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Proton lifetime and fermion masses in an SO(10) model
Nuclear Physics B181 (1981) 287-300 © North-Holland Publishing Company
PROTON LIFETIME AND FERMION MASSES IN AN SO(10) MODEL G. LAZARIDES and Q. SHAFI CERN, Genev~ Switzerland C. WETTERICH University of Freiburg, Germany Received 17 October 1980
Some consequences of an SO(10) gauge theory which breaks down to SU(3)c × U(l)e.m" via SU(4)c × SU(2)L × SU(2)R are presented. These include: (i) A proton lifetime estimate of (1-20) × 1031 yr. (ii) Mass relations involving quarks and charged leptons. The explanation of the violation of the asymptotic SU(5) relation m e ffi m d is linked to the new intermediate mass scale in SO(10). The top quark mass is estimated to be 20 + 2 GeV at the toponium mass. (iii) The possibility of a fourth generation (pc,,~', b', t') with m,, ~ 17-35 GeV, m b, ~ 42-118 GeV and mt,~ 150-230 GeV. (iv) Heaviest neutrino mass in the 1-10 eV range. In addition, the masses of the superheavy neutrinos are of order 5 × 10~2-5 × 1013 GeV. (v) Neutron-antineutron oscillations with %~ > 103s yr. 1. Introduction T h e unified SO(10) g a u g e m o d e l [1] possesses c e r t a i n features n o t s h a r e d b y the s i m p l e s t SU(5) m o d e l [2]. I n p a r t i c u l a r , B - L s y m m e t r y a p p e a r s in SO(10) as a s p o n t a n e o u s l y v i o l a t e d g a u g e s y m m e t r y . T h e SO(10) t h e o r y also possesses left-right s y m m e t r y which is s p o n t a n e o u s l y violated. M o r e o v e r , the existence of f i g h t - h a n d e d n e u t r i n o fields in SO(10) i m p l i e s t h a t n e u t r i n o s m u s t a c q u i r e a n o n - v a n i s h i n g mass. I n c o n t r a s t to SU(5), SO(10) a d m i t s various s y m m e t r y - b r e a k i n g p a t t e r n s , s o m e c o n t a i n i n g n e w i n t e r m e d i a t e m a s s scales, a n d all c o m p a t i b l e with p r e s e n t p h e n o m e n o l o g y [3]. H o w e v e r , it has r e c e n t l y b e e n a r g u e d [4] that c o s m o l o g i c a l c o n s i d e r a tions s e e m to select o u t o n e of these patterns. I n p a r t i c u l a r , it has b e e n s h o w n t h a t the s u p e r h e a v y m a g n e t i c m o n o p o l e d e n s i t y at the time of n u c l e o s y n t h e s i s c a n b e n a t u r a l l y s u p p r e s s e d o n l y if SO(10) b r e a k s d o w n to SU(3)c × U(1)e.m" via SU(4)c × SU(2)L X SU(2)R * as follows: Mx
SO(10) ---) SU(4)c × SU(2)L × SU(2)R MR
--~ SU(3)¢ × SU(2)L × U ( I ) y M--~SU(3)~ × U(1)c .m ."
* This subgroup of SO(10) was first considered in ref. [5]. 287
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In this note, we discuss some phenomenologlcal implications of this breaking scheme of SO(10). A renormalization-group analysis implies Mx---" 1015 GeV, MR--~ 1.5 X 1013 GeV and M L ~ ' 8 0 GeV for sin20w -- 0.23 and a s ( M L ) = 0.11. In this typical case, the proton lifetime is estimated to be ~p ~" (1-20) x 1031 )a'. The upper bound on the proton lifetime is, however, of the order of 1035 yr, for sin2#w = 0.24 and as(ME) ----0.14. Mass relations involving charged fermions are discussed. We find that the "good" SU(5) predictions [6] m b ( M R ) = m , ( g R ) and ms(MR)=m~,(MR)* are naturally maintained, whereas the bad relation rod(MR) = m e ( M R ) can be strongly violated. We estimate that m s (1 GeV) varies between 370 and 570 MeV and m b (10 GeV) between ,.6 and 5.5 GeV, for 0.11 < as(ME) < 0.14 (100 MeV < A < 400 MeV). We also obtain the relation m r ( M R ) = m c ( M R ) m , ( M R ) / m , ( M R ) * * . The top-quark mass at the toponium scale is then predicted to be 20 + 2 GeV. It is interesting to note that the SO(10) model considered here contains enough freedom so that one can obtain the correct values for the masses of the charged fermions of the first family as well as for the Cabibbo angle. The model allows for a fourth family v,,, T', b' and t'. The T' lepton has a mass between 17 and 35 GeV***. Then we find for as(ML)=O.11 that m t, at the "heavy" toponium scale varies between 150 GeV and 230 GeV, and m b, at the "heavy" bottomonium scale is 42-118 GeV. We discuss the problem of neutrino masses. The light physical neutrino masses are proportional to M ~ / M R ****[8]. If the B - L violation was connected with the scale M x (typically of order 1015 GeV) one would obtain physical neutrino masses of order 1 0 - 2 - 1 0 - 3 eV, which seem not to be favoured by recent experimental evidence [9]. Since B - L is violated in our model at M R < Mx, we can allow for bigger neutrino masses. The heaviest light neutrino mass can be of order 1-10 eV. The superheavy neutrino masses are of order 5 x 1012-5 × 10 '3 GeV. Finally, since B - L is violated, neutron-antineutron oscillations are allowed. One obtains ~'~ ~> 10 38 yr.
2. Symmetry-breaking pattern Let us consider the following symmetry-breaking pattern of SO(10):
SO(lO) SMx SU(4)~ x SU(2)L X SU(2)R SMR SU(3)c X SU(2)L X U(1)r
*ME SU(3)~ × U(1) .... •
(1)
* Note that in our case the asymptoticrelations hold at MR. ** An identical asymptotic relation, valid at Mx, has previously been considered in ref. [7], in the context of an E6 model. Their prediction for the top-quark mass is essentially the same as ours.
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The first step in the above breaking scheme is achieved by a 54-plet of Higgs fields [4]. The gauge bosons of SO(10) that can mediate proton decay in lowest order get masses of order M x. The second step requires a 12~6-plet of Higgs fields. In this case the gauge boson which couples to the B - L generator, the charge ~ bosons in SU(4)c and the gauge bosons that mediate charged fight-handed interactions, all get masses of order M R (typically, M R is about two orders of magnitude smaller than Mx). In addition, the right-handed neutrino fields pick up Majorana masses of order M R. For the final step of the breaking one needs a 10-plet of scalar fields. Thus, the minimal Higgs system which implements the above symmetry-breaking scheme employs 54, 12,~6and a IO of Higgs fields. For completeness, let us list the decompositions of these Higgs fields under SU(4)¢ x SU(2)L x SU(2)R: 2
54-- (20, 1, l) + ( 6 , 2 , 2 ) + (1,3,3) + (1, 1, 1), 126 = (TO, 1,3) + (10, 3, 1) + (15, 2, 2) + (6, 1, 1), 10 = (6, 1, 1) + (1,2, 2).
(2) u
Note that the VEV of the 126 that breaks B - L lies in the (10, 1, 3) component. The corresponding scalar transforms like a composite state of two right-handed neutrinos. Condensation of such composite states should lead to spontaneous breakdown of B - L, left-right symmetry and lepton-quark universality (SU(4)c).
3. Mass scales M x and M a and the proton lifetime In order to estimate the proton lifetime in the SO(10) model under discussion, we need to estimate the mass scale M x. Standard one-loop renormalization group arguments lead to the following relations:
sin2#w(ML) "~3 1
2¢r
ML
~ In "~L
'
a(ML) a ( M L ) { 22 M x MR} sin2 #w (ML) "~ as(ME) -~ 2~r 5- In M----L ~! l n ~ L "
(3)
Given a ( M L ) , OIs(ML) and sin2Ow, the mass scales M x and M R c a n be estimated. Some comments are in order: (a) In the limit M R = M x, one recovers the standard SU(5) predictions. (b) The Higgs contributions have been neglected since the scalar fields may be composite. However, if one assumes that only one SU(2)L Higgs doublet contributes to the renormalization group equations between M L and M x, the effect is to *** The reason for choosing this value of my, is explained below. **** Remember that the scale MR is connected to the violation of B - L.
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reduce the value of M x quoted below by a factor of 2. The scale M R remains essentially unchanged. (c) The effect of the two-loop corrections on the mass scales M x and Mr. as obtained from (3) is to reduce them by a factor of about three, just as in the SU(5) model [10]. For definiteness, let us take a(ML) -i as(ML) = 0.11 and sin 2 0 w = 0.23, where -- 12~,5' M L --38.5 G e V / s i n 0 w. This gives M x = 1015 GeV, M R ---- 1.7 × 1013
(4)
(5)
GeV.
The proton lifetime Zp can now be estimated*. One obtains 1-p'-~ (1-20) × l031 yr.
(6)
The effect of increasing (decreasing) as(ME) (keeping sin2 0 w fixed) is to increase (decrease) the mass scales M x and M R. The upper bound on the proton lifetime in the present scheme is about 10 35 yr, corresponding to a s(ML) -----0.14and sin2 0 w = 0.24. 4. Charged f e r m i o n m a s s e s
We know from experiment that the masses of the b quark and the ~"lepton satisfy rather well the minimal SU(5) prediction m b ( M x ) = m~(Mx). For the second generation, the relation m s ( M x ) = - m z ( M x) leads to a value m s (1 GeV)~>400 MeV which could be compatible with experiment. However, the relation for the first generation m d ( M x ) = m e ( M x) seems to be strongly violated, thus forcing us to have more than one scalar multiplet coupled to the fermions. In models based on SO(10) with only one 10-plet tPl0 responsible for the fermion masses, we again have the above relations. Since all the fermions belong to an irreducible representation of SO(10), one also has the following relations for the u-type quark masses and for the Dirac masses of the neutrinos: mt(nx)
=
D m~,(Mx )
,
mc(Mx)
mt(Mx) m c(Mx) m¢(Mx)
m.(Mx)
=
=
m ~D ( M x ) , m~(Mx) ms(Mx) '
= m~'(Mx)
m¢(Mx)
* For earlier estimates on proton lifetime see ref. [11].
m , ( M x ) __- m ,D. ( M x ) , (7) (8)
(9)
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All the Cabibbo-like mixing angles between the generations vanish. Again, relation (9) for the first generation is bad. However, we need a scalar ~ - p l e t q01e6 whose SU(2)L singlet and SU(2)L triplet vacuum expectation values provide the Majorana masses for the right-handed and left-handed neutrinos. Since there exist quartic scalar couplings involving three tP126, and one tpl o, the SU(2)L doublets in ~P126are forced to acquire non-vanishing VEV's and thus modify the above mass relations. Our main point here is the observation that the existence of an intermediate mass scale M R (MR<
.~¢y = V ~ f/j~/C~OlO~j + 1~hij~b; "F-¢~01261#j+ h.c.
(10)
Here i and j are generation indices. The relevant electrically neutral and colorless fields whose VEV's break the weak subgroup SU(2)L >( U(1)r belong to q~lO(1,2, 2) and q0126(15,2, 2). After the spontaneous breakdown of SU(4)c x SU(2)L × SU(2)R to SU(3)¢ )< SU(2)L >( U(1)r at the scale M R, the four colour singlet SU(2)L doublets contained in q0m(1, 2, 2) and q0126(15,2, 2) are mixed, and a linear combination of them acquires a non-vanishing SU(2)L breaking VEV. We denote the components of the VEV by <¢PlO(1,2,2)13L=_l/E,13a=l/2
> = W,
<~I26(15,2,2)I3L.--I/2,13R=l/2>
=y ,
<~lO(1,2,2)13L.l/2,1m=_l/2)
<~126(15,2,2)13L.l/2,13a._l/2>
~- X ,
= Z,
(11)
where ML2 _- 1- s g 2 (ML)(lwl2
+ [x[2+
ly12+ Iz12).
The most general mass matrix for u- and d-type quarks, charged leptons and neutrinos (only Dirac mass term) is M u =f,.jw ffi ~
h i j Y ffi otx i + [3Zij ,
g d =fijx "+V ~ h i j T ,
= X i + Zij,
e e ~ - f i j x -- 3VF~ h i j z ~- x i - 3zij , M~ =fijw
-
3~f~ hijY = ax i
assume that there is only one Y u k a w a coupling fermions. This can be achieved by a discrete symmetry.
* Here we
-
-
3flzij.
between the complex scalar ~10
(12) and the
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Without loss of generality we have chosen f,7 to be diagonal and we denote
a = w/x,
fl = y / z ,
zij ~" ~ 2 ~ hijz,
xi "~fii x
Consider now the mixing between the weak doublets ~1~6 and ¢p~20).This mixing is induced at the scale M R by the quartic coupling --A(~26)3~10, where the relevant term involves the square of the VEV of the singlet (¢p~12)6) = M R/g. Assuming that some not yet understood mechanism leads to a hierarchically small VEV (~20)), the leading terms in the effective potential for ¢p~22) 6 are given by V = -~[/-,~(1)\[2/¢"(2)\'~(2) "l-lp.2(fp(22)6)2. '"lN"iv126 / I N'f'lO/'11"126
(13)
Since/~2 has several contributions of order M x2, we denote/~2__ (~k,/g2)M~ and assume X ' ~ X. One obtains h M~/
(2)
<~0(22)6> = )k' M x2 \qOlO) (((qO(120)) "
(14)
Let us first concentrate on the value M R / M x ~ 10 -2 which is also needed in order to have neutrino masses in the eV region (see below). The VEWs y and z are then constrained to be about 15 MeV. Since we need contributions to the fermion mass matrix of this order to modify the first generation mass relations and to explain the measured value of the Cabibbo angle, the Yukawa couplings hi/have to be of order one, at least for the first generation. (Note also that a value M R / M x much smaller than 10 -2 would lead to unacceptably large Yukawa couplings and is therefore excluded). We are then left with the following situation. The relations mb(MR) = m , ( M g ) '
mr(MR) = m~,(Mg) D
(15)
are expected to hold with 1% accuracy~ whereas the relations ms(MR) ---- m . ( M R ) ,
mo(MR)=
D
me(MR) m t ( M a ) = me(MR) m~,(MR )
(16)
may have corrections of the order of 10% due to y and z. We may also consider a fourth generation p~,, ¢', b' and t' where the mass relations
mb,(MR) = m..(MR),
m,,(MR) = m
, . , m,,( MR) m,,(MR) = m,tMR)
(MR), (17)
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are expected to be even more accurate. For the first generation we find, from (12), +_ m o ( M R )
----
++_mu( M R) =
"C-me(MR) ----
sin20c
t- x I + z l l ,
+_
f12z~2 + mc(mg)
_.T.mr( M R)
=z~2
I- ctx I + flzn ,
F Xi - 3 z n ,
fl ~ 1 )2 mc(MR) _ ms(MR)
(18)
We have different possible choices of signs since the signs of the fermion masses cannot be measured. Since we have five free parameters (x l, z n, z]2, a, t ) for four measured quantities, it is not difficult to arrange for consistency with experiment. 5. Prediction of top quark mass Mass renormalization effects have to be included to derive the mass relations at present energies from the relations quoted at the scale M R (we only consider one-loop corrections). The top quark mass only depends on the value of as(ME) and is given by
m,( 2m,) ( as(40 GeV) l12/23( as(10 GeV) ) '2/25 mt(2mt)"~mc(Emc)m~,(2m~,) a~(10GeV)] ~a - ~
(19)
For ors(ME)----0.11 our prediction of the top quark mass is mt(40 GeV) ~ 20.7 GeV
(20)
Here we used me(3 GeV)--- 1.6 GeV as an input. For larger values of as(ME) we find lower values for m t (for example as(ME)-----0.14 gives m t (40 G e V ) = 18.2 GeV). There is a 10% inaccuracy in eq. (19) since rn~(2ma) stands here only for the q0~0 contribution to the muon mass which differs by about 10 MeV from the measured muon mass. For the d-type quark masses one has the relation
msb 0' msb' R' (Os 0')2/ 33 2.(
m,,,(O)
(21)
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Here f is the effective number of flavours in the appropriate energy region. The dependence of the s and b quark masses on sin2t~w is weak. We find, for three generations and for the above values of %(ML), a variation of m s (1 GeV) between 370 and 570 MeV and a variation of rn b (10 GeV) between 4.6 and 5.5 GeV. Inclusion of a fourth generation increases these predictions by about 6%.
6. Fourth generation
The renormalization effects in our SO(10) model permit a fourth generation (v,,, 7', t', b') of leptons and quarks. It is quite remarkable that in our scheme the masses of the charged lepton ~', and also of t' and b', can be fairly reliably estimated. This comes about as follows. From ongoing experiments at PETRA, it is presumably reasonable to assume that m,, > 17 GeV. However, m~, cannot be much larger than 35 GeV since, otherwise, the prediction for the t' mass at the heavy toponium (t'-I') scale violates the upper bound on fermion masses [12] obtained from the requirement that Yukawa couplings should not become strong anywhere in the range between M L and M x. Taking m~, = 35 GeV as input, we can predict the masses of t' and b'. Using similar renormalization relations as for the b and t quark mass, we find, for % ( M L ) = 0.11, mt,(2mt, ) ~---230 GeV,
mb,(2mb, ) ~ 118 GeV.
(22)
Thus, for m,, between 17 and 35 GeV we predict m b, between 42 and 118 GeV and m t, between 150 and 230 GeV. We also note that larger values of as(ML) give lower masses for the t' and higher masses for the b'. This effect is about 15% for a s ( M L ) = 0.14.
7. Possible modifications of mass relations
Let us discuss what happens for higher values of M R / M x. The couplings hij could now be somewhat smaller than one, say 0.1. On the other hand, the VEV's y and z become larger, and their contribution to the fermion masses could be in the range of 150 MeV, thus destroying the mass relations for the second generation and the prediction of the top quark mass [compare (12)]. In order to maintain the generation pattern, the Yukawa couplings hij of the first generation have to be of the o r d e r (M2x/M2)IO -4. For values of M R / M x ~ l0 -1, one has y and z ~ 1.5 GeV and the successful relation (15) is destroyed. Thus, if we want to keep this relation, only a narrow range M R
10-2 <77--, < 10 - l
Iv1x
(23)
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can be admitted. Furthermore, for our top quark mass prediction to hold, M s / M x has to be near 10-2. We finally remark that these results can be generalized to an arbitrary choice of Higgs fields. The only scalars that can couple to fermions in SO(10) are qolO, I]0126 and q012o. If SO(10) is broken down to SU(4)c × SU(2)L × SU(2)R by the VEV of a 54-plet, the VEV's of the SU(2)L doublets belonging to q~126 and q012o are suppressed by a factor M ~ / M 2 x .
8. Majorana masses of the right-handed neutrinos As has been pointed out in ref. [8], one needs large Majorana masses for the right-handed neutrinos in order to account for the smallness of the masses of the observed left-handed neutrinos within SO(10). The Majorana masses a i of the right-handed neutrinos violate B - L and left-right symmetry. They are generated by the VEV of the SU(2)L singlet (q0~6)= M g / g contained in (TO, 1,3). Corresponding Majorana masses c i of the left-handed neutrinos come from the VEV of the SU(2)L triplet (q0~6) contained in (10,3, 1). Neglecting for the moment the different generations, the neutrino mass matrix has eigenvalues a and c ( 1 / a ) ( m ~ ) 2, corresponding essentially to right-handed and left-handed Majorana masses. Consider first the right-handed neutrinos. Their mass eigenvalues are given by hi(MR) ai = ~ M
g(MR)
R,
(24)
where h i are the eigenvalues of the matrix hij. For M R / M x ~ 10 -2, at least one eigenvalue h i has to be of the order of one (see the discussion of the charged fermion masses). Thus, for Ors(ME)--0.11, we find for the largest right-handed neutrino mass m~R= (1--5) × 1013 GeV
(25)
For larger values of as(ML) this value increases (by a factor 5 for Ots(ML) = 0.14). If there is no generation pattern for the h i all the right-handed neutrino masses are expected to be in the range of about 5 × 1012-5 X 1013 GeV. We note, however, that a larger value M R / M x "~ 10 -1 requires a generation pattern of the h i leading to Majorana masses of the electron (muon) neutrino of order 5 x 1011-5 × 1012 GeV (5 X 1012-5 X 1013 GeV), whereas the mass of the heaviest neutrino may be of order 5 × 1013-5 × 1014 GeV. (The values have been quoted for 0ts(ML) between 0.11 and 0.14.) We also note that even admitting eigenvalues h i of order 10-4 (which cannot be related with the "normal" generation pattern) the Majorana mass of the lightest right-handed neutrino has to be larger than 109 GeV.
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9. B - L violation, neutron oscillations and baryon asymmetry
For MR<fi violate both B - L and B + L and are suppressed by powers both of E/2C-tx and E/;C/I R. The effective interaction leading to these oscillations has the form of a six-fermion operator and is therefore suppressed by five powers of the heavy mass scales. For oscillations mediated by X-bosons as well as a right-handed neutrinos, the suppression factor is m n5 / M x4M R. The n-fi oscillation time z~, is enhanced by a factor of order i('lR/m n compared with the proton lifetime. One finds, for a typical Majorana mass r ~ of order 1013 GeV, ~
'rn~ ,~, 10 42
yr.
(26)
Even with our smallest values of m~ ~ R ~> l0 9 GeV we find %a~> l03s yr. Let us finally comment on the baryon asymmetry in the universe. At temperatures of order M x a baryon asymmetry is created by the decay of the X-bosons. Since B - L is conserved at this temperature one has AB ----AL. At temperatures of order M R the B - L violating interactions become effective. These interactions conserve A B + A L and the baryon asymmetry created at temperatures below M x will not be washed out completely. It has been pointed out [16], however, that there is another interesting source of baryon asymmetry due to the decay of right-handed neutrinos at temperatures below M R. Our quoted values for Majorana masses and Yukawa couplings can account for a baryon asymmetry of order 1 0 - s to 10-9.
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10. Light neutrino masses and neutrino oscillations It has been suggested that the solar neutrino puzzle can be solved by neutrino oscillations [17]. Recent experiments [9] indicate neutrino oscillations compatible with neutrino mass differences in the eV range. How can these neutrino masses be obtained in unified gauge models? A crude estimate gives an upper bound on the light neutrino masses of (27)
m~L ~
where M R is the scale of B - L violation. Clearly, MR-----Mx,~ 1015 GeV leads to neutrino masses smaller than 10 -2 eV. However, an intermediate scale M R ~ 10-2M x predicts neutrino masses in the eV range. The light neutrino masses are given by m,L = C - ( 1 / a ) ( m D ) s. We consider the two contributions separately and start with the Majorana mass term c. Then the masses of the different neutrinos are given by (28)
c i = hi(mL)(cp(13)26).
As has been pointed out earlier [13], the SU(2)L triplet field cp(132)6belonging to (10, 3, 1) is forced to acquire a non-vanishing VEV by the quartic scalar couplings (cP126)4, (CPls6)3Cplo and (~126)2(9~10)s. The leading terms in the effective potential for the cp~6 can be written as V=
_~,. / ~ ( ! ) \ / ~ ( 2 ) \ 2 / ~ ( 3 ) \ "1X~126/\~i0 / \~F126/
+
(29)
_
This induces 2 g ( M R ) M2 ~1 g2(ML) MR r2
1.5 ×
1011
eV.
ML KI MR KS '
(30)
where
Here we use the fact that 1,2 cannot be much larger than M 2 without violating our symmetry-breaking pattern, since it is related by left-right symmetry (which is unbroken above M R) to the mass term of ~P~6. The value M L / M R strongly depends on sin20w . For as(ML) ----0.11 and sin20w between 0.21 and 0.24, it varies between 5 × 10 -13 and 1.5 × 10 - u . However, the supplementary constraint M R / M x ~ 10-2 derived from the charged fermion mass
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spectrum implies ML/MR<,~6 l0 2, we find
×
10 -12. Thus, for as(ML) ----0.11 and M R / M x --
-
KI
L
(31)
ci,-~leV'~2hi(m~ ) •
Larger values of as(ME) give somewhat lower neutrino masses. In general, KI and ~2 are both of order O(g 2) and their ratio is expected to be of order unity. For M R / M x ~ 10 -2, the largest eigenvalue among h i has to be of order one. Depending on x~/x2, we then predict the contribution c i to the mass of the heaviest neutrino to be in the range of a few eV. The contribution to the other neutrino masses can be of the same order of magnitude or much smaller, depending on the h i. If there is only one neutrino eigenstate with mass in the eV range, it must have a substantial contribution from the electron neutrino. For M R / M x > 10 -2, the neutrino masses are smaller due to smaller values of M E / M R and smaller values of h i for the electron (and muon) generation. If we take, for example, M R / M x ~ l0 -1 we would find the following neutrino mass spectrum: m L ~ 10 -3 eV,
m ~L
10 -2 eV,
L m ~,(~,)
10 -1 eV.
Neutrino masses in the eV range clearly favour M R / M x ~ 10 -2. Let us next discuss the contribution from the Dirac neutrino masses. The largest contribution would be from the Dirac mass of the neutrino of the fourth generation m I,T, O • Taking renormalization effects into account, we find (for m ~, -- 20 GeV) mD ( M L ) ~ 100 G e V .
(32)
Assuming that all the Majorana masses of right-handed neutrinos are of the same order M R, the contribution to the light neutrino masses is given by ( M R -- 1.5 × 1013 GeV)
Ac ~ 0.5 eV g( MR ) h(MR)
(33) "
In this case the fourth generation neutrino Dirac mass contributes less than 1 eV, since g ( M R ) / h ( M R ) has to be of order one. The contribution of the Dirac masses of the first three generations is much smaller, since it is suppressed by a factor D 2 D 2 (mr,) ~(mr,.) compared with the fourth generation. In summary, the light neutrino masses depend strongly on M R / M x and the values of Yukawa couplings. A lower bound for the mass of the heaviest neutrino
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of 10 - 2 eV is found (for M R / M x ~ 10-1). Neutrino masses in the 1-10 eV range are predicted for M R / M x ~ 10-2. We expect large mixing between neutrinos of different generations which lead to neutrino oscillations observable with present experiments. Since the oscillation length L is given by L = 4~rE/Am 2 we expect, for E - - 10 MeV, oscillation lengths of the order of 25 cm to 25 m. 11. Conclusion To conclude, the presence of an intermediate mass scale related to the violation of the B - L symmetry seems to be favoured by a number of considerations, both phenomenological and cosmological. Thus, sin2~w > 0.20 can easily be accommodated. The bad SU(5) asymptotic prediction m e = m d is avoided, while the good predictions for the second and third generations are maintained. The top quark mass is predicted to be 20 _+ 2 GeV. Neutrino masses of the order 1- l0 eV can be naturally obtained, as indicated by recent experiments. Moreover, the cosmological monopole problem can be avoided if SO(10) breaks via SU(4)c × SU(2)L × SU(2)R. Last, but by no means least, we urge experimentalists to make an all out effort to look for the t quark and the lepton of the fourth generation in the 20 GeV energy region. References [1] H. Fritzseh 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. Galliard, Nucl. Phys. B128 (1977) 506; H. Georgi and D.V. Nanopoulos, NucL Phys. B155 (1979) 52 [2] H. Georgi and S.L Glashow, Phys. Rev. Lett. 32 (1974) 438 [3] H. Georgi and D.V. Nanopoulos, NucL Phys. B159 (1979) 16; Q. Shaft, M. Sondermann and C. Wetterich, Phys. Lett. 92B (1980) 304; S. Rajpoot, Imperial College preprint (Dec., 1979) C. Vayonakis, Athens preprint (1978) [4] G. Lazarides, M. Magg and Q. Shaft, CERN preprint TH 2856 (1980) [5] J. Pail and A. Salam, Phys. Rev. DI0 (1974) 275 [6] A. Buras, J. Ellis, M. Galliard and D. Nanopoulos, Nucl. Phys. B135 (1978) 66 [7] R. Barbieri and D. Nanopoulos, CERN preprint TH 2810 (1980) [8] M. GelI-Mann, P. Ramond and IL Slansky, unpublished [9] F. Reines, E. Pasierb and H.W. Sobel, Univ. of California preprint (1980) [10] W. Marciano, Rockefeller preprint (1980); P. Binetruy and T. Schilcker, CERN preprint, to appear; J. Ellis, M. Gaillard, D. Nanopoulos and S. Rudaz, Nucl. Phys. B176 (1980) 61 [11] J. Ellis, M. GaiUard and D. Nanopoulos, CERN preprint TH 2749 (1979); C. Jarlskog and F. Ynduraln, Nucl. Phys. B149 (1979) 29; A. Din, G. Girardi and P. Sorba, Lapp preprint TH08 (1979); J. Donoghue, MIT preprint CTP 824 (1979); M. Machacek, Nucl. Phys. B159 (1979) 37; M.B. Gavela, A. Le Yaouanc, L. Oliver, O. P~ne and J.C. Raynal, Orsay preprint LPTHE 80/6 0980) [12] N. Cabibbo, L. Malani, G. Parisi and R. Petronzio, Nu¢l. Phys. B158 (1979) 295 [13] M. Magg and C. Wetterich, Phys. Lett. 94B (1980) 61
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[14] S. Weinberg, Phys. Rev. Latt. 43 (1979) 1566; F. Wilczek and A. Zee, Phys. Rev. Lett. 43 (1979) 1571 [15] R.N. Mohapatra and R. E. Marshal Virginia University preprint VIP-HEP 80/1 [16] J. Harvey, P. Ramond and D. Reiss, Caltech preprim CALT 68-758 0950); T. Yanagida and M. Yoshimura, KEK preprint TH8 0980) [17] B. Pontecorvo, JETP (Soy. Phys.) 53 (1967) 171; A. De R~jula, M. Lusignoli, L. Maiani, S. Petcov and R. Petronzio, Nucl. Phys. B168 0980) 54
PROTON LIFETIME AND FERMION MASSES IN AN SO(10) MODEL G. LAZARIDES and Q. SHAFI CERN, Genev~ Switzerland C. WETTERICH University of Freiburg, Germany Received 17 October 1980
Some consequences of an SO(10) gauge theory which breaks down to SU(3)c × U(l)e.m" via SU(4)c × SU(2)L × SU(2)R are presented. These include: (i) A proton lifetime estimate of (1-20) × 1031 yr. (ii) Mass relations involving quarks and charged leptons. The explanation of the violation of the asymptotic SU(5) relation m e ffi m d is linked to the new intermediate mass scale in SO(10). The top quark mass is estimated to be 20 + 2 GeV at the toponium mass. (iii) The possibility of a fourth generation (pc,,~', b', t') with m,, ~ 17-35 GeV, m b, ~ 42-118 GeV and mt,~ 150-230 GeV. (iv) Heaviest neutrino mass in the 1-10 eV range. In addition, the masses of the superheavy neutrinos are of order 5 × 10~2-5 × 1013 GeV. (v) Neutron-antineutron oscillations with %~ > 103s yr. 1. Introduction T h e unified SO(10) g a u g e m o d e l [1] possesses c e r t a i n features n o t s h a r e d b y the s i m p l e s t SU(5) m o d e l [2]. I n p a r t i c u l a r , B - L s y m m e t r y a p p e a r s in SO(10) as a s p o n t a n e o u s l y v i o l a t e d g a u g e s y m m e t r y . T h e SO(10) t h e o r y also possesses left-right s y m m e t r y which is s p o n t a n e o u s l y violated. M o r e o v e r , the existence of f i g h t - h a n d e d n e u t r i n o fields in SO(10) i m p l i e s t h a t n e u t r i n o s m u s t a c q u i r e a n o n - v a n i s h i n g mass. I n c o n t r a s t to SU(5), SO(10) a d m i t s various s y m m e t r y - b r e a k i n g p a t t e r n s , s o m e c o n t a i n i n g n e w i n t e r m e d i a t e m a s s scales, a n d all c o m p a t i b l e with p r e s e n t p h e n o m e n o l o g y [3]. H o w e v e r , it has r e c e n t l y b e e n a r g u e d [4] that c o s m o l o g i c a l c o n s i d e r a tions s e e m to select o u t o n e of these patterns. I n p a r t i c u l a r , it has b e e n s h o w n t h a t the s u p e r h e a v y m a g n e t i c m o n o p o l e d e n s i t y at the time of n u c l e o s y n t h e s i s c a n b e n a t u r a l l y s u p p r e s s e d o n l y if SO(10) b r e a k s d o w n to SU(3)c × U(1)e.m" via SU(4)c × SU(2)L X SU(2)R * as follows: Mx
SO(10) ---) SU(4)c × SU(2)L × SU(2)R MR
--~ SU(3)¢ × SU(2)L × U ( I ) y M--~SU(3)~ × U(1)c .m ."
* This subgroup of SO(10) was first considered in ref. [5]. 287
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In this note, we discuss some phenomenologlcal implications of this breaking scheme of SO(10). A renormalization-group analysis implies Mx---" 1015 GeV, MR--~ 1.5 X 1013 GeV and M L ~ ' 8 0 GeV for sin20w -- 0.23 and a s ( M L ) = 0.11. In this typical case, the proton lifetime is estimated to be ~p ~" (1-20) x 1031 )a'. The upper bound on the proton lifetime is, however, of the order of 1035 yr, for sin2#w = 0.24 and as(ME) ----0.14. Mass relations involving charged fermions are discussed. We find that the "good" SU(5) predictions [6] m b ( M R ) = m , ( g R ) and ms(MR)=m~,(MR)* are naturally maintained, whereas the bad relation rod(MR) = m e ( M R ) can be strongly violated. We estimate that m s (1 GeV) varies between 370 and 570 MeV and m b (10 GeV) between ,.6 and 5.5 GeV, for 0.11 < as(ME) < 0.14 (100 MeV < A < 400 MeV). We also obtain the relation m r ( M R ) = m c ( M R ) m , ( M R ) / m , ( M R ) * * . The top-quark mass at the toponium scale is then predicted to be 20 + 2 GeV. It is interesting to note that the SO(10) model considered here contains enough freedom so that one can obtain the correct values for the masses of the charged fermions of the first family as well as for the Cabibbo angle. The model allows for a fourth family v,,, T', b' and t'. The T' lepton has a mass between 17 and 35 GeV***. Then we find for as(ML)=O.11 that m t, at the "heavy" toponium scale varies between 150 GeV and 230 GeV, and m b, at the "heavy" bottomonium scale is 42-118 GeV. We discuss the problem of neutrino masses. The light physical neutrino masses are proportional to M ~ / M R ****[8]. If the B - L violation was connected with the scale M x (typically of order 1015 GeV) one would obtain physical neutrino masses of order 1 0 - 2 - 1 0 - 3 eV, which seem not to be favoured by recent experimental evidence [9]. Since B - L is violated in our model at M R < Mx, we can allow for bigger neutrino masses. The heaviest light neutrino mass can be of order 1-10 eV. The superheavy neutrino masses are of order 5 x 1012-5 × 10 '3 GeV. Finally, since B - L is violated, neutron-antineutron oscillations are allowed. One obtains ~'~ ~> 10 38 yr.
2. Symmetry-breaking pattern Let us consider the following symmetry-breaking pattern of SO(10):
SO(lO) SMx SU(4)~ x SU(2)L X SU(2)R SMR SU(3)c X SU(2)L X U(1)r
*ME SU(3)~ × U(1) .... •
(1)
* Note that in our case the asymptoticrelations hold at MR. ** An identical asymptotic relation, valid at Mx, has previously been considered in ref. [7], in the context of an E6 model. Their prediction for the top-quark mass is essentially the same as ours.
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The first step in the above breaking scheme is achieved by a 54-plet of Higgs fields [4]. The gauge bosons of SO(10) that can mediate proton decay in lowest order get masses of order M x. The second step requires a 12~6-plet of Higgs fields. In this case the gauge boson which couples to the B - L generator, the charge ~ bosons in SU(4)c and the gauge bosons that mediate charged fight-handed interactions, all get masses of order M R (typically, M R is about two orders of magnitude smaller than Mx). In addition, the right-handed neutrino fields pick up Majorana masses of order M R. For the final step of the breaking one needs a 10-plet of scalar fields. Thus, the minimal Higgs system which implements the above symmetry-breaking scheme employs 54, 12,~6and a IO of Higgs fields. For completeness, let us list the decompositions of these Higgs fields under SU(4)¢ x SU(2)L x SU(2)R: 2
54-- (20, 1, l) + ( 6 , 2 , 2 ) + (1,3,3) + (1, 1, 1), 126 = (TO, 1,3) + (10, 3, 1) + (15, 2, 2) + (6, 1, 1), 10 = (6, 1, 1) + (1,2, 2).
(2) u
Note that the VEV of the 126 that breaks B - L lies in the (10, 1, 3) component. The corresponding scalar transforms like a composite state of two right-handed neutrinos. Condensation of such composite states should lead to spontaneous breakdown of B - L, left-right symmetry and lepton-quark universality (SU(4)c).
3. Mass scales M x and M a and the proton lifetime In order to estimate the proton lifetime in the SO(10) model under discussion, we need to estimate the mass scale M x. Standard one-loop renormalization group arguments lead to the following relations:
sin2#w(ML) "~3 1
2¢r
ML
~ In "~L
'
a(ML) a ( M L ) { 22 M x MR} sin2 #w (ML) "~ as(ME) -~ 2~r 5- In M----L ~! l n ~ L "
(3)
Given a ( M L ) , OIs(ML) and sin2Ow, the mass scales M x and M R c a n be estimated. Some comments are in order: (a) In the limit M R = M x, one recovers the standard SU(5) predictions. (b) The Higgs contributions have been neglected since the scalar fields may be composite. However, if one assumes that only one SU(2)L Higgs doublet contributes to the renormalization group equations between M L and M x, the effect is to *** The reason for choosing this value of my, is explained below. **** Remember that the scale MR is connected to the violation of B - L.
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reduce the value of M x quoted below by a factor of 2. The scale M R remains essentially unchanged. (c) The effect of the two-loop corrections on the mass scales M x and Mr. as obtained from (3) is to reduce them by a factor of about three, just as in the SU(5) model [10]. For definiteness, let us take a(ML) -i as(ML) = 0.11 and sin 2 0 w = 0.23, where -- 12~,5' M L --38.5 G e V / s i n 0 w. This gives M x = 1015 GeV, M R ---- 1.7 × 1013
(4)
(5)
GeV.
The proton lifetime Zp can now be estimated*. One obtains 1-p'-~ (1-20) × l031 yr.
(6)
The effect of increasing (decreasing) as(ME) (keeping sin2 0 w fixed) is to increase (decrease) the mass scales M x and M R. The upper bound on the proton lifetime in the present scheme is about 10 35 yr, corresponding to a s(ML) -----0.14and sin2 0 w = 0.24. 4. Charged f e r m i o n m a s s e s
We know from experiment that the masses of the b quark and the ~"lepton satisfy rather well the minimal SU(5) prediction m b ( M x ) = m~(Mx). For the second generation, the relation m s ( M x ) = - m z ( M x) leads to a value m s (1 GeV)~>400 MeV which could be compatible with experiment. However, the relation for the first generation m d ( M x ) = m e ( M x) seems to be strongly violated, thus forcing us to have more than one scalar multiplet coupled to the fermions. In models based on SO(10) with only one 10-plet tPl0 responsible for the fermion masses, we again have the above relations. Since all the fermions belong to an irreducible representation of SO(10), one also has the following relations for the u-type quark masses and for the Dirac masses of the neutrinos: mt(nx)
=
D m~,(Mx )
,
mc(Mx)
mt(Mx) m c(Mx) m¢(Mx)
m.(Mx)
=
=
m ~D ( M x ) , m~(Mx) ms(Mx) '
= m~'(Mx)
m¢(Mx)
* For earlier estimates on proton lifetime see ref. [11].
m , ( M x ) __- m ,D. ( M x ) , (7) (8)
(9)
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All the Cabibbo-like mixing angles between the generations vanish. Again, relation (9) for the first generation is bad. However, we need a scalar ~ - p l e t q01e6 whose SU(2)L singlet and SU(2)L triplet vacuum expectation values provide the Majorana masses for the right-handed and left-handed neutrinos. Since there exist quartic scalar couplings involving three tP126, and one tpl o, the SU(2)L doublets in ~P126are forced to acquire non-vanishing VEV's and thus modify the above mass relations. Our main point here is the observation that the existence of an intermediate mass scale M R (MR<
.~¢y = V ~ f/j~/C~OlO~j + 1~hij~b; "F-¢~01261#j+ h.c.
(10)
Here i and j are generation indices. The relevant electrically neutral and colorless fields whose VEV's break the weak subgroup SU(2)L >( U(1)r belong to q~lO(1,2, 2) and q0126(15,2, 2). After the spontaneous breakdown of SU(4)c x SU(2)L × SU(2)R to SU(3)¢ )< SU(2)L >( U(1)r at the scale M R, the four colour singlet SU(2)L doublets contained in q0m(1, 2, 2) and q0126(15,2, 2) are mixed, and a linear combination of them acquires a non-vanishing SU(2)L breaking VEV. We denote the components of the VEV by <¢PlO(1,2,2)13L=_l/E,13a=l/2
> = W,
<~I26(15,2,2)I3L.--I/2,13R=l/2>
=y ,
<~lO(1,2,2)13L.l/2,1m=_l/2)
<~126(15,2,2)13L.l/2,13a._l/2>
~- X ,
= Z,
(11)
where ML2 _- 1- s g 2 (ML)(lwl2
+ [x[2+
ly12+ Iz12).
The most general mass matrix for u- and d-type quarks, charged leptons and neutrinos (only Dirac mass term) is M u =f,.jw ffi ~
h i j Y ffi otx i + [3Zij ,
g d =fijx "+V ~ h i j T ,
= X i + Zij,
e e ~ - f i j x -- 3VF~ h i j z ~- x i - 3zij , M~ =fijw
-
3~f~ hijY = ax i
assume that there is only one Y u k a w a coupling fermions. This can be achieved by a discrete symmetry.
* Here we
-
-
3flzij.
between the complex scalar ~10
(12) and the
G. Lazarides et al. / Proton hfetime
292
Without loss of generality we have chosen f,7 to be diagonal and we denote
a = w/x,
fl = y / z ,
zij ~" ~ 2 ~ hijz,
xi "~fii x
Consider now the mixing between the weak doublets ~1~6 and ¢p~20).This mixing is induced at the scale M R by the quartic coupling --A(~26)3~10, where the relevant term involves the square of the VEV of the singlet (¢p~12)6) = M R/g. Assuming that some not yet understood mechanism leads to a hierarchically small VEV (~20)), the leading terms in the effective potential for ¢p~22) 6 are given by V = -~[/-,~(1)\[2/¢"(2)\'~(2) "l-lp.2(fp(22)6)2. '"lN"iv126 / I N'f'lO/'11"126
(13)
Since/~2 has several contributions of order M x2, we denote/~2__ (~k,/g2)M~ and assume X ' ~ X. One obtains h M~/
(2)
<~0(22)6> = )k' M x2 \qOlO) (((qO(120)) "
(14)
Let us first concentrate on the value M R / M x ~ 10 -2 which is also needed in order to have neutrino masses in the eV region (see below). The VEWs y and z are then constrained to be about 15 MeV. Since we need contributions to the fermion mass matrix of this order to modify the first generation mass relations and to explain the measured value of the Cabibbo angle, the Yukawa couplings hi/have to be of order one, at least for the first generation. (Note also that a value M R / M x much smaller than 10 -2 would lead to unacceptably large Yukawa couplings and is therefore excluded). We are then left with the following situation. The relations mb(MR) = m , ( M g ) '
mr(MR) = m~,(Mg) D
(15)
are expected to hold with 1% accuracy~ whereas the relations ms(MR) ---- m . ( M R ) ,
mo(MR)=
D
me(MR) m t ( M a ) = me(MR) m~,(MR )
(16)
may have corrections of the order of 10% due to y and z. We may also consider a fourth generation p~,, ¢', b' and t' where the mass relations
mb,(MR) = m..(MR),
m,,(MR) = m
, . , m,,( MR) m,,(MR) = m,tMR)
(MR), (17)
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293
are expected to be even more accurate. For the first generation we find, from (12), +_ m o ( M R )
----
++_mu( M R) =
"C-me(MR) ----
sin20c
t- x I + z l l ,
+_
f12z~2 + mc(mg)
_.T.mr( M R)
=z~2
I- ctx I + flzn ,
F Xi - 3 z n ,
fl ~ 1 )2 mc(MR) _ ms(MR)
(18)
We have different possible choices of signs since the signs of the fermion masses cannot be measured. Since we have five free parameters (x l, z n, z]2, a, t ) for four measured quantities, it is not difficult to arrange for consistency with experiment. 5. Prediction of top quark mass Mass renormalization effects have to be included to derive the mass relations at present energies from the relations quoted at the scale M R (we only consider one-loop corrections). The top quark mass only depends on the value of as(ME) and is given by
m,( 2m,) ( as(40 GeV) l12/23( as(10 GeV) ) '2/25 mt(2mt)"~mc(Emc)m~,(2m~,) a~(10GeV)] ~a - ~
(19)
For ors(ME)----0.11 our prediction of the top quark mass is mt(40 GeV) ~ 20.7 GeV
(20)
Here we used me(3 GeV)--- 1.6 GeV as an input. For larger values of as(ME) we find lower values for m t (for example as(ME)-----0.14 gives m t (40 G e V ) = 18.2 GeV). There is a 10% inaccuracy in eq. (19) since rn~(2ma) stands here only for the q0~0 contribution to the muon mass which differs by about 10 MeV from the measured muon mass. For the d-type quark masses one has the relation
msb 0' msb' R' (Os 0')2/ 33 2.(
m,,,(O)
(21)
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G. Lazarides et al. / Proton lifetime
Here f is the effective number of flavours in the appropriate energy region. The dependence of the s and b quark masses on sin2t~w is weak. We find, for three generations and for the above values of %(ML), a variation of m s (1 GeV) between 370 and 570 MeV and a variation of rn b (10 GeV) between 4.6 and 5.5 GeV. Inclusion of a fourth generation increases these predictions by about 6%.
6. Fourth generation
The renormalization effects in our SO(10) model permit a fourth generation (v,,, 7', t', b') of leptons and quarks. It is quite remarkable that in our scheme the masses of the charged lepton ~', and also of t' and b', can be fairly reliably estimated. This comes about as follows. From ongoing experiments at PETRA, it is presumably reasonable to assume that m,, > 17 GeV. However, m~, cannot be much larger than 35 GeV since, otherwise, the prediction for the t' mass at the heavy toponium (t'-I') scale violates the upper bound on fermion masses [12] obtained from the requirement that Yukawa couplings should not become strong anywhere in the range between M L and M x. Taking m~, = 35 GeV as input, we can predict the masses of t' and b'. Using similar renormalization relations as for the b and t quark mass, we find, for % ( M L ) = 0.11, mt,(2mt, ) ~---230 GeV,
mb,(2mb, ) ~ 118 GeV.
(22)
Thus, for m,, between 17 and 35 GeV we predict m b, between 42 and 118 GeV and m t, between 150 and 230 GeV. We also note that larger values of as(ML) give lower masses for the t' and higher masses for the b'. This effect is about 15% for a s ( M L ) = 0.14.
7. Possible modifications of mass relations
Let us discuss what happens for higher values of M R / M x. The couplings hij could now be somewhat smaller than one, say 0.1. On the other hand, the VEV's y and z become larger, and their contribution to the fermion masses could be in the range of 150 MeV, thus destroying the mass relations for the second generation and the prediction of the top quark mass [compare (12)]. In order to maintain the generation pattern, the Yukawa couplings hij of the first generation have to be of the o r d e r (M2x/M2)IO -4. For values of M R / M x ~ l0 -1, one has y and z ~ 1.5 GeV and the successful relation (15) is destroyed. Thus, if we want to keep this relation, only a narrow range M R
10-2 <77--, < 10 - l
Iv1x
(23)
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295
can be admitted. Furthermore, for our top quark mass prediction to hold, M s / M x has to be near 10-2. We finally remark that these results can be generalized to an arbitrary choice of Higgs fields. The only scalars that can couple to fermions in SO(10) are qolO, I]0126 and q012o. If SO(10) is broken down to SU(4)c × SU(2)L × SU(2)R by the VEV of a 54-plet, the VEV's of the SU(2)L doublets belonging to q~126 and q012o are suppressed by a factor M ~ / M 2 x .
8. Majorana masses of the right-handed neutrinos As has been pointed out in ref. [8], one needs large Majorana masses for the right-handed neutrinos in order to account for the smallness of the masses of the observed left-handed neutrinos within SO(10). The Majorana masses a i of the right-handed neutrinos violate B - L and left-right symmetry. They are generated by the VEV of the SU(2)L singlet (q0~6)= M g / g contained in (TO, 1,3). Corresponding Majorana masses c i of the left-handed neutrinos come from the VEV of the SU(2)L triplet (q0~6) contained in (10,3, 1). Neglecting for the moment the different generations, the neutrino mass matrix has eigenvalues a and c ( 1 / a ) ( m ~ ) 2, corresponding essentially to right-handed and left-handed Majorana masses. Consider first the right-handed neutrinos. Their mass eigenvalues are given by hi(MR) ai = ~ M
g(MR)
R,
(24)
where h i are the eigenvalues of the matrix hij. For M R / M x ~ 10 -2, at least one eigenvalue h i has to be of the order of one (see the discussion of the charged fermion masses). Thus, for Ors(ME)--0.11, we find for the largest right-handed neutrino mass m~R= (1--5) × 1013 GeV
(25)
For larger values of as(ML) this value increases (by a factor 5 for Ots(ML) = 0.14). If there is no generation pattern for the h i all the right-handed neutrino masses are expected to be in the range of about 5 × 1012-5 X 1013 GeV. We note, however, that a larger value M R / M x "~ 10 -1 requires a generation pattern of the h i leading to Majorana masses of the electron (muon) neutrino of order 5 x 1011-5 × 1012 GeV (5 X 1012-5 X 1013 GeV), whereas the mass of the heaviest neutrino may be of order 5 × 1013-5 × 1014 GeV. (The values have been quoted for 0ts(ML) between 0.11 and 0.14.) We also note that even admitting eigenvalues h i of order 10-4 (which cannot be related with the "normal" generation pattern) the Majorana mass of the lightest right-handed neutrino has to be larger than 109 GeV.
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9. B - L violation, neutron oscillations and baryon asymmetry
For MR<
'rn~ ,~, 10 42
yr.
(26)
Even with our smallest values of m~ ~ R ~> l0 9 GeV we find %a~> l03s yr. Let us finally comment on the baryon asymmetry in the universe. At temperatures of order M x a baryon asymmetry is created by the decay of the X-bosons. Since B - L is conserved at this temperature one has AB ----AL. At temperatures of order M R the B - L violating interactions become effective. These interactions conserve A B + A L and the baryon asymmetry created at temperatures below M x will not be washed out completely. It has been pointed out [16], however, that there is another interesting source of baryon asymmetry due to the decay of right-handed neutrinos at temperatures below M R. Our quoted values for Majorana masses and Yukawa couplings can account for a baryon asymmetry of order 1 0 - s to 10-9.
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10. Light neutrino masses and neutrino oscillations It has been suggested that the solar neutrino puzzle can be solved by neutrino oscillations [17]. Recent experiments [9] indicate neutrino oscillations compatible with neutrino mass differences in the eV range. How can these neutrino masses be obtained in unified gauge models? A crude estimate gives an upper bound on the light neutrino masses of (27)
m~L ~
where M R is the scale of B - L violation. Clearly, MR-----Mx,~ 1015 GeV leads to neutrino masses smaller than 10 -2 eV. However, an intermediate scale M R ~ 10-2M x predicts neutrino masses in the eV range. The light neutrino masses are given by m,L = C - ( 1 / a ) ( m D ) s. We consider the two contributions separately and start with the Majorana mass term c. Then the masses of the different neutrinos are given by (28)
c i = hi(mL)(cp(13)26).
As has been pointed out earlier [13], the SU(2)L triplet field cp(132)6belonging to (10, 3, 1) is forced to acquire a non-vanishing VEV by the quartic scalar couplings (cP126)4, (CPls6)3Cplo and (~126)2(9~10)s. The leading terms in the effective potential for the cp~6 can be written as V=
_~,. / ~ ( ! ) \ / ~ ( 2 ) \ 2 / ~ ( 3 ) \ "1X~126/\~i0 / \~F126/
+
(29)
_
This induces 2 g ( M R ) M2 ~1 g2(ML) MR r2
1.5 ×
1011
eV.
ML KI MR KS '
(30)
where
Here we use the fact that 1,2 cannot be much larger than M 2 without violating our symmetry-breaking pattern, since it is related by left-right symmetry (which is unbroken above M R) to the mass term of ~P~6. The value M L / M R strongly depends on sin20w . For as(ML) ----0.11 and sin20w between 0.21 and 0.24, it varies between 5 × 10 -13 and 1.5 × 10 - u . However, the supplementary constraint M R / M x ~ 10-2 derived from the charged fermion mass
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spectrum implies ML/MR<,~6 l0 2, we find
×
10 -12. Thus, for as(ML) ----0.11 and M R / M x --
-
KI
L
(31)
ci,-~leV'~2hi(m~ ) •
Larger values of as(ME) give somewhat lower neutrino masses. In general, KI and ~2 are both of order O(g 2) and their ratio is expected to be of order unity. For M R / M x ~ 10 -2, the largest eigenvalue among h i has to be of order one. Depending on x~/x2, we then predict the contribution c i to the mass of the heaviest neutrino to be in the range of a few eV. The contribution to the other neutrino masses can be of the same order of magnitude or much smaller, depending on the h i. If there is only one neutrino eigenstate with mass in the eV range, it must have a substantial contribution from the electron neutrino. For M R / M x > 10 -2, the neutrino masses are smaller due to smaller values of M E / M R and smaller values of h i for the electron (and muon) generation. If we take, for example, M R / M x ~ l0 -1 we would find the following neutrino mass spectrum: m L ~ 10 -3 eV,
m ~L
10 -2 eV,
L m ~,(~,)
10 -1 eV.
Neutrino masses in the eV range clearly favour M R / M x ~ 10 -2. Let us next discuss the contribution from the Dirac neutrino masses. The largest contribution would be from the Dirac mass of the neutrino of the fourth generation m I,T, O • Taking renormalization effects into account, we find (for m ~, -- 20 GeV) mD ( M L ) ~ 100 G e V .
(32)
Assuming that all the Majorana masses of right-handed neutrinos are of the same order M R, the contribution to the light neutrino masses is given by ( M R -- 1.5 × 1013 GeV)
Ac ~ 0.5 eV g( MR ) h(MR)
(33) "
In this case the fourth generation neutrino Dirac mass contributes less than 1 eV, since g ( M R ) / h ( M R ) has to be of order one. The contribution of the Dirac masses of the first three generations is much smaller, since it is suppressed by a factor D 2 D 2 (mr,) ~(mr,.) compared with the fourth generation. In summary, the light neutrino masses depend strongly on M R / M x and the values of Yukawa couplings. A lower bound for the mass of the heaviest neutrino
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299
of 10 - 2 eV is found (for M R / M x ~ 10-1). Neutrino masses in the 1-10 eV range are predicted for M R / M x ~ 10-2. We expect large mixing between neutrinos of different generations which lead to neutrino oscillations observable with present experiments. Since the oscillation length L is given by L = 4~rE/Am 2 we expect, for E - - 10 MeV, oscillation lengths of the order of 25 cm to 25 m. 11. Conclusion To conclude, the presence of an intermediate mass scale related to the violation of the B - L symmetry seems to be favoured by a number of considerations, both phenomenological and cosmological. Thus, sin2~w > 0.20 can easily be accommodated. The bad SU(5) asymptotic prediction m e = m d is avoided, while the good predictions for the second and third generations are maintained. The top quark mass is predicted to be 20 _+ 2 GeV. Neutrino masses of the order 1- l0 eV can be naturally obtained, as indicated by recent experiments. Moreover, the cosmological monopole problem can be avoided if SO(10) breaks via SU(4)c × SU(2)L × SU(2)R. Last, but by no means least, we urge experimentalists to make an all out effort to look for the t quark and the lepton of the fourth generation in the 20 GeV energy region. References [1] H. Fritzseh 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. Galliard, Nucl. Phys. B128 (1977) 506; H. Georgi and D.V. Nanopoulos, NucL Phys. B155 (1979) 52 [2] H. Georgi and S.L Glashow, Phys. Rev. Lett. 32 (1974) 438 [3] H. Georgi and D.V. Nanopoulos, NucL Phys. B159 (1979) 16; Q. Shaft, M. Sondermann and C. Wetterich, Phys. Lett. 92B (1980) 304; S. Rajpoot, Imperial College preprint (Dec., 1979) C. Vayonakis, Athens preprint (1978) [4] G. Lazarides, M. Magg and Q. Shaft, CERN preprint TH 2856 (1980) [5] J. Pail and A. Salam, Phys. Rev. DI0 (1974) 275 [6] A. Buras, J. Ellis, M. Galliard and D. Nanopoulos, Nucl. Phys. B135 (1978) 66 [7] R. Barbieri and D. Nanopoulos, CERN preprint TH 2810 (1980) [8] M. GelI-Mann, P. Ramond and IL Slansky, unpublished [9] F. Reines, E. Pasierb and H.W. Sobel, Univ. of California preprint (1980) [10] W. Marciano, Rockefeller preprint (1980); P. Binetruy and T. Schilcker, CERN preprint, to appear; J. Ellis, M. Gaillard, D. Nanopoulos and S. Rudaz, Nucl. Phys. B176 (1980) 61 [11] J. Ellis, M. GaiUard and D. Nanopoulos, CERN preprint TH 2749 (1979); C. Jarlskog and F. Ynduraln, Nucl. Phys. B149 (1979) 29; A. Din, G. Girardi and P. Sorba, Lapp preprint TH08 (1979); J. Donoghue, MIT preprint CTP 824 (1979); M. Machacek, Nucl. Phys. B159 (1979) 37; M.B. Gavela, A. Le Yaouanc, L. Oliver, O. P~ne and J.C. Raynal, Orsay preprint LPTHE 80/6 0980) [12] N. Cabibbo, L. Malani, G. Parisi and R. Petronzio, Nu¢l. Phys. B158 (1979) 295 [13] M. Magg and C. Wetterich, Phys. Lett. 94B (1980) 61
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[14] S. Weinberg, Phys. Rev. Latt. 43 (1979) 1566; F. Wilczek and A. Zee, Phys. Rev. Lett. 43 (1979) 1571 [15] R.N. Mohapatra and R. E. Marshal Virginia University preprint VIP-HEP 80/1 [16] J. Harvey, P. Ramond and D. Reiss, Caltech preprim CALT 68-758 0950); T. Yanagida and M. Yoshimura, KEK preprint TH8 0980) [17] B. Pontecorvo, JETP (Soy. Phys.) 53 (1967) 171; A. De R~jula, M. Lusignoli, L. Maiani, S. Petcov and R. Petronzio, Nucl. Phys. B168 0980) 54