T H E N E U T R I N O S E E - S A W I N S O ( 1 0 )
R . J O H N S O N , S. R A N F O N E a n d J. S C H E C H T E R Phystcs Department, Syracuse Unwerstty, Syracuse, NY 13244-1130, USA
Received 15 July 1986
When a predictive and apparently reahstic model for the quark mass matrices is extended to the leptons by using an SO(10) framework, it is found that the mass matrix of heavy "right-handed" neutrinos is hkely to be close to a singular one. Thxs may result m a variety of interesting patterns for the neutrino masses which are analyzed and compared with ex- permaent.
The SO(10) grand unified m o d e l [1 ] provides a traditional guide for thinking about neutrino masses. In the case o f three generations the 6 × 6 neutrino mass m a t r i x M v has the form 4-1
MY = MvD 7S' '
where mvD and S ' are 3 × 3 matrices whose elements have magnitudes comparable to the charged fermion masses. The dtmensionless parameter 3 ( p r o p o r t i o n a l to the vacuum value o f an SU(2) L Higgs triplet) must be very small - say less than 1 0 - 8 or so - while the dimensionless parameter 3' ( p r o p o r t i o n a l to an SU(2)L Higgs singlet) should be very large, say o f the order o f 1013. Given these values offl and 7 it is reasonable to approximate the effective 3 × 3 m a t r i x for the ordinary light neutrinos (which must be o f " m a j o r a n a " t y p e in general) as
eff , _ - 1MT ,
M v ~-~S - 7 u D S - 1 M u D • (2)
The 7 - 1 term represents the "see-saw" mechamsm [3] and is generally considered to be the dominant one, al- though this is really a dynamical assumption. The a p p r o x i m a t i o n in (2) is justified unless
S'
is a singular m a t r i x (det S ' = 0).Curiously, m what seems to be g o o d guess for the matrices involved S ' m a y get close to, or actually become, singular. Of course, physicists are (perhaps justifiably) suspicious o f statements about the various fermion mass- matrices since t h e y clearly should come from some yet to be discovered deeper physical theory. Nevertheless, the semi-empirical approach to mass levels has an honorable history. Recently it was pointed out [4,5] that combm- mg two Ansatze - that o f Fritszch [6] and that o f Stech [7] - which had been at first glance considered incom- patible results in the possibility o f predicting (for any number o f generations) the K o b a y a s h i - M a s k a w a matrxx c o m p l e t e l y m terms o f the quark masses. In the case o f three generations the predictions are in agreement with our present expermaental information - t h e y also feature a " m a x i m a l " CP violation phase and require 25 GeV < mto p < 45 GeV. A natural way to extend the m o d e l to leptons is to embed it in SO(10). This has been suggested by Bottino, Kim, Nishiura and Sze [8] and we report here on our further study along that line. In this model the mass matrices are
M u = S + e S ' , M d = a M u +A + ( 1 - a e ) S ' , r M e = a M u + 6 A - ( 3 + a e ) S ' , (3a, b , c ) for the charge 2/3 quarks, charge - 1 / 3 quarks and charge - 1 leptons, respectively. S and S' are symmetric ma-
4-1 A review of neutrino masses m the SO(10) theory is gwen in ref. [2].
trices describing the couphng patterns o f the 10 and 126 Higgs fields while A is an antisymmetric matrix describ- ing the couplings o f the 120 Higgs field, a, 6 and e are dimensionless real constants very roughly of order unity. Finally, r is a renormalization factor appropriate to comparing the quark masses with the charged lepton masses at a low energy scale (say 1 GeV) rather than at the normal unified scale. It is traditionally [9] taken to be about 3 but, considering the uncertainties involved in its determination, one would be unhappy if any physical results were to change drastically as r varied over the region
3 < r < 4 . (4)
In this model the neutrino masses are given by (1) with S' determined from eqs. (3) and with the reasonable ap- proximation
r'MvD ~- S . (5)
r' is a renormalization group parameter which should be a little larger than r (say r' = 3.5 if r = 3.0). So far essen- tially SO(10) with one Hlggs field o f each relevant type has been assumed. The model is completed by requiring each matrix S, S' and A to be hermitian and the mass matrices to have the form, for example
0 A e 0 ,~
r M e = Q ( A e 0 Be ] Q - I ' Q = d i a g ( e i a l , e l a 2 , e l ~ 3 ) , (6) \ O Be C e
where
A e = ( m e m ~ m r / C e ) 1/2, B e = [ ( m r + m e ) ( m r - - m t ~ ) ( m ~ - - m e ) / C e ] 1/2, C e = m r - m u +m e.
A similar form holds f o r M d and (without any phases) f o r M u. Matrices of the form (6), though not symmetric m general, give the same predictions as the Fritzsch model. Notice that the choice ~e = 1 gives
M d = M ~ d = ~ 1 4 u + A , Mu=MTu --M u*, (7)
which is the original Stech model [7]. We shall adopt this choice here for convenience. If we relax ~e from unity we can accomodate a top quark mass as high as about 95 GeV but the predictions o f the model are otherwise very similar.
The possible singularity o f S' does not depend on any fine details of the model, from (3b) and (3c) we get without any assumptions
T r S ' = $33' = (TrM d - r T r M e ) / 4 = [(m b - m s + md) -- r(m r - m u + m e ) ] / 4 , (8) ' , ' c ' / " ~ 2
where the Fritzsch type form (6) was used in the first step. Since d e t S ' = -$33~.,,12 / we see that S' becomes sin- gular when r takes the "magic" value
m b - m s + m d
r = - 3 . 0 - 3 . 5 , (9)
m r - rn u + m e
depending on the precise values [10] o f the running masses, m s (1 GeV) and m b (1 GeV). Comparing (9) with (4) shows that the singularity could occur in the expected region for r. Notice that ~he above argument for the pos- sible singularity of S' would hold if there were any number of 10's and/or 120's present, assuming the coupling matrices to have the Frltzsch form. As we shall see the width o f the "magic canyon" (region where the effects o f the singularity are Important) is very small for the see-saw term in (2) (due to the large value o f ? ) so one could always avoid it with a clear conscience. However the fact that it occurs near the expected value of r suggests that Nature may be trying to tell us something. If the system were exactly at a singular point there would be only two superlight Majorana neutrinos and only two superheavy neutrinos. The other two (presumably related to the r neutrino) would coalesce to form a Dirac neutrino of mass in the range of tens to hundreds of MeV. Since the ex- perimental upper bound [ 11 ] ,2 on m (Vr) is as large as 5 0 - 8 5 MeV, such a picture is not a priori unreasonable. ,2 Ref. [11] refers to the earlier less restrictive bounds.
In order to study the neutrino masses and mixings for the singular case we must give up the approximation (2) and return to the original equation (1). At the singularity the effect of/3 xs not qualitatively important so we mxght as well set 3 = 0. With (5) and the notations
s=(O a o)
(o ~' o)
a 0 b , S ' = a' 0 b' ,
0 b c 0 b' c '
(10)
the secular equation for (1) then becomes 5 )k 6 + ~_J dn)t n = 0 , (11) n=0 where d o = - a 4 c 2 , d 1 = 2 a b c x - c'a2(a 2 + b 2) , d 2 = ( a 2 + b 2 ) 2 + x 2 + c 2 ( a ' 2 + 2 a 2 ) , d 3 = - 2 b c b ' + c ' ( a ' 2 + 2 a 2 + b 2 ) , d 4 = - 2 a 2 - 2 b 2 - c 2 - a ' 2 - b ' 2 , d 5 = - c ' , x = a b ' - b a ' .
Remember that the primed quantities are larger in order of magnitude than the unprimed ones by the huge factor
o f 3'- A singularity o f S' is achieved if c' = S~3 is zero which corresponds to r taking on the magic value given m
(9). When c' = 0 the equation determining the superheavy neutrino masses o f order 3' only involves the )t 6 and )t 4 terms and we find them to be [accurate to order (1/3')]
~t5, 6 = +(a '2 + b'2) 1/2 . (12)
We see that the two superheavies have the same mass ( o f course, a sign is irrelevant) so they can be considered as a single Dirac neutrino. I f c' 4= 0 the equation determining the superlight neutrinos ( o f order 3'-1) would be a cubic and we would get three of them. However, when c' = 0, only the 2, 2, )t 1 , and ~t 0 terms in (11) are relevant and we find only two superlights with masses
)~1,2 = [xac/( a'2c2 + x2)] ( - b + [b 2 + a2(1 + a ' 2 c 2 / x 2 ) ] 1/2} . (13)
Finally, noting that d e t M v = II6=12~ = - a 4 c 2 we get from (12) and (13) that
- ~ 3 X 4 = ) ~ = (X 2 + cZa'2)/(a '2 + b ' 2 ) , (14)
where we also used )~4 TM --~4 winch holds to excellent accuracy since Z6=1)~ = T r M v = 0. Eq. (14) shows that, as
mentioned before, the putative r-neutrino here is of Dirac type and has a mass o f the same order as the charged lepton masses. Notice that this mass is independent o f 3'. The expression (14) is rougly mlmmized by setting a' =0 which yields
X 3 ~ a ~ - ( m u m c ) l / 2 / r ' ~ 24 M e V . (15)
This seems promising for a realistic but exotic r-neutrino. However, the approximate third and fourth eigenvectors o f M v (obtained b y successively first diagonalizing the lower right 3 × 3 submatrix o f M v and then the upper left 4 X 4 submatrix) are
l0 )
i x
~3 4 ~ [2( x2 + a'2c2)]1/2 -ca'
, + ( x 2 + a , 2 c 2 ) l / . (16) 0 0 357F o r a' = 0 our 24 MeV neutrino is m o s t l y along the mu rather than the tau direction (the effect o f diagonalizing the charged l e p t o n mass m a t r i x turns out not to m o d i f y this feature much) which is unrealistic. A n evident way out is to consider x = 0. Then X3 -~
ca/r'b ~- (mumt)l/2/r'
~- 150 MeV, which is a little too large for the present b o u n dm (vr)
< 5 0 - 8 5 MeV. Thus, to achieve a reahstic theory, some detuning from magic r m a y be indicated. This is best handled b y a numerical rather than an analytical approach.We have carried out a detailed numerical analysis o f this m o d e l which will be reported elsewhere [12]. I f the masses o f the nme charged fermions were
precisely
known, all parameters o f the mass matrices M u , M d and M e would be fixed in theae
= 1 case, except for one in the lepton sector, which may be taken to be 6 (a measure o fCP
violation i n M e ) . Fixing 6 would then enable us to completely predict the neutrino masses and mixing m a t r i x for gwen choices o f the parameters/3 and 7 in (1). In practice, the fact that the quark masses and the K o b a y a s h i - Maskawa mixing angles I Uusl and I f c b l are k n o w n only to a certain accuracy must be taken into account to find the allowed regions in parameter space. F o r definiteness, we will here simply make the following consistent choices for the quark masses:mu/m d=0.57,
md/rn s=0.056,
m s = 0 . 1 6 G e V , m c = l . 3 5 G e V , m b = 5 . 3 G e V , m t = 5 3 G e V . (17) Let us illustrate a typical scenario for neutrino masses and mixmgs when r is in the magic canyon but not at the precise singular point. With (17), the magic value is r ~- 3.065707. Choosing for simplicity ~ = 0 (no leptonicCP
violation) and 3' = 1010 we detune slightly (since rn 3 = m 4 = 460 MeV at magic r here) to r ~- 3.065700 and f'md from a numerical diagnalization o f (1) in our model that the two mtermediate mass neutrinos have split to m 3 = 7.4 MeV and m 4 = 28 GeV. The two superlight neutrino masses are rn 1 = 0.0018 eV and rn 2 = 0.56 eV. Notice that the mixing matrix K - which is a 3 X 6 rectangular m a t r i x [13] defined b yL = (ig/x/~)~VI~-~L3'#K/3L+ h . c . ,
m a standard n o t a t i o n - is now completely predicted and turns out to be 1.00 0.012i 0.0098i 1.6×10 - 4
K - ~ ( 0.014i 0.99 0.13 - 2 . 1 × 1 0 - 3 i ) . (18)
0.0082i - 0 . 1 3 0.99 0.016i /
Here we have introduced some factors o f 1 in order to make all neutrino masses positive and have approximated the 3 × 6 b y a 3 × 4, neglecting the two superheavles. Let us focus on v 3 and ask whether the relevant particle- physics and astrophysical bounds ,3 are satisfied. The non-observation o f a v 3 around 10 MeV in the rr- -+ e - v 3 and rr- ~ - t , 3 reactions [15] results m the bounds IK131 < 0.01 and IK231 < 0.15 which are just consistent with (18). Astrophysical bounds [14] require v 3 to decay rapidly. Since re(v3) > 2 m ( e - ) here, the quick decay channel v 3 -+ e-e+Ve is open and one needs roughly [K131 > 0.005 which is again consistent. However, the non- observation o f neutrmoless double b e t a decay requires ,4
3
z~=lmt(Klz) 2
~ m 3 1 ( K 1 3 ) 2 ] < 1 0 e V or [ K 1 3 I < 0 . 0 0 1which is not satisfied. It is amusing, though, that precisely at magic r the contributions o f t, 3 and v 4 to neutrino- less double b e t a decay cancel each other. The sphtting o f m 3 and m 4 destroys this cancellation. A number o f op- tions remain to construct a viable t h e o r y b y complicating the model. Our main point here has been to show the existence o f a new scenario for the neutrmo mass spectrum. Thus one should not a priori rule out a tau neutrino mass as large as 10 MeV. A one-order-of-magnitude improvement in the measurement o f the tau neutrino mass just might reveal an interesting surprise.
4:3 A summary IS given m ref. [14]. See the dmgram on page 107. ,4 A revzew zs given mref. [16].
Smce this model is very predictive It is still quite interesting even if we imagine that r is well out o f the magic canyon. Then we may consider two obvious scenarios - the dominance o f the/3 term in (2) or the dominance o f the 7 term. The case where the/3 term is dominant was discussed in ref. [8] where it was found that the results were pretty much independent of the input parameters. This can be easily understood since S ' is close to the sin-
gular matrix
t00i
M e f f ' ~ / 3 S ' ~-/3 0 0 23 • 0 S~3
Note that the closeness to the singularity will be Important for the/3 term even if r differs from its magic value by as much as 0.5 or so. The singular limit has a zero-mass first neutrino and two degenerate heavier neutrinos which mix among themselves. This pattern is only slightly modified by the departure from the singularity. For example with r = 3.00 and 6 = 0.2 we find
( m l , m 2 , m3) = (109/3) × (2.2 X 10 - 5 , 0.10, 0.13) e V , and the mixing matrix
1.00 0.042@ - 62 ° 0 . 0 6 2 @ - 169 ° K = ( 0 . 0 7 @ - l 1 0 ° 0.82 0 . 5 7 @ - 9 1 ° ) ,
0.029@1 ° 0 . 5 7 @ - 8 9 ° 0.82 where p@0 - p e 1° .
(19)
This model features substantial u ~ - u r mLxing. It is interesting that an experunent on v u - u r oscillations
[ 17] then gives a strong bound on/3. From the con&tion that (m3)2 - (m2)2 < 5 eV 2 we obtain/3 < 2.5 × 10 - 8 so that the heaviest neutrino should weigh less than about 3 eV.
The case when the second (see-saw) term o f eq. (2) is dominant is the most orthodox possibility. With the same parameters as above and T = 1012 we fred the light neutrino masses
( m l , m2, m3) = (1.94 × 10 -5 , 0.0059, 7.90) e V , and the mixing matrix
1.00 0.0240@ 161 ° 0 . 0 1 0 @ - 6 6 °
K = ( 0 . 0 2 4 7 @ 21 ° 0.99 0 . 1 3 3 @ 1 8 5 ° ) . (20)
0 . 0 0 8 6 @ - 9 7 ° 0 . 1 3 3 @ - 5 ° 0.99
Ttus example is relevant for a recent model [18] which explains the lack of solar Ve'S observed on earth as result- ing from the transformation o f Ue'S produced at high density in the center o f the sun into v u ' s . For this purpose
one may have [ 19] m I negligible and m2 ~ 0.007 eV in agreement with this example. Furthermore it is necessary for the v e ~ u u transition to be adlabatm which implies IK121 > 0.007. This criterion is also seen to be satisfied
in our model. Actually the other two scenarios could (with suitably scaled 7 or/3) also solve the solar neutrino problem in this way.
We have seen that a plausible model for the mass matrices in SO(10) results in an interesting possible singular- Ity in the "right-handed" neutrino mass matrix which plays a crucial role in the see-saw mechanism. This singular- ity occurs if, at the grand unified scale, the relation m b - m s + m d = m r - m u + m e holds, which Is very close to
the usual expectation m b ~- m r . A signature for the system lying within the range o f influence o f this singularity
would be a tau neutrino mass o f the order o f several MeV or so. Even if the system is outside this "magic c a n y o n " the model is very predictwe. For example, the recent proposed solution o f the solar neutrmo problem requires knowledge o f the Pe-Uu mixing angle which is predicted here.
We w o u l d like t o t h a n k M. G r o n a u f o r h e l p f u l discussions. T h i s w o r k was s u p p o r t e d in p a r t b y t h e D e p a r t m e n t o f E n e r g y u n d e r C o n t r a c t N o . 0 2 - 8 5 E R 4 0 2 3 1 . O n e o f us ( S . R . ) w o u l d like t o t h a n k t h e " F o n d a z l o n e della R i c c i a " for p a r t i a l s u p p o r t .
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