E6 GUT and large neutrino mixing

PhysicsLettersB 296 (1992) 127-131

North-Holland

PHYSICS LETTERS B

E6 GUT and large neutrino mixing Yoav Achiman Department of Physics, University of Wuppertal, GauflstraJ3e20, W-5600 Wuppertal I, FRG

and Andr6 Lukas Physics Department, Technical University of Munich, James-Franck-Strafle, W-8046 Garching near Munich, FRG and Max-Planck-lnstitutfiir Physik, Werner-Heisenberg-lnstitut, P.O. Box 401212, W-8000 Munich, FRG

Received 31 August 1992

All experimental results concerning possible neutrino oscillations are naturally and simultaneously accounted for in a n E 6 GUT model. The fermionic mass matrices are dictated by the symmetry breaking and specific radiative corrections and not by the use of"Ans~itze" or discrete symmetries.

In a recent paper [1,2] (hereafter: "paper I " ) we presented in detail a n E 6 G U T with a very specific set of mass matrices. In particular, the "light" neutrino mass matrix is practically dictated by the quark sector and the scale o f the intermediate symmetry breaking. For an intermediate scale of 10 l o 1012 GeV, suggested by the recent values of sin2Ow [ 3 ], our "solutions" for the neutrino masses and mixing were concentrated exactly around the value, resulting from the latter announced G A L L E X experiment [4]. Moreover the experimental requirement of large neutrino mixing allowed us to fix the favored breaking chain. In this letter, we would like to emphasize this fact and use the exact G A L L E X results to limit the range of our solutions. This will enable us to fix the hierarchy of the heavy VEV's and in particular the allowed values of the intermediate breaking scale, which is also the scale of the right-handed ( R H ) neutrinos. Our model is based on the following considerations: ( 1 ) The superstrong breaking of E6 will be generated by one or several symmetric ~3sl. This dictates the direction of the breaking, it must go via SO (10), the only maximal subgroup with singlets in those representations. The further breaking goes then through

S U ( 5 ) or subgroups of G p s = S U c ( 4 ) × S U L ( 2 ) X SUR(2). (2) The low energy breaking into SUc (3) × UQ ( 1 ) will be generated via o n e H27 Higgs representation. In this case all the mass matrices of the standard fermions are proportional to each other on the tree level and can be diagonalized simultaneously

~o =~°,

~o =~o,

M- vo. D i r = a M- o ,

3710 = a~r ° ,

( 1)

with 37/° = diag (/h,/z2, ~t3),

/zi~ .

(2)

(3) The main requirement of our model is that the one-loop contributions to the mass matrices are dominated by the diagrams of fig. 1. Those diagrams involve in the loop superheavy gauge bosons and fermions but no scalars. This requirement can be justified using arguments of maximal calculability and predictability. Contributions involving scalar loops are less predictable than those involving gauge bosons with known couplings. Only in supersymmetric theories with SUSY broken at a low energy scale, the scalars may play an important role. In those theories, however, the non-renormalization theorems allow one

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V, All rights reserved.

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,,°v,s = O(Mw) v~,s = O(Mx) .

f

Meis the mass matrix

~

~

F

fc

Fig. 1. One-loop corrections to standard fermion masses containing gauge bosons. to avoid unwanted couplings in the superpotential, to all orders. This amounts then to fixing by hand the allowed radiative corrections. Our requirement assumes therefore that SUSY is broken at a relatively high scale and that all relevant Yukawa couplings are much smaller than the gauge couplings ~ Calculability dictates at the same time one more important requirement. It is well known that two-loop corrections to the diagram in fig. 1 diverge (see fig. 2). An obvious way to avoid this problem [ 2 ] is to require that the superheavy mass terms are "orthogonal" to the tree level masses of the light fermions, in the family space. In other words, one obtains a calculable theory if in the framework of diagonal tree level mass matrices the radiative contributions are pure off-diagonal. The off-diagonal corrections induced by the diagrams of fig. 1 are given by [ 5 ] ~MF, f = WF,fMF, 3Or/" ~r'ft~Ff ( m F f ) 2 r~F,ft,Lg /ag' wF.f= -4n 2 ~ 2 In - - ~ mF,f-- mF,f

mF,

(3)

f

~ This is well known phenomenologically for the Yukawa couplings of standard fermions except for the top.

(

f

10 December 1992

of the superheavy fermion F, mF, f and ?nF,f a r e the masses of the gauge b o s o n s X~'-* and .~F, 5. kF,r is the gauge boson mixing factor and otF,f, flF,f are the group theoretical coefficients o f the gauge couplings. Since one of the gauge bosons carries a weak isospin Iw = ½, one VEV contributing to the mixing kF,f is of order Mw:

kF,f=o(vOF.fVF.f), v ° , f = O ( M w )

(4)

The corrections are in general of the order of magnitude depending on the ratios of the superheavy masses in eq. (3). In paper I it is proven that in our breaking scheme F = N, v ¢ give the leading off-diagonal contributions ~2 where N is the SO ( 10)-singlet in 27E6. For group-theoretical reasons the graphs with F = N contribute to the masses of all standard fermions, whereas those induced by F = v c are limited to the u-mass matrix [ 6 ]. Taking these radiative corrections into account we get the following mass matrices:

aMw,

M~= 1 (57/O+E)

Md=~iO+pE

r

MvDir= 1

(a~O+sE) Mu=a)QO+qE+A

(5)

112 This is also true in the case ofa superheavy SO( 10)-invariant VEV in the 27E6 Higgs representation. AT the first sight this looks dangerous because such a VEV generates large masses for the fermions in 10so¢1o) [ 6 ]. But for group-theoretical reasons contributions from these fermions are restricted to the combinations (e, D) and (d, E) (where D and E are the charged exotic particles in the 10sort0) ). In both cases the large VEV v responsible for gauge boson mixing breaks SU (5) and consequently leads to a suppression by the large SO(10) invariant masses of the corresponding gauge bosons (see eq.

(3)).

F

F (~)

X Fig. 2. Infinite second order contributions to Higgs self-coupling. 128

.

fc

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The matrices E and d represent the contributions induced by F = N and F = v c respectively, while their relative strengths are given by p, q and s. The different renormalization behaviour of the quarks and leptons is taken into consideration by the factor ,3 1/r. (4) The matrix A is clearly proportional to the RHneutrino mass matrix My.R: Mv.R =r/A, r/= - -

(6)

1

(7)

Wvc,u

r/is very large and the see saw mechanism [ 7 ] is naturally "realized. The mass matrix of the light neutrinos is therefore 1

M . . . . . 11"~ ~

Mv,i~ird

-

IMv,Dir •

(8)

Now, because N and v c are Majorana particles the correction matrices must be symmetrical. They are off-diagonal and in general complex. We analyzed first the Mu, Md, M, matrices of eq. (5) for real entries. In this case, as explained in paper I, there is only one "free" parameter that obeys certain theoretical restrictions. We looked for "solutions" which give the best Z 2 fit to the known masses of the quarks and the charged leptons [ 8 ] as well as the CKM matrix [9]. We found solutions which allow top masses up to rnt ~ 250 GeV. This result is not trivial because it is very sensitive to the set of experimental input masses used. Our results were obtained using the by now standard set of masses due to Gasser and Leutwyler [ 8 ] while only a slightly different set of masses due to Barducci et al. [ 10 ] allowed for mt,phy s ~< 50 GeV only. We showed then in paper I that given A a phase, one can have CP violation without changing essentially the above results. By studying different breaking chains of E6 we can limit the freedom in the solutions drastically. We considered different possibilities, in particular: (a) One independent correction matrix, i.e. only o n e (I~3s I used or the superstrong breaking follows the chain E6 --* SO(10) ~ Gi . . . . . G~ --* SUc(3) ×

~3 For large top masses this is not a very good approximation, see the remark about that latter.

10 December 1992

S U L ( 2 ) X U ( 1 ) w h e r e Gi c G p s = S U c ( 4 ) X S U L ( 2 ) XSUR(2).

(b) Two independent corrections matrices, i.e. more than one ~ss~ used but the superstrong breaking going via E6-~SO(10)-~SU(5)--,SUc(3)X SUL(2) x U ( 1 ) . The essential result of this study is that these two breaking chains favor in our model top masses above 100 GeV! Case (a) is even more restricted: 105 ~
1H(1,1)1<<1£2#(1,1)1,

case (2):

[H(1,1)[>>122#(1,1)[,

where we used the following definitions: H( 1, 1 ) and ¢~i( 1, 1 ) denote the SO( 10)-invariant VEV's of H27 and ~ S l respectively and 2i the effective coupling constant for the mixing between H2T and ~ s l . Then 129

Volume 296, number 1,2 10-3

i

s

PHYSICS LETTERS B

I

,

J

I

'

,

I

10 December 1992 t O

•

-3

~

,

,

~

1

I

,

,

i

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, - -0~ - - = - --- -

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~o 10-5

12-0 I00 806040 2.0 ~ \

10- a

,08

,

l0 -3

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I0 -2

sin2. 20 Fig. 3. v~-v, mixing in the case ( 1) for an intermediate scale 1012 GeV. The curves describe iso-SNU lines for ~lGa detectors. I 0 -''5 - - ,

10-4

,

i

,

,'

j

,

,

,

. . . . 10-4

0.1

I I ~L'

[0 -3

IO-2 sin z

0.1 20

Fig. 5. ve-v. mixing in the case (2) for an intermediate scale ~ 1010 GeV. The curves describe iso-SNU lines for 71Ga detectors.

•

1

,-- . . . . . 0.1

10_5

0.01

10- z 10 "a

I o-8 10-4

I0 -3

I0 -2 sin z

0.I 20

Fig. 4. v~-v. missing in the case (2) for an intermediate scale

104

~ 1012GeV. The curves describe iso-SNU lines for 71Ga detectors. the p a r a m e t e r s in eq. ( 5 ) receive the following simple values: case(l):

p-~l,

q~s~-a,

case ( 2 ) :

p-~l,

q~s~l/a.

T h i s leads to definite p r e d i c t i o n s . Case ( 2 ) is especially interesting because could show q u a l i t a t i v e l y in 130

10 "5 0.01

,

, , , ;:;:',

.......

.

.

.

.

'

•

' ' 1

0.1 sina(20)

Fig. 6. v,-v, mixing in the case ( 1 ) for all solutions compatible with GALLEX data and experimental bounds by Kamiokande and Frrjus.

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PHYSICS LETTERS B

p a p e r I that large mt i n d u c e s large n e u t r i n o mixing. This is seen explicitly also in the detailed n u m e r i c a l calculations. T h e results for ve-v~ m i x i n g in the cases ( 1 ) a n d ( 2 ) for an i n t e r m e d i a t e scale o f ~ 1012 G e V are given in fig. 3 a n d fig. 4. T h e y ar s h o w n to lie in the p a r a m e t e r range n e e d e d for the M S W explanation [ 12 ] o f the solar n e u t r i n o p r o b l e m . Fig. 5 shows that in case ( 2 ) we can get nice M S W s o l u t i o n s for a n i n t e r m e d i a t e scale o f ~ 101° G e V as well. In figs. 3, 5 we see that o u r solutions c o r r e s p o n d exactly to the two regions #4 allowed b y the recently p u b l i s h e d ~4 Also fig. 4 corresponds actually to a possible small region in fig. 1 of the GALLEX paper. In fig. 7, however, it can be seen that then the depletion of atmospheric neutrinos measured in the Kamiokande experiment cannot be explained simultaneously (shift all solutions ~ 3 orders of magnitude down to small values in ArnZ). I

,I

t

i

*

°

0.1 ;oo& ¢'~2 0 0

0.01 % %

E 10 .3

1 0 .4

1 0 "s

0.01.

0.1 sin2(2O)

Fig. 7. v~,-v, mixing in the case (2) for all solutions compatible with GALLEX data and experimental bounds by Kamiokande and Fr6jus.

10 December 1992

results o f the G A L L E X C o l l a b o r a t i o n [ 4 ]: ( 8 3 ± 19 stat. + 8 syst.) S N U .

(9)

N o w we are in the p o s i t i o n to d e t e r m i n e all solutions o f our m o d e l compatible with the n e w G A L L E X data. F o r the cases ( 1 ) a n d ( 2 ) the vo-v~ m i x i n g o f these solutions is s h o w n in fig. 6 a n d fig. 7 together with the e x p e r i m e n t a l b o u n d s o f K a m i o k a n d e a n d Fr6jus [ 13]. We find that only in case ( 2 ) the d e p l e t i o n o f solar n e u t r i n o s as well as a t m o s p h e r i c n e u t r i n o s can be e x p l a i n e d simultaneously. A detailed study o f the solutions which obey all those r e q u i r e m e n t s leads to a value o f the i n t e r m e d i a t e scale b e t w e e n 101° G e V a n d l 0 II GeV.

References

[ 1] Y. Achiman and A. Lukas, Nucl. Phys. B 384 (1992) 78. A. Lukas, BUGH Wuppertal diploma thesis, University of Wuppertal report WUB 91-28 ( 1991 ). [2] Y. Achiman, J. Erler and W. Kalau, Nucl. Phys. B 331 (1990)213; Y. Achiman, Z. Phys. C 44 (1989) 103; Phys. Lett. B 131 (1983) 362. [3 ] U. Amaldi, W. de Boer and H. Ffirstenau, Phys. Lett. B 260 (1991) 443; P. Langacker and M.X. Lou, Phys. Rev. D 44 ( 1992 ) 817. [4] GALLEX Collab., P. Anselman et al., Phys. Lett. B 285 (1992) 390. [ 5 ] L.E. Ib,'ifiez,Nucl. Phys. B 193 ( 1981 ) 317. [6] R. Barbieri and D.V. Nanopoulos, Phys. Lett. B 91 (1980) 369; Nucl. Phys. B 95 (1980) 43. [ 7 ] M. GeU-Mann, P. Ramond and R. Slansky, in: Supergravity, eds. P. van Nieuwenhuizen and D.Z. Freedman (NorthHolland, Amsterdam, 1979 ). [8] J. Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 77. [91 F.J. Gilman and Y. Nir, Annu. Rev. Nucl. Part. Sci. 40 (1990) 213. [101 A. Barducei, R. Casalbuoni, S. De Curtis, D. Dominici and R. Gatto, Phys. Lett. B 193 (1987) 305. [111 S.A. Bludman, D.C. Kennedy and P.G. Langacker, Nuel. Phys. B 374 (1992) 373. [121 L. Wolfenstein, Phys. Rev. D 17 (1977) 2369; S.P. Mikheyev and A.Yu. Smirnov, Nuovo Cimento C 9 (1986) 17. [131 K.S. Hirata et al., Phys. Lett. B 280 (1992) 146; Frdjus Collab., Ch. Berger et al., Phys. Len. B 227 (1989) 489; B 245 (1990) 305.

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