, Phys. Lett. 90B (1980) 398 [20] J. Ellis, CERN preprint TH 2942 (1980) [21] R. Cowsik and J. McLelland, Phys. Rev. Lett. 29 (1972) 669; B.W. Lee and S. Weinberg, Phys. Rev. Lett. 39 (1977) 165; S. Tremaine and J. Gunn, Phys. Rev. Lett. 42 (1979) 407 [22] F. Reines, E. Pasierb and H. Sobel, Univ. of California preprint (1980) [23] T. Cheng, E. Eichten and L.F. Li, Phys. Rev. D9 (1974) 2259; N. Cabibbo, L. Maiani, G. Parisi and R. Petronzio, Nucl. Phys. B158 (1979) 295 [24] G. Lazarides and Q. Shaft, CERN preprint TH 2984 (1980)